id	sid	tid	token	lemma	pos
ejpam-4592	1	1	european	european	PROPN
ejpam-4592	1	2	journal	journal	PROPN
ejpam-4592	1	3	of	of	ADP
ejpam-4592	1	4	pure	pure	ADJ
ejpam-4592	1	5	and	and	CCONJ
ejpam-4592	1	6	applied	apply	VERB
ejpam-4592	1	7	mathematics	mathematic	NOUN
ejpam-4592	1	8	vol	vol	NOUN
ejpam-4592	1	9	.	.	PROPN
ejpam-4592	2	1	15	15	NUM
ejpam-4592	2	2	,	,	PUNCT
ejpam-4592	2	3	no	no	INTJ
ejpam-4592	2	4	.	.	NOUN
ejpam-4592	2	5	4	4	NUM
ejpam-4592	2	6	,	,	PUNCT
ejpam-4592	2	7	2022	2022	NUM
ejpam-4592	2	8	,	,	PUNCT
ejpam-4592	2	9	1869	1869	NUM
ejpam-4592	2	10	-	-	SYM
ejpam-4592	2	11	1886	1886	NUM
ejpam-4592	2	12	issn	issn	VERB
ejpam-4592	2	13	1307	1307	NUM
ejpam-4592	2	14	-	-	SYM
ejpam-4592	2	15	5543	5543	NUM
ejpam-4592	2	16	–	–	PUNCT
ejpam-4592	2	17	ejpam.com	ejpam.com	X
ejpam-4592	2	18	published	publish	VERB
ejpam-4592	2	19	by	by	ADP
ejpam-4592	2	20	new	new	PROPN
ejpam-4592	2	21	york	york	PROPN
ejpam-4592	2	22	business	business	PROPN
ejpam-4592	2	23	global	global	ADJ
ejpam-4592	2	24	special	special	ADJ
ejpam-4592	2	25	sequences	sequence	NOUN
ejpam-4592	2	26	on	on	ADP
ejpam-4592	2	27	generalized	generalized	ADJ
ejpam-4592	2	28	metric	metric	ADJ
ejpam-4592	2	29	spaces	space	NOUN
ejpam-4592	2	30	yasser	yasser	PROPN
ejpam-4592	2	31	farhat1	farhat1	PROPN
ejpam-4592	2	32	,	,	PUNCT
ejpam-4592	2	33	amel	amel	PROPN
ejpam-4592	2	34	souhail2	souhail2	PROPN
ejpam-4592	2	35	,	,	PUNCT
ejpam-4592	2	36	vadakasi	vadakasi	PROPN
ejpam-4592	2	37	subramanian3,∗	subramanian3,∗	PROPN
ejpam-4592	2	38	,	,	PUNCT
ejpam-4592	2	39	k.	k.	PROPN
ejpam-4592	2	40	muthumari3	muthumari3	PROPN
ejpam-4592	3	1	1	1	NUM
ejpam-4592	3	2	academic	academic	ADJ
ejpam-4592	3	3	support	support	NOUN
ejpam-4592	3	4	department	department	NOUN
ejpam-4592	3	5	,	,	PUNCT
ejpam-4592	3	6	abu	abu	PROPN
ejpam-4592	3	7	dhabi	dhabi	PROPN
ejpam-4592	3	8	polytechnic	polytechnic	PROPN
ejpam-4592	3	9	,	,	PUNCT
ejpam-4592	3	10	p.	p.	PROPN
ejpam-4592	3	11	o.	o.	PROPN
ejpam-4592	3	12	box	box	PROPN
ejpam-4592	3	13	111499	111499	NUM
ejpam-4592	3	14	,	,	PUNCT
ejpam-4592	3	15	abu	abu	PROPN
ejpam-4592	3	16	dhabi	dhabi	PROPN
ejpam-4592	3	17	,	,	PUNCT
ejpam-4592	3	18	uae	uae	PROPN
ejpam-4592	3	19	2	2	NUM
ejpam-4592	3	20	départment	départment	PROPN
ejpam-4592	3	21	de	de	PROPN
ejpam-4592	3	22	mathématiques	mathématiques	PROPN
ejpam-4592	3	23	,	,	PUNCT
ejpam-4592	3	24	faculté	faculté	PROPN
ejpam-4592	3	25	des	des	PROPN
ejpam-4592	3	26	sciences	sciences	PROPN
ejpam-4592	3	27	de	de	X
ejpam-4592	3	28	sfax	sfax	NOUN
ejpam-4592	3	29	,	,	PUNCT
ejpam-4592	3	30	université	université	ADJ
ejpam-4592	3	31	de	de	X
ejpam-4592	3	32	sfax	sfax	NOUN
ejpam-4592	3	33	,	,	PUNCT
ejpam-4592	3	34	route	route	NOUN
ejpam-4592	3	35	de	de	X
ejpam-4592	3	36	soukra	soukra	NOUN
ejpam-4592	3	37	km	km	PROPN
ejpam-4592	3	38	3.5	3.5	NUM
ejpam-4592	3	39	,	,	PUNCT
ejpam-4592	3	40	b.p	b.p	PROPN
ejpam-4592	3	41	.	.	PROPN
ejpam-4592	3	42	1171	1171	NUM
ejpam-4592	3	43	,	,	PUNCT
ejpam-4592	3	44	3000	3000	NUM
ejpam-4592	3	45	,	,	PUNCT
ejpam-4592	3	46	sfax	sfax	NOUN
ejpam-4592	3	47	,	,	PUNCT
ejpam-4592	3	48	tunisia	tunisia	PROPN
ejpam-4592	3	49	3	3	NUM
ejpam-4592	3	50	department	department	NOUN
ejpam-4592	3	51	of	of	ADP
ejpam-4592	3	52	mathematics	mathematics	PROPN
ejpam-4592	3	53	,	,	PUNCT
ejpam-4592	3	54	a.k.d	a.k.d	NOUN
ejpam-4592	3	55	.	.	PUNCT
ejpam-4592	3	56	dharma	dharma	PROPN
ejpam-4592	3	57	raja	raja	PROPN
ejpam-4592	3	58	women	women	PROPN
ejpam-4592	3	59	’s	’s	PART
ejpam-4592	3	60	college	college	PROPN
ejpam-4592	3	61	,	,	PUNCT
ejpam-4592	3	62	rajapalayam	rajapalayam	PROPN
ejpam-4592	3	63	,	,	PUNCT
ejpam-4592	3	64	tamilnadu	tamilnadu	ADJ
ejpam-4592	3	65	,	,	PUNCT
ejpam-4592	3	66	india	india	PROPN
ejpam-4592	3	67	abstract	abstract	NOUN
ejpam-4592	3	68	.	.	PUNCT
ejpam-4592	4	1	in	in	ADP
ejpam-4592	4	2	this	this	DET
ejpam-4592	4	3	article	article	NOUN
ejpam-4592	4	4	,	,	PUNCT
ejpam-4592	4	5	in	in	ADP
ejpam-4592	4	6	a	a	DET
ejpam-4592	4	7	generalized	generalized	ADJ
ejpam-4592	4	8	metric	metric	ADJ
ejpam-4592	4	9	space	space	NOUN
ejpam-4592	4	10	,	,	PUNCT
ejpam-4592	4	11	we	we	PRON
ejpam-4592	4	12	will	will	AUX
ejpam-4592	4	13	focus	focus	VERB
ejpam-4592	4	14	on	on	ADP
ejpam-4592	4	15	new	new	ADJ
ejpam-4592	4	16	types	type	NOUN
ejpam-4592	4	17	of	of	ADP
ejpam-4592	4	18	sequences	sequence	NOUN
ejpam-4592	4	19	.	.	PUNCT
ejpam-4592	5	1	we	we	PRON
ejpam-4592	5	2	introduce	introduce	VERB
ejpam-4592	5	3	three	three	NUM
ejpam-4592	5	4	new	new	ADJ
ejpam-4592	5	5	kinds	kind	NOUN
ejpam-4592	5	6	of	of	ADP
ejpam-4592	5	7	cauchy	cauchy	ADJ
ejpam-4592	5	8	sequences	sequence	NOUN
ejpam-4592	5	9	and	and	CCONJ
ejpam-4592	5	10	study	study	VERB
ejpam-4592	5	11	their	their	PRON
ejpam-4592	5	12	significance	significance	NOUN
ejpam-4592	5	13	in	in	ADP
ejpam-4592	5	14	generalized	generalized	ADJ
ejpam-4592	5	15	metric	metric	ADJ
ejpam-4592	5	16	spaces	space	NOUN
ejpam-4592	5	17	.	.	PUNCT
ejpam-4592	6	1	also	also	ADV
ejpam-4592	6	2	,	,	PUNCT
ejpam-4592	6	3	we	we	PRON
ejpam-4592	6	4	give	give	VERB
ejpam-4592	6	5	several	several	ADJ
ejpam-4592	6	6	interesting	interesting	ADJ
ejpam-4592	6	7	properties	property	NOUN
ejpam-4592	6	8	of	of	ADP
ejpam-4592	6	9	these	these	DET
ejpam-4592	6	10	sequences	sequence	NOUN
ejpam-4592	6	11	.	.	PUNCT
ejpam-4592	7	1	2020	2020	NUM
ejpam-4592	7	2	mathematics	mathematic	NOUN
ejpam-4592	7	3	subject	subject	NOUN
ejpam-4592	7	4	classifications	classification	NOUN
ejpam-4592	7	5	:	:	PUNCT
ejpam-4592	7	6	40a05	40a05	NUM
ejpam-4592	7	7	,	,	PUNCT
ejpam-4592	7	8	54e35	54e35	NUM
ejpam-4592	7	9	,	,	PUNCT
ejpam-4592	7	10	54e40	54e40	NUM
ejpam-4592	7	11	,	,	PUNCT
ejpam-4592	7	12	54e50	54e50	NUM
ejpam-4592	7	13	key	key	ADJ
ejpam-4592	7	14	words	word	NOUN
ejpam-4592	7	15	and	and	CCONJ
ejpam-4592	7	16	phrases	phrase	NOUN
ejpam-4592	7	17	:	:	PUNCT
ejpam-4592	7	18	kernel	kernel	PROPN
ejpam-4592	7	19	,	,	PUNCT
ejpam-4592	7	20	bourbaki	bourbaki	NOUN
ejpam-4592	7	21	-	-	PUNCT
ejpam-4592	7	22	cauchy	cauchy	ADJ
ejpam-4592	7	23	sequence	sequence	NOUN
ejpam-4592	7	24	,	,	PUNCT
ejpam-4592	7	25	pseudo	pseudo	NOUN
ejpam-4592	7	26	-	-	ADJ
ejpam-4592	7	27	cauchy	cauchy	ADJ
ejpam-4592	7	28	sequence	sequence	NOUN
ejpam-4592	7	29	,	,	PUNCT
ejpam-4592	7	30	bourbakicomplete	bourbakicomplete	ADJ
ejpam-4592	7	31	metric	metric	ADJ
ejpam-4592	7	32	space	space	NOUN
ejpam-4592	7	33	,	,	PUNCT
ejpam-4592	7	34	weakly	weakly	ADV
ejpam-4592	7	35	complete	complete	ADJ
ejpam-4592	7	36	metric	metric	ADJ
ejpam-4592	7	37	space	space	NOUN
ejpam-4592	7	38	and	and	CCONJ
ejpam-4592	7	39	asymptotic	asymptotic	ADJ
ejpam-4592	7	40	sequence	sequence	NOUN
ejpam-4592	7	41	1	1	NUM
ejpam-4592	7	42	.	.	PUNCT
ejpam-4592	7	43	introduction	introduction	NOUN
ejpam-4592	7	44	the	the	DET
ejpam-4592	7	45	notion	notion	NOUN
ejpam-4592	7	46	of	of	ADP
ejpam-4592	7	47	completeness	completeness	NOUN
ejpam-4592	7	48	plays	play	VERB
ejpam-4592	7	49	in	in	ADP
ejpam-4592	7	50	the	the	DET
ejpam-4592	7	51	theory	theory	NOUN
ejpam-4592	7	52	of	of	ADP
ejpam-4592	7	53	metric	metric	ADJ
ejpam-4592	7	54	spaces	space	NOUN
ejpam-4592	7	55	governs	govern	VERB
ejpam-4592	7	56	the	the	DET
ejpam-4592	7	57	importance	importance	NOUN
ejpam-4592	7	58	upgrade	upgrade	VERB
ejpam-4592	7	59	in	in	ADP
ejpam-4592	7	60	topological	topological	ADJ
ejpam-4592	7	61	metrics	metric	NOUN
ejpam-4592	7	62	.	.	PUNCT
ejpam-4592	8	1	few	few	ADJ
ejpam-4592	8	2	researchers	researcher	NOUN
ejpam-4592	8	3	had	have	AUX
ejpam-4592	8	4	examined	examine	VERB
ejpam-4592	8	5	the	the	DET
ejpam-4592	8	6	significance	significance	NOUN
ejpam-4592	8	7	of	of	ADP
ejpam-4592	8	8	complete	complete	ADJ
ejpam-4592	8	9	metric	metric	ADJ
ejpam-4592	8	10	spaces	space	NOUN
ejpam-4592	8	11	e.g.	e.g.	ADV
ejpam-4592	8	12	[	[	X
ejpam-4592	8	13	4	4	NUM
ejpam-4592	8	14	,	,	PUNCT
ejpam-4592	8	15	5	5	NUM
ejpam-4592	8	16	,	,	PUNCT
ejpam-4592	8	17	8	8	NUM
ejpam-4592	8	18	,	,	PUNCT
ejpam-4592	8	19	10–13	10–13	NUM
ejpam-4592	8	20	,	,	PUNCT
ejpam-4592	8	21	15–19	15–19	PROPN
ejpam-4592	8	22	]	]	PUNCT
ejpam-4592	8	23	.	.	PUNCT
ejpam-4592	9	1	inspiring	inspire	VERB
ejpam-4592	9	2	the	the	DET
ejpam-4592	9	3	earlier	early	ADJ
ejpam-4592	9	4	literature	literature	NOUN
ejpam-4592	9	5	,	,	PUNCT
ejpam-4592	9	6	korczak	korczak	PROPN
ejpam-4592	9	7	-	-	PUNCT
ejpam-4592	9	8	kubiak	kubiak	PROPN
ejpam-4592	9	9	et.al	et.al	PROPN
ejpam-4592	9	10	introduced	introduce	VERB
ejpam-4592	9	11	the	the	DET
ejpam-4592	9	12	concept	concept	NOUN
ejpam-4592	9	13	of	of	ADP
ejpam-4592	9	14	new	new	ADJ
ejpam-4592	9	15	metric	metric	ADJ
ejpam-4592	9	16	space	space	NOUN
ejpam-4592	9	17	namely	namely	ADV
ejpam-4592	9	18	,	,	PUNCT
ejpam-4592	9	19	generalized	generalize	VERB
ejpam-4592	9	20	metric	metric	ADJ
ejpam-4592	9	21	space	space	NOUN
ejpam-4592	9	22	during	during	ADP
ejpam-4592	9	23	2013	2013	NUM
ejpam-4592	9	24	and	and	CCONJ
ejpam-4592	9	25	then	then	ADV
ejpam-4592	9	26	defined	define	VERB
ejpam-4592	9	27	two	two	NUM
ejpam-4592	9	28	new	new	ADJ
ejpam-4592	9	29	things	thing	NOUN
ejpam-4592	9	30	in	in	ADP
ejpam-4592	9	31	the	the	DET
ejpam-4592	9	32	name	name	NOUN
ejpam-4592	9	33	of	of	ADP
ejpam-4592	9	34	kernel	kernel	NOUN
ejpam-4592	9	35	and	and	CCONJ
ejpam-4592	9	36	perfect	perfect	ADJ
ejpam-4592	9	37	kernel	kernel	NOUN
ejpam-4592	9	38	in	in	ADP
ejpam-4592	9	39	a	a	DET
ejpam-4592	9	40	generalized	generalized	ADJ
ejpam-4592	9	41	metric	metric	ADJ
ejpam-4592	9	42	space	space	NOUN
ejpam-4592	9	43	.	.	PUNCT
ejpam-4592	10	1	using	use	VERB
ejpam-4592	10	2	these	these	DET
ejpam-4592	10	3	tools	tool	NOUN
ejpam-4592	10	4	,	,	PUNCT
ejpam-4592	10	5	they	they	PRON
ejpam-4592	10	6	generated	generate	VERB
ejpam-4592	10	7	three	three	NUM
ejpam-4592	10	8	types	type	NOUN
ejpam-4592	10	9	of	of	ADP
ejpam-4592	10	10	complete	complete	ADJ
ejpam-4592	10	11	spaces	space	NOUN
ejpam-4592	10	12	which	which	PRON
ejpam-4592	10	13	has	have	AUX
ejpam-4592	10	14	been	be	AUX
ejpam-4592	10	15	defined	define	VERB
ejpam-4592	10	16	weakly	weakly	ADV
ejpam-4592	10	17	complete	complete	ADJ
ejpam-4592	10	18	,	,	PUNCT
ejpam-4592	10	19	complete	complete	ADJ
ejpam-4592	10	20	and	and	CCONJ
ejpam-4592	10	21	strongly	strongly	ADV
ejpam-4592	10	22	complete	complete	ADJ
ejpam-4592	10	23	spaces	space	NOUN
ejpam-4592	10	24	.	.	PUNCT
ejpam-4592	11	1	in	in	ADP
ejpam-4592	11	2	addition	addition	NOUN
ejpam-4592	11	3	,	,	PUNCT
ejpam-4592	11	4	they	they	PRON
ejpam-4592	11	5	have	have	AUX
ejpam-4592	11	6	analyzed	analyze	VERB
ejpam-4592	11	7	the	the	DET
ejpam-4592	11	8	nature	nature	NOUN
ejpam-4592	11	9	of	of	ADP
ejpam-4592	11	10	these	these	DET
ejpam-4592	11	11	three	three	NUM
ejpam-4592	11	12	types	type	NOUN
ejpam-4592	11	13	of	of	ADP
ejpam-4592	11	14	complete	complete	ADJ
ejpam-4592	11	15	spaces	space	NOUN
ejpam-4592	11	16	in	in	ADP
ejpam-4592	11	17	generalized	generalized	ADJ
ejpam-4592	11	18	metric	metric	ADJ
ejpam-4592	11	19	spaces	space	NOUN
ejpam-4592	11	20	.	.	PUNCT
ejpam-4592	12	1	similarly	similarly	ADV
ejpam-4592	12	2	,	,	PUNCT
ejpam-4592	12	3	the	the	DET
ejpam-4592	12	4	nature	nature	NOUN
ejpam-4592	12	5	of	of	ADP
ejpam-4592	12	6	bourbaki	bourbaki	NOUN
ejpam-4592	12	7	-	-	PUNCT
ejpam-4592	12	8	cauchy	cauchy	ADJ
ejpam-4592	12	9	sequences	sequence	NOUN
ejpam-4592	12	10	e.g.	e.g.	ADV
ejpam-4592	12	11	[	[	X
ejpam-4592	12	12	2	2	NUM
ejpam-4592	12	13	,	,	PUNCT
ejpam-4592	12	14	3	3	NUM
ejpam-4592	12	15	,	,	PUNCT
ejpam-4592	12	16	9	9	NUM
ejpam-4592	12	17	]	]	PUNCT
ejpam-4592	12	18	has	have	AUX
ejpam-4592	12	19	been	be	AUX
ejpam-4592	12	20	obtained	obtain	VERB
ejpam-4592	12	21	which	which	PRON
ejpam-4592	12	22	influenced	influence	VERB
ejpam-4592	12	23	some	some	DET
ejpam-4592	12	24	new	new	ADJ
ejpam-4592	12	25	types	type	NOUN
ejpam-4592	12	26	of	of	ADP
ejpam-4592	12	27	sequences	sequence	NOUN
ejpam-4592	12	28	namely	namely	ADV
ejpam-4592	12	29	bourbaki	bourbaki	VERB
ejpam-4592	12	30	-	-	PUNCT
ejpam-4592	12	31	cauchy	cauchy	NOUN
ejpam-4592	12	32	,	,	PUNCT
ejpam-4592	12	33	co	co	VERB
ejpam-4592	12	34	-	-	ADJ
ejpam-4592	12	35	finally	finally	ADV
ejpam-4592	12	36	-	-	PUNCT
ejpam-4592	12	37	cauchy	cauchy	ADJ
ejpam-4592	12	38	and	and	CCONJ
ejpam-4592	12	39	pseudo	pseudo	NOUN
ejpam-4592	12	40	-	-	NOUN
ejpam-4592	12	41	cauchy	cauchy	NOUN
ejpam-4592	12	42	in	in	ADP
ejpam-4592	12	43	a	a	DET
ejpam-4592	12	44	metric	metric	ADJ
ejpam-4592	12	45	space	space	NOUN
ejpam-4592	12	46	given	give	VERB
ejpam-4592	12	47	by	by	ADP
ejpam-4592	12	48	aggarwal	aggarwal	NOUN
ejpam-4592	12	49	et.al	et.al	PROPN
ejpam-4592	12	50	[	[	X
ejpam-4592	12	51	1	1	NUM
ejpam-4592	12	52	]	]	PUNCT
ejpam-4592	12	53	.	.	PUNCT
ejpam-4592	13	1	motivated	motivate	VERB
ejpam-4592	13	2	by	by	ADP
ejpam-4592	13	3	this	this	PRON
ejpam-4592	13	4	,	,	PUNCT
ejpam-4592	13	5	in	in	ADP
ejpam-4592	13	6	a	a	DET
ejpam-4592	13	7	generalized	generalized	ADJ
ejpam-4592	13	8	metric	metric	ADJ
ejpam-4592	13	9	space	space	NOUN
ejpam-4592	13	10	,	,	PUNCT
ejpam-4592	13	11	we	we	PRON
ejpam-4592	13	12	redefined	redefine	VERB
ejpam-4592	13	13	three	three	NUM
ejpam-4592	13	14	types	type	NOUN
ejpam-4592	13	15	of	of	ADP
ejpam-4592	13	16	sequences	sequence	NOUN
ejpam-4592	13	17	namely	namely	ADV
ejpam-4592	13	18	,	,	PUNCT
ejpam-4592	13	19	bourbaki	bourbaki	NOUN
ejpam-4592	13	20	-	-	PUNCT
ejpam-4592	13	21	cauchy	cauchy	ADJ
ejpam-4592	13	22	,	,	PUNCT
ejpam-4592	13	23	cofinally	cofinally	ADV
ejpam-4592	13	24	-	-	PUNCT
ejpam-4592	13	25	cauchy	cauchy	ADJ
ejpam-4592	13	26	and	and	CCONJ
ejpam-4592	13	27	pseudo	pseudo	NOUN
ejpam-4592	13	28	-	-	ADJ
ejpam-4592	13	29	cauchy	cauchy	ADJ
ejpam-4592	13	30	sequences	sequence	NOUN
ejpam-4592	13	31	as	as	ADV
ejpam-4592	13	32	well	well	ADV
ejpam-4592	13	33	as	as	ADP
ejpam-4592	13	34	complete	complete	ADJ
ejpam-4592	13	35	metric	metric	ADJ
ejpam-4592	13	36	spaces	space	NOUN
ejpam-4592	13	37	.	.	PUNCT
ejpam-4592	14	1	also	also	ADV
ejpam-4592	14	2	,	,	PUNCT
ejpam-4592	14	3	we	we	PRON
ejpam-4592	14	4	investigate	investigate	VERB
ejpam-4592	14	5	their	their	PRON
ejpam-4592	14	6	significance	significance	NOUN
ejpam-4592	14	7	and	and	CCONJ
ejpam-4592	14	8	give	give	VERB
ejpam-4592	14	9	some	some	DET
ejpam-4592	14	10	relationships	relationship	NOUN
ejpam-4592	14	11	between	between	ADP
ejpam-4592	14	12	these	these	DET
ejpam-4592	14	13	sequences	sequence	NOUN
ejpam-4592	14	14	in	in	ADP
ejpam-4592	14	15	a	a	DET
ejpam-4592	14	16	generalized	generalized	ADJ
ejpam-4592	14	17	metric	metric	ADJ
ejpam-4592	14	18	space	space	NOUN
ejpam-4592	14	19	.	.	PUNCT
ejpam-4592	15	1	∗corresponding	∗corresponde	VERB
ejpam-4592	15	2	author	author	NOUN
ejpam-4592	15	3	.	.	PUNCT
ejpam-4592	16	1	doi	doi	NOUN
ejpam-4592	16	2	:	:	PUNCT
ejpam-4592	16	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4592	https://doi.org/10.29020/nybg.ejpam.v15i4.4592	ADJ
ejpam-4592	16	4	email	email	NOUN
ejpam-4592	16	5	addresses	address	VERB
ejpam-4592	16	6	:	:	PUNCT
ejpam-4592	16	7	farhat.yasser.1@gmail.com	farhat.yasser.1@gmail.com	X
ejpam-4592	16	8	(	(	PUNCT
ejpam-4592	16	9	y.	y.	NOUN
ejpam-4592	16	10	farhat),vadakasisub1992@gmail.com	farhat),vadakasisub1992@gmail.com	X
ejpam-4592	16	11	(	(	PUNCT
ejpam-4592	16	12	s.	s.	PROPN
ejpam-4592	16	13	vadakasi	vadakasi	PROPN
ejpam-4592	16	14	)	)	PUNCT
ejpam-4592	16	15	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4592	16	16	1869	1869	NUM
ejpam-4592	17	1	©	©	ADP
ejpam-4592	17	2	2022	2022	NUM
ejpam-4592	17	3	ejpam	ejpam	VERB
ejpam-4592	17	4	all	all	DET
ejpam-4592	17	5	rights	right	NOUN
ejpam-4592	17	6	reserved	reserve	VERB
ejpam-4592	17	7	.	.	PUNCT
ejpam-4592	18	1	v.	v.	ADP
ejpam-4592	18	2	subramanian	subramanian	PROPN
ejpam-4592	18	3	et	et	PROPN
ejpam-4592	18	4	al	al	PROPN
ejpam-4592	18	5	.	.	PUNCT
ejpam-4592	18	6	/	/	SYM
ejpam-4592	18	7	eur	eur	PROPN
ejpam-4592	18	8	.	.	PUNCT
ejpam-4592	19	1	j.	j.	PROPN
ejpam-4592	19	2	pure	pure	PROPN
ejpam-4592	19	3	appl	appl	PROPN
ejpam-4592	19	4	.	.	PROPN
ejpam-4592	19	5	math	math	PROPN
ejpam-4592	19	6	,	,	PUNCT
ejpam-4592	19	7	15	15	NUM
ejpam-4592	19	8	(	(	PUNCT
ejpam-4592	19	9	4	4	NUM
ejpam-4592	19	10	)	)	PUNCT
ejpam-4592	19	11	(	(	PUNCT
ejpam-4592	19	12	2022	2022	NUM
ejpam-4592	19	13	)	)	PUNCT
ejpam-4592	19	14	,	,	PUNCT
ejpam-4592	19	15	1869	1869	NUM
ejpam-4592	19	16	-	-	SYM
ejpam-4592	19	17	1886	1886	NUM
ejpam-4592	19	18	1870	1870	NUM
ejpam-4592	19	19	throughout	throughout	ADP
ejpam-4592	19	20	this	this	DET
ejpam-4592	19	21	paper	paper	NOUN
ejpam-4592	19	22	,	,	PUNCT
ejpam-4592	19	23	r	r	NOUN
ejpam-4592	19	24	and	and	CCONJ
ejpam-4592	19	25	n	n	ADV
ejpam-4592	19	26	denote	denote	VERB
ejpam-4592	19	27	the	the	DET
ejpam-4592	19	28	set	set	NOUN
ejpam-4592	19	29	of	of	ADP
ejpam-4592	19	30	all	all	DET
ejpam-4592	19	31	real	real	ADJ
ejpam-4592	19	32	and	and	CCONJ
ejpam-4592	19	33	natural	natural	ADJ
ejpam-4592	19	34	numbers	number	NOUN
ejpam-4592	19	35	,	,	PUNCT
ejpam-4592	19	36	respectively	respectively	ADV
ejpam-4592	19	37	.	.	PUNCT
ejpam-4592	20	1	2	2	X
ejpam-4592	20	2	.	.	X
ejpam-4592	20	3	preliminaries	preliminary	NOUN
ejpam-4592	20	4	first	first	ADV
ejpam-4592	20	5	,	,	PUNCT
ejpam-4592	20	6	we	we	PRON
ejpam-4592	20	7	recall	recall	VERB
ejpam-4592	20	8	some	some	DET
ejpam-4592	20	9	basic	basic	ADJ
ejpam-4592	20	10	things	thing	NOUN
ejpam-4592	20	11	defined	define	VERB
ejpam-4592	20	12	in	in	ADP
ejpam-4592	20	13	a	a	DET
ejpam-4592	20	14	generalized	generalize	VERB
ejpam-4592	20	15	metric	metric	ADJ
ejpam-4592	20	16	space	space	NOUN
ejpam-4592	20	17	which	which	PRON
ejpam-4592	20	18	are	be	AUX
ejpam-4592	20	19	useful	useful	ADJ
ejpam-4592	20	20	to	to	ADP
ejpam-4592	20	21	the	the	DET
ejpam-4592	20	22	development	development	NOUN
ejpam-4592	20	23	of	of	ADP
ejpam-4592	20	24	the	the	DET
ejpam-4592	20	25	next	next	ADJ
ejpam-4592	20	26	sections	section	NOUN
ejpam-4592	20	27	.	.	PUNCT
ejpam-4592	21	1	let	let	VERB
ejpam-4592	21	2	x	x	PUNCT
ejpam-4592	21	3	̸=	̸=	PROPN
ejpam-4592	21	4	∅.	∅.	ADP
ejpam-4592	21	5	the	the	DET
ejpam-4592	21	6	symbol	symbol	NOUN
ejpam-4592	21	7	ω	ω	PROPN
ejpam-4592	21	8	to	to	PART
ejpam-4592	21	9	denote	denote	VERB
ejpam-4592	21	10	the	the	DET
ejpam-4592	21	11	family	family	NOUN
ejpam-4592	21	12	consisting	consist	VERB
ejpam-4592	21	13	of	of	ADP
ejpam-4592	21	14	metrics	metric	NOUN
ejpam-4592	21	15	defined	define	VERB
ejpam-4592	21	16	on	on	ADP
ejpam-4592	21	17	subsets	subset	NOUN
ejpam-4592	21	18	of	of	ADP
ejpam-4592	21	19	x	x	PRON
ejpam-4592	21	20	,	,	PUNCT
ejpam-4592	21	21	that	that	PRON
ejpam-4592	21	22	is	is	ADV
ejpam-4592	21	23	σ	σ	PROPN
ejpam-4592	21	24	∈	∈	PROPN
ejpam-4592	21	25	ω	ω	PROPN
ejpam-4592	21	26	,	,	PUNCT
ejpam-4592	21	27	then	then	ADV
ejpam-4592	21	28	there	there	PRON
ejpam-4592	21	29	exists	exist	VERB
ejpam-4592	21	30	a	a	DET
ejpam-4592	21	31	non	non	ADJ
ejpam-4592	21	32	-	-	ADJ
ejpam-4592	21	33	null	null	ADJ
ejpam-4592	21	34	set	set	ADJ
ejpam-4592	21	35	bσ	bσ	NOUN
ejpam-4592	21	36	⊂	⊂	PROPN
ejpam-4592	21	37	x	x	PUNCT
ejpam-4592	21	38	such	such	ADJ
ejpam-4592	21	39	that	that	SCONJ
ejpam-4592	21	40	σ	σ	PROPN
ejpam-4592	21	41	is	be	AUX
ejpam-4592	21	42	a	a	DET
ejpam-4592	21	43	metric	metric	NOUN
ejpam-4592	21	44	on	on	ADP
ejpam-4592	21	45	bσ	bσ	NOUN
ejpam-4592	21	46	where	where	SCONJ
ejpam-4592	21	47	bσ	bσ	PROPN
ejpam-4592	21	48	is	be	AUX
ejpam-4592	21	49	a	a	DET
ejpam-4592	21	50	domain	domain	NOUN
ejpam-4592	21	51	space	space	NOUN
ejpam-4592	21	52	of	of	ADP
ejpam-4592	21	53	σ	σ	PROPN
ejpam-4592	21	54	and	and	CCONJ
ejpam-4592	21	55	it	it	PRON
ejpam-4592	21	56	will	will	AUX
ejpam-4592	21	57	be	be	AUX
ejpam-4592	21	58	denoted	denote	VERB
ejpam-4592	21	59	by	by	ADP
ejpam-4592	21	60	dom(σ	dom(σ	PROPN
ejpam-4592	21	61	)	)	PUNCT
ejpam-4592	21	62	.	.	PUNCT
ejpam-4592	22	1	then	then	ADV
ejpam-4592	22	2	(	(	PUNCT
ejpam-4592	22	3	x	x	X
ejpam-4592	22	4	,	,	PUNCT
ejpam-4592	22	5	ω	ω	NUM
ejpam-4592	22	6	)	)	PUNCT
ejpam-4592	22	7	is	be	AUX
ejpam-4592	22	8	called	call	VERB
ejpam-4592	22	9	as	as	ADP
ejpam-4592	22	10	a	a	DET
ejpam-4592	22	11	generalized	generalized	ADJ
ejpam-4592	22	12	metric	metric	ADJ
ejpam-4592	22	13	space	space	NOUN
ejpam-4592	22	14	(	(	PUNCT
ejpam-4592	22	15	gms	gms	NOUN
ejpam-4592	22	16	)	)	PUNCT
ejpam-4592	23	1	[	[	X
ejpam-4592	23	2	7	7	NUM
ejpam-4592	23	3	]	]	PUNCT
ejpam-4592	23	4	.	.	PUNCT
ejpam-4592	24	1	if	if	SCONJ
ejpam-4592	24	2	σ	σ	PROPN
ejpam-4592	24	3	is	be	AUX
ejpam-4592	24	4	a	a	DET
ejpam-4592	24	5	metric	metric	NOUN
ejpam-4592	24	6	on	on	ADP
ejpam-4592	24	7	x	x	X
ejpam-4592	24	8	and	and	CCONJ
ejpam-4592	24	9	c	c	PROPN
ejpam-4592	24	10	is	be	AUX
ejpam-4592	24	11	a	a	DET
ejpam-4592	24	12	non	non	ADJ
ejpam-4592	24	13	-	-	ADJ
ejpam-4592	24	14	null	null	ADJ
ejpam-4592	24	15	subset	subset	NOUN
ejpam-4592	24	16	x	x	X
ejpam-4592	24	17	,	,	PUNCT
ejpam-4592	24	18	then	then	ADV
ejpam-4592	24	19	we	we	PRON
ejpam-4592	24	20	write	write	VERB
ejpam-4592	24	21	σ|c	σ|c	NOUN
ejpam-4592	24	22	for	for	ADP
ejpam-4592	24	23	the	the	DET
ejpam-4592	24	24	restriction	restriction	NOUN
ejpam-4592	24	25	of	of	ADP
ejpam-4592	24	26	the	the	DET
ejpam-4592	24	27	metric	metric	PROPN
ejpam-4592	24	28	σ	σ	PROPN
ejpam-4592	24	29	to	to	ADP
ejpam-4592	24	30	c	c	PROPN
ejpam-4592	24	31	×	×	PROPN
ejpam-4592	24	32	c.	c.	PROPN
ejpam-4592	24	33	moreover	moreover	ADV
ejpam-4592	24	34	,	,	PUNCT
ejpam-4592	24	35	put	put	VERB
ejpam-4592	24	36	ω|c	ω|c	PUNCT
ejpam-4592	25	1	=	=	PRON
ejpam-4592	25	2	{	{	PUNCT
ejpam-4592	25	3	σ|c	σ|c	NOUN
ejpam-4592	25	4	|	|	NOUN
ejpam-4592	25	5	σ	σ	PROPN
ejpam-4592	25	6	∈	∈	PROPN
ejpam-4592	25	7	ω	ω	PROPN
ejpam-4592	25	8	}	}	PUNCT
ejpam-4592	25	9	for	for	ADP
ejpam-4592	25	10	any	any	DET
ejpam-4592	25	11	ω	ω	NOUN
ejpam-4592	25	12	⊂	⊂	PROPN
ejpam-4592	25	13	ωx	ωx	PROPN
ejpam-4592	25	14	where	where	SCONJ
ejpam-4592	25	15	ωx	ωx	PROPN
ejpam-4592	25	16	is	be	AUX
ejpam-4592	25	17	the	the	DET
ejpam-4592	25	18	collection	collection	NOUN
ejpam-4592	25	19	of	of	ADP
ejpam-4592	25	20	all	all	DET
ejpam-4592	25	21	metrics	metric	NOUN
ejpam-4592	25	22	defined	define	VERB
ejpam-4592	25	23	on	on	ADP
ejpam-4592	25	24	x	x	PUNCT
ejpam-4592	26	1	[	[	X
ejpam-4592	26	2	7	7	NUM
ejpam-4592	26	3	]	]	PUNCT
ejpam-4592	26	4	.	.	PUNCT
ejpam-4592	27	1	obviously	obviously	ADV
ejpam-4592	27	2	,	,	PUNCT
ejpam-4592	27	3	if	if	SCONJ
ejpam-4592	27	4	(	(	PUNCT
ejpam-4592	27	5	x	x	NOUN
ejpam-4592	27	6	,	,	PUNCT
ejpam-4592	27	7	ωx	ωx	PRON
ejpam-4592	27	8	)	)	PUNCT
ejpam-4592	27	9	is	be	AUX
ejpam-4592	27	10	a	a	DET
ejpam-4592	27	11	generalized	generalized	ADJ
ejpam-4592	27	12	metric	metric	ADJ
ejpam-4592	27	13	space	space	NOUN
ejpam-4592	27	14	,	,	PUNCT
ejpam-4592	28	1	ω	ω	X
ejpam-4592	28	2	⊂	⊂	X
ejpam-4592	28	3	ωx	ωx	PROPN
ejpam-4592	28	4	and	and	CCONJ
ejpam-4592	28	5	c	c	PROPN
ejpam-4592	28	6	is	be	AUX
ejpam-4592	28	7	a	a	DET
ejpam-4592	28	8	non	non	ADJ
ejpam-4592	28	9	-	-	ADJ
ejpam-4592	28	10	null	null	ADJ
ejpam-4592	28	11	subset	subset	NOUN
ejpam-4592	28	12	of	of	ADP
ejpam-4592	28	13	x	x	PRON
ejpam-4592	28	14	,	,	PUNCT
ejpam-4592	28	15	then	then	ADV
ejpam-4592	28	16	(	(	PUNCT
ejpam-4592	28	17	c	c	X
ejpam-4592	28	18	,	,	PUNCT
ejpam-4592	28	19	ω|c	ω|c	NUM
ejpam-4592	28	20	)	)	PUNCT
ejpam-4592	28	21	is	be	AUX
ejpam-4592	28	22	a	a	DET
ejpam-4592	28	23	generalized	generalized	ADJ
ejpam-4592	28	24	metric	metric	ADJ
ejpam-4592	28	25	space	space	NOUN
ejpam-4592	28	26	[	[	X
ejpam-4592	28	27	7	7	NUM
ejpam-4592	28	28	]	]	PUNCT
ejpam-4592	28	29	.	.	PUNCT
ejpam-4592	29	1	denote	denote	VERB
ejpam-4592	29	2	µω	µω	PROPN
ejpam-4592	29	3	is	be	AUX
ejpam-4592	29	4	the	the	DET
ejpam-4592	29	5	family	family	NOUN
ejpam-4592	29	6	of	of	ADP
ejpam-4592	29	7	ω	ω	VERB
ejpam-4592	29	8	-	-	ADJ
ejpam-4592	29	9	open	open	ADJ
ejpam-4592	29	10	sets	set	NOUN
ejpam-4592	29	11	in	in	ADP
ejpam-4592	29	12	a	a	DET
ejpam-4592	29	13	generalized	generalized	ADJ
ejpam-4592	29	14	metric	metric	ADJ
ejpam-4592	29	15	space	space	NOUN
ejpam-4592	29	16	(	(	PUNCT
ejpam-4592	29	17	x	x	X
ejpam-4592	29	18	,	,	PUNCT
ejpam-4592	29	19	ω	ω	NOUN
ejpam-4592	29	20	)	)	PUNCT
ejpam-4592	29	21	,	,	PUNCT
ejpam-4592	29	22	more	more	ADV
ejpam-4592	29	23	precisely	precisely	ADV
ejpam-4592	29	24	,	,	PUNCT
ejpam-4592	29	25	h	h	NOUN
ejpam-4592	29	26	∈	∈	PROPN
ejpam-4592	29	27	µω	µω	ADV
ejpam-4592	29	28	if	if	SCONJ
ejpam-4592	29	29	and	and	CCONJ
ejpam-4592	29	30	only	only	ADV
ejpam-4592	29	31	if	if	SCONJ
ejpam-4592	29	32	for	for	ADP
ejpam-4592	29	33	each	each	DET
ejpam-4592	29	34	c	c	PROPN
ejpam-4592	29	35	∈	∈	PROPN
ejpam-4592	29	36	h	h	NOUN
ejpam-4592	29	37	,	,	PUNCT
ejpam-4592	29	38	there	there	PRON
ejpam-4592	29	39	exist	exist	VERB
ejpam-4592	29	40	σ	σ	PROPN
ejpam-4592	29	41	∈	∈	PROPN
ejpam-4592	29	42	ω	ω	PROPN
ejpam-4592	29	43	and	and	CCONJ
ejpam-4592	29	44	ε	ε	PROPN
ejpam-4592	29	45	>	>	X
ejpam-4592	29	46	0	0	NUM
ejpam-4592	29	47	such	such	ADJ
ejpam-4592	29	48	that	that	PRON
ejpam-4592	29	49	bσ(c	bσ(c	PROPN
ejpam-4592	29	50	,	,	PUNCT
ejpam-4592	29	51	ε	ε	PROPN
ejpam-4592	29	52	)	)	PUNCT
ejpam-4592	29	53	⊂	⊂	PROPN
ejpam-4592	30	1	h	h	NOUN
ejpam-4592	31	1	where	where	SCONJ
ejpam-4592	31	2	bσ(c	bσ(c	NUM
ejpam-4592	31	3	,	,	PUNCT
ejpam-4592	31	4	ε	ε	PROPN
ejpam-4592	31	5	)	)	PUNCT
ejpam-4592	31	6	=	=	PUNCT
ejpam-4592	31	7	{	{	PUNCT
ejpam-4592	31	8	d	d	X
ejpam-4592	31	9	∈	∈	PROPN
ejpam-4592	31	10	dom(σ	dom(σ	PROPN
ejpam-4592	31	11	)	)	PUNCT
ejpam-4592	31	12	|	|	ADV
ejpam-4592	31	13	σ(c	σ(c	ADJ
ejpam-4592	31	14	,	,	PUNCT
ejpam-4592	31	15	d	d	NOUN
ejpam-4592	31	16	)	)	PUNCT
ejpam-4592	31	17	<	<	X
ejpam-4592	31	18	ε	ε	X
ejpam-4592	31	19	}	}	PUNCT
ejpam-4592	31	20	[	[	X
ejpam-4592	31	21	7	7	NUM
ejpam-4592	31	22	]	]	PUNCT
ejpam-4592	31	23	.	.	PUNCT
ejpam-4592	32	1	let	let	VERB
ejpam-4592	32	2	x	x	PRON
ejpam-4592	32	3	be	be	AUX
ejpam-4592	32	4	any	any	DET
ejpam-4592	32	5	non	non	ADJ
ejpam-4592	32	6	-	-	ADJ
ejpam-4592	32	7	null	null	ADJ
ejpam-4592	32	8	set	set	NOUN
ejpam-4592	32	9	.	.	PUNCT
ejpam-4592	33	1	a	a	DET
ejpam-4592	33	2	collection	collection	NOUN
ejpam-4592	33	3	µ	µ	X
ejpam-4592	33	4	of	of	ADP
ejpam-4592	33	5	subsets	subset	NOUN
ejpam-4592	33	6	of	of	ADP
ejpam-4592	33	7	x	x	X
ejpam-4592	33	8	is	be	AUX
ejpam-4592	33	9	a	a	DET
ejpam-4592	33	10	generalized	generalized	ADJ
ejpam-4592	33	11	topology	topology	NOUN
ejpam-4592	34	1	[	[	X
ejpam-4592	34	2	7	7	X
ejpam-4592	34	3	]	]	PUNCT
ejpam-4592	34	4	in	in	ADP
ejpam-4592	34	5	x	x	SYM
ejpam-4592	34	6	if	if	SCONJ
ejpam-4592	34	7	it	it	PRON
ejpam-4592	34	8	contains	contain	VERB
ejpam-4592	34	9	the	the	DET
ejpam-4592	34	10	empty	empty	ADJ
ejpam-4592	34	11	set	set	NOUN
ejpam-4592	34	12	and	and	CCONJ
ejpam-4592	34	13	it	it	PRON
ejpam-4592	34	14	closed	close	VERB
ejpam-4592	34	15	under	under	ADP
ejpam-4592	34	16	arbitrary	arbitrary	ADJ
ejpam-4592	34	17	union	union	NOUN
ejpam-4592	34	18	.	.	PUNCT
ejpam-4592	35	1	then	then	ADV
ejpam-4592	35	2	the	the	DET
ejpam-4592	35	3	pair	pair	NOUN
ejpam-4592	35	4	(	(	PUNCT
ejpam-4592	35	5	x,µ	x,µ	NOUN
ejpam-4592	35	6	)	)	PUNCT
ejpam-4592	35	7	is	be	AUX
ejpam-4592	35	8	called	call	VERB
ejpam-4592	35	9	as	as	ADP
ejpam-4592	35	10	a	a	DET
ejpam-4592	35	11	generalized	generalized	ADJ
ejpam-4592	35	12	topological	topological	ADJ
ejpam-4592	35	13	space	space	NOUN
ejpam-4592	35	14	(	(	PUNCT
ejpam-4592	35	15	gts	gts	NOUN
ejpam-4592	35	16	)	)	PUNCT
ejpam-4592	36	1	[	[	X
ejpam-4592	36	2	7].the	7].the	DET
ejpam-4592	36	3	elements	element	NOUN
ejpam-4592	36	4	of	of	ADP
ejpam-4592	36	5	µ	µ	NOUN
ejpam-4592	36	6	are	be	AUX
ejpam-4592	36	7	called	call	VERB
ejpam-4592	36	8	µ-open	µ-open	NOUN
ejpam-4592	36	9	set	set	VERB
ejpam-4592	36	10	of	of	ADP
ejpam-4592	36	11	x.	x.	NOUN
ejpam-4592	36	12	moreover	moreover	ADV
ejpam-4592	36	13	,	,	PUNCT
ejpam-4592	36	14	if	if	SCONJ
ejpam-4592	36	15	(	(	PUNCT
ejpam-4592	36	16	x	x	X
ejpam-4592	36	17	,	,	PUNCT
ejpam-4592	36	18	ω	ω	NUM
ejpam-4592	36	19	)	)	PUNCT
ejpam-4592	36	20	is	be	AUX
ejpam-4592	36	21	a	a	DET
ejpam-4592	36	22	gms	gms	NOUN
ejpam-4592	36	23	,	,	PUNCT
ejpam-4592	36	24	then	then	ADV
ejpam-4592	36	25	(	(	PUNCT
ejpam-4592	36	26	x,µω	x,µω	NUM
ejpam-4592	36	27	)	)	PUNCT
ejpam-4592	36	28	is	be	AUX
ejpam-4592	36	29	a	a	DET
ejpam-4592	36	30	gts	gts	NOUN
ejpam-4592	36	31	[	[	X
ejpam-4592	36	32	7	7	NUM
ejpam-4592	36	33	]	]	PUNCT
ejpam-4592	36	34	.	.	PUNCT
ejpam-4592	37	1	now	now	ADV
ejpam-4592	37	2	we	we	PRON
ejpam-4592	37	3	remember	remember	VERB
ejpam-4592	37	4	some	some	DET
ejpam-4592	37	5	definitions	definition	NOUN
ejpam-4592	37	6	and	and	CCONJ
ejpam-4592	37	7	lemmas	lemma	NOUN
ejpam-4592	37	8	that	that	PRON
ejpam-4592	37	9	are	be	AUX
ejpam-4592	37	10	found	find	VERB
ejpam-4592	37	11	in	in	ADP
ejpam-4592	37	12	[	[	X
ejpam-4592	37	13	7	7	NUM
ejpam-4592	37	14	]	]	PUNCT
ejpam-4592	37	15	.	.	PUNCT
ejpam-4592	38	1	definition	definition	NOUN
ejpam-4592	38	2	1	1	NUM
ejpam-4592	38	3	.	.	PUNCT
ejpam-4592	39	1	a	a	DET
ejpam-4592	39	2	subset	subset	NOUN
ejpam-4592	39	3	q	q	NOUN
ejpam-4592	39	4	of	of	ADP
ejpam-4592	39	5	x	x	NOUN
ejpam-4592	39	6	is	be	AUX
ejpam-4592	39	7	said	say	VERB
ejpam-4592	39	8	to	to	PART
ejpam-4592	39	9	be	be	AUX
ejpam-4592	39	10	µ-dense	µ-dense	NOUN
ejpam-4592	39	11	if	if	SCONJ
ejpam-4592	39	12	cµ(q	cµ(q	NOUN
ejpam-4592	39	13	)	)	PUNCT
ejpam-4592	39	14	=	=	PUNCT
ejpam-4592	39	15	x.	x.	NOUN
ejpam-4592	39	16	definition	definition	NOUN
ejpam-4592	39	17	2	2	NUM
ejpam-4592	39	18	.	.	PUNCT
ejpam-4592	40	1	let	let	VERB
ejpam-4592	40	2	(	(	PUNCT
ejpam-4592	40	3	x	x	NOUN
ejpam-4592	40	4	,	,	PUNCT
ejpam-4592	40	5	ω	ω	NUM
ejpam-4592	40	6	)	)	PUNCT
ejpam-4592	40	7	be	be	AUX
ejpam-4592	40	8	a	a	DET
ejpam-4592	40	9	gms	gms	NOUN
ejpam-4592	40	10	.	.	PUNCT
ejpam-4592	41	1	a	a	DET
ejpam-4592	41	2	finite	finite	ADJ
ejpam-4592	41	3	family	family	NOUN
ejpam-4592	41	4	ω0	ω0	PROPN
ejpam-4592	41	5	⊂	⊂	PROPN
ejpam-4592	41	6	ω	ω	PROPN
ejpam-4592	41	7	is	be	AUX
ejpam-4592	41	8	called	call	VERB
ejpam-4592	41	9	a	a	DET
ejpam-4592	41	10	kernel	kernel	NOUN
ejpam-4592	41	11	of	of	ADP
ejpam-4592	41	12	the	the	DET
ejpam-4592	41	13	space	space	NOUN
ejpam-4592	41	14	x	x	NOUN
ejpam-4592	41	15	if	if	SCONJ
ejpam-4592	41	16	for	for	ADP
ejpam-4592	41	17	any	any	DET
ejpam-4592	41	18	j	j	PROPN
ejpam-4592	41	19	∈	∈	PROPN
ejpam-4592	41	20	µ̃ω	µ̃ω	NOUN
ejpam-4592	41	21	,	,	PUNCT
ejpam-4592	41	22	there	there	PRON
ejpam-4592	41	23	exists	exist	VERB
ejpam-4592	41	24	σ	σ	PROPN
ejpam-4592	41	25	∈	∈	PROPN
ejpam-4592	41	26	ω0	ω0	VERB
ejpam-4592	41	27	such	such	ADJ
ejpam-4592	41	28	that	that	SCONJ
ejpam-4592	41	29	iσ(j	iσ(j	X
ejpam-4592	41	30	)	)	PUNCT
ejpam-4592	41	31	̸=	̸=	NOUN
ejpam-4592	41	32	∅	∅	NOUN
ejpam-4592	41	33	where	where	SCONJ
ejpam-4592	41	34	µ̃ω	µ̃ω	ADV
ejpam-4592	41	35	=	=	PRON
ejpam-4592	41	36	{	{	PUNCT
ejpam-4592	41	37	k	k	PROPN
ejpam-4592	41	38	∈	∈	PROPN
ejpam-4592	41	39	µ̃ω|k	µ̃ω|k	PROPN
ejpam-4592	41	40	̸=	̸=	PROPN
ejpam-4592	41	41	∅	∅	NOUN
ejpam-4592	41	42	}	}	PUNCT
ejpam-4592	41	43	.	.	PUNCT
ejpam-4592	42	1	definition	definition	NOUN
ejpam-4592	42	2	3	3	X
ejpam-4592	42	3	.	.	PUNCT
ejpam-4592	43	1	let	let	VERB
ejpam-4592	43	2	(	(	PUNCT
ejpam-4592	43	3	x	x	NOUN
ejpam-4592	43	4	,	,	PUNCT
ejpam-4592	43	5	ω	ω	NUM
ejpam-4592	43	6	)	)	PUNCT
ejpam-4592	43	7	be	be	AUX
ejpam-4592	43	8	a	a	DET
ejpam-4592	43	9	gms	gms	NOUN
ejpam-4592	43	10	and	and	CCONJ
ejpam-4592	43	11	{	{	PUNCT
ejpam-4592	43	12	zn}n∈n	zn}n∈n	NUM
ejpam-4592	43	13	be	be	AUX
ejpam-4592	43	14	a	a	DET
ejpam-4592	43	15	sequence	sequence	NOUN
ejpam-4592	43	16	in	in	ADP
ejpam-4592	43	17	x.	x.	NOUN
ejpam-4592	43	18	then	then	ADV
ejpam-4592	43	19	{	{	PUNCT
ejpam-4592	43	20	zn	zn	X
ejpam-4592	43	21	}	}	PUNCT
ejpam-4592	43	22	is	be	AUX
ejpam-4592	43	23	said	say	VERB
ejpam-4592	43	24	to	to	PART
ejpam-4592	43	25	be	be	AUX
ejpam-4592	43	26	:	:	PUNCT
ejpam-4592	43	27	(	(	PUNCT
ejpam-4592	43	28	i	i	NOUN
ejpam-4592	43	29	)	)	PUNCT
ejpam-4592	43	30	cauchy	cauchy	VERB
ejpam-4592	43	31	if	if	SCONJ
ejpam-4592	43	32	there	there	PRON
ejpam-4592	43	33	is	be	VERB
ejpam-4592	43	34	a	a	DET
ejpam-4592	43	35	metric	metric	ADJ
ejpam-4592	43	36	σ	σ	X
ejpam-4592	43	37	∈	∈	PROPN
ejpam-4592	43	38	ω	ω	NOUN
ejpam-4592	43	39	and	and	CCONJ
ejpam-4592	43	40	for	for	ADP
ejpam-4592	43	41	every	every	DET
ejpam-4592	43	42	ε	ε	PROPN
ejpam-4592	43	43	>	>	X
ejpam-4592	43	44	0	0	PROPN
ejpam-4592	43	45	,	,	PUNCT
ejpam-4592	43	46	there	there	PRON
ejpam-4592	43	47	exists	exist	VERB
ejpam-4592	43	48	a	a	DET
ejpam-4592	43	49	positive	positive	ADJ
ejpam-4592	43	50	integer	integer	NOUN
ejpam-4592	43	51	n0	n0	NOUN
ejpam-4592	43	52	such	such	ADJ
ejpam-4592	43	53	that	that	SCONJ
ejpam-4592	43	54	σ(zn	σ(zn	PROPN
ejpam-4592	43	55	,	,	PUNCT
ejpam-4592	43	56	zm	zm	PROPN
ejpam-4592	43	57	)	)	PUNCT
ejpam-4592	43	58	<	<	X
ejpam-4592	43	59	ε	ε	PROPN
ejpam-4592	43	60	for	for	ADP
ejpam-4592	43	61	all	all	DET
ejpam-4592	43	62	n	n	CCONJ
ejpam-4592	43	63	,	,	PUNCT
ejpam-4592	43	64	m	m	PROPN
ejpam-4592	43	65	≥	≥	NOUN
ejpam-4592	43	66	n0	n0	NUM
ejpam-4592	43	67	.	.	PUNCT
ejpam-4592	44	1	(	(	PUNCT
ejpam-4592	44	2	ii	ii	NOUN
ejpam-4592	44	3	)	)	PUNCT
ejpam-4592	44	4	convergent	convergent	NOUN
ejpam-4592	44	5	to	to	ADP
ejpam-4592	44	6	z	z	PROPN
ejpam-4592	44	7	∈	∈	PROPN
ejpam-4592	44	8	x	x	INTJ
ejpam-4592	44	9	if	if	SCONJ
ejpam-4592	44	10	there	there	PRON
ejpam-4592	44	11	is	be	VERB
ejpam-4592	44	12	a	a	DET
ejpam-4592	44	13	metric	metric	ADJ
ejpam-4592	44	14	σ	σ	X
ejpam-4592	44	15	∈	∈	PROPN
ejpam-4592	44	16	ω	ω	NOUN
ejpam-4592	44	17	and	and	CCONJ
ejpam-4592	44	18	for	for	ADP
ejpam-4592	44	19	every	every	DET
ejpam-4592	44	20	ε	ε	PROPN
ejpam-4592	44	21	>	>	X
ejpam-4592	44	22	0	0	PROPN
ejpam-4592	44	23	,	,	PUNCT
ejpam-4592	44	24	there	there	PRON
ejpam-4592	44	25	exists	exist	VERB
ejpam-4592	44	26	a	a	DET
ejpam-4592	44	27	positive	positive	ADJ
ejpam-4592	44	28	integer	integer	NOUN
ejpam-4592	44	29	n1	n1	NOUN
ejpam-4592	44	30	such	such	ADJ
ejpam-4592	44	31	that	that	SCONJ
ejpam-4592	44	32	σ(zn	σ(zn	PROPN
ejpam-4592	44	33	,	,	PUNCT
ejpam-4592	44	34	z	z	NOUN
ejpam-4592	44	35	)	)	PUNCT
ejpam-4592	44	36	<	<	X
ejpam-4592	44	37	ε	ε	PROPN
ejpam-4592	44	38	for	for	ADP
ejpam-4592	44	39	all	all	DET
ejpam-4592	44	40	n	n	CCONJ
ejpam-4592	44	41	,	,	PUNCT
ejpam-4592	44	42	m	m	PROPN
ejpam-4592	44	43	≥	≥	NOUN
ejpam-4592	44	44	n1	n1	PROPN
ejpam-4592	44	45	.	.	PUNCT
ejpam-4592	45	1	then	then	ADV
ejpam-4592	45	2	z	z	PROPN
ejpam-4592	45	3	is	be	AUX
ejpam-4592	45	4	called	call	VERB
ejpam-4592	45	5	the	the	DET
ejpam-4592	45	6	limit	limit	NOUN
ejpam-4592	45	7	point	point	NOUN
ejpam-4592	45	8	of	of	ADP
ejpam-4592	45	9	the	the	DET
ejpam-4592	45	10	sequence	sequence	NOUN
ejpam-4592	45	11	{	{	PUNCT
ejpam-4592	45	12	zn}n∈n	zn}n∈n	NUM
ejpam-4592	45	13	.	.	PUNCT
ejpam-4592	45	14	definition	definition	NOUN
ejpam-4592	45	15	4	4	NUM
ejpam-4592	45	16	.	.	PUNCT
ejpam-4592	46	1	a	a	DET
ejpam-4592	46	2	generalized	generalized	ADJ
ejpam-4592	46	3	metric	metric	ADJ
ejpam-4592	46	4	space	space	NOUN
ejpam-4592	46	5	(	(	PUNCT
ejpam-4592	46	6	x	x	X
ejpam-4592	46	7	,	,	PUNCT
ejpam-4592	46	8	ω	ω	NUM
ejpam-4592	46	9	)	)	PUNCT
ejpam-4592	46	10	is	be	AUX
ejpam-4592	46	11	a	a	DET
ejpam-4592	46	12	weakly	weakly	ADJ
ejpam-4592	46	13	complete	complete	ADJ
ejpam-4592	46	14	metric	metric	ADJ
ejpam-4592	46	15	space	space	NOUN
ejpam-4592	46	16	if	if	SCONJ
ejpam-4592	46	17	there	there	PRON
ejpam-4592	46	18	exists	exist	VERB
ejpam-4592	46	19	a	a	DET
ejpam-4592	46	20	kernel	kernel	PROPN
ejpam-4592	46	21	ω0	ω0	PROPN
ejpam-4592	46	22	⊂	⊂	PROPN
ejpam-4592	46	23	ω	ω	PROPN
ejpam-4592	46	24	consisting	consist	VERB
ejpam-4592	46	25	of	of	ADP
ejpam-4592	46	26	complete	complete	ADJ
ejpam-4592	46	27	metrics	metric	NOUN
ejpam-4592	46	28	.	.	PUNCT
ejpam-4592	47	1	definition	definition	NOUN
ejpam-4592	47	2	5	5	NUM
ejpam-4592	47	3	.	.	PUNCT
ejpam-4592	48	1	a	a	DET
ejpam-4592	48	2	space	space	NOUN
ejpam-4592	48	3	x	x	PUNCT
ejpam-4592	48	4	is	be	AUX
ejpam-4592	48	5	called	call	VERB
ejpam-4592	48	6	as	as	ADP
ejpam-4592	48	7	a	a	DET
ejpam-4592	48	8	hyperconnected	hyperconnecte	VERB
ejpam-4592	48	9	space	space	NOUN
ejpam-4592	48	10	[	[	X
ejpam-4592	48	11	6	6	NUM
ejpam-4592	48	12	]	]	PUNCT
ejpam-4592	48	13	if	if	SCONJ
ejpam-4592	48	14	every	every	DET
ejpam-4592	48	15	non	non	ADJ
ejpam-4592	48	16	-	-	ADJ
ejpam-4592	48	17	null	null	ADJ
ejpam-4592	48	18	µ-open	µ-open	NOUN
ejpam-4592	48	19	subset	subset	VERB
ejpam-4592	48	20	h	h	NOUN
ejpam-4592	48	21	of	of	ADP
ejpam-4592	48	22	x	x	PUNCT
ejpam-4592	48	23	is	be	AUX
ejpam-4592	48	24	a	a	DET
ejpam-4592	48	25	µ-dense	µ-dense	NOUN
ejpam-4592	48	26	set	set	NOUN
ejpam-4592	48	27	.	.	PUNCT
ejpam-4592	49	1	v.	v.	ADP
ejpam-4592	49	2	subramanian	subramanian	PROPN
ejpam-4592	49	3	et	et	PROPN
ejpam-4592	49	4	al	al	PROPN
ejpam-4592	49	5	.	.	PUNCT
ejpam-4592	49	6	/	/	SYM
ejpam-4592	49	7	eur	eur	PROPN
ejpam-4592	49	8	.	.	PUNCT
ejpam-4592	50	1	j.	j.	PROPN
ejpam-4592	50	2	pure	pure	PROPN
ejpam-4592	50	3	appl	appl	PROPN
ejpam-4592	50	4	.	.	PROPN
ejpam-4592	50	5	math	math	PROPN
ejpam-4592	50	6	,	,	PUNCT
ejpam-4592	50	7	15	15	NUM
ejpam-4592	50	8	(	(	PUNCT
ejpam-4592	50	9	4	4	NUM
ejpam-4592	50	10	)	)	PUNCT
ejpam-4592	50	11	(	(	PUNCT
ejpam-4592	50	12	2022	2022	NUM
ejpam-4592	50	13	)	)	PUNCT
ejpam-4592	50	14	,	,	PUNCT
ejpam-4592	50	15	1869	1869	NUM
ejpam-4592	50	16	-	-	SYM
ejpam-4592	50	17	1886	1886	NUM
ejpam-4592	50	18	1871	1871	NUM
ejpam-4592	50	19	3	3	X
ejpam-4592	50	20	.	.	X
ejpam-4592	50	21	relationship	relationship	NOUN
ejpam-4592	50	22	among	among	ADP
ejpam-4592	50	23	sequences	sequence	NOUN
ejpam-4592	50	24	here	here	ADV
ejpam-4592	51	1	,	,	PUNCT
ejpam-4592	51	2	we	we	PRON
ejpam-4592	51	3	degenerate	degenerate	VERB
ejpam-4592	51	4	three	three	NUM
ejpam-4592	51	5	kinds	kind	NOUN
ejpam-4592	51	6	of	of	ADP
ejpam-4592	51	7	sequences	sequence	NOUN
ejpam-4592	51	8	namely	namely	ADV
ejpam-4592	51	9	,	,	PUNCT
ejpam-4592	51	10	bourbaki	bourbaki	NOUN
ejpam-4592	51	11	-	-	PUNCT
ejpam-4592	51	12	cauchy	cauchy	ADJ
ejpam-4592	51	13	,	,	PUNCT
ejpam-4592	51	14	pseudocauchy	pseudocauchy	NOUN
ejpam-4592	51	15	and	and	CCONJ
ejpam-4592	51	16	cofinally	cofinally	ADV
ejpam-4592	51	17	-	-	PUNCT
ejpam-4592	51	18	cauchy	cauchy	ADJ
ejpam-4592	51	19	sequences	sequence	NOUN
ejpam-4592	51	20	in	in	ADP
ejpam-4592	51	21	generalized	generalized	ADJ
ejpam-4592	51	22	metric	metric	ADJ
ejpam-4592	51	23	spaces	space	NOUN
ejpam-4592	51	24	.	.	PUNCT
ejpam-4592	52	1	in	in	ADP
ejpam-4592	52	2	a	a	DET
ejpam-4592	52	3	generalized	generalized	ADJ
ejpam-4592	52	4	metric	metric	ADJ
ejpam-4592	52	5	space	space	NOUN
ejpam-4592	52	6	,	,	PUNCT
ejpam-4592	52	7	the	the	DET
ejpam-4592	52	8	nature	nature	NOUN
ejpam-4592	52	9	of	of	ADP
ejpam-4592	52	10	these	these	DET
ejpam-4592	52	11	sequences	sequence	NOUN
ejpam-4592	52	12	is	be	AUX
ejpam-4592	52	13	analyzed	analyze	VERB
ejpam-4592	52	14	.	.	PUNCT
ejpam-4592	53	1	in	in	ADP
ejpam-4592	53	2	view	view	NOUN
ejpam-4592	53	3	of	of	ADP
ejpam-4592	53	4	complete	complete	ADJ
ejpam-4592	53	5	metric	metric	ADJ
ejpam-4592	53	6	spaces	space	NOUN
ejpam-4592	53	7	founded	found	VERB
ejpam-4592	53	8	in	in	ADP
ejpam-4592	53	9	[	[	X
ejpam-4592	53	10	7	7	NUM
ejpam-4592	53	11	]	]	PUNCT
ejpam-4592	53	12	,	,	PUNCT
ejpam-4592	53	13	we	we	PRON
ejpam-4592	53	14	define	define	VERB
ejpam-4592	53	15	three	three	NUM
ejpam-4592	53	16	types	type	NOUN
ejpam-4592	53	17	of	of	ADP
ejpam-4592	53	18	complete	complete	ADJ
ejpam-4592	53	19	metric	metric	ADJ
ejpam-4592	53	20	spaces	space	NOUN
ejpam-4592	53	21	,	,	PUNCT
ejpam-4592	53	22	namely	namely	ADV
ejpam-4592	53	23	bourbaki	bourbaki	VERB
ejpam-4592	53	24	-	-	PUNCT
ejpam-4592	53	25	complete	complete	ADJ
ejpam-4592	53	26	,	,	PUNCT
ejpam-4592	53	27	pseudo	pseudo	NOUN
ejpam-4592	53	28	-	-	NOUN
ejpam-4592	53	29	complete	complete	ADJ
ejpam-4592	53	30	and	and	CCONJ
ejpam-4592	53	31	cofinally	cofinally	ADV
ejpam-4592	53	32	-	-	PUNCT
ejpam-4592	53	33	complete	complete	ADJ
ejpam-4592	53	34	metric	metric	ADJ
ejpam-4592	53	35	spaces	space	NOUN
ejpam-4592	53	36	.	.	PUNCT
ejpam-4592	54	1	the	the	DET
ejpam-4592	54	2	relationship	relationship	NOUN
ejpam-4592	54	3	between	between	ADP
ejpam-4592	54	4	these	these	DET
ejpam-4592	54	5	complete	complete	ADJ
ejpam-4592	54	6	spaces	space	NOUN
ejpam-4592	54	7	is	be	AUX
ejpam-4592	54	8	explored	explore	VERB
ejpam-4592	54	9	.	.	PUNCT
ejpam-4592	55	1	definition	definition	NOUN
ejpam-4592	55	2	6	6	NUM
ejpam-4592	55	3	.	.	PUNCT
ejpam-4592	56	1	let	let	VERB
ejpam-4592	56	2	(	(	PUNCT
ejpam-4592	56	3	x	x	NOUN
ejpam-4592	56	4	,	,	PUNCT
ejpam-4592	56	5	ω	ω	NUM
ejpam-4592	56	6	)	)	PUNCT
ejpam-4592	56	7	be	be	AUX
ejpam-4592	56	8	a	a	DET
ejpam-4592	56	9	generalized	generalized	ADJ
ejpam-4592	56	10	metric	metric	ADJ
ejpam-4592	56	11	space	space	NOUN
ejpam-4592	56	12	and	and	CCONJ
ejpam-4592	56	13	ε	ε	PROPN
ejpam-4592	56	14	be	be	VERB
ejpam-4592	56	15	a	a	DET
ejpam-4592	56	16	positive	positive	ADJ
ejpam-4592	56	17	number	number	NOUN
ejpam-4592	56	18	.	.	PUNCT
ejpam-4592	57	1	an	an	DET
ejpam-4592	57	2	ordered	order	VERB
ejpam-4592	57	3	set	set	NOUN
ejpam-4592	57	4	of	of	ADP
ejpam-4592	57	5	points	point	NOUN
ejpam-4592	57	6	{	{	PUNCT
ejpam-4592	57	7	c0	c0	NOUN
ejpam-4592	57	8	,	,	PUNCT
ejpam-4592	57	9	c1	c1	PROPN
ejpam-4592	57	10	,	,	PUNCT
ejpam-4592	57	11	...	...	PUNCT
ejpam-4592	57	12	,	,	PUNCT
ejpam-4592	57	13	cm	cm	NOUN
ejpam-4592	57	14	}	}	PUNCT
ejpam-4592	57	15	is	be	AUX
ejpam-4592	57	16	said	say	VERB
ejpam-4592	57	17	to	to	PART
ejpam-4592	57	18	be	be	AUX
ejpam-4592	57	19	ε	ε	PROPN
ejpam-4592	57	20	-	-	PUNCT
ejpam-4592	57	21	chain	chain	NOUN
ejpam-4592	57	22	of	of	ADP
ejpam-4592	57	23	length	length	NOUN
ejpam-4592	57	24	of	of	ADP
ejpam-4592	57	25	m	m	PROPN
ejpam-4592	57	26	from	from	ADP
ejpam-4592	57	27	c0	c0	PROPN
ejpam-4592	57	28	to	to	ADP
ejpam-4592	57	29	cm	cm	PROPN
ejpam-4592	57	30	if	if	SCONJ
ejpam-4592	57	31	there	there	PRON
ejpam-4592	57	32	is	be	VERB
ejpam-4592	57	33	a	a	DET
ejpam-4592	57	34	metric	metric	ADJ
ejpam-4592	57	35	σ	σ	PROPN
ejpam-4592	57	36	∈	∈	PROPN
ejpam-4592	57	37	ω	ω	X
ejpam-4592	57	38	satisfying	satisfy	VERB
ejpam-4592	57	39	σ(ci−1	σ(ci−1	PROPN
ejpam-4592	57	40	,	,	PUNCT
ejpam-4592	57	41	ci	ci	NOUN
ejpam-4592	57	42	)	)	PUNCT
ejpam-4592	57	43	<	<	X
ejpam-4592	57	44	ε	ε	PROPN
ejpam-4592	57	45	where	where	SCONJ
ejpam-4592	57	46	i	i	PRON
ejpam-4592	57	47	=	=	NOUN
ejpam-4592	57	48	1	1	NUM
ejpam-4592	57	49	to	to	PART
ejpam-4592	57	50	m.	m.	VERB
ejpam-4592	57	51	example	example	NOUN
ejpam-4592	57	52	7	7	X
ejpam-4592	57	53	.	.	PUNCT
ejpam-4592	57	54	consider	consider	VERB
ejpam-4592	57	55	the	the	DET
ejpam-4592	57	56	generalized	generalized	ADJ
ejpam-4592	57	57	metric	metric	ADJ
ejpam-4592	57	58	space	space	NOUN
ejpam-4592	57	59	(	(	PUNCT
ejpam-4592	57	60	x	x	X
ejpam-4592	57	61	,	,	PUNCT
ejpam-4592	57	62	ω	ω	NOUN
ejpam-4592	57	63	)	)	PUNCT
ejpam-4592	57	64	where	where	SCONJ
ejpam-4592	57	65	x	x	X
ejpam-4592	57	66	=	=	SYM
ejpam-4592	57	67	r	r	PROPN
ejpam-4592	57	68	,	,	PUNCT
ejpam-4592	57	69	ω	ω	NOUN
ejpam-4592	57	70	=	=	SYM
ejpam-4592	57	71	{	{	PUNCT
ejpam-4592	57	72	σ1	σ1	PROPN
ejpam-4592	57	73	,	,	PUNCT
ejpam-4592	57	74	σ2	σ2	PROPN
ejpam-4592	57	75	,	,	PUNCT
ejpam-4592	57	76	σ3	σ3	PROPN
ejpam-4592	57	77	}	}	PUNCT
ejpam-4592	57	78	and	and	CCONJ
ejpam-4592	57	79	the	the	DET
ejpam-4592	57	80	metrics	metric	NOUN
ejpam-4592	57	81	are	be	AUX
ejpam-4592	57	82	defined	define	VERB
ejpam-4592	57	83	by	by	ADP
ejpam-4592	57	84	σ1(c	σ1(c	PRON
ejpam-4592	57	85	,	,	PUNCT
ejpam-4592	57	86	d	d	NOUN
ejpam-4592	57	87	)	)	PUNCT
ejpam-4592	58	1	=	=	SYM
ejpam-4592	58	2	|c−	|c−	VERB
ejpam-4592	58	3	d|	d|	PROPN
ejpam-4592	58	4	;	;	PUNCT
ejpam-4592	58	5	σ2(c	σ2(c	NOUN
ejpam-4592	58	6	,	,	PUNCT
ejpam-4592	58	7	d	d	NOUN
ejpam-4592	58	8	)	)	PUNCT
ejpam-4592	58	9	=	=	PRON
ejpam-4592	58	10	{	{	PUNCT
ejpam-4592	58	11	0	0	NUM
ejpam-4592	59	1	if	if	SCONJ
ejpam-4592	59	2	c	c	NOUN
ejpam-4592	59	3	=	=	SYM
ejpam-4592	59	4	d	d	PROPN
ejpam-4592	59	5	,	,	PUNCT
ejpam-4592	59	6	1	1	NUM
ejpam-4592	59	7	if	if	SCONJ
ejpam-4592	59	8	c	c	PROPN
ejpam-4592	59	9	̸=	̸=	PROPN
ejpam-4592	59	10	d.	d.	PROPN
ejpam-4592	59	11	and	and	CCONJ
ejpam-4592	59	12	σ3(c	σ3(c	PRON
ejpam-4592	59	13	,	,	PUNCT
ejpam-4592	59	14	d	d	NOUN
ejpam-4592	59	15	)	)	PUNCT
ejpam-4592	59	16	=	=	SYM
ejpam-4592	60	1	σ1(c	σ1(c	PROPN
ejpam-4592	60	2	,	,	PUNCT
ejpam-4592	60	3	d	d	NOUN
ejpam-4592	60	4	)	)	PUNCT
ejpam-4592	60	5	1+σ1(c	1+σ1(c	NOUN
ejpam-4592	60	6	,	,	PUNCT
ejpam-4592	60	7	d	d	NOUN
ejpam-4592	60	8	)	)	PUNCT
ejpam-4592	60	9	for	for	ADP
ejpam-4592	60	10	all	all	DET
ejpam-4592	60	11	c	c	NOUN
ejpam-4592	60	12	,	,	PUNCT
ejpam-4592	60	13	d	d	PROPN
ejpam-4592	60	14	∈	∈	PROPN
ejpam-4592	60	15	x.	x.	NOUN
ejpam-4592	60	16	(	(	PUNCT
ejpam-4592	60	17	a	a	X
ejpam-4592	60	18	)	)	PUNCT
ejpam-4592	60	19	.	.	PUNCT
ejpam-4592	61	1	let	let	VERB
ejpam-4592	61	2	ε	ε	PROPN
ejpam-4592	61	3	=	=	SYM
ejpam-4592	61	4	0.5	0.5	NUM
ejpam-4592	61	5	and	and	CCONJ
ejpam-4592	61	6	{	{	PUNCT
ejpam-4592	61	7	cm}m=13	cm}m=13	NOUN
ejpam-4592	61	8	m=0	m=0	PROPN
ejpam-4592	61	9	be	be	AUX
ejpam-4592	61	10	a	a	DET
ejpam-4592	61	11	set	set	NOUN
ejpam-4592	61	12	of	of	ADP
ejpam-4592	61	13	points	point	NOUN
ejpam-4592	61	14	where	where	SCONJ
ejpam-4592	61	15	c0	c0	NOUN
ejpam-4592	61	16	=	=	PROPN
ejpam-4592	61	17	1	1	NUM
ejpam-4592	61	18	;	;	PUNCT
ejpam-4592	61	19	cm	cm	NOUN
ejpam-4592	61	20	=	=	SYM
ejpam-4592	61	21	cm−1	cm−1	NOUN
ejpam-4592	61	22	+	+	CCONJ
ejpam-4592	61	23	0.01	0.01	NUM
ejpam-4592	61	24	for	for	ADP
ejpam-4592	61	25	m	m	NOUN
ejpam-4592	61	26	=	=	SYM
ejpam-4592	61	27	1	1	NUM
ejpam-4592	61	28	to	to	ADP
ejpam-4592	61	29	13	13	NUM
ejpam-4592	61	30	.	.	PUNCT
ejpam-4592	62	1	then	then	ADV
ejpam-4592	62	2	the	the	DET
ejpam-4592	62	3	set	set	NOUN
ejpam-4592	62	4	of	of	ADP
ejpam-4592	62	5	points	point	NOUN
ejpam-4592	62	6	{	{	PUNCT
ejpam-4592	62	7	cm}m=13	cm}m=13	NOUN
ejpam-4592	62	8	m=0	m=0	PROPN
ejpam-4592	62	9	have	have	VERB
ejpam-4592	62	10	a	a	DET
ejpam-4592	62	11	ε	ε	NOUN
ejpam-4592	62	12	-	-	PUNCT
ejpam-4592	62	13	chain	chain	NOUN
ejpam-4592	62	14	of	of	ADP
ejpam-4592	62	15	length	length	NOUN
ejpam-4592	62	16	of	of	ADP
ejpam-4592	62	17	m	m	PROPN
ejpam-4592	62	18	from	from	ADP
ejpam-4592	62	19	c0	c0	PROPN
ejpam-4592	62	20	to	to	ADP
ejpam-4592	62	21	cm	cm	PROPN
ejpam-4592	62	22	with	with	ADP
ejpam-4592	62	23	respect	respect	NOUN
ejpam-4592	62	24	to	to	ADP
ejpam-4592	62	25	σ1	σ1	PROPN
ejpam-4592	62	26	,	,	PUNCT
ejpam-4592	62	27	σ2	σ2	PROPN
ejpam-4592	62	28	and	and	CCONJ
ejpam-4592	62	29	σ3	σ3	PROPN
ejpam-4592	62	30	.	.	PUNCT
ejpam-4592	63	1	(	(	PUNCT
ejpam-4592	63	2	b	b	NOUN
ejpam-4592	63	3	)	)	PUNCT
ejpam-4592	63	4	.	.	PUNCT
ejpam-4592	64	1	let	let	VERB
ejpam-4592	64	2	ε	ε	PROPN
ejpam-4592	64	3	=	=	VERB
ejpam-4592	64	4	0.6	0.6	NUM
ejpam-4592	64	5	and	and	CCONJ
ejpam-4592	64	6	{	{	PUNCT
ejpam-4592	64	7	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	64	8	=	=	PRON
ejpam-4592	64	9	{	{	PUNCT
ejpam-4592	64	10	1	1	NUM
ejpam-4592	64	11	n}n∈n	n}n∈n	NOUN
ejpam-4592	64	12	be	be	AUX
ejpam-4592	64	13	a	a	DET
ejpam-4592	64	14	sequence	sequence	NOUN
ejpam-4592	64	15	in	in	ADP
ejpam-4592	64	16	x.	x.	NOUN
ejpam-4592	64	17	take	take	VERB
ejpam-4592	64	18	{	{	PUNCT
ejpam-4592	64	19	cm}m=13	cm}m=13	NOUN
ejpam-4592	64	20	m=0	m=0	PROPN
ejpam-4592	64	21	be	be	AUX
ejpam-4592	64	22	a	a	DET
ejpam-4592	64	23	set	set	NOUN
ejpam-4592	64	24	of	of	ADP
ejpam-4592	64	25	points	point	NOUN
ejpam-4592	64	26	where	where	SCONJ
ejpam-4592	64	27	c0	c0	NOUN
ejpam-4592	64	28	=	=	PROPN
ejpam-4592	64	29	1	1	NUM
ejpam-4592	64	30	;	;	PUNCT
ejpam-4592	64	31	cm	cm	NOUN
ejpam-4592	64	32	=	=	SYM
ejpam-4592	64	33	c0	c0	PROPN
ejpam-4592	64	34	m+1	m+1	NUM
ejpam-4592	64	35	for	for	ADP
ejpam-4592	64	36	m	m	PROPN
ejpam-4592	64	37	=	=	SYM
ejpam-4592	64	38	1	1	NUM
ejpam-4592	64	39	to	to	ADP
ejpam-4592	64	40	13	13	NUM
ejpam-4592	64	41	.	.	PUNCT
ejpam-4592	65	1	then	then	ADV
ejpam-4592	65	2	the	the	DET
ejpam-4592	65	3	set	set	NOUN
ejpam-4592	65	4	of	of	ADP
ejpam-4592	65	5	points	point	NOUN
ejpam-4592	65	6	{	{	PUNCT
ejpam-4592	65	7	cm}m=13	cm}m=13	NOUN
ejpam-4592	65	8	m=0	m=0	PROPN
ejpam-4592	65	9	have	have	VERB
ejpam-4592	65	10	a	a	DET
ejpam-4592	65	11	ε	ε	NOUN
ejpam-4592	65	12	-	-	PUNCT
ejpam-4592	65	13	chain	chain	NOUN
ejpam-4592	65	14	of	of	ADP
ejpam-4592	65	15	length	length	NOUN
ejpam-4592	65	16	of	of	ADP
ejpam-4592	65	17	m	m	PROPN
ejpam-4592	65	18	from	from	ADP
ejpam-4592	65	19	c0	c0	PROPN
ejpam-4592	65	20	to	to	ADP
ejpam-4592	65	21	cm	cm	PROPN
ejpam-4592	65	22	with	with	ADP
ejpam-4592	65	23	respect	respect	NOUN
ejpam-4592	65	24	to	to	ADP
ejpam-4592	65	25	σ1	σ1	PROPN
ejpam-4592	65	26	.	.	PUNCT
ejpam-4592	66	1	definition	definition	NOUN
ejpam-4592	66	2	8	8	NUM
ejpam-4592	66	3	.	.	PUNCT
ejpam-4592	67	1	let	let	VERB
ejpam-4592	67	2	(	(	PUNCT
ejpam-4592	67	3	x	x	NOUN
ejpam-4592	67	4	,	,	PUNCT
ejpam-4592	67	5	ω	ω	NUM
ejpam-4592	67	6	)	)	PUNCT
ejpam-4592	67	7	be	be	AUX
ejpam-4592	67	8	a	a	DET
ejpam-4592	67	9	generalized	generalized	ADJ
ejpam-4592	67	10	metric	metric	ADJ
ejpam-4592	67	11	space	space	NOUN
ejpam-4592	67	12	.	.	PUNCT
ejpam-4592	68	1	a	a	DET
ejpam-4592	68	2	sequence	sequence	NOUN
ejpam-4592	68	3	{	{	PUNCT
ejpam-4592	68	4	zn}n∈n	zn}n∈n	NUM
ejpam-4592	68	5	is	be	AUX
ejpam-4592	68	6	said	say	VERB
ejpam-4592	68	7	to	to	PART
ejpam-4592	68	8	be	be	AUX
ejpam-4592	68	9	:	:	PUNCT
ejpam-4592	68	10	(	(	PUNCT
ejpam-4592	68	11	a	a	X
ejpam-4592	68	12	)	)	PUNCT
ejpam-4592	68	13	bourbaki	bourbaki	NOUN
ejpam-4592	68	14	-	-	PUNCT
ejpam-4592	68	15	cauchy	cauchy	NOUN
ejpam-4592	68	16	with	with	ADP
ejpam-4592	68	17	respect	respect	NOUN
ejpam-4592	68	18	to	to	ADP
ejpam-4592	68	19	σ	σ	PROPN
ejpam-4592	68	20	∈	∈	PROPN
ejpam-4592	68	21	ω	ω	PROPN
ejpam-4592	68	22	in	in	ADP
ejpam-4592	68	23	x	x	PUNCT
ejpam-4592	68	24	if	if	SCONJ
ejpam-4592	68	25	for	for	ADP
ejpam-4592	68	26	every	every	DET
ejpam-4592	68	27	ε	ε	PROPN
ejpam-4592	68	28	>	>	X
ejpam-4592	68	29	0	0	PROPN
ejpam-4592	68	30	,	,	PUNCT
ejpam-4592	68	31	there	there	PRON
ejpam-4592	68	32	exist	exist	VERB
ejpam-4592	68	33	r	r	NOUN
ejpam-4592	68	34	∈	∈	PROPN
