id	sid	tid	token	lemma	pos
ejpam-5152	1	1	european	european	PROPN
ejpam-5152	1	2	journal	journal	PROPN
ejpam-5152	1	3	of	of	ADP
ejpam-5152	1	4	pure	pure	ADJ
ejpam-5152	1	5	and	and	CCONJ
ejpam-5152	1	6	applied	apply	VERB
ejpam-5152	1	7	mathematics	mathematic	NOUN
ejpam-5152	1	8	vol	vol	NOUN
ejpam-5152	1	9	.	.	PROPN
ejpam-5152	2	1	17	17	NUM
ejpam-5152	2	2	,	,	PUNCT
ejpam-5152	2	3	no	no	INTJ
ejpam-5152	2	4	.	.	NOUN
ejpam-5152	2	5	2	2	NUM
ejpam-5152	2	6	,	,	PUNCT
ejpam-5152	2	7	2024	2024	NUM
ejpam-5152	2	8	,	,	PUNCT
ejpam-5152	2	9	996	996	NUM
ejpam-5152	2	10	-	-	SYM
ejpam-5152	2	11	1008	1008	NUM
ejpam-5152	2	12	issn	issn	PROPN
ejpam-5152	2	13	1307	1307	NUM
ejpam-5152	2	14	-	-	SYM
ejpam-5152	2	15	5543	5543	NUM
ejpam-5152	2	16	–	–	PUNCT
ejpam-5152	3	1	ejpam.com	ejpam.com	X
ejpam-5152	3	2	published	publish	VERB
ejpam-5152	3	3	by	by	ADP
ejpam-5152	3	4	new	new	PROPN
ejpam-5152	3	5	york	york	PROPN
ejpam-5152	3	6	business	business	PROPN
ejpam-5152	3	7	global	global	ADJ
ejpam-5152	3	8	on	on	ADP
ejpam-5152	3	9	common	common	ADJ
ejpam-5152	3	10	fixed	fix	VERB
ejpam-5152	3	11	point	point	NOUN
ejpam-5152	3	12	for	for	ADP
ejpam-5152	3	13	contractive	contractive	ADJ
ejpam-5152	3	14	mappings	mapping	NOUN
ejpam-5152	3	15	in	in	ADP
ejpam-5152	3	16	p	p	NOUN
ejpam-5152	3	17	-	-	PUNCT
ejpam-5152	3	18	pompeiu	pompeiu	NOUN
ejpam-5152	3	19	-	-	PUNCT
ejpam-5152	3	20	hausdorff	hausdorff	NOUN
ejpam-5152	3	21	metric	metric	ADJ
ejpam-5152	3	22	spaces	space	NOUN
ejpam-5152	3	23	arta	arta	PROPN
ejpam-5152	3	24	ekayanti1,2,∗	ekayanti1,2,∗	PROPN
ejpam-5152	3	25	,	,	PUNCT
ejpam-5152	3	26	marjono	marjono	PROPN
ejpam-5152	3	27	marjono1	marjono1	PROPN
ejpam-5152	3	28	,	,	PUNCT
ejpam-5152	3	29	mohamad	mohamad	PROPN
ejpam-5152	3	30	muslikh1	muslikh1	PROPN
ejpam-5152	3	31	,	,	PUNCT
ejpam-5152	3	32	sa’adatul	sa’adatul	PUNCT
ejpam-5152	3	33	fitri1	fitri1	NOUN
ejpam-5152	3	34	1	1	NUM
ejpam-5152	3	35	department	department	NOUN
ejpam-5152	3	36	of	of	ADP
ejpam-5152	3	37	mathematics	mathematic	NOUN
ejpam-5152	3	38	,	,	PUNCT
ejpam-5152	3	39	faculty	faculty	NOUN
ejpam-5152	3	40	of	of	ADP
ejpam-5152	3	41	mathematics	mathematic	NOUN
ejpam-5152	3	42	and	and	CCONJ
ejpam-5152	3	43	natural	natural	ADJ
ejpam-5152	3	44	sciences	science	NOUN
ejpam-5152	3	45	,	,	PUNCT
ejpam-5152	3	46	university	university	NOUN
ejpam-5152	3	47	of	of	ADP
ejpam-5152	3	48	brawijaya	brawijaya	PROPN
ejpam-5152	3	49	,	,	PUNCT
ejpam-5152	3	50	veteran	veteran	NOUN
ejpam-5152	3	51	road	road	NOUN
ejpam-5152	3	52	,	,	PUNCT
ejpam-5152	3	53	malang	malang	PROPN
ejpam-5152	3	54	65145	65145	NUM
ejpam-5152	3	55	,	,	PUNCT
ejpam-5152	3	56	indonesia	indonesia	PROPN
ejpam-5152	3	57	2	2	NUM
ejpam-5152	3	58	department	department	NOUN
ejpam-5152	3	59	of	of	ADP
ejpam-5152	3	60	mathematics	mathematic	NOUN
ejpam-5152	3	61	educations	education	NOUN
ejpam-5152	3	62	,	,	PUNCT
ejpam-5152	3	63	faculty	faculty	NOUN
ejpam-5152	3	64	of	of	ADP
ejpam-5152	3	65	teacher	teacher	NOUN
ejpam-5152	3	66	training	training	NOUN
ejpam-5152	3	67	and	and	CCONJ
ejpam-5152	3	68	education	education	NOUN
ejpam-5152	3	69	,	,	PUNCT
ejpam-5152	3	70	universitas	universita	NOUN
ejpam-5152	3	71	muhammadiyah	muhammadiyah	ADP
ejpam-5152	3	72	ponorogo	ponorogo	PROPN
ejpam-5152	3	73	,	,	PUNCT
ejpam-5152	3	74	ponorogo	ponorogo	PROPN
ejpam-5152	3	75	,	,	PUNCT
ejpam-5152	3	76	east	east	PROPN
ejpam-5152	3	77	java	java	PROPN
ejpam-5152	3	78	63471	63471	NUM
ejpam-5152	3	79	,	,	PUNCT
ejpam-5152	3	80	indonesia	indonesia	PROPN
ejpam-5152	3	81	abstract	abstract	NOUN
ejpam-5152	3	82	.	.	PUNCT
ejpam-5152	4	1	in	in	ADP
ejpam-5152	4	2	this	this	DET
ejpam-5152	4	3	paper	paper	NOUN
ejpam-5152	4	4	we	we	PRON
ejpam-5152	4	5	establish	establish	VERB
ejpam-5152	4	6	the	the	DET
ejpam-5152	4	7	existence	existence	NOUN
ejpam-5152	4	8	of	of	ADP
ejpam-5152	4	9	a	a	DET
ejpam-5152	4	10	common	common	ADJ
ejpam-5152	4	11	fixed	fix	VERB
ejpam-5152	4	12	point	point	NOUN
ejpam-5152	4	13	from	from	ADP
ejpam-5152	4	14	a	a	DET
ejpam-5152	4	15	pair	pair	NOUN
ejpam-5152	4	16	of	of	ADP
ejpam-5152	4	17	setvalued	setvalue	VERB
ejpam-5152	4	18	mappings	mapping	NOUN
ejpam-5152	4	19	.	.	PUNCT
ejpam-5152	5	1	by	by	ADP
ejpam-5152	5	2	utilizing	utilize	VERB
ejpam-5152	5	3	the	the	DET
ejpam-5152	5	4	concept	concept	NOUN
ejpam-5152	5	5	of	of	ADP
ejpam-5152	5	6	convergence	convergence	NOUN
ejpam-5152	5	7	of	of	ADP
ejpam-5152	5	8	set	set	NOUN
ejpam-5152	5	9	-	-	PUNCT
ejpam-5152	5	10	valued	value	VERB
ejpam-5152	5	11	mappings	mapping	NOUN
ejpam-5152	5	12	’	'	PUNCT
ejpam-5152	5	13	sequences	sequence	NOUN
ejpam-5152	5	14	,	,	PUNCT
ejpam-5152	5	15	both	both	DET
ejpam-5152	5	16	ordinary	ordinary	ADJ
ejpam-5152	5	17	and	and	CCONJ
ejpam-5152	5	18	pointwise	pointwise	NOUN
ejpam-5152	5	19	convergence	convergence	NOUN
ejpam-5152	5	20	,	,	PUNCT
ejpam-5152	5	21	we	we	PRON
ejpam-5152	5	22	establish	establish	VERB
ejpam-5152	5	23	a	a	DET
ejpam-5152	5	24	common	common	ADJ
ejpam-5152	5	25	fixed	fix	VERB
ejpam-5152	5	26	point	point	NOUN
ejpam-5152	5	27	theorem	theorem	VERB
ejpam-5152	5	28	.	.	PUNCT
ejpam-5152	6	1	this	this	PRON
ejpam-5152	6	2	our	our	PRON
ejpam-5152	6	3	newly	newly	ADJ
ejpam-5152	6	4	result	result	NOUN
ejpam-5152	6	5	is	be	AUX
ejpam-5152	6	6	a	a	DET
ejpam-5152	6	7	generalization	generalization	NOUN
ejpam-5152	6	8	of	of	ADP
ejpam-5152	6	9	common	common	ADJ
ejpam-5152	6	10	fixed	fix	VERB
ejpam-5152	6	11	point	point	NOUN
ejpam-5152	6	12	theorem	theorem	NOUN
ejpam-5152	6	13	of	of	ADP
ejpam-5152	6	14	set	set	NOUN
ejpam-5152	6	15	-	-	PUNCT
ejpam-5152	6	16	valued	value	VERB
ejpam-5152	6	17	mappings	mapping	NOUN
ejpam-5152	6	18	on	on	ADP
ejpam-5152	6	19	partial	partial	ADJ
ejpam-5152	6	20	metric	metric	ADJ
ejpam-5152	6	21	spaces	space	NOUN
ejpam-5152	6	22	.	.	PUNCT
ejpam-5152	7	1	further	far	ADV
ejpam-5152	7	2	,	,	PUNCT
ejpam-5152	7	3	we	we	PRON
ejpam-5152	7	4	establish	establish	VERB
ejpam-5152	7	5	newly	newly	ADV
ejpam-5152	7	6	common	common	ADJ
ejpam-5152	7	7	fixed	fix	VERB
ejpam-5152	7	8	point	point	NOUN
ejpam-5152	7	9	theorem	theorem	VERB
ejpam-5152	7	10	under	under	ADP
ejpam-5152	7	11	ϕ-contraction	ϕ-contraction	NOUN
ejpam-5152	7	12	on	on	ADP
ejpam-5152	7	13	partial	partial	ADJ
ejpam-5152	7	14	metric	metric	ADJ
ejpam-5152	7	15	spaces	space	NOUN
ejpam-5152	7	16	.	.	PUNCT
ejpam-5152	8	1	2020	2020	NUM
ejpam-5152	8	2	mathematics	mathematic	NOUN
ejpam-5152	8	3	subject	subject	NOUN
ejpam-5152	8	4	classifications	classification	NOUN
ejpam-5152	8	5	:	:	PUNCT
ejpam-5152	8	6	47h10	47h10	NUM
ejpam-5152	8	7	,	,	PUNCT
ejpam-5152	8	8	26e25	26e25	NUM
ejpam-5152	8	9	key	key	ADJ
ejpam-5152	8	10	words	word	NOUN
ejpam-5152	8	11	and	and	CCONJ
ejpam-5152	8	12	phrases	phrase	NOUN
ejpam-5152	8	13	:	:	PUNCT
ejpam-5152	8	14	common	common	ADJ
ejpam-5152	8	15	fixed	fix	VERB
ejpam-5152	8	16	point	point	NOUN
ejpam-5152	8	17	,	,	PUNCT
ejpam-5152	8	18	set	set	NOUN
ejpam-5152	8	19	-	-	PUNCT
ejpam-5152	8	20	valued	value	VERB
ejpam-5152	8	21	mappings	mapping	NOUN
ejpam-5152	8	22	,	,	PUNCT
ejpam-5152	8	23	contractive	contractive	ADJ
ejpam-5152	8	24	mappings	mapping	NOUN
ejpam-5152	8	25	,	,	PUNCT
ejpam-5152	8	26	partial	partial	ADJ
ejpam-5152	8	27	metric	metric	ADJ
ejpam-5152	8	28	spaces	space	NOUN
ejpam-5152	8	29	,	,	PUNCT
ejpam-5152	8	30	p	p	NOUN
ejpam-5152	8	31	-	-	PUNCT
ejpam-5152	8	32	pompeiu	pompeiu	NOUN
ejpam-5152	8	33	-	-	PUNCT
ejpam-5152	8	34	hausdorff	hausdorff	NOUN
ejpam-5152	8	35	metric	metric	ADJ
ejpam-5152	8	36	spaces	space	NOUN
ejpam-5152	8	37	1	1	NUM
ejpam-5152	8	38	.	.	PUNCT
ejpam-5152	9	1	introduction	introduction	NOUN
ejpam-5152	9	2	discussions	discussion	NOUN
ejpam-5152	9	3	regarding	regard	VERB
ejpam-5152	9	4	banach	banach	NOUN
ejpam-5152	9	5	’s	’s	PART
ejpam-5152	9	6	principle	principle	NOUN
ejpam-5152	9	7	of	of	ADP
ejpam-5152	9	8	contraction	contraction	NOUN
ejpam-5152	9	9	often	often	ADV
ejpam-5152	9	10	appear	appear	VERB
ejpam-5152	9	11	in	in	ADP
ejpam-5152	9	12	various	various	ADJ
ejpam-5152	9	13	references	reference	NOUN
ejpam-5152	9	14	.	.	PUNCT
ejpam-5152	10	1	many	many	ADJ
ejpam-5152	10	2	generalizations	generalization	NOUN
ejpam-5152	10	3	are	be	AUX
ejpam-5152	10	4	also	also	ADV
ejpam-5152	10	5	given	give	VERB
ejpam-5152	10	6	for	for	ADP
ejpam-5152	10	7	which	which	PRON
ejpam-5152	10	8	a	a	DET
ejpam-5152	10	9	comparative	comparative	ADJ
ejpam-5152	10	10	study	study	NOUN
ejpam-5152	10	11	of	of	ADP
ejpam-5152	10	12	these	these	DET
ejpam-5152	10	13	generalizations	generalization	NOUN
ejpam-5152	10	14	is	be	AUX
ejpam-5152	10	15	given	give	VERB
ejpam-5152	10	16	by	by	ADP
ejpam-5152	10	17	rhoades	rhoade	NOUN
ejpam-5152	10	18	[	[	X
ejpam-5152	10	19	18	18	NUM
ejpam-5152	10	20	]	]	PUNCT
ejpam-5152	10	21	.	.	PUNCT
ejpam-5152	11	1	one	one	NUM
ejpam-5152	11	2	of	of	ADP
ejpam-5152	11	3	the	the	DET
ejpam-5152	11	4	generalizations	generalization	NOUN
ejpam-5152	11	5	of	of	ADP
ejpam-5152	11	6	the	the	DET
ejpam-5152	11	7	banach	banach	NOUN
ejpam-5152	11	8	contraction	contraction	NOUN
ejpam-5152	11	9	principle	principle	NOUN
ejpam-5152	11	10	that	that	PRON
ejpam-5152	11	11	is	be	AUX
ejpam-5152	11	12	also	also	ADV
ejpam-5152	11	13	quite	quite	ADV
ejpam-5152	11	14	widely	widely	ADV
ejpam-5152	11	15	discussed	discuss	VERB
ejpam-5152	11	16	is	be	AUX
ejpam-5152	11	17	in	in	ADP
ejpam-5152	11	18	the	the	DET
ejpam-5152	11	19	set	set	NOUN
ejpam-5152	11	20	-	-	PUNCT
ejpam-5152	11	21	valued	value	VERB
ejpam-5152	11	22	mapping	mapping	NOUN
ejpam-5152	11	23	.	.	PUNCT
ejpam-5152	12	1	various	various	ADJ
ejpam-5152	12	2	results	result	NOUN
ejpam-5152	12	3	of	of	ADP
ejpam-5152	12	4	the	the	DET
ejpam-5152	12	5	generalization	generalization	NOUN
ejpam-5152	12	6	of	of	ADP
ejpam-5152	12	7	banach	banach	NOUN
ejpam-5152	12	8	’s	’s	PART
ejpam-5152	12	9	contraction	contraction	NOUN
ejpam-5152	12	10	principle	principle	NOUN
ejpam-5152	12	11	can	can	AUX
ejpam-5152	12	12	be	be	AUX
ejpam-5152	12	13	found	find	VERB
ejpam-5152	12	14	in	in	ADP
ejpam-5152	12	15	[	[	X
ejpam-5152	12	16	8	8	NUM
ejpam-5152	12	17	,	,	PUNCT
ejpam-5152	12	18	13	13	NUM
ejpam-5152	12	19	,	,	PUNCT
ejpam-5152	12	20	14	14	NUM
ejpam-5152	12	21	,	,	PUNCT
ejpam-5152	12	22	17	17	NUM
ejpam-5152	12	23	,	,	PUNCT
ejpam-5152	12	24	20	20	NUM
ejpam-5152	12	25	]	]	PUNCT
ejpam-5152	12	26	and	and	CCONJ
ejpam-5152	12	27	reference	reference	NOUN
ejpam-5152	12	28	therein	therein	ADV
ejpam-5152	12	29	.	.	PUNCT
ejpam-5152	13	1	further	further	ADJ
ejpam-5152	13	2	results	result	NOUN
ejpam-5152	13	3	on	on	ADP
ejpam-5152	13	4	the	the	DET
ejpam-5152	13	5	general	general	ADJ
ejpam-5152	13	6	fixed	fix	VERB
ejpam-5152	13	7	point	point	NOUN
ejpam-5152	13	8	of	of	ADP
ejpam-5152	13	9	the	the	DET
ejpam-5152	13	10	set	set	NOUN
ejpam-5152	13	11	-	-	PUNCT
ejpam-5152	13	12	valued	value	VERB
ejpam-5152	13	13	mapping	mapping	NOUN
ejpam-5152	13	14	of	of	ADP
ejpam-5152	13	15	the	the	DET
ejpam-5152	13	16	contractive	contractive	ADJ
ejpam-5152	13	17	type	type	NOUN
ejpam-5152	13	18	may	may	AUX
ejpam-5152	13	19	be	be	AUX
ejpam-5152	13	20	found	find	VERB
ejpam-5152	13	21	in	in	ADP
ejpam-5152	13	22	kubiak	kubiak	PROPN
ejpam-5152	14	1	[	[	X
ejpam-5152	14	2	12	12	NUM
ejpam-5152	14	3	]	]	PUNCT
ejpam-5152	14	4	and	and	CCONJ
ejpam-5152	14	5	singh	singh	PROPN
ejpam-5152	15	1	[	[	X
ejpam-5152	15	2	19	19	NUM
ejpam-5152	15	3	]	]	PUNCT
ejpam-5152	15	4	.	.	PUNCT
ejpam-5152	16	1	on	on	ADP
ejpam-5152	16	2	the	the	DET
ejpam-5152	16	3	other	other	ADJ
ejpam-5152	16	4	hand	hand	NOUN
ejpam-5152	16	5	a	a	DET
ejpam-5152	16	6	generalization	generalization	NOUN
ejpam-5152	16	7	of	of	ADP
ejpam-5152	16	8	the	the	DET
ejpam-5152	16	9	principle	principle	NOUN
ejpam-5152	16	10	of	of	ADP
ejpam-5152	16	11	banach	banach	NOUN
ejpam-5152	16	12	contraction	contraction	NOUN
ejpam-5152	16	13	for	for	ADP
ejpam-5152	16	14	single	single	ADV
ejpam-5152	16	15	-	-	PUNCT
ejpam-5152	16	16	valued	value	VERB
ejpam-5152	16	17	mapping	mapping	NOUN
ejpam-5152	16	18	on	on	ADP
ejpam-5152	16	19	partial	partial	ADJ
ejpam-5152	16	20	metric	metric	ADJ
ejpam-5152	16	21	spaces	space	NOUN
ejpam-5152	16	22	can	can	AUX
ejpam-5152	16	23	be	be	AUX
ejpam-5152	16	24	seen	see	VERB
ejpam-5152	16	25	in	in	ADP
ejpam-5152	16	26	[	[	X
ejpam-5152	16	27	2	2	NUM
ejpam-5152	16	28	,	,	PUNCT
ejpam-5152	16	29	5	5	NUM
ejpam-5152	16	30	,	,	PUNCT
ejpam-5152	16	31	10	10	NUM
ejpam-5152	16	32	,	,	PUNCT
ejpam-5152	16	33	11	11	NUM
ejpam-5152	16	34	]	]	PUNCT
ejpam-5152	16	35	and	and	CCONJ
ejpam-5152	16	36	reference	reference	NOUN
ejpam-5152	16	37	therein	therein	ADV
ejpam-5152	16	38	.	.	PUNCT
ejpam-5152	17	1	furthermore	furthermore	ADV
ejpam-5152	17	2	,	,	PUNCT
ejpam-5152	17	3	a	a	DET
ejpam-5152	17	4	generalization	generalization	NOUN
ejpam-5152	17	5	of	of	ADP
ejpam-5152	17	6	the	the	DET
ejpam-5152	17	7	banach	banach	NOUN
ejpam-5152	17	8	contraction	contraction	NOUN
ejpam-5152	17	9	principle	principle	NOUN
ejpam-5152	17	10	for	for	ADP
ejpam-5152	17	11	set	set	NOUN
ejpam-5152	17	12	-	-	PUNCT
ejpam-5152	17	13	valued	value	VERB
ejpam-5152	17	14	mappings	mapping	NOUN
ejpam-5152	17	15	in	in	ADP
ejpam-5152	17	16	partial	partial	ADJ
ejpam-5152	17	17	∗corresponding	∗corresponding	NOUN
ejpam-5152	17	18	author	author	NOUN
ejpam-5152	17	19	.	.	PUNCT
ejpam-5152	18	1	doi	doi	NOUN
ejpam-5152	18	2	:	:	PUNCT
ejpam-5152	18	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5152	https://doi.org/10.29020/nybg.ejpam.v17i2.5152	PROPN
ejpam-5152	18	4	email	email	NOUN
ejpam-5152	18	5	addresses	address	VERB
ejpam-5152	18	6	:	:	PUNCT
ejpam-5152	18	7	arta	arta	PROPN
ejpam-5152	18	8	ekayanti@ub.ac.id	ekayanti@ub.ac.id	PROPN
ejpam-5152	18	9	(	(	PUNCT
ejpam-5152	18	10	a.	a.	NOUN
ejpam-5152	18	11	ekayanti	ekayanti	NOUN
ejpam-5152	18	12	)	)	PUNCT
ejpam-5152	18	13	,	,	PUNCT
ejpam-5152	18	14	marjono@ub.ac.id	marjono@ub.ac.id	PROPN
ejpam-5152	18	15	(	(	PUNCT
ejpam-5152	18	16	marjono	marjono	PROPN
ejpam-5152	18	17	)	)	PUNCT
ejpam-5152	18	18	,	,	PUNCT
ejpam-5152	18	19	mslk@ub.ac.id	mslk@ub.ac.id	PROPN
ejpam-5152	18	20	(	(	PUNCT
ejpam-5152	18	21	m.	m.	NOUN
ejpam-5152	18	22	muslikh	muslikh	PROPN
ejpam-5152	18	23	)	)	PUNCT
ejpam-5152	18	24	,	,	PUNCT
ejpam-5152	18	25	saadatulfitri@ub.ac.id	saadatulfitri@ub.ac.id	NOUN
ejpam-5152	18	26	(	(	PUNCT
ejpam-5152	18	27	s.	s.	PROPN
ejpam-5152	18	28	fitri	fitri	PROPN
ejpam-5152	18	29	)	)	PUNCT
ejpam-5152	18	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5152	18	31	996	996	NUM
ejpam-5152	18	32	©	©	ADP
ejpam-5152	18	33	2024	2024	NUM
ejpam-5152	18	34	ejpam	ejpam	NOUN
ejpam-5152	18	35	all	all	DET
ejpam-5152	18	36	rights	right	NOUN
ejpam-5152	18	37	reserved	reserve	VERB
ejpam-5152	18	38	.	.	PUNCT
ejpam-5152	19	1	a.	a.	PROPN
ejpam-5152	19	2	ekayanti	ekayanti	PROPN
ejpam-5152	19	3	et	et	PROPN
ejpam-5152	19	4	al	al	PROPN
ejpam-5152	19	5	.	.	PUNCT
ejpam-5152	19	6	/	/	SYM
ejpam-5152	19	7	eur	eur	PROPN
ejpam-5152	19	8	.	.	PUNCT
ejpam-5152	20	1	j.	j.	PROPN
ejpam-5152	20	2	pure	pure	PROPN
ejpam-5152	20	3	appl	appl	PROPN
ejpam-5152	20	4	.	.	PROPN
ejpam-5152	20	5	math	math	PROPN
ejpam-5152	20	6	,	,	PUNCT
ejpam-5152	20	7	17	17	NUM
ejpam-5152	20	8	(	(	PUNCT
ejpam-5152	20	9	2	2	NUM
ejpam-5152	20	10	)	)	PUNCT
ejpam-5152	20	11	(	(	PUNCT
ejpam-5152	20	12	2024	2024	NUM
ejpam-5152	20	13	)	)	PUNCT
ejpam-5152	20	14	,	,	PUNCT
ejpam-5152	20	15	996	996	NUM
ejpam-5152	20	16	-	-	SYM
ejpam-5152	20	17	1008	1008	NUM
ejpam-5152	20	18	997	997	NUM
ejpam-5152	20	19	metric	metric	ADJ
ejpam-5152	20	20	spaces	space	NOUN
ejpam-5152	20	21	can	can	AUX
ejpam-5152	20	22	be	be	AUX
ejpam-5152	20	23	found	find	VERB
ejpam-5152	20	24	in	in	ADP
ejpam-5152	20	25	[	[	X
ejpam-5152	20	26	1	1	NUM
ejpam-5152	20	27	,	,	PUNCT
ejpam-5152	20	28	4	4	NUM
ejpam-5152	20	29	,	,	PUNCT
ejpam-5152	20	30	6	6	NUM
ejpam-5152	20	31	]	]	PUNCT
ejpam-5152	20	32	.	.	PUNCT
ejpam-5152	21	1	this	this	DET
ejpam-5152	21	2	generalization	generalization	NOUN
ejpam-5152	21	3	builds	build	VERB
ejpam-5152	21	4	upon	upon	SCONJ
ejpam-5152	21	5	the	the	DET
ejpam-5152	21	6	banach	banach	NOUN
ejpam-5152	21	7	contraction	contraction	NOUN
ejpam-5152	21	8	principle	principle	NOUN
ejpam-5152	21	9	for	for	ADP
ejpam-5152	21	10	set	set	NOUN
ejpam-5152	21	11	-	-	PUNCT
ejpam-5152	21	12	valued	value	VERB
ejpam-5152	21	13	mappings	mapping	NOUN
ejpam-5152	21	14	,	,	PUNCT
ejpam-5152	21	15	which	which	PRON
ejpam-5152	21	16	was	be	AUX
ejpam-5152	21	17	initially	initially	ADV
ejpam-5152	21	18	introduced	introduce	VERB
ejpam-5152	21	19	by	by	ADP
ejpam-5152	21	20	nadler	nadler	PROPN
ejpam-5152	21	21	[	[	X
ejpam-5152	21	22	16	16	NUM
ejpam-5152	21	23	]	]	PUNCT
ejpam-5152	21	24	.	.	PUNCT
ejpam-5152	22	1	and	and	CCONJ
ejpam-5152	22	2	further	further	ADJ
ejpam-5152	22	3	results	result	NOUN
ejpam-5152	22	4	on	on	ADP
ejpam-5152	22	5	the	the	DET
ejpam-5152	22	6	general	general	ADJ
ejpam-5152	22	7	fixed	fix	VERB
ejpam-5152	22	8	point	point	NOUN
ejpam-5152	22	9	of	of	ADP
ejpam-5152	22	10	the	the	DET
ejpam-5152	22	11	set	set	NOUN
ejpam-5152	22	12	-	-	PUNCT
ejpam-5152	22	13	valued	value	VERB
ejpam-5152	22	14	mapping	mapping	NOUN
ejpam-5152	22	15	on	on	ADP
ejpam-5152	22	16	partial	partial	ADJ
ejpam-5152	22	17	metric	metric	ADJ
ejpam-5152	22	18	space	space	NOUN
ejpam-5152	22	19	be	be	AUX
ejpam-5152	22	20	found	find	VERB
ejpam-5152	22	21	in	in	ADP
ejpam-5152	22	22	aydi	aydi	VERB
ejpam-5152	22	23	et	et	NOUN
ejpam-5152	22	24	.	.	PUNCT
ejpam-5152	23	1	al	al	PROPN
ejpam-5152	23	2	.	.	PUNCT
ejpam-5152	24	1	[	[	X
ejpam-5152	24	2	7	7	X
ejpam-5152	24	3	]	]	PUNCT
ejpam-5152	24	4	and	and	CCONJ
ejpam-5152	24	5	ahmad	ahmad	PROPN
ejpam-5152	24	6	et	et	PROPN
ejpam-5152	24	7	.	.	PUNCT
ejpam-5152	25	1	al	al	PROPN
ejpam-5152	25	2	.	.	PUNCT
ejpam-5152	26	1	[	[	X
ejpam-5152	26	2	3	3	NUM
ejpam-5152	26	3	]	]	PUNCT
ejpam-5152	26	4	.	.	PUNCT
ejpam-5152	27	1	in	in	ADP
ejpam-5152	27	2	this	this	DET
ejpam-5152	27	3	paper	paper	NOUN
ejpam-5152	27	4	,	,	PUNCT
ejpam-5152	27	5	we	we	PRON
ejpam-5152	27	6	will	will	AUX
ejpam-5152	27	7	generalize	generalize	VERB
ejpam-5152	27	8	some	some	DET
ejpam-5152	27	9	results	result	NOUN
ejpam-5152	27	10	of	of	ADP
ejpam-5152	27	11	aydi	aydi	VERB
ejpam-5152	27	12	et	et	NOUN
ejpam-5152	27	13	.	.	PUNCT
ejpam-5152	28	1	al.[7	al.[7	PROPN
ejpam-5152	28	2	]	]	PUNCT
ejpam-5152	28	3	and	and	CCONJ
ejpam-5152	28	4	ahmad	ahmad	PROPN
ejpam-5152	28	5	et	et	PROPN
ejpam-5152	28	6	.	.	PUNCT
ejpam-5152	29	1	al	al	PROPN
ejpam-5152	29	2	.	.	PUNCT
ejpam-5152	30	1	[	[	X
ejpam-5152	30	2	3	3	NUM
ejpam-5152	30	3	]	]	PUNCT
ejpam-5152	30	4	.	.	PUNCT
ejpam-5152	31	1	referring	refer	VERB
ejpam-5152	31	2	to	to	ADP
ejpam-5152	31	3	kubiak	kubiak	PROPN
ejpam-5152	31	4	[	[	X
ejpam-5152	31	5	12	12	NUM
ejpam-5152	31	6	]	]	PUNCT
ejpam-5152	31	7	,	,	PUNCT
ejpam-5152	31	8	we	we	PRON
ejpam-5152	31	9	will	will	AUX
ejpam-5152	31	10	use	use	VERB
ejpam-5152	31	11	the	the	DET
ejpam-5152	31	12	common	common	ADJ
ejpam-5152	31	13	fixed	fix	VERB
ejpam-5152	31	14	point	point	NOUN
ejpam-5152	31	15	existence	existence	NOUN
ejpam-5152	31	16	of	of	ADP
ejpam-5152	31	17	a	a	DET
ejpam-5152	31	18	sequence	sequence	NOUN
ejpam-5152	31	19	of	of	ADP
ejpam-5152	31	20	set	set	NOUN
ejpam-5152	31	21	-	-	PUNCT
ejpam-5152	31	22	valued	value	VERB
ejpam-5152	31	23	mappings	mapping	NOUN
ejpam-5152	31	24	to	to	ADP
ejpam-5152	31	25	derived	derive	VERB
ejpam-5152	31	26	on	on	ADP
ejpam-5152	31	27	a	a	DET
ejpam-5152	31	28	pair	pair	NOUN
ejpam-5152	31	29	of	of	ADP
ejpam-5152	31	30	set	set	NOUN
ejpam-5152	31	31	-	-	PUNCT
ejpam-5152	31	32	valued	value	VERB
ejpam-5152	31	33	mappings	mapping	NOUN
ejpam-5152	31	34	so	so	SCONJ
ejpam-5152	31	35	that	that	SCONJ
ejpam-5152	31	36	the	the	DET
ejpam-5152	31	37	existence	existence	NOUN
ejpam-5152	31	38	of	of	ADP
ejpam-5152	31	39	a	a	DET
ejpam-5152	31	40	common	common	ADJ
ejpam-5152	31	41	fixed	fix	VERB
ejpam-5152	31	42	point	point	NOUN
ejpam-5152	31	43	is	be	AUX
ejpam-5152	31	44	guaranted	guarante	VERB
ejpam-5152	31	45	.	.	PUNCT
ejpam-5152	32	1	furthermore	furthermore	ADV
ejpam-5152	32	2	,	,	PUNCT
ejpam-5152	32	3	referring	refer	VERB
ejpam-5152	32	4	to	to	ADP
ejpam-5152	32	5	singh	singh	PROPN
ejpam-5152	32	6	[	[	X
ejpam-5152	32	7	19	19	NUM
ejpam-5152	32	8	]	]	X
ejpam-5152	32	9	we	we	PRON
ejpam-5152	32	10	will	will	AUX
ejpam-5152	32	11	use	use	VERB
ejpam-5152	32	12	some	some	DET
ejpam-5152	32	13	functions	function	NOUN
ejpam-5152	32	14	that	that	PRON
ejpam-5152	32	15	he	he	PRON
ejpam-5152	32	16	has	have	AUX
ejpam-5152	32	17	defined	define	VERB
ejpam-5152	32	18	to	to	PART
ejpam-5152	32	19	give	give	VERB
ejpam-5152	32	20	a	a	DET
ejpam-5152	32	21	new	new	ADJ
ejpam-5152	32	22	generalization	generalization	NOUN
ejpam-5152	32	23	of	of	ADP
ejpam-5152	32	24	the	the	DET
ejpam-5152	32	25	contraction	contraction	NOUN
ejpam-5152	32	26	of	of	ADP
ejpam-5152	32	27	banach	banach	NOUN
ejpam-5152	32	28	’s	’s	PART
ejpam-5152	32	29	principle	principle	NOUN
ejpam-5152	32	30	for	for	ADP
ejpam-5152	32	31	set	set	NOUN
ejpam-5152	32	32	-	-	PUNCT
ejpam-5152	32	33	valued	value	VERB
ejpam-5152	32	34	mappings	mapping	NOUN
ejpam-5152	32	35	on	on	ADP
ejpam-5152	32	36	partial	partial	ADJ
ejpam-5152	32	37	metric	metric	ADJ
ejpam-5152	32	38	spaces	space	NOUN
ejpam-5152	32	39	.	.	PUNCT
ejpam-5152	33	1	by	by	ADP
ejpam-5152	33	2	using	use	VERB
ejpam-5152	33	3	this	this	DET
ejpam-5152	33	4	contraction	contraction	NOUN
ejpam-5152	33	5	we	we	PRON
ejpam-5152	33	6	obtain	obtain	VERB
ejpam-5152	33	7	the	the	DET
ejpam-5152	33	8	common	common	ADJ
ejpam-5152	33	9	fixed	fix	VERB
ejpam-5152	33	10	points	point	NOUN
ejpam-5152	33	11	of	of	ADP
ejpam-5152	33	12	a	a	DET
ejpam-5152	33	13	pair	pair	NOUN
ejpam-5152	33	14	of	of	ADP
ejpam-5152	33	15	set	set	NOUN
ejpam-5152	33	16	-	-	PUNCT
ejpam-5152	33	17	valued	value	VERB
ejpam-5152	33	18	mappings	mapping	NOUN
ejpam-5152	33	19	.	.	PUNCT
ejpam-5152	34	1	2	2	X
ejpam-5152	34	2	.	.	X
ejpam-5152	34	3	preliminaries	preliminary	NOUN
ejpam-5152	34	4	let	let	VERB
ejpam-5152	34	5	(	(	PUNCT
ejpam-5152	34	6	x	x	X
ejpam-5152	34	7	,	,	PUNCT
ejpam-5152	34	8	p	p	X
ejpam-5152	34	9	)	)	PUNCT
ejpam-5152	34	10	be	be	AUX
ejpam-5152	34	11	a	a	DET
ejpam-5152	34	12	partial	partial	ADJ
ejpam-5152	34	13	metric	metric	ADJ
ejpam-5152	34	14	spaces	space	NOUN
ejpam-5152	34	15	.	.	PUNCT
ejpam-5152	35	1	suppose	suppose	VERB
ejpam-5152	35	2	that	that	SCONJ
ejpam-5152	35	3	cbp(x	cbp(x	PROPN
ejpam-5152	35	4	)	)	PUNCT
ejpam-5152	35	5	be	be	VERB
ejpam-5152	35	6	class	class	NOUN
ejpam-5152	35	7	of	of	ADP
ejpam-5152	35	8	all	all	DET
ejpam-5152	35	9	nonempty	nonempty	ADJ
ejpam-5152	35	10	,	,	PUNCT
ejpam-5152	35	11	closed	closed	ADJ
ejpam-5152	35	12	and	and	CCONJ
ejpam-5152	35	13	bounded	bound	VERB
ejpam-5152	35	14	subsets	subset	NOUN
ejpam-5152	35	15	of	of	ADP
ejpam-5152	35	16	x.	x.	NOUN
ejpam-5152	35	17	let	let	VERB
ejpam-5152	35	18	mapping	mapping	NOUN
ejpam-5152	35	19	hp	hp	VERB
ejpam-5152	35	20	:	:	PUNCT
ejpam-5152	35	21	x	x	X
ejpam-5152	35	22	→	→	SYM
ejpam-5152	35	23	cbp(x	cbp(x	PROPN
ejpam-5152	35	24	)	)	PUNCT
ejpam-5152	35	25	define	define	NOUN
ejpam-5152	35	26	hp(a	hp(a	PROPN
ejpam-5152	35	27	,	,	PUNCT
ejpam-5152	35	28	b	b	NOUN
ejpam-5152	35	29	)	)	PUNCT
ejpam-5152	35	30	=	=	SYM
ejpam-5152	35	31	max{sup{p(x	max{sup{p(x	PROPN
ejpam-5152	35	32	,	,	PUNCT
ejpam-5152	35	33	b	b	NOUN
ejpam-5152	35	34	)	)	PUNCT
ejpam-5152	35	35	:	:	PUNCT
ejpam-5152	35	36	x	x	PUNCT
ejpam-5152	35	37	∈	∈	PROPN
ejpam-5152	35	38	a	a	PRON
ejpam-5152	35	39	}	}	PUNCT
ejpam-5152	35	40	,	,	PUNCT
ejpam-5152	35	41	sup{p(y	sup{p(y	PROPN
ejpam-5152	35	42	,	,	PUNCT
ejpam-5152	35	43	a	a	PRON
ejpam-5152	35	44	)	)	PUNCT
ejpam-5152	35	45	:	:	PUNCT
ejpam-5152	35	46	y	y	PROPN
ejpam-5152	35	47	∈	∈	PROPN
ejpam-5152	35	48	b	b	PROPN
ejpam-5152	35	49	}	}	PUNCT
ejpam-5152	35	50	}	}	PUNCT
ejpam-5152	35	51	,	,	PUNCT
ejpam-5152	35	52	for	for	ADP
ejpam-5152	35	53	each	each	DET
ejpam-5152	35	54	a	a	NOUN
ejpam-5152	35	55	,	,	PUNCT
ejpam-5152	35	56	b	b	X
ejpam-5152	35	57	∈	∈	NOUN
ejpam-5152	35	58	x	x	X
ejpam-5152	35	59	and	and	CCONJ
ejpam-5152	35	60	p(x	p(x	PROPN
ejpam-5152	35	61	,	,	PUNCT
ejpam-5152	35	62	b	b	NOUN
ejpam-5152	35	63	)	)	PUNCT
ejpam-5152	35	64	=	=	SYM
ejpam-5152	36	1	inf{p(x	inf{p(x	PROPN
ejpam-5152	36	2	,	,	PUNCT
ejpam-5152	36	3	y	y	PROPN
ejpam-5152	36	4	)	)	PUNCT
ejpam-5152	36	5	:	:	PUNCT
ejpam-5152	37	1	y	y	PROPN
ejpam-5152	37	2	∈	∈	PROPN
ejpam-5152	37	3	b	b	PROPN
ejpam-5152	37	4	}	}	PUNCT
ejpam-5152	37	5	.	.	PUNCT
ejpam-5152	38	1	the	the	DET
ejpam-5152	38	2	mapping	mapping	NOUN
ejpam-5152	38	3	hp	hp	NOUN
ejpam-5152	38	4	is	be	AUX
ejpam-5152	38	5	p	p	NOUN
ejpam-5152	38	6	-	-	PUNCT
ejpam-5152	38	7	pompeiuhausdorff	pompeiuhausdorff	NOUN
ejpam-5152	38	8	(	(	PUNCT
ejpam-5152	38	9	partial	partial	ADJ
ejpam-5152	38	10	pompeiu	pompeiu	NOUN
ejpam-5152	38	11	hausdorff	hausdorff	NOUN
ejpam-5152	38	12	)	)	PUNCT
ejpam-5152	38	13	metric	metric	NOUN
ejpam-5152	38	14	,	,	PUNCT
ejpam-5152	38	15	and	and	CCONJ
ejpam-5152	38	16	the	the	DET
ejpam-5152	38	17	pairs	pair	NOUN
ejpam-5152	38	18	(	(	PUNCT
ejpam-5152	38	19	cbp(x	cbp(x	PROPN
ejpam-5152	38	20	)	)	PUNCT
ejpam-5152	38	21	,	,	PUNCT
ejpam-5152	38	22	hp	hp	PROPN
ejpam-5152	38	23	)	)	PUNCT
ejpam-5152	38	24	is	be	AUX
ejpam-5152	38	25	called	call	VERB
ejpam-5152	38	26	ppompeiu	ppompeiu	NOUN
ejpam-5152	38	27	-	-	PUNCT
ejpam-5152	38	28	hausdorff	hausdorff	NOUN
ejpam-5152	38	29	metric	metric	ADJ
ejpam-5152	38	30	spaces	space	NOUN
ejpam-5152	38	31	.	.	PUNCT
ejpam-5152	39	1	(	(	PUNCT
ejpam-5152	39	2	the	the	DET
ejpam-5152	39	3	use	use	NOUN
ejpam-5152	39	4	of	of	ADP
ejpam-5152	39	5	the	the	DET
ejpam-5152	39	6	term	term	NOUN
ejpam-5152	39	7	pompeiu	pompeiu	NOUN
ejpam-5152	39	8	-	-	PUNCT
ejpam-5152	39	9	hausdorf	hausdorf	PROPN
ejpam-5152	39	10	refers	refer	VERB
ejpam-5152	39	11	to	to	ADP
ejpam-5152	39	12	[	[	X
ejpam-5152	39	13	9	9	NUM
ejpam-5152	39	14	]	]	SYM
ejpam-5152	39	15	)	)	PUNCT