ejpam-4592	68	35	n	n	NOUN
ejpam-4592	68	36	and	and	CCONJ
ejpam-4592	68	37	n0	n0	NUM
ejpam-4592	68	38	∈	∈	PROPN
ejpam-4592	68	39	n	n	PRON
ejpam-4592	68	40	such	such	ADJ
ejpam-4592	68	41	that	that	SCONJ
ejpam-4592	68	42	whenever	whenever	SCONJ
ejpam-4592	68	43	n	n	PROPN
ejpam-4592	68	44	>	>	X
ejpam-4592	68	45	j	j	PROPN
ejpam-4592	68	46	≥	≥	PROPN
ejpam-4592	68	47	n0	n0	NUM
ejpam-4592	68	48	the	the	DET
ejpam-4592	68	49	points	point	NOUN
ejpam-4592	68	50	zj	zj	PROPN
ejpam-4592	68	51	and	and	CCONJ
ejpam-4592	68	52	zn	zn	PROPN
ejpam-4592	68	53	can	can	AUX
ejpam-4592	68	54	be	be	AUX
ejpam-4592	68	55	joined	join	VERB
ejpam-4592	68	56	by	by	ADP
ejpam-4592	68	57	an	an	DET
ejpam-4592	68	58	ε	ε	PROPN
ejpam-4592	68	59	-	-	PUNCT
ejpam-4592	68	60	chain	chain	NOUN
ejpam-4592	68	61	of	of	ADP
ejpam-4592	68	62	length	length	NOUN
ejpam-4592	68	63	r	r	NOUN
ejpam-4592	68	64	with	with	ADP
ejpam-4592	68	65	respect	respect	NOUN
ejpam-4592	68	66	to	to	ADP
ejpam-4592	68	67	the	the	DET
ejpam-4592	68	68	same	same	ADJ
ejpam-4592	68	69	metric	metric	NOUN
ejpam-4592	68	70	.	.	PUNCT
ejpam-4592	69	1	(	(	PUNCT
ejpam-4592	69	2	b	b	X
ejpam-4592	69	3	)	)	PUNCT
ejpam-4592	69	4	pseudo	pseudo	NOUN
ejpam-4592	69	5	-	-	NOUN
ejpam-4592	69	6	cauchy	cauchy	NOUN
ejpam-4592	69	7	with	with	ADP
ejpam-4592	69	8	respect	respect	NOUN
ejpam-4592	69	9	to	to	ADP
ejpam-4592	69	10	σ	σ	PROPN
ejpam-4592	69	11	∈	∈	PROPN
ejpam-4592	69	12	ω	ω	PROPN
ejpam-4592	69	13	in	in	ADP
ejpam-4592	69	14	x	x	PUNCT
ejpam-4592	69	15	if	if	SCONJ
ejpam-4592	69	16	for	for	ADP
ejpam-4592	69	17	every	every	DET
ejpam-4592	69	18	ε	ε	PROPN
ejpam-4592	69	19	>	>	X
ejpam-4592	69	20	0	0	PUNCT
ejpam-4592	69	21	and	and	CCONJ
ejpam-4592	69	22	for	for	ADP
ejpam-4592	69	23	every	every	DET
ejpam-4592	69	24	n	n	PRON
ejpam-4592	69	25	∈	∈	PROPN
ejpam-4592	69	26	n	n	CCONJ
ejpam-4592	69	27	,	,	PUNCT
ejpam-4592	69	28	there	there	PRON
ejpam-4592	69	29	exist	exist	VERB
ejpam-4592	69	30	s	s	PROPN
ejpam-4592	69	31	,	,	PUNCT
ejpam-4592	69	32	t	t	PROPN
ejpam-4592	69	33	∈	∈	PROPN
ejpam-4592	69	34	n	n	CCONJ
ejpam-4592	69	35	,	,	PUNCT
ejpam-4592	69	36	s	s	VERB
ejpam-4592	69	37	̸=	̸=	PROPN
ejpam-4592	69	38	t	t	NOUN
ejpam-4592	69	39	such	such	DET
ejpam-4592	69	40	that	that	DET
ejpam-4592	69	41	s	s	PROPN
ejpam-4592	69	42	,	,	PUNCT
ejpam-4592	69	43	t	t	PROPN
ejpam-4592	69	44	>	>	X
ejpam-4592	69	45	n	n	PROPN
ejpam-4592	69	46	and	and	CCONJ
ejpam-4592	69	47	σ(zs	σ(z	NOUN
ejpam-4592	69	48	,	,	PUNCT
ejpam-4592	69	49	zt	zt	PROPN
ejpam-4592	69	50	)	)	PUNCT
ejpam-4592	69	51	<	<	X
ejpam-4592	70	1	ε	ε	PROPN
ejpam-4592	70	2	.	.	PUNCT
ejpam-4592	70	3	(	(	PUNCT
ejpam-4592	70	4	c	c	X
ejpam-4592	70	5	)	)	PUNCT
ejpam-4592	70	6	cofinally	cofinally	ADV
ejpam-4592	70	7	-	-	PUNCT
ejpam-4592	70	8	cauchy	cauchy	NOUN
ejpam-4592	70	9	with	with	ADP
ejpam-4592	70	10	respect	respect	NOUN
ejpam-4592	70	11	to	to	ADP
ejpam-4592	70	12	σ	σ	PROPN
ejpam-4592	70	13	∈	∈	PROPN
ejpam-4592	70	14	ω	ω	PROPN
ejpam-4592	70	15	in	in	ADP
ejpam-4592	70	16	x	x	PUNCT
ejpam-4592	70	17	if	if	SCONJ
ejpam-4592	70	18	for	for	ADP
ejpam-4592	70	19	every	every	DET
ejpam-4592	70	20	ε	ε	PROPN
ejpam-4592	70	21	>	>	X
ejpam-4592	70	22	0	0	PROPN
ejpam-4592	70	23	,	,	PUNCT
ejpam-4592	70	24	there	there	PRON
ejpam-4592	70	25	exists	exist	VERB
ejpam-4592	70	26	an	an	DET
ejpam-4592	70	27	infinite	infinite	NOUN
ejpam-4592	70	28	subset	subset	NOUN
ejpam-4592	70	29	nε	nε	NOUN
ejpam-4592	70	30	of	of	ADP
ejpam-4592	70	31	n	n	PRON
ejpam-4592	70	32	such	such	ADJ
ejpam-4592	70	33	that	that	PRON
ejpam-4592	70	34	for	for	ADP
ejpam-4592	70	35	every	every	DET
ejpam-4592	70	36	l	l	NOUN
ejpam-4592	70	37	,	,	PUNCT
ejpam-4592	70	38	m	m	VERB
ejpam-4592	70	39	∈	∈	NOUN
ejpam-4592	70	40	nε	nε	ADJ
ejpam-4592	70	41	we	we	PRON
ejpam-4592	70	42	have	have	AUX
ejpam-4592	70	43	σ(zl	σ(zl	PROPN
ejpam-4592	70	44	,	,	PUNCT
ejpam-4592	70	45	zm	zm	PROPN
ejpam-4592	70	46	)	)	PUNCT
ejpam-4592	70	47	<	<	X
ejpam-4592	70	48	ε	ε	PROPN
ejpam-4592	70	49	.	.	PUNCT
ejpam-4592	71	1	we	we	PRON
ejpam-4592	71	2	notated	notate	VERB
ejpam-4592	71	3	by	by	ADP
ejpam-4592	71	4	,	,	PUNCT
ejpam-4592	71	5	b(σ	b(σ	PROPN
ejpam-4592	71	6	)	)	PUNCT
ejpam-4592	71	7	=	=	PRON
ejpam-4592	72	1	{	{	PUNCT
ejpam-4592	72	2	{	{	PUNCT
ejpam-4592	72	3	dn	dn	NOUN
ejpam-4592	72	4	}	}	PUNCT
ejpam-4592	72	5	|	|	ADV
ejpam-4592	72	6	{	{	PUNCT
ejpam-4592	72	7	dn	dn	VERB
ejpam-4592	72	8	}	}	PUNCT
ejpam-4592	72	9	is	be	AUX
ejpam-4592	72	10	a	a	DET
ejpam-4592	72	11	bourbaki	bourbaki	VERB
ejpam-4592	72	12	-	-	PUNCT
ejpam-4592	72	13	cauchy	cauchy	ADJ
ejpam-4592	72	14	sequence	sequence	NOUN
ejpam-4592	72	15	with	with	ADP
ejpam-4592	72	16	respect	respect	NOUN
ejpam-4592	72	17	to	to	ADP
ejpam-4592	72	18	σ	σ	PROPN
ejpam-4592	72	19	in	in	ADP
ejpam-4592	72	20	x	x	X
ejpam-4592	72	21	}	}	PUNCT
ejpam-4592	72	22	;	;	PUNCT
ejpam-4592	72	23	p(σ	p(σ	NOUN
ejpam-4592	72	24	)	)	PUNCT
ejpam-4592	72	25	=	=	PRON
ejpam-4592	72	26	{	{	PUNCT
ejpam-4592	72	27	{	{	PUNCT
ejpam-4592	72	28	ln	ln	NOUN
ejpam-4592	72	29	}	}	PUNCT
ejpam-4592	72	30	|	|	ADV
ejpam-4592	72	31	{	{	PUNCT
ejpam-4592	72	32	ln	ln	ADJ
ejpam-4592	72	33	}	}	PUNCT
ejpam-4592	72	34	is	be	AUX
ejpam-4592	72	35	a	a	DET
ejpam-4592	72	36	pseudo	pseudo	NOUN
ejpam-4592	72	37	-	-	ADJ
ejpam-4592	72	38	cauchy	cauchy	ADJ
ejpam-4592	72	39	sequence	sequence	NOUN
ejpam-4592	72	40	with	with	ADP
ejpam-4592	72	41	respect	respect	NOUN
ejpam-4592	72	42	to	to	ADP
ejpam-4592	72	43	σ	σ	PROPN
ejpam-4592	72	44	in	in	ADP
ejpam-4592	72	45	x	x	X
ejpam-4592	72	46	}	}	PUNCT
ejpam-4592	72	47	;	;	PUNCT
ejpam-4592	72	48	c(σ	c(σ	PROPN
ejpam-4592	72	49	)	)	PUNCT
ejpam-4592	72	50	=	=	PRON
ejpam-4592	72	51	{	{	PUNCT
ejpam-4592	72	52	{	{	PUNCT
ejpam-4592	72	53	ln	ln	NOUN
ejpam-4592	72	54	}	}	PUNCT
ejpam-4592	72	55	|	|	ADV
ejpam-4592	72	56	{	{	PUNCT
ejpam-4592	72	57	ln	ln	ADJ
ejpam-4592	72	58	}	}	PUNCT
ejpam-4592	72	59	is	be	AUX
ejpam-4592	72	60	a	a	DET
ejpam-4592	72	61	cofinally	cofinally	ADV
ejpam-4592	72	62	-	-	PUNCT
ejpam-4592	72	63	cauchy	cauchy	ADJ
ejpam-4592	72	64	sequence	sequence	NOUN
ejpam-4592	72	65	with	with	ADP
ejpam-4592	72	66	respect	respect	NOUN
ejpam-4592	72	67	to	to	ADP
ejpam-4592	72	68	σ	σ	PROPN
ejpam-4592	72	69	in	in	ADP
ejpam-4592	72	70	x	x	X
ejpam-4592	72	71	}	}	PUNCT
ejpam-4592	72	72	;	;	PUNCT
ejpam-4592	72	73	c(σ	c(σ	PROPN
ejpam-4592	72	74	)	)	PUNCT
ejpam-4592	72	75	=	=	PRON
ejpam-4592	72	76	{	{	PUNCT
ejpam-4592	72	77	{	{	PUNCT
ejpam-4592	72	78	un	un	PROPN
ejpam-4592	72	79	}	}	PUNCT
ejpam-4592	72	80	|	|	ADV
ejpam-4592	72	81	{	{	PUNCT
ejpam-4592	72	82	un	un	PROPN
ejpam-4592	72	83	}	}	PUNCT
ejpam-4592	72	84	is	be	AUX
ejpam-4592	72	85	a	a	DET
ejpam-4592	72	86	cauchy	cauchy	ADJ
ejpam-4592	72	87	sequence	sequence	NOUN
ejpam-4592	72	88	with	with	ADP
ejpam-4592	72	89	respect	respect	NOUN
ejpam-4592	72	90	to	to	ADP
ejpam-4592	72	91	σ	σ	PROPN
ejpam-4592	72	92	in	in	ADP
ejpam-4592	72	93	x	x	X
ejpam-4592	72	94	}	}	PUNCT
ejpam-4592	72	95	where	where	SCONJ
ejpam-4592	72	96	σ	σ	PROPN
ejpam-4592	72	97	∈	∈	PROPN
ejpam-4592	72	98	ω	ω	PROPN
ejpam-4592	72	99	.	.	PUNCT
ejpam-4592	73	1	v.	v.	ADP
ejpam-4592	73	2	subramanian	subramanian	PROPN
ejpam-4592	73	3	et	et	PROPN
ejpam-4592	73	4	al	al	PROPN
ejpam-4592	73	5	.	.	PUNCT
ejpam-4592	73	6	/	/	SYM
ejpam-4592	73	7	eur	eur	PROPN
ejpam-4592	73	8	.	.	PUNCT
ejpam-4592	74	1	j.	j.	PROPN
ejpam-4592	74	2	pure	pure	PROPN
ejpam-4592	74	3	appl	appl	PROPN
ejpam-4592	74	4	.	.	PROPN
ejpam-4592	74	5	math	math	PROPN
ejpam-4592	74	6	,	,	PUNCT
ejpam-4592	74	7	15	15	NUM
ejpam-4592	74	8	(	(	PUNCT
ejpam-4592	74	9	4	4	NUM
ejpam-4592	74	10	)	)	PUNCT
ejpam-4592	74	11	(	(	PUNCT
ejpam-4592	74	12	2022	2022	NUM
ejpam-4592	74	13	)	)	PUNCT
ejpam-4592	74	14	,	,	PUNCT
ejpam-4592	74	15	1869	1869	NUM
ejpam-4592	74	16	-	-	SYM
ejpam-4592	74	17	1886	1886	NUM
ejpam-4592	74	18	1872	1872	NUM
ejpam-4592	74	19	definition	definition	NOUN
ejpam-4592	74	20	9	9	NUM
ejpam-4592	74	21	.	.	PUNCT
ejpam-4592	75	1	let	let	VERB
ejpam-4592	75	2	(	(	PUNCT
ejpam-4592	75	3	x	x	NOUN
ejpam-4592	75	4	,	,	PUNCT
ejpam-4592	75	5	ω	ω	NUM
ejpam-4592	75	6	)	)	PUNCT
ejpam-4592	75	7	be	be	AUX
ejpam-4592	75	8	a	a	DET
ejpam-4592	75	9	gms	gms	NOUN
ejpam-4592	75	10	.	.	PUNCT
ejpam-4592	76	1	then	then	ADV
ejpam-4592	76	2	σ	σ	PROPN
ejpam-4592	76	3	∈	∈	PROPN
ejpam-4592	76	4	ω	ω	NOUN
ejpam-4592	76	5	is	be	AUX
ejpam-4592	76	6	called	call	VERB
ejpam-4592	76	7	bourbaki	bourbaki	NOUN
ejpam-4592	76	8	(	(	PUNCT
ejpam-4592	76	9	resp	resp	NOUN
ejpam-4592	76	10	.	.	PUNCT
ejpam-4592	77	1	pseudo	pseudo	NOUN
ejpam-4592	77	2	,	,	PUNCT
ejpam-4592	77	3	cofinally	cofinally	ADV
ejpam-4592	77	4	)	)	PUNCT
ejpam-4592	77	5	complete	complete	ADJ
ejpam-4592	77	6	metric	metric	NOUN
ejpam-4592	77	7	if	if	SCONJ
ejpam-4592	77	8	every	every	DET
ejpam-4592	77	9	bourbaki	bourbaki	NOUN
ejpam-4592	77	10	-	-	PUNCT
ejpam-4592	77	11	cauchy	cauchy	ADJ
ejpam-4592	77	12	(	(	PUNCT
ejpam-4592	77	13	resp	resp	NOUN
ejpam-4592	77	14	.	.	PUNCT
ejpam-4592	78	1	pseudo	pseudo	NOUN
ejpam-4592	78	2	-	-	NOUN
ejpam-4592	78	3	cauchy	cauchy	ADJ
ejpam-4592	78	4	,	,	PUNCT
ejpam-4592	78	5	cofinally	cofinally	ADV
ejpam-4592	78	6	-	-	PUNCT
ejpam-4592	78	7	cauchy	cauchy	ADJ
ejpam-4592	78	8	)	)	PUNCT
ejpam-4592	78	9	sequence	sequence	NOUN
ejpam-4592	78	10	with	with	ADP
ejpam-4592	78	11	respect	respect	NOUN
ejpam-4592	78	12	to	to	ADP
ejpam-4592	78	13	σ	σ	PROPN
ejpam-4592	78	14	is	be	AUX
ejpam-4592	78	15	a	a	DET
ejpam-4592	78	16	convergent	convergent	NOUN
ejpam-4592	78	17	sequence	sequence	NOUN
ejpam-4592	78	18	with	with	ADP
ejpam-4592	78	19	respect	respect	NOUN
ejpam-4592	78	20	to	to	ADP
ejpam-4592	78	21	σ	σ	PROPN
ejpam-4592	78	22	.	.	PUNCT
ejpam-4592	78	23	definition	definition	NOUN
ejpam-4592	78	24	10	10	NUM
ejpam-4592	78	25	.	.	PUNCT
ejpam-4592	79	1	a	a	DET
ejpam-4592	79	2	gms	gms	NOUN
ejpam-4592	79	3	(	(	PUNCT
ejpam-4592	79	4	x	x	X
ejpam-4592	79	5	,	,	PUNCT
ejpam-4592	79	6	ω	ω	NUM
ejpam-4592	79	7	)	)	PUNCT
ejpam-4592	79	8	is	be	AUX
ejpam-4592	79	9	said	say	VERB
ejpam-4592	79	10	to	to	PART
ejpam-4592	79	11	be	be	AUX
ejpam-4592	79	12	a	a	DET
ejpam-4592	79	13	bourbaki	bourbaki	NOUN
ejpam-4592	79	14	(	(	PUNCT
ejpam-4592	79	15	resp	resp	NOUN
ejpam-4592	79	16	.	.	PUNCT
ejpam-4592	80	1	pseudo	pseudo	NOUN
ejpam-4592	80	2	,	,	PUNCT
ejpam-4592	80	3	cofinally	cofinally	ADV
ejpam-4592	80	4	)	)	PUNCT
ejpam-4592	80	5	complete	complete	ADJ
ejpam-4592	80	6	metric	metric	ADJ
ejpam-4592	80	7	space	space	NOUN
ejpam-4592	80	8	if	if	SCONJ
ejpam-4592	80	9	there	there	PRON
ejpam-4592	80	10	exists	exist	VERB
ejpam-4592	80	11	a	a	DET
ejpam-4592	80	12	kernel	kernel	PROPN
ejpam-4592	80	13	ω0	ω0	PROPN
ejpam-4592	80	14	⊂	⊂	PROPN
ejpam-4592	81	1	ω	ω	PROPN
ejpam-4592	81	2	consisting	consist	VERB
ejpam-4592	81	3	of	of	ADP
ejpam-4592	81	4	all	all	DET
ejpam-4592	81	5	bourbaki	bourbaki	NOUN
ejpam-4592	81	6	(	(	PUNCT
ejpam-4592	81	7	resp	resp	NOUN
ejpam-4592	81	8	.	.	PUNCT
ejpam-4592	82	1	pseudo	pseudo	NOUN
ejpam-4592	82	2	,	,	PUNCT
ejpam-4592	82	3	cofinally	cofinally	ADV
ejpam-4592	82	4	)	)	PUNCT
ejpam-4592	82	5	complete	complete	ADJ
ejpam-4592	82	6	metrics	metric	NOUN
ejpam-4592	82	7	.	.	PUNCT
ejpam-4592	83	1	theorem	theorem	VERB
ejpam-4592	83	2	11	11	NUM
ejpam-4592	83	3	and	and	CCONJ
ejpam-4592	83	4	example	example	NOUN
ejpam-4592	84	1	12	12	NUM
ejpam-4592	84	2	are	be	AUX
ejpam-4592	84	3	describe	describe	NOUN
ejpam-4592	84	4	the	the	DET
ejpam-4592	84	5	below	below	ADJ
ejpam-4592	84	6	diagram	diagram	NOUN
ejpam-4592	84	7	.	.	PUNCT
ejpam-4592	85	1	cauchy	cauchy	PROPN
ejpam-4592	85	2	sequence	sequence	NOUN
ejpam-4592	85	3	bourbaki−	bourbaki−	NOUN
ejpam-4592	85	4	cauchy	cauchy	NOUN
ejpam-4592	85	5	sequence	sequence	NOUN
ejpam-4592	85	6	/	/	SYM
ejpam-4592	85	7	theorem	theorem	ADJ
ejpam-4592	85	8	11	11	NUM
ejpam-4592	85	9	provides	provide	VERB
ejpam-4592	85	10	an	an	DET
ejpam-4592	85	11	easy	easy	ADJ
ejpam-4592	85	12	way	way	NOUN
ejpam-4592	85	13	to	to	PART
ejpam-4592	85	14	check	check	VERB
ejpam-4592	85	15	,	,	PUNCT
ejpam-4592	85	16	in	in	ADP
ejpam-4592	85	17	a	a	DET
ejpam-4592	85	18	generalized	generalized	ADJ
ejpam-4592	85	19	metric	metric	ADJ
ejpam-4592	85	20	space	space	NOUN
ejpam-4592	85	21	,	,	PUNCT
ejpam-4592	85	22	whether	whether	SCONJ
ejpam-4592	85	23	a	a	DET
ejpam-4592	85	24	given	give	VERB
ejpam-4592	85	25	sequence	sequence	NOUN
ejpam-4592	85	26	is	be	AUX
ejpam-4592	85	27	bourbaki	bourbaki	NOUN
ejpam-4592	85	28	-	-	PUNCT
ejpam-4592	85	29	cauchy	cauchy	ADJ
ejpam-4592	85	30	or	or	CCONJ
ejpam-4592	85	31	not	not	PART
ejpam-4592	85	32	.	.	PUNCT
ejpam-4592	86	1	this	this	DET
ejpam-4592	86	2	theorem	theorem	NOUN
ejpam-4592	86	3	is	be	AUX
ejpam-4592	86	4	direct	direct	ADJ
ejpam-4592	86	5	impact	impact	NOUN
ejpam-4592	86	6	of	of	ADP
ejpam-4592	86	7	the	the	DET
ejpam-4592	86	8	definitions	definition	NOUN
ejpam-4592	86	9	(	(	PUNCT
ejpam-4592	86	10	definition	definition	NOUN
ejpam-4592	86	11	3	3	NUM
ejpam-4592	86	12	and	and	CCONJ
ejpam-4592	86	13	definition	definition	NOUN
ejpam-4592	86	14	8)	8)	NUM
ejpam-4592	86	15	so	so	CCONJ
ejpam-4592	86	16	the	the	DET
ejpam-4592	86	17	trivial	trivial	ADJ
ejpam-4592	86	18	proof	proof	NOUN
ejpam-4592	86	19	is	be	AUX
ejpam-4592	86	20	neglected	neglect	VERB
ejpam-4592	86	21	.	.	PUNCT
ejpam-4592	87	1	theorem	theorem	VERB
ejpam-4592	87	2	11	11	NUM
ejpam-4592	87	3	.	.	PUNCT
ejpam-4592	88	1	let	let	VERB
ejpam-4592	88	2	(	(	PUNCT
ejpam-4592	88	3	x	x	NOUN
ejpam-4592	88	4	,	,	PUNCT
ejpam-4592	88	5	ω	ω	NUM
ejpam-4592	88	6	)	)	PUNCT
ejpam-4592	88	7	be	be	AUX
ejpam-4592	88	8	a	a	DET
ejpam-4592	88	9	generalized	generalized	ADJ
ejpam-4592	88	10	metric	metric	ADJ
ejpam-4592	88	11	space	space	NOUN
ejpam-4592	88	12	.	.	PUNCT
ejpam-4592	89	1	then	then	ADV
ejpam-4592	89	2	the	the	DET
ejpam-4592	89	3	followings	following	NOUN
ejpam-4592	89	4	are	be	AUX
ejpam-4592	89	5	true	true	ADJ
ejpam-4592	89	6	.	.	PUNCT
ejpam-4592	90	1	(	(	PUNCT
ejpam-4592	90	2	a	a	X
ejpam-4592	90	3	)	)	PUNCT
ejpam-4592	90	4	every	every	DET
ejpam-4592	90	5	cauchy	cauchy	ADJ
ejpam-4592	90	6	sequence	sequence	NOUN
ejpam-4592	90	7	is	be	AUX
ejpam-4592	90	8	a	a	DET
ejpam-4592	90	9	bourbaki	bourbaki	VERB
ejpam-4592	90	10	-	-	PUNCT
ejpam-4592	90	11	cauchy	cauchy	ADJ
ejpam-4592	90	12	sequence	sequence	NOUN
ejpam-4592	90	13	with	with	ADP
ejpam-4592	90	14	the	the	DET
ejpam-4592	90	15	same	same	ADJ
ejpam-4592	90	16	metric	metric	NOUN
ejpam-4592	90	17	.	.	PUNCT
ejpam-4592	91	1	(	(	PUNCT
ejpam-4592	91	2	b	b	X
ejpam-4592	91	3	)	)	PUNCT
ejpam-4592	91	4	every	every	DET
ejpam-4592	91	5	bourbaki	bourbaki	NOUN
ejpam-4592	91	6	-complete	-complete	ADJ
ejpam-4592	91	7	metric	metric	ADJ
ejpam-4592	91	8	space	space	NOUN
ejpam-4592	91	9	is	be	AUX
ejpam-4592	91	10	a	a	DET
ejpam-4592	91	11	weakly	weakly	ADJ
ejpam-4592	91	12	complete	complete	ADJ
ejpam-4592	91	13	metric	metric	ADJ
ejpam-4592	91	14	space	space	NOUN
ejpam-4592	91	15	.	.	PUNCT
ejpam-4592	92	1	example	example	NOUN
ejpam-4592	92	2	12	12	NUM
ejpam-4592	92	3	.	.	PUNCT
ejpam-4592	93	1	consider	consider	VERB
ejpam-4592	93	2	the	the	DET
ejpam-4592	93	3	generalized	generalized	ADJ
ejpam-4592	93	4	metric	metric	ADJ
ejpam-4592	93	5	space	space	NOUN
ejpam-4592	93	6	(	(	PUNCT
ejpam-4592	93	7	x	x	X
ejpam-4592	93	8	,	,	PUNCT
ejpam-4592	93	9	ω	ω	NOUN
ejpam-4592	93	10	)	)	PUNCT
ejpam-4592	93	11	where	where	SCONJ
ejpam-4592	93	12	x	x	X
ejpam-4592	93	13	=	=	SYM
ejpam-4592	93	14	r	r	PROPN
ejpam-4592	93	15	,	,	PUNCT
ejpam-4592	93	16	ω	ω	NOUN
ejpam-4592	93	17	=	=	SYM
ejpam-4592	93	18	{	{	PUNCT
ejpam-4592	93	19	σm	σm	INTJ
ejpam-4592	93	20	|	|	ADV
ejpam-4592	93	21	m	m	VERB
ejpam-4592	93	22	∈	∈	PROPN
ejpam-4592	93	23	n	n	CCONJ
ejpam-4592	93	24	}	}	PUNCT
ejpam-4592	93	25	⊂	⊂	PROPN
ejpam-4592	93	26	ωx	ωx	VERB
ejpam-4592	93	27	with	with	ADP
ejpam-4592	93	28	σm(c	σm(c	NUM
ejpam-4592	93	29	,	,	PUNCT
ejpam-4592	93	30	d	d	NOUN
ejpam-4592	93	31	)	)	PUNCT
ejpam-4592	93	32	=	=	SYM
ejpam-4592	93	33	min{σe(c	min{σe(c	NOUN
ejpam-4592	93	34	,	,	PUNCT
ejpam-4592	93	35	d	d	NOUN
ejpam-4592	93	36	)	)	PUNCT
ejpam-4592	93	37	,	,	PUNCT
ejpam-4592	93	38	1	1	NUM
ejpam-4592	93	39	m	m	NOUN
ejpam-4592	93	40	}	}	PUNCT
ejpam-4592	93	41	where	where	SCONJ
ejpam-4592	93	42	σe(c	σe(c	X
ejpam-4592	93	43	,	,	PUNCT
ejpam-4592	93	44	d	d	X
ejpam-4592	93	45	)	)	PUNCT
ejpam-4592	93	46	=	=	SYM
ejpam-4592	93	47	|c−	|c−	VERB
ejpam-4592	93	48	d|	d|	PROPN
ejpam-4592	93	49	.	.	PUNCT
ejpam-4592	94	1	let	let	VERB
ejpam-4592	94	2	{	{	PUNCT
ejpam-4592	94	3	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	94	4	=	=	SYM
ejpam-4592	94	5	{	{	PUNCT
ejpam-4592	94	6	n+	n+	NOUN
ejpam-4592	94	7	0.1	0.1	NUM
ejpam-4592	94	8	8	8	NUM
ejpam-4592	94	9	}	}	PUNCT
ejpam-4592	94	10	and	and	CCONJ
ejpam-4592	94	11	ε	ε	PROPN
ejpam-4592	94	12	>	>	X
ejpam-4592	94	13	0	0	PUNCT
ejpam-4592	94	14	be	be	AUX
ejpam-4592	94	15	given	give	VERB
ejpam-4592	94	16	.	.	PUNCT
ejpam-4592	95	1	then	then	ADV
ejpam-4592	95	2	{	{	PUNCT
ejpam-4592	95	3	cn	cn	ADJ
ejpam-4592	95	4	}	}	PUNCT
ejpam-4592	95	5	∈	∈	PROPN
ejpam-4592	95	6	b(σ1	b(σ1	NOUN
ejpam-4592	95	7	)	)	PUNCT
ejpam-4592	95	8	.	.	PUNCT
ejpam-4592	96	1	but	but	CCONJ
ejpam-4592	96	2	{	{	PUNCT
ejpam-4592	96	3	cn	cn	ADJ
ejpam-4592	96	4	}	}	PUNCT
ejpam-4592	96	5	/∈	/∈	PUNCT
ejpam-4592	96	6	c(σ1	c(σ1	NOUN
ejpam-4592	96	7	)	)	PUNCT
ejpam-4592	96	8	.	.	PUNCT
ejpam-4592	97	1	cauchy	cauchy	PROPN
ejpam-4592	97	2	sequence	sequence	NOUN
ejpam-4592	97	3	cofinally	cofinally	ADV
ejpam-4592	97	4	−	−	ADP
ejpam-4592	97	5	cauchy	cauchy	NOUN
ejpam-4592	97	6	sequence	sequence	NOUN
ejpam-4592	97	7	/	/	PUNCT
ejpam-4592	97	8	the	the	DET
ejpam-4592	97	9	following	follow	VERB
ejpam-4592	97	10	theorem	theorem	VERB
ejpam-4592	97	11	13	13	NUM
ejpam-4592	97	12	and	and	CCONJ
ejpam-4592	97	13	example	example	NOUN
ejpam-4592	97	14	14	14	NUM
ejpam-4592	97	15	are	be	AUX
ejpam-4592	97	16	describe	describe	NOUN
ejpam-4592	97	17	the	the	DET
ejpam-4592	97	18	above	above	ADJ
ejpam-4592	97	19	diagram	diagram	NOUN
ejpam-4592	97	20	.	.	PUNCT
ejpam-4592	98	1	the	the	DET
ejpam-4592	98	2	following	follow	VERB
ejpam-4592	98	3	theorem	theorem	NOUN
ejpam-4592	98	4	13	13	NUM
ejpam-4592	98	5	gives	give	VERB
ejpam-4592	98	6	a	a	DET
ejpam-4592	98	7	shortcut	shortcut	NOUN
ejpam-4592	98	8	for	for	ADP
ejpam-4592	98	9	finding	find	VERB
ejpam-4592	98	10	the	the	DET
ejpam-4592	98	11	nature	nature	NOUN
ejpam-4592	98	12	of	of	ADP
ejpam-4592	98	13	a	a	DET
ejpam-4592	98	14	given	give	VERB
ejpam-4592	98	15	cauchy	cauchy	ADJ
ejpam-4592	98	16	sequence	sequence	NOUN
ejpam-4592	98	17	in	in	ADP
ejpam-4592	98	18	a	a	DET
ejpam-4592	98	19	generalized	generalized	ADJ
ejpam-4592	98	20	metric	metric	ADJ
ejpam-4592	98	21	space	space	NOUN
ejpam-4592	98	22	.	.	PUNCT
ejpam-4592	99	1	theorem	theorem	VERB
ejpam-4592	99	2	13	13	NUM
ejpam-4592	99	3	.	.	PUNCT
ejpam-4592	100	1	let	let	VERB
ejpam-4592	100	2	(	(	PUNCT
ejpam-4592	100	3	x	x	NOUN
ejpam-4592	100	4	,	,	PUNCT
ejpam-4592	100	5	ω	ω	NUM
ejpam-4592	100	6	)	)	PUNCT
ejpam-4592	100	7	be	be	AUX
ejpam-4592	100	8	a	a	DET
ejpam-4592	100	9	generalized	generalized	ADJ
ejpam-4592	100	10	metric	metric	ADJ
ejpam-4592	100	11	space	space	NOUN
ejpam-4592	100	12	.	.	PUNCT
ejpam-4592	101	1	then	then	ADV
ejpam-4592	101	2	the	the	DET
ejpam-4592	101	3	followings	following	NOUN
ejpam-4592	101	4	are	be	AUX
ejpam-4592	101	5	true	true	ADJ
ejpam-4592	101	6	.	.	PUNCT
ejpam-4592	102	1	(	(	PUNCT
ejpam-4592	102	2	a	a	X
ejpam-4592	102	3	)	)	PUNCT
ejpam-4592	102	4	every	every	DET
ejpam-4592	102	5	cauchy	cauchy	ADJ
ejpam-4592	102	6	sequence	sequence	NOUN
ejpam-4592	102	7	is	be	AUX
ejpam-4592	102	8	a	a	DET
ejpam-4592	102	9	cofinally	cofinally	ADV
ejpam-4592	102	10	-	-	PUNCT
ejpam-4592	102	11	cauchy	cauchy	ADJ
ejpam-4592	102	12	sequence	sequence	NOUN
ejpam-4592	102	13	with	with	ADP
ejpam-4592	102	14	the	the	DET
ejpam-4592	102	15	same	same	ADJ
ejpam-4592	102	16	metric	metric	NOUN
ejpam-4592	102	17	.	.	PUNCT
ejpam-4592	103	1	(	(	PUNCT
ejpam-4592	103	2	b	b	X
ejpam-4592	103	3	)	)	PUNCT
ejpam-4592	103	4	every	every	DET
ejpam-4592	103	5	cofinally	cofinally	ADV
ejpam-4592	103	6	-	-	PUNCT
ejpam-4592	103	7	complete	complete	ADJ
ejpam-4592	103	8	metric	metric	ADJ
ejpam-4592	103	9	space	space	NOUN
ejpam-4592	103	10	is	be	AUX
ejpam-4592	103	11	a	a	DET
ejpam-4592	103	12	weakly	weakly	ADJ
ejpam-4592	103	13	complete	complete	ADJ
ejpam-4592	103	14	metric	metric	ADJ
ejpam-4592	103	15	space	space	NOUN
ejpam-4592	103	16	.	.	PUNCT
ejpam-4592	104	1	proof	proof	NOUN
ejpam-4592	104	2	.	.	PUNCT
ejpam-4592	105	1	(	(	PUNCT
ejpam-4592	105	2	a	a	X
ejpam-4592	105	3	)	)	PUNCT
ejpam-4592	105	4	.	.	PUNCT
ejpam-4592	106	1	let	let	VERB
ejpam-4592	106	2	σ	σ	X
ejpam-4592	106	3	∈	∈	PROPN
ejpam-4592	106	4	ω	ω	PROPN
ejpam-4592	106	5	and	and	CCONJ
ejpam-4592	106	6	{	{	PUNCT
ejpam-4592	106	7	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	106	8	∈	∈	PROPN
ejpam-4592	106	9	c(σ	c(σ	PROPN
ejpam-4592	106	10	)	)	PUNCT
ejpam-4592	106	11	.	.	PUNCT
ejpam-4592	107	1	let	let	VERB
ejpam-4592	107	2	ε	ε	PROPN
ejpam-4592	107	3	>	>	X
ejpam-4592	107	4	0	0	PUNCT
ejpam-4592	107	5	be	be	AUX
ejpam-4592	107	6	given	give	VERB
ejpam-4592	107	7	.	.	PUNCT
ejpam-4592	108	1	then	then	ADV
ejpam-4592	108	2	there	there	PRON
ejpam-4592	108	3	exists	exist	VERB
ejpam-4592	108	4	a	a	DET
ejpam-4592	108	5	positive	positive	ADJ
ejpam-4592	108	6	integer	integer	NOUN
ejpam-4592	108	7	n0	n0	NOUN
ejpam-4592	108	8	such	such	ADJ
ejpam-4592	108	9	that	that	DET
ejpam-4592	108	10	σ(cn	σ(cn	NOUN
ejpam-4592	108	11	,	,	PUNCT
ejpam-4592	108	12	cm	cm	NOUN
ejpam-4592	108	13	)	)	PUNCT
ejpam-4592	108	14	<	<	X
ejpam-4592	108	15	ε	ε	PROPN
ejpam-4592	108	16	for	for	ADP
ejpam-4592	108	17	all	all	DET
ejpam-4592	108	18	n	n	CCONJ
ejpam-4592	108	19	,	,	PUNCT
ejpam-4592	108	20	m	m	PROPN
ejpam-4592	108	21	≥	≥	NOUN
ejpam-4592	108	22	n0	n0	NUM
ejpam-4592	108	23	.	.	PUNCT
ejpam-4592	109	1	take	take	VERB
ejpam-4592	109	2	nε	nε	PROPN
ejpam-4592	109	3	=	=	SYM
ejpam-4592	109	4	{	{	PUNCT
ejpam-4592	109	5	j	j	NOUN
ejpam-4592	109	6	:	:	PUNCT
ejpam-4592	109	7	j	j	PROPN
ejpam-4592	109	8	≥	≥	NUM
ejpam-4592	109	9	n0	n0	NUM
ejpam-4592	109	10	}	}	PUNCT
ejpam-4592	109	11	.	.	PUNCT
ejpam-4592	110	1	then	then	ADV
ejpam-4592	110	2	nε	nε	PROPN
ejpam-4592	110	3	is	be	AUX
ejpam-4592	110	4	an	an	DET
ejpam-4592	110	5	infinite	infinite	ADJ
ejpam-4592	110	6	subset	subset	NOUN
ejpam-4592	110	7	of	of	ADP
ejpam-4592	110	8	n	n	PROPN
ejpam-4592	110	9	and	and	CCONJ
ejpam-4592	110	10	σ(ck	σ(ck	NOUN
ejpam-4592	110	11	,	,	PUNCT
ejpam-4592	110	12	cl	cl	NOUN
ejpam-4592	110	13	)	)	PUNCT
ejpam-4592	110	14	<	<	X
ejpam-4592	110	15	ε	ε	PROPN
ejpam-4592	110	16	for	for	ADP
ejpam-4592	110	17	all	all	DET
ejpam-4592	110	18	k	k	PROPN
ejpam-4592	110	19	,	,	PUNCT
ejpam-4592	110	20	l	l	PROPN
ejpam-4592	110	21	∈	∈	PROPN
ejpam-4592	110	22	nε	nε	VERB
ejpam-4592	110	23	.	.	PUNCT
ejpam-4592	111	1	hence	hence	ADV
ejpam-4592	111	2	{	{	PUNCT
ejpam-4592	111	3	cn	cn	ADJ
ejpam-4592	111	4	}	}	PUNCT
ejpam-4592	111	5	∈	∈	PROPN
ejpam-4592	111	6	c(σ	c(σ	PROPN
ejpam-4592	111	7	)	)	PUNCT
ejpam-4592	111	8	.	.	PUNCT
ejpam-4592	112	1	(	(	PUNCT
ejpam-4592	112	2	b	b	NOUN
ejpam-4592	112	3	)	)	PUNCT
ejpam-4592	112	4	.	.	PUNCT
ejpam-4592	113	1	directly	directly	ADV
ejpam-4592	113	2	follows	follow	VERB
ejpam-4592	113	3	from	from	ADP
ejpam-4592	113	4	(	(	PUNCT
ejpam-4592	113	5	a	a	NOUN
ejpam-4592	113	6	)	)	PUNCT
ejpam-4592	113	7	and	and	CCONJ
ejpam-4592	113	8	the	the	DET
ejpam-4592	113	9	definition	definition	NOUN
ejpam-4592	113	10	of	of	ADP
ejpam-4592	113	11	weakly	weakly	ADJ
ejpam-4592	113	12	complete	complete	ADJ
ejpam-4592	113	13	metric	metric	ADJ
ejpam-4592	113	14	space	space	NOUN
ejpam-4592	113	15	.	.	PUNCT
ejpam-4592	114	1	v.	v.	ADP
ejpam-4592	114	2	subramanian	subramanian	PROPN
ejpam-4592	114	3	et	et	PROPN
ejpam-4592	114	4	al	al	PROPN
ejpam-4592	114	5	.	.	PUNCT
ejpam-4592	114	6	/	/	SYM
ejpam-4592	114	7	eur	eur	PROPN
ejpam-4592	114	8	.	.	PUNCT
ejpam-4592	115	1	j.	j.	PROPN
ejpam-4592	115	2	pure	pure	PROPN
ejpam-4592	115	3	appl	appl	PROPN
ejpam-4592	115	4	.	.	PROPN
ejpam-4592	115	5	math	math	PROPN
ejpam-4592	115	6	,	,	PUNCT
ejpam-4592	115	7	15	15	NUM
ejpam-4592	115	8	(	(	PUNCT
ejpam-4592	115	9	4	4	NUM
ejpam-4592	115	10	)	)	PUNCT
ejpam-4592	115	11	(	(	PUNCT
ejpam-4592	115	12	2022	2022	NUM
ejpam-4592	115	13	)	)	PUNCT
ejpam-4592	115	14	,	,	PUNCT
ejpam-4592	115	15	1869	1869	NUM
ejpam-4592	115	16	-	-	SYM
ejpam-4592	115	17	1886	1886	NUM
ejpam-4592	115	18	1873	1873	NUM
ejpam-4592	115	19	example	example	NOUN
ejpam-4592	115	20	14	14	NUM
ejpam-4592	115	21	.	.	PUNCT
ejpam-4592	116	1	consider	consider	VERB
ejpam-4592	116	2	the	the	DET
ejpam-4592	116	3	generalized	generalized	ADJ
ejpam-4592	116	4	metric	metric	ADJ
ejpam-4592	116	5	space	space	NOUN
ejpam-4592	116	6	(	(	PUNCT
ejpam-4592	116	7	x	x	NOUN
ejpam-4592	116	8	,	,	PUNCT
ejpam-4592	116	9	ω1	ω1	PROPN
ejpam-4592	116	10	)	)	PUNCT
ejpam-4592	116	11	where	where	SCONJ
ejpam-4592	116	12	x	x	X
ejpam-4592	116	13	=	=	PRON
ejpam-4592	116	14	r+	r+	NOUN
ejpam-4592	116	15	∪	∪	X
ejpam-4592	116	16	{	{	PUNCT
ejpam-4592	116	17	0},ω1	0},ω1	NOUN
ejpam-4592	116	18	=	=	SYM
ejpam-4592	116	19	{	{	PUNCT
ejpam-4592	116	20	σ1	σ1	PROPN
ejpam-4592	116	21	,	,	PUNCT
ejpam-4592	116	22	σ2	σ2	PROPN
ejpam-4592	116	23	,	,	PUNCT
ejpam-4592	116	24	σ3	σ3	PROPN
ejpam-4592	116	25	}	}	PUNCT
ejpam-4592	116	26	⊂	⊂	PROPN
ejpam-4592	116	27	ωx	ωx	PROPN
ejpam-4592	116	28	and	and	CCONJ
ejpam-4592	116	29	the	the	DET
ejpam-4592	116	30	metrics	metric	NOUN
ejpam-4592	116	31	are	be	AUX
ejpam-4592	116	32	defined	define	VERB
ejpam-4592	116	33	by	by	ADP
ejpam-4592	116	34	σ1(c	σ1(c	PRON
ejpam-4592	116	35	,	,	PUNCT
ejpam-4592	116	36	d	d	NOUN
ejpam-4592	116	37	)	)	PUNCT
ejpam-4592	116	38	=	=	SYM
ejpam-4592	116	39	|c−	|c−	VERB
ejpam-4592	116	40	d|	d|	PROPN
ejpam-4592	116	41	;	;	PUNCT
ejpam-4592	116	42	σ2(c	σ2(c	NOUN
ejpam-4592	116	43	,	,	PUNCT
ejpam-4592	116	44	d	d	NOUN
ejpam-4592	116	45	)	)	PUNCT
ejpam-4592	116	46	=	=	PRON
ejpam-4592	116	47	{	{	PUNCT
ejpam-4592	116	48	0	0	NUM
ejpam-4592	116	49	if	if	SCONJ
ejpam-4592	116	50	c	c	NOUN
ejpam-4592	116	51	=	=	SYM
ejpam-4592	116	52	d	d	PROPN
ejpam-4592	116	53	,	,	PUNCT
ejpam-4592	116	54	c+	c+	VERB
ejpam-4592	116	55	d	d	NOUN
ejpam-4592	116	56	if	if	SCONJ
ejpam-4592	116	57	c	c	PROPN
ejpam-4592	116	58	̸=	̸=	PROPN
ejpam-4592	116	59	d.	d.	PROPN
ejpam-4592	116	60	and	and	CCONJ
ejpam-4592	116	61	σ3(c	σ3(c	PRON
ejpam-4592	116	62	,	,	PUNCT
ejpam-4592	116	63	d	d	NOUN
ejpam-4592	116	64	)	)	PUNCT
ejpam-4592	116	65	=	=	SYM
ejpam-4592	116	66	σ1(c	σ1(c	PROPN
ejpam-4592	116	67	,	,	PUNCT
ejpam-4592	116	68	d	d	NOUN
ejpam-4592	116	69	)	)	PUNCT
ejpam-4592	116	70	1+σ1(c	1+σ1(c	NOUN
ejpam-4592	116	71	,	,	PUNCT
ejpam-4592	116	72	d	d	NOUN
ejpam-4592	116	73	)	)	PUNCT
ejpam-4592	116	74	for	for	ADP
ejpam-4592	116	75	all	all	DET
ejpam-4592	116	76	c	c	NOUN
ejpam-4592	116	77	,	,	PUNCT
ejpam-4592	116	78	d	d	PROPN
ejpam-4592	116	79	∈	∈	PROPN
ejpam-4592	116	80	x.	x.	NOUN
ejpam-4592	116	81	define	define	VERB
ejpam-4592	116	82	a	a	DET
ejpam-4592	116	83	sequence	sequence	NOUN
ejpam-4592	116	84	{	{	PUNCT
ejpam-4592	116	85	cn}n∈n	cn}n∈n	VERB
ejpam-4592	116	86	by	by	ADP
ejpam-4592	116	87	{	{	PUNCT
ejpam-4592	116	88	cn	cn	PROPN
ejpam-4592	116	89	}	}	PUNCT
ejpam-4592	116	90	=	=	SYM
ejpam-4592	116	91	{	{	PUNCT
ejpam-4592	116	92	log(n	log(n	NOUN
ejpam-4592	116	93	)	)	PUNCT
ejpam-4592	116	94	if	if	SCONJ
ejpam-4592	116	95	n	n	PRON
ejpam-4592	116	96	is	be	AUX
ejpam-4592	116	97	prime	prime	ADJ
ejpam-4592	116	98	,	,	PUNCT
ejpam-4592	116	99	0	0	NUM
ejpam-4592	116	100	otherwise	otherwise	ADV
ejpam-4592	116	101	.	.	PUNCT
ejpam-4592	117	1	in	in	ADP
ejpam-4592	117	2	x.	x.	PROPN
ejpam-4592	117	3	then	then	ADV
ejpam-4592	117	4	{	{	PUNCT
ejpam-4592	117	5	cn	cn	ADJ
ejpam-4592	117	6	}	}	PUNCT
ejpam-4592	117	7	∈	∈	PROPN
ejpam-4592	117	8	c(σ2	c(σ2	NUM
ejpam-4592	117	9	)	)	PUNCT
ejpam-4592	117	10	.	.	PUNCT
ejpam-4592	118	1	but	but	CCONJ
ejpam-4592	118	2	for	for	ADP
ejpam-4592	118	3	every	every	DET
ejpam-4592	118	4	ε	ε	PROPN
ejpam-4592	118	5	>	>	X
ejpam-4592	118	6	0	0	PROPN
ejpam-4592	118	7	,	,	PUNCT
ejpam-4592	118	8	there	there	PRON
ejpam-4592	118	9	is	be	VERB
ejpam-4592	118	10	no	no	DET
ejpam-4592	118	11	n0	n0	ADJ
ejpam-4592	118	12	∈	∈	PROPN
ejpam-4592	118	13	n	n	CCONJ
ejpam-4592	118	14	such	such	ADJ
ejpam-4592	118	15	that	that	SCONJ
ejpam-4592	118	16	σ2(cn	σ2(cn	PROPN
ejpam-4592	118	17	,	,	PUNCT
ejpam-4592	118	18	cm	cm	NOUN
ejpam-4592	118	19	)	)	PUNCT
ejpam-4592	118	20	<	<	X
ejpam-4592	118	21	ε	ε	PROPN
ejpam-4592	118	22	for	for	ADP
ejpam-4592	118	23	n	n	CCONJ
ejpam-4592	118	24	,	,	PUNCT
ejpam-4592	118	25	m	m	PROPN
ejpam-4592	118	26	≥	≥	NOUN
ejpam-4592	118	27	n0	n0	NUM
ejpam-4592	118	28	.	.	PUNCT
ejpam-4592	119	1	thus	thus	ADV
ejpam-4592	119	2	,	,	PUNCT
ejpam-4592	119	3	{	{	PUNCT
ejpam-4592	119	4	cn	cn	ADJ
ejpam-4592	119	5	}	}	PUNCT
ejpam-4592	119	6	/∈	/∈	PUNCT
ejpam-4592	119	7	c(σ2	c(σ2	NUM
ejpam-4592	119	8	)	)	PUNCT
ejpam-4592	119	9	.	.	PUNCT
ejpam-4592	120	1	cauchy	cauchy	PROPN
ejpam-4592	120	2	sequence	sequence	NOUN
ejpam-4592	120	3	pseudo−	pseudo−	PROPN
ejpam-4592	120	4	cauchy	cauchy	PROPN
ejpam-4592	120	5	sequence	sequence	NOUN
ejpam-4592	120	6	/	/	PUNCT
ejpam-4592	120	7	the	the	DET
ejpam-4592	120	8	following	follow	VERB
ejpam-4592	120	9	theorem	theorem	VERB
ejpam-4592	120	10	15	15	NUM
ejpam-4592	120	11	and	and	CCONJ
ejpam-4592	120	12	example	example	NOUN
ejpam-4592	120	13	16	16	NUM
ejpam-4592	120	14	are	be	AUX
ejpam-4592	120	15	describe	describe	NOUN
ejpam-4592	120	16	the	the	DET
ejpam-4592	120	17	above	above	ADJ
ejpam-4592	120	18	diagram	diagram	NOUN
ejpam-4592	120	19	.	.	PUNCT
ejpam-4592	121	1	theorem	theorem	NOUN
ejpam-4592	121	2	15	15	NUM
ejpam-4592	121	3	is	be	AUX
ejpam-4592	121	4	reduce	reduce	VERB
ejpam-4592	121	5	the	the	DET
ejpam-4592	121	6	complexity	complexity	NOUN
ejpam-4592	121	7	for	for	ADP
ejpam-4592	121	8	finding	finding	NOUN
ejpam-4592	121	9	,	,	PUNCT
ejpam-4592	121	10	in	in	ADP
ejpam-4592	121	11	a	a	DET
ejpam-4592	121	12	generalized	generalize	VERB
ejpam-4592	121	13	metric	metric	ADJ
ejpam-4592	121	14	space	space	NOUN
ejpam-4592	121	15	,	,	PUNCT
ejpam-4592	121	16	whether	whether	SCONJ
ejpam-4592	121	17	a	a	DET
ejpam-4592	121	18	given	give	VERB
ejpam-4592	121	19	cauchy	cauchy	PROPN
ejpam-4592	121	20	sequnce	sequnce	NOUN
ejpam-4592	121	21	is	be	AUX
ejpam-4592	121	22	pseudo	pseudo	ADJ
ejpam-4592	121	23	-	-	ADJ
ejpam-4592	121	24	cauchy	cauchy	ADJ
ejpam-4592	121	25	sequence	sequence	NOUN
ejpam-4592	121	26	or	or	CCONJ
ejpam-4592	121	27	not	not	PART
ejpam-4592	121	28	.	.	PUNCT
ejpam-4592	122	1	theorem	theorem	ADJ
ejpam-4592	122	2	15	15	NUM
ejpam-4592	122	3	.	.	PUNCT
ejpam-4592	123	1	let	let	VERB
ejpam-4592	123	2	(	(	PUNCT
ejpam-4592	123	3	x	x	NOUN
ejpam-4592	123	4	,	,	PUNCT
ejpam-4592	123	5	ω	ω	NUM
ejpam-4592	123	6	)	)	PUNCT
ejpam-4592	123	7	be	be	AUX
ejpam-4592	123	8	a	a	DET
ejpam-4592	123	9	generalized	generalized	ADJ
ejpam-4592	123	10	metric	metric	ADJ
ejpam-4592	123	11	space	space	NOUN
ejpam-4592	123	12	.	.	PUNCT
ejpam-4592	124	1	then	then	ADV
ejpam-4592	124	2	the	the	DET
ejpam-4592	124	3	followings	following	NOUN
ejpam-4592	124	4	are	be	AUX
ejpam-4592	124	5	true	true	ADJ
ejpam-4592	124	6	.	.	PUNCT
ejpam-4592	125	1	(	(	PUNCT
ejpam-4592	125	2	a	a	X
ejpam-4592	125	3	)	)	PUNCT
ejpam-4592	125	4	every	every	DET
ejpam-4592	125	5	cauchy	cauchy	ADJ
ejpam-4592	125	6	sequence	sequence	NOUN
ejpam-4592	125	7	is	be	AUX
ejpam-4592	125	8	a	a	DET
ejpam-4592	125	9	pseudo	pseudo	NOUN
ejpam-4592	125	10	-	-	ADJ
ejpam-4592	125	11	cauchy	cauchy	ADJ
ejpam-4592	125	12	sequence	sequence	NOUN
ejpam-4592	125	13	in	in	ADP
ejpam-4592	125	14	x	x	PUNCT
ejpam-4592	125	15	with	with	ADP
ejpam-4592	125	16	the	the	DET
ejpam-4592	125	17	same	same	ADJ
ejpam-4592	125	18	metric	metric	NOUN
ejpam-4592	125	19	.	.	PUNCT
ejpam-4592	126	1	(	(	PUNCT
ejpam-4592	126	2	b	b	X
ejpam-4592	126	3	)	)	PUNCT
ejpam-4592	126	4	every	every	DET
ejpam-4592	126	5	pseudo	pseudo	NOUN
ejpam-4592	126	6	-	-	ADJ
ejpam-4592	126	7	complete	complete	ADJ
ejpam-4592	126	8	metric	metric	ADJ
ejpam-4592	126	9	space	space	NOUN
ejpam-4592	126	10	is	be	AUX
ejpam-4592	126	11	a	a	DET
ejpam-4592	126	12	weakly	weakly	ADJ
ejpam-4592	126	13	complete	complete	ADJ
ejpam-4592	126	14	metric	metric	ADJ
ejpam-4592	126	15	space	space	NOUN
ejpam-4592	126	16	.	.	PUNCT
ejpam-4592	127	1	proof	proof	NOUN
ejpam-4592	127	2	.	.	PUNCT
ejpam-4592	128	1	(	(	PUNCT
ejpam-4592	128	2	a	a	X
ejpam-4592	128	3	)	)	PUNCT
ejpam-4592	128	4	.	.	PUNCT
ejpam-4592	129	1	let	let	VERB
ejpam-4592	129	2	{	{	PUNCT
ejpam-4592	129	3	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	129	4	∈	∈	PROPN
ejpam-4592	129	5	c(σ	c(σ	PROPN
ejpam-4592	129	6	)	)	PUNCT
ejpam-4592	129	7	.	.	PUNCT
ejpam-4592	130	1	then	then	ADV
ejpam-4592	130	2	there	there	PRON
ejpam-4592	130	3	is	be	VERB
ejpam-4592	130	4	n0	n0	NUM
ejpam-4592	130	5	∈	∈	PROPN
ejpam-4592	130	6	n	n	CCONJ
ejpam-4592	130	7	such	such	ADJ
ejpam-4592	130	8	that	that	DET
ejpam-4592	130	9	σ(cn	σ(cn	NOUN
ejpam-4592	130	10	,	,	PUNCT
ejpam-4592	130	11	cm	cm	NOUN
ejpam-4592	130	12	)	)	PUNCT
ejpam-4592	130	13	<	<	X
ejpam-4592	130	14	ε	ε	PROPN
ejpam-4592	130	15	for	for	ADP
ejpam-4592	130	16	all	all	DET
ejpam-4592	130	17	n	n	CCONJ
ejpam-4592	130	18	,	,	PUNCT
ejpam-4592	130	19	m	m	PROPN
ejpam-4592	130	20	≥	≥	NOUN
ejpam-4592	130	21	n0	n0	NUM
ejpam-4592	130	22	.	.	PUNCT
ejpam-4592	131	1	let	let	VERB
ejpam-4592	131	2	ε	ε	PROPN
ejpam-4592	131	3	>	>	X
ejpam-4592	131	4	0	0	PROPN
ejpam-4592	131	5	,	,	PUNCT
ejpam-4592	131	6	k	k	PROPN
ejpam-4592	131	7	∈	∈	PROPN
ejpam-4592	131	8	n	n	AUX
ejpam-4592	131	9	be	be	AUX
ejpam-4592	131	10	given	give	VERB
ejpam-4592	131	11	.	.	PUNCT
ejpam-4592	132	1	if	if	SCONJ
ejpam-4592	132	2	k	k	PROPN
ejpam-4592	132	3	≥	≥	X
ejpam-4592	132	4	n0	n0	PROPN
ejpam-4592	132	5	,	,	PUNCT
ejpam-4592	132	6	then	then	ADV
ejpam-4592	132	7	the	the	DET
ejpam-4592	132	8	result	result	NOUN
ejpam-4592	132	9	is	be	AUX
ejpam-4592	132	10	obvious	obvious	ADJ
ejpam-4592	132	11	.	.	PUNCT
ejpam-4592	133	1	suppose	suppose	VERB
ejpam-4592	133	2	that	that	SCONJ
ejpam-4592	133	3	k	k	PROPN
ejpam-4592	133	4	<	<	X
ejpam-4592	133	5	n0	n0	PROPN
ejpam-4592	133	6	.	.	PUNCT
ejpam-4592	134	1	since	since	SCONJ
ejpam-4592	134	2	n0	n0	PROPN
ejpam-4592	134	3	∈	∈	PROPN
ejpam-4592	134	4	n	n	CCONJ
ejpam-4592	134	5	,	,	PUNCT
ejpam-4592	134	6	we	we	PRON
ejpam-4592	134	7	get	get	VERB
ejpam-4592	134	8	elements	element	NOUN
ejpam-4592	134	9	l	l	NOUN
ejpam-4592	134	10	,	,	PUNCT
ejpam-4592	134	11	j	j	PROPN
ejpam-4592	134	12	∈	∈	PROPN
ejpam-4592	134	13	n	n	CCONJ
ejpam-4592	134	14	,	,	PUNCT
ejpam-4592	134	15	l	l	PROPN
ejpam-4592	134	16	̸=	̸=	PROPN
ejpam-4592	134	17	j	j	PROPN
ejpam-4592	134	18	such	such	ADJ
ejpam-4592	134	19	that	that	DET
ejpam-4592	134	20	l	l	NOUN
ejpam-4592	134	21	,	,	PUNCT
ejpam-4592	134	22	j	j	PROPN
ejpam-4592	134	23	>	>	X
ejpam-4592	134	24	n0	n0	PROPN
ejpam-4592	134	25	>	>	X
ejpam-4592	134	26	k.	k.	PROPN
ejpam-4592	134	27	also	also	ADV
ejpam-4592	134	28	,	,	PUNCT
ejpam-4592	134	29	σ(cl	σ(cl	PROPN
ejpam-4592	134	30	,	,	PUNCT
ejpam-4592	134	31	cj	cj	X
ejpam-4592	134	32	)	)	PUNCT
ejpam-4592	134	33	<	<	X
ejpam-4592	134	34	ε	ε	PROPN
ejpam-4592	134	35	.	.	PUNCT
ejpam-4592	135	1	therefore	therefore	ADV
ejpam-4592	135	2	,	,	PUNCT
ejpam-4592	135	3	{	{	PUNCT
ejpam-4592	135	4	cn	cn	ADJ
ejpam-4592	135	5	}	}	PUNCT
ejpam-4592	135	6	∈	∈	PROPN
ejpam-4592	135	7	p(σ	p(σ	NOUN
ejpam-4592	135	8	)	)	PUNCT
ejpam-4592	135	9	.	.	PUNCT
ejpam-4592	136	1	(	(	PUNCT
ejpam-4592	136	2	b	b	X
ejpam-4592	136	3	)	)	PUNCT
ejpam-4592	136	4	.	.	PUNCT
ejpam-4592	137	1	follows	follow	VERB
ejpam-4592	137	2	from	from	ADP
ejpam-4592	137	3	(	(	PUNCT
ejpam-4592	137	4	a	a	NOUN
ejpam-4592	137	5	)	)	PUNCT
ejpam-4592	137	6	.	.	PUNCT
ejpam-4592	138	1	example	example	NOUN
ejpam-4592	138	2	16	16	NUM
ejpam-4592	138	3	.	.	PUNCT
ejpam-4592	139	1	consider	consider	VERB
ejpam-4592	139	2	the	the	DET
ejpam-4592	139	3	generalized	generalized	ADJ
ejpam-4592	139	4	metric	metric	ADJ
ejpam-4592	139	5	space	space	NOUN
ejpam-4592	139	6	(	(	PUNCT
ejpam-4592	139	7	x	x	NOUN
ejpam-4592	139	8	,	,	PUNCT
ejpam-4592	139	9	ω1	ω1	PROPN
ejpam-4592	139	10	)	)	PUNCT
ejpam-4592	139	11	wherex	wherex	NOUN
ejpam-4592	139	12	=	=	SYM
ejpam-4592	139	13	r	r	PROPN
ejpam-4592	139	14	,	,	PUNCT
ejpam-4592	139	15	ω1	ω1	PROPN
ejpam-4592	139	16	=	=	SYM
ejpam-4592	139	17	{	{	PUNCT
ejpam-4592	139	18	σ1	σ1	PROPN
ejpam-4592	139	19	,	,	PUNCT
ejpam-4592	139	20	σ2	σ2	PROPN
ejpam-4592	139	21	,	,	PUNCT
ejpam-4592	139	22	σ3	σ3	PROPN
ejpam-4592	139	23	}	}	PUNCT
ejpam-4592	139	24	⊂	⊂	PROPN
ejpam-4592	139	25	ωx	ωx	PROPN
ejpam-4592	139	26	and	and	CCONJ
ejpam-4592	139	27	the	the	DET
ejpam-4592	139	28	metrics	metric	NOUN
ejpam-4592	139	29	are	be	AUX
ejpam-4592	139	30	defined	define	VERB
ejpam-4592	139	31	by	by	ADP
ejpam-4592	139	32	σ1(c	σ1(c	PRON
ejpam-4592	139	33	,	,	PUNCT
ejpam-4592	139	34	d	d	NOUN
ejpam-4592	139	35	)	)	PUNCT
ejpam-4592	140	1	=	=	PRON
ejpam-4592	140	2	{	{	PUNCT
ejpam-4592	140	3	0	0	NUM
ejpam-4592	140	4	if	if	SCONJ
ejpam-4592	140	5	c	c	NOUN
ejpam-4592	140	6	=	=	SYM
ejpam-4592	140	7	d	d	PROPN
ejpam-4592	140	8	,	,	PUNCT
ejpam-4592	140	9	1	1	NUM
ejpam-4592	140	10	if	if	SCONJ
ejpam-4592	140	11	c	c	PROPN
ejpam-4592	140	12	̸=	̸=	PROPN
ejpam-4592	140	13	d.	d.	PROPN
ejpam-4592	140	14	σ2(c	σ2(c	PROPN
ejpam-4592	140	15	,	,	PUNCT
ejpam-4592	140	16	d	d	NOUN
ejpam-4592	140	17	)	)	PUNCT
ejpam-4592	140	18	=	=	SYM
ejpam-4592	140	19	|c−	|c−	VERB
ejpam-4592	140	20	d|	d|	PROPN
ejpam-4592	140	21	and	and	CCONJ
ejpam-4592	140	22	σ3(c	σ3(c	PRON
ejpam-4592	140	23	,	,	PUNCT
ejpam-4592	140	24	d	d	NOUN
ejpam-4592	140	25	)	)	PUNCT
ejpam-4592	140	26	=	=	SYM
ejpam-4592	140	27	min{1	min{1	PROPN
ejpam-4592	140	28	,	,	PUNCT
ejpam-4592	140	29	σ1(c	σ1(c	PRON
ejpam-4592	140	30	,	,	PUNCT
ejpam-4592	140	31	d	d	NOUN
ejpam-4592	140	32	)	)	PUNCT
ejpam-4592	140	33	}	}	PUNCT
ejpam-4592	140	34	for	for	ADP
ejpam-4592	140	35	all	all	DET
ejpam-4592	140	36	c	c	NOUN
ejpam-4592	140	37	,	,	PUNCT
ejpam-4592	140	38	d	d	PROPN
ejpam-4592	140	39	∈	∈	PROPN
ejpam-4592	140	40	x.	x.	NOUN
ejpam-4592	140	41	define	define	VERB
ejpam-4592	140	42	a	a	DET
ejpam-4592	140	43	sequence	sequence	NOUN
ejpam-4592	140	44	{	{	PUNCT
ejpam-4592	140	45	cn}n∈n	cn}n∈n	VERB
ejpam-4592	140	46	by	by	ADP
ejpam-4592	140	47	{	{	PUNCT
ejpam-4592	140	48	cn	cn	PROPN
ejpam-4592	140	49	}	}	PUNCT
ejpam-4592	140	50	=	=	PUNCT
ejpam-4592	140	51	{	{	PUNCT
ejpam-4592	140	52	1	1	NUM
ejpam-4592	140	53	,	,	PUNCT
ejpam-4592	140	54	1	1	NUM
ejpam-4592	140	55	,	,	PUNCT
ejpam-4592	140	56	2	2	NUM
ejpam-4592	140	57	,	,	PUNCT
ejpam-4592	140	58	2	2	NUM
ejpam-4592	140	59	,	,	PUNCT
ejpam-4592	140	60	3	3	NUM
ejpam-4592	140	61	,	,	PUNCT
ejpam-4592	140	62	3	3	NUM
ejpam-4592	140	63	,	,	PUNCT
ejpam-4592	140	64	4	4	NUM
ejpam-4592	140	65	,	,	PUNCT
ejpam-4592	140	66	4	4	NUM
ejpam-4592	140	67	,	,	PUNCT
ejpam-4592	140	68	5	5	NUM
ejpam-4592	140	69	,	,	PUNCT
ejpam-4592	140	70	5	5	NUM
ejpam-4592	140	71	,	,	PUNCT
ejpam-4592	140	72	.....	.....	PUNCT
ejpam-4592	140	73	}	}	PUNCT
ejpam-4592	140	74	in	in	ADP
ejpam-4592	140	75	x.	x.	NOUN
ejpam-4592	140	76	then	then	ADV
ejpam-4592	140	77	{	{	PUNCT
ejpam-4592	140	78	cn	cn	PROPN
ejpam-4592	140	79	}	}	PUNCT
ejpam-4592	140	80	∈	∈	PROPN
ejpam-4592	140	81	p(σ2	p(σ2	VERB
ejpam-4592	140	82	)	)	PUNCT
ejpam-4592	140	83	.	.	PUNCT
ejpam-4592	141	1	but	but	CCONJ
ejpam-4592	141	2	for	for	ADP
ejpam-4592	141	3	every	every	DET
ejpam-4592	141	4	ε	ε	PROPN
ejpam-4592	141	5	>	>	X
ejpam-4592	141	6	0	0	PROPN
ejpam-4592	141	7	,	,	PUNCT
ejpam-4592	141	8	there	there	PRON
ejpam-4592	141	9	exists	exist	VERB
ejpam-4592	141	10	no	no	DET
ejpam-4592	141	11	n0	n0	NOUN
ejpam-4592	141	12	∈	∈	PROPN
ejpam-4592	141	13	n	n	CCONJ
ejpam-4592	141	14	such	such	ADJ
ejpam-4592	141	15	that	that	SCONJ
ejpam-4592	141	16	σ2(cn	σ2(cn	PROPN
ejpam-4592	141	17	,	,	PUNCT
ejpam-4592	141	18	cm	cm	NOUN
ejpam-4592	141	19	)	)	PUNCT
ejpam-4592	141	20	<	<	X
ejpam-4592	141	21	ε	ε	PROPN
ejpam-4592	141	22	for	for	ADP
ejpam-4592	141	23	n	n	CCONJ
ejpam-4592	141	24	,	,	PUNCT
ejpam-4592	141	25	m	m	PROPN
ejpam-4592	141	26	≥	≥	NOUN
ejpam-4592	141	27	n0	n0	NUM
ejpam-4592	141	28	.	.	PUNCT
ejpam-4592	142	1	thus	thus	ADV
ejpam-4592	142	2	,	,	PUNCT
ejpam-4592	142	3	{	{	PUNCT
ejpam-4592	142	4	cn	cn	ADJ
ejpam-4592	142	5	}	}	PUNCT
ejpam-4592	142	6	/∈	/∈	PUNCT
ejpam-4592	142	7	c(σ2	c(σ2	NUM
ejpam-4592	142	8	)	)	PUNCT
ejpam-4592	142	9	.	.	PUNCT
ejpam-4592	143	1	v.	v.	ADP
ejpam-4592	143	2	subramanian	subramanian	PROPN
ejpam-4592	143	3	et	et	PROPN
ejpam-4592	143	4	al	al	PROPN
ejpam-4592	143	5	.	.	PUNCT
ejpam-4592	143	6	/	/	SYM
ejpam-4592	143	7	eur	eur	PROPN
ejpam-4592	143	8	.	.	PUNCT
ejpam-4592	144	1	j.	j.	PROPN
ejpam-4592	144	2	pure	pure	PROPN
ejpam-4592	144	3	appl	appl	PROPN
ejpam-4592	144	4	.	.	PROPN
ejpam-4592	144	5	math	math	PROPN
ejpam-4592	144	6	,	,	PUNCT
ejpam-4592	144	7	15	15	NUM
ejpam-4592	144	8	(	(	PUNCT
ejpam-4592	144	9	4	4	NUM
ejpam-4592	144	10	)	)	PUNCT
ejpam-4592	144	11	(	(	PUNCT
ejpam-4592	144	12	2022	2022	NUM
ejpam-4592	144	13	)	)	PUNCT
ejpam-4592	144	14	,	,	PUNCT
ejpam-4592	144	15	1869	1869	NUM
ejpam-4592	144	16	-	-	SYM
ejpam-4592	144	17	1886	1886	NUM
ejpam-4592	144	18	1874	1874	NUM
ejpam-4592	144	19	theorem	theorem	VERB
ejpam-4592	144	20	17	17	NUM
ejpam-4592	144	21	and	and	CCONJ
ejpam-4592	144	22	example	example	NOUN
ejpam-4592	144	23	18	18	NUM
ejpam-4592	144	24	are	be	AUX
ejpam-4592	144	25	describe	describe	NOUN
ejpam-4592	144	26	the	the	DET
ejpam-4592	144	27	below	below	ADJ
ejpam-4592	144	28	diagram	diagram	NOUN
ejpam-4592	144	29	.	.	PUNCT
ejpam-4592	145	1	the	the	DET
ejpam-4592	145	2	below	below	ADJ
ejpam-4592	145	3	theorem	theorem	NOUN
ejpam-4592	145	4	17	17	NUM
ejpam-4592	145	5	gives	give	VERB
ejpam-4592	145	6	the	the	DET
ejpam-4592	145	7	relations	relation	NOUN
ejpam-4592	145	8	between	between	ADP
ejpam-4592	145	9	bourbaki	bourbaki	NOUN
ejpam-4592	145	10	-	-	PUNCT
ejpam-4592	145	11	cauchy	cauchy	ADJ
ejpam-4592	145	12	sequence	sequence	NOUN
ejpam-4592	145	13	and	and	CCONJ
ejpam-4592	145	14	cofinally	cofinally	ADV
ejpam-4592	145	15	-	-	PUNCT
ejpam-4592	145	16	cauchy	cauchy	ADJ
ejpam-4592	145	17	sequence	sequence	NOUN
ejpam-4592	145	18	in	in	ADP
ejpam-4592	145	19	a	a	DET
ejpam-4592	145	20	gms	gms	NOUN
ejpam-4592	145	21	.	.	PUNCT
ejpam-4592	145	22	bourbaki−	bourbaki−	NOUN
ejpam-4592	145	23	cauchy	cauchy	NOUN
ejpam-4592	145	24	sequence	sequence	NOUN
ejpam-4592	145	25	cofinally	cofinally	ADV
ejpam-4592	145	26	−	−	ADP
ejpam-4592	145	27	cauchy	cauchy	NOUN
ejpam-4592	145	28	sequence	sequence	NOUN
ejpam-4592	145	29	/	/	SYM
ejpam-4592	145	30	theorem	theorem	NOUN
ejpam-4592	145	31	17	17	NUM
ejpam-4592	145	32	.	.	PUNCT
ejpam-4592	146	1	let	let	AUX
ejpam-4592	146	2	(	(	PUNCT
ejpam-4592	146	3	x	x	NOUN
ejpam-4592	146	4	,	,	PUNCT
ejpam-4592	146	5	ω	ω	NUM
ejpam-4592	146	6	)	)	PUNCT
ejpam-4592	146	7	be	be	AUX
ejpam-4592	146	8	a	a	DET
ejpam-4592	146	9	gms	gms	NOUN
ejpam-4592	146	10	.	.	PUNCT
ejpam-4592	147	1	then	then	ADV
ejpam-4592	147	2	the	the	DET
ejpam-4592	147	3	followings	following	NOUN
ejpam-4592	147	4	are	be	AUX
ejpam-4592	147	5	true	true	ADJ
ejpam-4592	147	6	.	.	PUNCT
ejpam-4592	148	1	(	(	PUNCT
ejpam-4592	148	2	a	a	X
ejpam-4592	148	3	)	)	PUNCT
ejpam-4592	148	4	every	every	DET
ejpam-4592	148	5	bourbaki	bourbaki	NOUN
ejpam-4592	148	6	-	-	PUNCT
ejpam-4592	148	7	cauchy	cauchy	ADJ
ejpam-4592	148	8	sequence	sequence	NOUN
ejpam-4592	148	9	is	be	AUX
ejpam-4592	148	10	a	a	DET
ejpam-4592	148	11	cofinally	cofinally	ADV
ejpam-4592	148	12	-	-	PUNCT
ejpam-4592	148	13	cauchy	cauchy	ADJ
ejpam-4592	148	14	sequence	sequence	NOUN
ejpam-4592	148	15	in	in	ADP
ejpam-4592	148	16	x	x	PUNCT
ejpam-4592	148	17	with	with	ADP
ejpam-4592	148	18	the	the	DET
ejpam-4592	148	19	same	same	ADJ
ejpam-4592	148	20	metric	metric	NOUN
ejpam-4592	148	21	.	.	PUNCT
ejpam-4592	149	1	(	(	PUNCT
ejpam-4592	149	2	b	b	X
ejpam-4592	149	3	)	)	PUNCT
ejpam-4592	149	4	every	every	DET
ejpam-4592	149	5	cofinally	cofinally	ADV
ejpam-4592	149	6	-	-	PUNCT
ejpam-4592	149	7	complete	complete	ADJ
ejpam-4592	149	8	metric	metric	ADJ
ejpam-4592	149	9	space	space	NOUN
ejpam-4592	149	10	is	be	AUX
ejpam-4592	149	11	a	a	DET
ejpam-4592	149	12	bourbaki	bourbaki	NOUN
ejpam-4592	149	13	-complete	-complete	ADJ
ejpam-4592	149	14	metric	metric	ADJ
ejpam-4592	149	15	space	space	NOUN
ejpam-4592	149	16	.	.	PUNCT
ejpam-4592	150	1	proof	proof	NOUN
ejpam-4592	150	2	.	.	PUNCT
ejpam-4592	151	1	let	let	VERB
ejpam-4592	151	2	{	{	PUNCT
ejpam-4592	151	3	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	151	4	∈	∈	PROPN
ejpam-4592	151	5	b(σ	b(σ	PROPN
ejpam-4592	151	6	)	)	PUNCT
ejpam-4592	151	7	and	and	CCONJ
ejpam-4592	151	8	ε	ε	PROPN
ejpam-4592	151	9	>	>	X
ejpam-4592	151	10	0	0	PUNCT
ejpam-4592	151	11	be	be	AUX
ejpam-4592	151	12	given	give	VERB
ejpam-4592	151	13	.	.	PUNCT
ejpam-4592	152	1	then	then	ADV
ejpam-4592	152	2	there	there	PRON
ejpam-4592	152	3	is	be	VERB
ejpam-4592	152	4	m	m	PROPN
ejpam-4592	152	5	,	,	PUNCT
ejpam-4592	152	6	n0	n0	PROPN
ejpam-4592	152	7	∈	∈	PROPN
ejpam-4592	152	8	n	n	CCONJ
ejpam-4592	152	9	such	such	ADJ
ejpam-4592	152	10	that	that	SCONJ
ejpam-4592	152	11	whenever	whenever	SCONJ
ejpam-4592	152	12	n	n	PROPN
ejpam-4592	152	13	>	>	X
ejpam-4592	152	14	j	j	PROPN
ejpam-4592	152	15	≥	≥	PROPN
ejpam-4592	152	16	n0	n0	PROPN
ejpam-4592	152	17	,	,	PUNCT
ejpam-4592	152	18	the	the	DET
ejpam-4592	152	19	points	point	NOUN
ejpam-4592	152	20	cj	cj	NOUN
ejpam-4592	152	21	and	and	CCONJ
ejpam-4592	152	22	cn	cn	PROPN
ejpam-4592	152	23	can	can	AUX
ejpam-4592	152	24	be	be	AUX
ejpam-4592	152	25	joined	join	VERB
ejpam-4592	152	26	by	by	ADP
ejpam-4592	152	27	an	an	DET
ejpam-4592	152	28	ε	ε	PROPN
ejpam-4592	152	29	-	-	PUNCT
ejpam-4592	152	30	chain	chain	NOUN
ejpam-4592	152	31	of	of	ADP
ejpam-4592	152	32	length	length	NOUN
ejpam-4592	152	33	m.	m.	NOUN
ejpam-4592	152	34	let	let	VERB
ejpam-4592	152	35	nε	nε	PROPN
ejpam-4592	152	36	=	=	PRON
ejpam-4592	152	37	{	{	PUNCT
ejpam-4592	152	38	n	n	CCONJ
ejpam-4592	152	39	|	|	ADV
ejpam-4592	152	40	n	n	CCONJ
ejpam-4592	152	41	≥	≥	NOUN
ejpam-4592	152	42	n0	n0	NUM
ejpam-4592	152	43	and	and	CCONJ
ejpam-4592	152	44	cn	cn	PROPN
ejpam-4592	152	45	,	,	PUNCT
ejpam-4592	152	46	cj	cj	PROPN
ejpam-4592	152	47	can	can	AUX
ejpam-4592	152	48	be	be	AUX
ejpam-4592	152	49	joined	join	VERB
ejpam-4592	152	50	by	by	ADP
ejpam-4592	152	51	ε	ε	PROPN
ejpam-4592	152	52	m	m	VERB
ejpam-4592	152	53	-chain	-chain	NOUN
ejpam-4592	152	54	of	of	ADP
ejpam-4592	152	55	length	length	NOUN
ejpam-4592	152	56	m	m	PROPN
ejpam-4592	152	57	whenever	whenever	SCONJ
ejpam-4592	152	58	m	m	AUX
ejpam-4592	152	59	−	−	NUM
ejpam-4592	152	60	1	1	NUM
ejpam-4592	152	61	points	point	NOUN
ejpam-4592	152	62	lie	lie	VERB
ejpam-4592	152	63	between	between	ADP
ejpam-4592	152	64	cn	cn	PROPN
ejpam-4592	152	65	and	and	CCONJ
ejpam-4592	152	66	cj	cj	NOUN
ejpam-4592	152	67	where	where	SCONJ
ejpam-4592	152	68	n	n	X
ejpam-4592	152	69	̸=	̸=	PROPN
ejpam-4592	152	70	j	j	PROPN
ejpam-4592	152	71	;	;	PUNCT
ejpam-4592	152	72	j	j	PROPN
ejpam-4592	152	73	≥	≥	PROPN
ejpam-4592	152	74	n0;m	n0;m	PROPN
ejpam-4592	152	75	,	,	PUNCT
ejpam-4592	152	76	j	j	PROPN
ejpam-4592	152	77	∈	∈	PROPN
ejpam-4592	152	78	n	n	CCONJ
ejpam-4592	152	79	}	}	PUNCT
ejpam-4592	152	80	.	.	PUNCT
ejpam-4592	153	1	then	then	ADV
ejpam-4592	153	2	nε	nε	PROPN
ejpam-4592	153	3	is	be	AUX
ejpam-4592	153	4	an	an	DET
ejpam-4592	153	5	infinite	infinite	ADJ
ejpam-4592	153	6	subset	subset	NOUN
ejpam-4592	153	7	of	of	ADP
ejpam-4592	153	8	n.	n.	PROPN
ejpam-4592	153	9	let	let	VERB
ejpam-4592	153	10	k	k	NOUN
ejpam-4592	153	11	,	,	PUNCT
ejpam-4592	153	12	l	l	PROPN
ejpam-4592	153	13	∈	∈	PROPN
ejpam-4592	153	14	nε	nε	VERB
ejpam-4592	153	15	.	.	PUNCT
ejpam-4592	154	1	if	if	SCONJ
ejpam-4592	154	2	k	k	PROPN
ejpam-4592	154	3	and	and	CCONJ
ejpam-4592	154	4	l	l	NOUN
ejpam-4592	154	5	are	be	AUX
ejpam-4592	154	6	consecutive	consecutive	ADJ
ejpam-4592	154	7	numbers	number	NOUN
ejpam-4592	154	8	,	,	PUNCT
ejpam-4592	154	9	then	then	ADV
ejpam-4592	154	10	there	there	PRON
ejpam-4592	154	11	is	be	VERB
ejpam-4592	154	12	nothing	nothing	PRON
ejpam-4592	154	13	to	to	PART
ejpam-4592	154	14	prove	prove	VERB
ejpam-4592	154	15	.	.	PUNCT
ejpam-4592	155	1	suppose	suppose	VERB
ejpam-4592	155	2	that	that	SCONJ
ejpam-4592	155	3	there	there	PRON
ejpam-4592	155	4	is	be	VERB
ejpam-4592	155	5	some	some	DET
ejpam-4592	155	6	p	p	NOUN
ejpam-4592	155	7	points	point	NOUN
ejpam-4592	155	8	lies	lie	NOUN
ejpam-4592	155	9	between	between	ADP
ejpam-4592	155	10	k	k	PROPN
ejpam-4592	155	11	and	and	CCONJ
ejpam-4592	155	12	l.	l.	PROPN
ejpam-4592	156	1	then	then	ADV
ejpam-4592	156	2	ck	ck	PROPN
ejpam-4592	156	3	and	and	CCONJ
ejpam-4592	156	4	cl	cl	NOUN
ejpam-4592	156	5	can	can	AUX
ejpam-4592	156	6	be	be	AUX
ejpam-4592	156	7	joined	join	VERB
ejpam-4592	156	8	by	by	ADP
ejpam-4592	156	9	ε	ε	PROPN
ejpam-4592	156	10	p+1	p+1	NOUN
ejpam-4592	156	11	-chain	-chain	PROPN
ejpam-4592	156	12	of	of	ADP
ejpam-4592	156	13	length	length	NOUN
ejpam-4592	156	14	p+	p+	PROPN
ejpam-4592	156	15	1	1	NUM
ejpam-4592	156	16	and	and	CCONJ
ejpam-4592	156	17	so	so	ADV
ejpam-4592	156	18	the	the	DET
ejpam-4592	156	19	distance	distance	NOUN
ejpam-4592	156	20	between	between	ADP
ejpam-4592	156	21	the	the	DET
ejpam-4592	156	22	consecutive	consecutive	ADJ
ejpam-4592	156	23	numbers	number	NOUN
ejpam-4592	156	24	is	be	AUX
ejpam-4592	156	25	less	less	ADJ
ejpam-4592	156	26	than	than	ADP
ejpam-4592	156	27	ε	ε	PROPN
ejpam-4592	156	28	p+1	p+1	PROPN
ejpam-4592	156	29	.	.	PUNCT
ejpam-4592	157	1	thus	thus	ADV
ejpam-4592	157	2	,	,	PUNCT
ejpam-4592	157	3	σ(ck	σ(ck	PROPN
ejpam-4592	157	4	,	,	PUNCT
ejpam-4592	157	5	cl	cl	NOUN
ejpam-4592	157	6	)	)	PUNCT
ejpam-4592	157	7	<	<	X
ejpam-4592	157	8	ε	ε	PROPN
ejpam-4592	157	9	.	.	PUNCT
ejpam-4592	158	1	hence	hence	ADV
ejpam-4592	158	2	{	{	PUNCT
ejpam-4592	158	3	cn	cn	ADJ
ejpam-4592	158	4	}	}	PUNCT
ejpam-4592	158	5	∈	∈	PROPN