ejpam-5152	39	16	.	.	PUNCT
ejpam-5152	40	1	some	some	DET
ejpam-5152	40	2	properties	property	NOUN
ejpam-5152	40	3	of	of	ADP
ejpam-5152	40	4	metric	metric	ADJ
ejpam-5152	40	5	hp	hp	NOUN
ejpam-5152	40	6	can	can	AUX
ejpam-5152	40	7	be	be	AUX
ejpam-5152	40	8	found	find	VERB
ejpam-5152	40	9	in	in	ADP
ejpam-5152	40	10	[	[	X
ejpam-5152	40	11	6	6	NUM
ejpam-5152	40	12	,	,	PUNCT
ejpam-5152	40	13	7	7	NUM
ejpam-5152	40	14	,	,	PUNCT
ejpam-5152	40	15	15	15	NUM
ejpam-5152	40	16	]	]	PUNCT
ejpam-5152	40	17	.	.	PUNCT
ejpam-5152	41	1	definition	definition	NOUN
ejpam-5152	41	2	1	1	NUM
ejpam-5152	41	3	.	.	PUNCT
ejpam-5152	42	1	[	[	X
ejpam-5152	42	2	15	15	NUM
ejpam-5152	42	3	]	]	X
ejpam-5152	42	4	let	let	NOUN
ejpam-5152	42	5	(	(	PUNCT
ejpam-5152	42	6	cbp(x	cbp(x	PROPN
ejpam-5152	42	7	)	)	PUNCT
ejpam-5152	42	8	,	,	PUNCT
ejpam-5152	42	9	hp	hp	PROPN
ejpam-5152	42	10	)	)	PUNCT
ejpam-5152	42	11	be	be	VERB
ejpam-5152	42	12	a	a	DET
ejpam-5152	42	13	p	p	ADJ
ejpam-5152	42	14	-	-	PUNCT
ejpam-5152	42	15	pompeiu	pompeiu	NOUN
ejpam-5152	42	16	-	-	PUNCT
ejpam-5152	42	17	hausdorff	hausdorff	NOUN
ejpam-5152	42	18	metric	metric	ADJ
ejpam-5152	42	19	spaces	space	NOUN
ejpam-5152	42	20	.	.	PUNCT
ejpam-5152	43	1	a	a	DET
ejpam-5152	43	2	sequence	sequence	NOUN
ejpam-5152	43	3	(	(	PUNCT
ejpam-5152	43	4	fn	fn	NOUN
ejpam-5152	43	5	)	)	PUNCT
ejpam-5152	43	6	in	in	ADP
ejpam-5152	43	7	cbp(x	cbp(x	PROPN
ejpam-5152	43	8	)	)	PUNCT
ejpam-5152	43	9	converges	converge	VERB
ejpam-5152	43	10	to	to	PART
ejpam-5152	43	11	set	set	VERB
ejpam-5152	43	12	f	f	PROPN
ejpam-5152	43	13	∈	∈	PROPN
ejpam-5152	43	14	cbp(x	cbp(x	PROPN
ejpam-5152	43	15	)	)	PUNCT
ejpam-5152	43	16	if	if	SCONJ
ejpam-5152	43	17	lim	lim	PROPN
ejpam-5152	43	18	n→∞	n→∞	NUM
ejpam-5152	43	19	hp(fn	hp(fn	PROPN
ejpam-5152	43	20	,	,	PUNCT
ejpam-5152	43	21	f	f	PROPN
ejpam-5152	43	22	)	)	PUNCT
ejpam-5152	43	23	=	=	PUNCT
ejpam-5152	44	1	hp(f	hp(f	PROPN
ejpam-5152	44	2	,	,	PUNCT
ejpam-5152	44	3	f	f	PROPN
ejpam-5152	44	4	)	)	PUNCT
ejpam-5152	44	5	.	.	PUNCT
ejpam-5152	45	1	definition	definition	NOUN
ejpam-5152	45	2	2	2	NUM
ejpam-5152	45	3	.	.	PUNCT
ejpam-5152	46	1	[	[	X
ejpam-5152	46	2	15	15	NUM
ejpam-5152	46	3	]	]	X
ejpam-5152	46	4	let	let	NOUN
ejpam-5152	46	5	(	(	PUNCT
ejpam-5152	46	6	cbp(x	cbp(x	PROPN
ejpam-5152	46	7	)	)	PUNCT
ejpam-5152	46	8	,	,	PUNCT
ejpam-5152	46	9	hp	hp	PROPN
ejpam-5152	46	10	)	)	PUNCT
ejpam-5152	46	11	be	be	VERB
ejpam-5152	46	12	a	a	DET
ejpam-5152	46	13	p	p	ADJ
ejpam-5152	46	14	-	-	PUNCT
ejpam-5152	46	15	pompeiu	pompeiu	NOUN
ejpam-5152	46	16	-	-	PUNCT
ejpam-5152	46	17	hausdorff	hausdorff	NOUN
ejpam-5152	46	18	metric	metric	ADJ
ejpam-5152	46	19	spaces	space	NOUN
ejpam-5152	46	20	.	.	PUNCT
ejpam-5152	47	1	a	a	DET
ejpam-5152	47	2	sequence	sequence	NOUN
ejpam-5152	47	3	(	(	PUNCT
ejpam-5152	47	4	fn	fn	NOUN
ejpam-5152	47	5	)	)	PUNCT
ejpam-5152	47	6	in	in	ADP
ejpam-5152	47	7	cbp(x	cbp(x	PROPN
ejpam-5152	47	8	)	)	PUNCT
ejpam-5152	47	9	is	be	AUX
ejpam-5152	47	10	said	say	VERB
ejpam-5152	47	11	to	to	ADP
ejpam-5152	47	12	a	a	DET
ejpam-5152	47	13	cauchy	cauchy	ADJ
ejpam-5152	47	14	sequence	sequence	NOUN
ejpam-5152	47	15	if	if	SCONJ
ejpam-5152	47	16	lim	lim	PROPN
ejpam-5152	47	17	n	n	CCONJ
ejpam-5152	47	18	,	,	PUNCT
ejpam-5152	47	19	m→∞	m→∞	NOUN
ejpam-5152	47	20	hp(fn	hp(fn	NOUN
ejpam-5152	47	21	,	,	PUNCT
ejpam-5152	47	22	fm	fm	NOUN
ejpam-5152	47	23	)	)	PUNCT
ejpam-5152	47	24	exists	exist	VERB
ejpam-5152	47	25	and	and	CCONJ
ejpam-5152	47	26	finite	finite	NOUN
ejpam-5152	47	27	.	.	PUNCT
ejpam-5152	48	1	sequence	sequence	NOUN
ejpam-5152	48	2	(	(	PUNCT
ejpam-5152	48	3	fn	fn	NOUN
ejpam-5152	48	4	)	)	PUNCT
ejpam-5152	48	5	is	be	AUX
ejpam-5152	48	6	cauchy	cauchy	ADJ
ejpam-5152	48	7	sequence	sequence	NOUN
ejpam-5152	48	8	if	if	SCONJ
ejpam-5152	48	9	the	the	DET
ejpam-5152	48	10	sequence	sequence	NOUN
ejpam-5152	48	11	hp(fn	hp(fn	NOUN
ejpam-5152	48	12	,	,	PUNCT
ejpam-5152	48	13	fm	fm	PROPN
ejpam-5152	48	14	)	)	PUNCT
ejpam-5152	48	15	tends	tend	VERB
ejpam-5152	48	16	to	to	ADP
ejpam-5152	48	17	some	some	DET
ejpam-5152	48	18	λ	λ	NOUN
ejpam-5152	48	19	∈	∈	NOUN
ejpam-5152	48	20	r	r	NOUN
ejpam-5152	48	21	as	as	ADP
ejpam-5152	48	22	n	n	NUM
ejpam-5152	48	23	,	,	PUNCT
ejpam-5152	48	24	m	m	VERB
ejpam-5152	48	25	approach	approach	NOUN
ejpam-5152	48	26	to	to	ADP
ejpam-5152	48	27	infinity	infinity	NOUN
ejpam-5152	48	28	,	,	PUNCT
ejpam-5152	48	29	that	that	ADV
ejpam-5152	48	30	is	is	ADV
ejpam-5152	48	31	,	,	PUNCT
ejpam-5152	48	32	limn	limn	ADJ
ejpam-5152	48	33	,	,	PUNCT
ejpam-5152	48	34	m→∞hp(fn	m→∞hp(fn	NOUN
ejpam-5152	48	35	,	,	PUNCT
ejpam-5152	48	36	fm	fm	NOUN
ejpam-5152	48	37	)	)	PUNCT
ejpam-5152	48	38	=	=	SYM
ejpam-5152	49	1	λ	λ	X
ejpam-5152	49	2	<	<	X
ejpam-5152	49	3	∞	∞	PROPN
ejpam-5152	49	4	,	,	PUNCT
ejpam-5152	49	5	i.e.	i.e.	X
ejpam-5152	49	6	for	for	ADP
ejpam-5152	49	7	each	each	DET
ejpam-5152	49	8	ε	ε	PROPN
ejpam-5152	49	9	>	>	X
ejpam-5152	49	10	0	0	PUNCT
ejpam-5152	50	1	there	there	PRON
ejpam-5152	50	2	exists	exist	VERB
ejpam-5152	50	3	n	n	PRON
ejpam-5152	50	4	∈	∈	PROPN
ejpam-5152	50	5	n	n	PRON
ejpam-5152	50	6	such	such	ADJ
ejpam-5152	50	7	that	that	DET
ejpam-5152	50	8	|hp(fn	|hp(fn	NOUN
ejpam-5152	50	9	,	,	PUNCT
ejpam-5152	50	10	fm)−	fm)−	VERB
ejpam-5152	50	11	λ|	λ|	PROPN
ejpam-5152	50	12	<	<	X
ejpam-5152	50	13	ε	ε	PROPN
ejpam-5152	50	14	,	,	PUNCT
ejpam-5152	50	15	for	for	ADP
ejpam-5152	50	16	all	all	DET
ejpam-5152	50	17	n	n	CCONJ
ejpam-5152	50	18	,	,	PUNCT
ejpam-5152	50	19	m	m	PROPN
ejpam-5152	50	20	≥	≥	NOUN
ejpam-5152	50	21	n.	n.	PROPN
ejpam-5152	50	22	furthermore	furthermore	ADV
ejpam-5152	50	23	,	,	PUNCT
ejpam-5152	50	24	lets	lets	AUX
ejpam-5152	50	25	consider	consider	VERB
ejpam-5152	50	26	the	the	DET
ejpam-5152	50	27	properties	property	NOUN
ejpam-5152	50	28	of	of	ADP
ejpam-5152	50	29	cauchy	cauchy	ADJ
ejpam-5152	50	30	sequence	sequence	NOUN
ejpam-5152	50	31	(	(	PUNCT
ejpam-5152	50	32	fn	fn	NOUN
ejpam-5152	50	33	)	)	PUNCT
ejpam-5152	50	34	in	in	ADP
ejpam-5152	50	35	(	(	PUNCT
ejpam-5152	50	36	cbp(x	cbp(x	PROPN
ejpam-5152	50	37	)	)	PUNCT
ejpam-5152	50	38	,	,	PUNCT
ejpam-5152	50	39	hp	hp	PROPN
ejpam-5152	50	40	)	)	PUNCT
ejpam-5152	50	41	.	.	PUNCT
ejpam-5152	51	1	a.	a.	PROPN
ejpam-5152	51	2	ekayanti	ekayanti	PROPN
ejpam-5152	51	3	et	et	PROPN
ejpam-5152	51	4	al	al	PROPN
ejpam-5152	51	5	.	.	PUNCT
ejpam-5152	51	6	/	/	SYM
ejpam-5152	51	7	eur	eur	PROPN
ejpam-5152	51	8	.	.	PUNCT
ejpam-5152	52	1	j.	j.	PROPN
ejpam-5152	52	2	pure	pure	PROPN
ejpam-5152	52	3	appl	appl	PROPN
ejpam-5152	52	4	.	.	PROPN
ejpam-5152	52	5	math	math	PROPN
ejpam-5152	52	6	,	,	PUNCT
ejpam-5152	52	7	17	17	NUM
ejpam-5152	52	8	(	(	PUNCT
ejpam-5152	52	9	2	2	NUM
ejpam-5152	52	10	)	)	PUNCT
ejpam-5152	52	11	(	(	PUNCT
ejpam-5152	52	12	2024	2024	NUM
ejpam-5152	52	13	)	)	PUNCT
ejpam-5152	52	14	,	,	PUNCT
ejpam-5152	52	15	996	996	NUM
ejpam-5152	52	16	-	-	SYM
ejpam-5152	52	17	1008	1008	NUM
ejpam-5152	52	18	998	998	NUM
ejpam-5152	52	19	theorem	theorem	NOUN
ejpam-5152	52	20	1	1	NUM
ejpam-5152	52	21	.	.	PUNCT
ejpam-5152	53	1	[	[	X
ejpam-5152	53	2	15	15	NUM
ejpam-5152	53	3	]	]	X
ejpam-5152	53	4	a	a	DET
ejpam-5152	53	5	sequence	sequence	NOUN
ejpam-5152	53	6	(	(	PUNCT
ejpam-5152	53	7	fn	fn	NOUN
ejpam-5152	53	8	)	)	PUNCT
ejpam-5152	53	9	in	in	ADP
ejpam-5152	53	10	p	p	ADJ
ejpam-5152	53	11	-	-	PUNCT
ejpam-5152	53	12	pompeiu	pompeiu	NOUN
ejpam-5152	53	13	-	-	PUNCT
ejpam-5152	53	14	hausdorff	hausdorff	NOUN
ejpam-5152	53	15	metric	metric	ADJ
ejpam-5152	53	16	spaces	space	NOUN
ejpam-5152	53	17	(	(	PUNCT
ejpam-5152	53	18	cbp(x	cbp(x	PROPN
ejpam-5152	53	19	)	)	PUNCT
ejpam-5152	53	20	,	,	PUNCT
ejpam-5152	53	21	hp	hp	PROPN
ejpam-5152	53	22	)	)	PUNCT
ejpam-5152	53	23	is	be	AUX
ejpam-5152	53	24	cauchy	cauchy	ADJ
ejpam-5152	53	25	if	if	SCONJ
ejpam-5152	53	26	and	and	CCONJ
ejpam-5152	53	27	only	only	ADV
ejpam-5152	53	28	if	if	SCONJ
ejpam-5152	53	29	for	for	ADP
ejpam-5152	53	30	all	all	DET
ejpam-5152	53	31	ε	ε	PROPN
ejpam-5152	53	32	>	>	X
ejpam-5152	53	33	0	0	PUNCT
ejpam-5152	54	1	there	there	PRON
ejpam-5152	54	2	exists	exist	VERB
ejpam-5152	54	3	n	n	PRON
ejpam-5152	54	4	∈	∈	PROPN
ejpam-5152	54	5	n	n	PRON
ejpam-5152	54	6	such	such	ADJ
ejpam-5152	54	7	that	that	DET
ejpam-5152	54	8	hp(fn	hp(fn	NOUN
ejpam-5152	54	9	,	,	PUNCT
ejpam-5152	54	10	fm)−hp(fm	fm)−hp(fm	NUM
ejpam-5152	54	11	,	,	PUNCT
ejpam-5152	54	12	fm	fm	NOUN
ejpam-5152	54	13	)	)	PUNCT
ejpam-5152	54	14	<	<	X
ejpam-5152	54	15	ε	ε	PROPN
ejpam-5152	54	16	,	,	PUNCT
ejpam-5152	54	17	for	for	ADP
ejpam-5152	54	18	every	every	DET
ejpam-5152	54	19	n	n	CCONJ
ejpam-5152	54	20	,	,	PUNCT
ejpam-5152	54	21	m	m	VERB
ejpam-5152	54	22	≥	≥	NOUN
ejpam-5152	54	23	n	n	NOUN
ejpam-5152	54	24	.	.	PUNCT
ejpam-5152	55	1	definition	definition	NOUN
ejpam-5152	55	2	3	3	NUM
ejpam-5152	55	3	.	.	PUNCT
ejpam-5152	56	1	[	[	X
ejpam-5152	56	2	15	15	NUM
ejpam-5152	56	3	]	]	X
ejpam-5152	56	4	a	a	DET
ejpam-5152	56	5	p	p	ADJ
ejpam-5152	56	6	-	-	PUNCT
ejpam-5152	56	7	pompeiu	pompeiu	NOUN
ejpam-5152	56	8	-	-	PUNCT
ejpam-5152	56	9	hausdorff	hausdorff	NOUN
ejpam-5152	56	10	metric	metric	ADJ
ejpam-5152	56	11	spaces	space	NOUN
ejpam-5152	56	12	(	(	PUNCT
ejpam-5152	56	13	cbp(x	cbp(x	PROPN
ejpam-5152	56	14	)	)	PUNCT
ejpam-5152	56	15	,	,	PUNCT
ejpam-5152	56	16	hp	hp	PROPN
ejpam-5152	56	17	)	)	PUNCT
ejpam-5152	56	18	is	be	AUX
ejpam-5152	56	19	called	call	VERB
ejpam-5152	56	20	complete	complete	ADJ
ejpam-5152	56	21	if	if	SCONJ
ejpam-5152	56	22	every	every	DET
ejpam-5152	56	23	cauchy	cauchy	ADJ
ejpam-5152	56	24	sequences	sequence	NOUN
ejpam-5152	56	25	fn	fn	PROPN
ejpam-5152	56	26	∈	∈	PROPN
ejpam-5152	56	27	cbp(x	cbp(x	PROPN
ejpam-5152	56	28	)	)	PUNCT
ejpam-5152	56	29	converges	converge	NOUN
ejpam-5152	56	30	to	to	ADP
ejpam-5152	56	31	f	f	PROPN
ejpam-5152	56	32	∈	∈	PROPN
ejpam-5152	56	33	cbp(x	cbp(x	PROPN
ejpam-5152	56	34	)	)	PUNCT
ejpam-5152	56	35	and	and	CCONJ
ejpam-5152	56	36	lim	lim	PROPN
ejpam-5152	56	37	n→∞	n→∞	NUM
ejpam-5152	56	38	hp(fn	hp(fn	PROPN
ejpam-5152	56	39	,	,	PUNCT
ejpam-5152	56	40	f	f	PROPN
ejpam-5152	56	41	)	)	PUNCT
ejpam-5152	57	1	=	=	PUNCT
ejpam-5152	57	2	hp(f	hp(f	PROPN
ejpam-5152	57	3	,	,	PUNCT
ejpam-5152	57	4	f	f	PROPN
ejpam-5152	57	5	)	)	PUNCT
ejpam-5152	57	6	.	.	PUNCT
ejpam-5152	58	1	one	one	NUM
ejpam-5152	58	2	of	of	ADP
ejpam-5152	58	3	the	the	DET
ejpam-5152	58	4	relationships	relationship	NOUN
ejpam-5152	58	5	between	between	ADP
ejpam-5152	58	6	the	the	DET
ejpam-5152	58	7	partial	partial	ADJ
ejpam-5152	58	8	metric	metric	ADJ
ejpam-5152	58	9	space	space	NOUN
ejpam-5152	58	10	(	(	PUNCT
ejpam-5152	58	11	x	x	X
ejpam-5152	58	12	,	,	PUNCT
ejpam-5152	58	13	p	p	NOUN
ejpam-5152	58	14	)	)	PUNCT
ejpam-5152	58	15	and	and	CCONJ
ejpam-5152	58	16	the	the	DET
ejpam-5152	58	17	p	p	NOUN
ejpam-5152	58	18	-	-	PUNCT
ejpam-5152	58	19	pompeiuhausdorff	pompeiuhausdorff	NOUN
ejpam-5152	58	20	metric	metric	ADJ
ejpam-5152	58	21	space	space	NOUN
ejpam-5152	58	22	(	(	PUNCT
ejpam-5152	58	23	cbp(x	cbp(x	PROPN
ejpam-5152	58	24	)	)	PUNCT
ejpam-5152	58	25	,	,	PUNCT
ejpam-5152	58	26	hp	hp	PROPN
ejpam-5152	58	27	)	)	PUNCT
ejpam-5152	58	28	can	can	AUX
ejpam-5152	58	29	be	be	AUX
ejpam-5152	58	30	seen	see	VERB
ejpam-5152	58	31	in	in	ADP
ejpam-5152	58	32	its	its	PRON
ejpam-5152	58	33	completeness	completeness	NOUN
ejpam-5152	58	34	.	.	PUNCT
ejpam-5152	59	1	this	this	PRON
ejpam-5152	59	2	is	be	AUX
ejpam-5152	59	3	shown	show	VERB
ejpam-5152	59	4	in	in	ADP
ejpam-5152	59	5	the	the	DET
ejpam-5152	59	6	following	follow	VERB
ejpam-5152	59	7	theorem	theorem	ADJ
ejpam-5152	59	8	2	2	NUM
ejpam-5152	59	9	.	.	PUNCT
ejpam-5152	59	10	theorem	theorem	NOUN
ejpam-5152	59	11	2	2	NUM
ejpam-5152	59	12	.	.	PUNCT
ejpam-5152	60	1	[	[	X
ejpam-5152	60	2	15	15	NUM
ejpam-5152	60	3	]	]	X
ejpam-5152	60	4	if	if	SCONJ
ejpam-5152	60	5	(	(	PUNCT
ejpam-5152	60	6	cbp(x	cbp(x	PROPN
ejpam-5152	60	7	)	)	PUNCT
ejpam-5152	60	8	,	,	PUNCT
ejpam-5152	60	9	hp	hp	PROPN
ejpam-5152	60	10	)	)	PUNCT
ejpam-5152	60	11	be	be	VERB
ejpam-5152	60	12	a	a	DET
ejpam-5152	60	13	complete	complete	ADJ
ejpam-5152	60	14	partial	partial	ADJ
ejpam-5152	60	15	metric	metric	ADJ
ejpam-5152	60	16	spaces	space	NOUN
ejpam-5152	60	17	then	then	ADV
ejpam-5152	60	18	(	(	PUNCT
ejpam-5152	60	19	cbp(x	cbp(x	PROPN
ejpam-5152	60	20	)	)	PUNCT
ejpam-5152	60	21	,	,	PUNCT
ejpam-5152	60	22	hp	hp	PROPN
ejpam-5152	60	23	)	)	PUNCT
ejpam-5152	60	24	is	be	AUX
ejpam-5152	60	25	complete	complete	ADJ
ejpam-5152	60	26	.	.	PUNCT
ejpam-5152	61	1	for	for	ADP
ejpam-5152	61	2	set	set	NOUN
ejpam-5152	61	3	-	-	PUNCT
ejpam-5152	61	4	valued	value	VERB
ejpam-5152	61	5	mapping	mapping	NOUN
ejpam-5152	61	6	f	f	X
ejpam-5152	61	7	:	:	PUNCT
ejpam-5152	61	8	x	x	X
ejpam-5152	61	9	→	→	SYM
ejpam-5152	61	10	cbp(x	cbp(x	PROPN
ejpam-5152	61	11	)	)	PUNCT
ejpam-5152	61	12	,	,	PUNCT
ejpam-5152	61	13	a	a	DET
ejpam-5152	61	14	point	point	NOUN
ejpam-5152	61	15	x	x	X
ejpam-5152	61	16	∈	∈	NOUN
ejpam-5152	61	17	x	x	PUNCT
ejpam-5152	61	18	is	be	AUX
ejpam-5152	61	19	called	call	VERB