ejpam-4592	158	6	c(σ	c(σ	PROPN
ejpam-4592	158	7	)	)	PUNCT
ejpam-4592	158	8	.	.	PUNCT
ejpam-4592	159	1	(	(	PUNCT
ejpam-4592	159	2	b	b	NOUN
ejpam-4592	159	3	)	)	PUNCT
ejpam-4592	159	4	.	.	PUNCT
ejpam-4592	160	1	directly	directly	ADV
ejpam-4592	160	2	follows	follow	VERB
ejpam-4592	160	3	from	from	ADP
ejpam-4592	160	4	the	the	DET
ejpam-4592	160	5	definition	definition	NOUN
ejpam-4592	160	6	of	of	ADP
ejpam-4592	160	7	bourbaki	bourbaki	NOUN
ejpam-4592	160	8	-	-	PUNCT
ejpam-4592	160	9	complete	complete	ADJ
ejpam-4592	160	10	metric	metric	ADJ
ejpam-4592	160	11	space	space	NOUN
ejpam-4592	160	12	and	and	CCONJ
ejpam-4592	160	13	by	by	ADP
ejpam-4592	160	14	(	(	PUNCT
ejpam-4592	160	15	a	a	NOUN
ejpam-4592	160	16	)	)	PUNCT
ejpam-4592	160	17	.	.	PUNCT
ejpam-4592	160	18	example	example	NOUN
ejpam-4592	160	19	18	18	NUM
ejpam-4592	160	20	.	.	PUNCT
ejpam-4592	161	1	consider	consider	VERB
ejpam-4592	161	2	the	the	DET
ejpam-4592	161	3	generalized	generalized	ADJ
ejpam-4592	161	4	metric	metric	ADJ
ejpam-4592	161	5	space	space	NOUN
ejpam-4592	161	6	(	(	PUNCT
ejpam-4592	161	7	x	x	NOUN
ejpam-4592	161	8	,	,	PUNCT
ejpam-4592	161	9	ω1	ω1	PROPN
ejpam-4592	161	10	)	)	PUNCT
ejpam-4592	161	11	wherex	wherex	NOUN
ejpam-4592	161	12	=	=	SYM
ejpam-4592	161	13	r	r	PROPN
ejpam-4592	161	14	,	,	PUNCT
ejpam-4592	161	15	ω1	ω1	PROPN
ejpam-4592	161	16	=	=	SYM
ejpam-4592	161	17	{	{	PUNCT
ejpam-4592	161	18	σ1	σ1	PROPN
ejpam-4592	161	19	,	,	PUNCT
ejpam-4592	161	20	σ2	σ2	PROPN
ejpam-4592	161	21	,	,	PUNCT
ejpam-4592	161	22	σ3	σ3	PROPN
ejpam-4592	161	23	}	}	PUNCT
ejpam-4592	161	24	⊂	⊂	PROPN
ejpam-4592	161	25	ωx	ωx	PROPN
ejpam-4592	161	26	and	and	CCONJ
ejpam-4592	161	27	then	then	ADV
ejpam-4592	161	28	the	the	DET
ejpam-4592	161	29	metrics	metric	NOUN
ejpam-4592	161	30	are	be	AUX
ejpam-4592	161	31	defined	define	VERB
ejpam-4592	161	32	by	by	ADP
ejpam-4592	161	33	σ1(c	σ1(c	PRON
ejpam-4592	161	34	,	,	PUNCT
ejpam-4592	161	35	d	d	NOUN
ejpam-4592	161	36	)	)	PUNCT
ejpam-4592	162	1	=	=	SYM
ejpam-4592	162	2	min{|c−	min{|c−	PROPN
ejpam-4592	162	3	d|	d|	PROPN
ejpam-4592	162	4	,	,	PUNCT
ejpam-4592	162	5	1	1	NUM
ejpam-4592	162	6	}	}	PUNCT
ejpam-4592	162	7	;	;	PUNCT
ejpam-4592	162	8	σ2(c	σ2(c	X
ejpam-4592	162	9	,	,	PUNCT
ejpam-4592	162	10	d	d	NOUN
ejpam-4592	162	11	)	)	PUNCT
ejpam-4592	162	12	=	=	PRON
ejpam-4592	162	13	{	{	PUNCT
ejpam-4592	162	14	0	0	NUM
ejpam-4592	162	15	if	if	SCONJ
ejpam-4592	162	16	c	c	NOUN
ejpam-4592	162	17	=	=	SYM
ejpam-4592	162	18	d	d	PROPN
ejpam-4592	162	19	,	,	PUNCT
ejpam-4592	162	20	1	1	NUM
ejpam-4592	162	21	if	if	SCONJ
ejpam-4592	162	22	c	c	PROPN
ejpam-4592	162	23	̸=	̸=	PROPN
ejpam-4592	162	24	d	d	PROPN
ejpam-4592	162	25	and	and	CCONJ
ejpam-4592	162	26	hence	hence	ADV
ejpam-4592	162	27	σ3(c	σ3(c	PROPN
ejpam-4592	162	28	,	,	PUNCT
ejpam-4592	162	29	d	d	NOUN
ejpam-4592	162	30	)	)	PUNCT
ejpam-4592	162	31	=	=	SYM
ejpam-4592	162	32	σ1(c	σ1(c	PROPN
ejpam-4592	162	33	,	,	PUNCT
ejpam-4592	162	34	d)+σ2(c	d)+σ2(c	ADJ
ejpam-4592	162	35	,	,	PUNCT
ejpam-4592	162	36	d	d	NOUN
ejpam-4592	162	37	)	)	PUNCT
ejpam-4592	162	38	for	for	ADP
ejpam-4592	162	39	all	all	DET
ejpam-4592	162	40	c	c	NOUN
ejpam-4592	162	41	,	,	PUNCT
ejpam-4592	162	42	d	d	PROPN
ejpam-4592	162	43	∈	∈	PROPN
ejpam-4592	162	44	x.	x.	NOUN
ejpam-4592	162	45	define	define	VERB
ejpam-4592	162	46	a	a	DET
ejpam-4592	162	47	sequence	sequence	NOUN
ejpam-4592	162	48	{	{	PUNCT
ejpam-4592	162	49	cn}n∈n	cn}n∈n	VERB
ejpam-4592	162	50	by	by	ADP
ejpam-4592	162	51	{	{	PUNCT
ejpam-4592	162	52	cn	cn	PROPN
ejpam-4592	162	53	}	}	PUNCT
ejpam-4592	162	54	=	=	PUNCT
ejpam-4592	162	55	{	{	PUNCT
ejpam-4592	162	56	1,−1	1,−1	PROPN
ejpam-4592	162	57	,	,	PUNCT
ejpam-4592	162	58	2,−1	2,−1	NUM
ejpam-4592	162	59	,	,	PUNCT
ejpam-4592	162	60	3,−1	3,−1	NUM
ejpam-4592	162	61	,	,	PUNCT
ejpam-4592	162	62	4,−1	4,−1	PROPN
ejpam-4592	162	63	,	,	PUNCT
ejpam-4592	162	64	.....	.....	PUNCT
ejpam-4592	162	65	}	}	PUNCT
ejpam-4592	162	66	in	in	ADP
ejpam-4592	162	67	x.	x.	NOUN
ejpam-4592	162	68	then	then	ADV
ejpam-4592	162	69	{	{	PUNCT
ejpam-4592	162	70	cn	cn	ADJ
ejpam-4592	162	71	}	}	PUNCT
ejpam-4592	162	72	∈	∈	PROPN
ejpam-4592	162	73	c(σ1	c(σ1	NOUN
ejpam-4592	162	74	)	)	PUNCT
ejpam-4592	162	75	.	.	PUNCT
ejpam-4592	163	1	but	but	CCONJ
ejpam-4592	163	2	{	{	PUNCT
ejpam-4592	163	3	cn	cn	ADJ
ejpam-4592	163	4	}	}	PUNCT
ejpam-4592	163	5	/∈	/∈	PUNCT
ejpam-4592	163	6	b(σ1	b(σ1	NOUN
ejpam-4592	163	7	)	)	PUNCT
ejpam-4592	163	8	.	.	PUNCT
ejpam-4592	164	1	because	because	SCONJ
ejpam-4592	164	2	,	,	PUNCT
ejpam-4592	164	3	for	for	ADP
ejpam-4592	164	4	every	every	DET
ejpam-4592	164	5	ε	ε	PROPN
ejpam-4592	164	6	>	>	X
ejpam-4592	164	7	0	0	PROPN
ejpam-4592	164	8	,	,	PUNCT
ejpam-4592	164	9	there	there	PRON
ejpam-4592	164	10	is	be	VERB
ejpam-4592	164	11	no	no	DET
ejpam-4592	164	12	m	m	NOUN
ejpam-4592	164	13	,	,	PUNCT
ejpam-4592	164	14	n0	n0	PROPN
ejpam-4592	164	15	∈	∈	PROPN
ejpam-4592	164	16	n	n	CCONJ
ejpam-4592	164	17	such	such	ADJ
ejpam-4592	164	18	that	that	SCONJ
ejpam-4592	164	19	whenever	whenever	SCONJ
ejpam-4592	164	20	n	n	PROPN
ejpam-4592	164	21	>	>	X
ejpam-4592	164	22	j	j	PROPN
ejpam-4592	164	23	≥	≥	PROPN
ejpam-4592	164	24	n0	n0	PROPN
ejpam-4592	164	25	,	,	PUNCT
ejpam-4592	164	26	the	the	DET
ejpam-4592	164	27	points	point	NOUN
ejpam-4592	164	28	cj	cj	NOUN
ejpam-4592	164	29	and	and	CCONJ
ejpam-4592	164	30	cn	cn	PROPN
ejpam-4592	164	31	can	can	AUX
ejpam-4592	164	32	be	be	AUX
ejpam-4592	164	33	joined	join	VERB
ejpam-4592	164	34	by	by	ADP
ejpam-4592	164	35	an	an	DET
ejpam-4592	164	36	ε	ε	PROPN
ejpam-4592	164	37	-	-	PUNCT
ejpam-4592	164	38	chain	chain	NOUN
ejpam-4592	164	39	of	of	ADP
ejpam-4592	164	40	length	length	NOUN
ejpam-4592	164	41	m.	m.	NOUN
ejpam-4592	164	42	theorem	theorem	VERB
ejpam-4592	164	43	19	19	NUM
ejpam-4592	164	44	.	.	PUNCT
ejpam-4592	165	1	let	let	VERB
ejpam-4592	165	2	(	(	PUNCT
ejpam-4592	165	3	x	x	NOUN
ejpam-4592	165	4	,	,	PUNCT
ejpam-4592	165	5	ω	ω	NUM
ejpam-4592	165	6	)	)	PUNCT
ejpam-4592	165	7	be	be	AUX
ejpam-4592	165	8	a	a	DET
ejpam-4592	165	9	gms	gms	NOUN
ejpam-4592	165	10	.	.	PUNCT
ejpam-4592	166	1	then	then	ADV
ejpam-4592	166	2	every	every	DET
ejpam-4592	166	3	cofinally	cofinally	ADV
ejpam-4592	166	4	-	-	PUNCT
ejpam-4592	166	5	cauchy	cauchy	ADJ
ejpam-4592	166	6	sequence	sequence	NOUN
ejpam-4592	166	7	has	have	VERB
ejpam-4592	166	8	a	a	DET
ejpam-4592	166	9	cauchy	cauchy	ADJ
ejpam-4592	166	10	subsequence	subsequence	NOUN
ejpam-4592	166	11	with	with	ADP
ejpam-4592	166	12	the	the	DET
ejpam-4592	166	13	same	same	ADJ
ejpam-4592	166	14	metric	metric	NOUN
ejpam-4592	166	15	.	.	PUNCT
ejpam-4592	167	1	proof	proof	NOUN
ejpam-4592	167	2	.	.	PUNCT
ejpam-4592	168	1	let	let	VERB
ejpam-4592	168	2	σ	σ	X
ejpam-4592	168	3	∈	∈	PROPN
ejpam-4592	168	4	ω	ω	PROPN
ejpam-4592	168	5	and	and	CCONJ
ejpam-4592	168	6	{	{	PUNCT
ejpam-4592	168	7	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	168	8	∈	∈	PROPN
ejpam-4592	168	9	c(σ	c(σ	PROPN
ejpam-4592	168	10	)	)	PUNCT
ejpam-4592	168	11	.	.	PUNCT
ejpam-4592	169	1	given	give	VERB
ejpam-4592	169	2	ε	ε	PROPN
ejpam-4592	169	3	>	>	X
ejpam-4592	169	4	0	0	PROPN
ejpam-4592	169	5	.	.	PUNCT
ejpam-4592	170	1	then	then	ADV
ejpam-4592	170	2	there	there	PRON
ejpam-4592	170	3	is	be	VERB
ejpam-4592	170	4	an	an	DET
ejpam-4592	170	5	infinite	infinite	NOUN
ejpam-4592	170	6	subset	subset	NOUN
ejpam-4592	170	7	nε	nε	NOUN
ejpam-4592	170	8	of	of	ADP
ejpam-4592	170	9	n	n	PRON
ejpam-4592	170	10	such	such	ADJ
ejpam-4592	170	11	that	that	SCONJ
ejpam-4592	170	12	σ(cl	σ(cl	NOUN
ejpam-4592	170	13	,	,	PUNCT
ejpam-4592	170	14	ck	ck	ADJ
ejpam-4592	170	15	)	)	PUNCT
ejpam-4592	170	16	<	<	X
ejpam-4592	170	17	ε	ε	PROPN
ejpam-4592	170	18	for	for	ADP
ejpam-4592	170	19	each	each	DET
ejpam-4592	170	20	l	l	NOUN
ejpam-4592	170	21	,	,	PUNCT
ejpam-4592	170	22	k	k	PROPN
ejpam-4592	170	23	∈	∈	PROPN
ejpam-4592	170	24	nε	nε	VERB
ejpam-4592	170	25	.	.	PUNCT
ejpam-4592	171	1	let	let	VERB
ejpam-4592	171	2	a	a	DET
ejpam-4592	171	3	be	be	AUX
ejpam-4592	171	4	a	a	DET
ejpam-4592	171	5	partially	partially	ADV
ejpam-4592	171	6	ordered	order	VERB
ejpam-4592	171	7	subset	subset	NOUN
ejpam-4592	171	8	of	of	ADP
ejpam-4592	171	9	nε	nε	NOUN
ejpam-4592	171	10	with	with	ADP
ejpam-4592	171	11	<	<	X
ejpam-4592	171	12	ordering	ordering	NOUN
ejpam-4592	171	13	and	and	CCONJ
ejpam-4592	171	14	{	{	PUNCT
ejpam-4592	171	15	cnk	cnk	PROPN
ejpam-4592	171	16	}	}	PUNCT
ejpam-4592	171	17	be	be	AUX
ejpam-4592	171	18	a	a	DET
ejpam-4592	171	19	subsequence	subsequence	NOUN
ejpam-4592	171	20	of	of	ADP
ejpam-4592	171	21	{	{	PUNCT
ejpam-4592	171	22	cn	cn	PROPN
ejpam-4592	171	23	}	}	PUNCT
ejpam-4592	171	24	where	where	SCONJ
ejpam-4592	171	25	nk	nk	PROPN
ejpam-4592	171	26	∈	∈	PROPN
ejpam-4592	171	27	a.	a.	NOUN
ejpam-4592	171	28	choose	choose	VERB
ejpam-4592	171	29	m	m	NOUN
ejpam-4592	172	1	=	=	PRON
ejpam-4592	173	1	min{ni	min{ni	PUNCT
ejpam-4592	174	1	|	|	ADV
ejpam-4592	174	2	ni	ni	PROPN
ejpam-4592	174	3	∈	∈	PROPN
ejpam-4592	174	4	a	a	PRON
ejpam-4592	174	5	}	}	PUNCT
ejpam-4592	174	6	.	.	PUNCT
ejpam-4592	175	1	for	for	ADP
ejpam-4592	175	2	np	np	INTJ
ejpam-4592	175	3	,	,	PUNCT
ejpam-4592	175	4	nq	nq	PROPN
ejpam-4592	175	5	>	>	X
ejpam-4592	175	6	m	m	PROPN
ejpam-4592	175	7	,	,	PUNCT
ejpam-4592	175	8	we	we	PRON
ejpam-4592	175	9	get	get	VERB
ejpam-4592	175	10	σ(cnp	σ(cnp	NOUN
ejpam-4592	175	11	,	,	PUNCT
ejpam-4592	175	12	cnq	cnq	NOUN
ejpam-4592	175	13	)	)	PUNCT
ejpam-4592	175	14	<	<	X
ejpam-4592	175	15	ε	ε	PROPN
ejpam-4592	175	16	,	,	PUNCT
ejpam-4592	175	17	by	by	ADP
ejpam-4592	175	18	the	the	DET
ejpam-4592	175	19	definition	definition	NOUN
ejpam-4592	175	20	of	of	ADP
ejpam-4592	175	21	a.	a.	NOUN
ejpam-4592	175	22	thus	thus	ADV
ejpam-4592	175	23	,	,	PUNCT
ejpam-4592	175	24	σ(cnp	σ(cnp	NOUN
ejpam-4592	175	25	,	,	PUNCT
ejpam-4592	175	26	cnq	cnq	NOUN
ejpam-4592	175	27	)	)	PUNCT
ejpam-4592	175	28	<	<	X
ejpam-4592	175	29	ε	ε	PROPN
ejpam-4592	175	30	for	for	ADP
ejpam-4592	175	31	every	every	DET
ejpam-4592	175	32	np	np	NOUN
ejpam-4592	175	33	,	,	PUNCT
ejpam-4592	175	34	nq	nq	PROPN
ejpam-4592	175	35	>	>	X
ejpam-4592	175	36	m.	m.	NOUN
ejpam-4592	175	37	thus	thus	ADV
ejpam-4592	175	38	,	,	PUNCT
ejpam-4592	175	39	{	{	PUNCT
ejpam-4592	175	40	cnk	cnk	NOUN
ejpam-4592	175	41	}	}	PUNCT
ejpam-4592	175	42	∈	∈	PROPN
ejpam-4592	175	43	c(σ	c(σ	PROPN
ejpam-4592	175	44	)	)	PUNCT
ejpam-4592	175	45	.	.	PUNCT
ejpam-4592	176	1	therefore	therefore	ADV
ejpam-4592	176	2	,	,	PUNCT
ejpam-4592	176	3	{	{	PUNCT
ejpam-4592	176	4	cn	cn	VERB
ejpam-4592	176	5	}	}	PUNCT
ejpam-4592	176	6	has	have	VERB
ejpam-4592	176	7	a	a	DET
ejpam-4592	176	8	subsequence	subsequence	NOUN
ejpam-4592	176	9	in	in	ADP
ejpam-4592	176	10	c(σ	c(σ	PROPN
ejpam-4592	176	11	)	)	PUNCT
ejpam-4592	176	12	.	.	PUNCT
ejpam-4592	177	1	theorem	theorem	VERB
ejpam-4592	177	2	20	20	NUM
ejpam-4592	177	3	and	and	CCONJ
ejpam-4592	177	4	example	example	NOUN
ejpam-4592	177	5	21	21	NUM
ejpam-4592	177	6	are	be	AUX
ejpam-4592	177	7	describe	describe	NOUN
ejpam-4592	177	8	the	the	DET
ejpam-4592	177	9	below	below	ADJ
ejpam-4592	177	10	diagram	diagram	NOUN
ejpam-4592	177	11	.	.	PUNCT
ejpam-4592	178	1	v.	v.	ADP
ejpam-4592	178	2	subramanian	subramanian	PROPN
ejpam-4592	178	3	et	et	PROPN
ejpam-4592	178	4	al	al	PROPN
ejpam-4592	178	5	.	.	PUNCT
ejpam-4592	178	6	/	/	SYM
ejpam-4592	178	7	eur	eur	PROPN
ejpam-4592	178	8	.	.	PUNCT
ejpam-4592	179	1	j.	j.	PROPN
ejpam-4592	179	2	pure	pure	PROPN
ejpam-4592	179	3	appl	appl	PROPN
ejpam-4592	179	4	.	.	PROPN
ejpam-4592	179	5	math	math	PROPN
ejpam-4592	179	6	,	,	PUNCT
ejpam-4592	179	7	15	15	NUM
ejpam-4592	179	8	(	(	PUNCT
ejpam-4592	179	9	4	4	NUM
ejpam-4592	179	10	)	)	PUNCT
ejpam-4592	179	11	(	(	PUNCT
ejpam-4592	179	12	2022	2022	NUM
ejpam-4592	179	13	)	)	PUNCT
ejpam-4592	179	14	,	,	PUNCT
ejpam-4592	179	15	1869	1869	NUM
ejpam-4592	179	16	-	-	SYM
ejpam-4592	179	17	1886	1886	NUM
ejpam-4592	179	18	1875	1875	NUM
ejpam-4592	179	19	cofinally	cofinally	ADV
ejpam-4592	179	20	−	−	ADP
ejpam-4592	179	21	cauchy	cauchy	ADJ
ejpam-4592	179	22	sequence	sequence	NOUN
ejpam-4592	179	23	pseudo−	pseudo−	PROPN
ejpam-4592	179	24	cauchy	cauchy	PROPN
ejpam-4592	179	25	sequence	sequence	NOUN
ejpam-4592	179	26	/	/	SYM
ejpam-4592	179	27	theorem	theorem	ADJ
ejpam-4592	179	28	20	20	NUM
ejpam-4592	179	29	provides	provide	VERB
ejpam-4592	179	30	easier	easy	ADJ
ejpam-4592	179	31	way	way	NOUN
ejpam-4592	179	32	to	to	PART
ejpam-4592	179	33	explore	explore	VERB
ejpam-4592	179	34	,	,	PUNCT
ejpam-4592	179	35	in	in	ADP
ejpam-4592	179	36	a	a	DET
ejpam-4592	179	37	generalized	generalized	ADJ
ejpam-4592	179	38	metric	metric	ADJ
ejpam-4592	179	39	space	space	NOUN
ejpam-4592	179	40	,	,	PUNCT
ejpam-4592	179	41	a	a	DET
ejpam-4592	179	42	given	give	VERB
ejpam-4592	179	43	sequence	sequence	NOUN
ejpam-4592	179	44	is	be	AUX
ejpam-4592	179	45	pseudo	pseudo	NOUN
ejpam-4592	179	46	-	-	NOUN
ejpam-4592	179	47	cauchy	cauchy	ADJ
ejpam-4592	179	48	or	or	CCONJ
ejpam-4592	179	49	not	not	PART
ejpam-4592	179	50	.	.	PUNCT
ejpam-4592	180	1	theorem	theorem	ADJ
ejpam-4592	180	2	20	20	NUM
ejpam-4592	180	3	.	.	PUNCT
ejpam-4592	181	1	let	let	VERB
ejpam-4592	181	2	(	(	PUNCT
ejpam-4592	181	3	x	x	NOUN
ejpam-4592	181	4	,	,	PUNCT
ejpam-4592	181	5	ω	ω	NUM
ejpam-4592	181	6	)	)	PUNCT
ejpam-4592	181	7	be	be	AUX
ejpam-4592	181	8	a	a	DET
ejpam-4592	181	9	gms	gms	NOUN
ejpam-4592	181	10	.	.	PUNCT
ejpam-4592	182	1	then	then	ADV
ejpam-4592	182	2	every	every	DET
ejpam-4592	182	3	cofinally	cofinally	ADV
ejpam-4592	182	4	-	-	PUNCT
ejpam-4592	182	5	cauchy	cauchy	ADJ
ejpam-4592	182	6	sequence	sequence	NOUN
ejpam-4592	182	7	is	be	AUX
ejpam-4592	182	8	a	a	DET
ejpam-4592	182	9	pseudocauchy	pseudocauchy	ADJ
ejpam-4592	182	10	sequence	sequence	NOUN
ejpam-4592	182	11	with	with	ADP
ejpam-4592	182	12	the	the	DET
ejpam-4592	182	13	same	same	ADJ
ejpam-4592	182	14	metric	metric	NOUN
ejpam-4592	182	15	.	.	PUNCT
ejpam-4592	183	1	proof	proof	NOUN
ejpam-4592	183	2	.	.	PUNCT
ejpam-4592	184	1	let	let	VERB
ejpam-4592	184	2	{	{	PUNCT
ejpam-4592	184	3	cn	cn	ADJ
ejpam-4592	184	4	}	}	PUNCT
ejpam-4592	184	5	∈	∈	PROPN
ejpam-4592	184	6	c(σ	c(σ	PROPN
ejpam-4592	184	7	)	)	PUNCT
ejpam-4592	184	8	and	and	CCONJ
ejpam-4592	184	9	ε	ε	PROPN
ejpam-4592	184	10	>	>	X
ejpam-4592	184	11	0	0	PUNCT
ejpam-4592	184	12	be	be	AUX
ejpam-4592	184	13	given	give	VERB
ejpam-4592	184	14	.	.	PUNCT
ejpam-4592	185	1	then	then	ADV
ejpam-4592	185	2	there	there	PRON
ejpam-4592	185	3	exists	exist	VERB
ejpam-4592	185	4	an	an	DET
ejpam-4592	185	5	infinite	infinite	NOUN
ejpam-4592	185	6	subset	subset	NOUN
ejpam-4592	185	7	nε	nε	NOUN
ejpam-4592	185	8	of	of	ADP
ejpam-4592	185	9	n	n	PRON
ejpam-4592	185	10	such	such	ADJ
ejpam-4592	185	11	that	that	SCONJ
ejpam-4592	185	12	σ(ck	σ(ck	PROPN
ejpam-4592	185	13	,	,	PUNCT
ejpam-4592	185	14	cl	cl	NOUN
ejpam-4592	185	15	)	)	PUNCT
ejpam-4592	185	16	<	<	X
ejpam-4592	185	17	ε	ε	PROPN
ejpam-4592	185	18	for	for	ADP
ejpam-4592	185	19	every	every	DET
ejpam-4592	185	20	k	k	PROPN
ejpam-4592	185	21	,	,	PUNCT
ejpam-4592	185	22	l	l	PROPN
ejpam-4592	185	23	∈	∈	PROPN
ejpam-4592	185	24	nε	nε	VERB
ejpam-4592	185	25	.	.	PUNCT
ejpam-4592	186	1	let	let	VERB
ejpam-4592	186	2	n	n	PRON
ejpam-4592	186	3	∈	∈	PROPN
ejpam-4592	186	4	n.	n.	NOUN
ejpam-4592	186	5	since	since	SCONJ
ejpam-4592	186	6	nε	nε	PROPN
ejpam-4592	186	7	is	be	AUX
ejpam-4592	186	8	an	an	DET
ejpam-4592	186	9	infinite	infinite	ADJ
ejpam-4592	186	10	subset	subset	NOUN
ejpam-4592	186	11	of	of	ADP
ejpam-4592	186	12	n	n	CCONJ
ejpam-4592	186	13	,	,	PUNCT
ejpam-4592	186	14	there	there	PRON
ejpam-4592	186	15	exist	exist	VERB
ejpam-4592	186	16	p	p	PRON
ejpam-4592	186	17	,	,	PUNCT
ejpam-4592	186	18	q	q	SYM
ejpam-4592	186	19	∈	∈	NOUN
ejpam-4592	186	20	nε	nε	VERB
ejpam-4592	186	21	with	with	ADP
ejpam-4592	186	22	p	p	PROPN
ejpam-4592	186	23	̸=	̸=	PROPN
ejpam-4592	186	24	q	q	PROPN
ejpam-4592	186	25	and	and	CCONJ
ejpam-4592	186	26	p	p	X
ejpam-4592	186	27	,	,	PUNCT
ejpam-4592	186	28	q	q	PROPN
ejpam-4592	186	29	>	>	X
ejpam-4592	186	30	n.	n.	PROPN
ejpam-4592	186	31	also	also	ADV
ejpam-4592	186	32	,	,	PUNCT
ejpam-4592	186	33	σ(cp	σ(cp	PROPN
ejpam-4592	186	34	,	,	PUNCT
ejpam-4592	186	35	cq	cq	NOUN
ejpam-4592	186	36	)	)	PUNCT
ejpam-4592	186	37	<	<	X
ejpam-4592	187	1	ε	ε	PROPN
ejpam-4592	187	2	.	.	PUNCT
ejpam-4592	188	1	hence	hence	ADV
ejpam-4592	188	2	{	{	PUNCT
ejpam-4592	188	3	cn	cn	PROPN
ejpam-4592	188	4	}	}	PUNCT
ejpam-4592	188	5	∈	∈	PROPN
ejpam-4592	188	6	p(σ	p(σ	NOUN
ejpam-4592	188	7	)	)	PUNCT
ejpam-4592	188	8	.	.	PUNCT
ejpam-4592	189	1	example	example	NOUN
ejpam-4592	190	1	21	21	NUM
ejpam-4592	190	2	shows	show	VERB
ejpam-4592	190	3	that	that	SCONJ
ejpam-4592	190	4	the	the	DET
ejpam-4592	190	5	reverse	reverse	ADJ
ejpam-4592	190	6	part	part	NOUN
ejpam-4592	190	7	of	of	ADP
ejpam-4592	190	8	theorem	theorem	ADJ
ejpam-4592	190	9	20	20	NUM
ejpam-4592	190	10	is	be	AUX
ejpam-4592	190	11	need	need	AUX
ejpam-4592	190	12	not	not	PART
ejpam-4592	190	13	be	be	AUX
ejpam-4592	190	14	true	true	ADJ
ejpam-4592	190	15	.	.	PUNCT
ejpam-4592	190	16	example	example	NOUN
ejpam-4592	191	1	21	21	NUM
ejpam-4592	191	2	.	.	PUNCT
ejpam-4592	191	3	consider	consider	VERB
ejpam-4592	191	4	the	the	DET
ejpam-4592	191	5	generalized	generalized	ADJ
ejpam-4592	191	6	metric	metric	ADJ
ejpam-4592	191	7	space	space	NOUN
ejpam-4592	191	8	(	(	PUNCT
ejpam-4592	191	9	x	x	NOUN
ejpam-4592	191	10	,	,	PUNCT
ejpam-4592	191	11	ω1	ω1	PROPN
ejpam-4592	191	12	)	)	PUNCT
ejpam-4592	191	13	wherex	wherex	NOUN
ejpam-4592	191	14	=	=	SYM
ejpam-4592	191	15	r	r	PROPN
ejpam-4592	191	16	,	,	PUNCT
ejpam-4592	191	17	ω1	ω1	PROPN
ejpam-4592	191	18	=	=	SYM
ejpam-4592	191	19	{	{	PUNCT
ejpam-4592	191	20	σ1	σ1	PROPN
ejpam-4592	191	21	,	,	PUNCT
ejpam-4592	191	22	σ2	σ2	PROPN
ejpam-4592	191	23	,	,	PUNCT
ejpam-4592	191	24	σ3	σ3	PROPN
ejpam-4592	191	25	}	}	PUNCT
ejpam-4592	191	26	⊂	⊂	PROPN
ejpam-4592	191	27	ωx	ωx	VERB
ejpam-4592	191	28	such	such	ADJ
ejpam-4592	191	29	that	that	SCONJ
ejpam-4592	191	30	σ1(c	σ1(c	PRON
ejpam-4592	191	31	,	,	PUNCT
ejpam-4592	191	32	d	d	NOUN
ejpam-4592	191	33	)	)	PUNCT
ejpam-4592	191	34	=	=	PRON
ejpam-4592	191	35	{	{	PUNCT
ejpam-4592	191	36	0	0	NUM
ejpam-4592	191	37	if	if	SCONJ
ejpam-4592	191	38	c	c	NOUN
ejpam-4592	191	39	=	=	SYM
ejpam-4592	191	40	d	d	PROPN
ejpam-4592	191	41	,	,	PUNCT
ejpam-4592	191	42	1	1	NUM
ejpam-4592	191	43	if	if	SCONJ
ejpam-4592	191	44	c	c	PROPN
ejpam-4592	191	45	̸=	̸=	PROPN
ejpam-4592	191	46	d	d	PROPN
ejpam-4592	191	47	σ2(c	σ2(c	PROPN
ejpam-4592	191	48	,	,	PUNCT
ejpam-4592	191	49	d	d	NOUN
ejpam-4592	191	50	)	)	PUNCT
ejpam-4592	191	51	=	=	SYM
ejpam-4592	191	52	σe(c	σe(c	X
ejpam-4592	191	53	,	,	PUNCT
ejpam-4592	191	54	d	d	NOUN
ejpam-4592	191	55	)	)	PUNCT
ejpam-4592	191	56	where	where	SCONJ
ejpam-4592	191	57	σe(c	σe(c	X
ejpam-4592	191	58	,	,	PUNCT
ejpam-4592	191	59	d	d	X
ejpam-4592	191	60	)	)	PUNCT
ejpam-4592	191	61	=	=	SYM
ejpam-4592	191	62	|c	|c	VERB
ejpam-4592	191	63	−	−	PROPN
ejpam-4592	191	64	d|	d|	PROPN
ejpam-4592	191	65	and	and	CCONJ
ejpam-4592	191	66	then	then	ADV
ejpam-4592	191	67	σ3(c	σ3(c	NUM
ejpam-4592	191	68	,	,	PUNCT
ejpam-4592	191	69	d	d	NOUN
ejpam-4592	191	70	)	)	PUNCT
ejpam-4592	191	71	=	=	SYM
ejpam-4592	192	1	σ1(c	σ1(c	PROPN
ejpam-4592	192	2	,	,	PUNCT
ejpam-4592	192	3	d	d	NOUN
ejpam-4592	192	4	)	)	PUNCT
ejpam-4592	192	5	1+σ1(c	1+σ1(c	NOUN
ejpam-4592	192	6	,	,	PUNCT
ejpam-4592	192	7	d	d	NOUN
ejpam-4592	192	8	)	)	PUNCT
ejpam-4592	192	9	for	for	ADP
ejpam-4592	192	10	all	all	DET
ejpam-4592	192	11	c	c	NOUN
ejpam-4592	192	12	,	,	PUNCT
ejpam-4592	192	13	d	d	PROPN
ejpam-4592	192	14	∈	∈	PROPN
ejpam-4592	192	15	x.	x.	NOUN
ejpam-4592	192	16	define	define	VERB
ejpam-4592	192	17	a	a	DET
ejpam-4592	192	18	sequence	sequence	NOUN
ejpam-4592	192	19	{	{	PUNCT
ejpam-4592	192	20	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	192	21	=	=	SYM
ejpam-4592	192	22	{	{	PUNCT
ejpam-4592	192	23	1	1	NUM
ejpam-4592	192	24	,	,	PUNCT
ejpam-4592	192	25	1	1	NUM
ejpam-4592	192	26	,	,	PUNCT
ejpam-4592	192	27	1	1	NUM
ejpam-4592	192	28	,	,	PUNCT
ejpam-4592	192	29	2	2	NUM
ejpam-4592	192	30	,	,	PUNCT
ejpam-4592	192	31	2	2	NUM
ejpam-4592	192	32	,	,	PUNCT
ejpam-4592	192	33	2	2	NUM
ejpam-4592	192	34	,	,	PUNCT
ejpam-4592	192	35	2	2	NUM
ejpam-4592	192	36	,	,	PUNCT
ejpam-4592	192	37	3	3	NUM
ejpam-4592	192	38	,	,	PUNCT
ejpam-4592	192	39	3	3	NUM
ejpam-4592	192	40	,	,	PUNCT
ejpam-4592	192	41	3	3	NUM
ejpam-4592	192	42	,	,	PUNCT
ejpam-4592	192	43	3	3	NUM
ejpam-4592	192	44	,	,	PUNCT
ejpam-4592	192	45	3	3	NUM
ejpam-4592	192	46	,	,	PUNCT
ejpam-4592	192	47	....	....	PUNCT
ejpam-4592	192	48	}	}	PUNCT
ejpam-4592	192	49	.	.	PUNCT
ejpam-4592	193	1	then	then	ADV
ejpam-4592	193	2	{	{	PUNCT
ejpam-4592	193	3	cn	cn	ADJ
ejpam-4592	193	4	}	}	PUNCT
ejpam-4592	193	5	∈	∈	PROPN
ejpam-4592	193	6	p(σ2	p(σ2	VERB
ejpam-4592	193	7	)	)	PUNCT
ejpam-4592	193	8	.	.	PUNCT
ejpam-4592	194	1	but	but	CCONJ
ejpam-4592	194	2	there	there	PRON
ejpam-4592	194	3	is	be	VERB
ejpam-4592	194	4	no	no	DET
ejpam-4592	194	5	infinite	infinite	NOUN
ejpam-4592	194	6	subset	subset	NOUN
ejpam-4592	194	7	nε	nε	NOUN
ejpam-4592	194	8	of	of	ADP
ejpam-4592	194	9	n	n	PRON
ejpam-4592	194	10	such	such	ADJ
ejpam-4592	194	11	that	that	SCONJ
ejpam-4592	194	12	σ(ck	σ(ck	PROPN
ejpam-4592	194	13	,	,	PUNCT
ejpam-4592	194	14	cl	cl	NOUN
ejpam-4592	194	15	)	)	PUNCT
ejpam-4592	194	16	<	<	X
ejpam-4592	194	17	ε	ε	PROPN
ejpam-4592	194	18	for	for	ADP
ejpam-4592	194	19	every	every	DET
ejpam-4592	194	20	ε	ε	PROPN
ejpam-4592	194	21	>	>	X
ejpam-4592	194	22	0	0	PROPN
ejpam-4592	195	1	and	and	CCONJ
ejpam-4592	195	2	k	k	NOUN
ejpam-4592	195	3	,	,	PUNCT
ejpam-4592	195	4	l	l	PROPN
ejpam-4592	195	5	∈	∈	PROPN
ejpam-4592	195	6	nε	nε	VERB
ejpam-4592	195	7	.	.	PUNCT
ejpam-4592	196	1	thus	thus	ADV
ejpam-4592	196	2	,	,	PUNCT
ejpam-4592	196	3	{	{	PUNCT
ejpam-4592	196	4	cn	cn	ADJ
ejpam-4592	196	5	}	}	PUNCT
ejpam-4592	196	6	/∈	/∈	PUNCT
ejpam-4592	196	7	c(σ2	c(σ2	NUM
ejpam-4592	196	8	)	)	PUNCT
ejpam-4592	196	9	.	.	PUNCT
ejpam-4592	197	1	next	next	ADV
ejpam-4592	197	2	,	,	PUNCT
ejpam-4592	197	3	in	in	ADP
ejpam-4592	197	4	the	the	DET
ejpam-4592	197	5	rest	rest	NOUN
ejpam-4592	197	6	of	of	ADP
ejpam-4592	197	7	this	this	DET
ejpam-4592	197	8	section	section	NOUN
ejpam-4592	197	9	,	,	PUNCT
ejpam-4592	197	10	in	in	ADP
ejpam-4592	197	11	a	a	DET
ejpam-4592	197	12	generalized	generalized	ADJ
ejpam-4592	197	13	metric	metric	ADJ
ejpam-4592	197	14	space	space	NOUN
ejpam-4592	197	15	,	,	PUNCT
ejpam-4592	197	16	we	we	PRON
ejpam-4592	197	17	define	define	VERB
ejpam-4592	197	18	a	a	DET
ejpam-4592	197	19	rearrangement	rearrangement	NOUN
ejpam-4592	197	20	of	of	ADP
ejpam-4592	197	21	a	a	DET
ejpam-4592	197	22	sequence	sequence	NOUN
ejpam-4592	197	23	and	and	CCONJ
ejpam-4592	197	24	give	give	VERB
ejpam-4592	197	25	some	some	DET
ejpam-4592	197	26	interesting	interesting	ADJ
ejpam-4592	197	27	results	result	NOUN
ejpam-4592	197	28	based	base	VERB
ejpam-4592	197	29	on	on	ADP
ejpam-4592	197	30	this	this	DET
ejpam-4592	197	31	definition	definition	NOUN
ejpam-4592	197	32	.	.	PUNCT
ejpam-4592	198	1	definition	definition	NOUN
ejpam-4592	198	2	22	22	NUM
ejpam-4592	198	3	.	.	PUNCT
ejpam-4592	199	1	let	let	VERB
ejpam-4592	199	2	(	(	PUNCT
ejpam-4592	199	3	x	x	NOUN
ejpam-4592	199	4	,	,	PUNCT
ejpam-4592	199	5	ω	ω	NUM
ejpam-4592	199	6	)	)	PUNCT
ejpam-4592	199	7	be	be	AUX
ejpam-4592	199	8	a	a	DET
ejpam-4592	199	9	generalized	generalized	ADJ
ejpam-4592	199	10	metric	metric	ADJ
ejpam-4592	199	11	space	space	NOUN
ejpam-4592	199	12	.	.	PUNCT
ejpam-4592	200	1	then	then	ADV
ejpam-4592	200	2	the	the	DET
ejpam-4592	200	3	sequence	sequence	NOUN
ejpam-4592	200	4	{	{	PUNCT
ejpam-4592	200	5	dn	dn	VERB
ejpam-4592	200	6	}	}	PUNCT
ejpam-4592	200	7	is	be	AUX
ejpam-4592	200	8	a	a	DET
ejpam-4592	200	9	rearrangement	rearrangement	NOUN
ejpam-4592	200	10	of	of	ADP
ejpam-4592	200	11	a	a	DET
ejpam-4592	200	12	sequence	sequence	NOUN
ejpam-4592	200	13	{	{	PUNCT
ejpam-4592	200	14	cn	cn	NOUN
ejpam-4592	200	15	}	}	PUNCT
ejpam-4592	200	16	if	if	SCONJ
ejpam-4592	200	17	there	there	PRON
ejpam-4592	200	18	is	be	VERB
ejpam-4592	200	19	a	a	DET
ejpam-4592	200	20	1	1	NUM
ejpam-4592	200	21	-	-	SYM
ejpam-4592	200	22	1	1	NUM
ejpam-4592	200	23	correspondence	correspondence	NOUN
ejpam-4592	200	24	h	h	NOUN
ejpam-4592	201	1	:	:	PUNCT
ejpam-4592	201	2	n	n	X
ejpam-4592	201	3	→	→	PUNCT
ejpam-4592	201	4	n	n	X
ejpam-4592	201	5	such	such	ADJ
ejpam-4592	201	6	that	that	SCONJ
ejpam-4592	201	7	n	n	NUM
ejpam-4592	201	8	∈	∈	PROPN
ejpam-4592	201	9	n	n	CCONJ
ejpam-4592	201	10	,	,	PUNCT
ejpam-4592	201	11	dn	dn	NOUN
ejpam-4592	201	12	=	=	NOUN
ejpam-4592	201	13	ch(n	ch(n	X
ejpam-4592	201	14	)	)	PUNCT
ejpam-4592	201	15	.	.	PUNCT
ejpam-4592	202	1	theorem	theorem	NOUN
ejpam-4592	202	2	23	23	NUM
ejpam-4592	202	3	.	.	PUNCT
ejpam-4592	203	1	let	let	VERB
ejpam-4592	203	2	(	(	PUNCT
ejpam-4592	203	3	x	x	NOUN
ejpam-4592	203	4	,	,	PUNCT
ejpam-4592	203	5	ω	ω	NUM
ejpam-4592	203	6	)	)	PUNCT
ejpam-4592	203	7	be	be	AUX
ejpam-4592	203	8	a	a	DET
ejpam-4592	203	9	gms	gms	NOUN
ejpam-4592	203	10	.	.	PUNCT
ejpam-4592	204	1	if	if	SCONJ
ejpam-4592	204	2	{	{	PUNCT
ejpam-4592	204	3	cn	cn	ADJ
ejpam-4592	204	4	}	}	PUNCT
ejpam-4592	204	5	∈	∈	PROPN
ejpam-4592	204	6	c(σ	c(σ	PROPN
ejpam-4592	204	7	)	)	PUNCT
ejpam-4592	204	8	,	,	PUNCT
ejpam-4592	204	9	then	then	ADV
ejpam-4592	204	10	every	every	DET
ejpam-4592	204	11	rearrangement	rearrangement	ADJ
ejpam-4592	204	12	sequence	sequence	NOUN
ejpam-4592	204	13	of	of	ADP
ejpam-4592	204	14	{	{	PUNCT
ejpam-4592	204	15	cn	cn	PROPN
ejpam-4592	204	16	}	}	PUNCT
ejpam-4592	204	17	is	be	AUX
ejpam-4592	204	18	cofinally	cofinally	ADV
ejpam-4592	204	19	-	-	PUNCT
ejpam-4592	204	20	cauchy	cauchy	NOUN
ejpam-4592	204	21	with	with	ADP
ejpam-4592	204	22	the	the	DET
ejpam-4592	204	23	same	same	ADJ
ejpam-4592	204	24	metric	metric	NOUN
ejpam-4592	204	25	.	.	PUNCT
ejpam-4592	205	1	proof	proof	NOUN
ejpam-4592	205	2	.	.	PUNCT
ejpam-4592	206	1	suppose	suppose	VERB
ejpam-4592	206	2	{	{	PUNCT
ejpam-4592	206	3	cn	cn	ADJ
ejpam-4592	206	4	}	}	PUNCT
ejpam-4592	206	5	∈	∈	PROPN
ejpam-4592	206	6	c(σ	c(σ	PROPN
ejpam-4592	206	7	)	)	PUNCT
ejpam-4592	206	8	.	.	PUNCT
ejpam-4592	207	1	let	let	VERB
ejpam-4592	207	2	ε	ε	PROPN
ejpam-4592	207	3	>	>	X
ejpam-4592	207	4	0	0	PUNCT
ejpam-4592	207	5	be	be	AUX
ejpam-4592	207	6	given	give	VERB
ejpam-4592	207	7	.	.	PUNCT
ejpam-4592	208	1	then	then	ADV
ejpam-4592	208	2	there	there	PRON
ejpam-4592	208	3	exists	exist	VERB
ejpam-4592	208	4	an	an	DET
ejpam-4592	208	5	infinite	infinite	NOUN
ejpam-4592	208	6	subset	subset	NOUN
ejpam-4592	208	7	nε	nε	NOUN
ejpam-4592	208	8	of	of	ADP
ejpam-4592	208	9	n	n	PRON
ejpam-4592	208	10	such	such	ADJ
ejpam-4592	208	11	that	that	SCONJ
ejpam-4592	208	12	σ(ck	σ(ck	PROPN
ejpam-4592	208	13	,	,	PUNCT
ejpam-4592	208	14	cl	cl	NOUN
ejpam-4592	208	15	)	)	PUNCT
ejpam-4592	208	16	<	<	X
ejpam-4592	208	17	ε	ε	PROPN
ejpam-4592	208	18	for	for	ADP
ejpam-4592	208	19	every	every	DET
ejpam-4592	208	20	k	k	PROPN
ejpam-4592	208	21	,	,	PUNCT
ejpam-4592	208	22	l	l	PROPN
ejpam-4592	208	23	∈	∈	PROPN
ejpam-4592	208	24	nε	nε	VERB
ejpam-4592	208	25	.	.	PUNCT
ejpam-4592	209	1	let	let	AUX
ejpam-4592	209	2	{	{	PUNCT
ejpam-4592	209	3	dn	dn	AUX
ejpam-4592	209	4	}	}	PUNCT
ejpam-4592	209	5	be	be	AUX
ejpam-4592	209	6	a	a	DET
ejpam-4592	209	7	rearrangement	rearrangement	NOUN
ejpam-4592	209	8	of	of	ADP
ejpam-4592	209	9	a	a	DET
ejpam-4592	209	10	sequence	sequence	NOUN
ejpam-4592	209	11	{	{	PUNCT
ejpam-4592	209	12	cn	cn	NOUN
ejpam-4592	209	13	}	}	PUNCT
ejpam-4592	209	14	.	.	PUNCT
ejpam-4592	210	1	then	then	ADV
ejpam-4592	210	2	there	there	PRON
ejpam-4592	210	3	is	be	VERB
ejpam-4592	210	4	a	a	DET
ejpam-4592	210	5	1	1	NUM
ejpam-4592	210	6	-	-	SYM
ejpam-4592	210	7	1	1	NUM
ejpam-4592	210	8	correspondence	correspondence	NOUN
ejpam-4592	210	9	h	h	NOUN
ejpam-4592	210	10	:	:	PUNCT
ejpam-4592	210	11	n	n	X
ejpam-4592	210	12	→	→	PUNCT
ejpam-4592	210	13	n	n	X
ejpam-4592	210	14	such	such	ADJ
ejpam-4592	210	15	that	that	SCONJ
ejpam-4592	210	16	n	n	NUM
ejpam-4592	210	17	∈	∈	PROPN
ejpam-4592	210	18	n	n	CCONJ
ejpam-4592	210	19	,	,	PUNCT
ejpam-4592	210	20	dn	dn	NOUN
ejpam-4592	210	21	=	=	NOUN
ejpam-4592	210	22	ch(n	ch(n	X
ejpam-4592	210	23	)	)	PUNCT
ejpam-4592	210	24	.	.	PUNCT
ejpam-4592	211	1	take	take	VERB
ejpam-4592	211	2	nε0	nε0	NOUN
ejpam-4592	211	3	=	=	PUNCT
ejpam-4592	211	4	{	{	PUNCT
ejpam-4592	211	5	h−1(n)|n	h−1(n)|n	PROPN
ejpam-4592	211	6	∈	∈	PROPN
ejpam-4592	211	7	nε	nε	NOUN
ejpam-4592	211	8	}	}	PUNCT
ejpam-4592	211	9	.	.	PUNCT
ejpam-4592	212	1	then	then	ADV
ejpam-4592	212	2	nε0	nε0	NOUN
ejpam-4592	212	3	is	be	AUX
ejpam-4592	212	4	an	an	DET
ejpam-4592	212	5	infinite	infinite	ADJ
ejpam-4592	212	6	subset	subset	NOUN
ejpam-4592	212	7	of	of	ADP
ejpam-4592	212	8	n.	n.	PROPN
ejpam-4592	212	9	let	let	VERB
ejpam-4592	212	10	p	p	PRON
ejpam-4592	212	11	,	,	PUNCT
ejpam-4592	212	12	q	q	PROPN
ejpam-4592	212	13	∈	∈	PROPN
ejpam-4592	212	14	nε0	nε0	NOUN
ejpam-4592	212	15	.	.	PUNCT
ejpam-4592	213	1	then	then	ADV
ejpam-4592	213	2	p	p	X
ejpam-4592	213	3	=	=	X
ejpam-4592	213	4	h−1(s	h−1(s	PROPN
ejpam-4592	213	5	)	)	PUNCT
ejpam-4592	213	6	and	and	CCONJ
ejpam-4592	213	7	q	q	NOUN
ejpam-4592	213	8	=	=	SYM
ejpam-4592	213	9	h−1(t	h−1(t	PROPN
ejpam-4592	213	10	)	)	PUNCT
ejpam-4592	213	11	where	where	SCONJ
ejpam-4592	213	12	s	s	X
ejpam-4592	213	13	,	,	PUNCT
ejpam-4592	213	14	t	t	PROPN
ejpam-4592	213	15	∈	∈	PROPN
ejpam-4592	213	16	nε	nε	VERB
ejpam-4592	213	17	.	.	PUNCT
ejpam-4592	214	1	now	now	ADV
ejpam-4592	214	2	σ(dp	σ(dp	NUM
ejpam-4592	214	3	,	,	PUNCT
ejpam-4592	214	4	dq	dq	NOUN
ejpam-4592	214	5	)	)	PUNCT
ejpam-4592	214	6	=	=	SYM
ejpam-4592	215	1	σ(dh−1(s	σ(dh−1(s	PROPN
ejpam-4592	215	2	)	)	PUNCT
ejpam-4592	215	3	,	,	PUNCT
ejpam-4592	215	4	dh−1(t	dh−1(t	PROPN
ejpam-4592	215	5	)	)	PUNCT
ejpam-4592	215	6	)	)	PUNCT
ejpam-4592	216	1	=	=	SYM
ejpam-4592	216	2	σ(ch(h−1(s	σ(ch(h−1(s	PROPN
ejpam-4592	216	3	)	)	PUNCT
ejpam-4592	216	4	)	)	PUNCT
ejpam-4592	216	5	,	,	PUNCT
ejpam-4592	216	6	ch(h−1(t	ch(h−1(t	PROPN
ejpam-4592	216	7	)	)	PUNCT
ejpam-4592	216	8	)	)	PUNCT
ejpam-4592	216	9	)	)	PUNCT
ejpam-4592	217	1	which	which	PRON
ejpam-4592	217	2	implies	imply	VERB
ejpam-4592	217	3	σ(dp	σ(dp	NUM
ejpam-4592	217	4	,	,	PUNCT
ejpam-4592	217	5	dq	dq	NOUN
ejpam-4592	217	6	)	)	PUNCT
ejpam-4592	217	7	=	=	SYM
ejpam-4592	217	8	σ(cs	σ(cs	PROPN
ejpam-4592	217	9	,	,	PUNCT
ejpam-4592	217	10	ct	ct	PROPN
ejpam-4592	217	11	)	)	PUNCT
ejpam-4592	217	12	,	,	PUNCT
ejpam-4592	217	13	since	since	SCONJ
ejpam-4592	217	14	h	h	NOUN
ejpam-4592	217	15	is	be	AUX
ejpam-4592	217	16	bijective	bijective	ADJ
ejpam-4592	217	17	.	.	PUNCT
ejpam-4592	218	1	thus	thus	ADV
ejpam-4592	218	2	,	,	PUNCT
ejpam-4592	218	3	σ(dp	σ(dp	NUM
ejpam-4592	218	4	,	,	PUNCT
ejpam-4592	218	5	dq	dq	NOUN
ejpam-4592	218	6	)	)	PUNCT
ejpam-4592	218	7	<	<	X
ejpam-4592	218	8	ε	ε	PROPN
ejpam-4592	218	9	,	,	PUNCT
ejpam-4592	218	10	since	since	SCONJ
ejpam-4592	218	11	s	s	PROPN
ejpam-4592	218	12	,	,	PUNCT
ejpam-4592	218	13	t	t	PROPN
ejpam-4592	218	14	∈	∈	PROPN
ejpam-4592	218	15	nε	nε	VERB
ejpam-4592	218	16	.	.	PUNCT
ejpam-4592	219	1	therefore	therefore	ADV
ejpam-4592	219	2	,	,	PUNCT
ejpam-4592	219	3	{	{	PUNCT
ejpam-4592	219	4	dn	dn	ADJ
ejpam-4592	219	5	}	}	PUNCT
ejpam-4592	219	6	∈	∈	PROPN
ejpam-4592	219	7	c(σ	c(σ	PROPN
ejpam-4592	219	8	)	)	PUNCT
ejpam-4592	219	9	.	.	PUNCT
ejpam-4592	220	1	since	since	SCONJ
ejpam-4592	220	2	{	{	PUNCT
ejpam-4592	220	3	dn	dn	VERB
ejpam-4592	220	4	}	}	PUNCT
ejpam-4592	220	5	is	be	AUX
ejpam-4592	220	6	an	an	DET
ejpam-4592	220	7	arbitrary	arbitrary	ADJ
ejpam-4592	220	8	rearrangement	rearrangement	ADJ
ejpam-4592	220	9	sequence	sequence	NOUN
ejpam-4592	220	10	of	of	ADP
ejpam-4592	220	11	{	{	PUNCT
ejpam-4592	220	12	cn	cn	PROPN
ejpam-4592	220	13	}	}	PUNCT
ejpam-4592	220	14	we	we	PRON
ejpam-4592	220	15	have	have	VERB
ejpam-4592	220	16	every	every	DET
ejpam-4592	220	17	rearrangement	rearrangement	ADJ
ejpam-4592	220	18	sequence	sequence	NOUN
ejpam-4592	220	19	of	of	ADP
ejpam-4592	220	20	{	{	PUNCT
ejpam-4592	220	21	cn	cn	PROPN
ejpam-4592	220	22	}	}	PUNCT
ejpam-4592	220	23	is	be	AUX
ejpam-4592	220	24	cofinally	cofinally	ADV
ejpam-4592	220	25	-	-	PUNCT
ejpam-4592	220	26	cauchy	cauchy	NOUN
ejpam-4592	220	27	with	with	ADP
ejpam-4592	220	28	the	the	DET
ejpam-4592	220	29	same	same	ADJ
ejpam-4592	220	30	metric	metric	NOUN
ejpam-4592	220	31	.	.	PUNCT
ejpam-4592	221	1	v.	v.	ADP
ejpam-4592	221	2	subramanian	subramanian	PROPN
ejpam-4592	221	3	et	et	PROPN
ejpam-4592	221	4	al	al	PROPN
ejpam-4592	221	5	.	.	PUNCT
ejpam-4592	221	6	/	/	SYM
ejpam-4592	221	7	eur	eur	PROPN
ejpam-4592	221	8	.	.	PUNCT
ejpam-4592	222	1	j.	j.	PROPN
ejpam-4592	222	2	pure	pure	PROPN
ejpam-4592	222	3	appl	appl	PROPN
ejpam-4592	222	4	.	.	PROPN
ejpam-4592	222	5	math	math	PROPN
ejpam-4592	222	6	,	,	PUNCT
ejpam-4592	222	7	15	15	NUM
ejpam-4592	222	8	(	(	PUNCT
ejpam-4592	222	9	4	4	NUM
ejpam-4592	222	10	)	)	PUNCT
ejpam-4592	222	11	(	(	PUNCT
ejpam-4592	222	12	2022	2022	NUM
ejpam-4592	222	13	)	)	PUNCT
ejpam-4592	222	14	,	,	PUNCT
ejpam-4592	222	15	1869	1869	NUM
ejpam-4592	222	16	-	-	SYM
ejpam-4592	222	17	1886	1886	NUM
ejpam-4592	222	18	1876	1876	NUM
ejpam-4592	222	19	theorem	theorem	NOUN
ejpam-4592	222	20	24	24	NUM
ejpam-4592	222	21	.	.	PUNCT
ejpam-4592	223	1	let	let	VERB
ejpam-4592	223	2	(	(	PUNCT
ejpam-4592	223	3	x	x	NOUN
ejpam-4592	223	4	,	,	PUNCT
ejpam-4592	223	5	ω	ω	NUM
ejpam-4592	223	6	)	)	PUNCT
ejpam-4592	223	7	be	be	AUX
ejpam-4592	223	8	a	a	DET
ejpam-4592	223	9	gms	gms	NOUN
ejpam-4592	223	10	.	.	PUNCT
ejpam-4592	224	1	if	if	SCONJ
ejpam-4592	224	2	the	the	DET
ejpam-4592	224	3	rearrangement	rearrangement	ADJ
ejpam-4592	224	4	sequence	sequence	NOUN
ejpam-4592	224	5	of	of	ADP
ejpam-4592	224	6	{	{	PUNCT
ejpam-4592	224	7	cn	cn	NOUN
ejpam-4592	224	8	}	}	PUNCT
ejpam-4592	224	9	is	be	AUX
ejpam-4592	224	10	cofinallycauchy	cofinallycauchy	NOUN
ejpam-4592	224	11	,	,	PUNCT
ejpam-4592	224	12	then	then	ADV
ejpam-4592	224	13	{	{	PUNCT
ejpam-4592	224	14	cn	cn	NOUN
ejpam-4592	224	15	}	}	PUNCT
ejpam-4592	224	16	is	be	AUX
ejpam-4592	224	17	cofinally	cofinally	ADV
ejpam-4592	224	18	-	-	PUNCT
ejpam-4592	224	19	cauchy	cauchy	NOUN
ejpam-4592	224	20	with	with	ADP
ejpam-4592	224	21	the	the	DET
ejpam-4592	224	22	same	same	ADJ
ejpam-4592	224	23	metric	metric	NOUN
ejpam-4592	224	24	.	.	PUNCT
ejpam-4592	225	1	proof	proof	NOUN
ejpam-4592	225	2	.	.	PUNCT
ejpam-4592	226	1	assume	assume	VERB
ejpam-4592	226	2	that	that	SCONJ
ejpam-4592	226	3	,	,	PUNCT
ejpam-4592	226	4	the	the	DET
ejpam-4592	226	5	rearrangement	rearrangement	ADJ
ejpam-4592	226	6	sequence	sequence	NOUN
ejpam-4592	226	7	{	{	PUNCT
ejpam-4592	226	8	dn	dn	NOUN
ejpam-4592	226	9	}	}	PUNCT
ejpam-4592	226	10	of	of	ADP
ejpam-4592	226	11	{	{	PUNCT
ejpam-4592	226	12	cn	cn	NOUN
ejpam-4592	226	13	}	}	PUNCT
ejpam-4592	226	14	is	be	AUX
ejpam-4592	226	15	in	in	ADP
ejpam-4592	226	16	c(σ	c(σ	PROPN
ejpam-4592	226	17	)	)	PUNCT
ejpam-4592	226	18	.	.	PUNCT
ejpam-4592	227	1	then	then	ADV
ejpam-4592	227	2	there	there	PRON
ejpam-4592	227	3	exists	exist	VERB
ejpam-4592	227	4	an	an	DET
ejpam-4592	227	5	infinite	infinite	NOUN
ejpam-4592	227	6	subset	subset	NOUN
ejpam-4592	227	7	nε	nε	NOUN
ejpam-4592	227	8	of	of	ADP
ejpam-4592	227	9	n	n	PRON
ejpam-4592	227	10	such	such	ADJ
ejpam-4592	227	11	that	that	SCONJ
ejpam-4592	227	12	σ(dl	σ(dl	NOUN
ejpam-4592	227	13	,	,	PUNCT
ejpam-4592	227	14	dm	dm	NOUN
ejpam-4592	227	15	)	)	PUNCT
ejpam-4592	227	16	<	<	X
ejpam-4592	227	17	ε	ε	PROPN
ejpam-4592	227	18	for	for	ADP
ejpam-4592	227	19	every	every	DET
ejpam-4592	227	20	l	l	NOUN
ejpam-4592	227	21	,	,	PUNCT
ejpam-4592	227	22	m	m	VERB
ejpam-4592	227	23	∈	∈	NOUN
ejpam-4592	227	24	nε	nε	NOUN
ejpam-4592	227	25	.	.	PUNCT
ejpam-4592	228	1	by	by	ADP
ejpam-4592	228	2	assumption	assumption	NOUN
ejpam-4592	228	3	,	,	PUNCT
ejpam-4592	228	4	there	there	PRON
ejpam-4592	228	5	is	be	VERB
ejpam-4592	228	6	a	a	DET
ejpam-4592	228	7	1	1	NUM
ejpam-4592	228	8	-	-	SYM
ejpam-4592	228	9	1	1	NUM
ejpam-4592	228	10	correspondence	correspondence	NOUN
ejpam-4592	228	11	h	h	NOUN
ejpam-4592	228	12	:	:	PUNCT
ejpam-4592	228	13	n	n	X
ejpam-4592	228	14	→	→	PUNCT
ejpam-4592	228	15	n	n	X
ejpam-4592	228	16	such	such	ADJ
ejpam-4592	228	17	that	that	SCONJ
ejpam-4592	228	18	n	n	NUM
ejpam-4592	228	19	∈	∈	PROPN
ejpam-4592	228	20	n	n	CCONJ
ejpam-4592	228	21	,	,	PUNCT
ejpam-4592	228	22	dn	dn	NOUN
ejpam-4592	228	23	=	=	NOUN
ejpam-4592	228	24	ch(n	ch(n	X
ejpam-4592	228	25	)	)	PUNCT
ejpam-4592	228	26	.	.	PUNCT
ejpam-4592	229	1	take	take	VERB
ejpam-4592	229	2	nε1	nε1	NOUN
ejpam-4592	229	3	=	=	PUNCT
ejpam-4592	229	4	{	{	PUNCT
ejpam-4592	229	5	h(n)|n	h(n)|n	PROPN
ejpam-4592	229	6	∈	∈	PROPN
ejpam-4592	229	7	nε	nε	NOUN
ejpam-4592	229	8	}	}	PUNCT
ejpam-4592	229	9	.	.	PUNCT
ejpam-4592	230	1	then	then	ADV
ejpam-4592	230	2	nε1	nε1	PROPN
ejpam-4592	230	3	is	be	AUX
ejpam-4592	230	4	an	an	DET
ejpam-4592	230	5	infinite	infinite	ADJ
ejpam-4592	230	6	subset	subset	NOUN
ejpam-4592	230	7	of	of	ADP
ejpam-4592	230	8	n.	n.	PROPN
ejpam-4592	230	9	let	let	VERB
ejpam-4592	230	10	n	n	PRON
ejpam-4592	230	11	,	,	PUNCT
ejpam-4592	230	12	m	m	PROPN
ejpam-4592	230	13	∈	∈	ADJ
ejpam-4592	230	14	nε1	nε1	NOUN
ejpam-4592	230	15	.	.	PUNCT
ejpam-4592	231	1	then	then	ADV
ejpam-4592	231	2	n	n	PROPN
ejpam-4592	231	3	=	=	SYM
ejpam-4592	231	4	h(i	h(i	PROPN
ejpam-4592	231	5	)	)	PUNCT
ejpam-4592	231	6	and	and	CCONJ
ejpam-4592	231	7	m	m	NOUN
ejpam-4592	231	8	=	=	PUNCT
ejpam-4592	231	9	h(j	h(j	PROPN
ejpam-4592	231	10	)	)	PUNCT
ejpam-4592	231	11	where	where	SCONJ
ejpam-4592	231	12	i	i	PRON
ejpam-4592	231	13	,	,	PUNCT
ejpam-4592	231	14	j	j	PROPN
ejpam-4592	231	15	∈	∈	PROPN
ejpam-4592	231	16	nε	nε	VERB
ejpam-4592	231	17	.	.	PUNCT
ejpam-4592	232	1	now	now	ADV
ejpam-4592	232	2	σ(cn	σ(cn	PROPN
ejpam-4592	232	3	,	,	PUNCT
ejpam-4592	232	4	cm	cm	NOUN
ejpam-4592	232	5	)	)	PUNCT
ejpam-4592	233	1	=	=	SYM
ejpam-4592	233	2	σ(ch(i	σ(ch(i	NOUN
ejpam-4592	233	3	)	)	PUNCT
ejpam-4592	233	4	,	,	PUNCT
ejpam-4592	233	5	ch(j	ch(j	NOUN
ejpam-4592	233	6	)	)	PUNCT
ejpam-4592	233	7	)	)	PUNCT
ejpam-4592	233	8	which	which	PRON
ejpam-4592	233	9	implies	imply	VERB
ejpam-4592	233	10	σ(cn	σ(cn	PROPN
ejpam-4592	233	11	,	,	PUNCT
ejpam-4592	233	12	cm	cm	NOUN
ejpam-4592	233	13	)	)	PUNCT
ejpam-4592	233	14	=	=	SYM
ejpam-4592	233	15	σ(di	σ(di	PROPN
ejpam-4592	233	16	,	,	PUNCT
ejpam-4592	233	17	dj	dj	NOUN
ejpam-4592	233	18	)	)	PUNCT
ejpam-4592	233	19	.	.	PUNCT
ejpam-4592	234	1	thus	thus	ADV
ejpam-4592	234	2	,	,	PUNCT
ejpam-4592	234	3	σ(cn	σ(cn	PROPN
ejpam-4592	234	4	,	,	PUNCT
ejpam-4592	234	5	cm	cm	NOUN
ejpam-4592	234	6	)	)	PUNCT
ejpam-4592	234	7	<	<	X
ejpam-4592	234	8	ε	ε	PROPN
ejpam-4592	234	9	,	,	PUNCT
ejpam-4592	234	10	since	since	SCONJ
ejpam-4592	234	11	i	i	PRON
ejpam-4592	234	12	,	,	PUNCT
ejpam-4592	234	13	j	j	PROPN
ejpam-4592	234	14	∈	∈	PROPN
ejpam-4592	234	15	nε	nε	VERB
ejpam-4592	234	16	.	.	PUNCT
ejpam-4592	235	1	therefore	therefore	ADV
ejpam-4592	235	2	,	,	PUNCT
ejpam-4592	235	3	{	{	PUNCT
ejpam-4592	235	4	cn	cn	ADJ
ejpam-4592	235	5	}	}	PUNCT
ejpam-4592	235	6	∈	∈	PROPN
ejpam-4592	235	7	c(σ	c(σ	PROPN
ejpam-4592	235	8	)	)	PUNCT
ejpam-4592	235	9	.	.	PUNCT
ejpam-4592	236	1	theorem	theorem	VERB
ejpam-4592	236	2	25	25	NUM
ejpam-4592	236	3	.	.	PUNCT
ejpam-4592	237	1	let	let	AUX
ejpam-4592	237	2	(	(	PUNCT
ejpam-4592	237	3	x	x	NOUN
ejpam-4592	237	4	,	,	PUNCT
ejpam-4592	237	5	ω	ω	NUM
ejpam-4592	237	6	)	)	PUNCT
ejpam-4592	237	7	be	be	AUX
ejpam-4592	237	8	a	a	DET
ejpam-4592	237	9	gms	gms	NOUN
ejpam-4592	237	10	,	,	PUNCT
ejpam-4592	237	11	h	h	NOUN
ejpam-4592	237	12	:	:	PUNCT
ejpam-4592	237	13	n	n	X
ejpam-4592	237	14	→	→	SYM
ejpam-4592	237	15	n	n	CCONJ
ejpam-4592	237	16	be	be	AUX
ejpam-4592	237	17	a	a	DET
ejpam-4592	237	18	strictly	strictly	ADV
ejpam-4592	237	19	increasing	increase	VERB
ejpam-4592	237	20	,	,	PUNCT
ejpam-4592	237	21	bijective	bijective	ADJ
ejpam-4592	237	22	function	function	NOUN
ejpam-4592	237	23	.	.	PUNCT
ejpam-4592	238	1	if	if	SCONJ
ejpam-4592	238	2	the	the	DET
ejpam-4592	238	3	rearrangement	rearrangement	ADJ
ejpam-4592	238	4	sequence	sequence	NOUN
ejpam-4592	238	5	of	of	ADP
ejpam-4592	238	6	{	{	PUNCT
ejpam-4592	238	7	cn	cn	NOUN
ejpam-4592	238	8	}	}	PUNCT
ejpam-4592	238	9	under	under	ADP
ejpam-4592	238	10	h	h	NOUN
ejpam-4592	238	11	is	be	AUX
ejpam-4592	238	12	pseudo	pseudo	NOUN
ejpam-4592	238	13	-	-	NOUN
ejpam-4592	238	14	cauchy	cauchy	ADJ
ejpam-4592	238	15	,	,	PUNCT
ejpam-4592	238	16	then	then	ADV
ejpam-4592	238	17	{	{	PUNCT
ejpam-4592	238	18	cn	cn	NOUN
ejpam-4592	238	19	}	}	PUNCT
ejpam-4592	238	20	is	be	AUX
ejpam-4592	238	21	pseudocauchy	pseudocauchy	ADJ
ejpam-4592	238	22	with	with	ADP
ejpam-4592	238	23	the	the	DET
ejpam-4592	238	24	same	same	ADJ
ejpam-4592	238	25	metric	metric	NOUN
ejpam-4592	238	26	.	.	PUNCT
ejpam-4592	239	1	proof	proof	NOUN
ejpam-4592	239	2	.	.	PUNCT
ejpam-4592	240	1	let	let	VERB
ejpam-4592	240	2	{	{	PUNCT
ejpam-4592	240	3	dn	dn	AUX
ejpam-4592	240	4	}	}	PUNCT
ejpam-4592	240	5	be	be	AUX
ejpam-4592	240	6	a	a	DET
ejpam-4592	240	7	rearrangement	rearrangement	ADJ
ejpam-4592	240	8	sequence	sequence	NOUN
ejpam-4592	240	9	of	of	ADP
ejpam-4592	240	10	{	{	PUNCT
ejpam-4592	240	11	cn	cn	NOUN
ejpam-4592	240	12	}	}	PUNCT
ejpam-4592	240	13	under	under	ADP
ejpam-4592	240	14	h.	h.	PROPN
ejpam-4592	240	15	suppose	suppose	VERB
ejpam-4592	240	16	{	{	PUNCT
ejpam-4592	240	17	dn	dn	NOUN
ejpam-4592	240	18	}	}	PUNCT
ejpam-4592	240	19	∈	∈	PROPN
ejpam-4592	240	20	p(σ	p(σ	NOUN
ejpam-4592	240	21	)	)	PUNCT
ejpam-4592	240	22	.	.	PUNCT
ejpam-4592	241	1	let	let	VERB
ejpam-4592	241	2	ε	ε	PROPN
ejpam-4592	241	3	>	>	X
ejpam-4592	241	4	0	0	PUNCT
ejpam-4592	241	5	be	be	AUX
ejpam-4592	241	6	given	give	VERB
ejpam-4592	241	7	and	and	CCONJ
ejpam-4592	241	8	m	m	VERB
ejpam-4592	241	9	∈	∈	PROPN
ejpam-4592	241	10	n	n	PRON
ejpam-4592	241	11	where	where	SCONJ
ejpam-4592	241	12	m	m	VERB
ejpam-4592	241	13	=	=	PUNCT
ejpam-4592	241	14	h(n);n	h(n);n	SYM
ejpam-4592	241	15	∈	∈	PROPN
ejpam-4592	241	16	n.	n.	NOUN
ejpam-4592	241	17	since	since	SCONJ
ejpam-4592	241	18	n	n	PROPN
ejpam-4592	241	19	∈	∈	PROPN
ejpam-4592	241	20	n	n	NOUN
ejpam-4592	241	21	and	and	CCONJ
ejpam-4592	241	22	by	by	ADP
ejpam-4592	241	23	assumption	assumption	NOUN
ejpam-4592	241	24	,	,	PUNCT
ejpam-4592	241	25	there	there	PRON
ejpam-4592	241	26	exist	exist	VERB
ejpam-4592	241	27	k	k	PROPN
ejpam-4592	241	28	,	,	PUNCT
ejpam-4592	241	29	l	l	PROPN
ejpam-4592	241	30	∈	∈	PROPN
ejpam-4592	241	31	n	n	CCONJ
ejpam-4592	241	32	,	,	PUNCT
ejpam-4592	241	33	k	k	PROPN
ejpam-4592	241	34	̸=	̸=	PROPN
ejpam-4592	241	35	l	l	NOUN
ejpam-4592	241	36	such	such	ADJ
ejpam-4592	241	37	that	that	SCONJ
ejpam-4592	241	38	k	k	PROPN
ejpam-4592	241	39	>	>	X
ejpam-4592	241	40	n	n	PROPN
ejpam-4592	241	41	;	;	PUNCT
ejpam-4592	241	42	l	l	X
ejpam-4592	241	43	>	>	PUNCT
ejpam-4592	241	44	n	n	PROPN
ejpam-4592	241	45	and	and	CCONJ
ejpam-4592	241	46	σ(dk	σ(dk	PROPN
ejpam-4592	241	47	,	,	PUNCT
ejpam-4592	241	48	dl	dl	PROPN
ejpam-4592	241	49	)	)	PUNCT
ejpam-4592	241	50	<	<	X
ejpam-4592	241	51	ε	ε	PROPN
ejpam-4592	241	52	.	.	PUNCT
ejpam-4592	242	1	since	since	SCONJ
ejpam-4592	242	2	k	k	PROPN
ejpam-4592	242	3	,	,	PUNCT
ejpam-4592	242	4	l	l	PROPN
ejpam-4592	242	5	∈	∈	PROPN
ejpam-4592	242	6	n	n	CCONJ
ejpam-4592	242	7	,	,	PUNCT
ejpam-4592	242	8	k	k	PROPN
ejpam-4592	242	9	̸=	̸=	PROPN
ejpam-4592	242	10	l	l	NOUN
ejpam-4592	242	11	we	we	PRON
ejpam-4592	242	12	have	have	VERB
ejpam-4592	242	13	h(k	h(k	PROPN
ejpam-4592	242	14	)	)	PUNCT
ejpam-4592	242	15	,	,	PUNCT
ejpam-4592	242	16	h(l	h(l	PROPN
ejpam-4592	242	17	)	)	PUNCT
ejpam-4592	242	18	∈	∈	PROPN
ejpam-4592	242	19	n	n	CCONJ
ejpam-4592	242	20	,	,	PUNCT
ejpam-4592	242	21	h(k	h(k	PROPN
ejpam-4592	242	22	)	)	PUNCT
ejpam-4592	242	23	̸=	̸=	PROPN
ejpam-4592	242	24	h(l	h(l	NUM
ejpam-4592	242	25	)	)	PUNCT
ejpam-4592	242	26	,	,	PUNCT
ejpam-4592	242	27	by	by	ADP
ejpam-4592	242	28	h	h	NOUN
ejpam-4592	242	29	is	be	AUX
ejpam-4592	242	30	one	one	NUM
ejpam-4592	242	31	-	-	PUNCT
ejpam-4592	242	32	one	one	NUM
ejpam-4592	242	33	.	.	PUNCT
ejpam-4592	243	1	since	since	SCONJ
ejpam-4592	243	2	h	h	NOUN
ejpam-4592	243	3	is	be	AUX
ejpam-4592	243	4	strictly	strictly	ADV
ejpam-4592	243	5	increasing	increase	VERB
ejpam-4592	243	6	and	and	CCONJ
ejpam-4592	243	7	k	k	X
ejpam-4592	243	8	>	>	PUNCT
ejpam-4592	243	9	n	n	PROPN
ejpam-4592	243	10	;	;	PUNCT
ejpam-4592	243	11	l	l	X
ejpam-4592	243	12	>	>	PUNCT
ejpam-4592	243	13	n	n	CCONJ
ejpam-4592	243	14	we	we	PRON
ejpam-4592	243	15	have	have	VERB
ejpam-4592	243	16	h(k	h(k	PROPN
ejpam-4592	243	17	)	)	PUNCT
ejpam-4592	243	18	>	>	X
ejpam-4592	244	1	h(n);h(l	h(n);h(l	PROPN
ejpam-4592	244	2	)	)	PUNCT
ejpam-4592	244	3	>	>	X
ejpam-4592	245	1	h(n	h(n	PROPN
ejpam-4592	245	2	)	)	PUNCT
ejpam-4592	245	3	.	.	PUNCT
ejpam-4592	246	1	now	now	ADV
ejpam-4592	246	2	σ(ch(k	σ(ch(k	PROPN
ejpam-4592	246	3	)	)	PUNCT
ejpam-4592	246	4	,	,	PUNCT
ejpam-4592	246	5	ch(l	ch(l	NUM
ejpam-4592	246	6	)	)	PUNCT
ejpam-4592	246	7	)	)	PUNCT
ejpam-4592	247	1	=	=	PUNCT
ejpam-4592	247	2	σ(dk	σ(dk	X
ejpam-4592	247	3	,	,	PUNCT
ejpam-4592	247	4	dl	dl	PROPN
ejpam-4592	247	5	)	)	PUNCT
ejpam-4592	247	6	<	<	X
ejpam-4592	247	7	ε	ε	PROPN
ejpam-4592	247	8	.	.	PUNCT
ejpam-4592	248	1	thus	thus	ADV
ejpam-4592	248	2	,	,	PUNCT
ejpam-4592	248	3	there	there	PRON
ejpam-4592	248	4	exist	exist	VERB
ejpam-4592	248	5	h(k	h(k	NOUN
ejpam-4592	248	6	)	)	PUNCT
ejpam-4592	248	7	,	,	PUNCT
ejpam-4592	248	8	h(l	h(l	PROPN
ejpam-4592	248	9	)	)	PUNCT
ejpam-4592	248	10	∈	∈	PROPN
ejpam-4592	248	11	n	n	CCONJ
ejpam-4592	248	12	,	,	PUNCT
ejpam-4592	248	13	h(k	h(k	PROPN
ejpam-4592	248	14	)	)	PUNCT
ejpam-4592	248	15	̸=	̸=	PROPN
ejpam-4592	248	16	h(l	h(l	NUM
ejpam-4592	248	17	)	)	PUNCT
ejpam-4592	248	18	such	such	ADJ
ejpam-4592	248	19	that	that	SCONJ
ejpam-4592	248	20	h(k	h(k	PROPN
ejpam-4592	248	21	)	)	PUNCT
ejpam-4592	248	22	>	>	X
ejpam-4592	249	1	h(n	h(n	PROPN
ejpam-4592	249	2	)	)	PUNCT
ejpam-4592	249	3	=	=	SYM
ejpam-4592	249	4	m;h(l	m;h(l	PROPN
ejpam-4592	249	5	)	)	PUNCT
ejpam-4592	249	6	>	>	PUNCT
ejpam-4592	250	1	h(n	h(n	PROPN
ejpam-4592	250	2	)	)	PUNCT
ejpam-4592	251	1	=	=	SYM
ejpam-4592	251	2	m	m	PROPN
ejpam-4592	251	3	and	and	CCONJ
ejpam-4592	251	4	σ(ch(k	σ(ch(k	PROPN
ejpam-4592	251	5	)	)	PUNCT
ejpam-4592	251	6	,	,	PUNCT
ejpam-4592	251	7	ch(l	ch(l	NUM
ejpam-4592	251	8	)	)	PUNCT
ejpam-4592	251	9	)	)	PUNCT
ejpam-4592	251	10	<	<	X
ejpam-4592	251	11	ε	ε	PROPN
ejpam-4592	251	12	.	.	PUNCT
ejpam-4592	251	13	hence	hence	ADV
ejpam-4592	251	14	{	{	PUNCT
ejpam-4592	251	15	cn	cn	NOUN
ejpam-4592	251	16	}	}	PUNCT
ejpam-4592	251	17	is	be	AUX
ejpam-4592	251	18	pseudo	pseudo	NOUN
ejpam-4592	251	19	-	-	NOUN
ejpam-4592	251	20	cauchy	cauchy	ADJ
ejpam-4592	251	21	with	with	ADP
ejpam-4592	251	22	the	the	DET
ejpam-4592	251	23	same	same	ADJ
ejpam-4592	251	24	metric	metric	NOUN
ejpam-4592	251	25	.	.	PUNCT
ejpam-4592	252	1	4	4	X
ejpam-4592	252	2	.	.	X
ejpam-4592	252	3	asymptotic	asymptotic	ADJ
ejpam-4592	252	4	sequences	sequence	NOUN
ejpam-4592	252	5	here	here	ADV
ejpam-4592	252	6	,	,	PUNCT
ejpam-4592	252	7	in	in	ADP
ejpam-4592	252	8	a	a	DET
ejpam-4592	252	9	generalized	generalized	ADJ
ejpam-4592	252	10	metric	metric	ADJ
ejpam-4592	252	11	space	space	NOUN
ejpam-4592	252	12	,	,	PUNCT
ejpam-4592	252	13	two	two	NUM
ejpam-4592	252	14	kinds	kind	NOUN
ejpam-4592	252	15	of	of	ADP
ejpam-4592	252	16	new	new	ADJ
ejpam-4592	252	17	sequences	sequence	NOUN
ejpam-4592	252	18	are	be	AUX
ejpam-4592	252	19	defined	define	VERB
ejpam-4592	252	20	and	and	CCONJ
ejpam-4592	252	21	explore	explore	VERB
ejpam-4592	252	22	their	their	PRON
ejpam-4592	252	23	nature	nature	NOUN
ejpam-4592	252	24	.	.	PUNCT
ejpam-4592	253	1	we	we	PRON
ejpam-4592	253	2	begin	begin	VERB
ejpam-4592	253	3	with	with	ADP
ejpam-4592	253	4	a	a	DET
ejpam-4592	253	5	definition	definition	NOUN
ejpam-4592	253	6	of	of	ADP
ejpam-4592	253	7	asymptotic	asymptotic	ADJ
ejpam-4592	253	8	sequence	sequence	NOUN
ejpam-4592	253	9	in	in	ADP
ejpam-4592	253	10	a	a	DET
ejpam-4592	253	11	generalized	generalized	ADJ
ejpam-4592	253	12	metric	metric	ADJ
ejpam-4592	253	13	space	space	NOUN
ejpam-4592	253	14	.	.	PUNCT
ejpam-4592	254	1	definition	definition	NOUN
ejpam-4592	254	2	26	26	NUM
ejpam-4592	254	3	.	.	PUNCT
ejpam-4592	255	1	let	let	VERB
ejpam-4592	255	2	(	(	PUNCT
ejpam-4592	255	3	x	x	NOUN
ejpam-4592	255	4	,	,	PUNCT
ejpam-4592	255	5	ω	ω	NUM
ejpam-4592	255	6	)	)	PUNCT
ejpam-4592	255	7	be	be	AUX
ejpam-4592	255	8	a	a	DET
ejpam-4592	255	9	generalized	generalized	ADJ
ejpam-4592	255	10	metric	metric	ADJ
ejpam-4592	255	11	space	space	NOUN
ejpam-4592	255	12	.	.	PUNCT
ejpam-4592	256	1	a	a	DET
ejpam-4592	256	2	pair	pair	NOUN
ejpam-4592	256	3	of	of	ADP
ejpam-4592	256	4	sequence	sequence	NOUN
ejpam-4592	256	5	{	{	PUNCT
ejpam-4592	256	6	sn}n∈n	sn}n∈n	INTJ
ejpam-4592	256	7	and	and	CCONJ
ejpam-4592	256	8	{	{	PUNCT
ejpam-4592	256	9	tn}n∈n	tn}n∈n	X
ejpam-4592	256	10	in	in	ADP
ejpam-4592	256	11	x	x	PROPN
ejpam-4592	256	12	is	be	AUX
ejpam-4592	256	13	said	say	VERB
ejpam-4592	256	14	be	be	AUX
ejpam-4592	256	15	:	:	PUNCT
ejpam-4592	256	16	(	(	PUNCT
ejpam-4592	256	17	a	a	X
ejpam-4592	256	18	)	)	PUNCT
ejpam-4592	256	19	asymptotic	asymptotic	ADJ
ejpam-4592	256	20	with	with	ADP
ejpam-4592	256	21	respect	respect	NOUN
ejpam-4592	256	22	to	to	ADP
ejpam-4592	256	23	σ	σ	PROPN
ejpam-4592	256	24	∈	∈	PROPN
ejpam-4592	256	25	ω	ω	NOUN
ejpam-4592	256	26	if	if	SCONJ
ejpam-4592	256	27	for	for	ADP
ejpam-4592	256	28	every	every	DET
ejpam-4592	256	29	ε	ε	PROPN
ejpam-4592	256	30	>	>	X
ejpam-4592	256	31	0	0	PROPN
ejpam-4592	256	32	,	,	PUNCT
ejpam-4592	256	33	there	there	PRON
ejpam-4592	256	34	exists	exist	VERB
ejpam-4592	256	35	n0	n0	PROPN
ejpam-4592	256	36	∈	∈	PROPN
ejpam-4592	256	37	n	n	PRON
ejpam-4592	256	38	such	such	ADJ
ejpam-4592	256	39	that	that	SCONJ
ejpam-4592	256	40	σ(sn	σ(sn	PROPN
ejpam-4592	256	41	,	,	PUNCT
ejpam-4592	256	42	tn	tn	PROPN
ejpam-4592	256	43	)	)	PUNCT
ejpam-4592	256	44	<	<	X
ejpam-4592	256	45	ε	ε	PROPN
ejpam-4592	256	46	for	for	ADP
ejpam-4592	256	47	all	all	DET
ejpam-4592	256	48	n	n	PRON
ejpam-4592	256	49	≥	≥	NOUN
ejpam-4592	256	50	n0	n0	NUM
ejpam-4592	256	51	.	.	PUNCT
ejpam-4592	257	1	(	(	PUNCT
ejpam-4592	257	2	b	b	X
ejpam-4592	257	3	)	)	PUNCT
ejpam-4592	257	4	uniformly	uniformly	ADV
ejpam-4592	257	5	asymptotic	asymptotic	ADJ
ejpam-4592	257	6	with	with	ADP
ejpam-4592	257	7	respect	respect	NOUN
ejpam-4592	257	8	to	to	ADP
ejpam-4592	257	9	σ	σ	PROPN
ejpam-4592	257	10	∈	∈	PROPN
ejpam-4592	257	11	ω	ω	NOUN
ejpam-4592	257	12	if	if	SCONJ
ejpam-4592	257	13	for	for	ADP
ejpam-4592	257	14	every	every	DET
ejpam-4592	257	15	ε	ε	PROPN
ejpam-4592	257	16	>	>	X
ejpam-4592	257	17	0	0	PROPN
ejpam-4592	257	18	,	,	PUNCT
ejpam-4592	257	19	there	there	PRON
ejpam-4592	257	20	exists	exist	VERB
ejpam-4592	257	21	n0	n0	PROPN
ejpam-4592	257	22	∈	∈	PROPN
ejpam-4592	257	23	n	n	PRON
ejpam-4592	257	24	such	such	ADJ
ejpam-4592	257	25	that	that	PRON
ejpam-4592	257	26	σ(sm	σ(sm	PROPN
ejpam-4592	257	27	,	,	PUNCT
ejpam-4592	257	28	tn	tn	PROPN
ejpam-4592	257	29	)	)	PUNCT
ejpam-4592	257	30	<	<	X
ejpam-4592	257	31	ε	ε	PROPN
ejpam-4592	257	32	for	for	ADP
ejpam-4592	257	33	all	all	DET
ejpam-4592	257	34	m	m	PROPN
ejpam-4592	257	35	,	,	PUNCT
ejpam-4592	257	36	n	n	PRON
ejpam-4592	257	37	≥	≥	NOUN
ejpam-4592	257	38	n0	n0	NUM
ejpam-4592	257	39	.	.	PUNCT
ejpam-4592	257	40	theorem	theorem	VERB
ejpam-4592	257	41	27	27	NUM
ejpam-4592	257	42	gives	give	VERB
ejpam-4592	257	43	new	new	ADJ
ejpam-4592	257	44	tricks	trick	NOUN
ejpam-4592	257	45	to	to	PART
ejpam-4592	257	46	check	check	VERB
ejpam-4592	257	47	whether	whether	SCONJ
ejpam-4592	257	48	,	,	PUNCT
ejpam-4592	257	49	in	in	ADP
ejpam-4592	257	50	a	a	DET
ejpam-4592	257	51	generalized	generalized	ADJ
ejpam-4592	257	52	metric	metric	ADJ
ejpam-4592	257	53	space	space	NOUN
ejpam-4592	257	54	,	,	PUNCT
ejpam-4592	257	55	a	a	DET
ejpam-4592	257	56	given	give	VERB
ejpam-4592	257	57	pair	pair	NOUN
ejpam-4592	257	58	of	of	ADP
ejpam-4592	257	59	sequence	sequence	NOUN
ejpam-4592	257	60	is	be	AUX
ejpam-4592	257	61	in	in	ADP
ejpam-4592	257	62	c(σ	c(σ	PROPN
ejpam-4592	257	63	)	)	PUNCT
ejpam-4592	257	64	or	or	CCONJ
ejpam-4592	257	65	not	not	PART
ejpam-4592	257	66	.	.	PUNCT
ejpam-4592	258	1	theorem	theorem	ADJ
ejpam-4592	258	2	27	27	NUM
ejpam-4592	258	3	.	.	PUNCT
ejpam-4592	259	1	let	let	VERB
ejpam-4592	259	2	(	(	PUNCT
ejpam-4592	259	3	x	x	NOUN
ejpam-4592	259	4	,	,	PUNCT
ejpam-4592	259	5	ω	ω	NUM
ejpam-4592	259	6	)	)	PUNCT
ejpam-4592	259	7	be	be	AUX
ejpam-4592	259	8	a	a	DET
ejpam-4592	259	9	gms	gms	NOUN
ejpam-4592	259	10	.	.	PUNCT
ejpam-4592	260	1	if	if	SCONJ
ejpam-4592	260	2	the	the	DET
ejpam-4592	260	3	pair	pair	NOUN
ejpam-4592	260	4	of	of	ADP
ejpam-4592	260	5	sequence	sequence	NOUN
ejpam-4592	260	6	{	{	PUNCT
ejpam-4592	260	7	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	260	8	and	and	CCONJ
ejpam-4592	260	9	{	{	PUNCT
ejpam-4592	260	10	dn}n∈n	dn}n∈n	VERB
ejpam-4592	260	11	is	be	AUX
ejpam-4592	260	12	asymptotic	asymptotic	ADJ
ejpam-4592	260	13	with	with	ADP
ejpam-4592	260	14	respect	respect	NOUN
ejpam-4592	260	15	to	to	ADP
ejpam-4592	260	16	σ	σ	PROPN
ejpam-4592	260	17	∈	∈	PROPN
ejpam-4592	260	18	ω	ω	PROPN
ejpam-4592	260	19	,	,	PUNCT
ejpam-4592	260	20	then	then	ADV
ejpam-4592	260	21	the	the	DET
ejpam-4592	260	22	followings	following	NOUN
ejpam-4592	260	23	are	be	AUX
ejpam-4592	260	24	true	true	ADJ
ejpam-4592	260	25	.	.	PUNCT
ejpam-4592	261	1	(	(	PUNCT
ejpam-4592	261	2	a	a	X
ejpam-4592	261	3	)	)	PUNCT
ejpam-4592	261	4	if	if	SCONJ
ejpam-4592	261	5	{	{	PUNCT
ejpam-4592	261	6	dn	dn	NOUN
ejpam-4592	261	7	}	}	PUNCT
ejpam-4592	261	8	∈	∈	PROPN
ejpam-4592	261	9	c(σ	c(σ	PROPN
ejpam-4592	261	10	)	)	PUNCT
ejpam-4592	261	11	,	,	PUNCT
ejpam-4592	261	12	then	then	ADV
ejpam-4592	261	13	{	{	PUNCT
ejpam-4592	261	14	cn	cn	ADJ
ejpam-4592	261	15	}	}	PUNCT
ejpam-4592	261	16	∈	∈	PROPN
ejpam-4592	261	17	c(σ	c(σ	PROPN
ejpam-4592	261	18	)	)	PUNCT
ejpam-4592	261	19	.	.	PUNCT
ejpam-4592	262	1	(	(	PUNCT
ejpam-4592	262	2	b	b	X
ejpam-4592	262	3	)	)	PUNCT
ejpam-4592	262	4	if	if	SCONJ
ejpam-4592	262	5	{	{	PUNCT
ejpam-4592	262	6	dn	dn	NOUN
ejpam-4592	262	7	}	}	PUNCT
ejpam-4592	262	8	is	be	AUX
ejpam-4592	262	9	a	a	DET
ejpam-4592	262	10	convergent	convergent	NOUN
ejpam-4592	262	11	sequence	sequence	NOUN
ejpam-4592	262	12	with	with	ADP
ejpam-4592	262	13	respect	respect	NOUN
ejpam-4592	262	14	to	to	ADP
ejpam-4592	262	15	σ	σ	PROPN
ejpam-4592	262	16	∈	∈	PROPN
ejpam-4592	262	17	ω	ω	PROPN
ejpam-4592	262	18	,	,	PUNCT
ejpam-4592	262	19	then	then	ADV
ejpam-4592	262	20	{	{	PUNCT
ejpam-4592	262	21	cn	cn	NOUN
ejpam-4592	262	22	}	}	PUNCT
ejpam-4592	262	23	is	be	AUX
ejpam-4592	262	24	a	a	DET
ejpam-4592	262	25	convergent	convergent	NOUN
ejpam-4592	262	26	sequence	sequence	NOUN
ejpam-4592	262	27	with	with	ADP
ejpam-4592	262	28	the	the	DET
ejpam-4592	262	29	same	same	ADJ
ejpam-4592	262	30	metric	metric	NOUN
ejpam-4592	262	31	.	.	PUNCT
ejpam-4592	263	1	proof	proof	NOUN
ejpam-4592	263	2	.	.	PUNCT
ejpam-4592	264	1	assume	assume	VERB
ejpam-4592	264	2	that	that	SCONJ
ejpam-4592	264	3	,	,	PUNCT
ejpam-4592	264	4	{	{	PUNCT
ejpam-4592	264	5	cn}n∈n	cn}n∈n	X
ejpam-4592	264	6	and	and	CCONJ
ejpam-4592	264	7	{	{	PUNCT
ejpam-4592	264	8	dn}n∈n	dn}n∈n	VERB
ejpam-4592	264	9	is	be	AUX
ejpam-4592	264	10	asymptotic	asymptotic	ADJ
ejpam-4592	264	11	with	with	ADP
ejpam-4592	264	12	respect	respect	NOUN
ejpam-4592	264	13	to	to	ADP
ejpam-4592	264	14	σ	σ	PROPN
ejpam-4592	264	15	∈	∈	PROPN
ejpam-4592	264	16	ω	ω	PROPN
ejpam-4592	264	17	.	.	PUNCT
ejpam-4592	265	1	(	(	PUNCT
ejpam-4592	265	2	a	a	X
ejpam-4592	265	3	)	)	PUNCT
ejpam-4592	265	4	given	give	VERB
ejpam-4592	265	5	that	that	SCONJ
ejpam-4592	265	6	{	{	PUNCT
ejpam-4592	265	7	dn	dn	ADJ
ejpam-4592	265	8	}	}	PUNCT
ejpam-4592	265	9	∈	∈	PROPN
ejpam-4592	265	10	c(σ	c(σ	PROPN
ejpam-4592	265	11	)	)	PUNCT
ejpam-4592	265	12	.	.	PUNCT
ejpam-4592	266	1	let	let	VERB
ejpam-4592	266	2	ε	ε	PROPN
ejpam-4592	266	3	>	>	X
ejpam-4592	266	4	0	0	PROPN
ejpam-4592	266	5	.	.	PUNCT
ejpam-4592	267	1	then	then	ADV
ejpam-4592	267	2	there	there	PRON
ejpam-4592	267	3	is	be	VERB
ejpam-4592	267	4	n0	n0	NUM
ejpam-4592	267	5	∈	∈	PROPN
ejpam-4592	267	6	n	n	CCONJ
ejpam-4592	267	7	such	such	ADJ
ejpam-4592	267	8	that	that	SCONJ
ejpam-4592	267	9	σ(dn	σ(dn	PROPN
ejpam-4592	267	10	,	,	PUNCT
ejpam-4592	267	11	dm	dm	NOUN
ejpam-4592	267	12	)	)	PUNCT
ejpam-4592	267	13	<	<	X
ejpam-4592	267	14	ε	ε	PROPN
ejpam-4592	267	15	3	3	NUM
ejpam-4592	267	16	for	for	ADP
ejpam-4592	267	17	all	all	DET
ejpam-4592	267	18	n	n	CCONJ
ejpam-4592	267	19	,	,	PUNCT
ejpam-4592	267	20	m	m	PROPN
ejpam-4592	267	21	≥	≥	NOUN
ejpam-4592	267	22	n0	n0	NUM
ejpam-4592	267	23	.	.	PUNCT
ejpam-4592	268	1	by	by	ADP
ejpam-4592	268	2	hypothesis	hypothesis	NOUN
ejpam-4592	268	3	,	,	PUNCT
ejpam-4592	268	4	there	there	PRON
ejpam-4592	268	5	exists	exist	VERB
ejpam-4592	268	6	a	a	DET
ejpam-4592	268	7	positive	positive	ADJ
ejpam-4592	268	8	integer	integer	NOUN
ejpam-4592	268	9	n1	n1	NOUN
ejpam-4592	268	10	such	such	ADJ
ejpam-4592	268	11	that	that	DET
ejpam-4592	268	12	σ(cn	σ(cn	NOUN
ejpam-4592	268	13	,	,	PUNCT
ejpam-4592	268	14	dn	dn	NOUN
ejpam-4592	268	15	)	)	PUNCT
ejpam-4592	268	16	<	<	X
ejpam-4592	268	17	ε	ε	PROPN
ejpam-4592	268	18	3	3	NUM
ejpam-4592	268	19	v.	v.	ADP
ejpam-4592	268	20	subramanian	subramanian	PROPN
ejpam-4592	268	21	et	et	PROPN
ejpam-4592	268	22	al	al	PROPN
ejpam-4592	268	23	.	.	PUNCT
ejpam-4592	268	24	/	/	SYM
ejpam-4592	268	25	eur	eur	PROPN
ejpam-4592	268	26	.	.	PUNCT
ejpam-4592	269	1	j.	j.	PROPN
ejpam-4592	269	2	pure	pure	PROPN
ejpam-4592	269	3	appl	appl	PROPN
ejpam-4592	269	4	.	.	PROPN
ejpam-4592	269	5	math	math	PROPN
ejpam-4592	269	6	,	,	PUNCT
ejpam-4592	269	7	15	15	NUM
ejpam-4592	269	8	(	(	PUNCT
ejpam-4592	269	9	4	4	NUM
ejpam-4592	269	10	)	)	PUNCT
ejpam-4592	269	11	(	(	PUNCT
ejpam-4592	269	12	2022	2022	NUM
ejpam-4592	269	13	)	)	PUNCT
ejpam-4592	269	14	,	,	PUNCT
ejpam-4592	269	15	1869	1869	NUM
ejpam-4592	269	16	-	-	SYM
ejpam-4592	269	17	1886	1886	NUM
ejpam-4592	269	18	1877	1877	NUM
ejpam-4592	269	19	for	for	ADP
ejpam-4592	269	20	all	all	DET
ejpam-4592	269	21	n	n	PRON
ejpam-4592	269	22	≥	≥	NOUN
ejpam-4592	269	23	n1	n1	NOUN
ejpam-4592	269	24	.	.	PUNCT
ejpam-4592	270	1	take	take	VERB
ejpam-4592	270	2	n	n	NOUN
ejpam-4592	270	3	=	=	SYM
ejpam-4592	270	4	max{n0	max{n0	PROPN
ejpam-4592	270	5	,	,	PUNCT
ejpam-4592	270	6	n1	n1	NOUN
ejpam-4592	270	7	+	+	CCONJ
ejpam-4592	270	8	1	1	NUM
ejpam-4592	270	9	}	}	PUNCT
ejpam-4592	270	10	.	.	PUNCT
ejpam-4592	271	1	then	then	ADV
ejpam-4592	271	2	for	for	ADP
ejpam-4592	271	3	n	n	CCONJ
ejpam-4592	271	4	,	,	PUNCT
ejpam-4592	271	5	m	m	VERB
ejpam-4592	271	6	≥	≥	NOUN
ejpam-4592	271	7	n	n	CCONJ
ejpam-4592	271	8	,	,	PUNCT
ejpam-4592	271	9	σ(cn	σ(cn	PROPN
ejpam-4592	271	10	,	,	PUNCT
ejpam-4592	271	11	cm	cm	NOUN
ejpam-4592	271	12	)	)	PUNCT
ejpam-4592	271	13	≤	≤	NOUN
ejpam-4592	271	14	σ(cn	σ(cn	NOUN
ejpam-4592	271	15	,	,	PUNCT
ejpam-4592	271	16	dn	dn	NOUN
ejpam-4592	271	17	)	)	PUNCT
ejpam-4592	271	18	+	+	CCONJ
ejpam-4592	271	19	σ(dn	σ(dn	NOUN
ejpam-4592	271	20	,	,	PUNCT
ejpam-4592	271	21	dm	dm	NOUN
ejpam-4592	271	22	)	)	PUNCT
ejpam-4592	271	23	+	+	CCONJ
ejpam-4592	271	24	σ(dm	σ(dm	PROPN
ejpam-4592	271	25	,	,	PUNCT
ejpam-4592	271	26	cm	cm	NOUN
ejpam-4592	271	27	)	)	PUNCT
ejpam-4592	271	28	and	and	CCONJ
ejpam-4592	271	29	so	so	ADV
ejpam-4592	271	30	σ(cn	σ(cn	PROPN
ejpam-4592	271	31	,	,	PUNCT
ejpam-4592	271	32	cm	cm	NOUN
ejpam-4592	271	33	)	)	PUNCT
ejpam-4592	271	34	<	<	X
ejpam-4592	271	35	ε	ε	PROPN
ejpam-4592	271	36	for	for	ADP
ejpam-4592	271	37	all	all	DET
ejpam-4592	271	38	n	n	CCONJ
ejpam-4592	271	39	,	,	PUNCT
ejpam-4592	271	40	m	m	VERB
ejpam-4592	271	41	≥	≥	NOUN
ejpam-4592	271	42	n.	n.	NOUN
ejpam-4592	271	43	hence	hence	ADV
ejpam-4592	271	44	{	{	PUNCT
ejpam-4592	271	45	cn	cn	PROPN
ejpam-4592	271	46	}	}	PUNCT
ejpam-4592	271	47	∈	∈	PROPN
ejpam-4592	271	48	c(σ	c(σ	PROPN
ejpam-4592	271	49	)	)	PUNCT
ejpam-4592	271	50	.	.	PUNCT
ejpam-4592	272	1	(	(	PUNCT
ejpam-4592	272	2	b	b	X
ejpam-4592	272	3	)	)	PUNCT
ejpam-4592	272	4	suppose	suppose	VERB
ejpam-4592	272	5	that	that	SCONJ
ejpam-4592	272	6	{	{	PUNCT
ejpam-4592	272	7	dn	dn	X
ejpam-4592	272	8	}	}	PUNCT
ejpam-4592	272	9	is	be	AUX
ejpam-4592	272	10	a	a	DET
ejpam-4592	272	11	convergent	convergent	NOUN
ejpam-4592	272	12	sequence	sequence	NOUN
ejpam-4592	272	13	with	with	ADP
ejpam-4592	272	14	respect	respect	NOUN
ejpam-4592	272	15	to	to	ADP
ejpam-4592	272	16	σ	σ	PROPN
ejpam-4592	272	17	∈	∈	PROPN
ejpam-4592	272	18	ω	ω	X
ejpam-4592	272	19	.	.	PUNCT
ejpam-4592	273	1	let	let	VERB
ejpam-4592	273	2	ε	ε	PROPN
ejpam-4592	273	3	>	>	X
ejpam-4592	273	4	0	0	PUNCT
ejpam-4592	273	5	be	be	AUX
ejpam-4592	273	6	given	give	VERB
ejpam-4592	273	7	and	and	CCONJ
ejpam-4592	273	8	y	y	PROPN
ejpam-4592	273	9	∈	∈	PROPN
ejpam-4592	273	10	x	x	PUNCT
ejpam-4592	273	11	be	be	AUX
ejpam-4592	273	12	a	a	DET
ejpam-4592	273	13	limit	limit	NOUN
ejpam-4592	273	14	point	point	NOUN
ejpam-4592	273	15	of	of	ADP
ejpam-4592	273	16	{	{	PUNCT
ejpam-4592	273	17	dn	dn	NOUN
ejpam-4592	273	18	}	}	PUNCT
ejpam-4592	273	19	.	.	PUNCT
ejpam-4592	274	1	then	then	ADV
ejpam-4592	274	2	there	there	PRON
ejpam-4592	274	3	is	be	VERB
ejpam-4592	274	4	n0	n0	NUM
ejpam-4592	274	5	∈	∈	PROPN
ejpam-4592	274	6	n	n	CCONJ
ejpam-4592	274	7	such	such	ADJ
ejpam-4592	274	8	that	that	SCONJ
ejpam-4592	274	9	σ(dn	σ(dn	PROPN
ejpam-4592	274	10	,	,	PUNCT
ejpam-4592	274	11	y	y	NOUN
ejpam-4592	274	12	)	)	PUNCT
ejpam-4592	274	13	<	<	X
ejpam-4592	274	14	ε	ε	PROPN
ejpam-4592	274	15	2	2	NUM
ejpam-4592	274	16	for	for	ADP
ejpam-4592	274	17	all	all	DET
ejpam-4592	274	18	n	n	PRON
ejpam-4592	274	19	≥	≥	NOUN
ejpam-4592	274	20	n0	n0	NUM
ejpam-4592	274	21	.	.	PUNCT
ejpam-4592	275	1	by	by	ADP
ejpam-4592	275	2	hypothesis	hypothesis	NOUN
ejpam-4592	275	3	,	,	PUNCT
ejpam-4592	275	4	there	there	PRON
ejpam-4592	275	5	is	be	VERB
ejpam-4592	275	6	n1	n1	PROPN
ejpam-4592	275	7	∈	∈	NOUN
ejpam-4592	275	8	n	n	PRON
ejpam-4592	275	9	such	such	ADJ
ejpam-4592	275	10	that	that	DET
ejpam-4592	275	11	σ(cn	σ(cn	NOUN
ejpam-4592	275	12	,	,	PUNCT
ejpam-4592	275	13	dn	dn	NOUN
ejpam-4592	275	14	)	)	PUNCT
ejpam-4592	275	15	<	<	X
ejpam-4592	275	16	ε	ε	PROPN
ejpam-4592	275	17	2	2	NUM
ejpam-4592	275	18	for	for	ADP
ejpam-4592	275	19	all	all	PRON
ejpam-4592	275	20	n	n	PRON
ejpam-4592	275	21	>	>	X
ejpam-4592	275	22	n1	n1	PROPN
ejpam-4592	275	23	.	.	PUNCT
ejpam-4592	276	1	take	take	VERB
ejpam-4592	276	2	m0	m0	NOUN
ejpam-4592	276	3	=	=	SYM
ejpam-4592	276	4	max{n0	max{n0	PROPN
ejpam-4592	276	5	,	,	PUNCT
ejpam-4592	276	6	n1	n1	NOUN
ejpam-4592	276	7	+	+	CCONJ
ejpam-4592	276	8	1	1	NUM
ejpam-4592	276	9	}	}	PUNCT
ejpam-4592	276	10	.	.	PUNCT
ejpam-4592	277	1	then	then	ADV
ejpam-4592	277	2	σ(cn	σ(cn	PROPN
ejpam-4592	277	3	,	,	PUNCT
ejpam-4592	277	4	y	y	NOUN
ejpam-4592	277	5	)	)	PUNCT
ejpam-4592	277	6	≤	≤	NOUN
ejpam-4592	277	7	σ(cn	σ(cn	PROPN
ejpam-4592	277	8	,	,	PUNCT
ejpam-4592	277	9	dn	dn	NOUN
ejpam-4592	277	10	)	)	PUNCT
ejpam-4592	277	11	+	+	CCONJ
ejpam-4592	277	12	σ(dn	σ(dn	PROPN
ejpam-4592	277	13	,	,	PUNCT
ejpam-4592	277	14	y	y	NOUN
ejpam-4592	277	15	)	)	PUNCT
ejpam-4592	277	16	and	and	CCONJ
ejpam-4592	277	17	so	so	ADV
ejpam-4592	277	18	σ(cn	σ(cn	PROPN
ejpam-4592	277	19	,	,	PUNCT
ejpam-4592	277	20	y	y	NOUN
ejpam-4592	277	21	)	)	PUNCT
ejpam-4592	277	22	<	<	X
ejpam-4592	277	23	ε	ε	PROPN
ejpam-4592	277	24	for	for	ADP
ejpam-4592	277	25	all	all	DET
ejpam-4592	277	26	n	n	PRON
ejpam-4592	277	27	≥	≥	NOUN
ejpam-4592	277	28	m0	m0	NOUN
ejpam-4592	277	29	.	.	PUNCT
ejpam-4592	278	1	thus	thus	ADV
ejpam-4592	278	2	,	,	PUNCT
ejpam-4592	278	3	{	{	PUNCT
ejpam-4592	278	4	cn	cn	VERB
ejpam-4592	278	5	}	}	PUNCT
ejpam-4592	278	6	is	be	AUX
ejpam-4592	278	7	convergent	convergent	ADJ
ejpam-4592	278	8	to	to	ADP
ejpam-4592	278	9	y	y	PROPN
ejpam-4592	278	10	with	with	ADP
ejpam-4592	278	11	respect	respect	NOUN
ejpam-4592	278	12	to	to	ADP
ejpam-4592	278	13	σ	σ	PROPN
ejpam-4592	278	14	.	.	PUNCT
ejpam-4592	279	1	hence	hence	ADV
ejpam-4592	279	2	{	{	PUNCT
ejpam-4592	279	3	cn	cn	PROPN
ejpam-4592	279	4	}	}	PUNCT
ejpam-4592	279	5	is	be	AUX
ejpam-4592	279	6	a	a	DET
ejpam-4592	279	7	convergent	convergent	NOUN
ejpam-4592	279	8	sequence	sequence	NOUN
ejpam-4592	279	9	with	with	ADP
ejpam-4592	279	10	respect	respect	NOUN
ejpam-4592	279	11	to	to	ADP
ejpam-4592	279	12	σ	σ	PROPN
ejpam-4592	279	13	∈	∈	PROPN
ejpam-4592	279	14	ω	ω	PROPN
ejpam-4592	279	15	.	.	PUNCT
ejpam-4592	279	16	theorem	theorem	PROPN
ejpam-4592	279	17	28	28	NUM
ejpam-4592	279	18	.	.	PUNCT
ejpam-4592	280	1	let	let	VERB
ejpam-4592	280	2	(	(	PUNCT
ejpam-4592	280	3	x	x	NOUN
ejpam-4592	280	4	,	,	PUNCT
ejpam-4592	280	5	ω	ω	NUM
ejpam-4592	280	6	)	)	PUNCT
ejpam-4592	280	7	be	be	AUX
ejpam-4592	280	8	a	a	DET
ejpam-4592	280	9	gms	gms	NOUN
ejpam-4592	280	10	,	,	PUNCT
ejpam-4592	280	11	the	the	DET
ejpam-4592	280	12	pair	pair	NOUN
ejpam-4592	280	13	of	of	ADP
ejpam-4592	280	14	sequence	sequence	NOUN
ejpam-4592	280	15	{	{	PUNCT
ejpam-4592	280	16	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	280	17	and	and	CCONJ
ejpam-4592	280	18	{	{	PUNCT
ejpam-4592	280	19	dn}n∈n	dn}n∈n	VERB
ejpam-4592	280	20	be	be	AUX
ejpam-4592	280	21	asymptotic	asymptotic	ADJ
ejpam-4592	280	22	with	with	ADP
ejpam-4592	280	23	respect	respect	NOUN
ejpam-4592	280	24	to	to	ADP
ejpam-4592	280	25	the	the	DET
ejpam-4592	280	26	metric	metric	PROPN
ejpam-4592	280	27	σ	σ	PROPN
ejpam-4592	280	28	∈	∈	PROPN
ejpam-4592	280	29	ω	ω	NOUN
ejpam-4592	280	30	.	.	PUNCT
ejpam-4592	281	1	if	if	SCONJ
ejpam-4592	281	2	{	{	PUNCT
ejpam-4592	281	3	dn	dn	NOUN
ejpam-4592	281	4	}	}	PUNCT
ejpam-4592	281	5	∈	∈	PROPN
ejpam-4592	281	6	p(σ	p(σ	NOUN
ejpam-4592	281	7	)	)	PUNCT
ejpam-4592	281	8	,	,	PUNCT
ejpam-4592	281	9	then	then	ADV
ejpam-4592	281	10	{	{	PUNCT
ejpam-4592	281	11	cn	cn	ADJ
ejpam-4592	281	12	}	}	PUNCT
ejpam-4592	281	13	∈	∈	PROPN
ejpam-4592	281	14	p(σ	p(σ	NOUN
ejpam-4592	281	15	)	)	PUNCT
ejpam-4592	281	16	.	.	PUNCT
ejpam-4592	282	1	proof	proof	NOUN
ejpam-4592	282	2	.	.	PUNCT
ejpam-4592	283	1	assume	assume	VERB
ejpam-4592	283	2	that	that	SCONJ
ejpam-4592	283	3	,	,	PUNCT
ejpam-4592	283	4	{	{	PUNCT
ejpam-4592	283	5	cn	cn	X
ejpam-4592	283	6	}	}	PUNCT
ejpam-4592	283	7	and	and	CCONJ
ejpam-4592	283	8	{	{	PUNCT
ejpam-4592	283	9	dn	dn	VERB
ejpam-4592	283	10	}	}	PUNCT
ejpam-4592	283	11	is	be	AUX
ejpam-4592	283	12	asymptotic	asymptotic	ADJ
ejpam-4592	283	13	with	with	ADP
ejpam-4592	283	14	respect	respect	NOUN
ejpam-4592	283	15	to	to	ADP
ejpam-4592	283	16	the	the	DET
ejpam-4592	283	17	metric	metric	ADJ
ejpam-4592	283	18	σ	σ	PROPN
ejpam-4592	283	19	∈	∈	PROPN
ejpam-4592	283	20	ω	ω	PROPN
ejpam-4592	283	21	and	and	CCONJ
ejpam-4592	283	22	{	{	PUNCT
ejpam-4592	283	23	dn	dn	PROPN
ejpam-4592	283	24	}	}	PUNCT
ejpam-4592	283	25	∈	∈	PROPN
ejpam-4592	283	26	p(σ	p(σ	NOUN
ejpam-4592	283	27	)	)	PUNCT
ejpam-4592	283	28	.	.	PUNCT
ejpam-4592	284	1	let	let	VERB
ejpam-4592	284	2	ε	ε	PROPN
ejpam-4592	284	3	>	>	X
ejpam-4592	284	4	0	0	PUNCT
ejpam-4592	285	1	and	and	CCONJ
ejpam-4592	285	2	m	m	PROPN
ejpam-4592	285	3	∈	∈	PROPN
ejpam-4592	285	4	n.	n.	NOUN
ejpam-4592	285	5	then	then	ADV
ejpam-4592	285	6	there	there	PRON
ejpam-4592	285	7	exists	exist	VERB
ejpam-4592	285	8	n1	n1	PROPN
ejpam-4592	285	9	∈	∈	PROPN
ejpam-4592	285	10	n	n	PRON
ejpam-4592	285	11	such	such	ADJ
ejpam-4592	285	12	that	that	DET
ejpam-4592	285	13	σ(cn	σ(cn	NOUN
ejpam-4592	285	14	,	,	PUNCT
ejpam-4592	285	15	dn	dn	NOUN
ejpam-4592	285	16	)	)	PUNCT
ejpam-4592	285	17	<	<	X
ejpam-4592	285	18	ε	ε	PROPN
ejpam-4592	285	19	3	3	NUM
ejpam-4592	285	20	for	for	ADP
ejpam-4592	285	21	all	all	DET
ejpam-4592	285	22	n	n	PRON
ejpam-4592	285	23	≥	≥	NOUN
ejpam-4592	285	24	n1	n1	NOUN
ejpam-4592	285	25	and	and	CCONJ
ejpam-4592	285	26	there	there	PRON
ejpam-4592	285	27	exist	exist	VERB
ejpam-4592	285	28	k1	k1	NOUN
ejpam-4592	285	29	,	,	PUNCT
ejpam-4592	285	30	j1	j1	PROPN
ejpam-4592	285	31	∈	∈	PROPN
ejpam-4592	285	32	n	n	CCONJ
ejpam-4592	285	33	,	,	PUNCT
ejpam-4592	285	34	k1	k1	PROPN
ejpam-4592	285	35	̸=	̸=	PROPN
ejpam-4592	285	36	j1	j1	PROPN
ejpam-4592	285	37	such	such	ADJ
ejpam-4592	285	38	that	that	SCONJ
ejpam-4592	285	39	k1	k1	NOUN
ejpam-4592	285	40	,	,	PUNCT
ejpam-4592	285	41	j1	j1	PROPN
ejpam-4592	285	42	>	>	X
ejpam-4592	285	43	m	m	PROPN
ejpam-4592	285	44	and	and	CCONJ
ejpam-4592	285	45	σ(dk1	σ(dk1	PROPN
ejpam-4592	285	46	,	,	PUNCT
ejpam-4592	285	47	dj1	dj1	PROPN
ejpam-4592	285	48	)	)	PUNCT
ejpam-4592	285	49	<	<	X
ejpam-4592	285	50	ε	ε	PROPN
ejpam-4592	285	51	3	3	NUM
ejpam-4592	285	52	.	.	PUNCT
ejpam-4592	286	1	case	case	NOUN
ejpam-4592	286	2	1	1	NUM
ejpam-4592	286	3	:	:	PUNCT
ejpam-4592	286	4	suppose	suppose	VERB
ejpam-4592	286	5	that	that	SCONJ
ejpam-4592	286	6	,	,	PUNCT
ejpam-4592	287	1	n1	n1	PROPN
ejpam-4592	287	2	<	<	X
ejpam-4592	287	3	m.	m.	NOUN
ejpam-4592	287	4	take	take	VERB
ejpam-4592	287	5	k	k	PROPN
ejpam-4592	287	6	=	=	SYM
ejpam-4592	287	7	k1	k1	PROPN
ejpam-4592	287	8	,	,	PUNCT
ejpam-4592	287	9	j	j	PROPN
ejpam-4592	287	10	=	=	SYM
ejpam-4592	287	11	j1	j1	PROPN
ejpam-4592	287	12	.	.	PUNCT
ejpam-4592	288	1	then	then	ADV
ejpam-4592	288	2	k	k	PROPN
ejpam-4592	288	3	̸=	̸=	PROPN
ejpam-4592	288	4	j	j	PROPN
ejpam-4592	288	5	and	and	CCONJ
ejpam-4592	288	6	k	k	PROPN
ejpam-4592	288	7	,	,	PUNCT
ejpam-4592	288	8	j	j	PROPN
ejpam-4592	288	9	>	>	X
ejpam-4592	288	10	m.	m.	NOUN
ejpam-4592	288	11	now	now	ADV
ejpam-4592	288	12	σ(ck	σ(ck	PROPN
ejpam-4592	288	13	,	,	PUNCT
ejpam-4592	288	14	cj	cj	NOUN
ejpam-4592	288	15	)	)	PUNCT
ejpam-4592	288	16	≤	≤	NOUN
ejpam-4592	289	1	σ(ck	σ(ck	PROPN
ejpam-4592	289	2	,	,	PUNCT
ejpam-4592	289	3	dk	dk	PROPN
ejpam-4592	289	4	)	)	PUNCT
ejpam-4592	290	1	+	+	CCONJ
ejpam-4592	290	2	σ(dk	σ(dk	X
ejpam-4592	290	3	,	,	PUNCT
ejpam-4592	290	4	dj	dj	NOUN
ejpam-4592	290	5	)	)	PUNCT
ejpam-4592	291	1	+	+	CCONJ
ejpam-4592	291	2	σ(dj	σ(dj	PROPN
ejpam-4592	291	3	,	,	PUNCT
ejpam-4592	291	4	cj	cj	X
ejpam-4592	291	5	)	)	PUNCT
ejpam-4592	291	6	<	<	X
ejpam-4592	291	7	ε	ε	PROPN
ejpam-4592	291	8	3	3	NUM
ejpam-4592	291	9	+	+	CCONJ
ejpam-4592	291	10	ε	ε	PROPN
ejpam-4592	291	11	3	3	NUM
ejpam-4592	291	12	+	+	CCONJ
ejpam-4592	291	13	ε	ε	PROPN
ejpam-4592	291	14	3	3	NUM
ejpam-4592	291	15	=	=	SYM
ejpam-4592	291	16	ε	ε	PROPN
ejpam-4592	291	17	.	.	PUNCT
ejpam-4592	291	18	thus	thus	ADV
ejpam-4592	291	19	,	,	PUNCT
ejpam-4592	291	20	σ(ck	σ(ck	PROPN
ejpam-4592	291	21	,	,	PUNCT
ejpam-4592	291	22	cj	cj	X
ejpam-4592	291	23	)	)	PUNCT
ejpam-4592	291	24	<	<	X
ejpam-4592	291	25	ε	ε	PROPN
ejpam-4592	291	26	.	.	PUNCT
ejpam-4592	291	27	case	case	NOUN
ejpam-4592	291	28	2	2	NUM
ejpam-4592	291	29	:	:	PUNCT
ejpam-4592	291	30	assume	assume	VERB
ejpam-4592	291	31	that	that	SCONJ
ejpam-4592	291	32	,	,	PUNCT
ejpam-4592	291	33	n1	n1	PROPN
ejpam-4592	291	34	>	>	X
ejpam-4592	291	35	m.	m.	NOUN
ejpam-4592	291	36	if	if	SCONJ
ejpam-4592	291	37	k1	k1	PROPN
ejpam-4592	291	38	,	,	PUNCT
ejpam-4592	291	39	j1	j1	PROPN
ejpam-4592	291	40	>	>	X
ejpam-4592	291	41	n1	n1	PROPN
ejpam-4592	291	42	,	,	PUNCT
ejpam-4592	291	43	then	then	ADV
ejpam-4592	291	44	we	we	PRON
ejpam-4592	291	45	take	take	VERB
ejpam-4592	291	46	k	k	NOUN
ejpam-4592	291	47	=	=	SYM
ejpam-4592	291	48	k1	k1	PROPN
ejpam-4592	291	49	,	,	PUNCT
ejpam-4592	291	50	j	j	PROPN
ejpam-4592	291	51	=	=	SYM
ejpam-4592	291	52	j1	j1	PROPN
ejpam-4592	291	53	.	.	PUNCT
ejpam-4592	292	1	by	by	ADP
ejpam-4592	292	2	same	same	ADJ
ejpam-4592	292	3	argument	argument	NOUN
ejpam-4592	292	4	in	in	ADP
ejpam-4592	292	5	case	case	NOUN
ejpam-4592	292	6	1	1	NUM
ejpam-4592	292	7	,	,	PUNCT
ejpam-4592	292	8	we	we	PRON
ejpam-4592	292	9	get	get	VERB
ejpam-4592	292	10	σ(ck	σ(ck	NOUN
ejpam-4592	292	11	,	,	PUNCT
ejpam-4592	292	12	cj	cj	X
ejpam-4592	292	13	)	)	PUNCT
ejpam-4592	292	14	<	<	X
ejpam-4592	292	15	ε	ε	AUX
ejpam-4592	292	16	.	.	PROPN
ejpam-4592	292	17	suppose	suppose	VERB
ejpam-4592	292	18	that	that	SCONJ
ejpam-4592	292	19	,	,	PUNCT
ejpam-4592	292	20	k1	k1	PROPN
ejpam-4592	292	21	,	,	PUNCT
ejpam-4592	292	22	j1	j1	PROPN
ejpam-4592	292	23	<	<	X
ejpam-4592	292	24	n1	n1	PROPN
ejpam-4592	292	25	.	.	PUNCT
ejpam-4592	293	1	since	since	SCONJ
ejpam-4592	293	2	{	{	PUNCT
ejpam-4592	293	3	dn	dn	ADJ
ejpam-4592	293	4	}	}	PUNCT
ejpam-4592	293	5	∈	∈	PROPN
ejpam-4592	293	6	p(σ	p(σ	NOUN
ejpam-4592	293	7	)	)	PUNCT
ejpam-4592	293	8	and	and	CCONJ
ejpam-4592	293	9	n1	n1	PROPN
ejpam-4592	293	10	∈	∈	PROPN
ejpam-4592	293	11	n	n	CCONJ
ejpam-4592	293	12	,	,	PUNCT
ejpam-4592	293	13	there	there	PRON
ejpam-4592	293	14	is	be	VERB
ejpam-4592	293	15	k2	k2	ADJ
ejpam-4592	293	16	,	,	PUNCT
ejpam-4592	293	17	j2	j2	PROPN
ejpam-4592	293	18	∈	∈	PROPN
ejpam-4592	293	19	n	n	CCONJ
ejpam-4592	293	20	,	,	PUNCT
ejpam-4592	293	21	k2	k2	PROPN
ejpam-4592	293	22	̸=	̸=	PROPN
ejpam-4592	293	23	j2	j2	PROPN
ejpam-4592	293	24	such	such	ADJ
ejpam-4592	293	25	that	that	SCONJ
ejpam-4592	293	26	k2	k2	PROPN
ejpam-4592	293	27	,	,	PUNCT
ejpam-4592	293	28	j2	j2	PROPN
ejpam-4592	293	29	>	>	X
ejpam-4592	293	30	n1	n1	PROPN
ejpam-4592	293	31	and	and	CCONJ
ejpam-4592	293	32	σ(dk2	σ(dk2	PROPN
ejpam-4592	293	33	,	,	PUNCT
ejpam-4592	293	34	dj2	dj2	PROPN
ejpam-4592	293	35	)	)	PUNCT
ejpam-4592	293	36	<	<	X
ejpam-4592	293	37	ε	ε	PROPN
ejpam-4592	293	38	3	3	NUM
ejpam-4592	293	39	.	.	PUNCT
ejpam-4592	294	1	take	take	VERB
ejpam-4592	294	2	k	k	NOUN
ejpam-4592	294	3	=	=	SYM
ejpam-4592	294	4	k2	k2	PROPN
ejpam-4592	294	5	,	,	PUNCT
ejpam-4592	294	6	j	j	PROPN
ejpam-4592	294	7	=	=	PROPN
ejpam-4592	294	8	j2	j2	PROPN
ejpam-4592	294	9	.	.	PROPN
ejpam-4592	295	1	then	then	ADV
ejpam-4592	295	2	σ(ck	σ(ck	PROPN
ejpam-4592	295	3	,	,	PUNCT
ejpam-4592	295	4	cj	cj	NOUN
ejpam-4592	295	5	)	)	PUNCT
ejpam-4592	295	6	≤	≤	NOUN
ejpam-4592	295	7	σ(ck	σ(ck	PROPN
ejpam-4592	295	8	,	,	PUNCT
ejpam-4592	295	9	dk	dk	PROPN
ejpam-4592	295	10	)	)	PUNCT
ejpam-4592	295	11	+	+	CCONJ
ejpam-4592	295	12	σ(dk	σ(dk	X
ejpam-4592	295	13	,	,	PUNCT
ejpam-4592	295	14	dj	dj	NOUN
ejpam-4592	295	15	)	)	PUNCT
ejpam-4592	296	1	+	+	CCONJ
ejpam-4592	296	2	σ(dj	σ(dj	PROPN
ejpam-4592	296	3	,	,	PUNCT
ejpam-4592	296	4	cj	cj	X
ejpam-4592	296	5	)	)	PUNCT
ejpam-4592	296	6	and	and	CCONJ
ejpam-4592	296	7	so	so	ADV
ejpam-4592	296	8	σ(ck	σ(ck	PROPN
ejpam-4592	296	9	,	,	PUNCT
ejpam-4592	296	10	cj	cj	X
ejpam-4592	296	11	)	)	PUNCT
ejpam-4592	296	12	<	<	X
ejpam-4592	296	13	ε	ε	PROPN
ejpam-4592	296	14	.	.	PUNCT
ejpam-4592	297	1	if	if	SCONJ
ejpam-4592	297	2	k1	k1	NOUN
ejpam-4592	297	3	<	<	X
ejpam-4592	297	4	n1	n1	PROPN
ejpam-4592	297	5	<	<	X
ejpam-4592	297	6	j1	j1	PROPN
ejpam-4592	297	7	,	,	PUNCT
ejpam-4592	297	8	then	then	ADV
ejpam-4592	297	9	there	there	PRON
ejpam-4592	297	10	is	be	VERB
ejpam-4592	297	11	k2	k2	ADJ
ejpam-4592	297	12	,	,	PUNCT
ejpam-4592	297	13	j2	j2	PROPN
ejpam-4592	297	14	∈	∈	PROPN
ejpam-4592	297	15	n	n	CCONJ
ejpam-4592	297	16	,	,	PUNCT
ejpam-4592	297	17	k2	k2	PROPN
ejpam-4592	297	18	̸=	̸=	PROPN
ejpam-4592	297	19	j2	j2	PROPN
ejpam-4592	297	20	such	such	ADJ
ejpam-4592	297	21	that	that	SCONJ
ejpam-4592	297	22	k2	k2	PROPN
ejpam-4592	297	23	,	,	PUNCT
ejpam-4592	297	24	j2	j2	PROPN
ejpam-4592	297	25	>	>	X
ejpam-4592	297	26	j1	j1	PROPN
ejpam-4592	297	27	and	and	CCONJ
ejpam-4592	297	28	σ(dk2	σ(dk2	PROPN
ejpam-4592	297	29	,	,	PUNCT
ejpam-4592	297	30	dj2	dj2	PROPN
ejpam-4592	297	31	)	)	PUNCT
ejpam-4592	297	32	<	<	X
ejpam-4592	297	33	ε	ε	PROPN
ejpam-4592	297	34	3	3	NUM
ejpam-4592	297	35	,	,	PUNCT
ejpam-4592	297	36	since	since	SCONJ
ejpam-4592	297	37	{	{	PUNCT
ejpam-4592	297	38	dn	dn	VERB
ejpam-4592	297	39	}	}	PUNCT
ejpam-4592	297	40	is	be	AUX
ejpam-4592	297	41	a	a	DET
ejpam-4592	297	42	pseudo	pseudo	NOUN
ejpam-4592	297	43	-	-	ADJ
ejpam-4592	297	44	cauchy	cauchy	ADJ
ejpam-4592	297	45	sequence	sequence	NOUN
ejpam-4592	297	46	and	and	CCONJ
ejpam-4592	297	47	j1	j1	PROPN
ejpam-4592	297	48	∈	∈	PROPN
ejpam-4592	297	49	n.	n.	NOUN
ejpam-4592	297	50	take	take	VERB
ejpam-4592	297	51	k	k	PROPN
ejpam-4592	297	52	=	=	SYM
ejpam-4592	297	53	k2	k2	PROPN
ejpam-4592	297	54	,	,	PUNCT
ejpam-4592	297	55	j	j	PROPN
ejpam-4592	297	56	=	=	PROPN
ejpam-4592	297	57	j2	j2	PROPN
ejpam-4592	297	58	.	.	PROPN
ejpam-4592	298	1	then	then	ADV
ejpam-4592	298	2	σ(ck	σ(ck	PROPN
ejpam-4592	298	3	,	,	PUNCT
ejpam-4592	298	4	cj	cj	NOUN
ejpam-4592	298	5	)	)	PUNCT
ejpam-4592	298	6	≤	≤	NOUN
ejpam-4592	298	7	σ(ck	σ(ck	PROPN
ejpam-4592	298	8	,	,	PUNCT
ejpam-4592	298	9	dk	dk	PROPN
ejpam-4592	298	10	)	)	PUNCT
ejpam-4592	298	11	+	+	CCONJ
ejpam-4592	298	12	σ(dk	σ(dk	X
ejpam-4592	298	13	,	,	PUNCT
ejpam-4592	298	14	dj	dj	NOUN
ejpam-4592	298	15	)	)	PUNCT
ejpam-4592	299	1	+	+	CCONJ
ejpam-4592	299	2	σ(dj	σ(dj	PROPN
ejpam-4592	299	3	,	,	PUNCT
ejpam-4592	299	4	cj	cj	X
ejpam-4592	299	5	)	)	PUNCT
ejpam-4592	299	6	and	and	CCONJ
ejpam-4592	299	7	so	so	ADV
ejpam-4592	299	8	σ(ck	σ(ck	PROPN
ejpam-4592	299	9	,	,	PUNCT
ejpam-4592	299	10	cj	cj	X
ejpam-4592	299	11	)	)	PUNCT
ejpam-4592	299	12	<	<	X
ejpam-4592	299	13	ε	ε	PROPN
ejpam-4592	299	14	.	.	PROPN
ejpam-4592	300	1	from	from	ADP
ejpam-4592	300	2	all	all	DET
ejpam-4592	300	3	the	the	DET
ejpam-4592	300	4	cases	case	NOUN
ejpam-4592	300	5	,	,	PUNCT
ejpam-4592	300	6	we	we	PRON
ejpam-4592	300	7	get	get	VERB
ejpam-4592	300	8	for	for	ADP
ejpam-4592	300	9	every	every	DET
ejpam-4592	300	10	ε	ε	PROPN
ejpam-4592	300	11	>	>	X
ejpam-4592	300	12	0	0	NUM
ejpam-4592	301	1	and	and	CCONJ
ejpam-4592	301	2	m	m	PROPN
ejpam-4592	301	3	∈	∈	PROPN
ejpam-4592	301	4	n	n	CCONJ
ejpam-4592	301	5	,	,	PUNCT
ejpam-4592	301	6	there	there	PRON
ejpam-4592	301	7	exist	exist	VERB
ejpam-4592	301	8	k	k	PROPN
ejpam-4592	301	9	,	,	PUNCT
ejpam-4592	301	10	j	j	PROPN
ejpam-4592	301	11	∈	∈	PROPN
ejpam-4592	301	12	n	n	CCONJ
ejpam-4592	301	13	,	,	PUNCT
ejpam-4592	301	14	k	k	PROPN
ejpam-4592	301	15	̸=	̸=	PROPN
ejpam-4592	301	16	j	j	PROPN
ejpam-4592	302	1	such	such	ADJ
ejpam-4592	302	2	that	that	SCONJ
ejpam-4592	302	3	k	k	PROPN
ejpam-4592	302	4	,	,	PUNCT
ejpam-4592	302	5	j	j	PROPN
ejpam-4592	302	6	>	>	X
ejpam-4592	302	7	m	m	PROPN
ejpam-4592	302	8	and	and	CCONJ
ejpam-4592	302	9	σ(xk	σ(xk	NUM
ejpam-4592	302	10	,	,	PUNCT
ejpam-4592	302	11	xj	xj	NOUN
ejpam-4592	302	12	)	)	PUNCT
ejpam-4592	302	13	<	<	X
ejpam-4592	302	14	ε	ε	PROPN
ejpam-4592	302	15	.	.	PUNCT
ejpam-4592	302	16	therefore	therefore	ADV
ejpam-4592	302	17	,	,	PUNCT
ejpam-4592	302	18	{	{	PUNCT
ejpam-4592	302	19	cn	cn	ADJ
ejpam-4592	302	20	}	}	PUNCT
ejpam-4592	302	21	∈	∈	PROPN
ejpam-4592	302	22	p(σ	p(σ	NOUN
ejpam-4592	302	23	)	)	PUNCT
ejpam-4592	302	24	.	.	PUNCT
ejpam-4592	303	1	theorem	theorem	NOUN
ejpam-4592	303	2	29	29	NUM
ejpam-4592	303	3	provides	provide	VERB
ejpam-4592	303	4	an	an	DET
ejpam-4592	303	5	easier	easy	ADJ
ejpam-4592	303	6	way	way	NOUN
ejpam-4592	303	7	to	to	PART
ejpam-4592	303	8	check	check	VERB
ejpam-4592	303	9	the	the	DET
ejpam-4592	303	10	nature	nature	NOUN
ejpam-4592	303	11	of	of	ADP
ejpam-4592	303	12	a	a	DET
ejpam-4592	303	13	given	give	VERB
ejpam-4592	303	14	pair	pair	NOUN
ejpam-4592	303	15	of	of	ADP
ejpam-4592	303	16	sequence	sequence	NOUN
ejpam-4592	303	17	using	use	VERB
ejpam-4592	303	18	the	the	DET
ejpam-4592	303	19	tool	tool	NOUN
ejpam-4592	303	20	namely	namely	ADV
ejpam-4592	303	21	,	,	PUNCT
ejpam-4592	303	22	asymptotic	asymptotic	ADJ
ejpam-4592	303	23	.	.	PUNCT
ejpam-4592	304	1	theorem	theorem	NOUN
ejpam-4592	304	2	29	29	NUM
ejpam-4592	304	3	.	.	PUNCT
ejpam-4592	305	1	let	let	VERB
ejpam-4592	305	2	(	(	PUNCT
ejpam-4592	305	3	x	x	NOUN
ejpam-4592	305	4	,	,	PUNCT
ejpam-4592	305	5	ω	ω	NUM
ejpam-4592	305	6	)	)	PUNCT
ejpam-4592	305	7	be	be	AUX
ejpam-4592	305	8	a	a	DET
ejpam-4592	305	9	generalized	generalized	ADJ
ejpam-4592	305	10	metric	metric	ADJ
ejpam-4592	305	11	space	space	NOUN
ejpam-4592	305	12	,	,	PUNCT
ejpam-4592	305	13	the	the	DET
ejpam-4592	305	14	pair	pair	NOUN
ejpam-4592	305	15	of	of	ADP
ejpam-4592	305	16	sequence	sequence	NOUN
ejpam-4592	305	17	{	{	PUNCT
ejpam-4592	305	18	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	305	19	and	and	CCONJ
ejpam-4592	305	20	{	{	PUNCT
ejpam-4592	305	21	dn}n∈n	dn}n∈n	VERB
ejpam-4592	305	22	be	be	AUX
ejpam-4592	305	23	asymptotic	asymptotic	ADJ
ejpam-4592	305	24	with	with	ADP
ejpam-4592	305	25	respect	respect	NOUN
ejpam-4592	305	26	to	to	ADP
ejpam-4592	305	27	the	the	DET
ejpam-4592	305	28	metric	metric	PROPN
ejpam-4592	305	29	σ	σ	PROPN
ejpam-4592	305	30	∈	∈	PROPN
ejpam-4592	305	31	ω	ω	NOUN
ejpam-4592	305	32	.	.	PUNCT
ejpam-4592	306	1	if	if	SCONJ
ejpam-4592	306	2	{	{	PUNCT
ejpam-4592	306	3	dn	dn	NOUN
ejpam-4592	306	4	}	}	PUNCT
ejpam-4592	306	5	∈	∈	PROPN
ejpam-4592	306	6	b(σ	b(σ	PROPN
ejpam-4592	306	7	)	)	PUNCT
ejpam-4592	306	8	,	,	PUNCT
ejpam-4592	306	9	then	then	ADV
ejpam-4592	306	10	{	{	PUNCT
ejpam-4592	306	11	cn	cn	ADJ
ejpam-4592	306	12	}	}	PUNCT
ejpam-4592	306	13	∈	∈	PROPN
ejpam-4592	306	14	b(σ	b(σ	PROPN
ejpam-4592	306	15	)	)	PUNCT
ejpam-4592	306	16	.	.	PUNCT
ejpam-4592	307	1	proof	proof	NOUN
ejpam-4592	307	2	.	.	PUNCT
ejpam-4592	308	1	let	let	VERB
ejpam-4592	308	2	ε	ε	PROPN
ejpam-4592	308	3	>	>	X
ejpam-4592	308	4	0	0	PUNCT
ejpam-4592	308	5	be	be	AUX
ejpam-4592	308	6	given	give	VERB
ejpam-4592	308	7	.	.	PUNCT
ejpam-4592	309	1	then	then	ADV
ejpam-4592	309	2	there	there	PRON
ejpam-4592	309	3	exists	exist	VERB
ejpam-4592	309	4	n1	n1	PROPN
ejpam-4592	309	5	∈	∈	PROPN
ejpam-4592	309	6	n	n	PRON
ejpam-4592	309	7	such	such	ADJ
ejpam-4592	309	8	that	that	DET
ejpam-4592	309	9	σ(cn	σ(cn	NOUN
ejpam-4592	309	10	,	,	PUNCT
ejpam-4592	309	11	dn	dn	NOUN
ejpam-4592	309	12	)	)	PUNCT
ejpam-4592	309	13	<	<	X
ejpam-4592	309	14	ε	ε	PROPN
ejpam-4592	309	15	for	for	ADP
ejpam-4592	309	16	all	all	DET
ejpam-4592	309	17	n	n	DET
ejpam-4592	309	18	≥	≥	NOUN
ejpam-4592	309	19	n1	n1	NOUN
ejpam-4592	309	20	and	and	CCONJ
ejpam-4592	309	21	there	there	ADV
ejpam-4592	309	22	exist	exist	VERB
ejpam-4592	309	23	m1	m1	NOUN
ejpam-4592	309	24	,	,	PUNCT
ejpam-4592	309	25	n1	n1	PROPN
ejpam-4592	309	26	∈	∈	PROPN
ejpam-4592	309	27	n	n	CCONJ
ejpam-4592	309	28	such	such	ADJ
ejpam-4592	309	29	that	that	SCONJ
ejpam-4592	309	30	whenever	whenever	SCONJ
ejpam-4592	309	31	n	n	PROPN
ejpam-4592	309	32	>	>	X
ejpam-4592	309	33	j	j	PROPN
ejpam-4592	309	34	≥	≥	PROPN
ejpam-4592	309	35	n1	n1	PROPN
ejpam-4592	309	36	,	,	PUNCT
ejpam-4592	309	37	the	the	DET
ejpam-4592	309	38	points	point	NOUN
ejpam-4592	309	39	dj	dj	NOUN
ejpam-4592	310	1	and	and	CCONJ
ejpam-4592	310	2	dn	dn	NOUN
ejpam-4592	310	3	can	can	AUX
ejpam-4592	310	4	be	be	AUX
ejpam-4592	310	5	joined	join	VERB
ejpam-4592	310	6	by	by	ADP
ejpam-4592	310	7	an	an	DET
ejpam-4592	310	8	ε	ε	PROPN
ejpam-4592	310	9	-	-	PUNCT
ejpam-4592	310	10	chain	chain	NOUN
ejpam-4592	310	11	of	of	ADP
ejpam-4592	310	12	length	length	NOUN
ejpam-4592	310	13	m1	m1	NOUN
ejpam-4592	310	14	.	.	PUNCT
ejpam-4592	311	1	case	case	NOUN
ejpam-4592	311	2	1	1	NUM
ejpam-4592	311	3	:	:	PUNCT
ejpam-4592	311	4	if	if	SCONJ
ejpam-4592	311	5	n1	n1	PROPN
ejpam-4592	311	6	>	>	X
ejpam-4592	311	7	n1	n1	PROPN
ejpam-4592	311	8	,	,	PUNCT
ejpam-4592	311	9	then	then	ADV
ejpam-4592	311	10	take	take	VERB
ejpam-4592	311	11	k	k	NOUN
ejpam-4592	311	12	=	=	PUNCT
ejpam-4592	311	13	m1	m1	PROPN
ejpam-4592	312	1	+	+	PROPN
ejpam-4592	312	2	2	2	NUM
ejpam-4592	312	3	and	and	CCONJ
ejpam-4592	312	4	l	l	NOUN
ejpam-4592	312	5	=	=	SYM
ejpam-4592	312	6	n1	n1	PROPN
ejpam-4592	312	7	.	.	PUNCT
ejpam-4592	313	1	then	then	ADV
ejpam-4592	313	2	k	k	X
ejpam-4592	313	3	,	,	PUNCT
ejpam-4592	313	4	l	l	PROPN
ejpam-4592	313	5	∈	∈	PROPN
ejpam-4592	313	6	n.	n.	NOUN
ejpam-4592	313	7	if	if	SCONJ
ejpam-4592	313	8	n	n	PROPN
ejpam-4592	313	9	>	>	X
ejpam-4592	314	1	i	i	PRON
ejpam-4592	314	2	≥	≥	PROPN
ejpam-4592	314	3	l	l	NOUN
ejpam-4592	314	4	,	,	PUNCT
ejpam-4592	314	5	then	then	ADV
ejpam-4592	314	6	we	we	PRON
ejpam-4592	314	7	can	can	AUX
ejpam-4592	314	8	choose	choose	VERB
ejpam-4592	314	9	elements	element	NOUN
ejpam-4592	314	10	between	between	ADP
ejpam-4592	314	11	ci	ci	PROPN
ejpam-4592	314	12	and	and	CCONJ
ejpam-4592	314	13	cn	cn	PROPN
ejpam-4592	314	14	,	,	PUNCT
ejpam-4592	314	15	namely	namely	ADV
ejpam-4592	314	16	,	,	PUNCT
ejpam-4592	314	17	ci	ci	PROPN
ejpam-4592	314	18	,	,	PUNCT
ejpam-4592	314	19	di	di	NOUN
ejpam-4592	314	20	,	,	PUNCT
ejpam-4592	314	21	di+1	di+1	PROPN
ejpam-4592	314	22	,	,	PUNCT
ejpam-4592	314	23	...	...	PUNCT
ejpam-4592	314	24	,	,	PUNCT
ejpam-4592	314	25	dn−1	dn−1	PROPN
ejpam-4592	314	26	,	,	PUNCT
ejpam-4592	314	27	dn	dn	PROPN
ejpam-4592	314	28	,	,	PUNCT
ejpam-4592	314	29	cn	cn	PROPN
ejpam-4592	314	30	.	.	PUNCT
ejpam-4592	315	1	since	since	SCONJ
ejpam-4592	315	2	n1	n1	PROPN
ejpam-4592	315	3	≤	≤	PUNCT
ejpam-4592	315	4	i	i	PRON
ejpam-4592	315	5	<	<	X
ejpam-4592	315	6	n	n	CCONJ
ejpam-4592	315	7	,	,	PUNCT
ejpam-4592	315	8	the	the	DET
ejpam-4592	315	9	points	point	NOUN
ejpam-4592	315	10	di	di	NOUN
ejpam-4592	315	11	and	and	CCONJ
ejpam-4592	315	12	dn	dn	PROPN
ejpam-4592	315	13	can	can	AUX
ejpam-4592	315	14	be	be	AUX
ejpam-4592	315	15	joined	join	VERB
ejpam-4592	315	16	by	by	ADP
ejpam-4592	315	17	an	an	DET
ejpam-4592	315	18	ε	ε	PROPN
ejpam-4592	315	19	-	-	PUNCT
ejpam-4592	315	20	chain	chain	NOUN
ejpam-4592	315	21	of	of	ADP
ejpam-4592	315	22	length	length	NOUN
ejpam-4592	315	23	m1	m1	NOUN
ejpam-4592	315	24	,	,	PUNCT
ejpam-4592	315	25	by	by	ADP
ejpam-4592	315	26	assumption	assumption	NOUN
ejpam-4592	315	27	.	.	PUNCT
ejpam-4592	316	1	thus	thus	ADV
ejpam-4592	316	2	,	,	PUNCT
ejpam-4592	316	3	σ(di	σ(di	PROPN
ejpam-4592	316	4	,	,	PUNCT
ejpam-4592	316	5	di+1	di+1	NOUN
ejpam-4592	316	6	)	)	PUNCT
ejpam-4592	316	7	<	<	X
ejpam-4592	316	8	ε	ε	PROPN
ejpam-4592	316	9	,	,	PUNCT
ejpam-4592	316	10	σ(di+1	σ(di+1	ADV
ejpam-4592	316	11	,	,	PUNCT
ejpam-4592	316	12	di+2	di+2	NUM
ejpam-4592	316	13	)	)	PUNCT
ejpam-4592	316	14	<	<	X
ejpam-4592	316	15	ε	ε	PROPN
ejpam-4592	316	16	,	,	PUNCT
ejpam-4592	316	17	...	...	PUNCT
ejpam-4592	316	18	,	,	PUNCT
ejpam-4592	316	19	σ(dn−1	σ(dn−1	ADJ
ejpam-4592	316	20	,	,	PUNCT
ejpam-4592	316	21	dn	dn	PROPN
ejpam-4592	316	22	)	)	PUNCT
ejpam-4592	316	23	<	<	X
ejpam-4592	316	24	ε	ε	PROPN
ejpam-4592	316	25	.	.	PUNCT
ejpam-4592	317	1	also	also	ADV
ejpam-4592	317	2	,	,	PUNCT
ejpam-4592	317	3	σ(cn	σ(cn	PROPN
ejpam-4592	317	4	,	,	PUNCT
ejpam-4592	317	5	dn	dn	NOUN
ejpam-4592	317	6	)	)	PUNCT
ejpam-4592	317	7	<	<	X
ejpam-4592	317	8	ε	ε	PROPN
ejpam-4592	317	9	and	and	CCONJ
ejpam-4592	317	10	σ(ci	σ(ci	PROPN
ejpam-4592	317	11	,	,	PUNCT
ejpam-4592	317	12	di	di	NOUN
ejpam-4592	317	13	)	)	PUNCT
ejpam-4592	317	14	<	<	X
ejpam-4592	317	15	ε	ε	PROPN
ejpam-4592	317	16	.	.	PUNCT
ejpam-4592	318	1	therefore	therefore	ADV
ejpam-4592	318	2	,	,	PUNCT
ejpam-4592	318	3	the	the	DET
ejpam-4592	318	4	points	point	NOUN
ejpam-4592	318	5	ci	ci	NOUN
ejpam-4592	318	6	and	and	CCONJ
ejpam-4592	318	7	cn	cn	PROPN
ejpam-4592	318	8	can	can	AUX
ejpam-4592	318	9	be	be	AUX
ejpam-4592	318	10	joined	join	VERB
ejpam-4592	318	11	by	by	ADP
ejpam-4592	318	12	an	an	DET
ejpam-4592	318	13	ε	ε	PROPN
ejpam-4592	318	14	-	-	PUNCT
ejpam-4592	318	15	chain	chain	NOUN
ejpam-4592	318	16	of	of	ADP
ejpam-4592	318	17	length	length	PROPN
ejpam-4592	318	18	k.	k.	PROPN
ejpam-4592	318	19	case	case	NOUN
ejpam-4592	318	20	2	2	NUM
ejpam-4592	318	21	:	:	PUNCT
ejpam-4592	318	22	suppose	suppose	VERB
ejpam-4592	318	23	that	that	SCONJ
ejpam-4592	318	24	n1	n1	PROPN
ejpam-4592	318	25	<	<	X
ejpam-4592	318	26	n1	n1	PROPN
ejpam-4592	318	27	.	.	PUNCT
ejpam-4592	319	1	choose	choose	VERB
ejpam-4592	319	2	k	k	X
ejpam-4592	319	3	=	=	SYM
ejpam-4592	319	4	m1	m1	PROPN
ejpam-4592	319	5	+	+	CCONJ
ejpam-4592	319	6	2	2	NUM
ejpam-4592	319	7	and	and	CCONJ
ejpam-4592	319	8	l	l	NOUN
ejpam-4592	319	9	=	=	SYM
ejpam-4592	319	10	n1	n1	PROPN
ejpam-4592	319	11	.	.	PUNCT
ejpam-4592	320	1	then	then	ADV
ejpam-4592	320	2	by	by	ADP
ejpam-4592	320	3	same	same	ADJ
ejpam-4592	320	4	argument	argument	NOUN
ejpam-4592	320	5	as	as	ADP
ejpam-4592	320	6	in	in	ADP
ejpam-4592	320	7	case	case	NOUN
ejpam-4592	320	8	1	1	NUM
ejpam-4592	320	9	,	,	PUNCT
ejpam-4592	320	10	we	we	PRON
ejpam-4592	320	11	get	get	VERB
ejpam-4592	320	12	the	the	DET
ejpam-4592	320	13	points	point	NOUN
ejpam-4592	320	14	ci	ci	NOUN
ejpam-4592	320	15	,	,	PUNCT
ejpam-4592	320	16	cn	cn	PROPN
ejpam-4592	320	17	such	such	ADJ
ejpam-4592	320	18	that	that	DET
ejpam-4592	320	19	ci	ci	PROPN
ejpam-4592	320	20	and	and	CCONJ
ejpam-4592	320	21	cn	cn	PROPN
ejpam-4592	320	22	can	can	AUX
ejpam-4592	320	23	be	be	AUX
ejpam-4592	320	24	joined	join	VERB
ejpam-4592	320	25	by	by	ADP
ejpam-4592	320	26	an	an	DET
ejpam-4592	320	27	ε	ε	PROPN
ejpam-4592	320	28	-	-	PUNCT
ejpam-4592	320	29	chain	chain	NOUN
ejpam-4592	320	30	of	of	ADP
ejpam-4592	320	31	length	length	NOUN
ejpam-4592	320	32	k.	k.	PROPN
ejpam-4592	320	33	hence	hence	ADV
ejpam-4592	320	34	in	in	ADP
ejpam-4592	320	35	both	both	DET
ejpam-4592	320	36	cases	case	NOUN
ejpam-4592	320	37	,	,	PUNCT
ejpam-4592	320	38	we	we	PRON
ejpam-4592	320	39	get	get	VERB
ejpam-4592	320	40	that	that	SCONJ
ejpam-4592	320	41	the	the	DET
ejpam-4592	320	42	sequence	sequence	NOUN
ejpam-4592	320	43	{	{	PUNCT
ejpam-4592	320	44	cn	cn	PROPN
ejpam-4592	320	45	}	}	PUNCT
ejpam-4592	320	46	∈	∈	PROPN
ejpam-4592	320	47	b(σ	b(σ	PROPN
ejpam-4592	320	48	)	)	PUNCT
ejpam-4592	320	49	.	.	PUNCT
ejpam-4592	321	1	v.	v.	ADP
ejpam-4592	321	2	subramanian	subramanian	PROPN
ejpam-4592	321	3	et	et	PROPN
ejpam-4592	321	4	al	al	PROPN
ejpam-4592	321	5	.	.	PUNCT
ejpam-4592	321	6	/	/	SYM
ejpam-4592	321	7	eur	eur	PROPN
ejpam-4592	321	8	.	.	PUNCT
ejpam-4592	322	1	j.	j.	PROPN
ejpam-4592	322	2	pure	pure	PROPN
ejpam-4592	322	3	appl	appl	PROPN
ejpam-4592	322	4	.	.	PROPN
ejpam-4592	322	5	math	math	PROPN
ejpam-4592	322	6	,	,	PUNCT
ejpam-4592	322	7	15	15	NUM
ejpam-4592	322	8	(	(	PUNCT
ejpam-4592	322	9	4	4	NUM
ejpam-4592	322	10	)	)	PUNCT
ejpam-4592	322	11	(	(	PUNCT
ejpam-4592	322	12	2022	2022	NUM
ejpam-4592	322	13	)	)	PUNCT
ejpam-4592	322	14	,	,	PUNCT
ejpam-4592	322	15	1869	1869	NUM
ejpam-4592	322	16	-	-	SYM
ejpam-4592	322	17	1886	1886	NUM
ejpam-4592	322	18	1878	1878	NUM
ejpam-4592	322	19	the	the	DET
ejpam-4592	322	20	following	follow	VERB
ejpam-4592	322	21	theorem	theorem	NOUN
ejpam-4592	322	22	30	30	NUM
ejpam-4592	322	23	is	be	AUX
ejpam-4592	322	24	reduce	reduce	VERB
ejpam-4592	322	25	the	the	DET
ejpam-4592	322	26	complexity	complexity	NOUN
ejpam-4592	322	27	for	for	ADP
ejpam-4592	322	28	finding	find	VERB
ejpam-4592	322	29	whether	whether	SCONJ
ejpam-4592	322	30	,	,	PUNCT
ejpam-4592	322	31	in	in	ADP
ejpam-4592	322	32	a	a	DET
ejpam-4592	322	33	generalized	generalized	ADJ
ejpam-4592	322	34	metric	metric	ADJ
ejpam-4592	322	35	space	space	NOUN
ejpam-4592	322	36	,	,	PUNCT
ejpam-4592	322	37	a	a	DET
ejpam-4592	322	38	given	give	VERB
ejpam-4592	322	39	pair	pair	NOUN
ejpam-4592	322	40	of	of	ADP
ejpam-4592	322	41	sequence	sequence	NOUN
ejpam-4592	322	42	is	be	AUX
ejpam-4592	322	43	uniformly	uniformly	ADV
ejpam-4592	322	44	asymptotic	asymptotic	ADJ
ejpam-4592	322	45	or	or	CCONJ
ejpam-4592	322	46	not	not	PART
ejpam-4592	322	47	using	use	VERB
ejpam-4592	322	48	asymptotic	asymptotic	ADJ
ejpam-4592	322	49	.	.	PUNCT
ejpam-4592	323	1	theorem	theorem	NOUN
ejpam-4592	323	2	30	30	NUM
ejpam-4592	323	3	.	.	PUNCT
ejpam-4592	324	1	let	let	VERB
ejpam-4592	324	2	(	(	PUNCT
ejpam-4592	324	3	x	x	NOUN
ejpam-4592	324	4	,	,	PUNCT
ejpam-4592	324	5	ω	ω	NUM
ejpam-4592	324	6	)	)	PUNCT
ejpam-4592	324	7	be	be	AUX
ejpam-4592	324	8	a	a	DET
ejpam-4592	324	9	generalized	generalized	ADJ
ejpam-4592	324	10	metric	metric	ADJ
ejpam-4592	324	11	space	space	NOUN
ejpam-4592	324	12	,	,	PUNCT
ejpam-4592	324	13	the	the	DET
ejpam-4592	324	14	pair	pair	NOUN
ejpam-4592	324	15	of	of	ADP
ejpam-4592	324	16	sequence	sequence	NOUN
ejpam-4592	324	17	{	{	PUNCT
ejpam-4592	324	18	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	324	19	and	and	CCONJ
ejpam-4592	324	20	{	{	PUNCT
ejpam-4592	324	21	dn}n∈n	dn}n∈n	VERB
ejpam-4592	324	22	be	be	AUX
ejpam-4592	324	23	asymptotic	asymptotic	ADJ
ejpam-4592	324	24	with	with	ADP
ejpam-4592	324	25	respect	respect	NOUN
ejpam-4592	324	26	to	to	ADP
ejpam-4592	324	27	σ	σ	PROPN
ejpam-4592	324	28	∈	∈	PROPN
ejpam-4592	324	29	ω	ω	NOUN
ejpam-4592	324	30	.	.	PUNCT
ejpam-4592	325	1	if	if	SCONJ
ejpam-4592	325	2	either	either	CCONJ
ejpam-4592	325	3	{	{	PUNCT
ejpam-4592	325	4	cn	cn	NOUN
ejpam-4592	325	5	}	}	PUNCT
ejpam-4592	325	6	or	or	CCONJ
ejpam-4592	325	7	{	{	PUNCT
ejpam-4592	325	8	dn	dn	VERB
ejpam-4592	325	9	}	}	PUNCT
ejpam-4592	325	10	is	be	AUX
ejpam-4592	325	11	cauchy	cauchy	ADJ
ejpam-4592	325	12	sequence	sequence	NOUN
ejpam-4592	325	13	with	with	ADP
ejpam-4592	325	14	respect	respect	NOUN
ejpam-4592	325	15	to	to	ADP
ejpam-4592	325	16	σ	σ	PROPN
ejpam-4592	325	17	∈	∈	PROPN
ejpam-4592	325	18	ω	ω	PROPN
ejpam-4592	325	19	,	,	PUNCT
ejpam-4592	325	20	then	then	ADV
ejpam-4592	325	21	the	the	DET
ejpam-4592	325	22	pair	pair	NOUN
ejpam-4592	325	23	of	of	ADP
ejpam-4592	325	24	sequence	sequence	NOUN
ejpam-4592	325	25	{	{	PUNCT
ejpam-4592	325	26	cn	cn	NOUN
ejpam-4592	325	27	}	}	PUNCT
ejpam-4592	325	28	and	and	CCONJ
ejpam-4592	325	29	{	{	PUNCT
ejpam-4592	325	30	dn	dn	VERB
ejpam-4592	325	31	}	}	PUNCT
ejpam-4592	325	32	is	be	AUX
ejpam-4592	325	33	uniformly	uniformly	ADV
ejpam-4592	325	34	asymptotic	asymptotic	ADJ
ejpam-4592	325	35	with	with	ADP
ejpam-4592	325	36	respect	respect	NOUN
ejpam-4592	325	37	to	to	ADP
ejpam-4592	325	38	σ	σ	PROPN
ejpam-4592	325	39	.	.	PUNCT
ejpam-4592	326	1	proof	proof	NOUN
ejpam-4592	326	2	.	.	PUNCT
ejpam-4592	327	1	given	give	VERB
ejpam-4592	327	2	that	that	SCONJ
ejpam-4592	327	3	the	the	DET
ejpam-4592	327	4	pair	pair	NOUN
ejpam-4592	327	5	of	of	ADP
ejpam-4592	327	6	sequence	sequence	NOUN
ejpam-4592	327	7	{	{	PUNCT
ejpam-4592	327	8	cn	cn	NOUN
ejpam-4592	327	9	}	}	PUNCT
ejpam-4592	327	10	and	and	CCONJ
ejpam-4592	327	11	{	{	PUNCT
ejpam-4592	327	12	dn	dn	VERB
ejpam-4592	327	13	}	}	PUNCT
ejpam-4592	327	14	is	be	AUX
ejpam-4592	327	15	asymptotic	asymptotic	ADJ
ejpam-4592	327	16	with	with	ADP
ejpam-4592	327	17	respect	respect	NOUN
ejpam-4592	327	18	to	to	ADP
ejpam-4592	327	19	σ	σ	PROPN
ejpam-4592	327	20	∈	∈	PROPN
ejpam-4592	327	21	ω	ω	X
ejpam-4592	327	22	.	.	PUNCT
ejpam-4592	328	1	let	let	VERB
ejpam-4592	328	2	ε	ε	PROPN
ejpam-4592	328	3	>	>	X
ejpam-4592	328	4	0	0	PROPN
ejpam-4592	328	5	.	.	PUNCT
ejpam-4592	329	1	then	then	ADV
ejpam-4592	329	2	there	there	PRON
ejpam-4592	329	3	is	be	VERB
ejpam-4592	329	4	n1	n1	PROPN
ejpam-4592	329	5	∈	∈	NOUN
ejpam-4592	329	6	n	n	PRON
ejpam-4592	329	7	such	such	ADJ
ejpam-4592	329	8	that	that	DET
ejpam-4592	329	9	σ(cn	σ(cn	NOUN
ejpam-4592	329	10	,	,	PUNCT
ejpam-4592	329	11	dn	dn	NOUN
ejpam-4592	329	12	)	)	PUNCT
ejpam-4592	329	13	<	<	X
ejpam-4592	329	14	ε	ε	PROPN
ejpam-4592	329	15	2	2	NUM
ejpam-4592	329	16	for	for	ADP
ejpam-4592	329	17	all	all	DET
ejpam-4592	329	18	n	n	PRON
ejpam-4592	329	19	≥	≥	NOUN
ejpam-4592	329	20	n1	n1	NOUN
ejpam-4592	329	21	.	.	PUNCT
ejpam-4592	330	1	if	if	SCONJ
ejpam-4592	330	2	{	{	PUNCT
ejpam-4592	330	3	cn	cn	ADJ
ejpam-4592	330	4	}	}	PUNCT
ejpam-4592	330	5	∈	∈	PROPN
ejpam-4592	330	6	c(σ	c(σ	PROPN
ejpam-4592	330	7	)	)	PUNCT
ejpam-4592	330	8	,	,	PUNCT
ejpam-4592	330	9	then	then	ADV
ejpam-4592	330	10	there	there	PRON
ejpam-4592	330	11	is	be	VERB
ejpam-4592	330	12	n0	n0	NUM
ejpam-4592	330	13	∈	∈	PROPN
ejpam-4592	330	14	n	n	CCONJ
ejpam-4592	330	15	such	such	ADJ
ejpam-4592	330	16	that	that	DET
ejpam-4592	330	17	σ(cn	σ(cn	NOUN
ejpam-4592	330	18	,	,	PUNCT
ejpam-4592	330	19	cm	cm	NOUN
ejpam-4592	330	20	)	)	PUNCT
ejpam-4592	330	21	<	<	X
ejpam-4592	330	22	ε	ε	PROPN
ejpam-4592	330	23	2	2	NUM
ejpam-4592	330	24	for	for	ADP
ejpam-4592	330	25	all	all	DET
ejpam-4592	330	26	n	n	CCONJ
ejpam-4592	330	27	,	,	PUNCT
ejpam-4592	330	28	m	m	PROPN
ejpam-4592	330	29	≥	≥	NOUN
ejpam-4592	330	30	n0	n0	NUM
ejpam-4592	330	31	.	.	PUNCT
ejpam-4592	331	1	take	take	VERB
ejpam-4592	331	2	n	n	NOUN
ejpam-4592	331	3	=	=	SYM
ejpam-4592	331	4	max{n0	max{n0	PROPN
ejpam-4592	331	5	,	,	PUNCT
ejpam-4592	331	6	n1	n1	NOUN
ejpam-4592	331	7	+	+	CCONJ
ejpam-4592	331	8	1	1	NUM
ejpam-4592	331	9	}	}	PUNCT
ejpam-4592	331	10	.	.	PUNCT
ejpam-4592	332	1	then	then	ADV
ejpam-4592	332	2	for	for	ADP
ejpam-4592	332	3	n	n	CCONJ
ejpam-4592	332	4	,	,	PUNCT
ejpam-4592	332	5	m	m	VERB
ejpam-4592	332	6	≥	≥	NOUN
ejpam-4592	332	7	n	n	CCONJ
ejpam-4592	332	8	,	,	PUNCT
ejpam-4592	332	9	σ(cm	σ(cm	PROPN
ejpam-4592	332	10	,	,	PUNCT
ejpam-4592	332	11	dn	dn	ADJ
ejpam-4592	332	12	)	)	PUNCT
ejpam-4592	332	13	≤	≤	NOUN
ejpam-4592	332	14	σ(cm	σ(cm	PROPN
ejpam-4592	332	15	,	,	PUNCT
ejpam-4592	332	16	cn	cn	PROPN
ejpam-4592	332	17	)	)	PUNCT
ejpam-4592	332	18	+	+	NUM
ejpam-4592	332	19	σ(cn	σ(cn	PROPN
ejpam-4592	332	20	,	,	PUNCT
ejpam-4592	332	21	dn	dn	NOUN
ejpam-4592	332	22	)	)	PUNCT
ejpam-4592	332	23	and	and	CCONJ
ejpam-4592	332	24	so	so	ADV
ejpam-4592	332	25	σ(cm	σ(cm	PROPN
ejpam-4592	332	26	,	,	PUNCT
ejpam-4592	332	27	dn	dn	PROPN
ejpam-4592	332	28	)	)	PUNCT
ejpam-4592	332	29	<	<	X
ejpam-4592	332	30	ε	ε	PROPN
ejpam-4592	332	31	for	for	ADP
ejpam-4592	332	32	all	all	DET
ejpam-4592	332	33	n	n	CCONJ
ejpam-4592	332	34	,	,	PUNCT
ejpam-4592	332	35	m	m	VERB
ejpam-4592	332	36	≥	≥	NOUN
ejpam-4592	332	37	n.	n.	NOUN
ejpam-4592	332	38	hence	hence	ADV
ejpam-4592	332	39	the	the	DET
ejpam-4592	332	40	pair	pair	NOUN
ejpam-4592	332	41	of	of	ADP
ejpam-4592	332	42	sequence	sequence	NOUN
ejpam-4592	332	43	{	{	PUNCT
ejpam-4592	332	44	cn	cn	NOUN
ejpam-4592	332	45	}	}	PUNCT
ejpam-4592	332	46	and	and	CCONJ
ejpam-4592	332	47	{	{	PUNCT
ejpam-4592	332	48	dn	dn	VERB
ejpam-4592	332	49	}	}	PUNCT
ejpam-4592	332	50	is	be	AUX
ejpam-4592	332	51	uniformly	uniformly	ADV
ejpam-4592	332	52	asymptotic	asymptotic	ADJ
ejpam-4592	332	53	with	with	ADP
ejpam-4592	332	54	respect	respect	NOUN
ejpam-4592	332	55	to	to	ADP
ejpam-4592	332	56	σ	σ	PROPN
ejpam-4592	332	57	.	.	PUNCT
ejpam-4592	333	1	similarly	similarly	ADV
ejpam-4592	333	2	,	,	PUNCT
ejpam-4592	333	3	we	we	PRON
ejpam-4592	333	4	can	can	AUX
ejpam-4592	333	5	prove	prove	VERB
ejpam-4592	333	6	that	that	SCONJ
ejpam-4592	333	7	the	the	DET
ejpam-4592	333	8	pair	pair	NOUN
ejpam-4592	333	9	of	of	ADP
ejpam-4592	333	10	sequence	sequence	NOUN
ejpam-4592	333	11	{	{	PUNCT
ejpam-4592	333	12	cn	cn	NOUN
ejpam-4592	333	13	}	}	PUNCT
ejpam-4592	333	14	and	and	CCONJ
ejpam-4592	333	15	{	{	PUNCT
ejpam-4592	333	16	dn	dn	VERB
ejpam-4592	333	17	}	}	PUNCT
ejpam-4592	333	18	is	be	AUX
ejpam-4592	333	19	uniformly	uniformly	ADV
ejpam-4592	333	20	asymptotic	asymptotic	ADJ
ejpam-4592	333	21	if	if	SCONJ
ejpam-4592	333	22	{	{	PUNCT
ejpam-4592	333	23	dn	dn	NOUN
ejpam-4592	333	24	}	}	PUNCT
ejpam-4592	333	25	∈	∈	PROPN
ejpam-4592	333	26	c(σ	c(σ	PROPN
ejpam-4592	333	27	)	)	PUNCT
ejpam-4592	333	28	.	.	PUNCT
ejpam-4592	334	1	theorem	theorem	NOUN
ejpam-4592	334	2	31	31	NUM
ejpam-4592	334	3	provides	provide	VERB
ejpam-4592	334	4	the	the	DET
ejpam-4592	334	5	necessary	necessary	ADJ
ejpam-4592	334	6	condition	condition	NOUN
ejpam-4592	334	7	for	for	ADP
ejpam-4592	334	8	a	a	DET
ejpam-4592	334	9	given	give	VERB
ejpam-4592	334	10	pair	pair	NOUN
ejpam-4592	334	11	of	of	ADP
ejpam-4592	334	12	a	a	DET
ejpam-4592	334	13	convergent	convergent	NOUN
ejpam-4592	334	14	sequence	sequence	NOUN
ejpam-4592	334	15	is	be	AUX
ejpam-4592	334	16	asymptotic	asymptotic	ADJ
ejpam-4592	334	17	in	in	ADP
ejpam-4592	334	18	a	a	DET
ejpam-4592	334	19	generalized	generalize	VERB
ejpam-4592	334	20	metric	metric	ADJ
ejpam-4592	334	21	space	space	NOUN
ejpam-4592	334	22	.	.	PUNCT
ejpam-4592	335	1	theorem	theorem	NOUN
ejpam-4592	335	2	31	31	NUM
ejpam-4592	335	3	.	.	PUNCT
ejpam-4592	336	1	let	let	AUX
ejpam-4592	336	2	(	(	PUNCT
ejpam-4592	336	3	x	x	NOUN
ejpam-4592	336	4	,	,	PUNCT
ejpam-4592	336	5	ω	ω	NUM
ejpam-4592	336	6	)	)	PUNCT
ejpam-4592	336	7	be	be	AUX
ejpam-4592	336	8	a	a	DET
ejpam-4592	336	9	generalized	generalized	ADJ
ejpam-4592	336	10	metric	metric	ADJ
ejpam-4592	336	11	space	space	NOUN
ejpam-4592	336	12	and	and	CCONJ
ejpam-4592	336	13	{	{	PUNCT
ejpam-4592	336	14	cn	cn	PROPN
ejpam-4592	336	15	}	}	PUNCT
ejpam-4592	336	16	,	,	PUNCT
ejpam-4592	336	17	{	{	PUNCT
ejpam-4592	336	18	dn	dn	PART