ejpam-5152	61	20	a	a	DET
ejpam-5152	61	21	fixed	fix	VERB
ejpam-5152	61	22	point	point	NOUN
ejpam-5152	61	23	of	of	ADP
ejpam-5152	61	24	f	f	PROPN
ejpam-5152	61	25	if	if	SCONJ
ejpam-5152	61	26	x	x	PROPN
ejpam-5152	61	27	∈	∈	PROPN
ejpam-5152	61	28	f	f	X
ejpam-5152	61	29	(	(	PUNCT
ejpam-5152	61	30	x	x	NOUN
ejpam-5152	61	31	)	)	PUNCT
ejpam-5152	61	32	.	.	PUNCT
ejpam-5152	62	1	analogously	analogously	ADV
ejpam-5152	62	2	,	,	PUNCT
ejpam-5152	62	3	for	for	ADP
ejpam-5152	62	4	f	f	PROPN
ejpam-5152	62	5	and	and	CCONJ
ejpam-5152	62	6	g	g	PROPN
ejpam-5152	62	7	set	set	NOUN
ejpam-5152	62	8	-	-	PUNCT
ejpam-5152	62	9	valued	value	VERB
ejpam-5152	62	10	mappings	mapping	NOUN
ejpam-5152	62	11	from	from	ADP
ejpam-5152	62	12	x	x	PUNCT
ejpam-5152	62	13	into	into	ADP
ejpam-5152	62	14	cbp(x	cbp(x	PROPN
ejpam-5152	62	15	)	)	PUNCT
ejpam-5152	62	16	,	,	PUNCT
ejpam-5152	62	17	a	a	DET
ejpam-5152	62	18	point	point	NOUN
ejpam-5152	62	19	x	x	X
ejpam-5152	62	20	∈	∈	NOUN
ejpam-5152	62	21	x	x	PUNCT
ejpam-5152	62	22	is	be	AUX
ejpam-5152	62	23	called	call	VERB
ejpam-5152	62	24	as	as	ADP
ejpam-5152	62	25	a	a	DET
ejpam-5152	62	26	common	common	ADJ
ejpam-5152	62	27	fixed	fix	VERB
ejpam-5152	62	28	point	point	NOUN
ejpam-5152	62	29	of	of	ADP
ejpam-5152	62	30	f	f	PROPN
ejpam-5152	62	31	and	and	CCONJ
ejpam-5152	62	32	g	g	PROPN
ejpam-5152	62	33	if	if	SCONJ
ejpam-5152	62	34	x	x	PROPN
ejpam-5152	62	35	∈	∈	PROPN
ejpam-5152	62	36	f	f	X
ejpam-5152	62	37	(	(	PUNCT
ejpam-5152	62	38	x	x	NOUN
ejpam-5152	62	39	)	)	PUNCT
ejpam-5152	62	40	and	and	CCONJ
ejpam-5152	62	41	x	x	PUNCT
ejpam-5152	62	42	∈	∈	PROPN
ejpam-5152	62	43	g(x	g(x	NOUN
ejpam-5152	62	44	)	)	PUNCT
ejpam-5152	62	45	.	.	PUNCT
ejpam-5152	63	1	3	3	X
ejpam-5152	63	2	.	.	X
ejpam-5152	63	3	main	main	ADJ
ejpam-5152	63	4	results	result	NOUN
ejpam-5152	63	5	in	in	ADP
ejpam-5152	63	6	the	the	DET
ejpam-5152	63	7	following	follow	VERB
ejpam-5152	63	8	discussion	discussion	NOUN
ejpam-5152	63	9	,	,	PUNCT
ejpam-5152	63	10	we	we	PRON
ejpam-5152	63	11	assume	assume	VERB
ejpam-5152	63	12	that	that	SCONJ
ejpam-5152	63	13	(	(	PUNCT
ejpam-5152	63	14	x	x	X
ejpam-5152	63	15	,	,	PUNCT
ejpam-5152	63	16	p	p	NOUN
ejpam-5152	63	17	)	)	PUNCT
ejpam-5152	63	18	is	be	AUX
ejpam-5152	63	19	a	a	DET
ejpam-5152	63	20	complete	complete	ADJ
ejpam-5152	63	21	partial	partial	ADJ
ejpam-5152	63	22	metric	metric	ADJ
ejpam-5152	63	23	space	space	NOUN
ejpam-5152	63	24	.	.	PUNCT
ejpam-5152	64	1	theorem	theorem	NOUN
ejpam-5152	64	2	3	3	X
ejpam-5152	64	3	.	.	PUNCT
ejpam-5152	65	1	let	let	AUX
ejpam-5152	65	2	(	(	PUNCT
ejpam-5152	65	3	cbp(x	cbp(x	PROPN
ejpam-5152	65	4	)	)	PUNCT
ejpam-5152	65	5	,	,	PUNCT
ejpam-5152	65	6	hp	hp	PROPN
ejpam-5152	65	7	)	)	PUNCT
ejpam-5152	65	8	be	be	VERB
ejpam-5152	65	9	a	a	DET
ejpam-5152	65	10	p	p	ADJ
ejpam-5152	65	11	-	-	PUNCT
ejpam-5152	65	12	pompeiu	pompeiu	NOUN
ejpam-5152	65	13	-	-	PUNCT
ejpam-5152	65	14	hausdorff	hausdorff	NOUN
ejpam-5152	65	15	metric	metric	ADJ
ejpam-5152	65	16	spaces	space	NOUN
ejpam-5152	65	17	.	.	PUNCT
ejpam-5152	66	1	suppose	suppose	VERB
ejpam-5152	66	2	that	that	SCONJ
ejpam-5152	66	3	fn	fn	NOUN
ejpam-5152	66	4	,	,	PUNCT
ejpam-5152	66	5	gn	gn	INTJ
ejpam-5152	66	6	:	:	PUNCT
ejpam-5152	66	7	x	x	X
ejpam-5152	66	8	→	→	SYM
ejpam-5152	66	9	cbp(x	cbp(x	PROPN
ejpam-5152	66	10	)	)	PUNCT
ejpam-5152	66	11	,	,	PUNCT
ejpam-5152	66	12	n	n	PRON
ejpam-5152	66	13	∈	∈	PROPN
ejpam-5152	66	14	n	n	PRON
ejpam-5152	66	15	be	be	VERB
ejpam-5152	66	16	sequence	sequence	NOUN
ejpam-5152	66	17	of	of	ADP
ejpam-5152	66	18	set	set	NOUN
ejpam-5152	66	19	-	-	PUNCT
ejpam-5152	66	20	valued	value	VERB
ejpam-5152	66	21	mappings	mapping	NOUN
ejpam-5152	66	22	on	on	ADP
ejpam-5152	66	23	cbp(x	cbp(x	PROPN
ejpam-5152	66	24	)	)	PUNCT
ejpam-5152	66	25	,	,	PUNCT
ejpam-5152	66	26	there	there	PRON
ejpam-5152	66	27	exists	exist	VERB
ejpam-5152	66	28	κ	κ	X
ejpam-5152	66	29	where	where	SCONJ
ejpam-5152	66	30	0	0	NUM
ejpam-5152	66	31	≤	≤	NUM
ejpam-5152	66	32	κ	κ	X
ejpam-5152	66	33	<	<	X
ejpam-5152	66	34	1	1	NUM
ejpam-5152	66	35	such	such	ADJ
ejpam-5152	66	36	that	that	PRON
ejpam-5152	66	37	hp(fm(x	hp(fm(x	NUM
ejpam-5152	66	38	)	)	PUNCT
ejpam-5152	66	39	,	,	PUNCT
ejpam-5152	66	40	gn(y	gn(y	NOUN
ejpam-5152	66	41	)	)	PUNCT
ejpam-5152	66	42	)	)	PUNCT
ejpam-5152	66	43	≤	≤	NUM
ejpam-5152	66	44	κmax	κmax	VERB
ejpam-5152	66	45	{	{	PUNCT
ejpam-5152	66	46	p(x	p(x	PROPN
ejpam-5152	66	47	,	,	PUNCT
ejpam-5152	66	48	y	y	NOUN
ejpam-5152	66	49	)	)	PUNCT
ejpam-5152	66	50	,	,	PUNCT
ejpam-5152	66	51	p(x	p(x	PROPN
ejpam-5152	66	52	,	,	PUNCT
ejpam-5152	66	53	fm(x	fm(x	NUM
ejpam-5152	66	54	)	)	PUNCT
ejpam-5152	66	55	)	)	PUNCT
ejpam-5152	66	56	,	,	PUNCT
ejpam-5152	66	57	p(y	p(y	PROPN
ejpam-5152	66	58	,	,	PUNCT
ejpam-5152	66	59	gn(y	gn(y	NOUN
ejpam-5152	66	60	)	)	PUNCT
ejpam-5152	66	61	)	)	PUNCT
ejpam-5152	66	62	,	,	PUNCT
ejpam-5152	66	63	1	1	NUM
ejpam-5152	66	64	2	2	NUM
ejpam-5152	66	65	(	(	PUNCT
ejpam-5152	66	66	p(x	p(x	PROPN
ejpam-5152	66	67	,	,	PUNCT
ejpam-5152	66	68	gn(y	gn(y	NOUN
ejpam-5152	66	69	)	)	PUNCT
ejpam-5152	66	70	)	)	PUNCT
ejpam-5152	67	1	+	+	CCONJ
ejpam-5152	67	2	p(y	p(y	NOUN
ejpam-5152	67	3	,	,	PUNCT
ejpam-5152	67	4	fm(x	fm(x	NUM
ejpam-5152	67	5	)	)	PUNCT
ejpam-5152	67	6	)	)	PUNCT
ejpam-5152	67	7	)	)	PUNCT
ejpam-5152	67	8	}	}	PUNCT
ejpam-5152	67	9	,	,	PUNCT
ejpam-5152	67	10	for	for	ADP
ejpam-5152	67	11	each	each	DET
ejpam-5152	67	12	m	m	NOUN
ejpam-5152	67	13	,	,	PUNCT
ejpam-5152	67	14	n	n	PROPN
ejpam-5152	67	15	∈	∈	PROPN
ejpam-5152	67	16	n	n	NOUN
ejpam-5152	67	17	and	and	CCONJ
ejpam-5152	67	18	x	x	X
ejpam-5152	67	19	,	,	PUNCT
ejpam-5152	67	20	y	y	PROPN
ejpam-5152	67	21	∈	∈	PROPN
ejpam-5152	67	22	x	x	X
ejpam-5152	67	23	,	,	PUNCT
ejpam-5152	67	24	then	then	ADV
ejpam-5152	67	25	(	(	PUNCT
ejpam-5152	67	26	fn	fn	NOUN
ejpam-5152	67	27	)	)	PUNCT
ejpam-5152	67	28	and	and	CCONJ
ejpam-5152	67	29	(	(	PUNCT
ejpam-5152	67	30	gn	gn	X
ejpam-5152	67	31	)	)	PUNCT
ejpam-5152	67	32	have	have	VERB
ejpam-5152	67	33	a	a	DET
ejpam-5152	67	34	common	common	ADJ
ejpam-5152	67	35	fixed	fix	VERB
ejpam-5152	67	36	point	point	NOUN
ejpam-5152	67	37	,	,	PUNCT
ejpam-5152	67	38	i.e.	i.e.	X
ejpam-5152	67	39	there	there	PRON
ejpam-5152	67	40	exist	exist	VERB
ejpam-5152	67	41	a	a	DET
ejpam-5152	67	42	point	point	NOUN
ejpam-5152	67	43	x	x	SYM
ejpam-5152	67	44	∈	∈	NOUN
ejpam-5152	67	45	x	x	X
ejpam-5152	68	1	such	such	ADJ
ejpam-5152	68	2	that	that	SCONJ
ejpam-5152	68	3	x	x	SYM
ejpam-5152	68	4	∈	∈	NOUN
ejpam-5152	68	5	fm(x	fm(x	PRON
ejpam-5152	68	6	)	)	PUNCT
ejpam-5152	68	7	and	and	CCONJ
ejpam-5152	68	8	x	x	PUNCT
ejpam-5152	68	9	∈	∈	PROPN
ejpam-5152	68	10	gn(x	gn(x	PUNCT
ejpam-5152	68	11	)	)	PUNCT
ejpam-5152	68	12	for	for	ADP
ejpam-5152	68	13	each	each	DET
ejpam-5152	68	14	m	m	NOUN
ejpam-5152	68	15	,	,	PUNCT
ejpam-5152	68	16	n	n	PROPN
ejpam-5152	68	17	∈	∈	PROPN
ejpam-5152	68	18	n.	n.	NOUN
ejpam-5152	68	19	proof	proof	NOUN
ejpam-5152	68	20	.	.	PUNCT
ejpam-5152	69	1	let	let	VERB
ejpam-5152	69	2	we	we	PRON
ejpam-5152	69	3	consider	consider	VERB
ejpam-5152	69	4	that	that	DET
ejpam-5152	69	5	0	0	NUM
ejpam-5152	69	6	≤	≤	NUM
ejpam-5152	69	7	κ	κ	X
ejpam-5152	69	8	<	<	X
ejpam-5152	69	9	1	1	NUM
ejpam-5152	69	10	.	.	PUNCT
ejpam-5152	70	1	for	for	ADP
ejpam-5152	70	2	the	the	DET
ejpam-5152	70	3	first	first	ADJ
ejpam-5152	70	4	we	we	PRON
ejpam-5152	70	5	assume	assume	VERB
ejpam-5152	70	6	that	that	SCONJ
ejpam-5152	70	7	κ	κ	PROPN
ejpam-5152	70	8	=	=	NOUN
ejpam-5152	70	9	0	0	PROPN
ejpam-5152	70	10	.	.	PUNCT
ejpam-5152	70	11	suppose	suppose	VERB
ejpam-5152	70	12	that	that	SCONJ
ejpam-5152	70	13	x0	x0	PROPN
ejpam-5152	70	14	∈	∈	PROPN
ejpam-5152	70	15	x	x	X
ejpam-5152	70	16	and	and	CCONJ
ejpam-5152	70	17	x1	x1	PROPN
ejpam-5152	70	18	∈	∈	PROPN
ejpam-5152	70	19	f1(x0	f1(x0	PROPN
ejpam-5152	70	20	)	)	PUNCT
ejpam-5152	70	21	,	,	PUNCT
ejpam-5152	70	22	then	then	ADV
ejpam-5152	70	23	for	for	ADP
ejpam-5152	70	24	all	all	DET
ejpam-5152	70	25	n	n	PRON
ejpam-5152	70	26	∈	∈	NOUN
ejpam-5152	70	27	n	n	CCONJ
ejpam-5152	70	28	we	we	PRON
ejpam-5152	70	29	have	have	VERB
ejpam-5152	70	30	p(x1	p(x1	ADJ
ejpam-5152	70	31	,	,	PUNCT
ejpam-5152	70	32	gn(x1	gn(x1	NOUN
ejpam-5152	70	33	)	)	PUNCT
ejpam-5152	70	34	≤	≤	NUM
ejpam-5152	70	35	hp(f1(x0	hp(f1(x0	NOUN
ejpam-5152	70	36	)	)	PUNCT
ejpam-5152	70	37	,	,	PUNCT
ejpam-5152	70	38	gn(x1	gn(x1	NOUN
ejpam-5152	70	39	)	)	PUNCT
ejpam-5152	70	40	)	)	PUNCT
ejpam-5152	71	1	=	=	PUNCT
ejpam-5152	71	2	0	0	X
ejpam-5152	71	3	.	.	PUNCT
ejpam-5152	72	1	it	it	PRON
ejpam-5152	72	2	means	mean	VERB
ejpam-5152	72	3	p(x1	p(x1	ADJ
ejpam-5152	72	4	,	,	PUNCT
ejpam-5152	72	5	gn(x1	gn(x1	NOUN
ejpam-5152	72	6	)	)	PUNCT
ejpam-5152	72	7	)	)	PUNCT
ejpam-5152	73	1	=	=	PUNCT
ejpam-5152	73	2	0	0	X
ejpam-5152	73	3	.	.	PUNCT
ejpam-5152	74	1	since	since	SCONJ
ejpam-5152	74	2	gn	gn	PROPN
ejpam-5152	74	3	are	be	AUX
ejpam-5152	74	4	closed	close	VERB
ejpam-5152	74	5	for	for	ADP
ejpam-5152	74	6	each	each	DET
ejpam-5152	74	7	n	n	NOUN
ejpam-5152	74	8	then	then	ADV
ejpam-5152	74	9	x1	x1	PROPN
ejpam-5152	74	10	∈	∈	PROPN
ejpam-5152	74	11	gn(x1	gn(x1	NOUN
ejpam-5152	74	12	)	)	PUNCT
ejpam-5152	74	13	.	.	PUNCT
ejpam-5152	75	1	in	in	ADP
ejpam-5152	75	2	the	the	DET
ejpam-5152	75	3	similar	similar	ADJ
ejpam-5152	75	4	way	way	NOUN
ejpam-5152	75	5	,	,	PUNCT
ejpam-5152	75	6	we	we	PRON
ejpam-5152	75	7	can	can	AUX
ejpam-5152	75	8	obtain	obtain	VERB
ejpam-5152	75	9	that	that	PRON
ejpam-5152	75	10	for	for	ADP
ejpam-5152	75	11	x0	x0	PROPN
ejpam-5152	75	12	∈	∈	PROPN
ejpam-5152	75	13	x	x	X
ejpam-5152	75	14	and	and	CCONJ
ejpam-5152	75	15	x1	x1	PROPN
ejpam-5152	75	16	∈	∈	PROPN
ejpam-5152	75	17	g1(x0	g1(x0	NOUN
ejpam-5152	75	18	)	)	PUNCT
ejpam-5152	75	19	,	,	PUNCT
ejpam-5152	75	20	then	then	ADV
ejpam-5152	75	21	for	for	ADP
ejpam-5152	75	22	all	all	DET
ejpam-5152	75	23	n	n	PRON
ejpam-5152	75	24	∈	∈	NOUN
ejpam-5152	75	25	n	n	CCONJ
ejpam-5152	75	26	we	we	PRON
ejpam-5152	75	27	have	have	VERB
ejpam-5152	75	28	p(x1	p(x1	NOUN
ejpam-5152	75	29	,	,	PUNCT
ejpam-5152	75	30	fn(x1	fn(x1	NOUN
ejpam-5152	75	31	)	)	PUNCT
ejpam-5152	75	32	)	)	PUNCT
ejpam-5152	76	1	≤	≤	NUM
ejpam-5152	76	2	hp(g1(x0	hp(g1(x0	NOUN
ejpam-5152	76	3	)	)	PUNCT
ejpam-5152	76	4	,	,	PUNCT
ejpam-5152	76	5	fn(x1	fn(x1	NOUN
ejpam-5152	76	6	)	)	PUNCT
ejpam-5152	76	7	)	)	PUNCT
ejpam-5152	77	1	=	=	SYM
ejpam-5152	77	2	0	0	NUM
ejpam-5152	77	3	,	,	PUNCT
ejpam-5152	77	4	a.	a.	NOUN
ejpam-5152	77	5	ekayanti	ekayanti	PROPN
ejpam-5152	77	6	et	et	PROPN
ejpam-5152	77	7	al	al	PROPN
ejpam-5152	77	8	.	.	PUNCT
ejpam-5152	77	9	/	/	SYM
ejpam-5152	77	10	eur	eur	PROPN
ejpam-5152	77	11	.	.	PUNCT
ejpam-5152	78	1	j.	j.	PROPN
ejpam-5152	78	2	pure	pure	PROPN
ejpam-5152	78	3	appl	appl	PROPN
ejpam-5152	78	4	.	.	PROPN
ejpam-5152	78	5	math	math	PROPN
ejpam-5152	78	6	,	,	PUNCT
ejpam-5152	78	7	17	17	NUM
ejpam-5152	78	8	(	(	PUNCT
ejpam-5152	78	9	2	2	NUM
ejpam-5152	78	10	)	)	PUNCT
ejpam-5152	78	11	(	(	PUNCT
ejpam-5152	78	12	2024	2024	NUM
ejpam-5152	78	13	)	)	PUNCT
ejpam-5152	78	14	,	,	PUNCT
ejpam-5152	78	15	996	996	NUM
ejpam-5152	78	16	-	-	SYM
ejpam-5152	78	17	1008	1008	NUM
ejpam-5152	78	18	999	999	NUM
ejpam-5152	78	19	i.e.	i.e.	X
ejpam-5152	78	20	,	,	PUNCT
ejpam-5152	78	21	p(x1	p(x1	ADJ
ejpam-5152	78	22	,	,	PUNCT
ejpam-5152	78	23	fn(x1	fn(x1	NOUN
ejpam-5152	78	24	)	)	PUNCT
ejpam-5152	78	25	)	)	PUNCT
ejpam-5152	79	1	=	=	PUNCT
ejpam-5152	79	2	0	0	NUM
ejpam-5152	79	3	,	,	PUNCT
ejpam-5152	79	4	then	then	ADV
ejpam-5152	79	5	x1	x1	PROPN
ejpam-5152	79	6	∈	∈	PROPN
ejpam-5152	79	7	fn(x1	fn(x1	PROPN
ejpam-5152	79	8	)	)	PUNCT
ejpam-5152	79	9	.	.	PUNCT
ejpam-5152	80	1	from	from	ADP
ejpam-5152	80	2	this	this	DET
ejpam-5152	80	3	result	result	NOUN
ejpam-5152	80	4	,	,	PUNCT
ejpam-5152	80	5	it	it	PRON
ejpam-5152	80	6	can	can	AUX
ejpam-5152	80	7	be	be	AUX
ejpam-5152	80	8	seen	see	VERB
ejpam-5152	80	9	that	that	SCONJ
ejpam-5152	80	10	x1	x1	PROPN
ejpam-5152	80	11	is	be	AUX
ejpam-5152	80	12	the	the	DET
ejpam-5152	80	13	common	common	ADJ
ejpam-5152	80	14	fixed	fix	VERB
ejpam-5152	80	15	point	point	NOUN
ejpam-5152	80	16	of	of	ADP
ejpam-5152	80	17	fn	fn	PROPN
ejpam-5152	80	18	and	and	CCONJ
ejpam-5152	80	19	gn	gn	PROPN
ejpam-5152	80	20	.	.	PROPN
ejpam-5152	81	1	next	next	ADV
ejpam-5152	81	2	we	we	PRON
ejpam-5152	81	3	assume	assume	VERB
ejpam-5152	81	4	that	that	SCONJ
ejpam-5152	81	5	κ	κ	PROPN
ejpam-5152	81	6	̸=	̸=	PROPN
ejpam-5152	81	7	0	0	NUM
ejpam-5152	81	8	.	.	PUNCT
ejpam-5152	82	1	suppose	suppose	VERB
ejpam-5152	82	2	that	that	SCONJ
ejpam-5152	82	3	x0	x0	PROPN
ejpam-5152	82	4	∈	∈	PROPN
ejpam-5152	82	5	x	x	X
ejpam-5152	82	6	and	and	CCONJ
ejpam-5152	82	7	x1	x1	PROPN
ejpam-5152	82	8	∈	∈	PROPN
ejpam-5152	82	9	f1(x0	f1(x0	PROPN
ejpam-5152	82	10	)	)	PUNCT
ejpam-5152	82	11	.	.	PUNCT
ejpam-5152	83	1	furthermore	furthermore	ADV
ejpam-5152	83	2	,	,	PUNCT
ejpam-5152	83	3	define	define	VERB
ejpam-5152	83	4	the	the	DET
ejpam-5152	83	5	sequence	sequence	NOUN
ejpam-5152	83	6	(	(	PUNCT