ejpam-4592	336	19	}	}	PUNCT
ejpam-4592	336	20	be	be	AUX
ejpam-4592	336	21	convergent	convergent	ADJ
ejpam-4592	336	22	sequences	sequence	NOUN
ejpam-4592	336	23	with	with	ADP
ejpam-4592	336	24	respect	respect	NOUN
ejpam-4592	336	25	to	to	ADP
ejpam-4592	336	26	σ	σ	PROPN
ejpam-4592	336	27	∈	∈	PROPN
ejpam-4592	336	28	ω	ω	NOUN
ejpam-4592	336	29	.	.	PUNCT
ejpam-4592	337	1	if	if	SCONJ
ejpam-4592	337	2	they	they	PRON
ejpam-4592	337	3	have	have	VERB
ejpam-4592	337	4	the	the	DET
ejpam-4592	337	5	limit	limit	NOUN
ejpam-4592	337	6	points	point	NOUN
ejpam-4592	337	7	joined	join	VERB
ejpam-4592	337	8	by	by	ADP
ejpam-4592	337	9	an	an	DET
ejpam-4592	337	10	ε	ε	PROPN
ejpam-4592	337	11	-	-	PUNCT
ejpam-4592	337	12	chain	chain	NOUN
ejpam-4592	337	13	of	of	ADP
ejpam-4592	337	14	length	length	NOUN
ejpam-4592	337	15	1	1	NUM
ejpam-4592	337	16	for	for	ADP
ejpam-4592	337	17	every	every	DET
ejpam-4592	337	18	ε	ε	PROPN
ejpam-4592	337	19	>	>	X
ejpam-4592	337	20	0	0	PROPN
ejpam-4592	337	21	,	,	PUNCT
ejpam-4592	337	22	then	then	ADV
ejpam-4592	337	23	the	the	DET
ejpam-4592	337	24	pair	pair	NOUN
ejpam-4592	337	25	of	of	ADP
ejpam-4592	337	26	sequence	sequence	NOUN
ejpam-4592	337	27	is	be	AUX
ejpam-4592	337	28	asymptotic	asymptotic	ADJ
ejpam-4592	337	29	with	with	ADP
ejpam-4592	337	30	the	the	DET
ejpam-4592	337	31	same	same	ADJ
ejpam-4592	337	32	metric	metric	NOUN
ejpam-4592	337	33	.	.	PUNCT
ejpam-4592	338	1	proof	proof	NOUN
ejpam-4592	338	2	.	.	PUNCT
ejpam-4592	339	1	let	let	VERB
ejpam-4592	339	2	c	c	X
ejpam-4592	339	3	,	,	PUNCT
ejpam-4592	339	4	d	d	PROPN
ejpam-4592	339	5	∈	∈	PROPN
ejpam-4592	339	6	x	x	VERB
ejpam-4592	339	7	be	be	AUX
ejpam-4592	339	8	the	the	DET
ejpam-4592	339	9	limit	limit	NOUN
ejpam-4592	339	10	points	point	NOUN
ejpam-4592	339	11	of	of	ADP
ejpam-4592	339	12	the	the	DET
ejpam-4592	339	13	sequences	sequence	NOUN
ejpam-4592	339	14	{	{	PUNCT
ejpam-4592	339	15	cn	cn	NOUN
ejpam-4592	339	16	}	}	PUNCT
ejpam-4592	339	17	and	and	CCONJ
ejpam-4592	339	18	{	{	PUNCT
ejpam-4592	339	19	dn	dn	NOUN
ejpam-4592	339	20	}	}	PUNCT
ejpam-4592	339	21	,	,	PUNCT
ejpam-4592	339	22	respectively	respectively	ADV
ejpam-4592	339	23	with	with	ADP
ejpam-4592	339	24	respect	respect	NOUN
ejpam-4592	339	25	to	to	ADP
ejpam-4592	339	26	σ	σ	PROPN
ejpam-4592	339	27	.	.	PUNCT
ejpam-4592	340	1	let	let	VERB
ejpam-4592	340	2	ε	ε	PROPN
ejpam-4592	340	3	>	>	X
ejpam-4592	340	4	0	0	PUNCT
ejpam-4592	340	5	be	be	AUX
ejpam-4592	340	6	given	give	VERB
ejpam-4592	340	7	.	.	PUNCT
ejpam-4592	341	1	then	then	ADV
ejpam-4592	341	2	there	there	PRON
ejpam-4592	341	3	exists	exist	VERB
ejpam-4592	341	4	n0	n0	NUM
ejpam-4592	341	5	,	,	PUNCT
ejpam-4592	341	6	n1	n1	PROPN
ejpam-4592	341	7	∈	∈	PROPN
ejpam-4592	341	8	n	n	CCONJ
ejpam-4592	341	9	such	such	ADJ
ejpam-4592	341	10	that	that	DET
ejpam-4592	341	11	σ(cn	σ(cn	NOUN
ejpam-4592	341	12	,	,	PUNCT
ejpam-4592	341	13	c	c	NOUN
ejpam-4592	341	14	)	)	PUNCT
ejpam-4592	341	15	<	<	X
ejpam-4592	341	16	ε	ε	PROPN
ejpam-4592	341	17	3	3	NUM
ejpam-4592	341	18	for	for	ADP
ejpam-4592	341	19	every	every	DET
ejpam-4592	341	20	n	n	PRON
ejpam-4592	341	21	≥	≥	NOUN
ejpam-4592	341	22	n0	n0	NUM
ejpam-4592	341	23	and	and	CCONJ
ejpam-4592	341	24	σ(dn	σ(dn	NOUN
ejpam-4592	341	25	,	,	PUNCT
ejpam-4592	341	26	d	d	NOUN
ejpam-4592	341	27	)	)	PUNCT
ejpam-4592	341	28	<	<	X
ejpam-4592	341	29	ε	ε	PROPN
ejpam-4592	341	30	3	3	NUM
ejpam-4592	341	31	for	for	ADP
ejpam-4592	341	32	every	every	DET
ejpam-4592	341	33	n	n	PRON
ejpam-4592	341	34	≥	≥	NOUN
ejpam-4592	341	35	n1	n1	NOUN
ejpam-4592	341	36	.	.	PUNCT
ejpam-4592	342	1	take	take	VERB
ejpam-4592	342	2	m	m	NOUN
ejpam-4592	342	3	=	=	SYM
ejpam-4592	342	4	max{n0	max{n0	PROPN
ejpam-4592	342	5	,	,	PUNCT
ejpam-4592	342	6	n1	n1	NOUN
ejpam-4592	342	7	}	}	PUNCT
ejpam-4592	342	8	.	.	PUNCT
ejpam-4592	343	1	for	for	ADP
ejpam-4592	343	2	n	n	PRON
ejpam-4592	343	3	≥	≥	NOUN
ejpam-4592	343	4	m	m	PROPN
ejpam-4592	343	5	,	,	PUNCT
ejpam-4592	343	6	σ(cn	σ(cn	PROPN
ejpam-4592	343	7	,	,	PUNCT
ejpam-4592	343	8	dn	dn	NOUN
ejpam-4592	343	9	)	)	PUNCT
ejpam-4592	343	10	≤	≤	NOUN
ejpam-4592	343	11	σ(cn	σ(cn	NOUN
ejpam-4592	343	12	,	,	PUNCT
ejpam-4592	343	13	c	c	NOUN
ejpam-4592	343	14	)	)	PUNCT
ejpam-4592	344	1	+	+	CCONJ
ejpam-4592	344	2	σ(c	σ(c	ADJ
ejpam-4592	344	3	,	,	PUNCT
ejpam-4592	344	4	d	d	NOUN
ejpam-4592	344	5	)	)	PUNCT
ejpam-4592	344	6	+	+	CCONJ
ejpam-4592	344	7	σ(y	σ(y	NOUN
ejpam-4592	344	8	,	,	PUNCT
ejpam-4592	344	9	dn	dn	NOUN
ejpam-4592	344	10	)	)	PUNCT
ejpam-4592	344	11	and	and	CCONJ
ejpam-4592	344	12	so	so	ADV
ejpam-4592	344	13	σ(cn	σ(cn	PROPN
ejpam-4592	344	14	,	,	PUNCT
ejpam-4592	344	15	dn	dn	NOUN
ejpam-4592	344	16	)	)	PUNCT
ejpam-4592	344	17	<	<	X
ejpam-4592	344	18	ε	ε	PROPN
ejpam-4592	344	19	3	3	NUM
ejpam-4592	344	20	+	+	CCONJ
ejpam-4592	344	21	σ(c	σ(c	ADJ
ejpam-4592	344	22	,	,	PUNCT
ejpam-4592	344	23	d	d	NOUN
ejpam-4592	344	24	)	)	PUNCT
ejpam-4592	345	1	+	+	CCONJ
ejpam-4592	345	2	ε	ε	PROPN
ejpam-4592	345	3	3	3	NUM
ejpam-4592	345	4	.	.	PUNCT
ejpam-4592	346	1	by	by	ADP
ejpam-4592	346	2	assumption	assumption	NOUN
ejpam-4592	346	3	,	,	PUNCT
ejpam-4592	346	4	the	the	DET
ejpam-4592	346	5	points	point	NOUN
ejpam-4592	346	6	c	c	NOUN
ejpam-4592	346	7	and	and	CCONJ
ejpam-4592	346	8	d	d	NOUN
ejpam-4592	346	9	can	can	AUX
ejpam-4592	346	10	be	be	AUX
ejpam-4592	346	11	joined	join	VERB
ejpam-4592	346	12	by	by	ADP
ejpam-4592	346	13	an	an	DET
ejpam-4592	346	14	ε	ε	PROPN
ejpam-4592	346	15	3	3	NUM
ejpam-4592	346	16	-chain	-chain	NOUN
ejpam-4592	346	17	of	of	ADP
ejpam-4592	346	18	length	length	NOUN
ejpam-4592	346	19	1	1	NUM
ejpam-4592	346	20	.	.	PUNCT
ejpam-4592	347	1	thus	thus	ADV
ejpam-4592	347	2	,	,	PUNCT
ejpam-4592	347	3	σ(c	σ(c	PROPN
ejpam-4592	347	4	,	,	PUNCT
ejpam-4592	347	5	d	d	NOUN
ejpam-4592	347	6	)	)	PUNCT
ejpam-4592	347	7	<	<	X
ejpam-4592	347	8	ε	ε	PROPN
ejpam-4592	347	9	3	3	NUM
ejpam-4592	347	10	and	and	CCONJ
ejpam-4592	347	11	hence	hence	ADV
ejpam-4592	347	12	σ(cn	σ(cn	NUM
ejpam-4592	347	13	,	,	PUNCT
ejpam-4592	347	14	dn	dn	NOUN
ejpam-4592	347	15	)	)	PUNCT
ejpam-4592	347	16	<	<	X
ejpam-4592	347	17	ε	ε	PROPN
ejpam-4592	347	18	for	for	ADP
ejpam-4592	347	19	n	n	NUM
ejpam-4592	347	20	≥	≥	NOUN
ejpam-4592	347	21	m.	m.	NOUN
ejpam-4592	347	22	hence	hence	ADV
ejpam-4592	347	23	the	the	DET
ejpam-4592	347	24	pair	pair	NOUN
ejpam-4592	347	25	of	of	ADP
ejpam-4592	347	26	sequence	sequence	NOUN
ejpam-4592	347	27	{	{	PUNCT
ejpam-4592	347	28	cn	cn	NOUN
ejpam-4592	347	29	}	}	PUNCT
ejpam-4592	347	30	and	and	CCONJ
ejpam-4592	347	31	{	{	PUNCT
ejpam-4592	347	32	dn	dn	VERB
ejpam-4592	347	33	}	}	PUNCT
ejpam-4592	347	34	is	be	AUX
ejpam-4592	347	35	asymptotic	asymptotic	ADJ
ejpam-4592	347	36	with	with	ADP
ejpam-4592	347	37	respect	respect	NOUN
ejpam-4592	347	38	to	to	ADP
ejpam-4592	347	39	the	the	DET
ejpam-4592	347	40	metric	metric	PROPN
ejpam-4592	347	41	σ	σ	PROPN
ejpam-4592	347	42	.	.	PUNCT
ejpam-4592	348	1	the	the	DET
ejpam-4592	348	2	following	following	ADJ
ejpam-4592	348	3	example	example	NOUN
ejpam-4592	348	4	32	32	NUM
ejpam-4592	348	5	shows	show	VERB
ejpam-4592	348	6	that	that	SCONJ
ejpam-4592	348	7	the	the	DET
ejpam-4592	348	8	necessary	necessary	ADJ
ejpam-4592	348	9	condition	condition	NOUN
ejpam-4592	348	10	in	in	ADP
ejpam-4592	348	11	theorem	theorem	NOUN
ejpam-4592	348	12	31	31	NUM
ejpam-4592	348	13	can	can	AUX
ejpam-4592	348	14	not	not	PART
ejpam-4592	348	15	be	be	AUX
ejpam-4592	348	16	dropped	drop	VERB
ejpam-4592	348	17	.	.	PUNCT
ejpam-4592	348	18	example	example	NOUN
ejpam-4592	349	1	32	32	NUM
ejpam-4592	349	2	.	.	PUNCT
ejpam-4592	350	1	consider	consider	VERB
ejpam-4592	350	2	the	the	DET
ejpam-4592	350	3	generalized	generalized	ADJ
ejpam-4592	350	4	metric	metric	ADJ
ejpam-4592	350	5	space	space	NOUN
ejpam-4592	350	6	(	(	PUNCT
ejpam-4592	350	7	x	x	NOUN
ejpam-4592	350	8	,	,	PUNCT
ejpam-4592	350	9	ω1	ω1	PROPN
ejpam-4592	350	10	)	)	PUNCT
ejpam-4592	350	11	where	where	SCONJ
ejpam-4592	350	12	x	x	SYM
ejpam-4592	350	13	=	=	NOUN
ejpam-4592	350	14	n	n	NOUN
ejpam-4592	350	15	∪	∪	X
ejpam-4592	350	16	{	{	PUNCT
ejpam-4592	350	17	0},ω1	0},ω1	NOUN
ejpam-4592	350	18	=	=	SYM
ejpam-4592	350	19	{	{	PUNCT
ejpam-4592	350	20	σ1	σ1	PROPN
ejpam-4592	350	21	,	,	PUNCT
ejpam-4592	350	22	σ2	σ2	PROPN
ejpam-4592	350	23	,	,	PUNCT
ejpam-4592	350	24	σ3	σ3	PROPN
ejpam-4592	350	25	}	}	PUNCT
ejpam-4592	350	26	⊂	⊂	PROPN
ejpam-4592	350	27	ωx	ωx	PROPN
ejpam-4592	350	28	and	and	CCONJ
ejpam-4592	350	29	the	the	DET
ejpam-4592	350	30	metrics	metric	NOUN
ejpam-4592	350	31	are	be	AUX
ejpam-4592	350	32	defined	define	VERB
ejpam-4592	350	33	by	by	ADP
ejpam-4592	350	34	σ1(c	σ1(c	PRON
ejpam-4592	350	35	,	,	PUNCT
ejpam-4592	350	36	d	d	NOUN
ejpam-4592	350	37	)	)	PUNCT
ejpam-4592	350	38	=	=	SYM
ejpam-4592	350	39	|c−d|;σ2(c	|c−d|;σ2(c	PROPN
ejpam-4592	350	40	,	,	PUNCT
ejpam-4592	350	41	d	d	X
ejpam-4592	350	42	)	)	PUNCT
ejpam-4592	351	1	=	=	SYM
ejpam-4592	351	2	min{σ1(c	min{σ1(c	PROPN
ejpam-4592	351	3	,	,	PUNCT
ejpam-4592	351	4	d	d	NOUN
ejpam-4592	351	5	)	)	PUNCT
ejpam-4592	351	6	,	,	PUNCT
ejpam-4592	351	7	1	1	X
ejpam-4592	351	8	}	}	PUNCT
ejpam-4592	351	9	and	and	CCONJ
ejpam-4592	351	10	σ3(c	σ3(c	PRON
ejpam-4592	351	11	,	,	PUNCT
ejpam-4592	351	12	d	d	NOUN
ejpam-4592	351	13	)	)	PUNCT
ejpam-4592	351	14	=	=	SYM
ejpam-4592	352	1	σ1(c	σ1(c	PROPN
ejpam-4592	352	2	,	,	PUNCT
ejpam-4592	352	3	d	d	NOUN
ejpam-4592	352	4	)	)	PUNCT
ejpam-4592	352	5	1+σ1(c	1+σ1(c	NOUN
ejpam-4592	352	6	,	,	PUNCT
ejpam-4592	352	7	d	d	NOUN
ejpam-4592	352	8	)	)	PUNCT
ejpam-4592	352	9	for	for	ADP
ejpam-4592	352	10	all	all	DET
ejpam-4592	352	11	c	c	NOUN
ejpam-4592	352	12	,	,	PUNCT
ejpam-4592	352	13	d	d	PROPN
ejpam-4592	352	14	∈	∈	PROPN
ejpam-4592	352	15	x.	x.	NOUN
ejpam-4592	352	16	define	define	VERB
ejpam-4592	352	17	the	the	DET
ejpam-4592	352	18	sequence	sequence	NOUN
ejpam-4592	352	19	{	{	PUNCT
ejpam-4592	352	20	cn	cn	NOUN
ejpam-4592	352	21	}	}	PUNCT
ejpam-4592	352	22	and	and	CCONJ
ejpam-4592	352	23	{	{	PUNCT
ejpam-4592	352	24	dn	dn	VERB
ejpam-4592	352	25	}	}	PUNCT
ejpam-4592	352	26	by	by	ADP
ejpam-4592	352	27	{	{	PUNCT
ejpam-4592	352	28	cn	cn	PROPN
ejpam-4592	352	29	}	}	PUNCT
ejpam-4592	352	30	=	=	SYM
ejpam-4592	352	31	{	{	PUNCT
ejpam-4592	352	32	1	1	NUM
ejpam-4592	352	33	n	n	CCONJ
ejpam-4592	352	34	}	}	PUNCT
ejpam-4592	352	35	and	and	CCONJ
ejpam-4592	352	36	{	{	PUNCT
ejpam-4592	352	37	dn	dn	NOUN
ejpam-4592	352	38	}	}	PUNCT
ejpam-4592	352	39	=	=	PUNCT
ejpam-4592	352	40	{	{	PUNCT
ejpam-4592	352	41	1	1	NUM
ejpam-4592	352	42	+	+	SYM
ejpam-4592	352	43	1	1	NUM
ejpam-4592	352	44	n	n	CCONJ
ejpam-4592	352	45	}	}	PUNCT
ejpam-4592	352	46	.	.	PUNCT
ejpam-4592	353	1	then	then	ADV
ejpam-4592	353	2	{	{	PUNCT
ejpam-4592	353	3	cn	cn	ADJ
ejpam-4592	353	4	}	}	PUNCT
ejpam-4592	353	5	converges	converge	VERB
ejpam-4592	353	6	to	to	ADP
ejpam-4592	353	7	0	0	NUM
ejpam-4592	353	8	and	and	CCONJ
ejpam-4592	353	9	{	{	PUNCT
ejpam-4592	353	10	dn	dn	NOUN
ejpam-4592	353	11	}	}	PUNCT
ejpam-4592	353	12	converges	converge	NOUN
ejpam-4592	353	13	to	to	ADP
ejpam-4592	353	14	1	1	NUM
ejpam-4592	353	15	with	with	ADP
ejpam-4592	353	16	respect	respect	NOUN
ejpam-4592	353	17	to	to	ADP
ejpam-4592	353	18	σ1	σ1	PROPN
ejpam-4592	353	19	.	.	PUNCT
ejpam-4592	354	1	here	here	ADV
ejpam-4592	354	2	the	the	DET
ejpam-4592	354	3	points	point	NOUN
ejpam-4592	354	4	0	0	NUM
ejpam-4592	354	5	and	and	CCONJ
ejpam-4592	354	6	1	1	NUM
ejpam-4592	354	7	can	can	AUX
ejpam-4592	354	8	not	not	PART
ejpam-4592	354	9	be	be	AUX
ejpam-4592	354	10	joined	join	VERB
ejpam-4592	354	11	by	by	ADP
ejpam-4592	354	12	an	an	DET
ejpam-4592	354	13	ε	ε	PROPN
ejpam-4592	354	14	-	-	PUNCT
ejpam-4592	354	15	chain	chain	NOUN
ejpam-4592	354	16	of	of	ADP
ejpam-4592	354	17	length	length	NOUN
ejpam-4592	354	18	m	m	PROPN
ejpam-4592	354	19	where	where	SCONJ
ejpam-4592	354	20	ε	ε	PROPN
ejpam-4592	354	21	>	>	X
ejpam-4592	354	22	0,m	0,m	PROPN
ejpam-4592	354	23	is	be	AUX
ejpam-4592	354	24	a	a	DET
ejpam-4592	354	25	positive	positive	ADJ
ejpam-4592	354	26	integer	integer	NOUN
ejpam-4592	354	27	.	.	PUNCT
ejpam-4592	355	1	also	also	ADV
ejpam-4592	355	2	,	,	PUNCT
ejpam-4592	355	3	for	for	ADP
ejpam-4592	355	4	every	every	DET
ejpam-4592	355	5	ε	ε	PROPN
ejpam-4592	355	6	>	>	X
ejpam-4592	355	7	0	0	PROPN
ejpam-4592	355	8	,	,	PUNCT
ejpam-4592	355	9	there	there	PRON
ejpam-4592	355	10	is	be	VERB
ejpam-4592	355	11	no	no	DET
ejpam-4592	355	12	n0	n0	ADJ
ejpam-4592	355	13	∈	∈	PROPN
ejpam-4592	355	14	n	n	CCONJ
ejpam-4592	355	15	such	such	ADJ
ejpam-4592	355	16	that	that	DET
ejpam-4592	355	17	σ1(cn	σ1(cn	PROPN
ejpam-4592	355	18	,	,	PUNCT
ejpam-4592	355	19	dn	dn	NOUN
ejpam-4592	355	20	)	)	PUNCT
ejpam-4592	355	21	<	<	X
ejpam-4592	355	22	ε	ε	PROPN
ejpam-4592	355	23	for	for	ADP
ejpam-4592	355	24	all	all	PRON
ejpam-4592	355	25	n	n	PRON
ejpam-4592	355	26	>	>	X
ejpam-4592	355	27	n0	n0	X
ejpam-4592	355	28	theorem	theorem	ADJ
ejpam-4592	355	29	33	33	NUM
ejpam-4592	355	30	.	.	PUNCT
ejpam-4592	356	1	let	let	VERB
ejpam-4592	356	2	(	(	PUNCT
ejpam-4592	356	3	x	x	NOUN
ejpam-4592	356	4	,	,	PUNCT
ejpam-4592	356	5	ω	ω	NUM
ejpam-4592	356	6	)	)	PUNCT
ejpam-4592	356	7	be	be	AUX
ejpam-4592	356	8	a	a	DET
ejpam-4592	356	9	generalized	generalized	ADJ
ejpam-4592	356	10	metric	metric	ADJ
ejpam-4592	356	11	space	space	NOUN
ejpam-4592	356	12	.	.	PUNCT
ejpam-4592	357	1	then	then	ADV
ejpam-4592	357	2	every	every	DET
ejpam-4592	357	3	convergent	convergent	NOUN
ejpam-4592	357	4	sequence	sequence	NOUN
ejpam-4592	357	5	and	and	CCONJ
ejpam-4592	357	6	its	its	PRON
ejpam-4592	357	7	convergent	convergent	NOUN
ejpam-4592	357	8	subsequence	subsequence	NOUN
ejpam-4592	357	9	are	be	AUX
ejpam-4592	357	10	asymptotic	asymptotic	ADJ
ejpam-4592	357	11	with	with	ADP
ejpam-4592	357	12	the	the	DET
ejpam-4592	357	13	same	same	ADJ
ejpam-4592	357	14	metric	metric	NOUN
ejpam-4592	357	15	.	.	PUNCT
ejpam-4592	358	1	proof	proof	NOUN
ejpam-4592	358	2	.	.	PUNCT
ejpam-4592	359	1	let	let	VERB
ejpam-4592	359	2	{	{	PUNCT
ejpam-4592	359	3	cn	cn	AUX
ejpam-4592	359	4	}	}	PUNCT
ejpam-4592	359	5	be	be	AUX
ejpam-4592	359	6	a	a	DET
ejpam-4592	359	7	convergent	convergent	NOUN
ejpam-4592	359	8	sequence	sequence	NOUN
ejpam-4592	359	9	with	with	ADP
ejpam-4592	359	10	respect	respect	NOUN
ejpam-4592	359	11	to	to	ADP
ejpam-4592	359	12	the	the	DET
ejpam-4592	359	13	metric	metric	ADJ
ejpam-4592	359	14	σ	σ	PROPN
ejpam-4592	359	15	∈	∈	PROPN
ejpam-4592	359	16	ω	ω	PROPN
ejpam-4592	359	17	and	and	CCONJ
ejpam-4592	359	18	{	{	PUNCT
ejpam-4592	359	19	zn	zn	AUX
ejpam-4592	359	20	}	}	PUNCT
ejpam-4592	359	21	be	be	AUX
ejpam-4592	359	22	a	a	DET
ejpam-4592	359	23	convergent	convergent	NOUN
ejpam-4592	359	24	subsequence	subsequence	NOUN
ejpam-4592	359	25	of	of	ADP
ejpam-4592	359	26	{	{	PUNCT
ejpam-4592	359	27	cn	cn	PROPN
ejpam-4592	359	28	}	}	PUNCT
ejpam-4592	359	29	with	with	ADP
ejpam-4592	359	30	the	the	DET
ejpam-4592	359	31	metric	metric	PROPN
ejpam-4592	359	32	σ	σ	PROPN
ejpam-4592	359	33	.	.	PUNCT
ejpam-4592	360	1	let	let	VERB
ejpam-4592	360	2	x	x	PRON
ejpam-4592	360	3	be	be	AUX
ejpam-4592	360	4	a	a	DET
ejpam-4592	360	5	limit	limit	NOUN
ejpam-4592	360	6	point	point	NOUN
ejpam-4592	360	7	of	of	ADP
ejpam-4592	360	8	the	the	DET
ejpam-4592	360	9	sequence	sequence	NOUN
ejpam-4592	360	10	v.	v.	ADP
ejpam-4592	360	11	subramanian	subramanian	PROPN
ejpam-4592	360	12	et	et	PROPN
ejpam-4592	360	13	al	al	PROPN
ejpam-4592	360	14	.	.	PUNCT
ejpam-4592	360	15	/	/	SYM
ejpam-4592	360	16	eur	eur	PROPN
ejpam-4592	360	17	.	.	PUNCT
ejpam-4592	361	1	j.	j.	PROPN
ejpam-4592	361	2	pure	pure	PROPN
ejpam-4592	361	3	appl	appl	PROPN
ejpam-4592	361	4	.	.	PROPN
ejpam-4592	361	5	math	math	PROPN
ejpam-4592	361	6	,	,	PUNCT
ejpam-4592	361	7	15	15	NUM
ejpam-4592	361	8	(	(	PUNCT
ejpam-4592	361	9	4	4	NUM
ejpam-4592	361	10	)	)	PUNCT
ejpam-4592	361	11	(	(	PUNCT
ejpam-4592	361	12	2022	2022	NUM
ejpam-4592	361	13	)	)	PUNCT
ejpam-4592	361	14	,	,	PUNCT
ejpam-4592	361	15	1869	1869	NUM
ejpam-4592	361	16	-	-	SYM
ejpam-4592	361	17	1886	1886	NUM
ejpam-4592	361	18	1879	1879	NUM
ejpam-4592	361	19	{	{	PUNCT
ejpam-4592	361	20	cn	cn	NOUN
ejpam-4592	361	21	}	}	PUNCT
ejpam-4592	361	22	.	.	PUNCT
ejpam-4592	362	1	then	then	ADV
ejpam-4592	362	2	x	x	X
ejpam-4592	362	3	is	be	AUX
ejpam-4592	362	4	also	also	ADV
ejpam-4592	362	5	a	a	DET
ejpam-4592	362	6	limit	limit	NOUN
ejpam-4592	362	7	point	point	NOUN
ejpam-4592	362	8	of	of	ADP
ejpam-4592	362	9	the	the	DET
ejpam-4592	362	10	subsequence	subsequence	NOUN
ejpam-4592	362	11	{	{	PUNCT
ejpam-4592	362	12	zn	zn	NOUN
ejpam-4592	362	13	}	}	PUNCT
ejpam-4592	362	14	.	.	PUNCT
ejpam-4592	363	1	let	let	VERB
ejpam-4592	363	2	ε	ε	PROPN
ejpam-4592	363	3	>	>	X
ejpam-4592	363	4	0	0	PUNCT
ejpam-4592	363	5	be	be	AUX
ejpam-4592	363	6	given	give	VERB
ejpam-4592	363	7	.	.	PUNCT
ejpam-4592	364	1	then	then	ADV
ejpam-4592	364	2	there	there	PRON
ejpam-4592	364	3	exists	exist	VERB
ejpam-4592	364	4	n0	n0	NUM
ejpam-4592	364	5	,	,	PUNCT
ejpam-4592	364	6	n1	n1	PROPN
ejpam-4592	364	7	∈	∈	PROPN
ejpam-4592	364	8	n	n	CCONJ
ejpam-4592	364	9	such	such	ADJ
ejpam-4592	364	10	that	that	DET
ejpam-4592	364	11	σ(cn	σ(cn	NOUN
ejpam-4592	364	12	,	,	PUNCT
ejpam-4592	364	13	x	x	X
ejpam-4592	364	14	)	)	PUNCT
ejpam-4592	364	15	<	<	X
ejpam-4592	364	16	ε	ε	PROPN
ejpam-4592	364	17	2	2	NUM
ejpam-4592	364	18	for	for	ADP
ejpam-4592	364	19	all	all	DET
ejpam-4592	364	20	n	n	DET
ejpam-4592	364	21	≥	≥	NOUN
ejpam-4592	364	22	n0	n0	NUM
ejpam-4592	364	23	and	and	CCONJ
ejpam-4592	364	24	σ(zn	σ(zn	PROPN
ejpam-4592	364	25	,	,	PUNCT
ejpam-4592	364	26	x	x	X
ejpam-4592	364	27	)	)	PUNCT
ejpam-4592	364	28	<	<	X
ejpam-4592	364	29	ε	ε	PROPN
ejpam-4592	364	30	2	2	NUM
ejpam-4592	364	31	for	for	ADP
ejpam-4592	364	32	all	all	DET
ejpam-4592	364	33	n	n	PRON
ejpam-4592	364	34	≥	≥	NOUN
ejpam-4592	364	35	n1	n1	NOUN
ejpam-4592	364	36	.	.	PUNCT
ejpam-4592	365	1	choose	choose	VERB
ejpam-4592	365	2	m	m	PROPN
ejpam-4592	365	3	=	=	SYM
ejpam-4592	365	4	max{n0	max{n0	PROPN
ejpam-4592	365	5	,	,	PUNCT
ejpam-4592	365	6	n1	n1	NOUN
ejpam-4592	365	7	}	}	PUNCT
ejpam-4592	365	8	.	.	PUNCT
ejpam-4592	366	1	for	for	ADP
ejpam-4592	366	2	n	n	PRON
ejpam-4592	366	3	≥	≥	NOUN
ejpam-4592	366	4	m	m	PROPN
ejpam-4592	366	5	,	,	PUNCT
ejpam-4592	366	6	σ(cn	σ(cn	PROPN
ejpam-4592	366	7	,	,	PUNCT
ejpam-4592	366	8	zn	zn	NOUN
ejpam-4592	366	9	)	)	PUNCT
ejpam-4592	366	10	≤	≤	NOUN
ejpam-4592	366	11	σ(cn	σ(cn	PROPN
ejpam-4592	366	12	,	,	PUNCT
ejpam-4592	366	13	x	x	X
ejpam-4592	366	14	)	)	PUNCT
ejpam-4592	367	1	+	+	CCONJ
ejpam-4592	367	2	σ(x	σ(x	PROPN
ejpam-4592	367	3	,	,	PUNCT
ejpam-4592	367	4	zn	zn	NOUN
ejpam-4592	367	5	)	)	PUNCT
ejpam-4592	367	6	<	<	X
ejpam-4592	367	7	ε	ε	PROPN
ejpam-4592	367	8	2	2	NUM
ejpam-4592	367	9	+	+	CCONJ
ejpam-4592	367	10	ε	ε	PROPN
ejpam-4592	367	11	2	2	NUM
ejpam-4592	367	12	=	=	SYM
ejpam-4592	367	13	ε	ε	PROPN
ejpam-4592	367	14	and	and	CCONJ
ejpam-4592	367	15	so	so	ADV
ejpam-4592	367	16	σ(cn	σ(cn	PROPN
ejpam-4592	367	17	,	,	PUNCT
ejpam-4592	367	18	zn	zn	NOUN
ejpam-4592	367	19	)	)	PUNCT
ejpam-4592	367	20	<	<	X
ejpam-4592	367	21	ε	ε	PROPN
ejpam-4592	367	22	for	for	ADP
ejpam-4592	367	23	n	n	NUM
ejpam-4592	367	24	≥	≥	NOUN
ejpam-4592	367	25	m.	m.	NOUN
ejpam-4592	367	26	hence	hence	ADV
ejpam-4592	367	27	the	the	DET
ejpam-4592	367	28	pair	pair	NOUN
ejpam-4592	367	29	of	of	ADP
ejpam-4592	367	30	sequence	sequence	NOUN
ejpam-4592	367	31	{	{	PUNCT
ejpam-4592	367	32	cn	cn	NOUN
ejpam-4592	367	33	}	}	PUNCT
ejpam-4592	367	34	and	and	CCONJ
ejpam-4592	367	35	{	{	PUNCT
ejpam-4592	367	36	zn	zn	X
ejpam-4592	367	37	}	}	PUNCT
ejpam-4592	367	38	is	be	AUX
ejpam-4592	367	39	asymptotic	asymptotic	ADJ
ejpam-4592	367	40	with	with	ADP
ejpam-4592	367	41	respect	respect	NOUN
ejpam-4592	367	42	to	to	ADP
ejpam-4592	367	43	the	the	DET
ejpam-4592	367	44	metric	metric	PROPN
ejpam-4592	367	45	σ	σ	PROPN
ejpam-4592	367	46	.	.	PUNCT
ejpam-4592	368	1	in	in	ADP
ejpam-4592	368	2	a	a	DET
ejpam-4592	368	3	generalized	generalized	ADJ
ejpam-4592	368	4	metric	metric	ADJ
ejpam-4592	368	5	space	space	NOUN
ejpam-4592	368	6	,	,	PUNCT
ejpam-4592	368	7	the	the	DET
ejpam-4592	368	8	existence	existence	NOUN
ejpam-4592	368	9	of	of	ADP
ejpam-4592	368	10	a	a	DET
ejpam-4592	368	11	convergent	convergent	NOUN
ejpam-4592	368	12	sequence	sequence	NOUN
ejpam-4592	368	13	using	use	VERB
ejpam-4592	368	14	convergent	convergent	NOUN
ejpam-4592	368	15	subsequence	subsequence	NOUN
ejpam-4592	368	16	can	can	AUX
ejpam-4592	368	17	be	be	AUX
ejpam-4592	368	18	found	find	VERB
ejpam-4592	368	19	straightforwardly	straightforwardly	ADV
ejpam-4592	368	20	by	by	ADP
ejpam-4592	368	21	theorem	theorem	ADJ
ejpam-4592	368	22	34	34	NUM
ejpam-4592	368	23	.	.	PUNCT
ejpam-4592	368	24	theorem	theorem	NOUN
ejpam-4592	368	25	34	34	NUM
ejpam-4592	368	26	.	.	PUNCT
ejpam-4592	369	1	let	let	AUX
ejpam-4592	369	2	(	(	PUNCT
ejpam-4592	369	3	x	x	NOUN
ejpam-4592	369	4	,	,	PUNCT
ejpam-4592	369	5	ω	ω	NUM
ejpam-4592	369	6	)	)	PUNCT
ejpam-4592	369	7	be	be	AUX
ejpam-4592	369	8	a	a	DET
ejpam-4592	369	9	generalized	generalized	ADJ
ejpam-4592	369	10	metric	metric	ADJ
ejpam-4592	369	11	space	space	NOUN
ejpam-4592	369	12	and	and	CCONJ
ejpam-4592	369	13	{	{	PUNCT
ejpam-4592	369	14	cn}n∈n	cn}n∈n	X
ejpam-4592	369	15	has	have	VERB
ejpam-4592	369	16	a	a	DET
ejpam-4592	369	17	convergent	convergent	ADJ
ejpam-4592	369	18	subsequence	subsequence	NOUN
ejpam-4592	369	19	{	{	PUNCT
ejpam-4592	369	20	zn}n∈n	zn}n∈n	NUM
ejpam-4592	369	21	with	with	ADP
ejpam-4592	369	22	respect	respect	NOUN
ejpam-4592	369	23	to	to	ADP
ejpam-4592	369	24	σ	σ	PROPN
ejpam-4592	369	25	∈	∈	PROPN
ejpam-4592	369	26	ω	ω	NOUN
ejpam-4592	369	27	.	.	PUNCT
ejpam-4592	370	1	if	if	SCONJ
ejpam-4592	370	2	the	the	DET
ejpam-4592	370	3	pair	pair	NOUN
ejpam-4592	370	4	of	of	ADP
ejpam-4592	370	5	sequence	sequence	NOUN
ejpam-4592	370	6	{	{	PUNCT
ejpam-4592	370	7	cn	cn	NOUN
ejpam-4592	370	8	}	}	PUNCT
ejpam-4592	370	9	and	and	CCONJ
ejpam-4592	370	10	{	{	PUNCT
ejpam-4592	370	11	zn	zn	X
ejpam-4592	370	12	}	}	PUNCT
ejpam-4592	370	13	is	be	AUX
ejpam-4592	370	14	asymptotic	asymptotic	ADJ
ejpam-4592	370	15	with	with	ADP
ejpam-4592	370	16	respect	respect	NOUN
ejpam-4592	370	17	to	to	ADP
ejpam-4592	370	18	the	the	DET
ejpam-4592	370	19	metric	metric	PROPN
ejpam-4592	370	20	σ	σ	PROPN
ejpam-4592	370	21	,	,	PUNCT
ejpam-4592	370	22	then	then	ADV
ejpam-4592	370	23	{	{	PUNCT
ejpam-4592	370	24	cn	cn	NOUN
ejpam-4592	370	25	}	}	PUNCT
ejpam-4592	370	26	is	be	AUX
ejpam-4592	370	27	a	a	DET
ejpam-4592	370	28	convergent	convergent	NOUN
ejpam-4592	370	29	sequence	sequence	NOUN
ejpam-4592	370	30	with	with	ADP
ejpam-4592	370	31	the	the	DET
ejpam-4592	370	32	same	same	ADJ
ejpam-4592	370	33	metric	metric	NOUN
ejpam-4592	370	34	.	.	PUNCT
ejpam-4592	371	1	proof	proof	NOUN
ejpam-4592	371	2	.	.	PUNCT
ejpam-4592	372	1	given	give	VERB
ejpam-4592	372	2	that	that	SCONJ
ejpam-4592	372	3	{	{	PUNCT
ejpam-4592	372	4	zn	zn	X
ejpam-4592	372	5	}	}	PUNCT
ejpam-4592	372	6	is	be	AUX
ejpam-4592	372	7	a	a	DET
ejpam-4592	372	8	convergent	convergent	NOUN
ejpam-4592	372	9	subsequence	subsequence	NOUN
ejpam-4592	372	10	of	of	ADP
ejpam-4592	372	11	{	{	PUNCT
ejpam-4592	372	12	cn	cn	PROPN
ejpam-4592	372	13	}	}	PUNCT
ejpam-4592	372	14	and	and	CCONJ
ejpam-4592	372	15	z	z	PROPN
ejpam-4592	372	16	is	be	AUX
ejpam-4592	372	17	a	a	DET
ejpam-4592	372	18	limit	limit	NOUN
ejpam-4592	372	19	point	point	NOUN
ejpam-4592	372	20	of	of	ADP
ejpam-4592	372	21	the	the	DET
ejpam-4592	372	22	sequence	sequence	NOUN
ejpam-4592	372	23	{	{	PUNCT
ejpam-4592	372	24	zn	zn	NOUN
ejpam-4592	372	25	}	}	PUNCT
ejpam-4592	372	26	with	with	ADP
ejpam-4592	372	27	respect	respect	NOUN
ejpam-4592	372	28	to	to	ADP
ejpam-4592	372	29	σ	σ	PROPN
ejpam-4592	372	30	∈	∈	PROPN
ejpam-4592	372	31	ω	ω	X
ejpam-4592	372	32	.	.	PUNCT
ejpam-4592	373	1	let	let	VERB
ejpam-4592	373	2	ε	ε	PROPN
ejpam-4592	373	3	>	>	X
ejpam-4592	373	4	0	0	PUNCT
ejpam-4592	373	5	be	be	AUX
ejpam-4592	373	6	given	give	VERB
ejpam-4592	373	7	.	.	PUNCT
ejpam-4592	374	1	then	then	ADV
ejpam-4592	374	2	there	there	PRON
ejpam-4592	374	3	exists	exist	VERB
ejpam-4592	374	4	n0	n0	PROPN
ejpam-4592	374	5	∈	∈	PROPN
ejpam-4592	374	6	n	n	PRON
ejpam-4592	374	7	such	such	ADJ
ejpam-4592	374	8	that	that	SCONJ
ejpam-4592	374	9	σ(zn	σ(zn	PROPN
ejpam-4592	374	10	,	,	PUNCT
ejpam-4592	374	11	z	z	NOUN
ejpam-4592	374	12	)	)	PUNCT
ejpam-4592	374	13	<	<	X
ejpam-4592	374	14	ε	ε	PROPN
ejpam-4592	374	15	2	2	NUM
ejpam-4592	374	16	for	for	ADP
ejpam-4592	374	17	every	every	DET
ejpam-4592	374	18	n	n	PRON
ejpam-4592	374	19	≥	≥	NOUN
ejpam-4592	374	20	n0	n0	NUM
ejpam-4592	374	21	.	.	PUNCT
ejpam-4592	375	1	since	since	SCONJ
ejpam-4592	375	2	the	the	DET
ejpam-4592	375	3	pair	pair	NOUN
ejpam-4592	375	4	of	of	ADP
ejpam-4592	375	5	sequence	sequence	NOUN
ejpam-4592	375	6	{	{	PUNCT
ejpam-4592	375	7	cn	cn	NOUN
ejpam-4592	375	8	}	}	PUNCT
ejpam-4592	375	9	and	and	CCONJ
ejpam-4592	375	10	{	{	PUNCT
ejpam-4592	375	11	zn	zn	X
ejpam-4592	375	12	}	}	PUNCT
ejpam-4592	375	13	is	be	AUX
ejpam-4592	375	14	asymptotic	asymptotic	ADJ
ejpam-4592	375	15	with	with	ADP
ejpam-4592	375	16	respect	respect	NOUN
ejpam-4592	375	17	to	to	ADP
ejpam-4592	375	18	σ	σ	PROPN
ejpam-4592	375	19	,	,	PUNCT
ejpam-4592	375	20	there	there	PRON
ejpam-4592	375	21	exists	exist	VERB
ejpam-4592	375	22	n1	n1	PROPN
ejpam-4592	375	23	∈	∈	PROPN
ejpam-4592	375	24	n	n	PRON
ejpam-4592	375	25	such	such	ADJ
ejpam-4592	375	26	that	that	DET
ejpam-4592	375	27	σ(cn	σ(cn	NOUN
ejpam-4592	375	28	,	,	PUNCT
ejpam-4592	375	29	zn	zn	NOUN
ejpam-4592	375	30	)	)	PUNCT
ejpam-4592	375	31	<	<	X
ejpam-4592	375	32	ε	ε	PROPN
ejpam-4592	375	33	2	2	NUM
ejpam-4592	375	34	for	for	ADP
ejpam-4592	375	35	every	every	DET
ejpam-4592	375	36	n	n	PRON
ejpam-4592	375	37	≥	≥	NOUN
ejpam-4592	375	38	n1	n1	NOUN
ejpam-4592	375	39	.	.	PUNCT
ejpam-4592	376	1	take	take	VERB
ejpam-4592	376	2	m	m	NOUN
ejpam-4592	376	3	=	=	SYM
ejpam-4592	376	4	max{n0	max{n0	PROPN
ejpam-4592	376	5	,	,	PUNCT
ejpam-4592	376	6	n1	n1	NOUN
ejpam-4592	376	7	+	+	CCONJ
ejpam-4592	376	8	1	1	NUM
ejpam-4592	376	9	}	}	PUNCT
ejpam-4592	376	10	.	.	PUNCT
ejpam-4592	377	1	for	for	ADP
ejpam-4592	377	2	n	n	PRON
ejpam-4592	377	3	≥	≥	NOUN
ejpam-4592	377	4	m	m	PROPN
ejpam-4592	377	5	,	,	PUNCT
ejpam-4592	377	6	σ(cn	σ(cn	PROPN
ejpam-4592	377	7	,	,	PUNCT
ejpam-4592	377	8	z	z	NOUN
ejpam-4592	377	9	)	)	PUNCT
ejpam-4592	377	10	≤	≤	NOUN
ejpam-4592	377	11	σ(cn	σ(cn	PROPN
ejpam-4592	377	12	,	,	PUNCT
ejpam-4592	377	13	zn	zn	NOUN
ejpam-4592	377	14	)	)	PUNCT
ejpam-4592	378	1	+	+	CCONJ
ejpam-4592	378	2	σ(zn	σ(zn	PROPN
ejpam-4592	378	3	,	,	PUNCT
ejpam-4592	378	4	z	z	NOUN
ejpam-4592	378	5	)	)	PUNCT
ejpam-4592	378	6	<	<	X
ejpam-4592	378	7	ε	ε	PROPN
ejpam-4592	378	8	2	2	NUM
ejpam-4592	378	9	+	+	CCONJ
ejpam-4592	378	10	ε	ε	PROPN
ejpam-4592	378	11	2	2	NUM
ejpam-4592	378	12	=	=	SYM
ejpam-4592	378	13	ε	ε	PROPN
ejpam-4592	378	14	.	.	PUNCT
ejpam-4592	378	15	thus	thus	ADV
ejpam-4592	378	16	,	,	PUNCT
ejpam-4592	378	17	σ(cn	σ(cn	PROPN
ejpam-4592	378	18	,	,	PUNCT
ejpam-4592	378	19	z	z	NOUN
ejpam-4592	378	20	)	)	PUNCT
ejpam-4592	378	21	<	<	X
ejpam-4592	378	22	ε	ε	PROPN
ejpam-4592	378	23	for	for	ADP
ejpam-4592	378	24	every	every	DET
ejpam-4592	378	25	n	n	PRON
ejpam-4592	378	26	≥	≥	NOUN
ejpam-4592	378	27	m.	m.	NOUN
ejpam-4592	378	28	hence	hence	ADV
ejpam-4592	378	29	{	{	PUNCT
ejpam-4592	378	30	cn	cn	PROPN
ejpam-4592	378	31	}	}	PUNCT
ejpam-4592	378	32	is	be	AUX
ejpam-4592	378	33	a	a	DET
ejpam-4592	378	34	convergent	convergent	NOUN
ejpam-4592	378	35	sequence	sequence	NOUN
ejpam-4592	378	36	with	with	ADP
ejpam-4592	378	37	respect	respect	NOUN
ejpam-4592	378	38	to	to	ADP
ejpam-4592	378	39	σ	σ	PROPN
ejpam-4592	378	40	.	.	PUNCT
ejpam-4592	379	1	the	the	DET
ejpam-4592	379	2	condition	condition	NOUN
ejpam-4592	379	3	that	that	PRON
ejpam-4592	379	4	convergence	convergence	NOUN
ejpam-4592	379	5	on	on	ADP
ejpam-4592	379	6	the	the	DET
ejpam-4592	379	7	sequence	sequence	NOUN
ejpam-4592	379	8	{	{	PUNCT
ejpam-4592	379	9	cn	cn	PROPN
ejpam-4592	379	10	}	}	PUNCT
ejpam-4592	379	11	can	can	AUX
ejpam-4592	379	12	not	not	PART
ejpam-4592	379	13	be	be	AUX
ejpam-4592	379	14	dropped	drop	VERB
ejpam-4592	379	15	in	in	ADP
ejpam-4592	379	16	theorem	theorem	ADJ
ejpam-4592	379	17	33	33	NUM
ejpam-4592	379	18	as	as	SCONJ
ejpam-4592	379	19	shown	show	VERB
ejpam-4592	379	20	by	by	ADP
ejpam-4592	379	21	the	the	DET
ejpam-4592	379	22	following	following	ADJ
ejpam-4592	379	23	example	example	NOUN
ejpam-4592	379	24	35	35	NUM
ejpam-4592	379	25	.	.	PUNCT
ejpam-4592	380	1	also	also	ADV
ejpam-4592	380	2	,	,	PUNCT
ejpam-4592	380	3	it	it	PRON
ejpam-4592	380	4	shows	show	VERB
ejpam-4592	380	5	that	that	SCONJ
ejpam-4592	380	6	the	the	DET
ejpam-4592	380	7	condition	condition	NOUN
ejpam-4592	380	8	asymptotic	asymptotic	ADJ
ejpam-4592	380	9	is	be	AUX
ejpam-4592	380	10	necessary	necessary	ADJ
ejpam-4592	380	11	in	in	ADP
ejpam-4592	380	12	theorem	theorem	ADJ
ejpam-4592	380	13	34	34	NUM
ejpam-4592	380	14	.	.	PUNCT
ejpam-4592	380	15	example	example	NOUN
ejpam-4592	380	16	35	35	NUM
ejpam-4592	380	17	.	.	PUNCT
ejpam-4592	381	1	consider	consider	VERB
ejpam-4592	381	2	the	the	DET
ejpam-4592	381	3	generalized	generalized	ADJ
ejpam-4592	381	4	metric	metric	ADJ
ejpam-4592	381	5	space	space	NOUN
ejpam-4592	381	6	(	(	PUNCT
ejpam-4592	381	7	x	x	NOUN
ejpam-4592	381	8	,	,	PUNCT
ejpam-4592	381	9	ω1	ω1	PROPN
ejpam-4592	381	10	)	)	PUNCT
ejpam-4592	381	11	wherex	wherex	NOUN
ejpam-4592	381	12	=	=	SYM
ejpam-4592	381	13	r	r	PROPN
ejpam-4592	381	14	,	,	PUNCT
ejpam-4592	381	15	ω1	ω1	PROPN
ejpam-4592	381	16	=	=	SYM
ejpam-4592	381	17	{	{	PUNCT
ejpam-4592	381	18	σ1	σ1	PROPN
ejpam-4592	381	19	,	,	PUNCT
ejpam-4592	381	20	σ2	σ2	PROPN
ejpam-4592	381	21	,	,	PUNCT
ejpam-4592	381	22	σ3	σ3	PROPN
ejpam-4592	381	23	}	}	PUNCT
ejpam-4592	381	24	⊂	⊂	PROPN
ejpam-4592	381	25	ωx	ωx	PROPN
ejpam-4592	381	26	and	and	CCONJ
ejpam-4592	381	27	the	the	DET
ejpam-4592	381	28	metrics	metric	NOUN
ejpam-4592	381	29	are	be	AUX
ejpam-4592	381	30	defined	define	VERB
ejpam-4592	381	31	by	by	ADP
ejpam-4592	381	32	σ1(c	σ1(c	PRON
ejpam-4592	381	33	,	,	PUNCT
ejpam-4592	381	34	d	d	NOUN
ejpam-4592	381	35	)	)	PUNCT
ejpam-4592	382	1	=	=	SYM
ejpam-4592	382	2	|c−d|	|c−d|	PUNCT
ejpam-4592	382	3	1+|c−d|	1+|c−d|	INTJ
ejpam-4592	382	4	;	;	PUNCT
ejpam-4592	382	5	σ2(c	σ2(c	NOUN
ejpam-4592	382	6	,	,	PUNCT
ejpam-4592	382	7	d	d	NOUN
ejpam-4592	382	8	)	)	PUNCT
ejpam-4592	382	9	=	=	SYM
ejpam-4592	382	10	min{σ1(c	min{σ1(c	PROPN
ejpam-4592	382	11	,	,	PUNCT
ejpam-4592	382	12	d	d	NOUN
ejpam-4592	382	13	)	)	PUNCT
ejpam-4592	382	14	,	,	PUNCT
ejpam-4592	382	15	1	1	X
ejpam-4592	382	16	}	}	PUNCT
ejpam-4592	382	17	and	and	CCONJ
ejpam-4592	382	18	σ3(c	σ3(c	PRON
ejpam-4592	382	19	,	,	PUNCT
ejpam-4592	382	20	d	d	NOUN
ejpam-4592	382	21	)	)	PUNCT
ejpam-4592	382	22	=	=	PRON
ejpam-4592	382	23	{	{	PUNCT
ejpam-4592	382	24	0	0	NUM
ejpam-4592	382	25	if	if	SCONJ
ejpam-4592	382	26	c	c	NOUN
ejpam-4592	382	27	=	=	SYM
ejpam-4592	382	28	d	d	PROPN
ejpam-4592	382	29	,	,	PUNCT
ejpam-4592	382	30	1	1	NUM
ejpam-4592	382	31	if	if	SCONJ
ejpam-4592	382	32	c	c	PROPN
ejpam-4592	382	33	̸=	̸=	PROPN
ejpam-4592	382	34	d	d	PROPN
ejpam-4592	382	35	for	for	ADP
ejpam-4592	382	36	all	all	DET
ejpam-4592	382	37	c	c	NOUN
ejpam-4592	382	38	,	,	PUNCT
ejpam-4592	382	39	d	d	PROPN
ejpam-4592	382	40	∈	∈	PROPN
ejpam-4592	382	41	x.	x.	NOUN
ejpam-4592	382	42	let	let	VERB
ejpam-4592	382	43	{	{	PUNCT
ejpam-4592	382	44	cn	cn	VERB
ejpam-4592	382	45	}	}	PUNCT
ejpam-4592	382	46	=	=	SYM
ejpam-4592	382	47	{	{	PUNCT
ejpam-4592	382	48	(	(	PUNCT
ejpam-4592	382	49	−1)n}n∈n	−1)n}n∈n	PROPN
ejpam-4592	382	50	be	be	AUX
ejpam-4592	382	51	a	a	DET
ejpam-4592	382	52	sequence	sequence	NOUN
ejpam-4592	382	53	in	in	ADP
ejpam-4592	382	54	x	x	PUNCT
ejpam-4592	382	55	and	and	CCONJ
ejpam-4592	382	56	{	{	PUNCT
ejpam-4592	382	57	dn	dn	AUX
ejpam-4592	382	58	}	}	PUNCT
ejpam-4592	382	59	be	be	AUX
ejpam-4592	382	60	a	a	DET
ejpam-4592	382	61	subsequence	subsequence	NOUN
ejpam-4592	382	62	of	of	ADP
ejpam-4592	382	63	{	{	PUNCT
ejpam-4592	382	64	cn	cn	PROPN
ejpam-4592	382	65	}	}	PUNCT
ejpam-4592	382	66	defined	define	VERB
ejpam-4592	382	67	by	by	ADP
ejpam-4592	382	68	dn	dn	NOUN
ejpam-4592	382	69	=	=	SYM
ejpam-4592	382	70	1	1	NUM
ejpam-4592	382	71	for	for	ADP
ejpam-4592	382	72	all	all	DET
ejpam-4592	382	73	n	n	PRON
ejpam-4592	382	74	∈	∈	PROPN
ejpam-4592	382	75	n.	n.	NOUN
ejpam-4592	382	76	then	then	ADV
ejpam-4592	382	77	{	{	PUNCT
ejpam-4592	382	78	dn	dn	INTJ
ejpam-4592	382	79	}	}	PUNCT
ejpam-4592	382	80	is	be	AUX
ejpam-4592	382	81	a	a	DET
ejpam-4592	382	82	convergent	convergent	NOUN
ejpam-4592	382	83	subsequence	subsequence	NOUN
ejpam-4592	382	84	of	of	ADP
ejpam-4592	382	85	{	{	PUNCT
ejpam-4592	382	86	cn	cn	X
ejpam-4592	382	87	}	}	PUNCT
ejpam-4592	382	88	with	with	ADP
ejpam-4592	382	89	respect	respect	NOUN
ejpam-4592	382	90	to	to	ADP
ejpam-4592	382	91	the	the	DET
ejpam-4592	382	92	metric	metric	PROPN
ejpam-4592	382	93	σ1	σ1	PROPN
ejpam-4592	382	94	∈	∈	PROPN
ejpam-4592	382	95	ω1	ω1	PROPN
ejpam-4592	382	96	.	.	PUNCT
ejpam-4592	383	1	here	here	ADV
ejpam-4592	383	2	{	{	PUNCT
ejpam-4592	383	3	cn	cn	NOUN
ejpam-4592	383	4	}	}	PUNCT
ejpam-4592	383	5	is	be	AUX
ejpam-4592	383	6	not	not	PART
ejpam-4592	383	7	a	a	DET
ejpam-4592	383	8	convergent	convergent	NOUN
ejpam-4592	383	9	sequence	sequence	NOUN
ejpam-4592	383	10	.	.	PUNCT
ejpam-4592	384	1	also	also	ADV
ejpam-4592	384	2	,	,	PUNCT
ejpam-4592	384	3	{	{	PUNCT
ejpam-4592	384	4	cn	cn	NOUN
ejpam-4592	384	5	}	}	PUNCT
ejpam-4592	384	6	and	and	CCONJ
ejpam-4592	384	7	{	{	PUNCT
ejpam-4592	384	8	dn	dn	VERB
ejpam-4592	384	9	}	}	PUNCT
ejpam-4592	384	10	are	be	AUX
ejpam-4592	384	11	not	not	PART
ejpam-4592	384	12	asymptotic	asymptotic	ADJ
ejpam-4592	384	13	with	with	ADP
ejpam-4592	384	14	respect	respect	NOUN
ejpam-4592	384	15	to	to	ADP
ejpam-4592	384	16	the	the	DET
ejpam-4592	384	17	metric	metric	PROPN
ejpam-4592	384	18	σ1	σ1	PROPN
ejpam-4592	384	19	.	.	PUNCT
ejpam-4592	385	1	the	the	DET
ejpam-4592	385	2	following	following	ADJ
ejpam-4592	385	3	example	example	NOUN
ejpam-4592	385	4	36	36	NUM
ejpam-4592	385	5	shows	show	VERB
ejpam-4592	385	6	that	that	SCONJ
ejpam-4592	385	7	the	the	DET
ejpam-4592	385	8	condition	condition	NOUN
ejpam-4592	385	9	convergent	convergent	NOUN
ejpam-4592	385	10	on	on	ADP
ejpam-4592	385	11	{	{	PUNCT
ejpam-4592	385	12	zn	zn	NOUN
ejpam-4592	385	13	}	}	PUNCT
ejpam-4592	385	14	can	can	AUX
ejpam-4592	385	15	not	not	PART
ejpam-4592	385	16	be	be	AUX
ejpam-4592	385	17	dropped	drop	VERB
ejpam-4592	385	18	in	in	ADP
ejpam-4592	385	19	theorem	theorem	ADJ
ejpam-4592	385	20	34	34	NUM
ejpam-4592	385	21	.	.	PUNCT
ejpam-4592	385	22	example	example	NOUN
ejpam-4592	386	1	36	36	NUM
ejpam-4592	386	2	.	.	PUNCT
ejpam-4592	387	1	consider	consider	VERB
ejpam-4592	387	2	the	the	DET
ejpam-4592	387	3	generalized	generalized	ADJ
ejpam-4592	387	4	metric	metric	ADJ
ejpam-4592	387	5	space	space	NOUN
ejpam-4592	387	6	(	(	PUNCT
ejpam-4592	387	7	x	x	NOUN
ejpam-4592	387	8	,	,	PUNCT
ejpam-4592	387	9	ω1	ω1	PROPN
ejpam-4592	387	10	)	)	PUNCT
ejpam-4592	387	11	wherex	wherex	NOUN
ejpam-4592	387	12	=	=	SYM
ejpam-4592	387	13	r	r	PROPN
ejpam-4592	387	14	,	,	PUNCT
ejpam-4592	387	15	ω1	ω1	PROPN
ejpam-4592	387	16	=	=	SYM
ejpam-4592	387	17	{	{	PUNCT
ejpam-4592	387	18	σ1	σ1	PROPN
ejpam-4592	387	19	,	,	PUNCT
ejpam-4592	387	20	σ2	σ2	PROPN
ejpam-4592	387	21	,	,	PUNCT
ejpam-4592	387	22	σ3	σ3	PROPN
ejpam-4592	387	23	}	}	PUNCT
ejpam-4592	387	24	⊂	⊂	PROPN
ejpam-4592	387	25	ωx	ωx	PROPN
ejpam-4592	387	26	and	and	CCONJ
ejpam-4592	387	27	the	the	DET
ejpam-4592	387	28	metrics	metric	NOUN
ejpam-4592	387	29	are	be	AUX
ejpam-4592	387	30	defined	define	VERB
ejpam-4592	387	31	by	by	ADP
ejpam-4592	387	32	σ1(c	σ1(c	PRON
ejpam-4592	387	33	,	,	PUNCT
ejpam-4592	387	34	d	d	NOUN
ejpam-4592	387	35	)	)	PUNCT
ejpam-4592	388	1	=	=	PUNCT
ejpam-4592	388	2	|c−	|c−	NOUN
ejpam-4592	388	3	d|;σ2(c	d|;σ2(c	PROPN
ejpam-4592	388	4	,	,	PUNCT
ejpam-4592	388	5	d	d	NOUN
ejpam-4592	388	6	)	)	PUNCT
ejpam-4592	389	1	=	=	SYM
ejpam-4592	389	2	min{σ1(c	min{σ1(c	PROPN
ejpam-4592	389	3	,	,	PUNCT
ejpam-4592	389	4	d	d	NOUN
ejpam-4592	389	5	)	)	PUNCT
ejpam-4592	389	6	,	,	PUNCT
ejpam-4592	389	7	1	1	X
ejpam-4592	389	8	}	}	PUNCT
ejpam-4592	389	9	and	and	CCONJ
ejpam-4592	389	10	σ3(c	σ3(c	PRON
ejpam-4592	389	11	,	,	PUNCT
ejpam-4592	389	12	d	d	NOUN
ejpam-4592	389	13	)	)	PUNCT
ejpam-4592	389	14	=	=	PRON
ejpam-4592	389	15	{	{	PUNCT
ejpam-4592	389	16	0	0	NUM
ejpam-4592	390	1	if	if	SCONJ
ejpam-4592	390	2	c	c	NOUN
ejpam-4592	390	3	=	=	SYM
ejpam-4592	390	4	d	d	PROPN
ejpam-4592	390	5	,	,	PUNCT
ejpam-4592	390	6	1	1	NUM
ejpam-4592	390	7	if	if	SCONJ
ejpam-4592	390	8	c	c	PROPN
ejpam-4592	390	9	̸=	̸=	PROPN
ejpam-4592	390	10	d	d	PROPN
ejpam-4592	390	11	for	for	ADP
ejpam-4592	390	12	all	all	DET
ejpam-4592	390	13	c	c	NOUN
ejpam-4592	390	14	,	,	PUNCT
ejpam-4592	390	15	d	d	PROPN
ejpam-4592	390	16	∈	∈	PROPN
ejpam-4592	390	17	x.	x.	NOUN
ejpam-4592	390	18	let	let	VERB
ejpam-4592	390	19	v.	v.	INTJ
ejpam-4592	390	20	subramanian	subramanian	PROPN
ejpam-4592	390	21	et	et	PROPN
ejpam-4592	390	22	al	al	PROPN
ejpam-4592	390	23	.	.	PUNCT
ejpam-4592	390	24	/	/	SYM
ejpam-4592	390	25	eur	eur	PROPN
ejpam-4592	390	26	.	.	PUNCT
ejpam-4592	391	1	j.	j.	PROPN
ejpam-4592	391	2	pure	pure	PROPN
ejpam-4592	391	3	appl	appl	PROPN
ejpam-4592	391	4	.	.	PROPN
ejpam-4592	391	5	math	math	PROPN
ejpam-4592	391	6	,	,	PUNCT
ejpam-4592	391	7	15	15	NUM
ejpam-4592	391	8	(	(	PUNCT
ejpam-4592	391	9	4	4	NUM
ejpam-4592	391	10	)	)	PUNCT
ejpam-4592	391	11	(	(	PUNCT
ejpam-4592	391	12	2022	2022	NUM
ejpam-4592	391	13	)	)	PUNCT
ejpam-4592	391	14	,	,	PUNCT
ejpam-4592	391	15	1869	1869	NUM
ejpam-4592	391	16	-	-	SYM
ejpam-4592	391	17	1886	1886	NUM
ejpam-4592	391	18	1880	1880	NUM
ejpam-4592	391	19	{	{	PUNCT
ejpam-4592	391	20	cn	cn	PROPN
ejpam-4592	391	21	}	}	PUNCT
ejpam-4592	391	22	=	=	SYM
ejpam-4592	391	23	{	{	PUNCT
ejpam-4592	391	24	1	1	NUM
ejpam-4592	391	25	if	if	SCONJ
ejpam-4592	391	26	n	n	PRON
ejpam-4592	391	27	is	be	AUX
ejpam-4592	391	28	odd	odd	ADJ
ejpam-4592	391	29	2	2	NUM
ejpam-4592	391	30	if	if	SCONJ
ejpam-4592	391	31	n	n	PRON
ejpam-4592	391	32	is	be	AUX
ejpam-4592	391	33	even	even	ADV
ejpam-4592	391	34	be	be	AUX
ejpam-4592	391	35	a	a	DET
ejpam-4592	391	36	sequence	sequence	NOUN
ejpam-4592	391	37	in	in	ADP
ejpam-4592	391	38	x	x	PUNCT
ejpam-4592	391	39	and	and	CCONJ
ejpam-4592	391	40	{	{	PUNCT
ejpam-4592	391	41	zn	zn	NOUN
ejpam-4592	391	42	}	}	PUNCT
ejpam-4592	391	43	be	be	AUX
ejpam-4592	391	44	a	a	DET
ejpam-4592	391	45	subsequence	subsequence	NOUN
ejpam-4592	391	46	of	of	ADP
ejpam-4592	391	47	{	{	PUNCT
ejpam-4592	391	48	cn	cn	PROPN
ejpam-4592	391	49	}	}	PUNCT
ejpam-4592	391	50	defined	define	VERB
ejpam-4592	391	51	by	by	ADP
ejpam-4592	391	52	zn	zn	PROPN
ejpam-4592	391	53	=	=	SYM
ejpam-4592	391	54	cn+2	cn+2	PROPN
ejpam-4592	391	55	for	for	ADP
ejpam-4592	391	56	all	all	DET
ejpam-4592	391	57	n	n	PRON
ejpam-4592	391	58	∈	∈	PROPN
ejpam-4592	391	59	n.	n.	NOUN
ejpam-4592	391	60	then	then	ADV
ejpam-4592	391	61	the	the	DET
ejpam-4592	391	62	pair	pair	NOUN
ejpam-4592	391	63	of	of	ADP
ejpam-4592	391	64	sequence	sequence	NOUN
ejpam-4592	391	65	{	{	PUNCT
ejpam-4592	391	66	cn	cn	NOUN
ejpam-4592	391	67	}	}	PUNCT
ejpam-4592	391	68	and	and	CCONJ
ejpam-4592	391	69	{	{	PUNCT
ejpam-4592	391	70	zn	zn	X
ejpam-4592	391	71	}	}	PUNCT
ejpam-4592	391	72	is	be	AUX
ejpam-4592	391	73	asymptotic	asymptotic	ADJ
ejpam-4592	391	74	with	with	ADP
ejpam-4592	391	75	respect	respect	NOUN
ejpam-4592	391	76	to	to	ADP
ejpam-4592	391	77	the	the	DET
ejpam-4592	391	78	metric	metric	ADJ
ejpam-4592	391	79	σ1	σ1	PROPN
ejpam-4592	391	80	.	.	PUNCT
ejpam-4592	392	1	but	but	CCONJ
ejpam-4592	392	2	{	{	PUNCT
ejpam-4592	392	3	cn	cn	NOUN
ejpam-4592	392	4	}	}	PUNCT
ejpam-4592	392	5	is	be	AUX
ejpam-4592	392	6	not	not	PART
ejpam-4592	392	7	a	a	DET
ejpam-4592	392	8	convergent	convergent	NOUN
ejpam-4592	392	9	sequence	sequence	NOUN
ejpam-4592	392	10	with	with	ADP
ejpam-4592	392	11	respect	respect	NOUN
ejpam-4592	392	12	to	to	ADP
ejpam-4592	392	13	the	the	DET
ejpam-4592	392	14	metric	metric	ADJ
ejpam-4592	392	15	σ1	σ1	PROPN
ejpam-4592	392	16	.	.	PUNCT
ejpam-4592	393	1	because	because	SCONJ
ejpam-4592	393	2	,	,	PUNCT
ejpam-4592	393	3	{	{	PUNCT
ejpam-4592	393	4	zn	zn	X
ejpam-4592	393	5	}	}	PUNCT
ejpam-4592	393	6	is	be	AUX
ejpam-4592	393	7	not	not	PART
ejpam-4592	393	8	a	a	DET
ejpam-4592	393	9	convergent	convergent	NOUN
ejpam-4592	393	10	subsequence	subsequence	NOUN
ejpam-4592	393	11	of	of	ADP
ejpam-4592	393	12	{	{	PUNCT
ejpam-4592	393	13	cn	cn	X
ejpam-4592	393	14	}	}	PUNCT
ejpam-4592	393	15	with	with	ADP
ejpam-4592	393	16	respect	respect	NOUN
ejpam-4592	393	17	to	to	ADP
ejpam-4592	393	18	the	the	DET
ejpam-4592	393	19	metric	metric	PROPN
ejpam-4592	393	20	σ1	σ1	PROPN
ejpam-4592	393	21	∈	∈	PROPN
ejpam-4592	393	22	ω1	ω1	PROPN
ejpam-4592	393	23	.	.	PUNCT
ejpam-4592	394	1	{	{	PUNCT
ejpam-4592	394	2	dn	dn	NOUN
ejpam-4592	394	3	}	}	PUNCT
ejpam-4592	394	4	∈	∈	PROPN
ejpam-4592	394	5	b(σ	b(σ	PROPN
ejpam-4592	394	6	)	)	PUNCT
ejpam-4592	394	7	{	{	PUNCT
ejpam-4592	394	8	cn	cn	PROPN
ejpam-4592	394	9	}	}	PUNCT
ejpam-4592	394	10	∈	∈	PROPN
ejpam-4592	394	11	c(σ	c(σ	PROPN
ejpam-4592	394	12	)	)	PUNCT
ejpam-4592	394	13	{	{	PUNCT
ejpam-4592	394	14	dn	dn	NOUN
ejpam-4592	394	15	}	}	PUNCT
ejpam-4592	394	16	∈	∈	PROPN
ejpam-4592	394	17	p(σ	p(σ	NOUN
ejpam-4592	394	18	)	)	PUNCT
ejpam-4592	394	19	{	{	PUNCT
ejpam-4592	394	20	dn	dn	NOUN
ejpam-4592	394	21	}	}	PUNCT
ejpam-4592	394	22	∈	∈	PROPN
ejpam-4592	394	23	c(σ	c(σ	PROPN
ejpam-4592	394	24	)	)	PUNCT
ejpam-4592	394	25	the	the	DET
ejpam-4592	394	26	following	follow	VERB
ejpam-4592	394	27	theorem	theorem	VERB
ejpam-4592	394	28	37	37	NUM
ejpam-4592	394	29	describe	describe	VERB
ejpam-4592	394	30	the	the	DET
ejpam-4592	394	31	above	above	ADJ
ejpam-4592	394	32	diagram	diagram	NOUN
ejpam-4592	394	33	.	.	PUNCT
ejpam-4592	395	1	theorem	theorem	VERB
ejpam-4592	395	2	37	37	NUM
ejpam-4592	395	3	.	.	PUNCT
ejpam-4592	396	1	let	let	VERB
ejpam-4592	396	2	(	(	PUNCT
ejpam-4592	396	3	x	x	NOUN
ejpam-4592	396	4	,	,	PUNCT
ejpam-4592	396	5	ω	ω	NUM
ejpam-4592	396	6	)	)	PUNCT
ejpam-4592	396	7	be	be	AUX
ejpam-4592	396	8	a	a	DET
ejpam-4592	396	9	generalized	generalized	ADJ
ejpam-4592	396	10	metric	metric	ADJ
ejpam-4592	396	11	space	space	NOUN
ejpam-4592	396	12	,	,	PUNCT
ejpam-4592	396	13	the	the	DET
ejpam-4592	396	14	pair	pair	NOUN
ejpam-4592	396	15	of	of	ADP
ejpam-4592	396	16	sequence	sequence	NOUN
ejpam-4592	396	17	{	{	PUNCT
ejpam-4592	396	18	cn	cn	NOUN
ejpam-4592	396	19	}	}	PUNCT
ejpam-4592	396	20	and	and	CCONJ
ejpam-4592	396	21	{	{	PUNCT
ejpam-4592	396	22	dn	dn	ADP
ejpam-4592	396	23	}	}	PUNCT
ejpam-4592	396	24	where	where	SCONJ
ejpam-4592	396	25	cn	cn	PROPN
ejpam-4592	396	26	̸=	̸=	PROPN
ejpam-4592	396	27	dn	dn	VERB
ejpam-4592	396	28	for	for	ADP
ejpam-4592	396	29	all	all	DET
ejpam-4592	396	30	n	n	PRON
ejpam-4592	396	31	∈	∈	PRON
ejpam-4592	396	32	n	n	PRON
ejpam-4592	396	33	be	be	VERB
ejpam-4592	396	34	uniformly	uniformly	ADV
ejpam-4592	396	35	asymptotic	asymptotic	ADJ
ejpam-4592	396	36	with	with	ADP
ejpam-4592	396	37	respect	respect	NOUN
ejpam-4592	396	38	to	to	ADP
ejpam-4592	396	39	σ	σ	PROPN
ejpam-4592	396	40	∈	∈	PROPN
ejpam-4592	396	41	ω	ω	PROPN
ejpam-4592	396	42	.	.	PUNCT
ejpam-4592	397	1	then	then	ADV
ejpam-4592	397	2	{	{	PUNCT
ejpam-4592	397	3	cn	cn	ADJ
ejpam-4592	397	4	}	}	PUNCT
ejpam-4592	397	5	∈	∈	PROPN
ejpam-4592	397	6	c(σ	c(σ	PROPN
ejpam-4592	397	7	)	)	PUNCT
ejpam-4592	397	8	if	if	SCONJ
ejpam-4592	397	9	any	any	DET
ejpam-4592	397	10	one	one	NUM
ejpam-4592	397	11	of	of	ADP
ejpam-4592	397	12	the	the	DET
ejpam-4592	397	13	following	follow	VERB
ejpam-4592	397	14	hold	hold	NOUN
ejpam-4592	397	15	.	.	PUNCT
ejpam-4592	398	1	(	(	PUNCT
ejpam-4592	398	2	a	a	X
ejpam-4592	398	3	)	)	PUNCT
ejpam-4592	398	4	{	{	PUNCT
ejpam-4592	398	5	dn	dn	NOUN
ejpam-4592	398	6	}	}	PUNCT
ejpam-4592	398	7	∈	∈	PROPN
ejpam-4592	398	8	b(σ	b(σ	PROPN
ejpam-4592	398	9	)	)	PUNCT
ejpam-4592	398	10	.	.	PUNCT
ejpam-4592	399	1	(	(	PUNCT
ejpam-4592	399	2	b	b	X
ejpam-4592	399	3	)	)	PUNCT
ejpam-4592	399	4	{	{	PUNCT
ejpam-4592	399	5	dn	dn	NOUN
ejpam-4592	399	6	}	}	PUNCT
ejpam-4592	399	7	∈	∈	PROPN
ejpam-4592	399	8	c(σ	c(σ	PROPN
ejpam-4592	399	9	)	)	PUNCT
ejpam-4592	399	10	.	.	PUNCT
ejpam-4592	400	1	(	(	PUNCT
ejpam-4592	400	2	c	c	X
ejpam-4592	400	3	)	)	PUNCT
ejpam-4592	400	4	{	{	PUNCT
ejpam-4592	400	5	dn	dn	NOUN
ejpam-4592	400	6	}	}	PUNCT
ejpam-4592	400	7	∈	∈	PROPN
ejpam-4592	400	8	p(σ	p(σ	NOUN
ejpam-4592	400	9	)	)	PUNCT
ejpam-4592	400	10	.	.	PUNCT
ejpam-4592	401	1	proof	proof	NOUN
ejpam-4592	401	2	.	.	PUNCT
ejpam-4592	402	1	let	let	VERB
ejpam-4592	402	2	ε	ε	PROPN
ejpam-4592	402	3	>	>	X
ejpam-4592	402	4	0	0	PUNCT
ejpam-4592	402	5	be	be	AUX
ejpam-4592	402	6	given	give	VERB
ejpam-4592	402	7	.	.	PUNCT
ejpam-4592	403	1	then	then	ADV
ejpam-4592	403	2	there	there	PRON
ejpam-4592	403	3	exists	exist	VERB
ejpam-4592	403	4	n0	n0	PROPN
ejpam-4592	403	5	∈	∈	PROPN
ejpam-4592	403	6	n	n	PRON
ejpam-4592	403	7	such	such	ADJ
ejpam-4592	403	8	that	that	DET
ejpam-4592	403	9	σ(cm	σ(cm	PROPN
ejpam-4592	403	10	,	,	PUNCT
ejpam-4592	403	11	dn	dn	PROPN
ejpam-4592	403	12	)	)	PUNCT
ejpam-4592	403	13	<	<	X
ejpam-4592	403	14	ε	ε	PROPN
ejpam-4592	403	15	3	3	NUM
ejpam-4592	403	16	for	for	ADP
ejpam-4592	403	17	every	every	DET
ejpam-4592	403	18	m	m	NOUN
ejpam-4592	403	19	,	,	PUNCT
ejpam-4592	403	20	n	n	PRON
ejpam-4592	403	21	≥	≥	NOUN
ejpam-4592	403	22	n0	n0	NUM
ejpam-4592	403	23	,	,	PUNCT
ejpam-4592	403	24	since	since	SCONJ
ejpam-4592	403	25	the	the	DET
ejpam-4592	403	26	pair	pair	NOUN
ejpam-4592	403	27	of	of	ADP
ejpam-4592	403	28	sequence	sequence	NOUN
ejpam-4592	403	29	{	{	PUNCT
ejpam-4592	403	30	cn	cn	NOUN
ejpam-4592	403	31	}	}	PUNCT
ejpam-4592	403	32	and	and	CCONJ
ejpam-4592	403	33	{	{	PUNCT
ejpam-4592	403	34	dn	dn	VERB
ejpam-4592	403	35	}	}	PUNCT
ejpam-4592	403	36	is	be	AUX
ejpam-4592	403	37	uniformly	uniformly	ADV
ejpam-4592	403	38	asymptotic	asymptotic	ADJ
ejpam-4592	403	39	.	.	PUNCT
ejpam-4592	404	1	(	(	PUNCT
ejpam-4592	404	2	a	a	X
ejpam-4592	404	3	)	)	PUNCT
ejpam-4592	404	4	given	give	VERB
ejpam-4592	404	5	that	that	SCONJ
ejpam-4592	404	6	{	{	PUNCT
ejpam-4592	404	7	dn	dn	ADJ
ejpam-4592	404	8	}	}	PUNCT
ejpam-4592	404	9	∈	∈	PROPN
ejpam-4592	404	10	b(σ	b(σ	PROPN
ejpam-4592	404	11	)	)	PUNCT
ejpam-4592	404	12	.	.	PUNCT
ejpam-4592	405	1	then	then	ADV
ejpam-4592	405	2	there	there	PRON
ejpam-4592	405	3	exist	exist	VERB
ejpam-4592	405	4	l	l	NOUN
ejpam-4592	405	5	,	,	PUNCT
ejpam-4592	405	6	k0	k0	PROPN
ejpam-4592	405	7	∈	∈	PROPN
ejpam-4592	405	8	n	n	CCONJ
ejpam-4592	405	9	such	such	ADJ
ejpam-4592	405	10	that	that	SCONJ
ejpam-4592	405	11	whenever	whenever	SCONJ
ejpam-4592	405	12	n	n	PROPN
ejpam-4592	405	13	>	>	X
ejpam-4592	405	14	j	j	PROPN
ejpam-4592	405	15	≥	≥	PROPN
ejpam-4592	405	16	k0	k0	PROPN
ejpam-4592	405	17	,	,	PUNCT
ejpam-4592	405	18	the	the	DET
ejpam-4592	405	19	points	point	NOUN
ejpam-4592	405	20	dj	dj	NOUN
ejpam-4592	406	1	and	and	CCONJ
ejpam-4592	406	2	dn	dn	PROPN
ejpam-4592	406	3	are	be	AUX
ejpam-4592	406	4	joined	join	VERB
ejpam-4592	406	5	by	by	ADP
ejpam-4592	406	6	an	an	DET
ejpam-4592	406	7	ε	ε	PROPN
ejpam-4592	406	8	3	3	NUM
ejpam-4592	406	9	-chain	-chain	NOUN
ejpam-4592	406	10	of	of	ADP
ejpam-4592	406	11	length	length	NOUN
ejpam-4592	406	12	l.	l.	PROPN
ejpam-4592	406	13	choose	choose	PROPN
ejpam-4592	406	14	m	m	PROPN
ejpam-4592	406	15	=	=	PUNCT
ejpam-4592	406	16	max{n0	max{n0	PROPN
ejpam-4592	406	17	+	+	SYM
ejpam-4592	406	18	1	1	NUM
ejpam-4592	406	19	,	,	PUNCT
ejpam-4592	406	20	k0	k0	PROPN
ejpam-4592	406	21	}	}	PUNCT
ejpam-4592	406	22	.	.	PUNCT
ejpam-4592	407	1	for	for	ADP
ejpam-4592	407	2	n	n	CCONJ
ejpam-4592	407	3	,	,	PUNCT
ejpam-4592	407	4	m	m	VERB
ejpam-4592	407	5	≥	≥	NOUN
ejpam-4592	407	6	m	m	PROPN
ejpam-4592	407	7	,	,	PUNCT
ejpam-4592	407	8	σ(cn	σ(cn	PROPN
ejpam-4592	407	9	,	,	PUNCT
ejpam-4592	407	10	cm	cm	NOUN
ejpam-4592	407	11	)	)	PUNCT
ejpam-4592	407	12	≤	≤	NOUN
ejpam-4592	407	13	σ(cn	σ(cn	NOUN
ejpam-4592	407	14	,	,	PUNCT
ejpam-4592	407	15	dn+1	dn+1	NOUN
ejpam-4592	407	16	)	)	PUNCT
ejpam-4592	407	17	+	+	CCONJ
ejpam-4592	407	18	σ(dn+1	σ(dn+1	PROPN
ejpam-4592	407	19	,	,	PUNCT
ejpam-4592	407	20	dn+2	dn+2	ADV
ejpam-4592	407	21	)	)	PUNCT
ejpam-4592	407	22	+	+	CCONJ
ejpam-4592	407	23	σ(dn+2	σ(dn+2	NOUN
ejpam-4592	407	24	,	,	PUNCT
ejpam-4592	407	25	cm	cm	NOUN
ejpam-4592	407	26	)	)	PUNCT
ejpam-4592	407	27	<	<	X
ejpam-4592	407	28	ε	ε	PROPN
ejpam-4592	407	29	3	3	NUM
ejpam-4592	407	30	+	+	CCONJ
ejpam-4592	407	31	ε	ε	PROPN
ejpam-4592	407	32	3	3	NUM
ejpam-4592	407	33	+	+	CCONJ
ejpam-4592	407	34	ε	ε	PROPN
ejpam-4592	407	35	3	3	NUM
ejpam-4592	407	36	=	=	SYM
ejpam-4592	407	37	ε	ε	PROPN
ejpam-4592	407	38	.	.	PUNCT
ejpam-4592	407	39	thus	thus	ADV
ejpam-4592	407	40	,	,	PUNCT
ejpam-4592	407	41	σ(cn	σ(cn	PROPN
ejpam-4592	407	42	,	,	PUNCT
ejpam-4592	407	43	cm	cm	NOUN
ejpam-4592	407	44	)	)	PUNCT
ejpam-4592	407	45	<	<	X
ejpam-4592	407	46	ε	ε	PROPN
ejpam-4592	407	47	for	for	ADP
ejpam-4592	407	48	every	every	DET
ejpam-4592	407	49	n	n	CCONJ
ejpam-4592	407	50	,	,	PUNCT
ejpam-4592	407	51	m	m	PROPN
ejpam-4592	407	52	≥	≥	NOUN
ejpam-4592	407	53	m.	m.	NOUN
ejpam-4592	407	54	therefore	therefore	ADV
ejpam-4592	407	55	,	,	PUNCT
ejpam-4592	407	56	{	{	PUNCT
ejpam-4592	407	57	cn	cn	ADJ
ejpam-4592	407	58	}	}	PUNCT
ejpam-4592	407	59	∈	∈	PROPN
ejpam-4592	407	60	c(σ	c(σ	PROPN
ejpam-4592	407	61	)	)	PUNCT
ejpam-4592	407	62	.	.	PUNCT
ejpam-4592	408	1	(	(	PUNCT
ejpam-4592	408	2	b	b	X
ejpam-4592	408	3	)	)	PUNCT
ejpam-4592	408	4	let	let	VERB
ejpam-4592	408	5	{	{	PUNCT
ejpam-4592	408	6	dn	dn	VERB
ejpam-4592	408	7	}	}	PUNCT
ejpam-4592	408	8	∈	∈	PROPN
ejpam-4592	408	9	c(σ	c(σ	PROPN
ejpam-4592	408	10	)	)	PUNCT
ejpam-4592	408	11	.	.	PUNCT
ejpam-4592	409	1	then	then	ADV
ejpam-4592	409	2	there	there	PRON
ejpam-4592	409	3	exists	exist	VERB
ejpam-4592	409	4	an	an	DET
ejpam-4592	409	5	infinite	infinite	NOUN
ejpam-4592	409	6	subset	subset	NOUN
ejpam-4592	409	7	nε	nε	NOUN
ejpam-4592	409	8	of	of	ADP
ejpam-4592	409	9	n	n	PRON
ejpam-4592	409	10	such	such	ADJ
ejpam-4592	409	11	that	that	SCONJ
ejpam-4592	409	12	σ(dk	σ(dk	PROPN
ejpam-4592	409	13	,	,	PUNCT
ejpam-4592	409	14	dj	dj	NOUN
ejpam-4592	409	15	)	)	PUNCT
ejpam-4592	409	16	<	<	X
ejpam-4592	409	17	ε	ε	PROPN
ejpam-4592	409	18	3	3	NUM
ejpam-4592	409	19	for	for	ADP
ejpam-4592	409	20	all	all	DET
ejpam-4592	409	21	k	k	PROPN
ejpam-4592	409	22	,	,	PUNCT
ejpam-4592	409	23	j	j	PROPN
ejpam-4592	409	24	∈	∈	PROPN
ejpam-4592	409	25	nε	nε	VERB
ejpam-4592	409	26	.	.	PUNCT
ejpam-4592	410	1	let	let	VERB
ejpam-4592	410	2	nk	nk	PROPN
ejpam-4592	410	3	=	=	PUNCT
ejpam-4592	410	4	min{ni	min{ni	PUNCT
ejpam-4592	410	5	:	:	PUNCT
ejpam-4592	410	6	ni	ni	PROPN
ejpam-4592	410	7	∈	∈	PROPN
ejpam-4592	410	8	nε	nε	VERB
ejpam-4592	410	9	}	}	PUNCT
ejpam-4592	410	10	and	and	CCONJ
ejpam-4592	410	11	take	take	VERB
ejpam-4592	410	12	m	m	NOUN
ejpam-4592	410	13	=	=	PUNCT
ejpam-4592	410	14	max{n0	max{n0	X
ejpam-4592	410	15	+	+	SYM
ejpam-4592	410	16	1	1	NUM
ejpam-4592	410	17	,	,	PUNCT
ejpam-4592	410	18	nk	nk	PROPN
ejpam-4592	410	19	}	}	PUNCT
ejpam-4592	410	20	.	.	PUNCT
ejpam-4592	411	1	for	for	ADP
ejpam-4592	411	2	n	n	CCONJ
ejpam-4592	411	3	,	,	PUNCT
ejpam-4592	411	4	m	m	VERB
ejpam-4592	411	5	≥	≥	NOUN
ejpam-4592	411	6	m	m	PROPN
ejpam-4592	411	7	,	,	PUNCT
ejpam-4592	411	8	σ(cn	σ(cn	PROPN
ejpam-4592	411	9	,	,	PUNCT
ejpam-4592	411	10	cm	cm	NOUN
ejpam-4592	411	11	)	)	PUNCT
ejpam-4592	411	12	≤	≤	NOUN
ejpam-4592	411	13	σ(cn	σ(cn	NOUN
ejpam-4592	411	14	,	,	PUNCT
ejpam-4592	411	15	dj	dj	NOUN
ejpam-4592	411	16	)	)	PUNCT
ejpam-4592	411	17	+	+	CCONJ
ejpam-4592	411	18	σ(dj	σ(dj	PROPN
ejpam-4592	411	19	,	,	PUNCT
ejpam-4592	411	20	dk	dk	PROPN
ejpam-4592	411	21	)	)	PUNCT
ejpam-4592	411	22	+	+	CCONJ
ejpam-4592	411	23	σ(dk	σ(dk	X
ejpam-4592	411	24	,	,	PUNCT
ejpam-4592	411	25	cm	cm	NOUN
ejpam-4592	411	26	)	)	PUNCT
ejpam-4592	411	27	,	,	PUNCT
ejpam-4592	411	28	since	since	SCONJ
ejpam-4592	411	29	nε	nε	PROPN
ejpam-4592	411	30	is	be	AUX
ejpam-4592	411	31	an	an	DET
ejpam-4592	411	32	infinite	infinite	ADJ
ejpam-4592	411	33	subset	subset	NOUN
ejpam-4592	411	34	of	of	ADP
ejpam-4592	411	35	n	n	CCONJ
ejpam-4592	411	36	,	,	PUNCT
ejpam-4592	411	37	there	there	PRON
ejpam-4592	411	38	exist	exist	VERB
ejpam-4592	411	39	j	j	PROPN
ejpam-4592	411	40	,	,	PUNCT
ejpam-4592	411	41	k	k	PROPN
ejpam-4592	411	42	∈	∈	PROPN
ejpam-4592	411	43	nε	nε	VERB
ejpam-4592	411	44	such	such	ADJ
ejpam-4592	411	45	that	that	SCONJ
ejpam-4592	411	46	j	j	PROPN
ejpam-4592	411	47	,	,	PUNCT
ejpam-4592	411	48	k	k	PROPN
ejpam-4592	411	49	>	>	X
ejpam-4592	411	50	m.	m.	NOUN
ejpam-4592	411	51	thus	thus	ADV
ejpam-4592	411	52	,	,	PUNCT
ejpam-4592	411	53	σ(cn	σ(cn	PROPN
ejpam-4592	411	54	,	,	PUNCT
ejpam-4592	411	55	cm	cm	NOUN
ejpam-4592	411	56	)	)	PUNCT
ejpam-4592	411	57	<	<	X
ejpam-4592	411	58	ε	ε	PROPN
ejpam-4592	411	59	3	3	NUM
ejpam-4592	411	60	+	+	CCONJ
ejpam-4592	411	61	ε	ε	PROPN
ejpam-4592	411	62	3	3	NUM
ejpam-4592	411	63	+	+	CCONJ
ejpam-4592	411	64	ε	ε	PROPN
ejpam-4592	411	65	3	3	NUM
ejpam-4592	411	66	=	=	SYM
ejpam-4592	411	67	ε	ε	PROPN
ejpam-4592	411	68	.	.	PUNCT
ejpam-4592	411	69	hence	hence	ADV
ejpam-4592	411	70	σ(cn	σ(cn	PROPN
ejpam-4592	411	71	,	,	PUNCT
ejpam-4592	411	72	cm	cm	NOUN
ejpam-4592	411	73	)	)	PUNCT
ejpam-4592	411	74	<	<	X
ejpam-4592	411	75	ε	ε	PROPN
ejpam-4592	411	76	for	for	ADP
ejpam-4592	411	77	all	all	DET
ejpam-4592	411	78	n	n	CCONJ
ejpam-4592	411	79	,	,	PUNCT
ejpam-4592	411	80	m	m	PROPN
ejpam-4592	411	81	≥	≥	NOUN
ejpam-4592	411	82	m.	m.	NOUN
ejpam-4592	411	83	therefore	therefore	ADV
ejpam-4592	411	84	,	,	PUNCT
ejpam-4592	411	85	{	{	PUNCT
ejpam-4592	411	86	cn	cn	ADJ
ejpam-4592	411	87	}	}	PUNCT
ejpam-4592	411	88	∈	∈	PROPN
ejpam-4592	411	89	c(σ	c(σ	PROPN
ejpam-4592	411	90	)	)	PUNCT
ejpam-4592	411	91	.	.	PUNCT
ejpam-4592	412	1	(	(	PUNCT
ejpam-4592	412	2	c	c	X
ejpam-4592	412	3	)	)	PUNCT
ejpam-4592	412	4	suppose	suppose	VERB
ejpam-4592	412	5	that	that	SCONJ
ejpam-4592	412	6	{	{	PUNCT
ejpam-4592	412	7	dn	dn	ADJ
ejpam-4592	412	8	}	}	PUNCT
ejpam-4592	412	9	∈	∈	PROPN
ejpam-4592	412	10	p(σ	p(σ	NOUN
ejpam-4592	412	11	)	)	PUNCT
ejpam-4592	412	12	.	.	PUNCT
ejpam-4592	413	1	then	then	ADV
ejpam-4592	413	2	there	there	PRON
ejpam-4592	413	3	exists	exist	VERB
ejpam-4592	413	4	k	k	PROPN
ejpam-4592	413	5	,	,	PUNCT
ejpam-4592	413	6	l	l	PROPN
ejpam-4592	413	7	∈	∈	PROPN
ejpam-4592	413	8	n	n	X
ejpam-4592	413	9	with	with	ADP
ejpam-4592	413	10	k	k	PROPN
ejpam-4592	413	11	̸=	̸=	PROPN
ejpam-4592	413	12	l	l	NOUN
ejpam-4592	413	13	and	and	CCONJ
ejpam-4592	413	14	k	k	NOUN
ejpam-4592	413	15	,	,	PUNCT
ejpam-4592	413	16	l	l	NOUN
ejpam-4592	413	17	>	>	X
ejpam-4592	413	18	n0	n0	NUM
ejpam-4592	413	19	such	such	ADJ
ejpam-4592	413	20	that	that	SCONJ
ejpam-4592	413	21	σ(dk	σ(dk	PROPN
ejpam-4592	413	22	,	,	PUNCT
ejpam-4592	413	23	dl	dl	PROPN
ejpam-4592	413	24	)	)	PUNCT
ejpam-4592	413	25	<	<	X
ejpam-4592	413	26	ε	ε	PROPN
ejpam-4592	413	27	3	3	NUM
ejpam-4592	413	28	.	.	PUNCT
ejpam-4592	414	1	takem	takem	PROPN
ejpam-4592	414	2	=	=	PROPN
ejpam-4592	414	3	n0	n0	PROPN
ejpam-4592	415	1	+	+	PROPN
ejpam-4592	415	2	1	1	NUM
ejpam-4592	415	3	.	.	PUNCT
ejpam-4592	415	4	for	for	ADP
ejpam-4592	415	5	n	n	CCONJ
ejpam-4592	415	6	,	,	PUNCT
ejpam-4592	415	7	m	m	VERB
ejpam-4592	415	8	≥	≥	NOUN
ejpam-4592	415	9	m	m	PROPN
ejpam-4592	415	10	,	,	PUNCT
ejpam-4592	415	11	σ(cn	σ(cn	PROPN
ejpam-4592	415	12	,	,	PUNCT
ejpam-4592	415	13	cm	cm	NOUN
ejpam-4592	415	14	)	)	PUNCT
ejpam-4592	415	15	≤	≤	NOUN
ejpam-4592	415	16	σ(cn	σ(cn	PROPN
ejpam-4592	415	17	,	,	PUNCT
ejpam-4592	415	18	dk)+σ(dk	dk)+σ(dk	PROPN
ejpam-4592	415	19	,	,	PUNCT
ejpam-4592	415	20	dl)+σ(dk	dl)+σ(dk	PROPN
ejpam-4592	415	21	,	,	PUNCT
ejpam-4592	415	22	cm	cm	NOUN
ejpam-4592	415	23	)	)	PUNCT
ejpam-4592	415	24	.	.	PUNCT
ejpam-4592	416	1	thus	thus	ADV
ejpam-4592	416	2	,	,	PUNCT
ejpam-4592	416	3	σ(cn	σ(cn	PROPN
ejpam-4592	416	4	,	,	PUNCT
ejpam-4592	416	5	cm	cm	NOUN
ejpam-4592	416	6	)	)	PUNCT
ejpam-4592	416	7	<	<	X
ejpam-4592	416	8	ε	ε	PROPN
ejpam-4592	416	9	3	3	NUM
ejpam-4592	416	10	+	+	CCONJ
ejpam-4592	416	11	ε	ε	PROPN
ejpam-4592	416	12	3	3	NUM
ejpam-4592	416	13	+	+	CCONJ
ejpam-4592	416	14	ε	ε	PROPN
ejpam-4592	416	15	3	3	NUM
ejpam-4592	416	16	=	=	SYM
ejpam-4592	416	17	ε	ε	PROPN
ejpam-4592	416	18	.	.	PUNCT
ejpam-4592	416	19	therefore	therefore	ADV
ejpam-4592	416	20	,	,	PUNCT
ejpam-4592	416	21	σ(cn	σ(cn	PROPN
ejpam-4592	416	22	,	,	PUNCT
ejpam-4592	416	23	cm	cm	NOUN
ejpam-4592	416	24	)	)	PUNCT
ejpam-4592	416	25	<	<	X
ejpam-4592	416	26	ε	ε	PROPN
ejpam-4592	416	27	for	for	ADP
ejpam-4592	416	28	all	all	DET
ejpam-4592	416	29	n	n	CCONJ
ejpam-4592	416	30	,	,	PUNCT
ejpam-4592	416	31	m	m	PROPN
ejpam-4592	416	32	≥	≥	NOUN
ejpam-4592	416	33	m.	m.	NOUN
ejpam-4592	416	34	hence	hence	ADV
ejpam-4592	416	35	{	{	PUNCT
ejpam-4592	416	36	cn	cn	PROPN
ejpam-4592	416	37	}	}	PUNCT
ejpam-4592	416	38	∈	∈	PROPN
ejpam-4592	416	39	c(σ	c(σ	PROPN
ejpam-4592	416	40	)	)	PUNCT
ejpam-4592	416	41	.	.	PUNCT
ejpam-4592	417	1	theorem	theorem	VERB
ejpam-4592	417	2	38	38	NUM
ejpam-4592	417	3	.	.	PUNCT
ejpam-4592	418	1	let	let	AUX
ejpam-4592	418	2	(	(	PUNCT
ejpam-4592	418	3	x	x	NOUN
ejpam-4592	418	4	,	,	PUNCT
ejpam-4592	418	5	ω	ω	NUM
ejpam-4592	418	6	)	)	PUNCT
ejpam-4592	418	7	be	be	AUX
ejpam-4592	418	8	a	a	DET
ejpam-4592	418	9	gms	gms	NOUN
ejpam-4592	418	10	.	.	PUNCT
ejpam-4592	419	1	then	then	ADV
ejpam-4592	419	2	(	(	PUNCT
ejpam-4592	419	3	x	x	X
ejpam-4592	419	4	,	,	PUNCT
ejpam-4592	419	5	ω	ω	NUM
ejpam-4592	419	6	)	)	PUNCT
ejpam-4592	419	7	is	be	AUX
ejpam-4592	419	8	a	a	DET
ejpam-4592	419	9	weakly	weakly	ADJ
ejpam-4592	419	10	complete	complete	ADJ
ejpam-4592	419	11	space	space	NOUN
ejpam-4592	419	12	if	if	SCONJ
ejpam-4592	419	13	and	and	CCONJ
ejpam-4592	419	14	only	only	ADV
ejpam-4592	419	15	if	if	SCONJ
ejpam-4592	419	16	there	there	PRON
ejpam-4592	419	17	is	be	VERB
ejpam-4592	419	18	a	a	DET
ejpam-4592	419	19	kernel	kernel	NOUN
ejpam-4592	419	20	ω0	ω0	PROPN
ejpam-4592	419	21	⊂	⊂	PROPN
ejpam-4592	419	22	ω	ω	PROPN
ejpam-4592	419	23	and	and	CCONJ
ejpam-4592	419	24	if	if	SCONJ
ejpam-4592	419	25	the	the	DET
ejpam-4592	419	26	pair	pair	NOUN
ejpam-4592	419	27	of	of	ADP
ejpam-4592	419	28	sequence	sequence	NOUN
ejpam-4592	419	29	{	{	PUNCT
ejpam-4592	419	30	cn	cn	NOUN
ejpam-4592	419	31	}	}	PUNCT
ejpam-4592	419	32	and	and	CCONJ
ejpam-4592	419	33	{	{	PUNCT
ejpam-4592	419	34	dn	dn	VERB
ejpam-4592	419	35	}	}	PUNCT
ejpam-4592	419	36	is	be	AUX
ejpam-4592	419	37	asymptotic	asymptotic	ADJ
ejpam-4592	419	38	with	with	ADP
ejpam-4592	419	39	respect	respect	NOUN
ejpam-4592	419	40	to	to	ADP
ejpam-4592	419	41	σ	σ	PROPN
ejpam-4592	419	42	∈	∈	PROPN
ejpam-4592	419	43	ω0	ω0	NOUN
ejpam-4592	419	44	and	and	CCONJ
ejpam-4592	419	45	{	{	PUNCT
ejpam-4592	419	46	dn	dn	VERB
ejpam-4592	419	47	}	}	PUNCT
ejpam-4592	419	48	is	be	AUX
ejpam-4592	419	49	cauchy	cauchy	ADJ
ejpam-4592	419	50	with	with	ADP
ejpam-4592	419	51	respect	respect	NOUN
ejpam-4592	419	52	to	to	ADP
ejpam-4592	419	53	σ	σ	PROPN
ejpam-4592	419	54	,	,	PUNCT
ejpam-4592	419	55	then	then	ADV
ejpam-4592	419	56	{	{	PUNCT
ejpam-4592	419	57	cn	cn	NOUN
ejpam-4592	419	58	}	}	PUNCT
ejpam-4592	419	59	is	be	AUX
ejpam-4592	419	60	a	a	DET
ejpam-4592	419	61	convergent	convergent	NOUN
ejpam-4592	419	62	sequence	sequence	NOUN
ejpam-4592	419	63	with	with	ADP
ejpam-4592	419	64	the	the	DET
ejpam-4592	419	65	same	same	ADJ
ejpam-4592	419	66	metric	metric	NOUN
ejpam-4592	419	67	.	.	PUNCT
ejpam-4592	420	1	proof	proof	NOUN
ejpam-4592	420	2	.	.	PUNCT
ejpam-4592	421	1	given	give	VERB
ejpam-4592	421	2	that	that	SCONJ
ejpam-4592	421	3	(	(	PUNCT
ejpam-4592	421	4	x	x	X
ejpam-4592	421	5	,	,	PUNCT
ejpam-4592	421	6	ω	ω	NUM
ejpam-4592	421	7	)	)	PUNCT
ejpam-4592	421	8	is	be	AUX
ejpam-4592	421	9	a	a	DET
ejpam-4592	421	10	weakly	weakly	ADJ
ejpam-4592	421	11	complete	complete	ADJ
ejpam-4592	421	12	space	space	NOUN
ejpam-4592	421	13	.	.	PUNCT
ejpam-4592	422	1	then	then	ADV
ejpam-4592	422	2	there	there	PRON
ejpam-4592	422	3	exists	exist	VERB
ejpam-4592	422	4	a	a	DET
ejpam-4592	422	5	kernel	kernel	PROPN
ejpam-4592	422	6	ω0	ω0	PROPN
ejpam-4592	422	7	⊂	⊂	PROPN
ejpam-4592	422	8	ω	ω	PROPN
ejpam-4592	422	9	consisting	consist	VERB
ejpam-4592	422	10	of	of	ADP
ejpam-4592	422	11	complete	complete	ADJ
ejpam-4592	422	12	metrics	metric	NOUN
ejpam-4592	422	13	.	.	PUNCT
ejpam-4592	423	1	let	let	VERB
ejpam-4592	423	2	the	the	DET
ejpam-4592	423	3	pair	pair	NOUN
ejpam-4592	423	4	of	of	ADP
ejpam-4592	423	5	sequence	sequence	NOUN
ejpam-4592	423	6	{	{	PUNCT
ejpam-4592	423	7	cn	cn	NOUN
ejpam-4592	423	8	}	}	PUNCT
ejpam-4592	423	9	and	and	CCONJ
ejpam-4592	423	10	{	{	PUNCT
ejpam-4592	423	11	dn	dn	AUX
ejpam-4592	423	12	}	}	PUNCT
ejpam-4592	423	13	be	be	AUX
ejpam-4592	423	14	asymptotic	asymptotic	ADJ
ejpam-4592	423	15	with	with	ADP
ejpam-4592	423	16	v.	v.	ADP
ejpam-4592	423	17	subramanian	subramanian	PROPN
ejpam-4592	423	18	et	et	PROPN
ejpam-4592	423	19	al	al	PROPN
ejpam-4592	423	20	.	.	PUNCT
ejpam-4592	423	21	/	/	SYM
ejpam-4592	423	22	eur	eur	PROPN
ejpam-4592	423	23	.	.	PUNCT
ejpam-4592	424	1	j.	j.	PROPN
ejpam-4592	424	2	pure	pure	PROPN
ejpam-4592	424	3	appl	appl	PROPN
ejpam-4592	424	4	.	.	PROPN
ejpam-4592	424	5	math	math	PROPN
ejpam-4592	424	6	,	,	PUNCT
ejpam-4592	424	7	15	15	NUM
ejpam-4592	424	8	(	(	PUNCT
ejpam-4592	424	9	4	4	NUM
ejpam-4592	424	10	)	)	PUNCT
ejpam-4592	424	11	(	(	PUNCT
ejpam-4592	424	12	2022	2022	NUM
ejpam-4592	424	13	)	)	PUNCT
ejpam-4592	424	14	,	,	PUNCT
ejpam-4592	424	15	1869	1869	NUM
ejpam-4592	424	16	-	-	SYM
ejpam-4592	424	17	1886	1886	NUM
ejpam-4592	424	18	1881	1881	NUM
ejpam-4592	424	19	respect	respect	NOUN
ejpam-4592	424	20	to	to	ADP
ejpam-4592	424	21	σ	σ	PROPN
ejpam-4592	424	22	∈	∈	PROPN
ejpam-4592	424	23	ω0	ω0	NOUN
ejpam-4592	424	24	and	and	CCONJ
ejpam-4592	424	25	{	{	PUNCT
ejpam-4592	424	26	dn	dn	VERB
ejpam-4592	424	27	}	}	PUNCT
ejpam-4592	424	28	is	be	AUX
ejpam-4592	424	29	cauchy	cauchy	ADJ
ejpam-4592	424	30	with	with	ADP
ejpam-4592	424	31	the	the	DET
ejpam-4592	424	32	same	same	ADJ
ejpam-4592	424	33	metric	metric	NOUN
ejpam-4592	424	34	.	.	PUNCT
ejpam-4592	425	1	then	then	ADV
ejpam-4592	425	2	{	{	PUNCT
ejpam-4592	425	3	dn	dn	X
ejpam-4592	425	4	}	}	PUNCT
ejpam-4592	425	5	is	be	AUX
ejpam-4592	425	6	convergent	convergent	ADJ
ejpam-4592	425	7	with	with	ADP
ejpam-4592	425	8	respect	respect	NOUN
ejpam-4592	425	9	to	to	ADP
ejpam-4592	425	10	σ	σ	PROPN
ejpam-4592	425	11	.	.	PUNCT
ejpam-4592	426	1	by	by	ADP
ejpam-4592	426	2	theorem	theorem	ADJ
ejpam-4592	426	3	27	27	NUM
ejpam-4592	426	4	(	(	PUNCT
ejpam-4592	426	5	b	b	NOUN
ejpam-4592	426	6	)	)	PUNCT
ejpam-4592	426	7	,	,	PUNCT
ejpam-4592	426	8	{	{	PUNCT
ejpam-4592	426	9	cn	cn	VERB
ejpam-4592	426	10	}	}	PUNCT
ejpam-4592	426	11	is	be	AUX
ejpam-4592	426	12	a	a	DET
ejpam-4592	426	13	convergent	convergent	NOUN
ejpam-4592	426	14	sequence	sequence	NOUN
ejpam-4592	426	15	with	with	ADP
ejpam-4592	426	16	the	the	DET
ejpam-4592	426	17	same	same	ADJ
ejpam-4592	426	18	metric	metric	NOUN
ejpam-4592	426	19	.	.	PUNCT
ejpam-4592	427	1	conversely	conversely	ADV
ejpam-4592	427	2	,	,	PUNCT
ejpam-4592	427	3	let	let	VERB
ejpam-4592	427	4	{	{	PUNCT
ejpam-4592	427	5	zn	zn	PART
ejpam-4592	427	6	}	}	PUNCT
ejpam-4592	427	7	be	be	AUX
ejpam-4592	427	8	a	a	DET
ejpam-4592	427	9	cauchy	cauchy	ADJ
ejpam-4592	427	10	sequence	sequence	NOUN
ejpam-4592	427	11	with	with	ADP
ejpam-4592	427	12	respect	respect	NOUN
ejpam-4592	427	13	to	to	ADP
ejpam-4592	427	14	the	the	DET
ejpam-4592	427	15	metric	metric	ADJ
ejpam-4592	427	16	σ	σ	PROPN
ejpam-4592	427	17	∈	∈	PROPN
ejpam-4592	427	18	ω0	ω0	NOUN
ejpam-4592	427	19	.	.	PUNCT
ejpam-4592	428	1	define	define	VERB
ejpam-4592	428	2	the	the	DET
ejpam-4592	428	3	sequence	sequence	NOUN
ejpam-4592	428	4	{	{	PUNCT
ejpam-4592	428	5	tn	tn	NOUN
ejpam-4592	428	6	}	}	PUNCT
ejpam-4592	428	7	in	in	ADP
ejpam-4592	428	8	x	x	PUNCT
ejpam-4592	428	9	by	by	ADP
ejpam-4592	428	10	t1	t1	NOUN
ejpam-4592	428	11	=	=	SYM
ejpam-4592	428	12	2	2	NUM
ejpam-4592	428	13	,	,	PUNCT
ejpam-4592	428	14	t2	t2	NOUN
ejpam-4592	428	15	=	=	SYM
ejpam-4592	428	16	1	1	NUM
ejpam-4592	428	17	and	and	CCONJ
ejpam-4592	428	18	tn	tn	NOUN
ejpam-4592	428	19	=	=	SYM
ejpam-4592	428	20	zn−2	zn−2	PROPN
ejpam-4592	428	21	for	for	ADP
ejpam-4592	428	22	n	n	PRON
ejpam-4592	428	23	≥	≥	NOUN
ejpam-4592	428	24	3	3	NUM
ejpam-4592	428	25	.	.	PUNCT
ejpam-4592	429	1	since	since	SCONJ
ejpam-4592	429	2	{	{	PUNCT
ejpam-4592	429	3	zn	zn	X
ejpam-4592	429	4	}	}	PUNCT
ejpam-4592	429	5	is	be	AUX
ejpam-4592	429	6	cauchy	cauchy	NOUN
ejpam-4592	429	7	,	,	PUNCT
ejpam-4592	429	8	for	for	ADP
ejpam-4592	429	9	every	every	DET
ejpam-4592	429	10	ε	ε	PROPN
ejpam-4592	429	11	>	>	X
ejpam-4592	429	12	0	0	PROPN
ejpam-4592	429	13	,	,	PUNCT
ejpam-4592	429	14	there	there	PRON
ejpam-4592	429	15	exists	exist	VERB
ejpam-4592	429	16	n0	n0	PROPN
ejpam-4592	429	17	∈	∈	PROPN
ejpam-4592	429	18	n	n	PRON
ejpam-4592	429	19	such	such	ADJ
ejpam-4592	429	20	that	that	SCONJ
ejpam-4592	429	21	σ(zn	σ(zn	PROPN
ejpam-4592	429	22	,	,	PUNCT
ejpam-4592	429	23	tn	tn	PROPN
ejpam-4592	429	24	)	)	PUNCT
ejpam-4592	429	25	<	<	X
ejpam-4592	429	26	ε	ε	PROPN
ejpam-4592	429	27	for	for	ADP
ejpam-4592	429	28	every	every	DET
ejpam-4592	429	29	n	n	PRON
ejpam-4592	429	30	≥	≥	NOUN
ejpam-4592	429	31	n0	n0	NUM
ejpam-4592	429	32	.	.	PUNCT
ejpam-4592	430	1	thus	thus	ADV
ejpam-4592	430	2	,	,	PUNCT
ejpam-4592	430	3	the	the	DET
ejpam-4592	430	4	pair	pair	NOUN
ejpam-4592	430	5	of	of	ADP
ejpam-4592	430	6	sequence	sequence	NOUN
ejpam-4592	430	7	{	{	PUNCT
ejpam-4592	430	8	zn	zn	NOUN
ejpam-4592	430	9	}	}	PUNCT
ejpam-4592	430	10	and	and	CCONJ
ejpam-4592	430	11	{	{	PUNCT
ejpam-4592	430	12	tn	tn	NOUN
ejpam-4592	430	13	}	}	PUNCT
ejpam-4592	430	14	is	be	AUX
ejpam-4592	430	15	asymptotic	asymptotic	ADJ
ejpam-4592	430	16	with	with	ADP
ejpam-4592	430	17	respect	respect	NOUN
ejpam-4592	430	18	to	to	ADP
ejpam-4592	430	19	σ	σ	PROPN
ejpam-4592	430	20	∈	∈	PROPN
ejpam-4592	430	21	ω0	ω0	NOUN
ejpam-4592	430	22	.	.	PUNCT
ejpam-4592	431	1	by	by	ADP
ejpam-4592	431	2	assumption	assumption	NOUN
ejpam-4592	431	3	,	,	PUNCT
ejpam-4592	431	4	{	{	PUNCT
ejpam-4592	431	5	tn	tn	NOUN
ejpam-4592	431	6	}	}	PUNCT
ejpam-4592	431	7	is	be	AUX
ejpam-4592	431	8	a	a	DET
ejpam-4592	431	9	convergent	convergent	NOUN
ejpam-4592	431	10	sequence	sequence	NOUN
ejpam-4592	431	11	with	with	ADP
ejpam-4592	431	12	the	the	DET
ejpam-4592	431	13	same	same	ADJ
ejpam-4592	431	14	metric	metric	NOUN
ejpam-4592	431	15	.	.	PUNCT
ejpam-4592	432	1	by	by	ADP
ejpam-4592	432	2	theorem	theorem	ADJ
ejpam-4592	432	3	27	27	NUM
ejpam-4592	432	4	(	(	PUNCT
ejpam-4592	432	5	b	b	NOUN
ejpam-4592	432	6	)	)	PUNCT
ejpam-4592	432	7	,	,	PUNCT
ejpam-4592	432	8	{	{	PUNCT
ejpam-4592	432	9	zn	zn	X
ejpam-4592	432	10	}	}	PUNCT
ejpam-4592	432	11	is	be	AUX
ejpam-4592	432	12	a	a	DET
ejpam-4592	432	13	convergent	convergent	NOUN
ejpam-4592	432	14	sequence	sequence	NOUN
ejpam-4592	432	15	with	with	ADP
ejpam-4592	432	16	respect	respect	NOUN
ejpam-4592	432	17	to	to	ADP
ejpam-4592	432	18	σ	σ	PROPN
ejpam-4592	432	19	∈	∈	PROPN
ejpam-4592	432	20	ω0	ω0	NOUN
ejpam-4592	432	21	.	.	PUNCT
ejpam-4592	433	1	since	since	SCONJ
ejpam-4592	433	2	σ	σ	PROPN
ejpam-4592	433	3	is	be	AUX
ejpam-4592	433	4	arbitrary	arbitrary	ADJ
ejpam-4592	433	5	,	,	PUNCT
ejpam-4592	433	6	ω0	ω0	ADV
ejpam-4592	433	7	consisting	consist	VERB
ejpam-4592	433	8	of	of	ADP
ejpam-4592	433	9	complete	complete	ADJ
ejpam-4592	433	10	metrics	metric	NOUN
ejpam-4592	433	11	.	.	PUNCT
ejpam-4592	434	1	therefore	therefore	ADV
ejpam-4592	434	2	,	,	PUNCT
ejpam-4592	434	3	(	(	PUNCT
ejpam-4592	434	4	x	x	X
ejpam-4592	434	5	,	,	PUNCT
ejpam-4592	434	6	ω	ω	NUM
ejpam-4592	434	7	)	)	PUNCT
ejpam-4592	434	8	is	be	AUX
ejpam-4592	434	9	a	a	DET
ejpam-4592	434	10	weakly	weakly	ADJ
ejpam-4592	434	11	complete	complete	ADJ
ejpam-4592	434	12	space	space	NOUN
ejpam-4592	434	13	.	.	PUNCT
ejpam-4592	435	1	definition	definition	NOUN
ejpam-4592	435	2	39	39	NUM
ejpam-4592	435	3	.	.	PUNCT
ejpam-4592	436	1	let	let	VERB
ejpam-4592	436	2	(	(	PUNCT
ejpam-4592	436	3	x	x	NOUN
ejpam-4592	436	4	,	,	PUNCT
ejpam-4592	436	5	ω1	ω1	PROPN
ejpam-4592	436	6	)	)	PUNCT
ejpam-4592	436	7	and	and	CCONJ
ejpam-4592	436	8	(	(	PUNCT
ejpam-4592	436	9	y	y	PROPN
ejpam-4592	436	10	,	,	PUNCT
ejpam-4592	436	11	ω2	ω2	NUM
ejpam-4592	436	12	)	)	PUNCT
ejpam-4592	436	13	be	be	VERB
ejpam-4592	436	14	two	two	NUM
ejpam-4592	436	15	generalized	generalized	ADJ
ejpam-4592	436	16	metric	metric	ADJ
ejpam-4592	436	17	spaces	space	NOUN
ejpam-4592	436	18	.	.	PUNCT
ejpam-4592	437	1	a	a	DET
ejpam-4592	437	2	function	function	NOUN
ejpam-4592	437	3	h	h	NOUN
ejpam-4592	437	4	:	:	PUNCT
ejpam-4592	437	5	(	(	PUNCT
ejpam-4592	437	6	x	x	X
ejpam-4592	437	7	,	,	PUNCT
ejpam-4592	437	8	ω1	ω1	PROPN
ejpam-4592	437	9	)	)	PUNCT
ejpam-4592	437	10	→	→	SYM
ejpam-4592	437	11	(	(	PUNCT
ejpam-4592	437	12	y	y	PROPN
ejpam-4592	437	13	,	,	PUNCT
ejpam-4592	437	14	ω2	ω2	NUM
ejpam-4592	437	15	)	)	PUNCT
ejpam-4592	437	16	is	be	AUX
ejpam-4592	437	17	said	say	VERB
ejpam-4592	437	18	to	to	PART
ejpam-4592	437	19	be	be	AUX
ejpam-4592	437	20	cauchy	cauchy	NOUN
ejpam-4592	437	21	-	-	PUNCT
ejpam-4592	437	22	continuous	continuous	ADJ
ejpam-4592	437	23	if	if	SCONJ
ejpam-4592	437	24	{	{	PUNCT
ejpam-4592	437	25	h(zn)}n∈n	h(zn)}n∈n	PROPN
ejpam-4592	437	26	is	be	AUX
ejpam-4592	437	27	a	a	DET
ejpam-4592	437	28	cauchy	cauchy	ADJ
ejpam-4592	437	29	sequence	sequence	NOUN
ejpam-4592	437	30	in	in	ADP
ejpam-4592	437	31	(	(	PUNCT
ejpam-4592	437	32	y	y	PROPN
ejpam-4592	437	33	,	,	PUNCT
ejpam-4592	437	34	ω2	ω2	NUM
ejpam-4592	437	35	)	)	PUNCT
ejpam-4592	437	36	for	for	ADP
ejpam-4592	437	37	any	any	DET
ejpam-4592	437	38	cauchy	cauchy	ADJ
ejpam-4592	437	39	sequence	sequence	NOUN
ejpam-4592	437	40	{	{	PUNCT
ejpam-4592	437	41	zn}n∈n	zn}n∈n	NUM
ejpam-4592	437	42	in	in	ADP
ejpam-4592	437	43	(	(	PUNCT
ejpam-4592	437	44	x	x	NOUN
ejpam-4592	437	45	,	,	PUNCT
ejpam-4592	437	46	ω1	ω1	PROPN
ejpam-4592	437	47	)	)	PUNCT
ejpam-4592	437	48	.	.	PUNCT
ejpam-4592	438	1	that	that	PRON
ejpam-4592	438	2	is	be	AUX
ejpam-4592	438	3	,	,	PUNCT
ejpam-4592	438	4	h	h	PROPN
ejpam-4592	438	5	is	be	AUX
ejpam-4592	438	6	cauchy	cauchy	NOUN
ejpam-4592	438	7	-	-	PUNCT
ejpam-4592	438	8	continuous	continuous	ADJ
ejpam-4592	438	9	if	if	SCONJ
ejpam-4592	438	10	{	{	PUNCT
ejpam-4592	438	11	zn	zn	NOUN
ejpam-4592	438	12	}	}	PUNCT
ejpam-4592	438	13	is	be	AUX
ejpam-4592	438	14	cauchy	cauchy	ADJ
ejpam-4592	438	15	with	with	ADP
ejpam-4592	438	16	respect	respect	NOUN
ejpam-4592	438	17	to	to	ADP
ejpam-4592	438	18	σ	σ	PROPN
ejpam-4592	438	19	∈	∈	PROPN
ejpam-4592	438	20	ω1	ω1	PROPN
ejpam-4592	438	21	,	,	PUNCT
ejpam-4592	438	22	then	then	ADV
ejpam-4592	438	23	there	there	PRON
ejpam-4592	438	24	is	be	VERB
ejpam-4592	438	25	σ	σ	PROPN
ejpam-4592	438	26	∈	∈	PROPN
ejpam-4592	438	27	ω2	ω2	NOUN
ejpam-4592	438	28	such	such	ADJ
ejpam-4592	438	29	that	that	SCONJ
ejpam-4592	438	30	{	{	PUNCT
ejpam-4592	438	31	h(zn	h(zn	X
ejpam-4592	438	32	)	)	PUNCT
ejpam-4592	438	33	}	}	PUNCT
ejpam-4592	438	34	is	be	AUX
ejpam-4592	438	35	cauchy	cauchy	ADJ
ejpam-4592	438	36	with	with	ADP
ejpam-4592	438	37	respect	respect	NOUN
ejpam-4592	438	38	to	to	ADP
ejpam-4592	438	39	σ	σ	PROPN
ejpam-4592	438	40	.	.	PUNCT
ejpam-4592	438	41	theorem	theorem	PROPN
ejpam-4592	438	42	40	40	NUM
ejpam-4592	438	43	.	.	PUNCT
ejpam-4592	439	1	let	let	AUX
ejpam-4592	439	2	(	(	PUNCT
ejpam-4592	439	3	x	x	NOUN
ejpam-4592	439	4	,	,	PUNCT
ejpam-4592	439	5	ω1	ω1	PROPN
ejpam-4592	439	6	)	)	PUNCT
ejpam-4592	439	7	and	and	CCONJ
ejpam-4592	439	8	(	(	PUNCT
ejpam-4592	439	9	y	y	PROPN
ejpam-4592	439	10	,	,	PUNCT
ejpam-4592	439	11	ω2	ω2	NUM
ejpam-4592	439	12	)	)	PUNCT
ejpam-4592	439	13	be	be	AUX
ejpam-4592	439	14	two	two	NUM
ejpam-4592	439	15	gmss	gmss	NOUN
ejpam-4592	439	16	.	.	PUNCT
ejpam-4592	440	1	if	if	SCONJ
ejpam-4592	440	2	h	h	PRON
ejpam-4592	440	3	:	:	PUNCT
ejpam-4592	440	4	x	x	X
ejpam-4592	440	5	→	→	SYM
ejpam-4592	440	6	y	y	PROPN
ejpam-4592	440	7	is	be	AUX
ejpam-4592	440	8	a	a	DET
ejpam-4592	440	9	cauchy	cauchy	ADJ
ejpam-4592	440	10	continuous	continuous	ADJ
ejpam-4592	440	11	map	map	NOUN
ejpam-4592	440	12	and	and	CCONJ
ejpam-4592	440	13	if	if	SCONJ
ejpam-4592	440	14	{	{	PUNCT
ejpam-4592	440	15	cn	cn	NOUN
ejpam-4592	440	16	}	}	PUNCT
ejpam-4592	440	17	is	be	AUX
ejpam-4592	440	18	a	a	DET
ejpam-4592	440	19	cofinally	cofinally	ADV
ejpam-4592	440	20	-	-	PUNCT
ejpam-4592	440	21	cauchy	cauchy	ADJ
ejpam-4592	440	22	sequence	sequence	NOUN
ejpam-4592	440	23	in	in	ADP
ejpam-4592	440	24	x	x	NOUN
ejpam-4592	440	25	,	,	PUNCT
ejpam-4592	440	26	then	then	ADV
ejpam-4592	440	27	{	{	PUNCT
ejpam-4592	440	28	h(cn	h(cn	PROPN
ejpam-4592	440	29	)	)	PUNCT
ejpam-4592	440	30	}	}	PUNCT
ejpam-4592	440	31	has	have	VERB
ejpam-4592	440	32	a	a	DET
ejpam-4592	440	33	cofinally	cofinally	ADV
ejpam-4592	440	34	-	-	PUNCT
ejpam-4592	440	35	cauchy	cauchy	ADJ
ejpam-4592	440	36	subsequence	subsequence	NOUN
ejpam-4592	440	37	in	in	ADP
ejpam-4592	440	38	y.	y.	PROPN
ejpam-4592	440	39	proof	proof	NOUN
ejpam-4592	440	40	.	.	PUNCT
ejpam-4592	441	1	let	let	VERB
ejpam-4592	441	2	{	{	PUNCT
ejpam-4592	441	3	cn	cn	AUX
ejpam-4592	441	4	}	}	PUNCT
ejpam-4592	441	5	be	be	AUX
ejpam-4592	441	6	a	a	DET
ejpam-4592	441	7	cofinally	cofinally	ADV
ejpam-4592	441	8	-	-	PUNCT
ejpam-4592	441	9	cauchy	cauchy	ADJ
ejpam-4592	441	10	sequence	sequence	NOUN
ejpam-4592	441	11	in	in	ADP
ejpam-4592	441	12	x.	x.	NOUN
ejpam-4592	441	13	then	then	ADV
ejpam-4592	441	14	there	there	PRON
ejpam-4592	441	15	exists	exist	VERB
ejpam-4592	441	16	a	a	DET
ejpam-4592	441	17	metric	metric	ADJ
ejpam-4592	441	18	σ	σ	PROPN
ejpam-4592	441	19	∈	∈	PROPN
ejpam-4592	441	20	ω1	ω1	PROPN
ejpam-4592	441	21	such	such	ADJ
ejpam-4592	441	22	that	that	SCONJ
ejpam-4592	441	23	{	{	PUNCT
ejpam-4592	441	24	cn	cn	NOUN
ejpam-4592	441	25	}	}	PUNCT
ejpam-4592	441	26	is	be	AUX
ejpam-4592	441	27	a	a	DET
ejpam-4592	441	28	cofinally	cofinally	ADV
ejpam-4592	441	29	-	-	PUNCT
ejpam-4592	441	30	cauchy	cauchy	ADJ
ejpam-4592	441	31	sequence	sequence	NOUN
ejpam-4592	441	32	with	with	ADP
ejpam-4592	441	33	respect	respect	NOUN
ejpam-4592	441	34	to	to	ADP
ejpam-4592	441	35	this	this	DET
ejpam-4592	441	36	metric	metric	NOUN
ejpam-4592	441	37	in	in	ADP
ejpam-4592	441	38	x.	x.	NOUN
ejpam-4592	441	39	by	by	ADP
ejpam-4592	441	40	theorem	theorem	NOUN
ejpam-4592	441	41	19	19	NUM
ejpam-4592	441	42	,	,	PUNCT
ejpam-4592	441	43	{	{	PUNCT
ejpam-4592	441	44	cn	cn	VERB
ejpam-4592	441	45	}	}	PUNCT
ejpam-4592	441	46	has	have	AUX
ejpam-4592	441	47	a	a	DET
ejpam-4592	441	48	cauchy	cauchy	ADJ
ejpam-4592	441	49	subsequence	subsequence	NOUN
ejpam-4592	441	50	{	{	PUNCT
ejpam-4592	441	51	cnk	cnk	NOUN
ejpam-4592	441	52	}	}	PUNCT
ejpam-4592	441	53	with	with	ADP
ejpam-4592	441	54	respect	respect	NOUN
ejpam-4592	441	55	to	to	ADP
ejpam-4592	441	56	the	the	DET
ejpam-4592	441	57	same	same	ADJ
ejpam-4592	441	58	metric	metric	NOUN
ejpam-4592	441	59	in	in	ADP
ejpam-4592	441	60	x.	x.	NOUN
ejpam-4592	441	61	since	since	SCONJ
ejpam-4592	441	62	h	h	PROPN
ejpam-4592	441	63	is	be	AUX
ejpam-4592	441	64	cauchy	cauchy	PROPN
ejpam-4592	441	65	continuous	continuous	ADJ
ejpam-4592	441	66	,	,	PUNCT
ejpam-4592	441	67	there	there	PRON
ejpam-4592	441	68	exists	exist	VERB
ejpam-4592	441	69	a	a	DET
ejpam-4592	441	70	metric	metric	ADJ
ejpam-4592	441	71	d	d	PROPN
ejpam-4592	441	72	∈	∈	PROPN
ejpam-4592	441	73	ω2	ω2	NOUN
ejpam-4592	441	74	such	such	ADJ
ejpam-4592	441	75	that	that	SCONJ
ejpam-4592	441	76	{	{	PUNCT
ejpam-4592	441	77	h(cnk	h(cnk	PROPN
ejpam-4592	441	78	)	)	PUNCT
ejpam-4592	441	79	}	}	PUNCT
ejpam-4592	441	80	is	be	AUX
ejpam-4592	441	81	cauchy	cauchy	ADJ
ejpam-4592	441	82	with	with	ADP
ejpam-4592	441	83	respect	respect	NOUN
ejpam-4592	441	84	to	to	ADP
ejpam-4592	441	85	d.	d.	PROPN
ejpam-4592	441	86	by	by	ADP
ejpam-4592	441	87	theorem	theorem	VERB
ejpam-4592	441	88	13	13	NUM
ejpam-4592	441	89	(	(	PUNCT
ejpam-4592	441	90	a	a	NOUN
ejpam-4592	441	91	)	)	PUNCT
ejpam-4592	441	92	,	,	PUNCT
ejpam-4592	441	93	{	{	PUNCT
ejpam-4592	441	94	h(cnk	h(cnk	PROPN
ejpam-4592	441	95	)	)	PUNCT
ejpam-4592	441	96	}	}	PUNCT
ejpam-4592	441	97	is	be	AUX
ejpam-4592	441	98	a	a	DET
ejpam-4592	441	99	cofinally	cofinally	ADV
ejpam-4592	441	100	-	-	PUNCT
ejpam-4592	441	101	cauchy	cauchy	ADJ
ejpam-4592	441	102	sequence	sequence	NOUN
ejpam-4592	441	103	with	with	ADP
ejpam-4592	441	104	respect	respect	NOUN
ejpam-4592	441	105	to	to	ADP
ejpam-4592	441	106	the	the	DET
ejpam-4592	441	107	metric	metric	ADJ
ejpam-4592	441	108	d.	d.	PROPN
ejpam-4592	441	109	hence	hence	ADV
ejpam-4592	441	110	{	{	PUNCT
ejpam-4592	441	111	h(cn	h(cn	PROPN
ejpam-4592	441	112	)	)	PUNCT
ejpam-4592	441	113	}	}	PUNCT
ejpam-4592	441	114	has	have	VERB
ejpam-4592	441	115	a	a	DET
ejpam-4592	441	116	cofinally	cofinally	ADV
ejpam-4592	441	117	-	-	PUNCT
ejpam-4592	441	118	cauchy	cauchy	ADJ
ejpam-4592	441	119	subsequence	subsequence	NOUN
ejpam-4592	441	120	in	in	ADP
ejpam-4592	441	121	y.	y.	PROPN
ejpam-4592	441	122	definition	definition	NOUN
ejpam-4592	441	123	41	41	NUM
ejpam-4592	441	124	.	.	PUNCT
ejpam-4592	442	1	let	let	VERB
ejpam-4592	442	2	(	(	PUNCT
ejpam-4592	442	3	x	x	NOUN
ejpam-4592	442	4	,	,	PUNCT
ejpam-4592	442	5	ω1	ω1	PROPN
ejpam-4592	442	6	)	)	PUNCT
ejpam-4592	442	7	and	and	CCONJ
ejpam-4592	442	8	(	(	PUNCT
ejpam-4592	442	9	y	y	PROPN
ejpam-4592	442	10	,	,	PUNCT
ejpam-4592	442	11	ω2	ω2	NUM
ejpam-4592	442	12	)	)	PUNCT
ejpam-4592	442	13	be	be	VERB
ejpam-4592	442	14	two	two	NUM
ejpam-4592	442	15	generalized	generalized	ADJ
ejpam-4592	442	16	metric	metric	ADJ
ejpam-4592	442	17	spaces	space	NOUN
ejpam-4592	442	18	.	.	PUNCT
ejpam-4592	443	1	a	a	DET
ejpam-4592	443	2	function	function	NOUN
ejpam-4592	443	3	h	h	NOUN
ejpam-4592	443	4	:	:	PUNCT
ejpam-4592	443	5	x	x	X
ejpam-4592	443	6	→	→	SYM
ejpam-4592	443	7	y	y	PROPN
ejpam-4592	443	8	is	be	AUX
ejpam-4592	443	9	said	say	VERB
ejpam-4592	443	10	to	to	PART
ejpam-4592	443	11	be	be	AUX
ejpam-4592	443	12	:	:	PUNCT
ejpam-4592	443	13	i	i	PROPN
ejpam-4592	443	14	)	)	PUNCT
ejpam-4592	443	15	.	.	PUNCT
ejpam-4592	444	1	bourbaki	bourbaki	VERB
ejpam-4592	444	2	-	-	PUNCT
ejpam-4592	444	3	cauchy	cauchy	VERB
ejpam-4592	444	4	regular	regular	NOUN
ejpam-4592	444	5	(	(	PUNCT
ejpam-4592	444	6	in	in	ADP
ejpam-4592	444	7	short	short	ADJ
ejpam-4592	444	8	,	,	PUNCT
ejpam-4592	444	9	bc	bc	ADJ
ejpam-4592	444	10	-	-	NOUN
ejpam-4592	444	11	regular	regular	ADJ
ejpam-4592	444	12	)	)	PUNCT
ejpam-4592	444	13	if	if	SCONJ
ejpam-4592	444	14	{	{	PUNCT
ejpam-4592	444	15	h(zn)}n∈n	h(zn)}n∈n	PROPN
ejpam-4592	444	16	is	be	AUX
ejpam-4592	444	17	a	a	DET
ejpam-4592	444	18	bourbaki	bourbaki	NOUN
ejpam-4592	444	19	cauchy	cauchy	ADJ
ejpam-4592	444	20	sequence	sequence	NOUN
ejpam-4592	444	21	in	in	ADP
ejpam-4592	444	22	(	(	PUNCT
ejpam-4592	444	23	y	y	PROPN
ejpam-4592	444	24	,	,	PUNCT
ejpam-4592	444	25	ω2	ω2	NUM
ejpam-4592	444	26	)	)	PUNCT
ejpam-4592	444	27	whenever	whenever	SCONJ
ejpam-4592	444	28	{	{	PUNCT
ejpam-4592	444	29	zn}n∈n	zn}n∈n	VERB
ejpam-4592	444	30	is	be	AUX
ejpam-4592	444	31	bourbaki	bourbaki	NOUN
ejpam-4592	444	32	cauchy	cauchy	NOUN
ejpam-4592	444	33	in	in	ADP
ejpam-4592	444	34	(	(	PUNCT
ejpam-4592	444	35	x	x	NOUN
ejpam-4592	444	36	,	,	PUNCT
ejpam-4592	444	37	ω1	ω1	PROPN
ejpam-4592	444	38	)	)	PUNCT
ejpam-4592	444	39	.	.	PUNCT
ejpam-4592	445	1	ii	ii	PROPN
ejpam-4592	445	2	)	)	PUNCT
ejpam-4592	445	3	.	.	PUNCT
ejpam-4592	446	1	cofinally	cofinally	ADV
ejpam-4592	446	2	-	-	PUNCT
ejpam-4592	446	3	cauchy	cauchy	NOUN
ejpam-4592	446	4	regular	regular	NOUN
ejpam-4592	446	5	(	(	PUNCT
ejpam-4592	446	6	briefly	briefly	ADV
ejpam-4592	446	7	,	,	PUNCT
ejpam-4592	446	8	cc	cc	NOUN
ejpam-4592	446	9	-	-	NOUN
ejpam-4592	446	10	regular	regular	ADJ
ejpam-4592	446	11	)	)	PUNCT
ejpam-4592	446	12	if	if	SCONJ
ejpam-4592	446	13	{	{	PUNCT
ejpam-4592	446	14	h(zn)}n∈n	h(zn)}n∈n	NOUN
ejpam-4592	446	15	is	be	AUX
ejpam-4592	446	16	a	a	DET
ejpam-4592	446	17	cofinally	cofinally	ADV
ejpam-4592	446	18	cauchy	cauchy	ADJ
ejpam-4592	446	19	sequence	sequence	NOUN
ejpam-4592	446	20	in	in	ADP
ejpam-4592	446	21	(	(	PUNCT
ejpam-4592	446	22	y	y	PROPN
ejpam-4592	446	23	,	,	PUNCT
ejpam-4592	446	24	ω2	ω2	NUM
ejpam-4592	446	25	)	)	PUNCT
ejpam-4592	446	26	whenever	whenever	SCONJ
ejpam-4592	446	27	{	{	PUNCT
ejpam-4592	446	28	zn}n∈n	zn}n∈n	VERB
ejpam-4592	446	29	is	be	AUX
ejpam-4592	446	30	cofinally	cofinally	ADV
ejpam-4592	446	31	cauchy	cauchy	ADJ
ejpam-4592	446	32	in	in	ADP
ejpam-4592	446	33	(	(	PUNCT
ejpam-4592	446	34	x	x	NOUN
ejpam-4592	446	35	,	,	PUNCT
ejpam-4592	446	36	ω1	ω1	PROPN
ejpam-4592	446	37	)	)	PUNCT
ejpam-4592	446	38	.	.	PUNCT
ejpam-4592	447	1	iii	iii	X
ejpam-4592	447	2	)	)	PUNCT
ejpam-4592	447	3	.	.	PUNCT
ejpam-4592	448	1	pseudo	pseudo	NOUN
ejpam-4592	448	2	-	-	NOUN
ejpam-4592	448	3	cauchy	cauchy	ADJ
ejpam-4592	448	4	regular	regular	NOUN
ejpam-4592	448	5	(	(	PUNCT
ejpam-4592	448	6	in	in	ADP
ejpam-4592	448	7	short	short	ADJ
ejpam-4592	448	8	,	,	PUNCT
ejpam-4592	448	9	pc	pc	NOUN
ejpam-4592	448	10	-	-	NOUN
ejpam-4592	448	11	regular	regular	ADJ
ejpam-4592	448	12	)	)	PUNCT
ejpam-4592	448	13	if	if	SCONJ
ejpam-4592	448	14	{	{	PUNCT
ejpam-4592	448	15	h(zn)}n∈n	h(zn)}n∈n	NOUN
ejpam-4592	448	16	is	be	AUX
ejpam-4592	448	17	a	a	DET
ejpam-4592	448	18	pseudo	pseudo	NOUN
ejpam-4592	448	19	cauchy	cauchy	NOUN
ejpam-4592	448	20	sequence	sequence	NOUN
ejpam-4592	448	21	in	in	ADP
ejpam-4592	448	22	(	(	PUNCT
ejpam-4592	448	23	y	y	PROPN
ejpam-4592	448	24	,	,	PUNCT
ejpam-4592	448	25	ω2	ω2	NUM
ejpam-4592	448	26	)	)	PUNCT
ejpam-4592	448	27	whenever	whenever	SCONJ
ejpam-4592	448	28	{	{	PUNCT
ejpam-4592	448	29	zn}n∈n	zn}n∈n	VERB
ejpam-4592	448	30	is	be	AUX
ejpam-4592	448	31	pseudo	pseudo	NOUN
ejpam-4592	448	32	cauchy	cauchy	NOUN
ejpam-4592	448	33	in	in	ADP
ejpam-4592	448	34	(	(	PUNCT
ejpam-4592	448	35	x	x	NOUN
ejpam-4592	448	36	,	,	PUNCT
ejpam-4592	448	37	ω1	ω1	PROPN
ejpam-4592	448	38	)	)	PUNCT
ejpam-4592	448	39	.	.	PUNCT
ejpam-4592	449	1	that	that	PRON
ejpam-4592	449	2	is	be	AUX
ejpam-4592	449	3	,	,	PUNCT
ejpam-4592	449	4	h	h	PROPN
ejpam-4592	449	5	is	be	AUX
ejpam-4592	449	6	bourbaki	bourbaki	NOUN
ejpam-4592	449	7	-	-	PUNCT
ejpam-4592	449	8	cauchy	cauchy	ADJ
ejpam-4592	449	9	regular	regular	ADJ
ejpam-4592	449	10	(	(	PUNCT
ejpam-4592	449	11	resp	resp	NOUN
ejpam-4592	449	12	.	.	PUNCT
ejpam-4592	450	1	cc	cc	NOUN
ejpam-4592	450	2	-	-	ADJ
ejpam-4592	450	3	regular	regular	ADJ
ejpam-4592	450	4	,	,	PUNCT
ejpam-4592	450	5	pc	pc	NOUN
ejpam-4592	450	6	-	-	NOUN
ejpam-4592	450	7	regular	regular	ADJ
ejpam-4592	450	8	)	)	PUNCT
ejpam-4592	450	9	if	if	SCONJ
ejpam-4592	450	10	{	{	PUNCT
ejpam-4592	450	11	zn}n∈n	zn}n∈n	VERB
ejpam-4592	450	12	is	be	AUX
ejpam-4592	450	13	bourbaki	bourbaki	NOUN
ejpam-4592	450	14	(	(	PUNCT
ejpam-4592	450	15	resp	resp	NOUN
ejpam-4592	450	16	.	.	PUNCT
ejpam-4592	451	1	cofinally	cofinally	ADV
ejpam-4592	451	2	,	,	PUNCT
ejpam-4592	451	3	pseudo	pseudo	NOUN
ejpam-4592	451	4	)	)	PUNCT
ejpam-4592	451	5	cauchy	cauchy	NOUN
ejpam-4592	451	6	with	with	ADP
ejpam-4592	451	7	respect	respect	NOUN
ejpam-4592	451	8	to	to	ADP
ejpam-4592	451	9	σ1	σ1	PROPN
ejpam-4592	451	10	∈	∈	PROPN
ejpam-4592	451	11	ω1	ω1	PROPN
ejpam-4592	451	12	,	,	PUNCT
ejpam-4592	451	13	then	then	ADV
ejpam-4592	451	14	there	there	PRON
ejpam-4592	451	15	is	be	VERB
ejpam-4592	451	16	d	d	PROPN
ejpam-4592	451	17	∈	∈	PROPN
ejpam-4592	451	18	ω2	ω2	NOUN
ejpam-4592	451	19	such	such	ADJ
ejpam-4592	451	20	that	that	SCONJ
ejpam-4592	451	21	{	{	PUNCT
ejpam-4592	451	22	h(zn)}n∈n	h(zn)}n∈n	PROPN
ejpam-4592	451	23	is	be	AUX
ejpam-4592	451	24	bourbaki	bourbaki	NOUN
ejpam-4592	451	25	(	(	PUNCT
ejpam-4592	451	26	resp	resp	NOUN
ejpam-4592	451	27	.	.	PUNCT
ejpam-4592	452	1	cofinally	cofinally	ADV
ejpam-4592	452	2	,	,	PUNCT
ejpam-4592	452	3	pseudo	pseudo	NOUN
ejpam-4592	452	4	)	)	PUNCT
ejpam-4592	452	5	cauchy	cauchy	NOUN
ejpam-4592	452	6	with	with	ADP
ejpam-4592	452	7	respect	respect	NOUN
ejpam-4592	452	8	to	to	ADP
ejpam-4592	452	9	d.	d.	NOUN
ejpam-4592	452	10	the	the	DET
ejpam-4592	452	11	following	follow	VERB
ejpam-4592	452	12	three	three	NUM
ejpam-4592	452	13	theorems	theorem	NOUN
ejpam-4592	452	14	(	(	PUNCT
ejpam-4592	452	15	theorem	theorem	VERB
ejpam-4592	452	16	42	42	NUM
ejpam-4592	452	17	,	,	PUNCT
ejpam-4592	452	18	theorem	theorem	VERB
ejpam-4592	452	19	43	43	NUM
ejpam-4592	452	20	and	and	CCONJ
ejpam-4592	452	21	theorem	theorem	VERB
ejpam-4592	452	22	44	44	NUM
ejpam-4592	452	23	)	)	PUNCT
ejpam-4592	452	24	are	be	AUX
ejpam-4592	452	25	give	give	VERB
ejpam-4592	452	26	some	some	DET
ejpam-4592	452	27	relations	relation	NOUN
ejpam-4592	452	28	between	between	ADP
ejpam-4592	452	29	the	the	DET
ejpam-4592	452	30	functions	function	NOUN
ejpam-4592	452	31	defined	define	VERB
ejpam-4592	452	32	in	in	ADP
ejpam-4592	452	33	above	above	ADP
ejpam-4592	452	34	definition	definition	NOUN
ejpam-4592	452	35	41	41	NUM
ejpam-4592	452	36	.	.	PUNCT
ejpam-4592	453	1	v.	v.	ADP
ejpam-4592	453	2	subramanian	subramanian	PROPN
ejpam-4592	453	3	et	et	PROPN
ejpam-4592	453	4	al	al	PROPN
ejpam-4592	453	5	.	.	PUNCT
ejpam-4592	453	6	/	/	SYM
ejpam-4592	453	7	eur	eur	PROPN
ejpam-4592	453	8	.	.	PUNCT
ejpam-4592	454	1	j.	j.	PROPN
ejpam-4592	454	2	pure	pure	PROPN
ejpam-4592	454	3	appl	appl	PROPN
ejpam-4592	454	4	.	.	PROPN
ejpam-4592	454	5	math	math	PROPN
ejpam-4592	454	6	,	,	PUNCT
ejpam-4592	454	7	15	15	NUM
ejpam-4592	454	8	(	(	PUNCT
ejpam-4592	454	9	4	4	NUM
ejpam-4592	454	10	)	)	PUNCT
ejpam-4592	454	11	(	(	PUNCT
ejpam-4592	454	12	2022	2022	NUM
ejpam-4592	454	13	)	)	PUNCT
ejpam-4592	454	14	,	,	PUNCT
ejpam-4592	454	15	1869	1869	NUM
ejpam-4592	454	16	-	-	SYM
ejpam-4592	454	17	1886	1886	NUM
ejpam-4592	454	18	1882	1882	NUM
ejpam-4592	454	19	theorem	theorem	VERB
ejpam-4592	454	20	42	42	NUM
ejpam-4592	454	21	.	.	PUNCT
ejpam-4592	455	1	let	let	AUX
ejpam-4592	455	2	(	(	PUNCT
ejpam-4592	455	3	x	x	NOUN
ejpam-4592	455	4	,	,	PUNCT
ejpam-4592	455	5	ω1	ω1	PROPN
ejpam-4592	455	6	)	)	PUNCT
ejpam-4592	455	7	and	and	CCONJ
ejpam-4592	455	8	(	(	PUNCT
ejpam-4592	455	9	y	y	PROPN
ejpam-4592	455	10	,	,	PUNCT
ejpam-4592	455	11	ω2	ω2	NUM
ejpam-4592	455	12	)	)	PUNCT
ejpam-4592	455	13	be	be	VERB
ejpam-4592	455	14	two	two	NUM
ejpam-4592	455	15	gmss	gmss	NOUN
ejpam-4592	455	16	,	,	PUNCT
ejpam-4592	455	17	h	h	NOUN
ejpam-4592	455	18	:	:	PUNCT
ejpam-4592	455	19	x	x	X
ejpam-4592	455	20	→	→	SYM
ejpam-4592	455	21	y	y	PROPN
ejpam-4592	455	22	be	be	AUX
ejpam-4592	455	23	a	a	DET
ejpam-4592	455	24	bc	bc	ADJ
ejpam-4592	455	25	-	-	ADJ
ejpam-4592	455	26	regular	regular	ADJ
ejpam-4592	455	27	map	map	NOUN
ejpam-4592	455	28	.	.	PUNCT
ejpam-4592	456	1	if	if	SCONJ
ejpam-4592	456	2	every	every	DET
ejpam-4592	456	3	cofinally	cofinally	ADV
ejpam-4592	456	4	-	-	PUNCT
ejpam-4592	456	5	cauchy	cauchy	ADJ
ejpam-4592	456	6	sequence	sequence	NOUN
ejpam-4592	456	7	{	{	PUNCT
ejpam-4592	456	8	cn	cn	NOUN
ejpam-4592	456	9	}	}	PUNCT
ejpam-4592	456	10	in	in	ADP
ejpam-4592	456	11	x	x	NOUN
ejpam-4592	456	12	,	,	PUNCT
ejpam-4592	456	13	there	there	PRON
ejpam-4592	456	14	exists	exist	VERB
ejpam-4592	456	15	a	a	DET
ejpam-4592	456	16	sequence	sequence	NOUN
ejpam-4592	456	17	{	{	PUNCT
ejpam-4592	456	18	dn	dn	NOUN
ejpam-4592	456	19	}	}	PUNCT
ejpam-4592	456	20	in	in	ADP
ejpam-4592	456	21	x	x	PUNCT
ejpam-4592	456	22	with	with	ADP
ejpam-4592	456	23	cn	cn	PROPN
ejpam-4592	456	24	̸=	̸=	PROPN
ejpam-4592	456	25	dn	dn	NOUN
ejpam-4592	456	26	for	for	ADP
ejpam-4592	456	27	every	every	DET
ejpam-4592	456	28	n	n	PRON
ejpam-4592	456	29	∈	∈	NOUN
ejpam-4592	456	30	n	n	PRON
ejpam-4592	456	31	such	such	ADJ
ejpam-4592	456	32	that	that	SCONJ
ejpam-4592	456	33	the	the	DET
ejpam-4592	456	34	pair	pair	NOUN
ejpam-4592	456	35	of	of	ADP
ejpam-4592	456	36	sequence	sequence	NOUN
ejpam-4592	456	37	{	{	PUNCT
ejpam-4592	456	38	cn	cn	NOUN
ejpam-4592	456	39	}	}	PUNCT
ejpam-4592	456	40	and	and	CCONJ
ejpam-4592	456	41	{	{	PUNCT
ejpam-4592	456	42	dn	dn	VERB
ejpam-4592	456	43	}	}	PUNCT
ejpam-4592	456	44	is	be	AUX
ejpam-4592	456	45	uniformly	uniformly	ADV
ejpam-4592	456	46	asymptotic	asymptotic	ADJ
ejpam-4592	456	47	with	with	ADP
ejpam-4592	456	48	respect	respect	NOUN
ejpam-4592	456	49	to	to	ADP
ejpam-4592	456	50	the	the	DET
ejpam-4592	456	51	same	same	ADJ
ejpam-4592	456	52	metric	metric	NOUN
ejpam-4592	456	53	,	,	PUNCT
ejpam-4592	456	54	then	then	ADV
ejpam-4592	456	55	h	h	PROPN
ejpam-4592	456	56	is	be	AUX
ejpam-4592	456	57	a	a	DET
ejpam-4592	456	58	cc	cc	NOUN
ejpam-4592	456	59	-	-	ADJ
ejpam-4592	456	60	regular	regular	ADJ
ejpam-4592	456	61	map	map	NOUN
ejpam-4592	456	62	.	.	PUNCT
ejpam-4592	457	1	proof	proof	NOUN
ejpam-4592	457	2	.	.	PUNCT
ejpam-4592	458	1	let	let	VERB
ejpam-4592	458	2	{	{	PUNCT
ejpam-4592	458	3	an	an	PRON
ejpam-4592	458	4	}	}	PUNCT
ejpam-4592	458	5	∈	∈	PROPN
ejpam-4592	458	6	c(σ	c(σ	PROPN
ejpam-4592	458	7	)	)	PUNCT
ejpam-4592	458	8	where	where	SCONJ
ejpam-4592	458	9	σ	σ	PROPN
ejpam-4592	458	10	∈	∈	PROPN
ejpam-4592	458	11	ω1	ω1	PROPN
ejpam-4592	458	12	.	.	PUNCT
ejpam-4592	459	1	then	then	ADV
ejpam-4592	459	2	by	by	ADP
ejpam-4592	459	3	hypothesis	hypothesis	NOUN
ejpam-4592	459	4	,	,	PUNCT
ejpam-4592	459	5	there	there	PRON
ejpam-4592	459	6	exists	exist	VERB
ejpam-4592	459	7	a	a	DET
ejpam-4592	459	8	sequence	sequence	NOUN
ejpam-4592	459	9	{	{	PUNCT
ejpam-4592	459	10	bn	bn	ADP
ejpam-4592	459	11	}	}	PUNCT
ejpam-4592	459	12	in	in	ADP
ejpam-4592	459	13	x	x	PUNCT
ejpam-4592	459	14	with	with	ADP
ejpam-4592	459	15	an	an	DET
ejpam-4592	459	16	̸=	̸=	PROPN
ejpam-4592	459	17	bn	bn	NOUN
ejpam-4592	459	18	for	for	ADP
ejpam-4592	459	19	every	every	DET
ejpam-4592	459	20	n	n	NOUN
ejpam-4592	459	21	∈	∈	NOUN
ejpam-4592	459	22	n	n	PRON
ejpam-4592	459	23	such	such	ADJ
ejpam-4592	459	24	that	that	SCONJ
ejpam-4592	459	25	the	the	DET
ejpam-4592	459	26	pair	pair	NOUN
ejpam-4592	459	27	of	of	ADP
ejpam-4592	459	28	sequence	sequence	NOUN
ejpam-4592	459	29	{	{	PUNCT
ejpam-4592	459	30	an	an	NOUN
ejpam-4592	459	31	}	}	PUNCT
ejpam-4592	459	32	and	and	CCONJ
ejpam-4592	459	33	{	{	PUNCT
ejpam-4592	459	34	bn	bn	X
ejpam-4592	459	35	}	}	PUNCT
ejpam-4592	459	36	is	be	AUX
ejpam-4592	459	37	uniformly	uniformly	ADV
ejpam-4592	459	38	asymptotic	asymptotic	ADJ
ejpam-4592	459	39	with	with	ADP
ejpam-4592	459	40	respect	respect	NOUN
ejpam-4592	459	41	to	to	ADP
ejpam-4592	459	42	σ	σ	PROPN
ejpam-4592	459	43	.	.	PUNCT
ejpam-4592	460	1	by	by	ADP
ejpam-4592	460	2	theorem	theorem	NOUN
ejpam-4592	460	3	37	37	NUM
ejpam-4592	460	4	,	,	PUNCT
ejpam-4592	460	5	{	{	PUNCT
ejpam-4592	460	6	bn	bn	NOUN
ejpam-4592	460	7	}	}	PUNCT
ejpam-4592	460	8	∈	∈	PROPN
ejpam-4592	460	9	c(σ	c(σ	PROPN
ejpam-4592	460	10	)	)	PUNCT
ejpam-4592	460	11	and	and	CCONJ
ejpam-4592	460	12	so	so	ADV
ejpam-4592	460	13	{	{	PUNCT
ejpam-4592	460	14	bn	bn	NOUN
ejpam-4592	460	15	}	}	PUNCT
ejpam-4592	460	16	∈	∈	PROPN
ejpam-4592	460	17	b(σ	b(σ	PROPN
ejpam-4592	460	18	)	)	PUNCT
ejpam-4592	460	19	,	,	PUNCT
ejpam-4592	460	20	by	by	ADP
ejpam-4592	460	21	theorem	theorem	NOUN
ejpam-4592	460	22	11(a	11(a	NUM
ejpam-4592	460	23	)	)	PUNCT
ejpam-4592	460	24	.	.	PUNCT
ejpam-4592	461	1	then	then	ADV
ejpam-4592	461	2	{	{	PUNCT
ejpam-4592	461	3	an	an	DET
ejpam-4592	461	4	}	}	PUNCT
ejpam-4592	461	5	∈	∈	PROPN
ejpam-4592	461	6	b(σ	b(σ	PROPN
ejpam-4592	461	7	)	)	PUNCT
ejpam-4592	461	8	,	,	PUNCT
ejpam-4592	461	9	by	by	ADP
ejpam-4592	461	10	theorem	theorem	NOUN
ejpam-4592	461	11	29	29	NUM
ejpam-4592	461	12	.	.	PUNCT
ejpam-4592	462	1	since	since	SCONJ
ejpam-4592	462	2	h	h	NOUN
ejpam-4592	462	3	is	be	AUX
ejpam-4592	462	4	a	a	DET
ejpam-4592	462	5	bc	bc	ADJ
ejpam-4592	462	6	-	-	ADJ
ejpam-4592	462	7	regular	regular	ADJ
ejpam-4592	462	8	map	map	NOUN
ejpam-4592	462	9	,	,	PUNCT
ejpam-4592	462	10	there	there	PRON
ejpam-4592	462	11	is	be	VERB
ejpam-4592	462	12	a	a	DET
ejpam-4592	462	13	metric	metric	ADJ
ejpam-4592	462	14	d	d	PROPN
ejpam-4592	462	15	∈	∈	PROPN
ejpam-4592	462	16	ω2	ω2	NOUN
ejpam-4592	462	17	such	such	ADJ
ejpam-4592	462	18	that	that	SCONJ
ejpam-4592	462	19	{	{	PUNCT
ejpam-4592	462	20	h(an	h(an	NOUN
ejpam-4592	462	21	)	)	PUNCT
ejpam-4592	462	22	}	}	PUNCT
ejpam-4592	462	23	∈	∈	PROPN
ejpam-4592	462	24	b(d	b(d	PROPN
ejpam-4592	462	25	)	)	PUNCT
ejpam-4592	462	26	where	where	SCONJ
ejpam-4592	462	27	d	d	PROPN
ejpam-4592	462	28	∈	∈	PROPN
ejpam-4592	462	29	ω2	ω2	PROPN
ejpam-4592	462	30	.	.	PUNCT
ejpam-4592	463	1	by	by	ADP
ejpam-4592	463	2	theorem	theorem	NOUN
ejpam-4592	463	3	17	17	NUM
ejpam-4592	463	4	,	,	PUNCT
ejpam-4592	463	5	{	{	PUNCT
ejpam-4592	463	6	h(an	h(an	NOUN
ejpam-4592	463	7	)	)	PUNCT
ejpam-4592	463	8	}	}	PUNCT
ejpam-4592	463	9	∈	∈	PROPN
ejpam-4592	463	10	c(d	c(d	PROPN
ejpam-4592	463	11	)	)	PUNCT
ejpam-4592	463	12	where	where	SCONJ
ejpam-4592	463	13	d	d	PROPN
ejpam-4592	463	14	∈	∈	PROPN
ejpam-4592	463	15	ω2	ω2	PROPN
ejpam-4592	463	16	.	.	PUNCT
ejpam-4592	464	1	hence	hence	ADV
ejpam-4592	464	2	h	h	PROPN
ejpam-4592	464	3	is	be	AUX
ejpam-4592	464	4	a	a	DET
ejpam-4592	464	5	cc	cc	NOUN
ejpam-4592	464	6	-	-	ADJ
ejpam-4592	464	7	regular	regular	ADJ
ejpam-4592	464	8	map	map	NOUN
ejpam-4592	464	9	.	.	PUNCT
ejpam-4592	465	1	theorem	theorem	VERB
ejpam-4592	465	2	43	43	NUM
ejpam-4592	465	3	.	.	PUNCT
ejpam-4592	466	1	let	let	AUX
ejpam-4592	466	2	(	(	PUNCT
ejpam-4592	466	3	x	x	NOUN
ejpam-4592	466	4	,	,	PUNCT
ejpam-4592	466	5	ω1	ω1	PROPN
ejpam-4592	466	6	)	)	PUNCT
ejpam-4592	466	7	and	and	CCONJ
ejpam-4592	466	8	(	(	PUNCT
ejpam-4592	466	9	y	y	PROPN
ejpam-4592	466	10	,	,	PUNCT
ejpam-4592	466	11	ω2	ω2	NUM
ejpam-4592	466	12	)	)	PUNCT
ejpam-4592	466	13	be	be	VERB
ejpam-4592	466	14	two	two	NUM
ejpam-4592	466	15	generalized	generalized	ADJ
ejpam-4592	466	16	metric	metric	ADJ
ejpam-4592	466	17	spaces	space	NOUN
ejpam-4592	466	18	,	,	PUNCT
ejpam-4592	466	19	h	h	NOUN
ejpam-4592	466	20	:	:	PUNCT
ejpam-4592	466	21	x	x	X
ejpam-4592	466	22	→	→	SYM
ejpam-4592	466	23	y	y	X
ejpam-4592	466	24	be	be	AUX
ejpam-4592	466	25	a	a	DET
ejpam-4592	466	26	cauchy	cauchy	ADJ
ejpam-4592	466	27	-	-	PUNCT
ejpam-4592	466	28	continuous	continuous	ADJ
ejpam-4592	466	29	map	map	NOUN
ejpam-4592	466	30	.	.	PUNCT
ejpam-4592	467	1	if	if	SCONJ
ejpam-4592	467	2	for	for	ADP
ejpam-4592	467	3	every	every	DET
ejpam-4592	467	4	bourbaki	bourbaki	NOUN
ejpam-4592	467	5	-	-	PUNCT
ejpam-4592	467	6	cauchy	cauchy	ADJ
ejpam-4592	467	7	sequence	sequence	NOUN
ejpam-4592	467	8	{	{	PUNCT
ejpam-4592	467	9	cn	cn	NOUN
ejpam-4592	467	10	}	}	PUNCT
ejpam-4592	467	11	in	in	ADP
ejpam-4592	467	12	x	x	NOUN
ejpam-4592	467	13	,	,	PUNCT
ejpam-4592	467	14	there	there	PRON
ejpam-4592	467	15	exists	exist	VERB
ejpam-4592	467	16	a	a	DET
ejpam-4592	467	17	sequence	sequence	NOUN
ejpam-4592	467	18	{	{	PUNCT
ejpam-4592	467	19	dn	dn	NOUN
ejpam-4592	467	20	}	}	PUNCT
ejpam-4592	467	21	in	in	ADP
ejpam-4592	467	22	x	x	PUNCT
ejpam-4592	467	23	with	with	ADP
ejpam-4592	467	24	cn	cn	PROPN
ejpam-4592	467	25	̸=	̸=	PROPN
ejpam-4592	467	26	dn	dn	NOUN
ejpam-4592	467	27	for	for	ADP
ejpam-4592	467	28	every	every	DET
ejpam-4592	467	29	n	n	PRON
ejpam-4592	467	30	∈	∈	NOUN
ejpam-4592	467	31	n	n	PRON
ejpam-4592	467	32	such	such	ADJ
ejpam-4592	467	33	that	that	SCONJ
ejpam-4592	467	34	the	the	DET
ejpam-4592	467	35	pair	pair	NOUN
ejpam-4592	467	36	of	of	ADP
ejpam-4592	467	37	sequence	sequence	NOUN
ejpam-4592	467	38	{	{	PUNCT
ejpam-4592	467	39	cn	cn	NOUN
ejpam-4592	467	40	}	}	PUNCT
ejpam-4592	467	41	and	and	CCONJ
ejpam-4592	467	42	{	{	PUNCT
ejpam-4592	467	43	dn	dn	VERB
ejpam-4592	467	44	}	}	PUNCT
ejpam-4592	467	45	is	be	AUX
ejpam-4592	467	46	uniformly	uniformly	ADV
ejpam-4592	467	47	asymptotic	asymptotic	ADJ
ejpam-4592	467	48	with	with	ADP
ejpam-4592	467	49	the	the	DET
ejpam-4592	467	50	same	same	ADJ
ejpam-4592	467	51	metric	metric	NOUN
ejpam-4592	467	52	,	,	PUNCT
ejpam-4592	467	53	then	then	ADV
ejpam-4592	467	54	h	h	PROPN
ejpam-4592	467	55	is	be	AUX
ejpam-4592	467	56	a	a	DET
ejpam-4592	467	57	bc	bc	ADJ
ejpam-4592	467	58	-	-	ADJ
ejpam-4592	467	59	regular	regular	ADJ
ejpam-4592	467	60	map	map	NOUN
ejpam-4592	467	61	.	.	PUNCT
ejpam-4592	468	1	proof	proof	NOUN
ejpam-4592	468	2	.	.	PUNCT
ejpam-4592	469	1	let	let	VERB
ejpam-4592	469	2	{	{	PUNCT
ejpam-4592	469	3	an	an	DET
ejpam-4592	469	4	}	}	PUNCT
ejpam-4592	469	5	∈	∈	PROPN
ejpam-4592	469	6	b(σ	b(σ	PROPN
ejpam-4592	469	7	)	)	PUNCT
ejpam-4592	469	8	where	where	SCONJ
ejpam-4592	469	9	σ	σ	PROPN
ejpam-4592	469	10	∈	∈	PROPN
ejpam-4592	469	11	ω1	ω1	PROPN
ejpam-4592	469	12	.	.	PUNCT
ejpam-4592	470	1	then	then	ADV
ejpam-4592	470	2	by	by	ADP
ejpam-4592	470	3	hypothesis	hypothesis	NOUN
ejpam-4592	470	4	,	,	PUNCT
ejpam-4592	470	5	there	there	PRON
ejpam-4592	470	6	exists	exist	VERB
ejpam-4592	470	7	a	a	DET
ejpam-4592	470	8	sequence	sequence	NOUN
ejpam-4592	470	9	{	{	PUNCT
ejpam-4592	470	10	bn	bn	ADP
ejpam-4592	470	11	}	}	PUNCT
ejpam-4592	470	12	in	in	ADP
ejpam-4592	470	13	x	x	PUNCT
ejpam-4592	470	14	with	with	ADP
ejpam-4592	470	15	an	an	DET
ejpam-4592	470	16	̸=	̸=	PROPN
ejpam-4592	470	17	bn	bn	NOUN
ejpam-4592	470	18	for	for	ADP
ejpam-4592	470	19	every	every	DET
ejpam-4592	470	20	n	n	NOUN
ejpam-4592	470	21	∈	∈	NOUN
ejpam-4592	470	22	n	n	PRON
ejpam-4592	470	23	such	such	ADJ
ejpam-4592	470	24	that	that	SCONJ
ejpam-4592	470	25	the	the	DET
ejpam-4592	470	26	pair	pair	NOUN
ejpam-4592	470	27	of	of	ADP
ejpam-4592	470	28	sequence	sequence	NOUN
ejpam-4592	470	29	{	{	PUNCT
ejpam-4592	470	30	an	an	NOUN
ejpam-4592	470	31	}	}	PUNCT
ejpam-4592	470	32	and	and	CCONJ
ejpam-4592	470	33	{	{	PUNCT
ejpam-4592	470	34	bn	bn	X
ejpam-4592	470	35	}	}	PUNCT
ejpam-4592	470	36	is	be	AUX
ejpam-4592	470	37	uniformly	uniformly	ADV
ejpam-4592	470	38	asymptotic	asymptotic	ADJ
ejpam-4592	470	39	with	with	ADP
ejpam-4592	470	40	the	the	DET
ejpam-4592	470	41	same	same	ADJ
ejpam-4592	470	42	metric	metric	NOUN
ejpam-4592	470	43	.	.	PUNCT
ejpam-4592	471	1	by	by	ADP
ejpam-4592	471	2	theorem	theorem	NOUN
ejpam-4592	471	3	37	37	NUM
ejpam-4592	471	4	,	,	PUNCT
ejpam-4592	471	5	{	{	PUNCT
ejpam-4592	471	6	bn	bn	NOUN
ejpam-4592	471	7	}	}	PUNCT
ejpam-4592	471	8	∈	∈	PROPN
ejpam-4592	471	9	c(σ	c(σ	PROPN
ejpam-4592	471	10	)	)	PUNCT
ejpam-4592	471	11	.	.	PUNCT
ejpam-4592	472	1	then	then	ADV
ejpam-4592	472	2	{	{	PUNCT
ejpam-4592	472	3	an	an	DET
ejpam-4592	472	4	}	}	PUNCT
ejpam-4592	472	5	∈	∈	PROPN
ejpam-4592	472	6	c(σ	c(σ	PROPN
ejpam-4592	472	7	)	)	PUNCT
ejpam-4592	472	8	,	,	PUNCT
ejpam-4592	472	9	by	by	ADP
ejpam-4592	472	10	theorem	theorem	NOUN
ejpam-4592	472	11	27	27	NUM
ejpam-4592	472	12	.	.	PUNCT
ejpam-4592	473	1	since	since	SCONJ
ejpam-4592	473	2	h	h	NOUN
ejpam-4592	473	3	is	be	AUX
ejpam-4592	473	4	a	a	DET
ejpam-4592	473	5	cauchy	cauchy	ADJ
ejpam-4592	473	6	-	-	PUNCT
ejpam-4592	473	7	continuous	continuous	ADJ
ejpam-4592	473	8	map	map	NOUN
ejpam-4592	473	9	,	,	PUNCT
ejpam-4592	473	10	there	there	PRON
ejpam-4592	473	11	is	be	VERB
ejpam-4592	473	12	a	a	DET
ejpam-4592	473	13	metric	metric	ADJ
ejpam-4592	473	14	d	d	PROPN
ejpam-4592	473	15	∈	∈	PROPN
ejpam-4592	473	16	ω2	ω2	NOUN
ejpam-4592	473	17	such	such	ADJ
ejpam-4592	473	18	that	that	SCONJ
ejpam-4592	473	19	{	{	PUNCT
ejpam-4592	473	20	h(an	h(an	NOUN
ejpam-4592	473	21	)	)	PUNCT
ejpam-4592	473	22	}	}	PUNCT
ejpam-4592	473	23	∈	∈	PROPN
ejpam-4592	473	24	c(d	c(d	PROPN
ejpam-4592	473	25	)	)	PUNCT
ejpam-4592	473	26	where	where	SCONJ
ejpam-4592	473	27	d	d	PROPN
ejpam-4592	473	28	∈	∈	PROPN
ejpam-4592	473	29	ω2	ω2	PROPN
ejpam-4592	473	30	.	.	PUNCT
ejpam-4592	474	1	by	by	ADP
ejpam-4592	474	2	theorem	theorem	NOUN
ejpam-4592	474	3	11(a	11(a	NUM
ejpam-4592	474	4	)	)	PUNCT
ejpam-4592	474	5	,	,	PUNCT
ejpam-4592	474	6	{	{	PUNCT
ejpam-4592	474	7	h(an	h(an	NOUN
ejpam-4592	474	8	)	)	PUNCT
ejpam-4592	474	9	}	}	PUNCT
ejpam-4592	474	10	∈	∈	PROPN
ejpam-4592	474	11	b(d	b(d	PROPN
ejpam-4592	474	12	)	)	PUNCT
ejpam-4592	474	13	where	where	SCONJ
ejpam-4592	474	14	d	d	PROPN
ejpam-4592	474	15	∈	∈	PROPN
ejpam-4592	474	16	ω2	ω2	PROPN
ejpam-4592	474	17	.	.	PUNCT
ejpam-4592	475	1	hence	hence	ADV
ejpam-4592	475	2	h	h	PROPN
ejpam-4592	475	3	is	be	AUX