ejpam-5152	83	7	xn	xn	PROPN
ejpam-5152	83	8	)	)	PUNCT
ejpam-5152	83	9	where	where	SCONJ
ejpam-5152	83	10	x2n	x2n	PROPN
ejpam-5152	83	11	∈	∈	PROPN
ejpam-5152	83	12	gn(x2n−1	gn(x2n−1	VERB
ejpam-5152	83	13	)	)	PUNCT
ejpam-5152	83	14	and	and	CCONJ
ejpam-5152	83	15	x2n−1	x2n−1	PROPN
ejpam-5152	83	16	∈	∈	PROPN
ejpam-5152	83	17	fn(x2n−2	fn(x2n−2	PROPN
ejpam-5152	83	18	)	)	PUNCT
ejpam-5152	83	19	are	be	AUX
ejpam-5152	83	20	such	such	ADJ
ejpam-5152	83	21	that	that	DET
ejpam-5152	83	22	p(x2n−1	p(x2n−1	NOUN
ejpam-5152	83	23	,	,	PUNCT
ejpam-5152	83	24	x2n	x2n	NOUN
ejpam-5152	83	25	)	)	PUNCT
ejpam-5152	83	26	≤	≤	NUM
ejpam-5152	83	27	1√	1√	NUM
ejpam-5152	83	28	κ	κ	PROPN
ejpam-5152	83	29	hp(fn(x2n−2	hp(fn(x2n−2	NUM
ejpam-5152	83	30	)	)	PUNCT
ejpam-5152	83	31	,	,	PUNCT
ejpam-5152	83	32	gn(x2n−1	gn(x2n−1	X
ejpam-5152	83	33	)	)	PUNCT
ejpam-5152	83	34	)	)	PUNCT
ejpam-5152	83	35	p(x2n	p(x2n	PROPN
ejpam-5152	83	36	,	,	PUNCT
ejpam-5152	83	37	x2n+1	x2n+1	PROPN
ejpam-5152	83	38	)	)	PUNCT
ejpam-5152	83	39	≤	≤	NUM
ejpam-5152	83	40	1√	1√	NUM
ejpam-5152	83	41	κ	κ	ADP
ejpam-5152	83	42	hp(fn(x2n	hp(fn(x2n	NOUN
ejpam-5152	83	43	)	)	PUNCT
ejpam-5152	83	44	,	,	PUNCT
ejpam-5152	83	45	gn(x2n−1	gn(x2n−1	NOUN
ejpam-5152	83	46	)	)	PUNCT
ejpam-5152	83	47	)	)	PUNCT
ejpam-5152	83	48	,	,	PUNCT
ejpam-5152	83	49	for	for	ADP
ejpam-5152	83	50	n	n	NOUN
ejpam-5152	83	51	=	=	SYM
ejpam-5152	83	52	1	1	NUM
ejpam-5152	83	53	,	,	PUNCT
ejpam-5152	83	54	2	2	NUM
ejpam-5152	83	55	,	,	PUNCT
ejpam-5152	83	56	3	3	NUM
ejpam-5152	83	57	,	,	PUNCT
ejpam-5152	83	58	.	.	PUNCT
ejpam-5152	83	59	.	.	PUNCT
ejpam-5152	83	60	.	.	PUNCT
ejpam-5152	84	1	.	.	PUNCT
ejpam-5152	85	1	suppose	suppose	VERB
ejpam-5152	85	2	that	that	SCONJ
ejpam-5152	85	3	xn	xn	PROPN
ejpam-5152	85	4	̸=	̸=	PROPN
ejpam-5152	85	5	xn+1	xn+1	PROPN
ejpam-5152	85	6	for	for	ADP
ejpam-5152	85	7	all	all	DET
ejpam-5152	85	8	n	n	PRON
ejpam-5152	85	9	∈	∈	PROPN
ejpam-5152	85	10	n.	n.	NOUN
ejpam-5152	85	11	for	for	ADP
ejpam-5152	85	12	n	n	PRON
ejpam-5152	85	13	being	be	AUX
ejpam-5152	85	14	even	even	ADV
ejpam-5152	85	15	,	,	PUNCT
ejpam-5152	85	16	we	we	PRON
ejpam-5152	85	17	have	have	VERB
ejpam-5152	85	18	x2n	x2n	PROPN
ejpam-5152	85	19	∈	∈	PROPN
ejpam-5152	85	20	fn+1(x2n	fn+1(x2n	NOUN
ejpam-5152	85	21	)	)	PUNCT
ejpam-5152	85	22	thus	thus	ADV
ejpam-5152	85	23	for	for	ADP
ejpam-5152	85	24	each	each	DET
ejpam-5152	85	25	m	m	PROPN
ejpam-5152	85	26	∈	∈	PROPN
ejpam-5152	85	27	n	n	PRON
ejpam-5152	85	28	p(x2n	p(x2n	PROPN
ejpam-5152	85	29	,	,	PUNCT
ejpam-5152	85	30	gm(x2n	gm(x2n	NOUN
ejpam-5152	85	31	)	)	PUNCT
ejpam-5152	85	32	)	)	PUNCT
ejpam-5152	86	1	≤	≤	PUNCT
ejpam-5152	87	1	hp(fn+1(x2n	hp(fn+1(x2n	PROPN
ejpam-5152	87	2	)	)	PUNCT
ejpam-5152	87	3	,	,	PUNCT
ejpam-5152	87	4	gm(x2n	gm(x2n	NOUN
ejpam-5152	87	5	)	)	PUNCT
ejpam-5152	87	6	)	)	PUNCT
ejpam-5152	88	1	≤	≤	NUM
ejpam-5152	88	2	κmax{p(x2n	κmax{p(x2n	PROPN
ejpam-5152	88	3	,	,	PUNCT
ejpam-5152	88	4	x2n	x2n	PROPN
ejpam-5152	88	5	)	)	PUNCT
ejpam-5152	88	6	,	,	PUNCT
ejpam-5152	88	7	p(x2n	p(x2n	NOUN
ejpam-5152	88	8	,	,	PUNCT
ejpam-5152	88	9	fn+1(x2n	fn+1(x2n	NOUN
ejpam-5152	88	10	)	)	PUNCT
ejpam-5152	88	11	)	)	PUNCT
ejpam-5152	88	12	,	,	PUNCT
ejpam-5152	88	13	p(x2n	p(x2n	NOUN
ejpam-5152	88	14	,	,	PUNCT
ejpam-5152	88	15	gm(x2n	gm(x2n	NOUN
ejpam-5152	88	16	)	)	PUNCT
ejpam-5152	88	17	)	)	PUNCT
ejpam-5152	88	18	,	,	PUNCT
ejpam-5152	88	19	1	1	NUM
ejpam-5152	88	20	2(p(x2n	2(p(x2n	NUM
ejpam-5152	88	21	,	,	PUNCT
ejpam-5152	88	22	fn+1(x2n	fn+1(x2n	NOUN
ejpam-5152	88	23	)	)	PUNCT
ejpam-5152	88	24	)	)	PUNCT
ejpam-5152	89	1	+	+	PUNCT
ejpam-5152	89	2	p(x2n	p(x2n	NOUN
ejpam-5152	89	3	,	,	PUNCT
ejpam-5152	89	4	gm(x2n	gm(x2n	NOUN
ejpam-5152	89	5	)	)	PUNCT
ejpam-5152	89	6	)	)	PUNCT
ejpam-5152	89	7	}	}	PUNCT
ejpam-5152	89	8	≤	≤	NUM
ejpam-5152	89	9	κp(x2n	κp(x2n	PROPN
ejpam-5152	89	10	,	,	PUNCT
ejpam-5152	89	11	gm(x2n	gm(x2n	NOUN
ejpam-5152	89	12	)	)	PUNCT
ejpam-5152	89	13	)	)	PUNCT
ejpam-5152	89	14	.	.	PUNCT
ejpam-5152	90	1	since	since	SCONJ
ejpam-5152	90	2	0	0	NUM
ejpam-5152	90	3	<	<	X
ejpam-5152	90	4	κ	κ	X
ejpam-5152	90	5	<	<	X
ejpam-5152	90	6	1	1	NUM
ejpam-5152	90	7	then	then	ADV
ejpam-5152	90	8	p(x2n	p(x2n	NOUN
ejpam-5152	90	9	,	,	PUNCT
ejpam-5152	90	10	gm(x2n	gm(x2n	NOUN
ejpam-5152	90	11	)	)	PUNCT
ejpam-5152	90	12	=	=	SYM
ejpam-5152	91	1	0	0	X
ejpam-5152	91	2	.	.	PUNCT
ejpam-5152	92	1	therefore	therefore	ADV
ejpam-5152	92	2	,	,	PUNCT
ejpam-5152	92	3	we	we	PRON
ejpam-5152	92	4	have	have	VERB
ejpam-5152	92	5	x2n	x2n	PROPN
ejpam-5152	92	6	∈	∈	PROPN
ejpam-5152	92	7	gm(x2n	gm(x2n	NOUN
ejpam-5152	92	8	)	)	PUNCT
ejpam-5152	92	9	for	for	ADP
ejpam-5152	92	10	each	each	DET
ejpam-5152	92	11	m	m	PROPN
ejpam-5152	92	12	∈	∈	PROPN
ejpam-5152	92	13	n.	n.	NOUN
ejpam-5152	92	14	similarly	similarly	ADV
ejpam-5152	92	15	,	,	PUNCT
ejpam-5152	92	16	for	for	ADP
ejpam-5152	92	17	n	n	PRON
ejpam-5152	92	18	being	be	AUX
ejpam-5152	92	19	odd	odd	ADJ
ejpam-5152	92	20	numbers	number	NOUN
ejpam-5152	92	21	,	,	PUNCT
ejpam-5152	92	22	we	we	PRON
ejpam-5152	92	23	have	have	VERB
ejpam-5152	92	24	x2n+1	x2n+1	PROPN
ejpam-5152	92	25	∈	∈	PROPN
ejpam-5152	92	26	gn+1(x2n+1	gn+1(x2n+1	PROPN
ejpam-5152	92	27	)	)	PUNCT
ejpam-5152	92	28	,	,	PUNCT
ejpam-5152	92	29	and	and	CCONJ
ejpam-5152	92	30	for	for	ADP
ejpam-5152	92	31	every	every	DET
ejpam-5152	92	32	m	m	NOUN
ejpam-5152	92	33	implies	imply	VERB
ejpam-5152	92	34	p(x2n+1	p(x2n+1	ADV
ejpam-5152	92	35	,	,	PUNCT
ejpam-5152	92	36	fm(x2n+1	fm(x2n+1	NOUN
ejpam-5152	92	37	)	)	PUNCT
ejpam-5152	92	38	)	)	PUNCT
ejpam-5152	93	1	≤	≤	NUM
ejpam-5152	93	2	hp(gn+1(x2n+1	hp(gn+1(x2n+1	NOUN
ejpam-5152	93	3	)	)	PUNCT
ejpam-5152	93	4	,	,	PUNCT
ejpam-5152	93	5	fm(x2n+1	fm(x2n+1	NOUN
ejpam-5152	93	6	)	)	PUNCT
ejpam-5152	93	7	)	)	PUNCT
ejpam-5152	93	8	≤	≤	NUM
ejpam-5152	93	9	κmax{p(x2n+1	κmax{p(x2n+1	X
ejpam-5152	93	10	,	,	PUNCT
ejpam-5152	93	11	x2n+1	x2n+1	PROPN
ejpam-5152	93	12	)	)	PUNCT
ejpam-5152	93	13	,	,	PUNCT
ejpam-5152	93	14	p(x2n+1	p(x2n+1	PROPN
ejpam-5152	93	15	,	,	PUNCT
ejpam-5152	93	16	gn+1(x2n+1	gn+1(x2n+1	PROPN
ejpam-5152	93	17	)	)	PUNCT
ejpam-5152	93	18	)	)	PUNCT
ejpam-5152	93	19	,	,	PUNCT
ejpam-5152	93	20	p(x2n+1	p(x2n+1	PROPN
ejpam-5152	93	21	,	,	PUNCT
ejpam-5152	93	22	fm(x2n+1	fm(x2n+1	NOUN
ejpam-5152	93	23	)	)	PUNCT
ejpam-5152	93	24	)	)	PUNCT
ejpam-5152	93	25	,	,	PUNCT
ejpam-5152	93	26	1	1	NUM
ejpam-5152	93	27	2(p(x2n+1	2(p(x2n+1	NUM
ejpam-5152	93	28	,	,	PUNCT
ejpam-5152	93	29	fm(x2n+1	fm(x2n+1	NOUN
ejpam-5152	93	30	)	)	PUNCT
ejpam-5152	93	31	)	)	PUNCT
ejpam-5152	94	1	+	+	PUNCT
ejpam-5152	94	2	p(x2n+1	p(x2n+1	ADJ
ejpam-5152	94	3	,	,	PUNCT
ejpam-5152	94	4	gn+1(x2n+1	gn+1(x2n+1	PROPN
ejpam-5152	94	5	)	)	PUNCT
ejpam-5152	94	6	)	)	PUNCT
ejpam-5152	94	7	}	}	PUNCT
ejpam-5152	94	8	≤	≤	NUM
ejpam-5152	94	9	κp(x2n+1	κp(x2n+1	NOUN
ejpam-5152	94	10	,	,	PUNCT
ejpam-5152	94	11	fm(x2n+1	fm(x2n+1	NOUN
ejpam-5152	94	12	)	)	PUNCT
ejpam-5152	94	13	)	)	PUNCT
ejpam-5152	94	14	.	.	PUNCT
ejpam-5152	95	1	analogous	analogous	ADJ
ejpam-5152	95	2	to	to	ADP
ejpam-5152	95	3	n	n	PROPN
ejpam-5152	95	4	is	be	AUX
ejpam-5152	95	5	even	even	ADV
ejpam-5152	95	6	,	,	PUNCT
ejpam-5152	95	7	it	it	PRON
ejpam-5152	95	8	can	can	AUX
ejpam-5152	95	9	be	be	AUX
ejpam-5152	95	10	concluded	conclude	VERB
ejpam-5152	95	11	that	that	SCONJ
ejpam-5152	95	12	p(x2n+1	p(x2n+1	ADJ
ejpam-5152	95	13	,	,	PUNCT
ejpam-5152	95	14	fm(x2n+1	fm(x2n+1	NOUN
ejpam-5152	95	15	)	)	PUNCT
ejpam-5152	95	16	)	)	PUNCT
ejpam-5152	96	1	=	=	SYM
ejpam-5152	96	2	0	0	X
ejpam-5152	96	3	,	,	PUNCT
ejpam-5152	96	4	it	it	PRON
ejpam-5152	96	5	means	mean	VERB
ejpam-5152	96	6	x2n+1	x2n+1	PROPN
ejpam-5152	96	7	∈	∈	PROPN
ejpam-5152	96	8	fm(x2n+1	fm(x2n+1	PROPN
ejpam-5152	96	9	)	)	PUNCT
ejpam-5152	96	10	.	.	PUNCT
ejpam-5152	97	1	for	for	ADP
ejpam-5152	97	2	the	the	DET
ejpam-5152	97	3	next	next	ADJ
ejpam-5152	97	4	step	step	NOUN
ejpam-5152	97	5	,	,	PUNCT
ejpam-5152	97	6	we	we	PRON
ejpam-5152	97	7	will	will	AUX
ejpam-5152	97	8	show	show	VERB
ejpam-5152	97	9	that	that	SCONJ
ejpam-5152	97	10	(	(	PUNCT
ejpam-5152	97	11	xn	xn	X
ejpam-5152	97	12	)	)	PUNCT
ejpam-5152	97	13	is	be	AUX
ejpam-5152	97	14	cauchy	cauchy	ADJ
ejpam-5152	97	15	sequence	sequence	NOUN
ejpam-5152	97	16	in	in	ADP
ejpam-5152	97	17	(	(	PUNCT
ejpam-5152	97	18	x	x	X
ejpam-5152	97	19	,	,	PUNCT
ejpam-5152	97	20	p	p	NOUN
ejpam-5152	97	21	)	)	PUNCT
ejpam-5152	97	22	.	.	PUNCT
ejpam-5152	98	1	let	let	VERB
ejpam-5152	98	2	we	we	PRON
ejpam-5152	98	3	consider	consider	VERB
ejpam-5152	98	4	p(x2n	p(x2n	NOUN
ejpam-5152	98	5	,	,	PUNCT
ejpam-5152	98	6	x2n+1	x2n+1	PROPN
ejpam-5152	98	7	)	)	PUNCT
ejpam-5152	98	8	)	)	PUNCT
ejpam-5152	99	1	≤	≤	NUM
ejpam-5152	99	2	1√	1√	NUM
ejpam-5152	99	3	κ	κ	PROPN
ejpam-5152	99	4	hp(fn+1(x2n	hp(fn+1(x2n	PROPN
ejpam-5152	99	5	)	)	PUNCT
ejpam-5152	99	6	,	,	PUNCT
ejpam-5152	99	7	gn(x2n−1	gn(x2n−1	NOUN
ejpam-5152	99	8	)	)	PUNCT
ejpam-5152	99	9	)	)	PUNCT
ejpam-5152	100	1	≤	≤	NUM
ejpam-5152	100	2	1√	1√	PROPN
ejpam-5152	100	3	κ	κ	PROPN
ejpam-5152	100	4	κmax{p(x2n	κmax{p(x2n	PROPN
ejpam-5152	100	5	,	,	PUNCT
ejpam-5152	100	6	x2n−1	x2n−1	PROPN
ejpam-5152	100	7	)	)	PUNCT
ejpam-5152	100	8	,	,	PUNCT
ejpam-5152	100	9	p(x2n	p(x2n	NOUN
ejpam-5152	100	10	,	,	PUNCT
ejpam-5152	100	11	fn+1(x2n	fn+1(x2n	NOUN
ejpam-5152	100	12	)	)	PUNCT
ejpam-5152	100	13	)	)	PUNCT
ejpam-5152	100	14	,	,	PUNCT
ejpam-5152	100	15	p(x2n−1	p(x2n−1	PROPN
ejpam-5152	100	16	,	,	PUNCT
ejpam-5152	100	17	gn(x2n−1	gn(x2n−1	NOUN
ejpam-5152	100	18	)	)	PUNCT
ejpam-5152	100	19	)	)	PUNCT
ejpam-5152	100	20	,	,	PUNCT
ejpam-5152	100	21	1	1	NUM
ejpam-5152	100	22	2(p(x2n	2(p(x2n	NUM
ejpam-5152	100	23	,	,	PUNCT
ejpam-5152	100	24	gn(x2n−1	gn(x2n−1	NOUN
ejpam-5152	100	25	)	)	PUNCT
ejpam-5152	100	26	)	)	PUNCT
ejpam-5152	101	1	+	+	CCONJ
ejpam-5152	101	2	p(x2n−1	p(x2n−1	ADJ
ejpam-5152	101	3	,	,	PUNCT
ejpam-5152	101	4	fn+1(x2n	fn+1(x2n	NOUN
ejpam-5152	101	5	)	)	PUNCT
ejpam-5152	101	6	)	)	PUNCT
ejpam-5152	101	7	}	}	PUNCT
ejpam-5152	101	8	≤	≤	NOUN
ejpam-5152	101	9	√	√	NUM
ejpam-5152	101	10	κmax{p(x2n	κmax{p(x2n	PROPN
ejpam-5152	101	11	,	,	PUNCT
ejpam-5152	101	12	x2n−1	x2n−1	PROPN
ejpam-5152	101	13	)	)	PUNCT
ejpam-5152	101	14	,	,	PUNCT
ejpam-5152	101	15	p(x2n	p(x2n	NOUN
ejpam-5152	101	16	,	,	PUNCT
ejpam-5152	101	17	x2n+1	x2n+1	PROPN
ejpam-5152	101	18	)	)	PUNCT
ejpam-5152	101	19	,	,	PUNCT
ejpam-5152	101	20	p(x2n−1	p(x2n−1	PROPN
ejpam-5152	101	21	,	,	PUNCT
ejpam-5152	101	22	x2n	x2n	PROPN
ejpam-5152	101	23	)	)	PUNCT
ejpam-5152	101	24	,	,	PUNCT
ejpam-5152	101	25	1	1	NUM
ejpam-5152	101	26	2(p(x2n	2(p(x2n	NUM
ejpam-5152	101	27	,	,	PUNCT
ejpam-5152	101	28	x2n+1	x2n+1	PUNCT
ejpam-5152	101	29	)	)	PUNCT
ejpam-5152	102	1	+	+	CCONJ
ejpam-5152	103	1	p(x2n−1	p(x2n−1	ADJ
ejpam-5152	103	2	,	,	PUNCT
ejpam-5152	103	3	x2n	x2n	PROPN
ejpam-5152	103	4	)	)	PUNCT
ejpam-5152	103	5	)	)	PUNCT
ejpam-5152	103	6	}	}	PUNCT
ejpam-5152	103	7	≤	≤	NOUN
ejpam-5152	104	1	√	√	VERB
ejpam-5152	104	2	κmax{p(x2n−1	κmax{p(x2n−1	PROPN
ejpam-5152	104	3	,	,	PUNCT
ejpam-5152	104	4	x2n	x2n	PROPN
ejpam-5152	104	5	)	)	PUNCT
ejpam-5152	104	6	,	,	PUNCT
ejpam-5152	104	7	p(x2n	p(x2n	NOUN
ejpam-5152	104	8	,	,	PUNCT
ejpam-5152	104	9	x2n+1	x2n+1	PROPN
ejpam-5152	104	10	)	)	PUNCT
ejpam-5152	104	11	}	}	PUNCT
ejpam-5152	104	12	,	,	PUNCT
ejpam-5152	104	13	when	when	SCONJ
ejpam-5152	104	14	p(x2n	p(x2n	NOUN
ejpam-5152	104	15	,	,	PUNCT
ejpam-5152	104	16	x2n+1	x2n+1	PROPN
ejpam-5152	104	17	)	)	PUNCT
ejpam-5152	104	18	is	be	AUX
ejpam-5152	104	19	the	the	DET
ejpam-5152	104	20	maximum	maximum	NOUN
ejpam-5152	104	21	then	then	ADV
ejpam-5152	104	22	we	we	PRON
ejpam-5152	104	23	have	have	VERB
ejpam-5152	104	24	x2n	x2n	PUNCT
ejpam-5152	104	25	=	=	SYM
ejpam-5152	104	26	x2n+1	x2n+1	PROPN
ejpam-5152	104	27	.	.	PUNCT
ejpam-5152	105	1	since	since	SCONJ
ejpam-5152	105	2	xn	xn	PROPN
ejpam-5152	105	3	̸=	̸=	PROPN
ejpam-5152	105	4	xx+1	xx+1	NUM
ejpam-5152	105	5	for	for	ADP
ejpam-5152	105	6	each	each	DET
ejpam-5152	105	7	n	n	CCONJ
ejpam-5152	105	8	thus	thus	ADV
ejpam-5152	105	9	we	we	PRON
ejpam-5152	105	10	get	get	VERB
ejpam-5152	105	11	a	a	DET
ejpam-5152	105	12	contradiction	contradiction	NOUN
ejpam-5152	105	13	.	.	PUNCT
ejpam-5152	106	1	therefore	therefore	ADV
ejpam-5152	106	2	,	,	PUNCT
ejpam-5152	106	3	we	we	PRON
ejpam-5152	106	4	have	have	VERB
ejpam-5152	106	5	the	the	DET
ejpam-5152	106	6	maximum	maximum	NOUN
ejpam-5152	106	7	is	be	AUX
ejpam-5152	106	8	p(x2n−1	p(x2n−1	ADJ
ejpam-5152	106	9	,	,	PUNCT
ejpam-5152	106	10	x2n	x2n	PROPN
ejpam-5152	106	11	)	)	PUNCT
ejpam-5152	106	12	.	.	PUNCT
ejpam-5152	107	1	it	it	PRON
ejpam-5152	107	2	implies	imply	VERB
ejpam-5152	107	3	p(x2n	p(x2n	NOUN
ejpam-5152	107	4	,	,	PUNCT
ejpam-5152	107	5	x2n+1	x2n+1	PROPN
ejpam-5152	107	6	)	)	PUNCT
ejpam-5152	107	7	≤	≤	NOUN
ejpam-5152	107	8	√	√	PUNCT
ejpam-5152	107	9	κp(x2n−1	κp(x2n−1	ADJ
ejpam-5152	107	10	,	,	PUNCT
ejpam-5152	107	11	x2n	x2n	PROPN
ejpam-5152	107	12	)	)	PUNCT
ejpam-5152	107	13	.	.	PUNCT
ejpam-5152	108	1	a.	a.	PROPN