ejpam-4592	475	4	a	a	DET
ejpam-4592	475	5	bc	bc	ADJ
ejpam-4592	475	6	-	-	ADJ
ejpam-4592	475	7	regular	regular	ADJ
ejpam-4592	475	8	map	map	NOUN
ejpam-4592	475	9	.	.	PUNCT
ejpam-4592	476	1	theorem	theorem	VERB
ejpam-4592	476	2	44	44	NUM
ejpam-4592	476	3	.	.	PUNCT
ejpam-4592	477	1	let	let	VERB
ejpam-4592	477	2	(	(	PUNCT
ejpam-4592	477	3	x	x	NOUN
ejpam-4592	477	4	,	,	PUNCT
ejpam-4592	477	5	ω1	ω1	PROPN
ejpam-4592	477	6	)	)	PUNCT
ejpam-4592	477	7	and	and	CCONJ
ejpam-4592	477	8	(	(	PUNCT
ejpam-4592	477	9	y	y	PROPN
ejpam-4592	477	10	,	,	PUNCT
ejpam-4592	477	11	ω2	ω2	NUM
ejpam-4592	477	12	)	)	PUNCT
ejpam-4592	477	13	be	be	VERB
ejpam-4592	477	14	two	two	NUM
ejpam-4592	477	15	generalized	generalized	ADJ
ejpam-4592	477	16	metric	metric	ADJ
ejpam-4592	477	17	spaces	space	NOUN
ejpam-4592	477	18	,	,	PUNCT
ejpam-4592	477	19	h	h	NOUN
ejpam-4592	477	20	:	:	PUNCT
ejpam-4592	477	21	x	x	X
ejpam-4592	477	22	→	→	SYM
ejpam-4592	477	23	y	y	PROPN
ejpam-4592	477	24	be	be	AUX
ejpam-4592	477	25	a	a	DET
ejpam-4592	477	26	cc	cc	NOUN
ejpam-4592	477	27	-	-	ADJ
ejpam-4592	477	28	regular	regular	ADJ
ejpam-4592	477	29	map	map	NOUN
ejpam-4592	477	30	.	.	PUNCT
ejpam-4592	478	1	if	if	SCONJ
ejpam-4592	478	2	for	for	ADP
ejpam-4592	478	3	every	every	DET
ejpam-4592	478	4	cofinally	cofinally	ADV
ejpam-4592	478	5	-	-	PUNCT
ejpam-4592	478	6	cauchy	cauchy	ADJ
ejpam-4592	478	7	sequence	sequence	NOUN
ejpam-4592	478	8	{	{	PUNCT
ejpam-4592	478	9	sn	sn	NOUN
ejpam-4592	478	10	}	}	PUNCT
ejpam-4592	478	11	in	in	ADP
ejpam-4592	478	12	y	y	PROPN
ejpam-4592	478	13	,	,	PUNCT
ejpam-4592	478	14	there	there	PRON
ejpam-4592	478	15	exists	exist	VERB
ejpam-4592	478	16	a	a	DET
ejpam-4592	478	17	sequence	sequence	NOUN
ejpam-4592	478	18	{	{	PUNCT
ejpam-4592	478	19	tn	tn	NOUN
ejpam-4592	478	20	}	}	PUNCT
ejpam-4592	478	21	in	in	ADP
ejpam-4592	478	22	y	y	PROPN
ejpam-4592	478	23	with	with	ADP
ejpam-4592	478	24	sn	sn	PROPN
ejpam-4592	478	25	̸=	̸=	PROPN
ejpam-4592	478	26	tn	tn	NOUN
ejpam-4592	478	27	for	for	ADP
ejpam-4592	478	28	every	every	DET
ejpam-4592	478	29	n	n	NOUN
ejpam-4592	478	30	∈	∈	NOUN
ejpam-4592	478	31	n	n	PRON
ejpam-4592	478	32	such	such	ADJ
ejpam-4592	478	33	that	that	SCONJ
ejpam-4592	478	34	the	the	DET
ejpam-4592	478	35	pair	pair	NOUN
ejpam-4592	478	36	of	of	ADP
ejpam-4592	478	37	sequence	sequence	NOUN
ejpam-4592	478	38	{	{	PUNCT
ejpam-4592	478	39	sn	sn	NOUN
ejpam-4592	478	40	}	}	PUNCT
ejpam-4592	478	41	and	and	CCONJ
ejpam-4592	478	42	{	{	PUNCT
ejpam-4592	478	43	tn	tn	NOUN
ejpam-4592	478	44	}	}	PUNCT
ejpam-4592	478	45	is	be	AUX
ejpam-4592	478	46	uniformly	uniformly	ADV
ejpam-4592	478	47	asymptotic	asymptotic	ADJ
ejpam-4592	478	48	with	with	ADP
ejpam-4592	478	49	the	the	DET
ejpam-4592	478	50	same	same	ADJ
ejpam-4592	478	51	metric	metric	NOUN
ejpam-4592	478	52	,	,	PUNCT
ejpam-4592	478	53	then	then	ADV
ejpam-4592	478	54	h	h	PROPN
ejpam-4592	478	55	is	be	AUX
ejpam-4592	478	56	a	a	DET
ejpam-4592	478	57	cauchy	cauchy	ADJ
ejpam-4592	478	58	-	-	PUNCT
ejpam-4592	478	59	continuous	continuous	ADJ
ejpam-4592	478	60	map	map	NOUN
ejpam-4592	478	61	.	.	PUNCT
ejpam-4592	479	1	proof	proof	NOUN
ejpam-4592	479	2	.	.	PUNCT
ejpam-4592	480	1	let	let	VERB
ejpam-4592	480	2	{	{	PUNCT
ejpam-4592	480	3	an	an	PRON
ejpam-4592	480	4	}	}	PUNCT
ejpam-4592	480	5	∈	∈	PROPN
ejpam-4592	480	6	c(σ	c(σ	PROPN
ejpam-4592	480	7	)	)	PUNCT
ejpam-4592	480	8	for	for	ADP
ejpam-4592	480	9	σ	σ	PROPN
ejpam-4592	480	10	∈	∈	PROPN
ejpam-4592	480	11	ω1	ω1	PROPN
ejpam-4592	480	12	.	.	PUNCT
ejpam-4592	481	1	by	by	ADP
ejpam-4592	481	2	theorem	theorem	NOUN
ejpam-4592	481	3	13	13	NUM
ejpam-4592	481	4	,	,	PUNCT
ejpam-4592	481	5	{	{	PUNCT
ejpam-4592	481	6	an	an	DET
ejpam-4592	481	7	}	}	PUNCT
ejpam-4592	481	8	∈	∈	PROPN
ejpam-4592	481	9	c(σ	c(σ	PROPN
ejpam-4592	481	10	)	)	PUNCT
ejpam-4592	481	11	..	..	PUNCT
ejpam-4592	481	12	since	since	SCONJ
ejpam-4592	481	13	h	h	NOUN
ejpam-4592	481	14	is	be	AUX
ejpam-4592	481	15	a	a	DET
ejpam-4592	481	16	cc	cc	NOUN
ejpam-4592	481	17	-	-	ADJ
ejpam-4592	481	18	regular	regular	ADJ
ejpam-4592	481	19	map	map	NOUN
ejpam-4592	481	20	,	,	PUNCT
ejpam-4592	481	21	there	there	PRON
ejpam-4592	481	22	is	be	VERB
ejpam-4592	481	23	a	a	DET
ejpam-4592	481	24	metric	metric	ADJ
ejpam-4592	481	25	d	d	PROPN
ejpam-4592	481	26	∈	∈	PROPN
ejpam-4592	481	27	ω2	ω2	NOUN
ejpam-4592	481	28	such	such	ADJ
ejpam-4592	481	29	that	that	SCONJ
ejpam-4592	481	30	{	{	PUNCT
ejpam-4592	481	31	h(an	h(an	NOUN
ejpam-4592	481	32	)	)	PUNCT
ejpam-4592	481	33	}	}	PUNCT
ejpam-4592	481	34	∈	∈	PROPN
ejpam-4592	481	35	c(d	c(d	PROPN
ejpam-4592	481	36	)	)	PUNCT
ejpam-4592	481	37	where	where	SCONJ
ejpam-4592	481	38	d	d	PROPN
ejpam-4592	481	39	∈	∈	PROPN
ejpam-4592	481	40	ω2	ω2	PROPN
ejpam-4592	481	41	.	.	PUNCT
ejpam-4592	482	1	by	by	ADP
ejpam-4592	482	2	hypothesis	hypothesis	NOUN
ejpam-4592	482	3	,	,	PUNCT
ejpam-4592	482	4	there	there	PRON
ejpam-4592	482	5	exists	exist	VERB
ejpam-4592	482	6	a	a	DET
ejpam-4592	482	7	sequence	sequence	NOUN
ejpam-4592	482	8	{	{	PUNCT
ejpam-4592	482	9	bn	bn	ADP
ejpam-4592	482	10	}	}	PUNCT
ejpam-4592	482	11	in	in	ADP
ejpam-4592	482	12	y	y	PROPN
ejpam-4592	482	13	with	with	ADP
ejpam-4592	482	14	h(an	h(an	NOUN
ejpam-4592	482	15	)	)	PUNCT
ejpam-4592	482	16	̸=	̸=	PROPN
ejpam-4592	482	17	bn	bn	NOUN
ejpam-4592	482	18	for	for	ADP
ejpam-4592	482	19	every	every	DET
ejpam-4592	482	20	n	n	NOUN
ejpam-4592	482	21	∈	∈	NOUN
ejpam-4592	482	22	n	n	PRON
ejpam-4592	482	23	such	such	ADJ
ejpam-4592	482	24	that	that	SCONJ
ejpam-4592	482	25	the	the	DET
ejpam-4592	482	26	pair	pair	NOUN
ejpam-4592	482	27	of	of	ADP
ejpam-4592	482	28	sequence	sequence	NOUN
ejpam-4592	482	29	{	{	PUNCT
ejpam-4592	482	30	h(an	h(an	NOUN
ejpam-4592	482	31	)	)	PUNCT
ejpam-4592	482	32	}	}	PUNCT
ejpam-4592	482	33	and	and	CCONJ
ejpam-4592	482	34	{	{	PUNCT
ejpam-4592	482	35	bn	bn	X
ejpam-4592	482	36	}	}	PUNCT
ejpam-4592	482	37	is	be	AUX
ejpam-4592	482	38	uniformly	uniformly	ADV
ejpam-4592	482	39	asymptotic	asymptotic	ADJ
ejpam-4592	482	40	with	with	ADP
ejpam-4592	482	41	the	the	DET
ejpam-4592	482	42	same	same	ADJ
ejpam-4592	482	43	metric	metric	NOUN
ejpam-4592	482	44	.	.	PUNCT
ejpam-4592	483	1	by	by	ADP
ejpam-4592	483	2	theorem	theorem	NOUN
ejpam-4592	483	3	37	37	NUM
ejpam-4592	483	4	,	,	PUNCT
ejpam-4592	483	5	{	{	PUNCT
ejpam-4592	483	6	bn	bn	NOUN
ejpam-4592	483	7	}	}	PUNCT
ejpam-4592	483	8	∈	∈	PROPN
ejpam-4592	483	9	c(d	c(d	PROPN
ejpam-4592	483	10	)	)	PUNCT
ejpam-4592	483	11	for	for	ADP
ejpam-4592	483	12	d	d	PROPN
ejpam-4592	483	13	∈	∈	PROPN
ejpam-4592	483	14	ω2	ω2	PROPN
ejpam-4592	483	15	.	.	PUNCT
ejpam-4592	484	1	then	then	ADV
ejpam-4592	484	2	{	{	PUNCT
ejpam-4592	484	3	h(an	h(an	NOUN
ejpam-4592	484	4	)	)	PUNCT
ejpam-4592	484	5	}	}	PUNCT
ejpam-4592	484	6	∈	∈	PROPN
ejpam-4592	484	7	c(d	c(d	PROPN
ejpam-4592	484	8	)	)	PUNCT
ejpam-4592	484	9	where	where	SCONJ
ejpam-4592	484	10	d	d	PROPN
ejpam-4592	484	11	∈	∈	PROPN
ejpam-4592	484	12	ω2	ω2	PROPN
ejpam-4592	484	13	,	,	PUNCT
ejpam-4592	484	14	by	by	ADP
ejpam-4592	484	15	theorem	theorem	NOUN
ejpam-4592	484	16	27	27	NUM
ejpam-4592	484	17	.	.	PUNCT
ejpam-4592	485	1	hence	hence	ADV
ejpam-4592	485	2	h	h	PROPN
ejpam-4592	485	3	is	be	AUX
ejpam-4592	485	4	a	a	DET
ejpam-4592	485	5	cauchy	cauchy	ADJ
ejpam-4592	485	6	-	-	PUNCT
ejpam-4592	485	7	continuous	continuous	ADJ
ejpam-4592	485	8	map	map	NOUN
ejpam-4592	485	9	.	.	PUNCT
ejpam-4592	486	1	the	the	DET
ejpam-4592	486	2	following	follow	VERB
ejpam-4592	486	3	theorem	theorem	VERB
ejpam-4592	486	4	45	45	NUM
ejpam-4592	486	5	gives	give	VERB
ejpam-4592	486	6	a	a	DET
ejpam-4592	486	7	necessary	necessary	ADJ
ejpam-4592	486	8	condition	condition	NOUN
ejpam-4592	486	9	for	for	ADP
ejpam-4592	486	10	a	a	DET
ejpam-4592	486	11	weakly	weakly	ADJ
ejpam-4592	486	12	complete	complete	ADJ
ejpam-4592	486	13	metric	metric	ADJ
ejpam-4592	486	14	space	space	NOUN
ejpam-4592	486	15	to	to	PART
ejpam-4592	486	16	be	be	AUX
ejpam-4592	486	17	a	a	DET
ejpam-4592	486	18	cofinally	cofinally	ADV
ejpam-4592	486	19	-	-	PUNCT
ejpam-4592	486	20	complete	complete	ADJ
ejpam-4592	486	21	metric	metric	ADJ
ejpam-4592	486	22	space	space	NOUN
ejpam-4592	486	23	.	.	PUNCT
ejpam-4592	487	1	theorem	theorem	VERB
ejpam-4592	487	2	45	45	NUM
ejpam-4592	487	3	.	.	PUNCT
ejpam-4592	488	1	let	let	AUX
ejpam-4592	488	2	(	(	PUNCT
ejpam-4592	488	3	x	x	NOUN
ejpam-4592	488	4	,	,	PUNCT
ejpam-4592	488	5	ω	ω	NUM
ejpam-4592	488	6	)	)	PUNCT
ejpam-4592	488	7	be	be	AUX
ejpam-4592	488	8	a	a	DET
ejpam-4592	488	9	weakly	weakly	ADJ
ejpam-4592	488	10	complete	complete	ADJ
ejpam-4592	488	11	space	space	NOUN
ejpam-4592	488	12	.	.	PUNCT
ejpam-4592	489	1	if	if	SCONJ
ejpam-4592	489	2	for	for	ADP
ejpam-4592	489	3	every	every	DET
ejpam-4592	489	4	cofinally	cofinally	ADV
ejpam-4592	489	5	-	-	PUNCT
ejpam-4592	489	6	cauchy	cauchy	ADJ
ejpam-4592	489	7	sequence	sequence	NOUN
ejpam-4592	489	8	{	{	PUNCT
ejpam-4592	489	9	sn	sn	NOUN
ejpam-4592	489	10	}	}	PUNCT
ejpam-4592	489	11	in	in	ADP
ejpam-4592	489	12	x	x	NOUN
ejpam-4592	489	13	,	,	PUNCT
ejpam-4592	489	14	there	there	PRON
ejpam-4592	489	15	exists	exist	VERB
ejpam-4592	489	16	a	a	DET
ejpam-4592	489	17	sequence	sequence	NOUN
ejpam-4592	489	18	{	{	PUNCT
ejpam-4592	489	19	tn	tn	NOUN
ejpam-4592	489	20	}	}	PUNCT
ejpam-4592	489	21	in	in	ADP
ejpam-4592	489	22	x	x	PUNCT
ejpam-4592	489	23	with	with	ADP
ejpam-4592	489	24	sn	sn	NOUN
ejpam-4592	489	25	̸=	̸=	PROPN
ejpam-4592	489	26	tn	tn	NOUN
ejpam-4592	489	27	for	for	ADP
ejpam-4592	489	28	every	every	DET
ejpam-4592	489	29	n	n	NOUN
ejpam-4592	489	30	∈	∈	NOUN
ejpam-4592	489	31	n	n	PRON
ejpam-4592	489	32	such	such	ADJ
ejpam-4592	489	33	that	that	SCONJ
ejpam-4592	489	34	the	the	DET
ejpam-4592	489	35	pair	pair	NOUN
ejpam-4592	489	36	of	of	ADP
ejpam-4592	489	37	sequence	sequence	NOUN
ejpam-4592	489	38	{	{	PUNCT
ejpam-4592	489	39	sn	sn	NOUN
ejpam-4592	489	40	}	}	PUNCT
ejpam-4592	489	41	and	and	CCONJ
ejpam-4592	489	42	{	{	PUNCT
ejpam-4592	489	43	tn	tn	NOUN
ejpam-4592	489	44	}	}	PUNCT
ejpam-4592	489	45	is	be	AUX
ejpam-4592	489	46	uniformly	uniformly	ADV
ejpam-4592	489	47	asymptotic	asymptotic	ADJ
ejpam-4592	489	48	with	with	ADP
ejpam-4592	489	49	the	the	DET
ejpam-4592	489	50	same	same	ADJ
ejpam-4592	489	51	metric	metric	NOUN
ejpam-4592	489	52	,	,	PUNCT
ejpam-4592	489	53	then	then	ADV
ejpam-4592	489	54	(	(	PUNCT
ejpam-4592	489	55	x	x	X
ejpam-4592	489	56	,	,	PUNCT
ejpam-4592	489	57	ω	ω	NUM
ejpam-4592	489	58	)	)	PUNCT
ejpam-4592	489	59	is	be	AUX
ejpam-4592	489	60	a	a	DET
ejpam-4592	489	61	cofinally	cofinally	ADV
ejpam-4592	489	62	-	-	PUNCT
ejpam-4592	489	63	complete	complete	ADJ
ejpam-4592	489	64	space	space	NOUN
ejpam-4592	489	65	.	.	PUNCT
ejpam-4592	490	1	proof	proof	NOUN
ejpam-4592	490	2	.	.	PUNCT
ejpam-4592	491	1	given	give	VERB
ejpam-4592	491	2	that	that	SCONJ
ejpam-4592	491	3	(	(	PUNCT
ejpam-4592	491	4	x	x	X
ejpam-4592	491	5	,	,	PUNCT
ejpam-4592	491	6	ω	ω	NUM
ejpam-4592	491	7	)	)	PUNCT
ejpam-4592	491	8	is	be	AUX
ejpam-4592	491	9	a	a	DET
ejpam-4592	491	10	weakly	weakly	ADJ
ejpam-4592	491	11	complete	complete	ADJ
ejpam-4592	491	12	space	space	NOUN
ejpam-4592	491	13	.	.	PUNCT
ejpam-4592	492	1	then	then	ADV
ejpam-4592	492	2	there	there	PRON
ejpam-4592	492	3	exists	exist	VERB
ejpam-4592	492	4	a	a	DET
ejpam-4592	492	5	kernel	kernel	PROPN
ejpam-4592	492	6	ω0	ω0	PROPN
ejpam-4592	492	7	⊂	⊂	PROPN
ejpam-4592	492	8	ω	ω	PROPN
ejpam-4592	492	9	consisting	consist	VERB
ejpam-4592	492	10	of	of	ADP
ejpam-4592	492	11	complete	complete	ADJ
ejpam-4592	492	12	metrics	metric	NOUN
ejpam-4592	492	13	.	.	PUNCT
ejpam-4592	493	1	let	let	VERB
ejpam-4592	493	2	σ0	σ0	PROPN
ejpam-4592	493	3	∈	∈	PROPN
ejpam-4592	493	4	ω0	ω0	NOUN
ejpam-4592	493	5	and	and	CCONJ
ejpam-4592	493	6	{	{	PUNCT
ejpam-4592	493	7	an	an	DET
ejpam-4592	493	8	}	}	PUNCT
ejpam-4592	493	9	∈	∈	PROPN
ejpam-4592	493	10	c(σ0	c(σ0	NOUN
ejpam-4592	493	11	)	)	PUNCT
ejpam-4592	493	12	.	.	PUNCT
ejpam-4592	494	1	by	by	ADP
ejpam-4592	494	2	hypothesis	hypothesis	NOUN
ejpam-4592	494	3	,	,	PUNCT
ejpam-4592	494	4	there	there	PRON
ejpam-4592	494	5	exists	exist	VERB
ejpam-4592	494	6	a	a	DET
ejpam-4592	494	7	sequence	sequence	NOUN
ejpam-4592	494	8	{	{	PUNCT
ejpam-4592	494	9	tn	tn	NOUN
ejpam-4592	494	10	}	}	PUNCT
ejpam-4592	494	11	in	in	ADP
ejpam-4592	494	12	x	x	PUNCT
ejpam-4592	494	13	with	with	ADP
ejpam-4592	494	14	an	an	DET
ejpam-4592	494	15	̸=	̸=	PROPN
ejpam-4592	494	16	tn	tn	NOUN
ejpam-4592	494	17	for	for	ADP
ejpam-4592	494	18	every	every	DET
ejpam-4592	494	19	n	n	NOUN
ejpam-4592	494	20	∈	∈	NOUN
ejpam-4592	494	21	n	n	PRON
ejpam-4592	494	22	such	such	ADJ
ejpam-4592	494	23	that	that	SCONJ
ejpam-4592	494	24	the	the	DET
ejpam-4592	494	25	pair	pair	NOUN
ejpam-4592	494	26	of	of	ADP
ejpam-4592	494	27	sequence	sequence	NOUN
ejpam-4592	494	28	{	{	PUNCT
ejpam-4592	494	29	an	an	PROPN
ejpam-4592	494	30	}	}	PUNCT
ejpam-4592	494	31	v.	v.	PROPN
ejpam-4592	494	32	subramanian	subramanian	PROPN
ejpam-4592	494	33	et	et	PROPN
ejpam-4592	494	34	al	al	PROPN
ejpam-4592	494	35	.	.	PUNCT
ejpam-4592	494	36	/	/	SYM
ejpam-4592	494	37	eur	eur	PROPN
ejpam-4592	494	38	.	.	PUNCT
ejpam-4592	495	1	j.	j.	PROPN
ejpam-4592	495	2	pure	pure	PROPN
ejpam-4592	495	3	appl	appl	PROPN
ejpam-4592	495	4	.	.	PROPN
ejpam-4592	495	5	math	math	PROPN
ejpam-4592	495	6	,	,	PUNCT
ejpam-4592	495	7	15	15	NUM
ejpam-4592	495	8	(	(	PUNCT
ejpam-4592	495	9	4	4	NUM
ejpam-4592	495	10	)	)	PUNCT
ejpam-4592	495	11	(	(	PUNCT
ejpam-4592	495	12	2022	2022	NUM
ejpam-4592	495	13	)	)	PUNCT
ejpam-4592	495	14	,	,	PUNCT
ejpam-4592	495	15	1869	1869	NUM
ejpam-4592	495	16	-	-	SYM
ejpam-4592	495	17	1886	1886	NUM
ejpam-4592	495	18	1883	1883	NUM
ejpam-4592	495	19	and	and	CCONJ
ejpam-4592	495	20	{	{	PUNCT
ejpam-4592	495	21	tn	tn	NOUN
ejpam-4592	495	22	}	}	PUNCT
ejpam-4592	495	23	is	be	AUX
ejpam-4592	495	24	uniformly	uniformly	ADV
ejpam-4592	495	25	asymptotic	asymptotic	ADJ
ejpam-4592	495	26	with	with	ADP
ejpam-4592	495	27	the	the	DET
ejpam-4592	495	28	same	same	ADJ
ejpam-4592	495	29	metric	metric	NOUN
ejpam-4592	495	30	.	.	PUNCT
ejpam-4592	496	1	by	by	ADP
ejpam-4592	496	2	theorem	theorem	NOUN
ejpam-4592	496	3	37	37	NUM
ejpam-4592	496	4	,	,	PUNCT
ejpam-4592	496	5	{	{	PUNCT
ejpam-4592	496	6	tn	tn	NOUN
ejpam-4592	496	7	}	}	PUNCT
ejpam-4592	496	8	∈	∈	PROPN
ejpam-4592	496	9	c(σ0	c(σ0	NOUN
ejpam-4592	496	10	)	)	PUNCT
ejpam-4592	496	11	and	and	CCONJ
ejpam-4592	496	12	hence	hence	ADV
ejpam-4592	496	13	it	it	PRON
ejpam-4592	496	14	is	be	AUX
ejpam-4592	496	15	a	a	DET
ejpam-4592	496	16	convergent	convergent	NOUN
ejpam-4592	496	17	sequence	sequence	NOUN
ejpam-4592	496	18	,	,	PUNCT
ejpam-4592	496	19	by	by	ADP
ejpam-4592	496	20	assumption	assumption	NOUN
ejpam-4592	496	21	.	.	PUNCT
ejpam-4592	497	1	hence	hence	ADV
ejpam-4592	497	2	{	{	PUNCT
ejpam-4592	497	3	an	an	PRON
ejpam-4592	497	4	}	}	PUNCT
ejpam-4592	497	5	is	be	AUX
ejpam-4592	497	6	a	a	DET
ejpam-4592	497	7	convergent	convergent	NOUN
ejpam-4592	497	8	sequence	sequence	NOUN
ejpam-4592	497	9	with	with	ADP
ejpam-4592	497	10	respect	respect	NOUN
ejpam-4592	497	11	to	to	ADP
ejpam-4592	497	12	σ0	σ0	PROPN
ejpam-4592	497	13	,	,	PUNCT
ejpam-4592	497	14	by	by	ADP
ejpam-4592	497	15	theorem	theorem	NOUN
ejpam-4592	497	16	27(b	27(b	NUM
ejpam-4592	497	17	)	)	PUNCT
ejpam-4592	497	18	.	.	PUNCT
ejpam-4592	498	1	thus	thus	ADV
ejpam-4592	498	2	,	,	PUNCT
ejpam-4592	498	3	ω0	ω0	ADV
ejpam-4592	498	4	consisting	consist	VERB
ejpam-4592	498	5	of	of	ADP
ejpam-4592	498	6	complete	complete	ADJ
ejpam-4592	498	7	metrics	metric	NOUN
ejpam-4592	498	8	.	.	PUNCT
ejpam-4592	499	1	therefore	therefore	ADV
ejpam-4592	499	2	,	,	PUNCT
ejpam-4592	499	3	(	(	PUNCT
ejpam-4592	499	4	x	x	X
ejpam-4592	499	5	,	,	PUNCT
ejpam-4592	499	6	ω	ω	NUM
ejpam-4592	499	7	)	)	PUNCT
ejpam-4592	499	8	is	be	AUX
ejpam-4592	499	9	a	a	DET
ejpam-4592	499	10	cofinally	cofinally	ADV
ejpam-4592	499	11	-	-	PUNCT
ejpam-4592	499	12	complete	complete	ADJ
ejpam-4592	499	13	space	space	NOUN
ejpam-4592	499	14	.	.	PUNCT
ejpam-4592	500	1	in	in	ADP
ejpam-4592	500	2	theorem	theorem	NOUN
ejpam-4592	500	3	45	45	NUM
ejpam-4592	500	4	,	,	PUNCT
ejpam-4592	500	5	if	if	SCONJ
ejpam-4592	500	6	we	we	PRON
ejpam-4592	500	7	replace	replace	VERB
ejpam-4592	500	8	“	"	PUNCT
ejpam-4592	500	9	cofinally	cofinally	ADV
ejpam-4592	500	10	-	-	PUNCT
ejpam-4592	500	11	cauchy	cauchy	NOUN
ejpam-4592	500	12	”	"	PUNCT
ejpam-4592	500	13	by	by	ADP
ejpam-4592	500	14	“	"	PUNCT
ejpam-4592	500	15	pseudo	pseudo	NOUN
ejpam-4592	500	16	-	-	NOUN
ejpam-4592	500	17	cauchy	cauchy	NOUN
ejpam-4592	500	18	”	"	PUNCT
ejpam-4592	500	19	we	we	PRON
ejpam-4592	500	20	get	get	VERB
ejpam-4592	500	21	that	that	PRON
ejpam-4592	500	22	(	(	PUNCT
ejpam-4592	500	23	x	x	X
ejpam-4592	500	24	,	,	PUNCT
ejpam-4592	500	25	ω	ω	NUM
ejpam-4592	500	26	)	)	PUNCT
ejpam-4592	500	27	is	be	AUX
ejpam-4592	500	28	a	a	DET
ejpam-4592	500	29	pseudo	pseudo	NOUN
ejpam-4592	500	30	-	-	ADJ
ejpam-4592	500	31	complete	complete	ADJ
ejpam-4592	500	32	space	space	NOUN
ejpam-4592	500	33	.	.	PUNCT
ejpam-4592	501	1	theorem	theorem	NOUN
ejpam-4592	501	2	46	46	NUM
ejpam-4592	501	3	.	.	PUNCT
ejpam-4592	502	1	let	let	VERB
ejpam-4592	502	2	(	(	PUNCT
ejpam-4592	502	3	x	x	NOUN
ejpam-4592	502	4	,	,	PUNCT
ejpam-4592	502	5	ω	ω	NUM
ejpam-4592	502	6	)	)	PUNCT
ejpam-4592	502	7	be	be	AUX
ejpam-4592	502	8	a	a	DET
ejpam-4592	502	9	gms	gms	NOUN
ejpam-4592	502	10	,	,	PUNCT
ejpam-4592	502	11	the	the	DET
ejpam-4592	502	12	pair	pair	NOUN
ejpam-4592	502	13	of	of	ADP
ejpam-4592	502	14	sequence	sequence	NOUN
ejpam-4592	502	15	{	{	PUNCT
ejpam-4592	502	16	cn	cn	NOUN
ejpam-4592	502	17	}	}	PUNCT
ejpam-4592	502	18	and	and	CCONJ
ejpam-4592	502	19	{	{	PUNCT
ejpam-4592	502	20	dn	dn	VERB
ejpam-4592	502	21	}	}	PUNCT
ejpam-4592	502	22	is	be	AUX
ejpam-4592	502	23	asymptotic	asymptotic	ADJ
ejpam-4592	502	24	,	,	PUNCT
ejpam-4592	502	25	{	{	PUNCT
ejpam-4592	502	26	dn	dn	ADJ
ejpam-4592	502	27	}	}	PUNCT
ejpam-4592	502	28	∈	∈	PROPN
ejpam-4592	502	29	b(σ	b(σ	PROPN
ejpam-4592	502	30	)	)	PUNCT
ejpam-4592	502	31	.	.	PUNCT
ejpam-4592	503	1	if	if	SCONJ
ejpam-4592	503	2	every	every	DET
ejpam-4592	503	3	ε	ε	PROPN
ejpam-4592	503	4	k	k	PROPN
ejpam-4592	503	5	-chain	-chain	PROPN
ejpam-4592	503	6	have	have	VERB
ejpam-4592	503	7	length	length	NOUN
ejpam-4592	504	1	k	k	PROPN
ejpam-4592	504	2	−	−	PROPN
ejpam-4592	504	3	1	1	NUM
ejpam-4592	505	1	where	where	SCONJ
ejpam-4592	505	2	k	k	PROPN
ejpam-4592	505	3	∈	∈	PROPN
ejpam-4592	505	4	n	n	CCONJ
ejpam-4592	505	5	,	,	PUNCT
ejpam-4592	505	6	ε	ε	PROPN
ejpam-4592	505	7	>	>	X
ejpam-4592	505	8	0	0	PROPN
ejpam-4592	505	9	,	,	PUNCT
ejpam-4592	505	10	then	then	ADV
ejpam-4592	505	11	the	the	DET
ejpam-4592	505	12	pair	pair	NOUN
ejpam-4592	505	13	of	of	ADP
ejpam-4592	505	14	sequence	sequence	NOUN
ejpam-4592	505	15	{	{	PUNCT
ejpam-4592	505	16	cn	cn	NOUN
ejpam-4592	505	17	}	}	PUNCT
ejpam-4592	505	18	and	and	CCONJ
ejpam-4592	505	19	{	{	PUNCT
ejpam-4592	505	20	dn	dn	VERB
ejpam-4592	505	21	}	}	PUNCT
ejpam-4592	505	22	is	be	AUX
ejpam-4592	505	23	uniformly	uniformly	ADV
ejpam-4592	505	24	asymptotic	asymptotic	ADJ
ejpam-4592	505	25	with	with	ADP
ejpam-4592	505	26	respect	respect	NOUN
ejpam-4592	505	27	to	to	ADP
ejpam-4592	505	28	the	the	DET
ejpam-4592	505	29	same	same	ADJ
ejpam-4592	505	30	metric	metric	NOUN
ejpam-4592	505	31	.	.	PUNCT
ejpam-4592	506	1	proof	proof	NOUN
ejpam-4592	506	2	.	.	PUNCT
ejpam-4592	507	1	let	let	VERB
ejpam-4592	507	2	ε	ε	PROPN
ejpam-4592	507	3	>	>	X
ejpam-4592	507	4	0	0	PUNCT
ejpam-4592	507	5	be	be	AUX
ejpam-4592	507	6	given	give	VERB
ejpam-4592	507	7	and	and	CCONJ
ejpam-4592	507	8	m	m	NOUN
ejpam-4592	507	9	∈	∈	PROPN
ejpam-4592	507	10	n.	n.	NOUN
ejpam-4592	507	11	since	since	SCONJ
ejpam-4592	507	12	ε	ε	PROPN
ejpam-4592	507	13	m	m	VERB
ejpam-4592	507	14	>	>	X
ejpam-4592	507	15	0	0	PUNCT
ejpam-4592	508	1	and	and	CCONJ
ejpam-4592	508	2	{	{	PUNCT
ejpam-4592	508	3	dn	dn	PART
ejpam-4592	508	4	}	}	PUNCT
ejpam-4592	508	5	∈	∈	PROPN
ejpam-4592	508	6	b(σ	b(σ	PROPN
ejpam-4592	508	7	)	)	PUNCT
ejpam-4592	508	8	,	,	PUNCT
ejpam-4592	508	9	there	there	PRON
ejpam-4592	508	10	exist	exist	VERB
ejpam-4592	508	11	n1,m1	n1,m1	PROPN
ejpam-4592	508	12	∈	∈	PROPN
ejpam-4592	508	13	n	n	CCONJ
ejpam-4592	508	14	whenever	whenever	SCONJ
ejpam-4592	508	15	n	n	CCONJ
ejpam-4592	508	16	>	>	X
ejpam-4592	508	17	j	j	PROPN
ejpam-4592	508	18	≥	≥	PROPN
ejpam-4592	508	19	n1	n1	PROPN
ejpam-4592	508	20	such	such	ADJ
ejpam-4592	508	21	that	that	SCONJ
ejpam-4592	508	22	the	the	DET
ejpam-4592	508	23	points	point	NOUN
ejpam-4592	508	24	dj	dj	NOUN
ejpam-4592	508	25	and	and	CCONJ
ejpam-4592	508	26	dn	dn	NOUN
ejpam-4592	508	27	can	can	AUX
ejpam-4592	508	28	be	be	AUX
ejpam-4592	508	29	joined	join	VERB
ejpam-4592	508	30	by	by	ADP
ejpam-4592	508	31	an	an	DET
ejpam-4592	508	32	ε	ε	PROPN
ejpam-4592	508	33	m	m	VERB
ejpam-4592	508	34	-chain	-chain	NOUN
ejpam-4592	508	35	of	of	ADP
ejpam-4592	508	36	length	length	NOUN
ejpam-4592	508	37	m1	m1	NOUN
ejpam-4592	508	38	.	.	PUNCT
ejpam-4592	509	1	by	by	ADP
ejpam-4592	509	2	hypothesis	hypothesis	NOUN
ejpam-4592	509	3	,	,	PUNCT
ejpam-4592	509	4	m1	m1	PROPN
ejpam-4592	509	5	=	=	PUNCT
ejpam-4592	509	6	m−	m−	PROPN
ejpam-4592	509	7	1	1	NUM
ejpam-4592	509	8	.	.	PUNCT
ejpam-4592	510	1	since	since	SCONJ
ejpam-4592	510	2	ε	ε	PROPN
ejpam-4592	510	3	m	m	VERB
ejpam-4592	510	4	>	>	X
ejpam-4592	510	5	0	0	PUNCT
ejpam-4592	511	1	and	and	CCONJ
ejpam-4592	511	2	the	the	DET
ejpam-4592	511	3	pair	pair	NOUN
ejpam-4592	511	4	of	of	ADP
ejpam-4592	511	5	sequence	sequence	NOUN
ejpam-4592	511	6	{	{	PUNCT
ejpam-4592	511	7	cn	cn	NOUN
ejpam-4592	511	8	}	}	PUNCT
ejpam-4592	511	9	,	,	PUNCT
ejpam-4592	511	10	{	{	PUNCT
ejpam-4592	511	11	dn	dn	VERB
ejpam-4592	511	12	}	}	PUNCT
ejpam-4592	511	13	is	be	AUX
ejpam-4592	511	14	asymptotic	asymptotic	ADJ
ejpam-4592	511	15	,	,	PUNCT
ejpam-4592	511	16	there	there	PRON
ejpam-4592	511	17	exists	exist	VERB
ejpam-4592	511	18	n0	n0	PROPN
ejpam-4592	511	19	∈	∈	PROPN
ejpam-4592	511	20	n	n	PRON
ejpam-4592	511	21	such	such	ADJ
ejpam-4592	511	22	that	that	DET
ejpam-4592	511	23	σ(cn	σ(cn	NOUN
ejpam-4592	511	24	,	,	PUNCT
ejpam-4592	511	25	dn	dn	NOUN
ejpam-4592	511	26	)	)	PUNCT
ejpam-4592	511	27	<	<	X
ejpam-4592	511	28	ε	ε	PROPN
ejpam-4592	511	29	m	m	VERB
ejpam-4592	511	30	for	for	ADP
ejpam-4592	511	31	all	all	DET
ejpam-4592	511	32	n	n	PRON
ejpam-4592	511	33	≥	≥	NOUN
ejpam-4592	511	34	n0	n0	NUM
ejpam-4592	511	35	.	.	PUNCT
ejpam-4592	512	1	take	take	VERB
ejpam-4592	512	2	m	m	NOUN
ejpam-4592	512	3	=	=	NOUN
ejpam-4592	512	4	max{n1	max{n1	NOUN
ejpam-4592	512	5	,	,	PUNCT
ejpam-4592	512	6	n0	n0	NUM
ejpam-4592	512	7	}	}	PUNCT
ejpam-4592	512	8	.	.	PUNCT
ejpam-4592	513	1	for	for	ADP
ejpam-4592	513	2	n	n	CCONJ
ejpam-4592	513	3	,	,	PUNCT
ejpam-4592	513	4	m	m	VERB
ejpam-4592	513	5	≥	≥	NOUN
ejpam-4592	513	6	m	m	PROPN
ejpam-4592	513	7	,	,	PUNCT
ejpam-4592	513	8	case	case	NOUN
ejpam-4592	513	9	1	1	NUM
ejpam-4592	513	10	:	:	PUNCT
ejpam-4592	513	11	if	if	SCONJ
ejpam-4592	513	12	m	m	VERB
ejpam-4592	513	13	<	<	X
ejpam-4592	513	14	n	n	CCONJ
ejpam-4592	513	15	,	,	PUNCT
ejpam-4592	513	16	then	then	ADV
ejpam-4592	513	17	m	m	VERB
ejpam-4592	513	18	≥	≥	NOUN
ejpam-4592	513	19	m	m	VERB
ejpam-4592	513	20	<	<	X
ejpam-4592	513	21	n.	n.	NOUN
ejpam-4592	513	22	by	by	ADP
ejpam-4592	513	23	assumption	assumption	NOUN
ejpam-4592	513	24	,	,	PUNCT
ejpam-4592	513	25	the	the	DET
ejpam-4592	513	26	points	point	NOUN
ejpam-4592	513	27	dm	dm	NOUN
ejpam-4592	513	28	and	and	CCONJ
ejpam-4592	513	29	dn	dn	PROPN
ejpam-4592	513	30	can	can	AUX
ejpam-4592	513	31	be	be	AUX
ejpam-4592	513	32	joined	join	VERB
ejpam-4592	513	33	by	by	ADP
ejpam-4592	513	34	an	an	DET
ejpam-4592	513	35	ε	ε	PROPN
ejpam-4592	513	36	m	m	VERB
ejpam-4592	513	37	-chain	-chain	NOUN
ejpam-4592	513	38	of	of	ADP
ejpam-4592	513	39	length	length	NOUN
ejpam-4592	513	40	m−	m−	PROPN
ejpam-4592	513	41	1	1	NUM
ejpam-4592	513	42	.	.	PUNCT
ejpam-4592	514	1	thus	thus	ADV
ejpam-4592	514	2	,	,	PUNCT
ejpam-4592	514	3	σ(dm	σ(dm	PROPN
ejpam-4592	514	4	,	,	PUNCT
ejpam-4592	514	5	dn	dn	PROPN
ejpam-4592	514	6	)	)	PUNCT
ejpam-4592	514	7	<	<	X
ejpam-4592	514	8	(	(	PUNCT
ejpam-4592	514	9	m−1)ε	m−1)ε	PROPN
ejpam-4592	514	10	m	m	NOUN
ejpam-4592	514	11	.	.	PUNCT
ejpam-4592	515	1	now	now	ADV
ejpam-4592	515	2	σ(cm	σ(cm	PROPN
ejpam-4592	515	3	,	,	PUNCT
ejpam-4592	515	4	dn	dn	ADJ
ejpam-4592	515	5	)	)	PUNCT
ejpam-4592	515	6	≤	≤	NOUN
ejpam-4592	515	7	σ(cm	σ(cm	PROPN
ejpam-4592	515	8	,	,	PUNCT
ejpam-4592	515	9	dm	dm	PROPN
ejpam-4592	515	10	)	)	PUNCT
ejpam-4592	516	1	+	+	CCONJ
ejpam-4592	516	2	σ(dm	σ(dm	PROPN
ejpam-4592	516	3	,	,	PUNCT
ejpam-4592	516	4	dn	dn	PROPN
ejpam-4592	516	5	)	)	PUNCT
ejpam-4592	516	6	<	<	X
ejpam-4592	516	7	ε	ε	PROPN
ejpam-4592	516	8	m	m	VERB
ejpam-4592	516	9	+	+	PUNCT
ejpam-4592	516	10	(	(	PUNCT
ejpam-4592	516	11	m−1)ε	m−1)ε	NUM
ejpam-4592	516	12	m	m	PROPN
ejpam-4592	516	13	=	=	SYM
ejpam-4592	516	14	ε	ε	PROPN
ejpam-4592	516	15	.	.	PUNCT
ejpam-4592	516	16	case	case	NOUN
ejpam-4592	516	17	2	2	NUM
ejpam-4592	516	18	:	:	PUNCT
ejpam-4592	516	19	suppose	suppose	VERB
ejpam-4592	516	20	that	that	SCONJ
ejpam-4592	516	21	m	m	VERB
ejpam-4592	516	22	>	>	X
ejpam-4592	516	23	n.	n.	NOUN
ejpam-4592	516	24	then	then	ADV
ejpam-4592	516	25	by	by	ADP
ejpam-4592	516	26	similar	similar	ADJ
ejpam-4592	516	27	argument	argument	NOUN
ejpam-4592	516	28	as	as	ADP
ejpam-4592	516	29	in	in	ADP
ejpam-4592	516	30	case-1	case-1	NUM
ejpam-4592	516	31	,	,	PUNCT
ejpam-4592	516	32	we	we	PRON
ejpam-4592	516	33	get	get	VERB
ejpam-4592	516	34	σ(cm	σ(cm	PROPN
ejpam-4592	516	35	,	,	PUNCT
ejpam-4592	516	36	dn	dn	ADJ
ejpam-4592	516	37	)	)	PUNCT
ejpam-4592	516	38	<	<	X
ejpam-4592	517	1	ε	ε	PROPN
ejpam-4592	517	2	.	.	PROPN
ejpam-4592	518	1	from	from	ADP
ejpam-4592	518	2	case	case	NOUN
ejpam-4592	518	3	1	1	NUM
ejpam-4592	518	4	and	and	CCONJ
ejpam-4592	518	5	case	case	NOUN
ejpam-4592	518	6	2	2	NUM
ejpam-4592	518	7	,	,	PUNCT
ejpam-4592	518	8	we	we	PRON
ejpam-4592	518	9	get	get	VERB
ejpam-4592	518	10	,	,	PUNCT
ejpam-4592	518	11	σ(cm	σ(cm	PROPN
ejpam-4592	518	12	,	,	PUNCT
ejpam-4592	518	13	dn	dn	PROPN
ejpam-4592	518	14	)	)	PUNCT
ejpam-4592	518	15	<	<	X
ejpam-4592	518	16	ε	ε	PROPN
ejpam-4592	518	17	for	for	ADP
ejpam-4592	518	18	all	all	DET
ejpam-4592	518	19	m	m	PROPN
ejpam-4592	518	20	,	,	PUNCT
ejpam-4592	518	21	n	n	PRON
ejpam-4592	518	22	≥	≥	NOUN
ejpam-4592	518	23	m.	m.	NOUN
ejpam-4592	518	24	hence	hence	ADV
ejpam-4592	518	25	the	the	DET
ejpam-4592	518	26	pair	pair	NOUN
ejpam-4592	518	27	of	of	ADP
ejpam-4592	518	28	sequence	sequence	NOUN
ejpam-4592	518	29	{	{	PUNCT
ejpam-4592	518	30	cn	cn	NOUN
ejpam-4592	518	31	}	}	PUNCT
ejpam-4592	518	32	and	and	CCONJ
ejpam-4592	518	33	{	{	PUNCT
ejpam-4592	518	34	dn	dn	VERB
ejpam-4592	518	35	}	}	PUNCT
ejpam-4592	518	36	is	be	AUX
ejpam-4592	518	37	uniformly	uniformly	ADV
ejpam-4592	518	38	asymptotic	asymptotic	ADJ
ejpam-4592	518	39	with	with	ADP
ejpam-4592	518	40	respect	respect	NOUN
ejpam-4592	518	41	to	to	ADP
ejpam-4592	518	42	the	the	DET
ejpam-4592	518	43	metric	metric	PROPN
ejpam-4592	518	44	σ	σ	PROPN
ejpam-4592	518	45	∈	∈	PROPN
ejpam-4592	518	46	ω	ω	PROPN
ejpam-4592	518	47	.	.	PUNCT
ejpam-4592	519	1	next	next	ADJ
ejpam-4592	519	2	,	,	PUNCT
ejpam-4592	519	3	the	the	DET
ejpam-4592	519	4	rest	rest	NOUN
ejpam-4592	519	5	of	of	ADP
ejpam-4592	519	6	this	this	DET
ejpam-4592	519	7	section	section	NOUN
ejpam-4592	519	8	,	,	PUNCT
ejpam-4592	519	9	in	in	ADP
ejpam-4592	519	10	a	a	DET
ejpam-4592	519	11	hyperconnected	hyperconnecte	VERB
ejpam-4592	519	12	space	space	NOUN
ejpam-4592	519	13	,	,	PUNCT
ejpam-4592	519	14	some	some	DET
ejpam-4592	519	15	interesting	interesting	ADJ
ejpam-4592	519	16	results	result	NOUN
ejpam-4592	519	17	for	for	ADP
ejpam-4592	519	18	uniformly	uniformly	ADV
ejpam-4592	519	19	asymptotic	asymptotic	ADJ
ejpam-4592	519	20	sequence	sequence	NOUN
ejpam-4592	519	21	are	be	AUX
ejpam-4592	519	22	proven	prove	VERB
ejpam-4592	519	23	.	.	PUNCT
ejpam-4592	520	1	theorem	theorem	PROPN
ejpam-4592	520	2	47	47	NUM
ejpam-4592	520	3	.	.	PUNCT
ejpam-4592	521	1	let	let	VERB
ejpam-4592	521	2	(	(	PUNCT
ejpam-4592	521	3	x	x	NOUN
ejpam-4592	521	4	,	,	PUNCT
ejpam-4592	521	5	ω	ω	NUM
ejpam-4592	521	6	)	)	PUNCT
ejpam-4592	521	7	be	be	AUX
ejpam-4592	521	8	a	a	DET
ejpam-4592	521	9	gms	gms	NOUN
ejpam-4592	521	10	and	and	CCONJ
ejpam-4592	521	11	{	{	PUNCT
ejpam-4592	521	12	cn	cn	PROPN
ejpam-4592	521	13	}	}	PUNCT
ejpam-4592	521	14	,	,	PUNCT
ejpam-4592	521	15	{	{	PUNCT
ejpam-4592	521	16	dn	dn	PART
ejpam-4592	521	17	}	}	PUNCT
ejpam-4592	521	18	be	be	AUX
ejpam-4592	521	19	cauchy	cauchy	NOUN
ejpam-4592	521	20	,	,	PUNCT
ejpam-4592	521	21	cofinally	cofinally	ADV
ejpam-4592	521	22	-	-	PUNCT
ejpam-4592	521	23	cauchy	cauchy	ADJ
ejpam-4592	521	24	sequences	sequence	NOUN
ejpam-4592	521	25	with	with	ADP
ejpam-4592	521	26	respect	respect	NOUN
ejpam-4592	521	27	to	to	ADP
ejpam-4592	521	28	σ	σ	PROPN
ejpam-4592	521	29	∈	∈	PROPN
ejpam-4592	521	30	ω	ω	PROPN
ejpam-4592	521	31	in	in	ADP
ejpam-4592	521	32	x	x	PROPN
ejpam-4592	521	33	,	,	PUNCT
ejpam-4592	521	34	respectively	respectively	ADV
ejpam-4592	521	35	.	.	PUNCT
ejpam-4592	522	1	if	if	SCONJ
ejpam-4592	522	2	(	(	PUNCT
ejpam-4592	522	3	x,µω	x,µω	NUM
ejpam-4592	522	4	)	)	PUNCT
ejpam-4592	522	5	is	be	AUX
ejpam-4592	522	6	a	a	DET
ejpam-4592	522	7	hyperconnected	hyperconnected	ADJ
ejpam-4592	522	8	space	space	NOUN
ejpam-4592	522	9	,	,	PUNCT
ejpam-4592	522	10	then	then	ADV
ejpam-4592	522	11	the	the	DET
ejpam-4592	522	12	pair	pair	NOUN
ejpam-4592	522	13	of	of	ADP
ejpam-4592	522	14	sequence	sequence	NOUN
ejpam-4592	522	15	{	{	PUNCT
ejpam-4592	522	16	cn	cn	NOUN
ejpam-4592	522	17	}	}	PUNCT
ejpam-4592	522	18	and	and	CCONJ
ejpam-4592	522	19	{	{	PUNCT
ejpam-4592	522	20	dn	dn	VERB
ejpam-4592	522	21	}	}	PUNCT
ejpam-4592	522	22	is	be	AUX
ejpam-4592	522	23	uniformly	uniformly	ADV
ejpam-4592	522	24	asymptotic	asymptotic	ADJ
ejpam-4592	522	25	with	with	ADP
ejpam-4592	522	26	the	the	DET
ejpam-4592	522	27	same	same	ADJ
ejpam-4592	522	28	metric	metric	NOUN
ejpam-4592	522	29	.	.	PUNCT
ejpam-4592	523	1	proof	proof	NOUN
ejpam-4592	523	2	.	.	PUNCT
ejpam-4592	524	1	let	let	VERB
ejpam-4592	524	2	ε	ε	PROPN
ejpam-4592	524	3	>	>	X
ejpam-4592	524	4	0	0	PUNCT
ejpam-4592	524	5	be	be	AUX
ejpam-4592	524	6	given	give	VERB
ejpam-4592	524	7	.	.	PUNCT
ejpam-4592	525	1	then	then	ADV
ejpam-4592	525	2	there	there	PRON
ejpam-4592	525	3	is	be	VERB
ejpam-4592	525	4	n0	n0	NUM
ejpam-4592	525	5	∈	∈	PROPN
ejpam-4592	525	6	n	n	CCONJ
ejpam-4592	525	7	such	such	ADJ
ejpam-4592	525	8	that	that	DET
ejpam-4592	525	9	σ(cn	σ(cn	NOUN
ejpam-4592	525	10	,	,	PUNCT
ejpam-4592	525	11	cm	cm	NOUN
ejpam-4592	525	12	)	)	PUNCT
ejpam-4592	525	13	<	<	X
ejpam-4592	525	14	ε	ε	PROPN
ejpam-4592	525	15	4	4	NUM
ejpam-4592	525	16	for	for	ADP
ejpam-4592	525	17	all	all	DET
ejpam-4592	525	18	n	n	CCONJ
ejpam-4592	525	19	,	,	PUNCT
ejpam-4592	525	20	m	m	PROPN
ejpam-4592	525	21	≥	≥	NOUN
ejpam-4592	525	22	n0	n0	ADJ
ejpam-4592	525	23	and	and	CCONJ
ejpam-4592	525	24	there	there	PRON
ejpam-4592	525	25	is	be	VERB
ejpam-4592	525	26	an	an	DET
ejpam-4592	525	27	infinite	infinite	NOUN
ejpam-4592	525	28	subset	subset	NOUN
ejpam-4592	525	29	nε	nε	NOUN
ejpam-4592	525	30	of	of	ADP
ejpam-4592	525	31	n	n	PRON
ejpam-4592	525	32	such	such	ADJ
ejpam-4592	525	33	that	that	SCONJ
ejpam-4592	525	34	σ(dk	σ(dk	PROPN
ejpam-4592	525	35	,	,	PUNCT
ejpam-4592	525	36	dl	dl	PROPN
ejpam-4592	525	37	)	)	PUNCT
ejpam-4592	525	38	<	<	X
ejpam-4592	525	39	ε	ε	PROPN
ejpam-4592	525	40	4	4	NUM
ejpam-4592	525	41	for	for	ADP
ejpam-4592	525	42	every	every	DET
ejpam-4592	525	43	k	k	PROPN
ejpam-4592	525	44	,	,	PUNCT
ejpam-4592	525	45	l	l	PROPN
ejpam-4592	525	46	∈	∈	PROPN
ejpam-4592	525	47	nε	nε	VERB
ejpam-4592	525	48	.	.	PUNCT
ejpam-4592	526	1	take	take	VERB
ejpam-4592	526	2	m0	m0	NOUN
ejpam-4592	526	3	=	=	SYM
ejpam-4592	526	4	max{n0	max{n0	PROPN
ejpam-4592	526	5	,	,	PUNCT
ejpam-4592	526	6	n1	n1	NOUN
ejpam-4592	526	7	|	|	ADV
ejpam-4592	526	8	n1	n1	NOUN
ejpam-4592	526	9	=	=	SYM
ejpam-4592	526	10	inf{nε	inf{nε	NOUN
ejpam-4592	526	11	}	}	PUNCT
ejpam-4592	526	12	}	}	PUNCT
ejpam-4592	526	13	.	.	PUNCT
ejpam-4592	527	1	if	if	SCONJ
ejpam-4592	527	2	bσ(cn	bσ(cn	NOUN
ejpam-4592	527	3	,	,	PUNCT
ejpam-4592	527	4	ε	ε	PROPN
ejpam-4592	527	5	4	4	NUM
ejpam-4592	527	6	)	)	PUNCT
ejpam-4592	527	7	∩	∩	NOUN
ejpam-4592	527	8	bσ(dl	bσ(dl	NOUN
ejpam-4592	527	9	,	,	PUNCT
ejpam-4592	527	10	ε	ε	PROPN
ejpam-4592	527	11	4	4	NUM
ejpam-4592	527	12	)	)	PUNCT
ejpam-4592	527	13	=	=	NOUN
ejpam-4592	527	14	∅	∅	NOUN
ejpam-4592	527	15	,	,	PUNCT
ejpam-4592	527	16	then	then	ADV
ejpam-4592	527	17	bσ(cn	bσ(cn	NOUN
ejpam-4592	527	18	,	,	PUNCT
ejpam-4592	527	19	ε	ε	PROPN
ejpam-4592	527	20	4	4	NUM
ejpam-4592	527	21	)	)	PUNCT
ejpam-4592	527	22	⊂	⊂	PROPN
ejpam-4592	527	23	x−bσ(dl	x−bσ(dl	PROPN
ejpam-4592	527	24	,	,	PUNCT
ejpam-4592	527	25	ε	ε	PROPN
ejpam-4592	527	26	4	4	NUM
ejpam-4592	527	27	)	)	PUNCT
ejpam-4592	527	28	.	.	PUNCT
ejpam-4592	528	1	this	this	PRON
ejpam-4592	528	2	implies	imply	VERB
ejpam-4592	528	3	iµω(x−bσ(dl	iµω(x−bσ(dl	PROPN
ejpam-4592	528	4	,	,	PUNCT
ejpam-4592	528	5	ε	ε	PROPN
ejpam-4592	528	6	4	4	NUM
ejpam-4592	528	7	)	)	PUNCT
ejpam-4592	528	8	)	)	PUNCT
ejpam-4592	529	1	̸=	̸=	PROPN
ejpam-4592	529	2	∅	∅	NOUN
ejpam-4592	529	3	which	which	PRON
ejpam-4592	529	4	implies	imply	VERB
ejpam-4592	529	5	that	that	SCONJ
ejpam-4592	529	6	cµω(bσ(dl	cµω(bσ(dl	PROPN
ejpam-4592	529	7	,	,	PUNCT
ejpam-4592	529	8	ε	ε	PROPN
ejpam-4592	529	9	4	4	NUM
ejpam-4592	529	10	)	)	PUNCT
ejpam-4592	529	11	)	)	PUNCT
ejpam-4592	529	12	̸=	̸=	PROPN
ejpam-4592	529	13	x.	x.	NOUN
ejpam-4592	529	14	but	but	CCONJ
ejpam-4592	529	15	bσ(dl	bσ(dl	PROPN
ejpam-4592	529	16	,	,	PUNCT
ejpam-4592	529	17	ε	ε	PROPN
ejpam-4592	529	18	4	4	NUM
ejpam-4592	529	19	)	)	PUNCT
ejpam-4592	529	20	is	be	AUX
ejpam-4592	529	21	µω	µω	ADJ
ejpam-4592	529	22	-	-	PUNCT
ejpam-4592	529	23	dense	dense	ADJ
ejpam-4592	529	24	,	,	PUNCT
ejpam-4592	529	25	since	since	SCONJ
ejpam-4592	529	26	(	(	PUNCT
ejpam-4592	529	27	x,µω	x,µω	X
ejpam-4592	529	28	)	)	PUNCT
ejpam-4592	529	29	is	be	AUX
ejpam-4592	529	30	a	a	DET
ejpam-4592	529	31	hyperconnected	hyperconnected	ADJ
ejpam-4592	529	32	space	space	NOUN
ejpam-4592	529	33	.	.	PUNCT
ejpam-4592	530	1	therefore	therefore	ADV
ejpam-4592	530	2	,	,	PUNCT
ejpam-4592	530	3	bσ(cn	bσ(cn	NOUN
ejpam-4592	530	4	,	,	PUNCT
ejpam-4592	530	5	ε	ε	PROPN
ejpam-4592	530	6	4)∩	4)∩	PROPN
ejpam-4592	530	7	bσ(dl	bσ(dl	PROPN
ejpam-4592	530	8	,	,	PUNCT
ejpam-4592	530	9	ε	ε	PROPN
ejpam-4592	530	10	4	4	NUM
ejpam-4592	530	11	)	)	PUNCT
ejpam-4592	530	12	̸=	̸=	PROPN
ejpam-4592	530	13	∅.	∅.	ADV
ejpam-4592	530	14	let	let	VERB
ejpam-4592	530	15	t	t	PROPN
ejpam-4592	530	16	∈	∈	PROPN
ejpam-4592	530	17	bσ(cn	bσ(cn	PROPN
ejpam-4592	530	18	,	,	PUNCT
ejpam-4592	530	19	ε	ε	PROPN
ejpam-4592	530	20	4	4	NUM
ejpam-4592	530	21	)	)	PUNCT
ejpam-4592	530	22	∩	∩	NOUN
ejpam-4592	530	23	bσ(dl	bσ(dl	NOUN
ejpam-4592	530	24	,	,	PUNCT
ejpam-4592	530	25	ε	ε	PROPN
ejpam-4592	530	26	4	4	NUM
ejpam-4592	530	27	)	)	PUNCT
ejpam-4592	530	28	.	.	PUNCT
ejpam-4592	531	1	then	then	ADV
ejpam-4592	531	2	σ(cn	σ(cn	PROPN
ejpam-4592	531	3	,	,	PUNCT
ejpam-4592	531	4	t	t	PROPN
ejpam-4592	531	5	)	)	PUNCT
ejpam-4592	531	6	<	<	X
ejpam-4592	531	7	ε	ε	PROPN
ejpam-4592	531	8	4	4	NUM
ejpam-4592	531	9	and	and	CCONJ
ejpam-4592	531	10	σ(dl	σ(dl	NOUN
ejpam-4592	531	11	,	,	PUNCT
ejpam-4592	531	12	t	t	NOUN
ejpam-4592	531	13	)	)	PUNCT
ejpam-4592	531	14	<	<	X
ejpam-4592	531	15	ε	ε	PROPN
ejpam-4592	531	16	4	4	NUM
ejpam-4592	531	17	.	.	PUNCT
ejpam-4592	532	1	for	for	SCONJ
ejpam-4592	532	2	l1,m1	l1,m1	PROPN
ejpam-4592	532	3	≥	≥	NUM
ejpam-4592	532	4	m0	m0	NOUN
ejpam-4592	532	5	,	,	PUNCT
ejpam-4592	532	6	σ(cl1	σ(cl1	PROPN
ejpam-4592	532	7	,	,	PUNCT
ejpam-4592	532	8	dm1	dm1	NOUN
ejpam-4592	532	9	)	)	PUNCT
ejpam-4592	532	10	≤	≤	NUM
ejpam-4592	532	11	σ(cl1	σ(cl1	PROPN
ejpam-4592	532	12	,	,	PUNCT
ejpam-4592	532	13	cn	cn	ADJ
ejpam-4592	532	14	)	)	PUNCT
ejpam-4592	532	15	+	+	NUM
ejpam-4592	532	16	σ(cn	σ(cn	PROPN
ejpam-4592	532	17	,	,	PUNCT
ejpam-4592	532	18	t	t	PROPN
ejpam-4592	532	19	)	)	PUNCT
ejpam-4592	532	20	+	+	CCONJ
ejpam-4592	532	21	σ(t	σ(t	PROPN
ejpam-4592	532	22	,	,	PUNCT
ejpam-4592	532	23	dl	dl	NOUN
ejpam-4592	532	24	)	)	PUNCT
ejpam-4592	532	25	+	+	CCONJ
ejpam-4592	532	26	σ(dl	σ(dl	NOUN
ejpam-4592	532	27	,	,	PUNCT
ejpam-4592	532	28	dm1	dm1	PROPN
ejpam-4592	532	29	)	)	PUNCT
ejpam-4592	532	30	.	.	PUNCT
ejpam-4592	533	1	if	if	SCONJ
ejpam-4592	533	2	m1	m1	PROPN
ejpam-4592	533	3	∈	∈	PROPN
ejpam-4592	533	4	nε	nε	VERB
ejpam-4592	533	5	,	,	PUNCT
ejpam-4592	533	6	then	then	ADV
ejpam-4592	533	7	we	we	PRON
ejpam-4592	533	8	get	get	VERB
ejpam-4592	533	9	σ(dl	σ(dl	NOUN
ejpam-4592	533	10	,	,	PUNCT
ejpam-4592	533	11	dm1	dm1	PROPN
ejpam-4592	533	12	)	)	PUNCT
ejpam-4592	533	13	<	<	X
ejpam-4592	533	14	ε	ε	PROPN
ejpam-4592	533	15	4	4	NUM
ejpam-4592	533	16	.	.	PUNCT
ejpam-4592	534	1	therefore	therefore	ADV
ejpam-4592	534	2	,	,	PUNCT
ejpam-4592	534	3	σ(cl1	σ(cl1	PROPN
ejpam-4592	534	4	,	,	PUNCT
ejpam-4592	534	5	dm1	dm1	PROPN
ejpam-4592	534	6	)	)	PUNCT
ejpam-4592	534	7	<	<	X
ejpam-4592	534	8	ε	ε	PROPN
ejpam-4592	534	9	for	for	ADP
ejpam-4592	534	10	all	all	PRON
ejpam-4592	534	11	l1,m1	l1,m1	PROPN
ejpam-4592	534	12	≥	≥	NUM
ejpam-4592	534	13	m0	m0	PROPN
ejpam-4592	534	14	.	.	PUNCT
ejpam-4592	535	1	assume	assume	VERB
ejpam-4592	535	2	that	that	SCONJ
ejpam-4592	535	3	,	,	PUNCT
ejpam-4592	535	4	m1	m1	PROPN
ejpam-4592	535	5	/∈	/∈	PUNCT
ejpam-4592	536	1	nε	nε	PROPN
ejpam-4592	536	2	.	.	PUNCT
ejpam-4592	537	1	since	since	SCONJ
ejpam-4592	537	2	(	(	PUNCT
ejpam-4592	537	3	x,µω	x,µω	X
ejpam-4592	537	4	)	)	PUNCT
ejpam-4592	537	5	is	be	AUX
ejpam-4592	537	6	a	a	DET
ejpam-4592	537	7	hyperconnected	hyperconnected	ADJ
ejpam-4592	537	8	space	space	NOUN
ejpam-4592	537	9	,	,	PUNCT
ejpam-4592	537	10	bσ(dl	bσ(dl	PROPN
ejpam-4592	537	11	,	,	PUNCT
ejpam-4592	537	12	ε	ε	PROPN
ejpam-4592	537	13	8)	8)	NUM
ejpam-4592	537	14	is	be	AUX
ejpam-4592	537	15	µω	µω	ADJ
ejpam-4592	537	16	-	-	PUNCT
ejpam-4592	537	17	dense	dense	ADJ
ejpam-4592	537	18	.	.	PUNCT
ejpam-4592	538	1	then	then	ADV
ejpam-4592	538	2	bσ(dl	bσ(dl	PROPN
ejpam-4592	538	3	,	,	PUNCT
ejpam-4592	538	4	ε	ε	PROPN
ejpam-4592	538	5	8)∩	8)∩	NUM
ejpam-4592	538	6	bσ(dm1	bσ(dm1	NOUN
ejpam-4592	538	7	,	,	PUNCT
ejpam-4592	538	8	ε	ε	PROPN
ejpam-4592	538	9	8)	8)	NUM
ejpam-4592	538	10	̸=	̸=	PROPN
ejpam-4592	538	11	∅	∅	NOUN
ejpam-4592	538	12	,	,	PUNCT
ejpam-4592	538	13	since	since	SCONJ
ejpam-4592	538	14	bσ(dm1	bσ(dm1	NOUN
ejpam-4592	538	15	,	,	PUNCT
ejpam-4592	538	16	ε	ε	PROPN
ejpam-4592	538	17	8)	8)	NUM
ejpam-4592	538	18	∈	∈	PROPN
ejpam-4592	538	19	µ̃ω	µ̃ω	NOUN
ejpam-4592	538	20	.	.	PUNCT
ejpam-4592	539	1	let	let	VERB
ejpam-4592	539	2	z	z	NOUN
ejpam-4592	539	3	∈	∈	PROPN
ejpam-4592	539	4	bσ(dl	bσ(dl	PROPN
ejpam-4592	539	5	,	,	PUNCT
ejpam-4592	539	6	ε	ε	PROPN
ejpam-4592	539	7	8)∩bσ(dm1	8)∩bσ(dm1	NUM
ejpam-4592	539	8	,	,	PUNCT
ejpam-4592	539	9	ε	ε	PROPN
ejpam-4592	539	10	8)	8)	NUM
ejpam-4592	539	11	.	.	PUNCT
ejpam-4592	540	1	then	then	ADV
ejpam-4592	540	2	σ(dl	σ(dl	NUM
ejpam-4592	540	3	,	,	PUNCT
ejpam-4592	540	4	dm1	dm1	NOUN
ejpam-4592	540	5	)	)	PUNCT
ejpam-4592	540	6	≤	≤	NOUN
ejpam-4592	540	7	σ(dl	σ(dl	NOUN
ejpam-4592	540	8	,	,	PUNCT
ejpam-4592	540	9	z)+σ(z	z)+σ(z	PROPN
ejpam-4592	540	10	,	,	PUNCT
ejpam-4592	540	11	dm1	dm1	PROPN
ejpam-4592	540	12	)	)	PUNCT
ejpam-4592	540	13	<	<	X
ejpam-4592	540	14	ε	ε	PROPN
ejpam-4592	540	15	8	8	NUM
ejpam-4592	540	16	+	+	CCONJ
ejpam-4592	540	17	ε	ε	PROPN
ejpam-4592	540	18	8	8	NUM
ejpam-4592	540	19	=	=	SYM
ejpam-4592	540	20	ε	ε	PROPN
ejpam-4592	540	21	4	4	NUM
ejpam-4592	540	22	.	.	PUNCT
ejpam-4592	541	1	therefore	therefore	ADV
ejpam-4592	541	2	,	,	PUNCT
ejpam-4592	541	3	σ(cl1	σ(cl1	PROPN
ejpam-4592	541	4	,	,	PUNCT
ejpam-4592	541	5	dm1	dm1	PROPN
ejpam-4592	541	6	)	)	PUNCT
ejpam-4592	541	7	<	<	X
ejpam-4592	541	8	ε	ε	PROPN
ejpam-4592	541	9	for	for	ADP
ejpam-4592	541	10	all	all	DET
ejpam-4592	541	11	l1,m1	l1,m1	PROPN
ejpam-4592	541	12	≥	≥	NUM
ejpam-4592	541	13	m0	m0	NOUN
ejpam-4592	541	14	.	.	PUNCT
ejpam-4592	542	1	hence	hence	ADV
ejpam-4592	542	2	the	the	DET
ejpam-4592	542	3	pair	pair	NOUN
ejpam-4592	542	4	of	of	ADP
ejpam-4592	542	5	sequence	sequence	NOUN
ejpam-4592	542	6	{	{	PUNCT
ejpam-4592	542	7	cn	cn	NOUN
ejpam-4592	542	8	}	}	PUNCT
ejpam-4592	542	9	and	and	CCONJ
ejpam-4592	542	10	{	{	PUNCT
ejpam-4592	542	11	dn	dn	VERB
ejpam-4592	542	12	}	}	PUNCT
ejpam-4592	542	13	is	be	AUX
ejpam-4592	542	14	uniformly	uniformly	ADV
ejpam-4592	542	15	asymptotic	asymptotic	ADJ
ejpam-4592	542	16	sequence	sequence	NOUN
ejpam-4592	542	17	with	with	ADP
ejpam-4592	542	18	respect	respect	NOUN
ejpam-4592	542	19	to	to	ADP
ejpam-4592	542	20	σ	σ	PROPN
ejpam-4592	542	21	∈	∈	PROPN
ejpam-4592	542	22	ω	ω	PROPN
ejpam-4592	542	23	in	in	ADP
ejpam-4592	542	24	x.	x.	PROPN
ejpam-4592	542	25	v.	v.	PROPN
ejpam-4592	542	26	subramanian	subramanian	PROPN
ejpam-4592	542	27	et	et	PROPN
ejpam-4592	542	28	al	al	PROPN
ejpam-4592	542	29	.	.	PUNCT
ejpam-4592	542	30	/	/	SYM
ejpam-4592	542	31	eur	eur	PROPN
ejpam-4592	542	32	.	.	PUNCT
ejpam-4592	543	1	j.	j.	PROPN
ejpam-4592	543	2	pure	pure	PROPN
ejpam-4592	543	3	appl	appl	PROPN
ejpam-4592	543	4	.	.	PROPN
ejpam-4592	543	5	math	math	PROPN
ejpam-4592	543	6	,	,	PUNCT
ejpam-4592	543	7	15	15	NUM
ejpam-4592	543	8	(	(	PUNCT
ejpam-4592	543	9	4	4	NUM
ejpam-4592	543	10	)	)	PUNCT
ejpam-4592	543	11	(	(	PUNCT
ejpam-4592	543	12	2022	2022	NUM
ejpam-4592	543	13	)	)	PUNCT
ejpam-4592	543	14	,	,	PUNCT
ejpam-4592	543	15	1869	1869	NUM
ejpam-4592	543	16	-	-	SYM
ejpam-4592	543	17	1886	1886	NUM
ejpam-4592	543	18	1884	1884	NUM
ejpam-4592	543	19	corollary	corollary	NOUN
ejpam-4592	543	20	48	48	NUM
ejpam-4592	543	21	.	.	PUNCT
ejpam-4592	544	1	let	let	VERB
ejpam-4592	544	2	(	(	PUNCT
ejpam-4592	544	3	x	x	NOUN
ejpam-4592	544	4	,	,	PUNCT
ejpam-4592	544	5	ω	ω	NUM
ejpam-4592	544	6	)	)	PUNCT
ejpam-4592	544	7	be	be	AUX
ejpam-4592	544	8	a	a	DET
ejpam-4592	544	9	gms	gms	NOUN
ejpam-4592	544	10	and	and	CCONJ
ejpam-4592	544	11	{	{	PUNCT
ejpam-4592	544	12	cn	cn	PROPN
ejpam-4592	544	13	}	}	PUNCT
ejpam-4592	544	14	,	,	PUNCT
ejpam-4592	544	15	{	{	PUNCT
ejpam-4592	544	16	dn	dn	PART
ejpam-4592	544	17	}	}	PUNCT
ejpam-4592	544	18	be	be	AUX
ejpam-4592	544	19	cauchy	cauchy	ADJ
ejpam-4592	544	20	sequences	sequence	NOUN
ejpam-4592	544	21	with	with	ADP
ejpam-4592	544	22	respect	respect	NOUN
ejpam-4592	544	23	to	to	ADP
ejpam-4592	544	24	σ	σ	PROPN
ejpam-4592	544	25	∈	∈	PROPN
ejpam-4592	544	26	ω	ω	PROPN
ejpam-4592	544	27	in	in	ADP
ejpam-4592	544	28	x.	x.	NOUN
ejpam-4592	544	29	if	if	SCONJ
ejpam-4592	544	30	(	(	PUNCT
ejpam-4592	544	31	x,µω	x,µω	NUM
ejpam-4592	544	32	)	)	PUNCT
ejpam-4592	544	33	is	be	AUX
ejpam-4592	544	34	a	a	DET
ejpam-4592	544	35	hyperconnected	hyperconnected	ADJ
ejpam-4592	544	36	space	space	NOUN
ejpam-4592	544	37	,	,	PUNCT
ejpam-4592	544	38	then	then	ADV
ejpam-4592	544	39	the	the	DET
ejpam-4592	544	40	pair	pair	NOUN
ejpam-4592	544	41	of	of	ADP
ejpam-4592	544	42	sequence	sequence	NOUN
ejpam-4592	544	43	{	{	PUNCT
ejpam-4592	544	44	cn	cn	NOUN
ejpam-4592	544	45	}	}	PUNCT
ejpam-4592	544	46	and	and	CCONJ
ejpam-4592	544	47	{	{	PUNCT
ejpam-4592	544	48	dn	dn	VERB
ejpam-4592	544	49	}	}	PUNCT
ejpam-4592	544	50	is	be	AUX
ejpam-4592	544	51	uniformly	uniformly	ADV
ejpam-4592	544	52	asymptotic	asymptotic	ADJ
ejpam-4592	544	53	sequence	sequence	NOUN
ejpam-4592	544	54	with	with	ADP
ejpam-4592	544	55	respect	respect	NOUN
ejpam-4592	544	56	to	to	ADP
ejpam-4592	544	57	the	the	DET
ejpam-4592	544	58	same	same	ADJ
ejpam-4592	544	59	metric	metric	NOUN
ejpam-4592	544	60	.	.	PUNCT
ejpam-4592	545	1	theorem	theorem	VERB
ejpam-4592	545	2	49	49	NUM
ejpam-4592	545	3	.	.	PUNCT
ejpam-4592	546	1	let	let	VERB
ejpam-4592	546	2	(	(	PUNCT
ejpam-4592	546	3	x	x	NOUN
ejpam-4592	546	4	,	,	PUNCT
ejpam-4592	546	5	ω	ω	NUM
ejpam-4592	546	6	)	)	PUNCT
ejpam-4592	546	7	be	be	AUX
ejpam-4592	546	8	a	a	DET
ejpam-4592	546	9	gms	gms	NOUN
ejpam-4592	546	10	and	and	CCONJ
ejpam-4592	546	11	{	{	PUNCT
ejpam-4592	546	12	cn	cn	PROPN
ejpam-4592	546	13	}	}	PUNCT
ejpam-4592	546	14	,	,	PUNCT
ejpam-4592	546	15	{	{	PUNCT
ejpam-4592	546	16	dn	dn	PART
ejpam-4592	546	17	}	}	PUNCT
ejpam-4592	546	18	be	be	AUX
ejpam-4592	546	19	cauchy	cauchy	NOUN
ejpam-4592	546	20	,	,	PUNCT
ejpam-4592	546	21	convergent	convergent	NOUN
ejpam-4592	546	22	sequences	sequence	NOUN
ejpam-4592	546	23	with	with	ADP
ejpam-4592	546	24	respect	respect	NOUN
ejpam-4592	546	25	to	to	ADP
ejpam-4592	546	26	σ	σ	PROPN
ejpam-4592	546	27	∈	∈	PROPN
ejpam-4592	546	28	ω	ω	PROPN
ejpam-4592	546	29	in	in	ADP
ejpam-4592	546	30	x	x	PROPN
ejpam-4592	546	31	,	,	PUNCT
ejpam-4592	546	32	respectively	respectively	ADV
ejpam-4592	546	33	.	.	PUNCT
ejpam-4592	547	1	if	if	SCONJ
ejpam-4592	547	2	(	(	PUNCT
ejpam-4592	547	3	x,µω	x,µω	NUM
ejpam-4592	547	4	)	)	PUNCT
ejpam-4592	547	5	is	be	AUX
ejpam-4592	547	6	a	a	DET
ejpam-4592	547	7	hyperconnected	hyperconnected	ADJ
ejpam-4592	547	8	space	space	NOUN
ejpam-4592	547	9	,	,	PUNCT
ejpam-4592	547	10	then	then	ADV
ejpam-4592	547	11	{	{	PUNCT
ejpam-4592	547	12	cn	cn	NOUN
ejpam-4592	547	13	}	}	PUNCT
ejpam-4592	547	14	is	be	AUX
ejpam-4592	547	15	a	a	DET
ejpam-4592	547	16	convergent	convergent	NOUN
ejpam-4592	547	17	sequence	sequence	NOUN
ejpam-4592	547	18	with	with	ADP
ejpam-4592	547	19	respect	respect	NOUN
ejpam-4592	547	20	to	to	ADP
ejpam-4592	547	21	σ	σ	PROPN
ejpam-4592	547	22	.	.	PUNCT
ejpam-4592	548	1	proof	proof	NOUN
ejpam-4592	548	2	.	.	PUNCT
ejpam-4592	549	1	let	let	VERB
ejpam-4592	549	2	ε	ε	PROPN
ejpam-4592	549	3	>	>	X
ejpam-4592	549	4	0	0	PUNCT
ejpam-4592	549	5	be	be	AUX
ejpam-4592	549	6	given	give	VERB
ejpam-4592	549	7	and	and	CCONJ
ejpam-4592	549	8	y	y	PRON
ejpam-4592	549	9	be	be	AUX
ejpam-4592	549	10	a	a	DET
ejpam-4592	549	11	limit	limit	NOUN
ejpam-4592	549	12	point	point	NOUN
ejpam-4592	549	13	of	of	ADP
ejpam-4592	549	14	{	{	PUNCT
ejpam-4592	549	15	dn	dn	NOUN
ejpam-4592	549	16	}	}	PUNCT
ejpam-4592	549	17	.	.	PUNCT
ejpam-4592	550	1	then	then	ADV
ejpam-4592	550	2	there	there	PRON
ejpam-4592	550	3	exist	exist	VERB
ejpam-4592	550	4	positive	positive	ADJ
ejpam-4592	550	5	integers	integer	NOUN
ejpam-4592	550	6	n0	n0	ADJ
ejpam-4592	550	7	,	,	PUNCT
ejpam-4592	550	8	n1	n1	PROPN
ejpam-4592	550	9	∈	∈	PROPN
ejpam-4592	550	10	n	n	CCONJ
ejpam-4592	550	11	such	such	ADJ
ejpam-4592	550	12	that	that	DET
ejpam-4592	550	13	σ(cn	σ(cn	NOUN
ejpam-4592	550	14	,	,	PUNCT
ejpam-4592	550	15	cm	cm	NOUN
ejpam-4592	550	16	)	)	PUNCT
ejpam-4592	550	17	<	<	X
ejpam-4592	550	18	ε	ε	PROPN
ejpam-4592	550	19	2	2	NUM
ejpam-4592	550	20	for	for	ADP
ejpam-4592	550	21	all	all	DET
ejpam-4592	550	22	n	n	CCONJ
ejpam-4592	550	23	,	,	PUNCT
ejpam-4592	550	24	m	m	PROPN
ejpam-4592	550	25	≥	≥	NOUN
ejpam-4592	550	26	n0	n0	NUM
ejpam-4592	550	27	and	and	CCONJ
ejpam-4592	550	28	σ(dn	σ(dn	PROPN
ejpam-4592	550	29	,	,	PUNCT
ejpam-4592	550	30	y	y	NOUN
ejpam-4592	550	31	)	)	PUNCT
ejpam-4592	550	32	<	<	X
ejpam-4592	550	33	ε	ε	PROPN
ejpam-4592	550	34	2	2	NUM
ejpam-4592	550	35	for	for	ADP
ejpam-4592	550	36	all	all	DET
ejpam-4592	550	37	n	n	PRON
ejpam-4592	550	38	≥	≥	NOUN
ejpam-4592	550	39	n1	n1	NOUN
ejpam-4592	550	40	.	.	PUNCT
ejpam-4592	551	1	take	take	VERB
ejpam-4592	551	2	m	m	NOUN
ejpam-4592	551	3	=	=	SYM
ejpam-4592	551	4	max{n0	max{n0	PROPN
ejpam-4592	551	5	,	,	PUNCT
ejpam-4592	551	6	n1	n1	NOUN
ejpam-4592	551	7	}	}	PUNCT
ejpam-4592	551	8	.	.	PUNCT
ejpam-4592	552	1	fix	fix	VERB
ejpam-4592	552	2	n	n	PRON
ejpam-4592	552	3	≥	≥	NOUN
ejpam-4592	552	4	m.	m.	NOUN
ejpam-4592	552	5	since	since	SCONJ
ejpam-4592	552	6	(	(	PUNCT
ejpam-4592	552	7	x,µω	x,µω	X
ejpam-4592	552	8	)	)	PUNCT
ejpam-4592	552	9	is	be	AUX
ejpam-4592	552	10	a	a	DET
ejpam-4592	552	11	hyperconnected	hyperconnected	ADJ
ejpam-4592	552	12	space	space	NOUN
ejpam-4592	552	13	,	,	PUNCT
ejpam-4592	552	14	bσ(cn	bσ(cn	NOUN
ejpam-4592	552	15	,	,	PUNCT
ejpam-4592	552	16	ε	ε	PROPN
ejpam-4592	552	17	2)∩bσ(y	2)∩bσ(y	NUM
ejpam-4592	552	18	,	,	PUNCT
ejpam-4592	552	19	ε	ε	PROPN
ejpam-4592	552	20	2	2	NUM
ejpam-4592	552	21	)	)	PUNCT
ejpam-4592	552	22	̸=	̸=	PROPN
ejpam-4592	552	23	∅.	∅.	ADV
ejpam-4592	552	24	let	let	VERB
ejpam-4592	552	25	t	t	PROPN
ejpam-4592	552	26	∈	∈	PROPN
ejpam-4592	552	27	bσ(cn	bσ(cn	NOUN
ejpam-4592	552	28	,	,	PUNCT
ejpam-4592	552	29	ε	ε	PROPN
ejpam-4592	552	30	2)∩bσ(y	2)∩bσ(y	NUM
ejpam-4592	552	31	,	,	PUNCT
ejpam-4592	552	32	ε	ε	PROPN
ejpam-4592	552	33	2	2	NUM
ejpam-4592	552	34	)	)	PUNCT
ejpam-4592	552	35	.	.	PUNCT
ejpam-4592	553	1	then	then	ADV
ejpam-4592	553	2	σ(cn	σ(cn	PROPN
ejpam-4592	553	3	,	,	PUNCT
ejpam-4592	553	4	y	y	NOUN
ejpam-4592	553	5	)	)	PUNCT
ejpam-4592	553	6	≤	≤	NOUN
ejpam-4592	553	7	σ(cn	σ(cn	PROPN
ejpam-4592	553	8	,	,	PUNCT
ejpam-4592	553	9	t)+σ(t	t)+σ(t	PROPN
ejpam-4592	553	10	,	,	PUNCT
ejpam-4592	553	11	y	y	NOUN
ejpam-4592	553	12	)	)	PUNCT
ejpam-4592	553	13	<	<	X
ejpam-4592	553	14	ε	ε	PROPN
ejpam-4592	553	15	2	2	NUM
ejpam-4592	553	16	+	+	CCONJ
ejpam-4592	553	17	ε	ε	PROPN
ejpam-4592	553	18	2	2	NUM
ejpam-4592	553	19	=	=	SYM
ejpam-4592	553	20	ε	ε	PROPN
ejpam-4592	553	21	.	.	PUNCT
ejpam-4592	553	22	therefore	therefore	ADV
ejpam-4592	553	23	,	,	PUNCT
ejpam-4592	553	24	σ(cn	σ(cn	PROPN
ejpam-4592	553	25	,	,	PUNCT
ejpam-4592	553	26	y	y	NOUN
ejpam-4592	553	27	)	)	PUNCT
ejpam-4592	553	28	<	<	X
ejpam-4592	553	29	ε	ε	PROPN
ejpam-4592	553	30	for	for	ADP
ejpam-4592	553	31	all	all	DET
ejpam-4592	553	32	n	n	DET
ejpam-4592	553	33	≥	≥	NOUN
ejpam-4592	553	34	m.	m.	NOUN
ejpam-4592	553	35	hence	hence	ADV
ejpam-4592	553	36	{	{	PUNCT
ejpam-4592	553	37	cn	cn	PROPN
ejpam-4592	553	38	}	}	PUNCT
ejpam-4592	553	39	is	be	AUX
ejpam-4592	553	40	a	a	DET
ejpam-4592	553	41	convergent	convergent	NOUN
ejpam-4592	553	42	sequence	sequence	NOUN
ejpam-4592	553	43	with	with	ADP
ejpam-4592	553	44	respect	respect	NOUN
ejpam-4592	553	45	to	to	ADP
ejpam-4592	553	46	σ	σ	PROPN
ejpam-4592	553	47	.	.	PUNCT
ejpam-4592	553	48	definition	definition	NOUN
ejpam-4592	553	49	50	50	NUM