ejpam-5152	108	2	ekayanti	ekayanti	PROPN
ejpam-5152	108	3	et	et	PROPN
ejpam-5152	108	4	al	al	PROPN
ejpam-5152	108	5	.	.	PUNCT
ejpam-5152	108	6	/	/	SYM
ejpam-5152	108	7	eur	eur	PROPN
ejpam-5152	108	8	.	.	PUNCT
ejpam-5152	109	1	j.	j.	PROPN
ejpam-5152	109	2	pure	pure	PROPN
ejpam-5152	109	3	appl	appl	PROPN
ejpam-5152	109	4	.	.	PROPN
ejpam-5152	109	5	math	math	PROPN
ejpam-5152	109	6	,	,	PUNCT
ejpam-5152	109	7	17	17	NUM
ejpam-5152	109	8	(	(	PUNCT
ejpam-5152	109	9	2	2	NUM
ejpam-5152	109	10	)	)	PUNCT
ejpam-5152	109	11	(	(	PUNCT
ejpam-5152	109	12	2024	2024	NUM
ejpam-5152	109	13	)	)	PUNCT
ejpam-5152	109	14	,	,	PUNCT
ejpam-5152	109	15	996	996	NUM
ejpam-5152	109	16	-	-	SYM
ejpam-5152	109	17	1008	1008	NUM
ejpam-5152	109	18	1000	1000	NUM
ejpam-5152	109	19	in	in	ADP
ejpam-5152	109	20	the	the	DET
ejpam-5152	109	21	similar	similar	ADJ
ejpam-5152	109	22	way	way	NOUN
ejpam-5152	109	23	,	,	PUNCT
ejpam-5152	109	24	we	we	PRON
ejpam-5152	109	25	have	have	AUX
ejpam-5152	109	26	p(x2n+1	p(x2n+1	ADJ
ejpam-5152	109	27	,	,	PUNCT
ejpam-5152	109	28	x2n+2	x2n+2	PROPN
ejpam-5152	109	29	)	)	PUNCT
ejpam-5152	109	30	≤	≤	NOUN
ejpam-5152	109	31	√	√	NUM
ejpam-5152	110	1	κp(x2n	κp(x2n	PROPN
ejpam-5152	110	2	,	,	PUNCT
ejpam-5152	110	3	x2n+1	x2n+1	PROPN
ejpam-5152	110	4	)	)	PUNCT
ejpam-5152	110	5	.	.	PUNCT
ejpam-5152	111	1	therefore	therefore	ADV
ejpam-5152	111	2	,	,	PUNCT
ejpam-5152	111	3	we	we	PRON
ejpam-5152	111	4	obtain	obtain	VERB
ejpam-5152	111	5	p(x2n	p(x2n	NOUN
ejpam-5152	111	6	,	,	PUNCT
ejpam-5152	111	7	x2n+1	x2n+1	PROPN
ejpam-5152	111	8	)	)	PUNCT
ejpam-5152	111	9	)	)	PUNCT
ejpam-5152	112	1	≤	≤	NOUN
ejpam-5152	112	2	√	√	PUNCT
ejpam-5152	112	3	κp(x2n−1	κp(x2n−1	ADJ
ejpam-5152	112	4	,	,	PUNCT
ejpam-5152	112	5	x2n	x2n	NOUN
ejpam-5152	112	6	)	)	PUNCT
ejpam-5152	112	7	≤	≤	NOUN
ejpam-5152	112	8	√	√	ADP
ejpam-5152	112	9	κ	κ	ADP
ejpam-5152	112	10	√	√	PROPN
ejpam-5152	112	11	κp(x2n−2	κp(x2n−2	ADP
ejpam-5152	112	12	,	,	PUNCT
ejpam-5152	112	13	x2n−1	x2n−1	PROPN
ejpam-5152	112	14	)	)	PUNCT
ejpam-5152	112	15	=	=	PRON
ejpam-5152	112	16	(	(	PUNCT
ejpam-5152	112	17	√	√	NUM
ejpam-5152	112	18	κ)2p(x2n−2	κ)2p(x2n−2	PROPN
ejpam-5152	112	19	,	,	PUNCT
ejpam-5152	112	20	x2n−1	x2n−1	PROPN
ejpam-5152	112	21	)	)	PUNCT
ejpam-5152	112	22	≤	≤	NOUN
ejpam-5152	112	23	(	(	PUNCT
ejpam-5152	112	24	√	√	NUM
ejpam-5152	112	25	κ)2	κ)2	NOUN
ejpam-5152	112	26	√	√	PUNCT
ejpam-5152	112	27	κp(x2n−3	κp(x2n−3	PROPN
ejpam-5152	112	28	,	,	PUNCT
ejpam-5152	112	29	x2n−2	x2n−2	PROPN
ejpam-5152	112	30	)	)	PUNCT
ejpam-5152	112	31	=	=	PRON
ejpam-5152	112	32	(	(	PUNCT
ejpam-5152	112	33	√	√	PROPN
ejpam-5152	112	34	κ)3p(x2n−3	κ)3p(x2n−3	NUM
ejpam-5152	112	35	,	,	PUNCT
ejpam-5152	112	36	x2n−2	x2n−2	PROPN
ejpam-5152	112	37	)	)	PUNCT
ejpam-5152	112	38	...	...	PUNCT
ejpam-5152	113	1	≤	≤	NUM
ejpam-5152	113	2	(	(	PUNCT
ejpam-5152	113	3	√	√	NUM
ejpam-5152	113	4	κ)2np(x0	κ)2np(x0	PROPN
ejpam-5152	113	5	,	,	PUNCT
ejpam-5152	113	6	x1	x1	NUM
ejpam-5152	113	7	)	)	PUNCT
ejpam-5152	114	1	=	=	SYM
ejpam-5152	115	1	κnp(x0	κnp(x0	PROPN
ejpam-5152	115	2	,	,	PUNCT
ejpam-5152	115	3	x1	x1	PROPN
ejpam-5152	115	4	)	)	PUNCT
ejpam-5152	115	5	.	.	PUNCT
ejpam-5152	116	1	and	and	CCONJ
ejpam-5152	116	2	also	also	ADV
ejpam-5152	116	3	we	we	PRON
ejpam-5152	116	4	have	have	VERB
ejpam-5152	116	5	p(x2n+1	p(x2n+1	ADJ
ejpam-5152	116	6	,	,	PUNCT
ejpam-5152	116	7	x2n+2	x2n+2	PROPN
ejpam-5152	116	8	)	)	PUNCT
ejpam-5152	116	9	)	)	PUNCT
ejpam-5152	117	1	≤	≤	NUM
ejpam-5152	117	2	κnp(x1	κnp(x1	PROPN
ejpam-5152	117	3	,	,	PUNCT
ejpam-5152	117	4	x2	x2	PROPN
ejpam-5152	117	5	)	)	PUNCT
ejpam-5152	117	6	.	.	PUNCT
ejpam-5152	118	1	let	let	VERB
ejpam-5152	118	2	t(x0	t(x0	NOUN
ejpam-5152	118	3	)	)	PUNCT
ejpam-5152	119	1	:	:	PUNCT
ejpam-5152	119	2	=	=	SYM
ejpam-5152	119	3	max{p(x0	max{p(x0	X
ejpam-5152	119	4	,	,	PUNCT
ejpam-5152	119	5	x1	x1	PROPN
ejpam-5152	119	6	)	)	PUNCT
ejpam-5152	119	7	,	,	PUNCT
ejpam-5152	119	8	p(x1	p(x1	NOUN
ejpam-5152	119	9	,	,	PUNCT
ejpam-5152	119	10	x2	x2	PROPN
ejpam-5152	119	11	)	)	PUNCT
ejpam-5152	119	12	}	}	PUNCT
ejpam-5152	119	13	,	,	PUNCT
ejpam-5152	119	14	then	then	ADV
ejpam-5152	119	15	for	for	SCONJ
ejpam-5152	119	16	m	m	PROPN
ejpam-5152	119	17	>	>	X
ejpam-5152	119	18	n	n	PROPN
ejpam-5152	119	19	w	w	PROPN
ejpam-5152	119	20	have	have	VERB
ejpam-5152	119	21	p(xm	p(xm	PROPN
ejpam-5152	119	22	,	,	PUNCT
ejpam-5152	119	23	xn	xn	PROPN
ejpam-5152	119	24	)	)	PUNCT
ejpam-5152	119	25	)	)	PUNCT
ejpam-5152	119	26	≤	≤	NOUN
ejpam-5152	120	1	m−(n+1)∑	m−(n+1)∑	ADJ
ejpam-5152	120	2	i=0	i=0	PROPN
ejpam-5152	120	3	p(xn+i	p(xn+i	NOUN
ejpam-5152	120	4	,	,	PUNCT
ejpam-5152	120	5	xn+1+i	xn+1+i	NOUN
ejpam-5152	120	6	)	)	PUNCT
ejpam-5152	120	7	≤	≤	NUM
ejpam-5152	120	8	m−(n+1)∑	m−(n+1)∑	ADJ
ejpam-5152	120	9	i=0	i=0	PROPN
ejpam-5152	120	10	hn+it(x0	hn+it(x0	NOUN
ejpam-5152	120	11	)	)	PUNCT
ejpam-5152	120	12	=	=	SYM
ejpam-5152	120	13	t(x0	t(x0	NOUN
ejpam-5152	120	14	)	)	PUNCT
ejpam-5152	120	15	m−(n+1)∑	m−(n+1)∑	ADJ
ejpam-5152	120	16	i=0	i=0	PROPN
ejpam-5152	120	17	hn+i	hn+i	PROPN
ejpam-5152	120	18	=	=	SYM
ejpam-5152	120	19	thn	thn	PROPN
ejpam-5152	120	20	m−(n+1)∑	m−(n+1)∑	PROPN
ejpam-5152	120	21	i=0	i=0	PROPN
ejpam-5152	120	22	hi	hi	ADJ
ejpam-5152	120	23	≤	≤	VERB
ejpam-5152	120	24	t(x0)hn	t(x0)hn	ADJ
ejpam-5152	120	25	1−h	1−h	NUM
ejpam-5152	120	26	since	since	SCONJ
ejpam-5152	120	27	t(x0)hn	t(x0)hn	ADJ
ejpam-5152	120	28	1−h	1−h	NUM
ejpam-5152	120	29	→	→	SYM
ejpam-5152	120	30	0	0	NUM
ejpam-5152	120	31	as	as	ADP
ejpam-5152	120	32	n	n	NUM
ejpam-5152	120	33	→	→	SYM
ejpam-5152	120	34	∞	∞	PROPN
ejpam-5152	120	35	,	,	PUNCT
ejpam-5152	120	36	it	it	PRON
ejpam-5152	120	37	means	mean	VERB
ejpam-5152	120	38	we	we	PRON
ejpam-5152	120	39	are	be	AUX
ejpam-5152	120	40	already	already	ADV
ejpam-5152	120	41	shown	show	VERB
ejpam-5152	120	42	that	that	SCONJ
ejpam-5152	120	43	(	(	PUNCT
ejpam-5152	120	44	xn	xn	X
ejpam-5152	120	45	)	)	PUNCT
ejpam-5152	120	46	is	be	AUX
ejpam-5152	120	47	a	a	DET
ejpam-5152	120	48	cauchy	cauchy	ADJ
ejpam-5152	120	49	sequence	sequence	NOUN
ejpam-5152	120	50	in	in	ADP
ejpam-5152	120	51	x.	x.	NOUN
ejpam-5152	120	52	since	since	SCONJ
ejpam-5152	120	53	(	(	PUNCT
ejpam-5152	120	54	x	x	X
ejpam-5152	120	55	,	,	PUNCT
ejpam-5152	120	56	p	p	NOUN
ejpam-5152	120	57	)	)	PUNCT
ejpam-5152	120	58	is	be	AUX
ejpam-5152	120	59	complete	complete	ADJ
ejpam-5152	120	60	partial	partial	ADJ
ejpam-5152	120	61	metric	metric	ADJ
ejpam-5152	120	62	space	space	NOUN
ejpam-5152	120	63	then	then	ADV
ejpam-5152	120	64	there	there	PRON
ejpam-5152	120	65	exists	exist	VERB
ejpam-5152	120	66	x	x	X
ejpam-5152	120	67	∈	∈	PROPN
ejpam-5152	120	68	x	x	X
ejpam-5152	120	69	such	such	ADJ
ejpam-5152	120	70	that	that	PRON
ejpam-5152	120	71	xn	xn	PUNCT
ejpam-5152	121	1	→	→	SYM
ejpam-5152	121	2	x	x	SYM
ejpam-5152	121	3	whereas	whereas	SCONJ
ejpam-5152	121	4	n	n	NOUN
ejpam-5152	121	5	→	→	SYM
ejpam-5152	121	6	∞.	∞.	PROPN
ejpam-5152	121	7	let	let	VERB
ejpam-5152	121	8	we	we	PRON
ejpam-5152	121	9	observe	observe	VERB
ejpam-5152	121	10	the	the	DET
ejpam-5152	121	11	following	follow	VERB
ejpam-5152	121	12	condition	condition	NOUN
ejpam-5152	121	13	p(x2n−1	p(x2n−1	NOUN
ejpam-5152	121	14	,	,	PUNCT
ejpam-5152	121	15	gm(x	gm(x	NUM
ejpam-5152	121	16	)	)	PUNCT
ejpam-5152	121	17	)	)	PUNCT
ejpam-5152	122	1	≤	≤	NUM
ejpam-5152	122	2	hp(fn(x2n−2	hp(fn(x2n−2	NUM
ejpam-5152	122	3	)	)	PUNCT
ejpam-5152	122	4	,	,	PUNCT
ejpam-5152	122	5	gm(x	gm(x	NUM
ejpam-5152	122	6	)	)	PUNCT
ejpam-5152	122	7	)	)	PUNCT
ejpam-5152	123	1	≤	≤	PROPN
ejpam-5152	124	1	κmax{p(x2n−2	κmax{p(x2n−2	PROPN
ejpam-5152	124	2	,	,	PUNCT
ejpam-5152	124	3	x	x	NOUN
ejpam-5152	124	4	)	)	PUNCT
ejpam-5152	124	5	,	,	PUNCT
ejpam-5152	124	6	p(x2n−2	p(x2n−2	PROPN
ejpam-5152	124	7	,	,	PUNCT
ejpam-5152	124	8	fn(x2n−2	fn(x2n−2	NOUN
ejpam-5152	124	9	)	)	PUNCT
ejpam-5152	124	10	)	)	PUNCT
ejpam-5152	124	11	,	,	PUNCT
ejpam-5152	124	12	p(x	p(x	NOUN
ejpam-5152	124	13	,	,	PUNCT
ejpam-5152	124	14	gm(x	gm(x	NUM
ejpam-5152	124	15	)	)	PUNCT
ejpam-5152	124	16	)	)	PUNCT
ejpam-5152	124	17	,	,	PUNCT
ejpam-5152	124	18	1	1	NUM
ejpam-5152	124	19	2(p(x2n−2	2(p(x2n−2	NUM
ejpam-5152	124	20	,	,	PUNCT
ejpam-5152	124	21	gm(x	gm(x	NUM
ejpam-5152	124	22	)	)	PUNCT
ejpam-5152	124	23	)	)	PUNCT
ejpam-5152	125	1	+	+	CCONJ
ejpam-5152	125	2	p(x	p(x	PROPN
ejpam-5152	125	3	,	,	PUNCT
ejpam-5152	125	4	fn(x2n−2	fn(x2n−2	NOUN
ejpam-5152	125	5	)	)	PUNCT
ejpam-5152	125	6	)	)	PUNCT
ejpam-5152	125	7	)	)	PUNCT
ejpam-5152	125	8	}	}	PUNCT
ejpam-5152	125	9	≤	≤	X
ejpam-5152	126	1	κmax{p(x2n−2	κmax{p(x2n−2	PROPN
ejpam-5152	126	2	,	,	PUNCT
ejpam-5152	126	3	x	x	NOUN
ejpam-5152	126	4	)	)	PUNCT
ejpam-5152	126	5	,	,	PUNCT
ejpam-5152	126	6	p(x2n−2	p(x2n−2	PROPN
ejpam-5152	126	7	,	,	PUNCT
ejpam-5152	126	8	x2n−1	x2n−1	PROPN
ejpam-5152	126	9	)	)	PUNCT
ejpam-5152	126	10	,	,	PUNCT
ejpam-5152	126	11	p(x	p(x	NOUN
ejpam-5152	126	12	,	,	PUNCT
ejpam-5152	126	13	gm(x	gm(x	NUM
ejpam-5152	126	14	)	)	PUNCT
ejpam-5152	126	15	)	)	PUNCT
ejpam-5152	126	16	,	,	PUNCT
ejpam-5152	126	17	1	1	NUM
ejpam-5152	126	18	2(p(x2n−2	2(p(x2n−2	NUM
ejpam-5152	126	19	,	,	PUNCT
ejpam-5152	126	20	gm(x	gm(x	NUM
ejpam-5152	126	21	)	)	PUNCT
ejpam-5152	126	22	)	)	PUNCT
ejpam-5152	127	1	+	+	CCONJ
ejpam-5152	127	2	p(x	p(x	PROPN
ejpam-5152	127	3	,	,	PUNCT
ejpam-5152	127	4	x2n−1	x2n−1	PROPN
ejpam-5152	127	5	)	)	PUNCT
ejpam-5152	127	6	)	)	PUNCT
ejpam-5152	127	7	)	)	PUNCT
ejpam-5152	127	8	}	}	PUNCT
ejpam-5152	127	9	by	by	ADP
ejpam-5152	127	10	taking	take	VERB
ejpam-5152	127	11	n	n	PRON
ejpam-5152	127	12	→	→	SYM
ejpam-5152	127	13	∞	∞	NUM
ejpam-5152	127	14	we	we	PRON
ejpam-5152	127	15	obtain	obtain	VERB
ejpam-5152	127	16	p(x	p(x	NOUN
ejpam-5152	127	17	,	,	PUNCT
ejpam-5152	127	18	gm(x	gm(x	NUM
ejpam-5152	127	19	)	)	PUNCT
ejpam-5152	127	20	)	)	PUNCT
ejpam-5152	128	1	≤	≤	NUM
ejpam-5152	128	2	κmax	κmax	VERB
ejpam-5152	128	3	{	{	PUNCT
ejpam-5152	128	4	p(x	p(x	PROPN
ejpam-5152	128	5	,	,	PUNCT
ejpam-5152	128	6	x	x	NOUN
ejpam-5152	128	7	)	)	PUNCT
ejpam-5152	128	8	,	,	PUNCT
ejpam-5152	128	9	p(x	p(x	PROPN
ejpam-5152	128	10	,	,	PUNCT
ejpam-5152	128	11	x	x	NOUN
ejpam-5152	128	12	)	)	PUNCT
ejpam-5152	128	13	,	,	PUNCT
ejpam-5152	128	14	p(x	p(x	NOUN
ejpam-5152	128	15	,	,	PUNCT
ejpam-5152	128	16	gm(x	gm(x	NUM
ejpam-5152	128	17	)	)	PUNCT
ejpam-5152	128	18	)	)	PUNCT
ejpam-5152	128	19	,	,	PUNCT
ejpam-5152	128	20	1	1	NUM
ejpam-5152	128	21	2	2	NUM
ejpam-5152	128	22	(	(	PUNCT
ejpam-5152	128	23	p(x	p(x	NOUN
ejpam-5152	128	24	,	,	PUNCT
ejpam-5152	128	25	gm(x	gm(x	NUM
ejpam-5152	128	26	)	)	PUNCT
ejpam-5152	128	27	)	)	PUNCT
ejpam-5152	129	1	+	+	CCONJ
ejpam-5152	129	2	p(x	p(x	PROPN
ejpam-5152	129	3	,	,	PUNCT
ejpam-5152	129	4	x	x	NOUN
ejpam-5152	129	5	)	)	PUNCT
ejpam-5152	129	6	)	)	PUNCT
ejpam-5152	129	7	}	}	PUNCT
ejpam-5152	129	8	,	,	PUNCT
ejpam-5152	129	9	for	for	ADP
ejpam-5152	129	10	each	each	DET
ejpam-5152	129	11	m.	m.	NOUN
ejpam-5152	129	12	therefore	therefore	ADV
ejpam-5152	129	13	,	,	PUNCT
ejpam-5152	129	14	we	we	PRON
ejpam-5152	129	15	have	have	VERB
ejpam-5152	129	16	p(x	p(x	NOUN
ejpam-5152	129	17	,	,	PUNCT
ejpam-5152	129	18	gm(x	gm(x	NUM
ejpam-5152	129	19	)	)	PUNCT
ejpam-5152	129	20	)	)	PUNCT
ejpam-5152	130	1	≤	≤	NOUN
ejpam-5152	130	2	κp(x	κp(x	NOUN
ejpam-5152	130	3	,	,	PUNCT
ejpam-5152	130	4	gm(x	gm(x	NUM
ejpam-5152	130	5	)	)	PUNCT
ejpam-5152	130	6	)	)	PUNCT
ejpam-5152	130	7	.	.	PUNCT
ejpam-5152	131	1	a.	a.	PROPN
ejpam-5152	131	2	ekayanti	ekayanti	PROPN
ejpam-5152	131	3	et	et	PROPN
ejpam-5152	131	4	al	al	PROPN
ejpam-5152	131	5	.	.	PUNCT
ejpam-5152	131	6	/	/	SYM
ejpam-5152	131	7	eur	eur	PROPN
ejpam-5152	131	8	.	.	PUNCT
ejpam-5152	132	1	j.	j.	PROPN
ejpam-5152	132	2	pure	pure	PROPN
ejpam-5152	132	3	appl	appl	PROPN
ejpam-5152	132	4	.	.	PROPN
ejpam-5152	132	5	math	math	PROPN
ejpam-5152	132	6	,	,	PUNCT
ejpam-5152	132	7	17	17	NUM
ejpam-5152	132	8	(	(	PUNCT
ejpam-5152	132	9	2	2	NUM
ejpam-5152	132	10	)	)	PUNCT
ejpam-5152	132	11	(	(	PUNCT
ejpam-5152	132	12	2024	2024	NUM
ejpam-5152	132	13	)	)	PUNCT
ejpam-5152	132	14	,	,	PUNCT
ejpam-5152	132	15	996	996	NUM
ejpam-5152	132	16	-	-	SYM
ejpam-5152	132	17	1008	1008	NUM
ejpam-5152	132	18	1001	1001	NUM
ejpam-5152	132	19	since	since	SCONJ
ejpam-5152	132	20	0	0	NUM
ejpam-5152	132	21	≤	≤	NUM
ejpam-5152	132	22	κ	κ	NOUN
ejpam-5152	132	23	<	<	X
ejpam-5152	132	24	1	1	NUM
ejpam-5152	132	25	then	then	ADV
ejpam-5152	132	26	p(x	p(x	PROPN
ejpam-5152	132	27	,	,	PUNCT
ejpam-5152	132	28	gm(x	gm(x	NUM
ejpam-5152	132	29	)	)	PUNCT
ejpam-5152	132	30	)	)	PUNCT
ejpam-5152	133	1	=	=	PUNCT
ejpam-5152	133	2	0	0	X
ejpam-5152	133	3	.	.	PUNCT
ejpam-5152	134	1	it	it	PRON
ejpam-5152	134	2	implies	imply	VERB
ejpam-5152	134	3	that	that	SCONJ
ejpam-5152	134	4	x	x	SYM
ejpam-5152	134	5	∈	∈	NOUN
ejpam-5152	134	6	gm(x	gm(x	PUNCT
ejpam-5152	134	7	)	)	PUNCT
ejpam-5152	134	8	because	because	SCONJ
ejpam-5152	134	9	gm(x	gm(x	NUM
ejpam-5152	134	10	)	)	PUNCT
ejpam-5152	134	11	is	be	AUX
ejpam-5152	134	12	closed	close	VERB
ejpam-5152	134	13	.	.	PUNCT
ejpam-5152	135	1	similarly	similarly	ADV
ejpam-5152	135	2	,	,	PUNCT
ejpam-5152	135	3	we	we	PRON
ejpam-5152	135	4	can	can	AUX
ejpam-5152	135	5	show	show	VERB
ejpam-5152	135	6	that	that	SCONJ
ejpam-5152	135	7	x	x	SYM
ejpam-5152	135	8	∈	∈	PROPN
ejpam-5152	135	9	fn(x	fn(x	NOUN
ejpam-5152	135	10	)	)	PUNCT
ejpam-5152	135	11	for	for	ADP
ejpam-5152	135	12	each	each	DET
ejpam-5152	135	13	n.	n.	NOUN
ejpam-5152	135	14	so	so	ADV
ejpam-5152	135	15	,	,	PUNCT
ejpam-5152	135	16	we	we	PRON
ejpam-5152	135	17	obtain	obtain	VERB
ejpam-5152	135	18	x	x	SYM
ejpam-5152	135	19	∈	∈	NOUN
ejpam-5152	135	20	gm(x	gm(x	PUNCT
ejpam-5152	135	21	)	)	PUNCT
ejpam-5152	135	22	and	and	CCONJ
ejpam-5152	135	23	x	x	PUNCT
ejpam-5152	135	24	∈	∈	PROPN
ejpam-5152	135	25	fn(x	fn(x	NOUN
ejpam-5152	135	26	)	)	PUNCT
ejpam-5152	135	27	for	for	ADP
ejpam-5152	135	28	each	each	DET
ejpam-5152	135	29	m	m	NOUN
ejpam-5152	135	30	,	,	PUNCT
ejpam-5152	135	31	n.	n.	PROPN
ejpam-5152	135	32	it	it	PRON
ejpam-5152	135	33	means	mean	VERB
ejpam-5152	135	34	x	x	SYM
ejpam-5152	135	35	is	be	AUX
ejpam-5152	135	36	common	common	ADJ
ejpam-5152	135	37	fixed	fix	VERB
ejpam-5152	135	38	point	point	NOUN
ejpam-5152	135	39	of	of	ADP
ejpam-5152	135	40	gm	gm	PROPN
ejpam-5152	135	41	and	and	CCONJ
ejpam-5152	135	42	fn	fn	PROPN
ejpam-5152	135	43	for	for	ADP
ejpam-5152	135	44	every	every	DET
ejpam-5152	135	45	m	m	NOUN
ejpam-5152	135	46	and	and	CCONJ
ejpam-5152	135	47	n.	n.	VERB
ejpam-5152	135	48	this	this	PRON
ejpam-5152	135	49	complete	complete	ADJ
ejpam-5152	135	50	the	the	DET
ejpam-5152	135	51	proof	proof	NOUN
ejpam-5152	135	52	.	.	PUNCT
ejpam-5152	136	1	in	in	ADP
ejpam-5152	136	2	theorem	theorem	NOUN
ejpam-5152	136	3	3	3	NUM
ejpam-5152	136	4	above	above	ADV
ejpam-5152	136	5	,	,	PUNCT
ejpam-5152	136	6	we	we	PRON
ejpam-5152	136	7	have	have	VERB
ejpam-5152	136	8	the	the	DET
ejpam-5152	136	9	principle	principle	NOUN
ejpam-5152	136	10	of	of	ADP
ejpam-5152	136	11	contraction	contraction	NOUN
ejpam-5152	136	12	on	on	ADP
ejpam-5152	136	13	set	set	NOUN
ejpam-5152	136	14	-	-	PUNCT
ejpam-5152	136	15	valued	value	VERB
ejpam-5152	136	16	mapping	mapping	NOUN
ejpam-5152	136	17	sequences	sequence	NOUN
ejpam-5152	136	18	as	as	SCONJ
ejpam-5152	136	19	follows	follow	VERB
ejpam-5152	136	20	:	:	PUNCT
ejpam-5152	136	21	hp(fn(x	hp(fn(x	PROPN
ejpam-5152	136	22	)	)	PUNCT
ejpam-5152	136	23	,	,	PUNCT
ejpam-5152	136	24	gn(y	gn(y	NOUN
ejpam-5152	136	25	)	)	PUNCT
ejpam-5152	136	26	)	)	PUNCT
ejpam-5152	137	1	≤	≤	NUM
ejpam-5152	137	2	κmax	κmax	VERB
ejpam-5152	137	3	{	{	PUNCT
ejpam-5152	137	4	p(x	p(x	PROPN
ejpam-5152	137	5	,	,	PUNCT
ejpam-5152	137	6	y	y	NOUN
ejpam-5152	137	7	)	)	PUNCT
ejpam-5152	137	8	,	,	PUNCT
ejpam-5152	137	9	p(x	p(x	PROPN
ejpam-5152	137	10	,	,	PUNCT
ejpam-5152	137	11	fn(x	fn(x	NOUN
ejpam-5152	137	12	)	)	PUNCT
ejpam-5152	137	13	)	)	PUNCT
ejpam-5152	137	14	,	,	PUNCT
ejpam-5152	137	15	p(y	p(y	PROPN
ejpam-5152	137	16	,	,	PUNCT
ejpam-5152	137	17	gn(y	gn(y	NOUN
ejpam-5152	137	18	)	)	PUNCT
ejpam-5152	137	19	)	)	PUNCT
ejpam-5152	137	20	,	,	PUNCT
ejpam-5152	137	21	1	1	NUM
ejpam-5152	137	22	2	2	NUM
ejpam-5152	137	23	(	(	PUNCT
ejpam-5152	137	24	p(x	p(x	PROPN
ejpam-5152	137	25	,	,	PUNCT
ejpam-5152	137	26	gn(y	gn(y	NOUN
ejpam-5152	137	27	)	)	PUNCT
ejpam-5152	137	28	)	)	PUNCT
ejpam-5152	138	1	+	+	CCONJ
ejpam-5152	138	2	p(y	p(y	NOUN
ejpam-5152	138	3	,	,	PUNCT
ejpam-5152	138	4	fn(x	fn(x	NOUN
ejpam-5152	138	5	)	)	PUNCT
ejpam-5152	138	6	)	)	PUNCT
ejpam-5152	138	7	)	)	PUNCT
ejpam-5152	138	8	}	}	PUNCT
ejpam-5152	138	9	,	,	PUNCT
ejpam-5152	138	10	for	for	ADP
ejpam-5152	138	11	each	each	DET
ejpam-5152	138	12	n	n	PRON
ejpam-5152	138	13	∈	∈	PROPN
ejpam-5152	138	14	n	n	NOUN
ejpam-5152	138	15	and	and	CCONJ
ejpam-5152	138	16	x	x	X
ejpam-5152	138	17	,	,	PUNCT
ejpam-5152	138	18	y	y	PROPN
ejpam-5152	138	19	∈	∈	PROPN
ejpam-5152	138	20	x	x	X
ejpam-5152	138	21	and	and	CCONJ
ejpam-5152	138	22	κ	κ	PROPN
ejpam-5152	138	23	∈	∈	PROPN
ejpam-5152	139	1	[	[	X
ejpam-5152	139	2	0	0	NUM
ejpam-5152	139	3	,	,	PUNCT
ejpam-5152	139	4	1	1	NUM
ejpam-5152	139	5	)	)	PUNCT
ejpam-5152	139	6	.	.	PUNCT
ejpam-5152	140	1	by	by	ADP
ejpam-5152	140	2	looking	look	VERB
ejpam-5152	140	3	at	at	ADP
ejpam-5152	140	4	the	the	DET
ejpam-5152	140	5	sequences	sequence	NOUN
ejpam-5152	140	6	fn	fn	NOUN
ejpam-5152	140	7	and	and	CCONJ
ejpam-5152	140	8	gn	gn	PROPN
ejpam-5152	140	9	in	in	ADP
ejpam-5152	140	10	theorem	theorem	NOUN
ejpam-5152	140	11	3	3	NUM
ejpam-5152	140	12	respectively	respectively	ADV
ejpam-5152	140	13	as	as	ADP
ejpam-5152	140	14	constant	constant	ADJ
ejpam-5152	140	15	sequences	sequence	NOUN
ejpam-5152	140	16	,	,	PUNCT
ejpam-5152	140	17	corollary	corollary	ADJ
ejpam-5152	140	18	1	1	NUM
ejpam-5152	140	19	can	can	AUX
ejpam-5152	140	20	be	be	AUX
ejpam-5152	140	21	obtained	obtain	VERB
ejpam-5152	140	22	as	as	ADP
ejpam-5152	140	23	follows	follow	VERB
ejpam-5152	140	24	.	.	PUNCT
ejpam-5152	141	1	this	this	DET
ejpam-5152	141	2	corollary	corollary	ADJ
ejpam-5152	141	3	1	1	NUM
ejpam-5152	141	4	is	be	AUX
ejpam-5152	141	5	a	a	DET
ejpam-5152	141	6	generalization	generalization	NOUN
ejpam-5152	141	7	of	of	ADP
ejpam-5152	141	8	the	the	DET
ejpam-5152	141	9	result	result	NOUN
ejpam-5152	141	10	[	[	X
ejpam-5152	141	11	12	12	NUM
ejpam-5152	141	12	]	]	PUNCT
ejpam-5152	141	13	on	on	ADP
ejpam-5152	141	14	the	the	DET
ejpam-5152	141	15	partial	partial	ADJ
ejpam-5152	141	16	metric	metric	ADJ
ejpam-5152	141	17	space	space	NOUN
ejpam-5152	141	18	.	.	PUNCT
ejpam-5152	142	1	corollary	corollary	ADJ
ejpam-5152	142	2	1	1	NUM
ejpam-5152	142	3	.	.	PUNCT
ejpam-5152	143	1	let	let	VERB
ejpam-5152	143	2	(	(	PUNCT
ejpam-5152	143	3	cbp(x	cbp(x	PROPN
ejpam-5152	143	4	)	)	PUNCT
ejpam-5152	143	5	,	,	PUNCT
ejpam-5152	143	6	hp	hp	PROPN
ejpam-5152	143	7	)	)	PUNCT
ejpam-5152	143	8	be	be	VERB
ejpam-5152	143	9	a	a	DET
ejpam-5152	143	10	p	p	ADJ
ejpam-5152	143	11	-	-	PUNCT
ejpam-5152	143	12	pompeiu	pompeiu	NOUN
ejpam-5152	143	13	-	-	PUNCT
ejpam-5152	143	14	hausdorff	hausdorff	NOUN
ejpam-5152	143	15	metric	metric	ADJ
ejpam-5152	143	16	spaces	space	NOUN
ejpam-5152	143	17	.	.	PUNCT
ejpam-5152	144	1	suppose	suppose	VERB
ejpam-5152	144	2	that	that	SCONJ
ejpam-5152	144	3	f	f	X
ejpam-5152	144	4	,	,	PUNCT
ejpam-5152	144	5	g	g	NOUN
ejpam-5152	144	6	:	:	PUNCT
ejpam-5152	144	7	x	x	X
ejpam-5152	144	8	→	→	SYM
ejpam-5152	144	9	cbp(x	cbp(x	PROPN
ejpam-5152	144	10	)	)	PUNCT
ejpam-5152	144	11	with	with	ADP
ejpam-5152	144	12	the	the	DET
ejpam-5152	144	13	following	follow	VERB
ejpam-5152	144	14	condition	condition	NOUN
ejpam-5152	144	15	hp(f	hp(f	PUNCT
ejpam-5152	144	16	(	(	PUNCT
ejpam-5152	144	17	x	x	NOUN
ejpam-5152	144	18	)	)	PUNCT
ejpam-5152	144	19	,	,	PUNCT
ejpam-5152	144	20	g(y	g(y	PROPN
ejpam-5152	144	21	)	)	PUNCT
ejpam-5152	144	22	)	)	PUNCT
ejpam-5152	144	23	≤	≤	NUM
ejpam-5152	144	24	κmax	κmax	VERB
ejpam-5152	144	25	{	{	PUNCT
ejpam-5152	144	26	p(x	p(x	PROPN
ejpam-5152	144	27	,	,	PUNCT
ejpam-5152	144	28	y	y	NOUN
ejpam-5152	144	29	)	)	PUNCT
ejpam-5152	144	30	,	,	PUNCT
ejpam-5152	144	31	p(x	p(x	PROPN
ejpam-5152	144	32	,	,	PUNCT
ejpam-5152	144	33	f	f	PROPN
ejpam-5152	144	34	(	(	PUNCT
ejpam-5152	144	35	x	x	NOUN
ejpam-5152	144	36	)	)	PUNCT
ejpam-5152	144	37	)	)	PUNCT
ejpam-5152	144	38	,	,	PUNCT
ejpam-5152	144	39	p(y	p(y	PROPN
ejpam-5152	144	40	,	,	PUNCT
ejpam-5152	144	41	g(y	g(y	NOUN
ejpam-5152	144	42	)	)	PUNCT
ejpam-5152	144	43	)	)	PUNCT
ejpam-5152	144	44	,	,	PUNCT
ejpam-5152	144	45	1	1	NUM
ejpam-5152	144	46	2	2	NUM
ejpam-5152	144	47	(	(	PUNCT
ejpam-5152	144	48	p(x	p(x	NOUN
ejpam-5152	144	49	,	,	PUNCT
ejpam-5152	144	50	g(y	g(y	NOUN
ejpam-5152	144	51	)	)	PUNCT
ejpam-5152	144	52	)	)	PUNCT
ejpam-5152	145	1	+	+	CCONJ
ejpam-5152	145	2	p(y	p(y	PROPN
ejpam-5152	145	3	,	,	PUNCT
ejpam-5152	145	4	f	f	PROPN
ejpam-5152	145	5	(	(	PUNCT
ejpam-5152	145	6	x	x	NOUN
ejpam-5152	145	7	)	)	PUNCT
ejpam-5152	145	8	)	)	PUNCT
ejpam-5152	145	9	)	)	PUNCT
ejpam-5152	145	10	}	}	PUNCT
ejpam-5152	145	11	,	,	PUNCT
ejpam-5152	145	12	(	(	PUNCT
ejpam-5152	145	13	1	1	X
ejpam-5152	145	14	)	)	PUNCT
ejpam-5152	145	15	for	for	ADP
ejpam-5152	145	16	each	each	DET
ejpam-5152	145	17	x	x	NOUN
ejpam-5152	145	18	,	,	PUNCT
ejpam-5152	145	19	y	y	PROPN
ejpam-5152	145	20	∈	∈	PROPN
ejpam-5152	145	21	x	x	X
ejpam-5152	145	22	where	where	SCONJ
ejpam-5152	145	23	0	0	NUM
ejpam-5152	145	24	≤	≤	NOUN
ejpam-5152	145	25	κ	κ	X
ejpam-5152	145	26	<	<	X
ejpam-5152	145	27	1	1	NUM
ejpam-5152	145	28	,	,	PUNCT
ejpam-5152	145	29	then	then	ADV
ejpam-5152	145	30	f	f	PROPN
ejpam-5152	145	31	and	and	CCONJ
ejpam-5152	145	32	g	g	PROPN
ejpam-5152	145	33	have	have	VERB
ejpam-5152	145	34	a	a	DET
ejpam-5152	145	35	common	common	ADJ
ejpam-5152	145	36	fixed	fix	VERB
ejpam-5152	145	37	point	point	NOUN
ejpam-5152	145	38	.	.	PUNCT
ejpam-5152	146	1	by	by	ADP
ejpam-5152	146	2	utilizing	utilize	VERB
ejpam-5152	146	3	the	the	DET
ejpam-5152	146	4	concept	concept	NOUN
ejpam-5152	146	5	of	of	ADP
ejpam-5152	146	6	pointwise	pointwise	ADJ
ejpam-5152	146	7	convergence	convergence	NOUN
ejpam-5152	146	8	of	of	ADP
ejpam-5152	146	9	set	set	NOUN
ejpam-5152	146	10	-	-	PUNCT
ejpam-5152	146	11	valued	value	VERB
ejpam-5152	146	12	sequences	sequence	NOUN
ejpam-5152	146	13	,	,	PUNCT
ejpam-5152	146	14	we	we	PRON
ejpam-5152	146	15	can	can	AUX
ejpam-5152	146	16	also	also	ADV
ejpam-5152	146	17	investigate	investigate	VERB
ejpam-5152	146	18	the	the	DET
ejpam-5152	146	19	existence	existence	NOUN
ejpam-5152	146	20	of	of	ADP
ejpam-5152	146	21	a	a	DET
ejpam-5152	146	22	common	common	ADJ
ejpam-5152	146	23	fixed	fix	VERB
ejpam-5152	146	24	point	point	NOUN
ejpam-5152	146	25	of	of	ADP
ejpam-5152	146	26	set	set	NOUN
ejpam-5152	146	27	-	-	PUNCT
ejpam-5152	146	28	valued	value	VERB
ejpam-5152	146	29	mappings	mapping	NOUN
ejpam-5152	146	30	.	.	PUNCT
ejpam-5152	147	1	let	let	AUX
ejpam-5152	147	2	see	see	VERB
ejpam-5152	147	3	on	on	ADP
ejpam-5152	147	4	the	the	DET
ejpam-5152	147	5	following	follow	VERB
ejpam-5152	147	6	theorem	theorem	NOUN
ejpam-5152	147	7	.	.	PUNCT
ejpam-5152	147	8	theorem	theorem	NOUN
ejpam-5152	147	9	4	4	NUM
ejpam-5152	147	10	.	.	PUNCT
ejpam-5152	148	1	let	let	VERB
ejpam-5152	148	2	(	(	PUNCT
ejpam-5152	148	3	cbp(x	cbp(x	PROPN
ejpam-5152	148	4	)	)	PUNCT
ejpam-5152	148	5	,	,	PUNCT
ejpam-5152	148	6	hp	hp	PROPN
ejpam-5152	148	7	)	)	PUNCT
ejpam-5152	148	8	be	be	VERB
ejpam-5152	148	9	a	a	DET
ejpam-5152	148	10	p	p	ADJ
ejpam-5152	148	11	-	-	PUNCT
ejpam-5152	148	12	pompeiu	pompeiu	NOUN
ejpam-5152	148	13	-	-	PUNCT
ejpam-5152	148	14	hausdorff	hausdorff	NOUN
ejpam-5152	148	15	metric	metric	ADJ
ejpam-5152	148	16	spaces	space	NOUN
ejpam-5152	148	17	.	.	PUNCT
ejpam-5152	149	1	suppose	suppose	VERB
ejpam-5152	149	2	that	that	SCONJ
ejpam-5152	149	3	fn	fn	NOUN
ejpam-5152	149	4	,	,	PUNCT
ejpam-5152	149	5	gn	gn	INTJ
ejpam-5152	149	6	:	:	PUNCT
ejpam-5152	149	7	x	x	X
ejpam-5152	149	8	→	→	SYM
ejpam-5152	149	9	cbp(x	cbp(x	PROPN
ejpam-5152	149	10	)	)	PUNCT
ejpam-5152	149	11	sequences	sequence	NOUN
ejpam-5152	149	12	in	in	ADP
ejpam-5152	149	13	cbp(x	cbp(x	PROPN
ejpam-5152	149	14	)	)	PUNCT
ejpam-5152	149	15	.	.	PUNCT
ejpam-5152	150	1	sequences	sequence	NOUN
ejpam-5152	150	2	fn	fn	PROPN
ejpam-5152	150	3	,	,	PUNCT
ejpam-5152	151	1	gn	gn	PROPN
ejpam-5152	151	2	converging	converge	VERB
ejpam-5152	151	3	pointwise	pointwise	PROPN
ejpam-5152	151	4	to	to	ADP
ejpam-5152	151	5	f	f	PROPN
ejpam-5152	151	6	,	,	PUNCT
ejpam-5152	151	7	g	g	NOUN
ejpam-5152	151	8	:	:	PUNCT
ejpam-5152	151	9	x	x	X
ejpam-5152	151	10	→	→	SYM
ejpam-5152	151	11	cbp(x	cbp(x	PROPN
ejpam-5152	151	12	)	)	PUNCT
ejpam-5152	151	13	respectively	respectively	ADV
ejpam-5152	151	14	.	.	PUNCT
ejpam-5152	152	1	if	if	SCONJ
ejpam-5152	152	2	the	the	DET
ejpam-5152	152	3	following	follow	VERB
ejpam-5152	152	4	condition	condition	NOUN
ejpam-5152	152	5	holds	hold	VERB
ejpam-5152	152	6	hp(fn(x	hp(fn(x	NOUN
ejpam-5152	152	7	)	)	PUNCT
ejpam-5152	152	8	,	,	PUNCT
ejpam-5152	152	9	gn(y	gn(y	NOUN
ejpam-5152	152	10	)	)	PUNCT
ejpam-5152	152	11	)	)	PUNCT
ejpam-5152	152	12	≤	≤	NUM
ejpam-5152	152	13	κmax	κmax	VERB
ejpam-5152	152	14	{	{	PUNCT
ejpam-5152	152	15	p(x	p(x	PROPN
ejpam-5152	152	16	,	,	PUNCT
ejpam-5152	152	17	y	y	NOUN
ejpam-5152	152	18	)	)	PUNCT
ejpam-5152	152	19	,	,	PUNCT
ejpam-5152	152	20	p(x	p(x	PROPN
ejpam-5152	152	21	,	,	PUNCT
ejpam-5152	152	22	fn(x	fn(x	NOUN
ejpam-5152	152	23	)	)	PUNCT
ejpam-5152	152	24	)	)	PUNCT
ejpam-5152	152	25	,	,	PUNCT
ejpam-5152	152	26	p(y	p(y	PROPN
ejpam-5152	152	27	,	,	PUNCT
ejpam-5152	152	28	gn(y	gn(y	NOUN
ejpam-5152	152	29	)	)	PUNCT
ejpam-5152	152	30	)	)	PUNCT
ejpam-5152	152	31	,	,	PUNCT
ejpam-5152	152	32	1	1	NUM
ejpam-5152	152	33	2	2	NUM
ejpam-5152	152	34	(	(	PUNCT
ejpam-5152	152	35	p(x	p(x	PROPN
ejpam-5152	152	36	,	,	PUNCT
ejpam-5152	152	37	gn(y	gn(y	NOUN
ejpam-5152	152	38	)	)	PUNCT
ejpam-5152	152	39	)	)	PUNCT
ejpam-5152	153	1	+	+	CCONJ
ejpam-5152	153	2	p(y	p(y	NOUN
ejpam-5152	153	3	,	,	PUNCT