ejpam-4592	553	50	.	.	PUNCT
ejpam-4592	554	1	let	let	VERB
ejpam-4592	554	2	(	(	PUNCT
ejpam-4592	554	3	x	x	NOUN
ejpam-4592	554	4	,	,	PUNCT
ejpam-4592	554	5	ω	ω	NUM
ejpam-4592	554	6	)	)	PUNCT
ejpam-4592	554	7	be	be	AUX
ejpam-4592	554	8	a	a	DET
ejpam-4592	554	9	gms	gms	NOUN
ejpam-4592	554	10	.	.	PUNCT
ejpam-4592	555	1	then	then	ADV
ejpam-4592	555	2	ω	ω	PROPN
ejpam-4592	555	3	is	be	AUX
ejpam-4592	555	4	said	say	VERB
ejpam-4592	555	5	to	to	PART
ejpam-4592	555	6	satisfy	satisfy	VERB
ejpam-4592	555	7	the	the	DET
ejpam-4592	555	8	s	s	NOUN
ejpam-4592	555	9	-	-	PUNCT
ejpam-4592	555	10	property	property	NOUN
ejpam-4592	555	11	if	if	SCONJ
ejpam-4592	555	12	σi(c	σi(c	NOUN
ejpam-4592	555	13	,	,	PUNCT
ejpam-4592	555	14	d	d	NOUN
ejpam-4592	555	15	)	)	PUNCT
ejpam-4592	555	16	≤	≤	NOUN
ejpam-4592	555	17	σj(c	σj(c	NOUN
ejpam-4592	555	18	,	,	PUNCT
ejpam-4592	555	19	d	d	NOUN
ejpam-4592	555	20	)	)	PUNCT
ejpam-4592	555	21	or	or	CCONJ
ejpam-4592	555	22	σj(c	σj(c	PRON
ejpam-4592	555	23	,	,	PUNCT
ejpam-4592	555	24	d	d	NOUN
ejpam-4592	555	25	)	)	PUNCT
ejpam-4592	555	26	≤	≤	NOUN
ejpam-4592	555	27	σi(c	σi(c	NOUN
ejpam-4592	555	28	,	,	PUNCT
ejpam-4592	555	29	d	d	NOUN
ejpam-4592	555	30	)	)	PUNCT
ejpam-4592	555	31	for	for	ADP
ejpam-4592	555	32	any	any	DET
ejpam-4592	555	33	σi	σi	NOUN
ejpam-4592	555	34	,	,	PUNCT
ejpam-4592	555	35	σj	σj	VERB
ejpam-4592	555	36	∈	∈	PROPN
ejpam-4592	555	37	ω	ω	PROPN
ejpam-4592	555	38	and	and	CCONJ
ejpam-4592	555	39	c	c	NOUN
ejpam-4592	555	40	,	,	PUNCT
ejpam-4592	555	41	d	d	PROPN
ejpam-4592	555	42	∈	∈	PROPN
ejpam-4592	555	43	x.	x.	NOUN
ejpam-4592	555	44	theorem	theorem	VERB
ejpam-4592	555	45	51	51	NUM
ejpam-4592	555	46	.	.	PUNCT
ejpam-4592	556	1	let	let	VERB
ejpam-4592	556	2	(	(	PUNCT
ejpam-4592	556	3	x	x	NOUN
ejpam-4592	556	4	,	,	PUNCT
ejpam-4592	556	5	ω	ω	NUM
ejpam-4592	556	6	)	)	PUNCT
ejpam-4592	556	7	be	be	AUX
ejpam-4592	556	8	a	a	DET
ejpam-4592	556	9	gms	gms	NOUN
ejpam-4592	556	10	,	,	PUNCT
ejpam-4592	556	11	ω	ω	NUM
ejpam-4592	556	12	satisfies	satisfy	VERB
ejpam-4592	556	13	the	the	DET
ejpam-4592	556	14	s	s	NOUN
ejpam-4592	556	15	-	-	PUNCT
ejpam-4592	556	16	property	property	NOUN
ejpam-4592	556	17	and	and	CCONJ
ejpam-4592	556	18	the	the	DET
ejpam-4592	556	19	pair	pair	NOUN
ejpam-4592	556	20	of	of	ADP
ejpam-4592	556	21	sequences	sequence	NOUN
ejpam-4592	556	22	{	{	PUNCT
ejpam-4592	556	23	cn	cn	NOUN
ejpam-4592	556	24	}	}	PUNCT
ejpam-4592	556	25	and	and	CCONJ
ejpam-4592	556	26	{	{	PUNCT
ejpam-4592	556	27	dn	dn	AUX
ejpam-4592	556	28	}	}	PUNCT
ejpam-4592	556	29	be	be	AUX
ejpam-4592	556	30	asymptotic	asymptotic	ADJ
ejpam-4592	556	31	with	with	ADP
ejpam-4592	556	32	respect	respect	NOUN
ejpam-4592	556	33	to	to	PART
ejpam-4592	556	34	σj	σj	VERB
ejpam-4592	556	35	∈	∈	PROPN
ejpam-4592	556	36	ω	ω	PROPN
ejpam-4592	556	37	.	.	PUNCT
ejpam-4592	557	1	then	then	ADV
ejpam-4592	557	2	the	the	DET
ejpam-4592	557	3	following	follow	VERB
ejpam-4592	557	4	hold	hold	NOUN
ejpam-4592	557	5	.	.	PUNCT
ejpam-4592	558	1	(	(	PUNCT
ejpam-4592	558	2	a	a	X
ejpam-4592	558	3	)	)	PUNCT
ejpam-4592	558	4	if	if	SCONJ
ejpam-4592	558	5	{	{	PUNCT
ejpam-4592	558	6	cn	cn	ADJ
ejpam-4592	558	7	}	}	PUNCT
ejpam-4592	558	8	∈	∈	PROPN
ejpam-4592	558	9	c(σi	c(σi	PROPN
ejpam-4592	558	10	)	)	PUNCT
ejpam-4592	558	11	,	,	PUNCT
ejpam-4592	558	12	then	then	ADV
ejpam-4592	558	13	{	{	PUNCT
ejpam-4592	558	14	dn	dn	ADJ
ejpam-4592	558	15	}	}	PUNCT
ejpam-4592	558	16	∈	∈	PROPN
ejpam-4592	558	17	c(σi	c(σi	PROPN
ejpam-4592	558	18	)	)	PUNCT
ejpam-4592	558	19	.	.	PUNCT
ejpam-4592	559	1	(	(	PUNCT
ejpam-4592	559	2	b	b	X
ejpam-4592	559	3	)	)	PUNCT
ejpam-4592	559	4	if	if	SCONJ
ejpam-4592	559	5	{	{	PUNCT
ejpam-4592	559	6	cn	cn	NOUN
ejpam-4592	559	7	}	}	PUNCT
ejpam-4592	559	8	is	be	AUX
ejpam-4592	559	9	convergent	convergent	ADJ
ejpam-4592	559	10	with	with	ADP
ejpam-4592	559	11	respect	respect	NOUN
ejpam-4592	559	12	to	to	ADP
ejpam-4592	559	13	σi	σi	PROPN
ejpam-4592	559	14	∈	∈	PROPN
ejpam-4592	559	15	ω	ω	PROPN
ejpam-4592	559	16	in	in	ADP
ejpam-4592	559	17	x	x	PRON
ejpam-4592	559	18	,	,	PUNCT
ejpam-4592	559	19	then	then	ADV
ejpam-4592	559	20	{	{	PUNCT
ejpam-4592	559	21	dn	dn	ADJ
ejpam-4592	559	22	}	}	PUNCT
ejpam-4592	559	23	is	be	AUX
ejpam-4592	559	24	convergent	convergent	ADJ
ejpam-4592	559	25	with	with	ADP
ejpam-4592	559	26	respect	respect	NOUN
ejpam-4592	559	27	to	to	ADP
ejpam-4592	559	28	σi	σi	PRON
ejpam-4592	559	29	.	.	PUNCT
ejpam-4592	560	1	proof	proof	NOUN
ejpam-4592	560	2	.	.	PUNCT
ejpam-4592	561	1	(	(	PUNCT
ejpam-4592	561	2	a	a	X
ejpam-4592	561	3	)	)	PUNCT
ejpam-4592	561	4	.	.	PUNCT
ejpam-4592	562	1	suppose	suppose	VERB
ejpam-4592	562	2	that	that	SCONJ
ejpam-4592	562	3	{	{	PUNCT
ejpam-4592	562	4	cn	cn	ADJ
ejpam-4592	562	5	}	}	PUNCT
ejpam-4592	562	6	∈	∈	PROPN
ejpam-4592	562	7	c(σi	c(σi	PROPN
ejpam-4592	562	8	)	)	PUNCT
ejpam-4592	562	9	.	.	PUNCT
ejpam-4592	563	1	let	let	VERB
ejpam-4592	563	2	ε	ε	PROPN
ejpam-4592	563	3	>	>	X
ejpam-4592	563	4	0	0	PUNCT
ejpam-4592	563	5	be	be	AUX
ejpam-4592	563	6	given	give	VERB
ejpam-4592	563	7	.	.	PUNCT
ejpam-4592	564	1	then	then	ADV
ejpam-4592	564	2	there	there	PRON
ejpam-4592	564	3	is	be	VERB
ejpam-4592	564	4	positive	positive	ADJ
ejpam-4592	564	5	integers	integer	NOUN
ejpam-4592	564	6	n0	n0	ADJ
ejpam-4592	564	7	,	,	PUNCT
ejpam-4592	564	8	n1	n1	NOUN
ejpam-4592	564	9	such	such	ADJ
ejpam-4592	564	10	that	that	DET
ejpam-4592	564	11	σj(cn	σj(cn	PROPN
ejpam-4592	564	12	,	,	PUNCT
ejpam-4592	564	13	dn	dn	PROPN
ejpam-4592	564	14	)	)	PUNCT
ejpam-4592	564	15	<	<	X
ejpam-4592	564	16	ε	ε	PROPN
ejpam-4592	564	17	3	3	NUM
ejpam-4592	564	18	for	for	ADP
ejpam-4592	564	19	all	all	PRON
ejpam-4592	564	20	n	n	PRON
ejpam-4592	564	21	>	>	X
ejpam-4592	564	22	n0	n0	PROPN
ejpam-4592	564	23	and	and	CCONJ
ejpam-4592	564	24	σi(cn	σi(cn	PROPN
ejpam-4592	564	25	,	,	PUNCT
ejpam-4592	564	26	cm	cm	NOUN
ejpam-4592	564	27	)	)	PUNCT
ejpam-4592	564	28	<	<	X
ejpam-4592	564	29	ε	ε	PROPN
ejpam-4592	564	30	3	3	NUM
ejpam-4592	564	31	for	for	ADP
ejpam-4592	564	32	all	all	DET
ejpam-4592	564	33	n	n	CCONJ
ejpam-4592	564	34	,	,	PUNCT
ejpam-4592	564	35	m	m	PROPN
ejpam-4592	564	36	≥	≥	NOUN
ejpam-4592	564	37	n1	n1	NOUN
ejpam-4592	564	38	.	.	PUNCT
ejpam-4592	565	1	by	by	ADP
ejpam-4592	565	2	hypothesis	hypothesis	NOUN
ejpam-4592	565	3	,	,	PUNCT
ejpam-4592	565	4	σi(c	σi(c	X
ejpam-4592	565	5	,	,	PUNCT
ejpam-4592	565	6	d	d	NOUN
ejpam-4592	565	7	)	)	PUNCT
ejpam-4592	565	8	≤	≤	NOUN
ejpam-4592	565	9	σj(c	σj(c	NOUN
ejpam-4592	565	10	,	,	PUNCT
ejpam-4592	565	11	d	d	NOUN
ejpam-4592	565	12	)	)	PUNCT
ejpam-4592	565	13	or	or	CCONJ
ejpam-4592	565	14	σj(c	σj(c	PRON
ejpam-4592	565	15	,	,	PUNCT
ejpam-4592	565	16	d	d	NOUN
ejpam-4592	565	17	)	)	PUNCT
ejpam-4592	565	18	≤	≤	NOUN
ejpam-4592	565	19	σi(c	σi(c	NOUN
ejpam-4592	565	20	,	,	PUNCT
ejpam-4592	565	21	d	d	NOUN
ejpam-4592	565	22	)	)	PUNCT
ejpam-4592	565	23	.	.	PUNCT
ejpam-4592	565	24	case	case	NOUN
ejpam-4592	565	25	1	1	NUM
ejpam-4592	565	26	:	:	PUNCT
ejpam-4592	565	27	if	if	SCONJ
ejpam-4592	565	28	σi(c	σi(c	NOUN
ejpam-4592	565	29	,	,	PUNCT
ejpam-4592	565	30	d	d	X
ejpam-4592	565	31	)	)	PUNCT
ejpam-4592	565	32	<	<	X
ejpam-4592	565	33	σj(c	σj(c	X
ejpam-4592	565	34	,	,	PUNCT
ejpam-4592	565	35	d	d	NOUN
ejpam-4592	565	36	)	)	PUNCT
ejpam-4592	565	37	,	,	PUNCT
ejpam-4592	565	38	then	then	ADV
ejpam-4592	565	39	σi(cn	σi(cn	NOUN
ejpam-4592	565	40	,	,	PUNCT
ejpam-4592	565	41	dn	dn	PROPN
ejpam-4592	565	42	)	)	PUNCT
ejpam-4592	565	43	<	<	X
ejpam-4592	565	44	ε	ε	PROPN
ejpam-4592	565	45	3	3	NUM
ejpam-4592	565	46	for	for	ADP
ejpam-4592	565	47	all	all	PRON
ejpam-4592	565	48	n	n	PROPN
ejpam-4592	565	49	>	>	X
ejpam-4592	565	50	n0	n0	PROPN
ejpam-4592	565	51	.	.	PUNCT
ejpam-4592	566	1	choose	choose	VERB
ejpam-4592	566	2	n0	n0	X
ejpam-4592	566	3	=	=	SYM
ejpam-4592	566	4	max{n0	max{n0	PROPN
ejpam-4592	566	5	,	,	PUNCT
ejpam-4592	566	6	n1	n1	NOUN
ejpam-4592	566	7	}	}	PUNCT
ejpam-4592	566	8	.	.	PUNCT
ejpam-4592	567	1	for	for	ADP
ejpam-4592	567	2	n	n	CCONJ
ejpam-4592	567	3	,	,	PUNCT
ejpam-4592	567	4	m	m	VERB
ejpam-4592	567	5	>	>	X
ejpam-4592	567	6	n0	n0	PROPN
ejpam-4592	567	7	,	,	PUNCT
ejpam-4592	567	8	σi(dn	σi(dn	PROPN
ejpam-4592	567	9	,	,	PUNCT
ejpam-4592	567	10	dm	dm	NOUN
ejpam-4592	567	11	)	)	PUNCT
ejpam-4592	567	12	≤	≤	PROPN
ejpam-4592	567	13	σi(dn	σi(dn	PROPN
ejpam-4592	567	14	,	,	PUNCT
ejpam-4592	567	15	cn)+σi(cn	cn)+σi(cn	NOUN
ejpam-4592	567	16	,	,	PUNCT
ejpam-4592	567	17	cm)+σi(cm	cm)+σi(cm	NOUN
ejpam-4592	567	18	,	,	PUNCT
ejpam-4592	567	19	dm	dm	PROPN
ejpam-4592	567	20	)	)	PUNCT
ejpam-4592	567	21	<	<	X
ejpam-4592	567	22	ε	ε	PROPN
ejpam-4592	567	23	3	3	NUM
ejpam-4592	567	24	+	+	CCONJ
ejpam-4592	567	25	ε	ε	PROPN
ejpam-4592	567	26	3	3	NUM
ejpam-4592	567	27	+	+	CCONJ
ejpam-4592	567	28	ε	ε	PROPN
ejpam-4592	567	29	3	3	NUM
ejpam-4592	567	30	=	=	SYM
ejpam-4592	567	31	ε	ε	PROPN
ejpam-4592	567	32	.	.	PUNCT
ejpam-4592	567	33	therefore	therefore	ADV
ejpam-4592	567	34	,	,	PUNCT
ejpam-4592	567	35	σi(dn	σi(dn	PROPN
ejpam-4592	567	36	,	,	PUNCT
ejpam-4592	567	37	dm	dm	NOUN
ejpam-4592	567	38	)	)	PUNCT
ejpam-4592	567	39	<	<	X
ejpam-4592	567	40	ε	ε	PROPN
ejpam-4592	567	41	for	for	ADP
ejpam-4592	567	42	all	all	DET
ejpam-4592	567	43	n	n	CCONJ
ejpam-4592	567	44	,	,	PUNCT
ejpam-4592	567	45	m	m	VERB
ejpam-4592	567	46	>	>	X
ejpam-4592	567	47	n0	n0	PROPN
ejpam-4592	567	48	.	.	PUNCT
ejpam-4592	568	1	hence	hence	ADV
ejpam-4592	568	2	{	{	PUNCT
ejpam-4592	568	3	dn	dn	NOUN
ejpam-4592	568	4	}	}	PUNCT
ejpam-4592	568	5	∈	∈	PROPN
ejpam-4592	568	6	c(σi	c(σi	PROPN
ejpam-4592	568	7	)	)	PUNCT
ejpam-4592	568	8	.	.	PUNCT
ejpam-4592	569	1	case	case	NOUN
ejpam-4592	569	2	2	2	NUM
ejpam-4592	569	3	:	:	PUNCT
ejpam-4592	569	4	if	if	SCONJ
ejpam-4592	569	5	σj(c	σj(c	NOUN
ejpam-4592	569	6	,	,	PUNCT
ejpam-4592	569	7	d	d	NOUN
ejpam-4592	569	8	)	)	PUNCT
ejpam-4592	569	9	<	<	X
ejpam-4592	569	10	σi(c	σi(c	X
ejpam-4592	569	11	,	,	PUNCT
ejpam-4592	569	12	d	d	NOUN
ejpam-4592	569	13	)	)	PUNCT
ejpam-4592	569	14	,	,	PUNCT
ejpam-4592	569	15	then	then	ADV
ejpam-4592	569	16	the	the	DET
ejpam-4592	569	17	proof	proof	NOUN
ejpam-4592	569	18	is	be	AUX
ejpam-4592	569	19	completed	complete	VERB
ejpam-4592	569	20	by	by	ADP
ejpam-4592	569	21	replacing	replace	VERB
ejpam-4592	569	22	i	i	PRON
ejpam-4592	569	23	by	by	ADP
ejpam-4592	569	24	j	j	PROPN
ejpam-4592	569	25	as	as	ADP
ejpam-4592	569	26	in	in	ADP
ejpam-4592	569	27	the	the	DET
ejpam-4592	569	28	proof	proof	NOUN
ejpam-4592	569	29	of	of	ADP
ejpam-4592	569	30	case-1	case-1	PROPN
ejpam-4592	569	31	.	.	PUNCT
ejpam-4592	570	1	(	(	PUNCT
ejpam-4592	570	2	b	b	NOUN
ejpam-4592	570	3	)	)	PUNCT
ejpam-4592	570	4	.	.	PUNCT
ejpam-4592	571	1	it	it	PRON
ejpam-4592	571	2	is	be	AUX
ejpam-4592	571	3	obvious	obvious	ADJ
ejpam-4592	571	4	.	.	PUNCT
ejpam-4592	572	1	theorem	theorem	NOUN
ejpam-4592	572	2	52	52	NUM
ejpam-4592	572	3	.	.	PUNCT
ejpam-4592	573	1	let	let	VERB
ejpam-4592	573	2	(	(	PUNCT
ejpam-4592	573	3	x	x	NOUN
ejpam-4592	573	4	,	,	PUNCT
ejpam-4592	573	5	ω	ω	NUM
ejpam-4592	573	6	)	)	PUNCT
ejpam-4592	573	7	be	be	AUX
ejpam-4592	573	8	a	a	DET
ejpam-4592	573	9	gms	gms	NOUN
ejpam-4592	573	10	,	,	PUNCT
ejpam-4592	573	11	ω	ω	NUM
ejpam-4592	573	12	satisfy	satisfy	NOUN
ejpam-4592	573	13	the	the	DET
ejpam-4592	573	14	s	s	NOUN
ejpam-4592	573	15	-	-	PUNCT
ejpam-4592	573	16	property	property	NOUN
ejpam-4592	573	17	and	and	CCONJ
ejpam-4592	573	18	the	the	DET
ejpam-4592	573	19	pair	pair	NOUN
ejpam-4592	573	20	of	of	ADP
ejpam-4592	573	21	sequences	sequence	NOUN
ejpam-4592	573	22	{	{	PUNCT
ejpam-4592	573	23	cn	cn	NOUN
ejpam-4592	573	24	}	}	PUNCT
ejpam-4592	573	25	and	and	CCONJ
ejpam-4592	573	26	{	{	PUNCT
ejpam-4592	573	27	dn	dn	AUX
ejpam-4592	573	28	}	}	PUNCT
ejpam-4592	573	29	be	be	AUX
ejpam-4592	573	30	uniformly	uniformly	ADV
ejpam-4592	573	31	asymptotic	asymptotic	ADJ
ejpam-4592	573	32	with	with	ADP
ejpam-4592	573	33	respect	respect	NOUN
ejpam-4592	573	34	to	to	PART
ejpam-4592	573	35	σj	σj	VERB
ejpam-4592	573	36	∈	∈	PROPN
ejpam-4592	573	37	ω	ω	PROPN
ejpam-4592	573	38	.	.	PUNCT
ejpam-4592	574	1	then	then	ADV
ejpam-4592	574	2	the	the	DET
ejpam-4592	574	3	following	follow	VERB
ejpam-4592	574	4	hold	hold	NOUN
ejpam-4592	574	5	.	.	PUNCT
ejpam-4592	575	1	(	(	PUNCT
ejpam-4592	575	2	a	a	X
ejpam-4592	575	3	)	)	PUNCT
ejpam-4592	575	4	if	if	SCONJ
ejpam-4592	575	5	{	{	PUNCT
ejpam-4592	575	6	dn	dn	NOUN
ejpam-4592	575	7	}	}	PUNCT
ejpam-4592	575	8	∈	∈	PROPN
ejpam-4592	575	9	c(σi	c(σi	PROPN
ejpam-4592	575	10	)	)	PUNCT
ejpam-4592	575	11	,	,	PUNCT
ejpam-4592	575	12	then	then	ADV
ejpam-4592	575	13	{	{	PUNCT
ejpam-4592	575	14	dn	dn	ADJ
ejpam-4592	575	15	}	}	PUNCT
ejpam-4592	575	16	∈	∈	PROPN
ejpam-4592	575	17	c(σi	c(σi	PROPN
ejpam-4592	575	18	)	)	PUNCT
ejpam-4592	575	19	.	.	PUNCT
ejpam-4592	576	1	(	(	PUNCT
ejpam-4592	576	2	b	b	X
ejpam-4592	576	3	)	)	PUNCT
ejpam-4592	576	4	if	if	SCONJ
ejpam-4592	576	5	{	{	PUNCT
ejpam-4592	576	6	dn	dn	NOUN
ejpam-4592	576	7	}	}	PUNCT
ejpam-4592	576	8	∈	∈	PROPN
ejpam-4592	576	9	p(σi	p(σi	NOUN
ejpam-4592	576	10	)	)	PUNCT
ejpam-4592	576	11	,	,	PUNCT
ejpam-4592	576	12	then	then	ADV
ejpam-4592	576	13	{	{	PUNCT
ejpam-4592	576	14	dn	dn	ADJ
ejpam-4592	576	15	}	}	PUNCT
ejpam-4592	576	16	∈	∈	PROPN
ejpam-4592	576	17	c(σi	c(σi	PROPN
ejpam-4592	576	18	)	)	PUNCT
ejpam-4592	576	19	.	.	PUNCT
ejpam-4592	577	1	in	in	ADP
ejpam-4592	577	2	[	[	X
ejpam-4592	577	3	14	14	NUM
ejpam-4592	577	4	]	]	PUNCT
ejpam-4592	577	5	,	,	PUNCT
ejpam-4592	577	6	preecha	preecha	PROPN
ejpam-4592	577	7	yupapin	yupapin	PROPN
ejpam-4592	577	8	et	et	PROPN
ejpam-4592	577	9	.	.	PUNCT
ejpam-4592	578	1	al	al	PROPN
ejpam-4592	578	2	have	have	AUX
ejpam-4592	578	3	obtained	obtain	VERB
ejpam-4592	578	4	new	new	ADJ
ejpam-4592	578	5	set	set	NOUN
ejpam-4592	578	6	of	of	ADP
ejpam-4592	578	7	properties	property	NOUN
ejpam-4592	578	8	of	of	ADP
ejpam-4592	578	9	nowhere	nowhere	ADJ
ejpam-4592	578	10	dense	dense	ADJ
ejpam-4592	578	11	sets	set	NOUN
ejpam-4592	578	12	.	.	PUNCT
ejpam-4592	579	1	inspired	inspire	VERB
ejpam-4592	579	2	by	by	ADP
ejpam-4592	579	3	this	this	PRON
ejpam-4592	579	4	,	,	PUNCT
ejpam-4592	579	5	rest	rest	NOUN
ejpam-4592	579	6	of	of	ADP
ejpam-4592	579	7	this	this	DET
ejpam-4592	579	8	section	section	NOUN
ejpam-4592	579	9	,	,	PUNCT
ejpam-4592	579	10	we	we	PRON
ejpam-4592	579	11	check	check	VERB
ejpam-4592	579	12	some	some	PRON
ejpam-4592	579	13	of	of	ADP
ejpam-4592	579	14	the	the	DET
ejpam-4592	579	15	collections	collection	NOUN
ejpam-4592	579	16	are	be	AUX
ejpam-4592	579	17	whether	whether	SCONJ
ejpam-4592	579	18	satisfies	satisfie	NOUN
ejpam-4592	579	19	the	the	DET
ejpam-4592	579	20	stack	stack	NOUN
ejpam-4592	579	21	property	property	NOUN
ejpam-4592	579	22	or	or	CCONJ
ejpam-4592	579	23	not	not	PART
ejpam-4592	579	24	,	,	PUNCT
ejpam-4592	579	25	in	in	ADP
ejpam-4592	579	26	a	a	DET
ejpam-4592	579	27	generalized	generalized	ADJ
ejpam-4592	579	28	metric	metric	ADJ
ejpam-4592	579	29	space	space	NOUN
ejpam-4592	579	30	.	.	PUNCT
ejpam-4592	580	1	let	let	VERB
ejpam-4592	580	2	(	(	PUNCT
ejpam-4592	580	3	x,µ	x,µ	NOUN
ejpam-4592	580	4	)	)	PUNCT
ejpam-4592	580	5	be	be	AUX
ejpam-4592	580	6	a	a	DET
ejpam-4592	580	7	gts	gts	NOUN
ejpam-4592	580	8	.	.	PUNCT
ejpam-4592	581	1	a	a	DET
ejpam-4592	581	2	family	family	NOUN
ejpam-4592	581	3	e	e	NOUN
ejpam-4592	581	4	of	of	ADP
ejpam-4592	581	5	subsets	subset	NOUN
ejpam-4592	581	6	of	of	ADP
ejpam-4592	581	7	x	x	PRON
ejpam-4592	581	8	is	be	AUX
ejpam-4592	581	9	called	call	VERB
ejpam-4592	581	10	a	a	DET
ejpam-4592	581	11	stack	stack	NOUN
ejpam-4592	581	12	[	[	X
ejpam-4592	581	13	13	13	NUM
ejpam-4592	581	14	]	]	PUNCT
ejpam-4592	581	15	if	if	SCONJ
ejpam-4592	581	16	c	c	PROPN
ejpam-4592	581	17	∈	∈	PROPN
ejpam-4592	581	18	e	e	X
ejpam-4592	581	19	whenever	whenever	SCONJ
ejpam-4592	581	20	d	d	PROPN
ejpam-4592	581	21	∈	∈	PROPN
ejpam-4592	581	22	e	e	PROPN
ejpam-4592	581	23	and	and	CCONJ
ejpam-4592	581	24	d	d	PROPN
ejpam-4592	581	25	⊂	⊂	PROPN
ejpam-4592	581	26	c.	c.	PROPN
ejpam-4592	581	27	theorem	theorem	VERB
ejpam-4592	581	28	53	53	NUM
ejpam-4592	581	29	.	.	PUNCT
ejpam-4592	582	1	let	let	VERB
ejpam-4592	582	2	(	(	PUNCT
ejpam-4592	582	3	x	x	NOUN
ejpam-4592	582	4	,	,	PUNCT
ejpam-4592	582	5	ω	ω	NUM
ejpam-4592	582	6	)	)	PUNCT
ejpam-4592	582	7	be	be	AUX
ejpam-4592	582	8	a	a	DET
ejpam-4592	582	9	gms	gms	NOUN
ejpam-4592	582	10	and	and	CCONJ
ejpam-4592	582	11	η	η	NOUN
ejpam-4592	582	12	=	=	PRON
ejpam-4592	582	13	{	{	PUNCT
ejpam-4592	582	14	{	{	PUNCT
ejpam-4592	582	15	cn}n∈n	cn}n∈n	INTJ
ejpam-4592	582	16	|	|	ADV
ejpam-4592	582	17	{	{	PUNCT
ejpam-4592	582	18	cn}n∈n	cn}n∈n	INTJ
ejpam-4592	582	19	is	be	AUX
ejpam-4592	582	20	a	a	DET
ejpam-4592	582	21	cofinally	cofinally	ADV
ejpam-4592	582	22	-	-	PUNCT
ejpam-4592	582	23	cauchy	cauchy	ADJ
ejpam-4592	582	24	sequence	sequence	NOUN
ejpam-4592	582	25	in	in	ADP
ejpam-4592	582	26	x	x	NOUN
ejpam-4592	582	27	}	}	PUNCT
ejpam-4592	582	28	.	.	PUNCT
ejpam-4592	583	1	then	then	ADV
ejpam-4592	583	2	η	η	PROPN
ejpam-4592	583	3	is	be	AUX
ejpam-4592	583	4	a	a	DET
ejpam-4592	583	5	stack	stack	NOUN
ejpam-4592	583	6	.	.	PUNCT
ejpam-4592	584	1	references	reference	NOUN
ejpam-4592	584	2	1885	1885	NUM
ejpam-4592	584	3	proof	proof	NOUN
ejpam-4592	584	4	.	.	PUNCT
ejpam-4592	585	1	let	let	VERB
ejpam-4592	585	2	{	{	PUNCT
ejpam-4592	585	3	dn}n∈n	dn}n∈n	VERB
ejpam-4592	585	4	be	be	AUX
ejpam-4592	585	5	a	a	DET
ejpam-4592	585	6	subsequence	subsequence	NOUN
ejpam-4592	585	7	of	of	ADP
ejpam-4592	585	8	{	{	PUNCT
ejpam-4592	585	9	cn}n∈n	cn}n∈n	INTJ
ejpam-4592	585	10	in	in	ADP
ejpam-4592	585	11	x	x	SYM
ejpam-4592	585	12	where	where	SCONJ
ejpam-4592	585	13	{	{	PUNCT
ejpam-4592	585	14	dn}n∈n	dn}n∈n	PROPN
ejpam-4592	585	15	∈	∈	PROPN
ejpam-4592	585	16	η	η	PROPN
ejpam-4592	585	17	.	.	PROPN
ejpam-4592	585	18	then	then	ADV
ejpam-4592	585	19	there	there	PRON
ejpam-4592	585	20	is	be	VERB
ejpam-4592	585	21	a	a	DET
ejpam-4592	585	22	metric	metric	ADJ
ejpam-4592	585	23	σ	σ	PROPN
ejpam-4592	585	24	∈	∈	PROPN
ejpam-4592	585	25	ω	ω	NOUN
ejpam-4592	585	26	such	such	ADJ
ejpam-4592	585	27	that	that	SCONJ
ejpam-4592	585	28	{	{	PUNCT
ejpam-4592	585	29	dn}n∈n	dn}n∈n	PUNCT
ejpam-4592	585	30	∈	∈	PROPN
ejpam-4592	585	31	c(σ	c(σ	PROPN
ejpam-4592	585	32	)	)	PUNCT
ejpam-4592	585	33	and	and	CCONJ
ejpam-4592	585	34	so	so	ADV
ejpam-4592	585	35	for	for	ADP
ejpam-4592	585	36	every	every	DET
ejpam-4592	585	37	ε	ε	PROPN
ejpam-4592	585	38	>	>	X
ejpam-4592	585	39	0	0	PROPN
ejpam-4592	585	40	,	,	PUNCT
ejpam-4592	585	41	there	there	PRON
ejpam-4592	585	42	exists	exist	VERB
ejpam-4592	585	43	an	an	DET
ejpam-4592	585	44	infinite	infinite	NOUN
ejpam-4592	585	45	subset	subset	NOUN
ejpam-4592	585	46	nε	nε	NOUN
ejpam-4592	585	47	of	of	ADP
ejpam-4592	585	48	n	n	PRON
ejpam-4592	585	49	such	such	ADJ
ejpam-4592	585	50	that	that	PRON
ejpam-4592	585	51	for	for	ADP
ejpam-4592	585	52	every	every	DET
ejpam-4592	585	53	n	n	NOUN
ejpam-4592	585	54	,	,	PUNCT
ejpam-4592	585	55	j	j	PROPN
ejpam-4592	585	56	∈	∈	PROPN
ejpam-4592	585	57	nε	nε	VERB
ejpam-4592	585	58	we	we	PRON
ejpam-4592	585	59	have	have	VERB
ejpam-4592	585	60	σ(dn	σ(dn	NOUN
ejpam-4592	585	61	,	,	PUNCT
ejpam-4592	585	62	dj	dj	NOUN
ejpam-4592	585	63	)	)	PUNCT
ejpam-4592	585	64	<	<	X
ejpam-4592	586	1	ε	ε	PROPN
ejpam-4592	586	2	.	.	PUNCT
ejpam-4592	586	3	by	by	ADP
ejpam-4592	586	4	hypothesis	hypothesis	NOUN
ejpam-4592	586	5	,	,	PUNCT
ejpam-4592	586	6	σ(cn	σ(cn	PROPN
ejpam-4592	586	7	,	,	PUNCT
ejpam-4592	586	8	cj	cj	X
ejpam-4592	586	9	)	)	PUNCT
ejpam-4592	586	10	<	<	X
ejpam-4592	586	11	ε	ε	PROPN
ejpam-4592	586	12	for	for	ADP
ejpam-4592	586	13	every	every	DET
ejpam-4592	586	14	n	n	NOUN
ejpam-4592	586	15	,	,	PUNCT
ejpam-4592	586	16	j	j	PROPN
ejpam-4592	586	17	∈	∈	PROPN
ejpam-4592	586	18	nε	nε	VERB
ejpam-4592	586	19	.	.	PUNCT
ejpam-4592	587	1	thus	thus	ADV
ejpam-4592	587	2	,	,	PUNCT
ejpam-4592	587	3	for	for	ADP
ejpam-4592	587	4	every	every	DET
ejpam-4592	587	5	ε	ε	PROPN
ejpam-4592	587	6	>	>	X
ejpam-4592	587	7	0	0	PROPN
ejpam-4592	587	8	,	,	PUNCT
ejpam-4592	587	9	there	there	PRON
ejpam-4592	587	10	exists	exist	VERB
ejpam-4592	587	11	an	an	DET
ejpam-4592	587	12	infinite	infinite	NOUN
ejpam-4592	587	13	subset	subset	NOUN
ejpam-4592	587	14	nε	nε	NOUN
ejpam-4592	587	15	of	of	ADP
ejpam-4592	587	16	n	n	PRON
ejpam-4592	587	17	such	such	ADJ
ejpam-4592	587	18	that	that	PRON
ejpam-4592	587	19	for	for	ADP
ejpam-4592	587	20	every	every	DET
ejpam-4592	587	21	n	n	NOUN
ejpam-4592	587	22	,	,	PUNCT
ejpam-4592	587	23	j	j	PROPN
ejpam-4592	587	24	∈	∈	PROPN
ejpam-4592	587	25	nε	nε	VERB
ejpam-4592	587	26	we	we	PRON
ejpam-4592	587	27	have	have	VERB
ejpam-4592	587	28	σ(cn	σ(cn	PROPN
ejpam-4592	587	29	,	,	PUNCT
ejpam-4592	587	30	cj	cj	X
ejpam-4592	587	31	)	)	PUNCT
ejpam-4592	587	32	<	<	X
ejpam-4592	587	33	ε	ε	PROPN
ejpam-4592	587	34	.	.	PUNCT
ejpam-4592	588	1	therefore	therefore	ADV
ejpam-4592	588	2	,	,	PUNCT
ejpam-4592	588	3	{	{	PUNCT
ejpam-4592	588	4	cn}n∈n	cn}n∈n	NOUN
ejpam-4592	588	5	∈	∈	PROPN
ejpam-4592	588	6	c(σ	c(σ	PROPN
ejpam-4592	588	7	)	)	PUNCT
ejpam-4592	588	8	.	.	PUNCT
ejpam-4592	589	1	hence	hence	ADV
ejpam-4592	589	2	{	{	PUNCT
ejpam-4592	589	3	cn}n∈n	cn}n∈n	PROPN
ejpam-4592	589	4	∈	∈	PROPN
ejpam-4592	589	5	η	η	PROPN
ejpam-4592	589	6	.	.	PROPN
ejpam-4592	589	7	thus	thus	ADV
ejpam-4592	589	8	,	,	PUNCT
ejpam-4592	589	9	η	η	PROPN
ejpam-4592	589	10	is	be	AUX
ejpam-4592	589	11	a	a	DET
ejpam-4592	589	12	stack	stack	NOUN
ejpam-4592	589	13	in	in	ADP
ejpam-4592	589	14	(	(	PUNCT
ejpam-4592	589	15	x	x	X
ejpam-4592	589	16	,	,	PUNCT
ejpam-4592	589	17	ω	ω	NOUN
ejpam-4592	589	18	)	)	PUNCT
ejpam-4592	589	19	.	.	PUNCT
ejpam-4592	590	1	theorem	theorem	VERB
ejpam-4592	590	2	54	54	NUM
ejpam-4592	590	3	.	.	PUNCT
ejpam-4592	591	1	let	let	VERB
ejpam-4592	591	2	(	(	PUNCT
ejpam-4592	591	3	x	x	NOUN
ejpam-4592	591	4	,	,	PUNCT
ejpam-4592	591	5	ω	ω	NUM
ejpam-4592	591	6	)	)	PUNCT
ejpam-4592	591	7	be	be	AUX
ejpam-4592	591	8	a	a	DET
ejpam-4592	591	9	generalized	generalized	ADJ
ejpam-4592	591	10	metric	metric	ADJ
ejpam-4592	591	11	space	space	NOUN
ejpam-4592	591	12	and	and	CCONJ
ejpam-4592	591	13	η	η	PROPN
ejpam-4592	591	14	=	=	PRON
ejpam-4592	591	15	{	{	PUNCT
ejpam-4592	591	16	{	{	PUNCT
ejpam-4592	591	17	cn}n∈n	cn}n∈n	INTJ
ejpam-4592	592	1	|	|	ADV
ejpam-4592	592	2	{	{	PUNCT
ejpam-4592	592	3	cn}n∈n	cn}n∈n	INTJ
ejpam-4592	592	4	is	be	AUX
ejpam-4592	592	5	a	a	DET
ejpam-4592	592	6	pseudo	pseudo	NOUN
ejpam-4592	592	7	-	-	ADJ
ejpam-4592	592	8	cauchy	cauchy	ADJ
ejpam-4592	592	9	sequence	sequence	NOUN
ejpam-4592	592	10	in	in	ADP
ejpam-4592	592	11	x	x	NOUN
ejpam-4592	592	12	}	}	PUNCT
ejpam-4592	592	13	.	.	PUNCT
ejpam-4592	593	1	then	then	ADV
ejpam-4592	593	2	η	η	PROPN
ejpam-4592	593	3	is	be	AUX
ejpam-4592	593	4	a	a	DET
ejpam-4592	593	5	stack	stack	NOUN
ejpam-4592	593	6	.	.	PUNCT
ejpam-4592	594	1	proof	proof	NOUN
ejpam-4592	594	2	.	.	PUNCT
ejpam-4592	595	1	let	let	VERB
ejpam-4592	595	2	{	{	PUNCT
ejpam-4592	595	3	ln}n∈n	ln}n∈n	VERB
ejpam-4592	595	4	be	be	AUX
ejpam-4592	595	5	a	a	DET
ejpam-4592	595	6	subsequence	subsequence	NOUN
ejpam-4592	595	7	of	of	ADP
ejpam-4592	595	8	{	{	PUNCT
ejpam-4592	595	9	kn}n∈n	kn}n∈n	PROPN
ejpam-4592	595	10	in	in	ADP
ejpam-4592	595	11	x	x	SYM
ejpam-4592	595	12	where	where	SCONJ
ejpam-4592	595	13	{	{	PUNCT
ejpam-4592	595	14	ln}n∈n	ln}n∈n	PROPN
ejpam-4592	595	15	∈	∈	PROPN
ejpam-4592	595	16	η	η	PROPN
ejpam-4592	595	17	.	.	PROPN
ejpam-4592	596	1	then	then	ADV
ejpam-4592	596	2	there	there	PRON
ejpam-4592	596	3	is	be	VERB
ejpam-4592	596	4	a	a	DET
ejpam-4592	596	5	metric	metric	ADJ
ejpam-4592	596	6	σ	σ	PROPN
ejpam-4592	596	7	∈	∈	PROPN
ejpam-4592	596	8	ω	ω	NOUN
ejpam-4592	596	9	such	such	ADJ
ejpam-4592	596	10	that	that	SCONJ
ejpam-4592	596	11	{	{	PUNCT
ejpam-4592	596	12	ln}n∈n	ln}n∈n	PROPN
ejpam-4592	596	13	∈	∈	PROPN
ejpam-4592	596	14	p(σ	p(σ	NOUN
ejpam-4592	596	15	)	)	PUNCT
ejpam-4592	596	16	and	and	CCONJ
ejpam-4592	596	17	so	so	ADV
ejpam-4592	596	18	for	for	ADP
ejpam-4592	596	19	every	every	DET
ejpam-4592	596	20	ε	ε	PROPN
ejpam-4592	596	21	>	>	X
ejpam-4592	596	22	0	0	PUNCT
ejpam-4592	596	23	and	and	CCONJ
ejpam-4592	596	24	for	for	ADP
ejpam-4592	596	25	every	every	DET
ejpam-4592	596	26	n	n	PRON
ejpam-4592	596	27	∈	∈	PROPN
ejpam-4592	596	28	n	n	CCONJ
ejpam-4592	596	29	,	,	PUNCT
ejpam-4592	596	30	there	there	PRON
ejpam-4592	596	31	exist	exist	VERB
ejpam-4592	596	32	s	s	PROPN
ejpam-4592	596	33	,	,	PUNCT
ejpam-4592	596	34	t	t	PROPN
ejpam-4592	596	35	∈	∈	PROPN
ejpam-4592	596	36	n	n	CCONJ
ejpam-4592	596	37	,	,	PUNCT
ejpam-4592	596	38	s	s	VERB
ejpam-4592	596	39	̸=	̸=	PROPN
ejpam-4592	596	40	t	t	NOUN
ejpam-4592	596	41	such	such	DET
ejpam-4592	596	42	that	that	DET
ejpam-4592	596	43	s	s	PROPN
ejpam-4592	596	44	,	,	PUNCT
ejpam-4592	596	45	t	t	PROPN
ejpam-4592	596	46	>	>	X
ejpam-4592	596	47	n	n	PROPN
ejpam-4592	596	48	and	and	CCONJ
ejpam-4592	596	49	σ(ls	σ(ls	NOUN
ejpam-4592	596	50	,	,	PUNCT
ejpam-4592	596	51	lt	lt	NOUN
ejpam-4592	596	52	)	)	PUNCT
ejpam-4592	596	53	<	<	X
ejpam-4592	596	54	ε	ε	PROPN
ejpam-4592	596	55	.	.	PUNCT
ejpam-4592	596	56	by	by	ADP
ejpam-4592	596	57	hypothesis	hypothesis	NOUN
ejpam-4592	596	58	,	,	PUNCT
ejpam-4592	596	59	σ(ks	σ(ks	PROPN
ejpam-4592	596	60	,	,	PUNCT
ejpam-4592	596	61	kt	kt	PROPN
ejpam-4592	596	62	)	)	PUNCT
ejpam-4592	596	63	<	<	X
ejpam-4592	596	64	ε	ε	PROPN
ejpam-4592	596	65	.	.	PUNCT
ejpam-4592	597	1	thus	thus	ADV
ejpam-4592	597	2	,	,	PUNCT
ejpam-4592	597	3	for	for	ADP
ejpam-4592	597	4	every	every	DET
ejpam-4592	597	5	ε	ε	PROPN
ejpam-4592	597	6	>	>	X
ejpam-4592	597	7	0	0	PUNCT
ejpam-4592	597	8	and	and	CCONJ
ejpam-4592	597	9	for	for	ADP
ejpam-4592	597	10	every	every	DET
ejpam-4592	597	11	n	n	PRON
ejpam-4592	597	12	∈	∈	PROPN
ejpam-4592	597	13	n	n	CCONJ
ejpam-4592	597	14	,	,	PUNCT
ejpam-4592	597	15	there	there	PRON
ejpam-4592	597	16	exist	exist	VERB
ejpam-4592	597	17	s	s	PROPN
ejpam-4592	597	18	,	,	PUNCT
ejpam-4592	597	19	t	t	PROPN
ejpam-4592	597	20	∈	∈	PROPN
ejpam-4592	597	21	n	n	CCONJ
ejpam-4592	597	22	,	,	PUNCT
ejpam-4592	597	23	s	s	VERB
ejpam-4592	597	24	̸=	̸=	PROPN
ejpam-4592	597	25	t	t	NOUN
ejpam-4592	597	26	such	such	DET
ejpam-4592	597	27	that	that	DET
ejpam-4592	597	28	s	s	PROPN
ejpam-4592	597	29	,	,	PUNCT
ejpam-4592	597	30	t	t	PROPN
ejpam-4592	597	31	>	>	X
ejpam-4592	597	32	n	n	PROPN
ejpam-4592	597	33	and	and	CCONJ
ejpam-4592	597	34	σ(ks	σ(ks	PROPN
ejpam-4592	597	35	,	,	PUNCT
ejpam-4592	597	36	kt	kt	PROPN
ejpam-4592	597	37	)	)	PUNCT
ejpam-4592	597	38	<	<	X
ejpam-4592	597	39	ε	ε	PROPN
ejpam-4592	597	40	.	.	PUNCT
ejpam-4592	597	41	therefore	therefore	ADV
ejpam-4592	597	42	,	,	PUNCT
ejpam-4592	597	43	{	{	PUNCT
ejpam-4592	597	44	kn}n∈n	kn}n∈n	PROPN
ejpam-4592	597	45	∈	∈	PROPN
ejpam-4592	597	46	p(σ	p(σ	PROPN
ejpam-4592	597	47	)	)	PUNCT
ejpam-4592	597	48	.	.	PUNCT
ejpam-4592	598	1	hence	hence	ADV
ejpam-4592	598	2	{	{	PUNCT
ejpam-4592	598	3	kn}n∈n	kn}n∈n	PROPN
ejpam-4592	598	4	∈	∈	PROPN
ejpam-4592	598	5	η	η	PROPN
ejpam-4592	598	6	.	.	PROPN
ejpam-4592	598	7	thus	thus	ADV
ejpam-4592	598	8	,	,	PUNCT
ejpam-4592	598	9	η	η	PROPN
ejpam-4592	598	10	is	be	AUX
ejpam-4592	598	11	a	a	DET
ejpam-4592	598	12	stack	stack	NOUN
ejpam-4592	598	13	in	in	ADP
ejpam-4592	598	14	(	(	PUNCT
ejpam-4592	598	15	x	x	X
ejpam-4592	598	16	,	,	PUNCT
ejpam-4592	598	17	ω	ω	NOUN
ejpam-4592	598	18	)	)	PUNCT
ejpam-4592	598	19	.	.	PUNCT
ejpam-4592	599	1	theorem	theorem	VERB
ejpam-4592	599	2	55	55	NUM
ejpam-4592	599	3	.	.	PUNCT
ejpam-4592	600	1	let	let	VERB
ejpam-4592	600	2	(	(	PUNCT
ejpam-4592	600	3	x	x	NOUN
ejpam-4592	600	4	,	,	PUNCT
ejpam-4592	600	5	ω	ω	NUM
ejpam-4592	600	6	)	)	PUNCT
ejpam-4592	600	7	be	be	AUX
ejpam-4592	600	8	a	a	DET
ejpam-4592	600	9	generalized	generalized	ADJ
ejpam-4592	600	10	metric	metric	ADJ
ejpam-4592	600	11	space	space	NOUN
ejpam-4592	600	12	.	.	PUNCT
ejpam-4592	601	1	if	if	SCONJ
ejpam-4592	601	2	η	η	PROPN
ejpam-4592	601	3	=	=	SYM
ejpam-4592	601	4	{	{	PUNCT
ejpam-4592	601	5	ω0	ω0	ADV
ejpam-4592	601	6	|	|	ADV
ejpam-4592	601	7	ω0	ω0	PROPN
ejpam-4592	601	8	is	be	AUX
ejpam-4592	601	9	a	a	DET
ejpam-4592	601	10	kernel	kernel	NOUN
ejpam-4592	601	11	in	in	ADP
ejpam-4592	601	12	x	x	NOUN
ejpam-4592	601	13	}	}	PUNCT
ejpam-4592	601	14	,	,	PUNCT
ejpam-4592	601	15	then	then	ADV
ejpam-4592	601	16	η	η	PROPN
ejpam-4592	601	17	is	be	AUX
ejpam-4592	601	18	a	a	DET
ejpam-4592	601	19	stack	stack	NOUN
ejpam-4592	601	20	.	.	PUNCT
ejpam-4592	602	1	proof	proof	NOUN
ejpam-4592	602	2	.	.	PUNCT
ejpam-4592	603	1	let	let	VERB
ejpam-4592	603	2	ω1	ω1	PROPN
ejpam-4592	603	3	⊂	⊂	PROPN
ejpam-4592	603	4	ω0	ω0	ADV
ejpam-4592	603	5	where	where	SCONJ
ejpam-4592	603	6	ω1	ω1	PROPN
ejpam-4592	603	7	∈	∈	PROPN
ejpam-4592	603	8	η	η	PROPN
ejpam-4592	603	9	.	.	PROPN
ejpam-4592	603	10	then	then	PROPN
ejpam-4592	603	11	ω1	ω1	PROPN
ejpam-4592	603	12	is	be	AUX
ejpam-4592	603	13	a	a	DET
ejpam-4592	603	14	kernel	kernel	NOUN
ejpam-4592	603	15	.	.	PUNCT
ejpam-4592	604	1	let	let	VERB
ejpam-4592	604	2	k	k	PROPN
ejpam-4592	604	3	∈	∈	PROPN
ejpam-4592	604	4	µ̃ω	µ̃ω	PROPN
ejpam-4592	604	5	.	.	PUNCT
ejpam-4592	605	1	then	then	ADV
ejpam-4592	605	2	there	there	PRON
ejpam-4592	605	3	exists	exist	VERB
ejpam-4592	605	4	σ	σ	PROPN
ejpam-4592	605	5	∈	∈	PROPN
ejpam-4592	605	6	ω1	ω1	PROPN
ejpam-4592	605	7	such	such	ADJ
ejpam-4592	605	8	that	that	DET
ejpam-4592	605	9	iσk	iσk	PROPN
ejpam-4592	605	10	̸=	̸=	PROPN
ejpam-4592	605	11	∅.	∅.	NOUN
ejpam-4592	605	12	since	since	SCONJ
ejpam-4592	605	13	ω1	ω1	PROPN
ejpam-4592	605	14	⊂	⊂	PROPN
ejpam-4592	605	15	ω0	ω0	ADV
ejpam-4592	605	16	we	we	PRON
ejpam-4592	605	17	have	have	VERB
ejpam-4592	605	18	there	there	PRON
ejpam-4592	605	19	exists	exist	VERB
ejpam-4592	605	20	a	a	DET
ejpam-4592	605	21	metric	metric	ADJ
ejpam-4592	605	22	σ	σ	X
ejpam-4592	605	23	∈	∈	PROPN
ejpam-4592	605	24	ω0	ω0	ADV
ejpam-4592	605	25	such	such	ADJ
ejpam-4592	605	26	that	that	SCONJ
ejpam-4592	605	27	iσk	iσk	PROPN
ejpam-4592	605	28	̸=	̸=	PROPN
ejpam-4592	605	29	∅.	∅.	PRON
ejpam-4592	605	30	hence	hence	ADV
ejpam-4592	605	31	ω0	ω0	ADV
ejpam-4592	605	32	is	be	AUX
ejpam-4592	605	33	a	a	DET
ejpam-4592	605	34	kernel	kernel	NOUN
ejpam-4592	605	35	.	.	PUNCT
ejpam-4592	606	1	5	5	X
ejpam-4592	606	2	.	.	X
ejpam-4592	606	3	conclusions	conclusion	NOUN
ejpam-4592	606	4	the	the	DET
ejpam-4592	606	5	various	various	ADJ
ejpam-4592	606	6	relations	relation	NOUN
ejpam-4592	606	7	between	between	ADP
ejpam-4592	606	8	the	the	DET
ejpam-4592	606	9	three	three	NUM
ejpam-4592	606	10	types	type	NOUN
ejpam-4592	606	11	of	of	ADP
ejpam-4592	606	12	sequences	sequence	NOUN
ejpam-4592	606	13	in	in	ADP
ejpam-4592	606	14	a	a	DET
ejpam-4592	606	15	generalized	generalized	ADJ
ejpam-4592	606	16	metric	metric	ADJ
ejpam-4592	606	17	space	space	NOUN
ejpam-4592	606	18	has	have	AUX
ejpam-4592	606	19	evaluated	evaluate	VERB
ejpam-4592	606	20	.	.	PUNCT
ejpam-4592	607	1	further	far	ADV
ejpam-4592	607	2	,	,	PUNCT
ejpam-4592	607	3	the	the	DET
ejpam-4592	607	4	necessity	necessity	NOUN
ejpam-4592	607	5	of	of	ADP
ejpam-4592	607	6	uniformly	uniformly	ADV
ejpam-4592	607	7	asymptotic	asymptotic	ADJ
ejpam-4592	607	8	condition	condition	NOUN
ejpam-4592	607	9	has	have	AUX
ejpam-4592	607	10	been	be	AUX
ejpam-4592	607	11	examined	examine	VERB
ejpam-4592	607	12	to	to	PART
ejpam-4592	607	13	prove	prove	VERB
ejpam-4592	607	14	some	some	DET
ejpam-4592	607	15	new	new	ADJ
ejpam-4592	607	16	relationship	relationship	NOUN
ejpam-4592	607	17	between	between	ADP
ejpam-4592	607	18	the	the	DET
ejpam-4592	607	19	functions	function	NOUN
ejpam-4592	607	20	defined	define	VERB
ejpam-4592	607	21	in	in	ADP
ejpam-4592	607	22	definition	definition	NOUN
ejpam-4592	607	23	41	41	NUM
ejpam-4592	607	24	.	.	PUNCT
ejpam-4592	608	1	hence	hence	ADV
ejpam-4592	608	2	the	the	DET
ejpam-4592	608	3	meaning	meaning	NOUN
ejpam-4592	608	4	of	of	ADP
ejpam-4592	608	5	the	the	DET
ejpam-4592	608	6	collection	collection	NOUN
ejpam-4592	608	7	of	of	ADP
ejpam-4592	608	8	sequences	sequence	NOUN
ejpam-4592	608	9	has	have	AUX
ejpam-4592	608	10	been	be	AUX
ejpam-4592	608	11	analyzed	analyze	VERB
ejpam-4592	608	12	.	.	PUNCT
ejpam-4592	609	1	references	reference	NOUN
ejpam-4592	609	2	[	[	X
ejpam-4592	609	3	1	1	NUM
ejpam-4592	609	4	]	]	PUNCT
ejpam-4592	609	5	m.	m.	NOUN
ejpam-4592	609	6	aggarwal	aggarwal	PROPN
ejpam-4592	609	7	and	and	CCONJ
ejpam-4592	609	8	s.	s.	PROPN
ejpam-4592	609	9	kundu	kundu	PROPN
ejpam-4592	609	10	.	.	PUNCT
ejpam-4592	610	1	more	more	ADJ
ejpam-4592	610	2	on	on	ADP
ejpam-4592	610	3	variants	variant	NOUN
ejpam-4592	610	4	of	of	ADP
ejpam-4592	610	5	complete	complete	ADJ
ejpam-4592	610	6	metric	metric	ADJ
ejpam-4592	610	7	spaces	space	NOUN
ejpam-4592	610	8	.	.	PUNCT
ejpam-4592	611	1	acta	acta	PROPN
ejpam-4592	611	2	math	math	PROPN
ejpam-4592	611	3	.	.	PUNCT
ejpam-4592	612	1	hungar	hungar	PROPN
ejpam-4592	612	2	.	.	PUNCT
ejpam-4592	612	3	,	,	PUNCT
ejpam-4592	612	4	151(2):391–408	151(2):391–408	NUM
ejpam-4592	612	5	,	,	PUNCT
ejpam-4592	612	6	2017	2017	NUM
ejpam-4592	612	7	.	.	PUNCT
ejpam-4592	613	1	[	[	X
ejpam-4592	613	2	2	2	NUM
ejpam-4592	613	3	]	]	X
ejpam-4592	613	4	g.	g.	NOUN
ejpam-4592	613	5	beer	beer	PROPN
ejpam-4592	613	6	.	.	PUNCT
ejpam-4592	614	1	between	between	ADP
ejpam-4592	614	2	compactness	compactness	NOUN
ejpam-4592	614	3	and	and	CCONJ
ejpam-4592	614	4	completeness	completeness	NOUN
ejpam-4592	614	5	.	.	PUNCT
ejpam-4592	615	1	topology	topology	NOUN
ejpam-4592	615	2	appl	appl	PROPN
ejpam-4592	615	3	.	.	PROPN
ejpam-4592	615	4	,	,	PUNCT
ejpam-4592	615	5	155:503	155:503	NUM
ejpam-4592	615	6	–	–	PUNCT
ejpam-4592	615	7	514	514	NUM
ejpam-4592	615	8	,	,	PUNCT
ejpam-4592	615	9	2008	2008	NUM
ejpam-4592	615	10	.	.	PUNCT
ejpam-4592	616	1	[	[	X
ejpam-4592	616	2	3	3	X
ejpam-4592	616	3	]	]	X
ejpam-4592	616	4	g.	g.	NOUN
ejpam-4592	616	5	beer	beer	PROPN
ejpam-4592	616	6	.	.	PUNCT
ejpam-4592	617	1	between	between	ADP
ejpam-4592	617	2	the	the	DET
ejpam-4592	617	3	cofinally	cofinally	ADV
ejpam-4592	617	4	complete	complete	ADJ
ejpam-4592	617	5	spaces	space	NOUN
ejpam-4592	617	6	and	and	CCONJ
ejpam-4592	617	7	the	the	DET
ejpam-4592	617	8	uc	uc	PROPN
ejpam-4592	617	9	spaces	space	NOUN
ejpam-4592	617	10	.	.	PUNCT
ejpam-4592	618	1	houston	houston	PROPN
ejpam-4592	618	2	j.	j.	PROPN
ejpam-4592	618	3	math	math	PROPN
ejpam-4592	618	4	.	.	PUNCT
ejpam-4592	619	1	,	,	PUNCT
ejpam-4592	619	2	,	,	PUNCT
ejpam-4592	619	3	38:999	38:999	NUM
ejpam-4592	619	4	–	–	PUNCT
ejpam-4592	619	5	1015	1015	NUM
ejpam-4592	619	6	,	,	PUNCT
ejpam-4592	619	7	2012	2012	NUM
ejpam-4592	619	8	.	.	PUNCT
ejpam-4592	620	1	[	[	X
ejpam-4592	620	2	4	4	NUM
ejpam-4592	620	3	]	]	PUNCT
ejpam-4592	620	4	m.	m.	NOUN
ejpam-4592	620	5	m.	m.	PROPN
ejpam-4592	620	6	bonsangue	bonsangue	PROPN
ejpam-4592	620	7	,	,	PUNCT
ejpam-4592	620	8	f.	f.	PROPN
ejpam-4592	620	9	van	van	PROPN
ejpam-4592	620	10	breugel	breugel	PROPN
ejpam-4592	620	11	,	,	PUNCT
ejpam-4592	620	12	and	and	CCONJ
ejpam-4592	620	13	j.j.m.m	j.j.m.m	PROPN
ejpam-4592	620	14	.	.	PROPN
ejpam-4592	620	15	rutten	rutten	PROPN
ejpam-4592	620	16	.	.	PUNCT
ejpam-4592	621	1	generalized	generalize	VERB
ejpam-4592	621	2	metric	metric	ADJ
ejpam-4592	621	3	spaces	space	NOUN
ejpam-4592	621	4	;	;	PUNCT
ejpam-4592	621	5	completion	completion	NOUN
ejpam-4592	621	6	,	,	PUNCT
ejpam-4592	621	7	topology	topology	NOUN
ejpam-4592	621	8	,	,	PUNCT
ejpam-4592	621	9	and	and	CCONJ
ejpam-4592	621	10	power	power	NOUN
ejpam-4592	621	11	domains	domain	NOUN
ejpam-4592	621	12	via	via	ADP
ejpam-4592	621	13	the	the	DET
ejpam-4592	621	14	yoneda	yoneda	PROPN
ejpam-4592	621	15	embedding	embed	VERB
ejpam-4592	621	16	.	.	PUNCT
ejpam-4592	622	1	theoretical	theoretical	ADJ
ejpam-4592	622	2	computer	computer	NOUN
ejpam-4592	622	3	science	science	NOUN
ejpam-4592	622	4	,	,	PUNCT
ejpam-4592	622	5	193	193	NUM
ejpam-4592	622	6	,	,	PUNCT
ejpam-4592	622	7	1998	1998	NUM
ejpam-4592	622	8	.	.	PUNCT
ejpam-4592	623	1	references	reference	NOUN
ejpam-4592	623	2	1886	1886	NUM
ejpam-4592	624	1	[	[	X
ejpam-4592	624	2	5	5	X
ejpam-4592	624	3	]	]	PUNCT
ejpam-4592	624	4	t.	t.	PROPN
ejpam-4592	624	5	dosenovi	dosenovi	PROPN
ejpam-4592	624	6	,	,	PUNCT
ejpam-4592	624	7	s.	s.	PROPN
ejpam-4592	624	8	radenovi	radenovi	PROPN
ejpam-4592	624	9	,	,	PUNCT
ejpam-4592	624	10	and	and	CCONJ
ejpam-4592	624	11	s.	s.	PROPN
ejpam-4592	624	12	sedghi	sedghi	PROPN
ejpam-4592	624	13	.	.	PUNCT
ejpam-4592	625	1	generalized	generalize	VERB
ejpam-4592	625	2	metric	metric	ADJ
ejpam-4592	625	3	spaces	space	NOUN
ejpam-4592	625	4	:	:	PUNCT
ejpam-4592	625	5	survey	survey	NOUN
ejpam-4592	625	6	,	,	PUNCT
ejpam-4592	625	7	twms	twms	PROPN
ejpam-4592	625	8	.	.	PUNCT
ejpam-4592	626	1	j.	j.	PROPN
ejpam-4592	626	2	pure	pure	PROPN
ejpam-4592	626	3	appl	appl	PROPN
ejpam-4592	626	4	.	.	PUNCT
ejpam-4592	626	5	math	math	PROPN
ejpam-4592	626	6	.	.	PUNCT
ejpam-4592	626	7	,	,	PUNCT
ejpam-4592	626	8	9(1):3–17	9(1):3–17	NUM
ejpam-4592	626	9	,	,	PUNCT
ejpam-4592	626	10	2018	2018	NUM
ejpam-4592	626	11	.	.	PUNCT
ejpam-4592	627	1	[	[	X
ejpam-4592	627	2	6	6	NUM
ejpam-4592	627	3	]	]	SYM
ejpam-4592	627	4	ekici	ekici	PROPN
ejpam-4592	627	5	erdal	erdal	PROPN
ejpam-4592	627	6	.	.	PUNCT
ejpam-4592	628	1	generalized	generalized	ADJ
ejpam-4592	628	2	hyperconnectedness	hyperconnectedness	NOUN
ejpam-4592	628	3	.	.	PUNCT
ejpam-4592	629	1	acta	acta	PROPN
ejpam-4592	629	2	mathematica	mathematica	PROPN
ejpam-4592	629	3	hungarica	hungarica	PROPN
ejpam-4592	629	4	,	,	PUNCT
ejpam-4592	629	5	133:140	133:140	PROPN
ejpam-4592	629	6	–	–	PUNCT
ejpam-4592	629	7	147	147	NUM
ejpam-4592	629	8	,	,	PUNCT
ejpam-4592	629	9	2011	2011	NUM
ejpam-4592	629	10	.	.	PUNCT
ejpam-4592	630	1	[	[	X
ejpam-4592	630	2	7	7	X
ejpam-4592	630	3	]	]	X
ejpam-4592	630	4	korczak	korczak	PROPN
ejpam-4592	630	5	-	-	PUNCT
ejpam-4592	630	6	kubiak	kubiak	PROPN
ejpam-4592	630	7	ewa	ewa	PROPN
ejpam-4592	630	8	,	,	PUNCT
ejpam-4592	630	9	loranty	loranty	PROPN
ejpam-4592	630	10	anna	anna	NOUN
ejpam-4592	630	11	,	,	PUNCT
ejpam-4592	630	12	and	and	CCONJ
ejpam-4592	630	13	pawlak	pawlak	ADJ
ejpam-4592	630	14	ryszard	ryszard	PROPN
ejpam-4592	630	15	j.	j.	PROPN
ejpam-4592	630	16	baire	baire	PROPN
ejpam-4592	630	17	generalized	generalize	VERB
ejpam-4592	630	18	topological	topological	ADJ
ejpam-4592	630	19	spaces	space	NOUN
ejpam-4592	630	20	,	,	PUNCT
ejpam-4592	630	21	generalized	generalize	VERB
ejpam-4592	630	22	metric	metric	ADJ
ejpam-4592	630	23	spaces	space	NOUN
ejpam-4592	630	24	and	and	CCONJ
ejpam-4592	630	25	infinite	infinite	ADJ
ejpam-4592	630	26	games	game	NOUN
ejpam-4592	630	27	.	.	PUNCT
ejpam-4592	631	1	acta	acta	PROPN
ejpam-4592	631	2	mathematica	mathematica	PROPN
ejpam-4592	631	3	hungarica	hungarica	PROPN
ejpam-4592	631	4	,	,	PUNCT
ejpam-4592	631	5	140(3):203–231	140(3):203–231	NUM
ejpam-4592	631	6	,	,	PUNCT
ejpam-4592	631	7	2013	2013	NUM
ejpam-4592	631	8	.	.	PUNCT
ejpam-4592	632	1	[	[	X
ejpam-4592	632	2	8	8	NUM
ejpam-4592	632	3	]	]	SYM
ejpam-4592	632	4	zdeněk	zdeněk	PUNCT
ejpam-4592	632	5	froĺık	froĺık	PROPN
ejpam-4592	632	6	and	and	CCONJ
ejpam-4592	632	7	praha	praha	PROPN
ejpam-4592	632	8	.	.	PUNCT
ejpam-4592	633	1	baire	baire	NOUN
ejpam-4592	633	2	spaces	space	NOUN
ejpam-4592	633	3	and	and	CCONJ
ejpam-4592	633	4	some	some	DET
ejpam-4592	633	5	generalizations	generalization	NOUN
ejpam-4592	633	6	of	of	ADP
ejpam-4592	633	7	complete	complete	ADJ
ejpam-4592	633	8	metric	metric	ADJ
ejpam-4592	633	9	spaces	space	NOUN
ejpam-4592	633	10	.	.	PUNCT
ejpam-4592	634	1	czechoslov	czechoslov	PROPN
ejpam-4592	634	2	.	.	PUNCT
ejpam-4592	635	1	math	math	NOUN
ejpam-4592	635	2	.	.	PUNCT
ejpam-4592	636	1	j.	j.	PROPN
ejpam-4592	636	2	,	,	PUNCT
ejpam-4592	636	3	,	,	PUNCT
ejpam-4592	636	4	11(2):237–248	11(2):237–248	PROPN
ejpam-4592	636	5	,	,	PUNCT
ejpam-4592	636	6	1961	1961	NUM
ejpam-4592	636	7	.	.	PUNCT
ejpam-4592	637	1	[	[	X
ejpam-4592	637	2	9	9	NUM
ejpam-4592	637	3	]	]	PUNCT
ejpam-4592	637	4	m.	m.	NOUN
ejpam-4592	637	5	isabel	isabel	PROPN
ejpam-4592	637	6	garrido	garrido	PROPN
ejpam-4592	637	7	and	and	CCONJ
ejpam-4592	637	8	ana	ana	PROPN
ejpam-4592	637	9	s.	s.	PROPN
ejpam-4592	637	10	mero	mero	PROPN
ejpam-4592	637	11	noi	noi	PROPN
ejpam-4592	637	12	.	.	PUNCT
ejpam-4592	638	1	new	new	ADJ
ejpam-4592	638	2	types	type	NOUN
ejpam-4592	638	3	of	of	ADP
ejpam-4592	638	4	completeness	completeness	NOUN
ejpam-4592	638	5	in	in	ADP
ejpam-4592	638	6	metric	metric	ADJ
ejpam-4592	638	7	spaces	space	NOUN
ejpam-4592	638	8	.	.	PUNCT
ejpam-4592	639	1	ann	ann	PROPN
ejpam-4592	639	2	.	.	PUNCT
ejpam-4592	639	3	acad	acad	PROPN
ejpam-4592	639	4	.	.	PUNCT
ejpam-4592	640	1	sci	sci	PROPN
ejpam-4592	640	2	.	.	PUNCT
ejpam-4592	640	3	fenn	fenn	PROPN
ejpam-4592	640	4	.	.	PROPN
ejpam-4592	640	5	,	,	PUNCT
ejpam-4592	640	6	39:733–758	39:733–758	NOUN
ejpam-4592	640	7	,	,	PUNCT
ejpam-4592	640	8	2014	2014	NUM
ejpam-4592	640	9	.	.	PUNCT
ejpam-4592	641	1	[	[	X
ejpam-4592	641	2	10	10	NUM
ejpam-4592	641	3	]	]	X
ejpam-4592	641	4	lee	lee	PROPN
ejpam-4592	641	5	j.-g	j.-g	PROPN
ejpam-4592	641	6	.	.	PROPN
ejpam-4592	641	7	,	,	PUNCT
ejpam-4592	641	8	şenel	şenel	PROPN
ejpam-4592	641	9	g.	g.	NOUN
ejpam-4592	641	10	,	,	PUNCT
ejpam-4592	641	11	baek	baek	VERB
ejpam-4592	641	12	j.-i	j.-i	PROPN
ejpam-4592	641	13	.	.	PUNCT
ejpam-4592	641	14	,	,	PUNCT
ejpam-4592	641	15	han	han	PROPN
ejpam-4592	641	16	s.h	s.h	PROPN
ejpam-4592	641	17	.	.	PROPN
ejpam-4592	641	18	,	,	PUNCT
ejpam-4592	641	19	and	and	CCONJ
ejpam-4592	641	20	hur	hur	PROPN
ejpam-4592	641	21	k.	k.	PROPN
ejpam-4592	641	22	neighborhood	neighborhood	PROPN
ejpam-4592	641	23	structures	structure	NOUN
ejpam-4592	641	24	and	and	CCONJ
ejpam-4592	641	25	continuities	continuity	NOUN
ejpam-4592	641	26	via	via	ADP
ejpam-4592	641	27	cubic	cubic	ADJ
ejpam-4592	641	28	sets	set	NOUN
ejpam-4592	641	29	.	.	PUNCT
ejpam-4592	642	1	axioms	axiom	NOUN
ejpam-4592	642	2	,	,	PUNCT
ejpam-4592	642	3	11	11	NUM
ejpam-4592	642	4	,	,	PUNCT
ejpam-4592	642	5	2022	2022	NUM
ejpam-4592	642	6	.	.	PUNCT
ejpam-4592	643	1	[	[	X
ejpam-4592	643	2	11	11	NUM
ejpam-4592	643	3	]	]	X
ejpam-4592	643	4	tanvi	tanvi	NOUN
ejpam-4592	643	5	jain	jain	NOUN
ejpam-4592	643	6	and	and	CCONJ
ejpam-4592	643	7	s.	s.	PROPN
ejpam-4592	643	8	kundu	kundu	PROPN
ejpam-4592	643	9	.	.	PUNCT
ejpam-4592	644	1	atsuji	atsuji	PROPN
ejpam-4592	644	2	completions	completion	NOUN
ejpam-4592	644	3	:	:	PUNCT
ejpam-4592	644	4	equivalent	equivalent	ADJ
ejpam-4592	644	5	characterisations	characterisation	NOUN
ejpam-4592	644	6	.	.	PUNCT
ejpam-4592	645	1	topol	topol	PROPN
ejpam-4592	645	2	.	.	PUNCT
ejpam-4592	645	3	appl	appl	PROPN
ejpam-4592	645	4	.	.	PROPN
ejpam-4592	645	5	,	,	PUNCT
ejpam-4592	646	1	154:28–38	154:28–38	NUM
ejpam-4592	646	2	,	,	PUNCT
ejpam-4592	646	3	2007	2007	NUM
ejpam-4592	646	4	.	.	PUNCT
ejpam-4592	647	1	[	[	X
ejpam-4592	647	2	12	12	NUM
ejpam-4592	647	3	]	]	PUNCT
ejpam-4592	647	4	m.	m.	NOUN
ejpam-4592	647	5	jleli	jleli	PROPN
ejpam-4592	647	6	and	and	CCONJ
ejpam-4592	647	7	b.	b.	PROPN
ejpam-4592	647	8	samet	samet	PROPN
ejpam-4592	647	9	.	.	PUNCT
ejpam-4592	648	1	on	on	ADP
ejpam-4592	648	2	a	a	DET
ejpam-4592	648	3	new	new	ADJ
ejpam-4592	648	4	generalized	generalize	VERB
ejpam-4592	648	5	of	of	ADP
ejpam-4592	648	6	metric	metric	ADJ
ejpam-4592	648	7	spaces	space	NOUN
ejpam-4592	648	8	.	.	PUNCT
ejpam-4592	649	1	arxiv	arxiv	PROPN
ejpam-4592	649	2	:	:	PUNCT
ejpam-4592	649	3	,	,	PUNCT
ejpam-4592	649	4	pages	page	NOUN
ejpam-4592	649	5	1	1	NUM
ejpam-4592	649	6	–	–	SYM
ejpam-4592	649	7	20	20	NUM
ejpam-4592	649	8	,	,	PUNCT
ejpam-4592	649	9	2018	2018	NUM
ejpam-4592	649	10	.	.	PUNCT
ejpam-4592	650	1	[	[	X
ejpam-4592	650	2	13	13	NUM
ejpam-4592	650	3	]	]	PUNCT
ejpam-4592	650	4	w.	w.	PROPN
ejpam-4592	650	5	k.	k.	PROPN
ejpam-4592	650	6	min	min	PROPN
ejpam-4592	650	7	.	.	PROPN
ejpam-4592	651	1	on	on	ADP
ejpam-4592	651	2	weak	weak	ADJ
ejpam-4592	651	3	neighborhood	neighborhood	NOUN
ejpam-4592	651	4	systems	system	NOUN
ejpam-4592	651	5	and	and	CCONJ
ejpam-4592	651	6	spaces	space	NOUN
ejpam-4592	651	7	.	.	PUNCT
ejpam-4592	652	1	acta	acta	PROPN
ejpam-4592	652	2	mathematica	mathematica	PROPN
ejpam-4592	652	3	hungarica	hungarica	PROPN
ejpam-4592	652	4	,	,	PUNCT
ejpam-4592	652	5	121(3):283–292	121(3):283–292	NUM
ejpam-4592	652	6	,	,	PUNCT
ejpam-4592	652	7	2008	2008	NUM
ejpam-4592	652	8	.	.	PUNCT
ejpam-4592	653	1	[	[	X
ejpam-4592	653	2	14	14	NUM
ejpam-4592	653	3	]	]	X
ejpam-4592	653	4	preecha	preecha	NOUN
ejpam-4592	653	5	yupapin	yupapin	NOUN
ejpam-4592	653	6	,	,	PUNCT
ejpam-4592	653	7	vadakasi	vadakasi	PROPN
ejpam-4592	653	8	subramanian	subramanian	PROPN
ejpam-4592	653	9	,	,	PUNCT
ejpam-4592	653	10	and	and	CCONJ
ejpam-4592	653	11	yasser	yasser	PROPN
ejpam-4592	653	12	farhat	farhat	PROPN
ejpam-4592	653	13	.	.	PUNCT
ejpam-4592	654	1	on	on	ADP
ejpam-4592	654	2	nowhere	nowhere	PRON
ejpam-4592	654	3	dense	dense	ADJ
ejpam-4592	654	4	sets	set	NOUN
ejpam-4592	654	5	.	.	PUNCT
ejpam-4592	655	1	european	european	ADJ
ejpam-4592	655	2	journal	journal	PROPN
ejpam-4592	655	3	of	of	ADP
ejpam-4592	655	4	pure	pure	ADJ
ejpam-4592	655	5	and	and	CCONJ
ejpam-4592	655	6	applied	applied	ADJ
ejpam-4592	655	7	mathematics	mathematic	NOUN
ejpam-4592	655	8	,	,	PUNCT
ejpam-4592	655	9	15(2):403–414	15(2):403–414	NUM
ejpam-4592	655	10	,	,	PUNCT
ejpam-4592	655	11	2022	2022	NUM
ejpam-4592	655	12	.	.	PUNCT
ejpam-4592	656	1	[	[	X
ejpam-4592	656	2	15	15	NUM
ejpam-4592	656	3	]	]	X
ejpam-4592	656	4	şenel	şenel	NOUN
ejpam-4592	656	5	guzide	guzide	NOUN
ejpam-4592	656	6	and	and	CCONJ
ejpam-4592	656	7	cagman	cagman	PROPN
ejpam-4592	656	8	naim	naim	PROPN
ejpam-4592	656	9	.	.	PUNCT
ejpam-4592	657	1	soft	soft	ADJ
ejpam-4592	657	2	closed	closed	ADJ
ejpam-4592	657	3	sets	set	NOUN
ejpam-4592	657	4	on	on	ADP
ejpam-4592	657	5	soft	soft	ADJ
ejpam-4592	657	6	bitopological	bitopological	ADJ
ejpam-4592	657	7	space	space	NOUN
ejpam-4592	657	8	.	.	PUNCT
ejpam-4592	658	1	journal	journal	NOUN
ejpam-4592	658	2	of	of	ADP
ejpam-4592	658	3	new	new	ADJ
ejpam-4592	658	4	results	result	NOUN
ejpam-4592	658	5	in	in	ADP
ejpam-4592	658	6	science	science	NOUN
ejpam-4592	658	7	,	,	PUNCT
ejpam-4592	658	8	3(5):57–66	3(5):57–66	NUM
ejpam-4592	658	9	,	,	PUNCT
ejpam-4592	658	10	2014	2014	NUM
ejpam-4592	658	11	.	.	PUNCT
ejpam-4592	659	1	[	[	X
ejpam-4592	659	2	16	16	NUM
ejpam-4592	659	3	]	]	X
ejpam-4592	659	4	şenel	şenel	NOUN
ejpam-4592	659	5	guzide	guzide	NOUN
ejpam-4592	659	6	and	and	CCONJ
ejpam-4592	659	7	cagman	cagman	PROPN
ejpam-4592	659	8	naim	naim	PROPN
ejpam-4592	659	9	.	.	PUNCT
ejpam-4592	660	1	soft	soft	ADJ
ejpam-4592	660	2	topological	topological	ADJ
ejpam-4592	660	3	subspaces	subspace	NOUN
ejpam-4592	660	4	.	.	PUNCT
ejpam-4592	661	1	annals	annal	NOUN
ejpam-4592	661	2	of	of	ADP
ejpam-4592	661	3	fuzzy	fuzzy	ADJ
ejpam-4592	661	4	mathematics	mathematic	NOUN
ejpam-4592	661	5	and	and	CCONJ
ejpam-4592	661	6	informatics	informatic	NOUN
ejpam-4592	661	7	,	,	PUNCT
ejpam-4592	661	8	10(4):525–535	10(4):525–535	NUM
ejpam-4592	661	9	,	,	PUNCT
ejpam-4592	661	10	2015	2015	NUM
ejpam-4592	661	11	.	.	PUNCT
ejpam-4592	662	1	[	[	X
ejpam-4592	662	2	17	17	NUM
ejpam-4592	662	3	]	]	X
ejpam-4592	662	4	şenel	şenel	NOUN
ejpam-4592	662	5	güzide	güzide	NOUN
ejpam-4592	662	6	.	.	PUNCT
ejpam-4592	663	1	a	a	DET
ejpam-4592	663	2	new	new	ADJ
ejpam-4592	663	3	approach	approach	NOUN
ejpam-4592	663	4	to	to	ADP
ejpam-4592	663	5	hausdorff	hausdorff	NOUN
ejpam-4592	663	6	space	space	NOUN
ejpam-4592	663	7	theory	theory	NOUN
ejpam-4592	663	8	via	via	ADP
ejpam-4592	663	9	the	the	DET
ejpam-4592	663	10	soft	soft	ADJ
ejpam-4592	663	11	sets	set	NOUN
ejpam-4592	663	12	.	.	PUNCT
ejpam-4592	664	1	mathematical	mathematical	ADJ
ejpam-4592	664	2	problems	problem	NOUN
ejpam-4592	664	3	in	in	ADP
ejpam-4592	664	4	engineering	engineering	NOUN
ejpam-4592	664	5	,	,	PUNCT
ejpam-4592	664	6	9:1–6	9:1–6	NUM
ejpam-4592	664	7	,	,	PUNCT
ejpam-4592	664	8	2016	2016	NUM
ejpam-4592	664	9	.	.	PUNCT
ejpam-4592	665	1	[	[	X
ejpam-4592	665	2	18	18	NUM
ejpam-4592	665	3	]	]	X
ejpam-4592	665	4	şenel	şenel	NOUN
ejpam-4592	665	5	güzide	güzide	PROPN
ejpam-4592	665	6	,	,	PUNCT
ejpam-4592	665	7	lee	lee	PROPN
ejpam-4592	665	8	jeong	jeong	PROPN
ejpam-4592	665	9	gon	gon	PROPN
ejpam-4592	665	10	,	,	PUNCT
ejpam-4592	665	11	and	and	CCONJ
ejpam-4592	665	12	kul	kul	PROPN
ejpam-4592	665	13	hur	hur	PROPN
ejpam-4592	665	14	.	.	PROPN
ejpam-4592	665	15	distance	distance	NOUN
ejpam-4592	665	16	and	and	CCONJ
ejpam-4592	665	17	similarity	similarity	NOUN
ejpam-4592	665	18	measures	measure	NOUN
ejpam-4592	665	19	for	for	ADP
ejpam-4592	665	20	octahedron	octahedron	NOUN
ejpam-4592	665	21	sets	set	NOUN
ejpam-4592	665	22	and	and	CCONJ
ejpam-4592	665	23	their	their	PRON
ejpam-4592	665	24	application	application	NOUN
ejpam-4592	665	25	to	to	ADP
ejpam-4592	665	26	mcgdm	mcgdm	ADJ
ejpam-4592	665	27	problems	problem	NOUN
ejpam-4592	665	28	.	.	PUNCT
ejpam-4592	666	1	mathematics	mathematic	NOUN
ejpam-4592	666	2	,	,	PUNCT
ejpam-4592	666	3	8	8	NUM
ejpam-4592	666	4	,	,	PUNCT
ejpam-4592	666	5	2020	2020	NUM
ejpam-4592	666	6	.	.	PUNCT
ejpam-4592	667	1	[	[	X
ejpam-4592	667	2	19	19	NUM
ejpam-4592	667	3	]	]	X
ejpam-4592	667	4	şenel	şenel	NOUN
ejpam-4592	667	5	güzide	güzide	PROPN
ejpam-4592	667	6	,	,	PUNCT
ejpam-4592	667	7	lee	lee	PROPN
ejpam-4592	667	8	jeong	jeong	PROPN
ejpam-4592	667	9	gon	gon	PROPN
ejpam-4592	667	10	,	,	PUNCT
ejpam-4592	667	11	and	and	CCONJ
ejpam-4592	667	12	kul	kul	PROPN
ejpam-4592	667	13	hur	hur	PROPN
ejpam-4592	667	14	.	.	PROPN
ejpam-4592	667	15	advanced	advanced	ADJ
ejpam-4592	667	16	soft	soft	ADJ
ejpam-4592	667	17	relation	relation	NOUN
ejpam-4592	667	18	and	and	CCONJ
ejpam-4592	667	19	soft	soft	ADJ
ejpam-4592	667	20	mapping	mapping	NOUN
ejpam-4592	667	21	.	.	PUNCT
ejpam-4592	668	1	international	international	ADJ
ejpam-4592	668	2	journal	journal	NOUN
ejpam-4592	668	3	of	of	ADP
ejpam-4592	668	4	computational	computational	ADJ
ejpam-4592	668	5	intelligence	intelligence	NOUN
ejpam-4592	668	6	systems	system	NOUN
ejpam-4592	668	7	,	,	PUNCT
ejpam-4592	668	8	14(1):461–470	14(1):461–470	PROPN
ejpam-4592	668	9	,	,	PUNCT
ejpam-4592	668	10	2021	2021	NUM
ejpam-4592	668	11	.	.	PUNCT