ejpam-5152	153	4	fn(x	fn(x	NOUN
ejpam-5152	153	5	)	)	PUNCT
ejpam-5152	153	6	)	)	PUNCT
ejpam-5152	153	7	)	)	PUNCT
ejpam-5152	153	8	}	}	PUNCT
ejpam-5152	153	9	,	,	PUNCT
ejpam-5152	153	10	(	(	PUNCT
ejpam-5152	153	11	2	2	X
ejpam-5152	153	12	)	)	PUNCT
ejpam-5152	153	13	for	for	ADP
ejpam-5152	153	14	every	every	DET
ejpam-5152	153	15	x	x	PROPN
ejpam-5152	153	16	,	,	PUNCT
ejpam-5152	153	17	y	y	PROPN
ejpam-5152	153	18	∈	∈	PROPN
ejpam-5152	153	19	x	x	X
ejpam-5152	153	20	and	and	CCONJ
ejpam-5152	153	21	n	n	CCONJ
ejpam-5152	153	22	∈	∈	PROPN
ejpam-5152	153	23	n	n	CCONJ
ejpam-5152	153	24	where	where	SCONJ
ejpam-5152	153	25	0	0	NUM
ejpam-5152	153	26	≤	≤	NUM
ejpam-5152	153	27	κ	κ	X
ejpam-5152	153	28	<	<	X
ejpam-5152	153	29	1	1	NUM
ejpam-5152	153	30	then	then	ADV
ejpam-5152	153	31	f	f	PROPN
ejpam-5152	153	32	and	and	CCONJ
ejpam-5152	153	33	g	g	PROPN
ejpam-5152	153	34	have	have	VERB
ejpam-5152	153	35	a	a	DET
ejpam-5152	153	36	common	common	ADJ
ejpam-5152	153	37	fixed	fix	VERB
ejpam-5152	153	38	point	point	NOUN
ejpam-5152	153	39	.	.	PUNCT
ejpam-5152	154	1	proof	proof	NOUN
ejpam-5152	154	2	.	.	PUNCT
ejpam-5152	155	1	take	take	VERB
ejpam-5152	155	2	any	any	DET
ejpam-5152	155	3	point	point	NOUN
ejpam-5152	155	4	x	x	NOUN
ejpam-5152	155	5	,	,	PUNCT
ejpam-5152	155	6	y	y	PROPN
ejpam-5152	155	7	∈	∈	PROPN
ejpam-5152	155	8	x.	x.	NOUN
ejpam-5152	155	9	let	let	VERB
ejpam-5152	155	10	u	u	PRON
ejpam-5152	155	11	∈	∈	PROPN
ejpam-5152	155	12	fn(x	fn(x	X
ejpam-5152	155	13	)	)	PUNCT
ejpam-5152	155	14	and	and	CCONJ
ejpam-5152	155	15	v	v	ADP
ejpam-5152	155	16	∈	∈	PROPN
ejpam-5152	155	17	f	f	X
ejpam-5152	155	18	(	(	PUNCT
ejpam-5152	155	19	x	x	NOUN
ejpam-5152	155	20	)	)	PUNCT
ejpam-5152	155	21	,	,	PUNCT
ejpam-5152	155	22	then	then	ADV
ejpam-5152	155	23	we	we	PRON
ejpam-5152	155	24	have	have	VERB
ejpam-5152	155	25	p(y	p(y	NOUN
ejpam-5152	155	26	,	,	PUNCT
ejpam-5152	155	27	u	u	NOUN
ejpam-5152	155	28	)	)	PUNCT
ejpam-5152	155	29	≤	≤	NOUN
ejpam-5152	155	30	p(y	p(y	PROPN
ejpam-5152	155	31	,	,	PUNCT
ejpam-5152	155	32	v	v	NOUN
ejpam-5152	155	33	)	)	PUNCT
ejpam-5152	156	1	+	+	CCONJ
ejpam-5152	156	2	p(v	p(v	NOUN
ejpam-5152	156	3	,	,	PUNCT
ejpam-5152	156	4	u)−	u)−	PROPN
ejpam-5152	156	5	p(v	p(v	PROPN
ejpam-5152	156	6	,	,	PUNCT
ejpam-5152	156	7	v	v	NOUN
ejpam-5152	156	8	)	)	PUNCT
ejpam-5152	156	9	≤	≤	NUM
ejpam-5152	156	10	p(y	p(y	PROPN
ejpam-5152	156	11	,	,	PUNCT
ejpam-5152	156	12	v	v	NOUN
ejpam-5152	156	13	)	)	PUNCT
ejpam-5152	157	1	+	+	NUM
ejpam-5152	157	2	p(v	p(v	NOUN
ejpam-5152	157	3	,	,	PUNCT
ejpam-5152	157	4	u	u	NOUN
ejpam-5152	157	5	)	)	PUNCT
ejpam-5152	157	6	≤	≤	NOUN
ejpam-5152	157	7	p(y	p(y	PROPN
ejpam-5152	157	8	,	,	PUNCT
ejpam-5152	157	9	f	f	PROPN
ejpam-5152	157	10	(	(	PUNCT
ejpam-5152	157	11	x	x	NOUN
ejpam-5152	157	12	)	)	PUNCT
ejpam-5152	157	13	)	)	PUNCT
ejpam-5152	158	1	+	+	PUNCT
ejpam-5152	158	2	p(u	p(u	ADJ
ejpam-5152	158	3	,	,	PUNCT
ejpam-5152	158	4	f	f	PROPN
ejpam-5152	158	5	(	(	PUNCT
ejpam-5152	158	6	x	x	NOUN
ejpam-5152	158	7	)	)	PUNCT
ejpam-5152	158	8	)	)	PUNCT
ejpam-5152	158	9	.	.	PUNCT
ejpam-5152	159	1	consequently	consequently	ADV
ejpam-5152	159	2	p(y	p(y	NOUN
ejpam-5152	159	3	,	,	PUNCT
ejpam-5152	159	4	fn(x	fn(x	X
ejpam-5152	159	5	)	)	PUNCT
ejpam-5152	159	6	)	)	PUNCT
ejpam-5152	160	1	≤	≤	NUM
ejpam-5152	160	2	p(y	p(y	PROPN
ejpam-5152	160	3	,	,	PUNCT
ejpam-5152	160	4	f	f	PROPN
ejpam-5152	160	5	(	(	PUNCT
ejpam-5152	160	6	x	x	NOUN
ejpam-5152	160	7	)	)	PUNCT
ejpam-5152	160	8	)	)	PUNCT
ejpam-5152	161	1	+	+	ADJ
ejpam-5152	161	2	hp(fn(x	hp(fn(x	X
ejpam-5152	161	3	)	)	PUNCT
ejpam-5152	161	4	,	,	PUNCT
ejpam-5152	161	5	f	f	PROPN
ejpam-5152	161	6	(	(	PUNCT
ejpam-5152	161	7	x	x	NOUN
ejpam-5152	161	8	)	)	PUNCT
ejpam-5152	161	9	)	)	PUNCT
ejpam-5152	161	10	.	.	PUNCT
ejpam-5152	162	1	on	on	ADP
ejpam-5152	162	2	the	the	DET
ejpam-5152	162	3	other	other	ADJ
ejpam-5152	162	4	side	side	NOUN
ejpam-5152	162	5	we	we	PRON
ejpam-5152	162	6	also	also	ADV
ejpam-5152	162	7	have	have	VERB
ejpam-5152	162	8	the	the	DET
ejpam-5152	162	9	following	follow	VERB
ejpam-5152	162	10	condition	condition	NOUN
ejpam-5152	162	11	p(y	p(y	NOUN
ejpam-5152	162	12	,	,	PUNCT
ejpam-5152	162	13	v	v	NOUN
ejpam-5152	162	14	)	)	PUNCT
ejpam-5152	162	15	≤	≤	NUM
ejpam-5152	162	16	p(y	p(y	PROPN
ejpam-5152	162	17	,	,	PUNCT
ejpam-5152	162	18	u	u	NOUN
ejpam-5152	162	19	)	)	PUNCT
ejpam-5152	162	20	+	+	CCONJ
ejpam-5152	163	1	p(u	p(u	ADJ
ejpam-5152	163	2	,	,	PUNCT
ejpam-5152	163	3	v)−	v)−	PROPN
ejpam-5152	163	4	p(u	p(u	NOUN
ejpam-5152	163	5	,	,	PUNCT
ejpam-5152	163	6	u	u	NOUN
ejpam-5152	163	7	)	)	PUNCT
ejpam-5152	163	8	≤	≤	NOUN
ejpam-5152	163	9	p(y	p(y	PROPN
ejpam-5152	163	10	,	,	PUNCT
ejpam-5152	163	11	u	u	NOUN
ejpam-5152	163	12	)	)	PUNCT
ejpam-5152	163	13	+	+	CCONJ
ejpam-5152	163	14	p(u	p(u	ADJ
ejpam-5152	163	15	,	,	PUNCT
ejpam-5152	163	16	v	v	NOUN
ejpam-5152	163	17	)	)	PUNCT
ejpam-5152	163	18	≤	≤	NUM
ejpam-5152	163	19	p(y	p(y	NOUN
ejpam-5152	163	20	,	,	PUNCT
ejpam-5152	163	21	fn(x	fn(x	NOUN
ejpam-5152	163	22	)	)	PUNCT
ejpam-5152	163	23	)	)	PUNCT
ejpam-5152	164	1	+	+	CCONJ
ejpam-5152	164	2	p(v	p(v	NOUN
ejpam-5152	164	3	,	,	PUNCT
ejpam-5152	164	4	fn(x	fn(x	NOUN
ejpam-5152	164	5	)	)	PUNCT
ejpam-5152	164	6	)	)	PUNCT
ejpam-5152	164	7	.	.	PUNCT
ejpam-5152	165	1	a.	a.	PROPN
ejpam-5152	165	2	ekayanti	ekayanti	PROPN
ejpam-5152	165	3	et	et	PROPN
ejpam-5152	165	4	al	al	PROPN
ejpam-5152	165	5	.	.	PUNCT
ejpam-5152	165	6	/	/	SYM
ejpam-5152	165	7	eur	eur	PROPN
ejpam-5152	165	8	.	.	PUNCT
ejpam-5152	166	1	j.	j.	PROPN
ejpam-5152	166	2	pure	pure	PROPN
ejpam-5152	166	3	appl	appl	PROPN
ejpam-5152	166	4	.	.	PROPN
ejpam-5152	166	5	math	math	PROPN
ejpam-5152	166	6	,	,	PUNCT
ejpam-5152	166	7	17	17	NUM
ejpam-5152	166	8	(	(	PUNCT
ejpam-5152	166	9	2	2	NUM
ejpam-5152	166	10	)	)	PUNCT
ejpam-5152	166	11	(	(	PUNCT
ejpam-5152	166	12	2024	2024	NUM
ejpam-5152	166	13	)	)	PUNCT
ejpam-5152	166	14	,	,	PUNCT
ejpam-5152	166	15	996	996	NUM
ejpam-5152	166	16	-	-	SYM
ejpam-5152	166	17	1008	1008	NUM
ejpam-5152	166	18	1002	1002	NUM
ejpam-5152	166	19	it	it	PRON
ejpam-5152	166	20	implies	imply	VERB
ejpam-5152	166	21	p(y	p(y	PROPN
ejpam-5152	166	22	,	,	PUNCT
ejpam-5152	166	23	f	f	PROPN
ejpam-5152	166	24	(	(	PUNCT
ejpam-5152	166	25	x	x	NOUN
ejpam-5152	166	26	)	)	PUNCT
ejpam-5152	166	27	)	)	PUNCT
ejpam-5152	167	1	≤	≤	NUM
ejpam-5152	167	2	p(y	p(y	PROPN
ejpam-5152	167	3	,	,	PUNCT
ejpam-5152	167	4	fn(x	fn(x	NOUN
ejpam-5152	167	5	)	)	PUNCT
ejpam-5152	167	6	)	)	PUNCT
ejpam-5152	168	1	+	+	X
ejpam-5152	168	2	hp(f	hp(f	X
ejpam-5152	168	3	(	(	PUNCT
ejpam-5152	168	4	x	x	NOUN
ejpam-5152	168	5	)	)	PUNCT
ejpam-5152	168	6	,	,	PUNCT
ejpam-5152	168	7	fn(x	fn(x	X
ejpam-5152	168	8	)	)	PUNCT
ejpam-5152	168	9	)	)	PUNCT
ejpam-5152	168	10	.	.	PUNCT
ejpam-5152	169	1	therefore	therefore	ADV
ejpam-5152	169	2	,	,	PUNCT
ejpam-5152	169	3	we	we	PRON
ejpam-5152	169	4	have	have	VERB
ejpam-5152	169	5	|p(y	|p(y	PROPN
ejpam-5152	169	6	,	,	PUNCT
ejpam-5152	169	7	fn(x))−	fn(x))−	ADJ
ejpam-5152	169	8	p(y	p(y	PROPN
ejpam-5152	169	9	,	,	PUNCT
ejpam-5152	169	10	f	f	PROPN
ejpam-5152	169	11	(	(	PUNCT
ejpam-5152	169	12	x))|	x))|	PROPN
ejpam-5152	169	13	≤	≤	PROPN
ejpam-5152	169	14	hp(fn(x	hp(fn(x	X
ejpam-5152	169	15	)	)	PUNCT
ejpam-5152	169	16	,	,	PUNCT
ejpam-5152	169	17	f	f	PROPN
ejpam-5152	169	18	(	(	PUNCT
ejpam-5152	169	19	x	x	NOUN
ejpam-5152	169	20	)	)	PUNCT
ejpam-5152	169	21	)	)	PUNCT
ejpam-5152	169	22	.	.	PUNCT
ejpam-5152	170	1	(	(	PUNCT
ejpam-5152	170	2	3	3	X
ejpam-5152	170	3	)	)	PUNCT
ejpam-5152	170	4	on	on	ADP
ejpam-5152	170	5	the	the	DET
ejpam-5152	170	6	similar	similar	ADJ
ejpam-5152	170	7	way	way	NOUN
ejpam-5152	170	8	we	we	PRON
ejpam-5152	170	9	can	can	AUX
ejpam-5152	170	10	also	also	ADV
ejpam-5152	170	11	show	show	VERB
ejpam-5152	170	12	that	that	SCONJ
ejpam-5152	170	13	|p(x	|p(x	PROPN
ejpam-5152	170	14	,	,	PUNCT
ejpam-5152	170	15	gn(y))−	gn(y))−	NOUN
ejpam-5152	170	16	p(x	p(x	NOUN
ejpam-5152	170	17	,	,	PUNCT
ejpam-5152	170	18	g(y))|	g(y))|	VERB
ejpam-5152	170	19	≤	≤	NUM
ejpam-5152	170	20	hp(gn(y	hp(gn(y	PROPN
ejpam-5152	170	21	)	)	PUNCT
ejpam-5152	170	22	,	,	PUNCT
ejpam-5152	170	23	g(y	g(y	PROPN
ejpam-5152	170	24	)	)	PUNCT
ejpam-5152	170	25	)	)	PUNCT
ejpam-5152	170	26	.	.	PUNCT
ejpam-5152	171	1	(	(	PUNCT
ejpam-5152	171	2	4	4	X
ejpam-5152	171	3	)	)	PUNCT
ejpam-5152	171	4	furthermore	furthermore	ADV
ejpam-5152	171	5	,	,	PUNCT
ejpam-5152	171	6	by	by	ADP
ejpam-5152	171	7	using	use	VERB
ejpam-5152	171	8	inequality	inequality	NOUN
ejpam-5152	171	9	(	(	PUNCT
ejpam-5152	171	10	3	3	NUM
ejpam-5152	171	11	)	)	PUNCT
ejpam-5152	171	12	and	and	CCONJ
ejpam-5152	171	13	(	(	PUNCT
ejpam-5152	171	14	4	4	NUM
ejpam-5152	171	15	)	)	PUNCT
ejpam-5152	171	16	and	and	CCONJ
ejpam-5152	171	17	also	also	ADV
ejpam-5152	171	18	the	the	DET
ejpam-5152	171	19	continuity	continuity	NOUN
ejpam-5152	171	20	of	of	ADP
ejpam-5152	171	21	hp	hp	NOUN
ejpam-5152	171	22	then	then	ADV
ejpam-5152	171	23	by	by	ADP
ejpam-5152	171	24	taking	take	VERB
ejpam-5152	171	25	n	n	PRON
ejpam-5152	171	26	→	→	SYM
ejpam-5152	171	27	∞	∞	NUM
ejpam-5152	171	28	in	in	ADP
ejpam-5152	171	29	inequality	inequality	NOUN
ejpam-5152	171	30	(	(	PUNCT
ejpam-5152	171	31	2	2	X
ejpam-5152	171	32	)	)	PUNCT
ejpam-5152	171	33	we	we	PRON
ejpam-5152	171	34	obtain	obtain	VERB
ejpam-5152	171	35	hp(f	hp(f	PUNCT
ejpam-5152	171	36	(	(	PUNCT
ejpam-5152	171	37	x	x	NOUN
ejpam-5152	171	38	)	)	PUNCT
ejpam-5152	171	39	,	,	PUNCT
ejpam-5152	171	40	g(y	g(y	PROPN
ejpam-5152	171	41	)	)	PUNCT
ejpam-5152	171	42	)	)	PUNCT
ejpam-5152	172	1	≤	≤	NUM
ejpam-5152	172	2	κmax	κmax	VERB
ejpam-5152	172	3	{	{	PUNCT
ejpam-5152	172	4	p(x	p(x	PROPN
ejpam-5152	172	5	,	,	PUNCT
ejpam-5152	172	6	y	y	NOUN
ejpam-5152	172	7	)	)	PUNCT
ejpam-5152	172	8	,	,	PUNCT
ejpam-5152	172	9	p(x	p(x	PROPN
ejpam-5152	172	10	,	,	PUNCT
ejpam-5152	172	11	f	f	PROPN
ejpam-5152	172	12	(	(	PUNCT
ejpam-5152	172	13	x	x	NOUN
ejpam-5152	172	14	)	)	PUNCT
ejpam-5152	172	15	)	)	PUNCT
ejpam-5152	172	16	,	,	PUNCT
ejpam-5152	172	17	p(y	p(y	PROPN
ejpam-5152	172	18	,	,	PUNCT
ejpam-5152	172	19	g(y	g(y	NOUN
ejpam-5152	172	20	)	)	PUNCT
ejpam-5152	172	21	)	)	PUNCT
ejpam-5152	172	22	,	,	PUNCT
ejpam-5152	172	23	1	1	NUM
ejpam-5152	172	24	2	2	NUM
ejpam-5152	172	25	(	(	PUNCT
ejpam-5152	172	26	p(x	p(x	NOUN
ejpam-5152	172	27	,	,	PUNCT
ejpam-5152	172	28	g(y	g(y	NOUN
ejpam-5152	172	29	)	)	PUNCT
ejpam-5152	172	30	)	)	PUNCT
ejpam-5152	172	31	+	+	CCONJ
ejpam-5152	173	1	p(y	p(y	PROPN
ejpam-5152	173	2	,	,	PUNCT
ejpam-5152	173	3	f	f	PROPN
ejpam-5152	173	4	(	(	PUNCT
ejpam-5152	173	5	x	x	NOUN
ejpam-5152	173	6	)	)	PUNCT
ejpam-5152	173	7	)	)	PUNCT
ejpam-5152	173	8	)	)	PUNCT
ejpam-5152	173	9	}	}	PUNCT
ejpam-5152	173	10	,	,	PUNCT
ejpam-5152	173	11	these	these	DET
ejpam-5152	173	12	conditions	condition	NOUN
ejpam-5152	173	13	show	show	VERB
ejpam-5152	173	14	that	that	SCONJ
ejpam-5152	173	15	the	the	DET
ejpam-5152	173	16	set	set	NOUN
ejpam-5152	173	17	-	-	PUNCT
ejpam-5152	173	18	valued	value	VERB
ejpam-5152	173	19	mapping	mapping	NOUN
ejpam-5152	173	20	f	f	NOUN
ejpam-5152	173	21	and	and	CCONJ
ejpam-5152	173	22	g	g	PROPN
ejpam-5152	173	23	satisfies	satisfy	VERB
ejpam-5152	173	24	the	the	DET
ejpam-5152	173	25	hypothesis	hypothesis	NOUN
ejpam-5152	173	26	on	on	ADP
ejpam-5152	173	27	corollary	corollary	ADJ
ejpam-5152	173	28	1	1	NUM
ejpam-5152	173	29	.	.	PUNCT
ejpam-5152	173	30	thus	thus	ADV
ejpam-5152	173	31	,	,	PUNCT
ejpam-5152	173	32	based	base	VERB
ejpam-5152	173	33	on	on	ADP
ejpam-5152	173	34	corollary	corollary	ADJ
ejpam-5152	173	35	1	1	NUM
ejpam-5152	173	36	it	it	PRON
ejpam-5152	173	37	can	can	AUX
ejpam-5152	173	38	be	be	AUX
ejpam-5152	173	39	concluded	conclude	VERB
ejpam-5152	173	40	that	that	SCONJ
ejpam-5152	173	41	f	f	PROPN
ejpam-5152	173	42	and	and	CCONJ
ejpam-5152	173	43	g	g	PROPN
ejpam-5152	173	44	have	have	VERB
ejpam-5152	173	45	a	a	DET
ejpam-5152	173	46	common	common	ADJ
ejpam-5152	173	47	fixed	fix	VERB
ejpam-5152	173	48	point	point	NOUN
ejpam-5152	173	49	.	.	PUNCT
ejpam-5152	174	1	this	this	PRON
ejpam-5152	174	2	complete	complete	ADJ
ejpam-5152	174	3	the	the	DET
ejpam-5152	174	4	proof	proof	NOUN
ejpam-5152	174	5	.	.	PUNCT
ejpam-5152	175	1	let	let	VERB
ejpam-5152	175	2	we	we	PRON
ejpam-5152	175	3	consider	consider	VERB
ejpam-5152	175	4	that	that	PRON
ejpam-5152	175	5	for	for	ADP
ejpam-5152	175	6	any	any	DET
ejpam-5152	175	7	positive	positive	ADJ
ejpam-5152	175	8	real	real	ADJ
ejpam-5152	175	9	numbers	number	NOUN
ejpam-5152	175	10	s	s	PART
ejpam-5152	175	11	and	and	CCONJ
ejpam-5152	175	12	t	t	PROPN
ejpam-5152	175	13	holds	hold	VERB
ejpam-5152	175	14	1	1	NUM
ejpam-5152	175	15	2	2	NUM
ejpam-5152	175	16	(	(	PUNCT
ejpam-5152	175	17	s+	s+	NOUN
ejpam-5152	175	18	t	t	PROPN
ejpam-5152	175	19	)	)	PUNCT
ejpam-5152	175	20	≤	≤	NOUN
ejpam-5152	175	21	max{s	max{	NOUN
ejpam-5152	175	22	,	,	PUNCT
ejpam-5152	175	23	t	t	PROPN
ejpam-5152	175	24	}	}	PUNCT
ejpam-5152	175	25	.	.	PUNCT
ejpam-5152	176	1	it	it	PRON
ejpam-5152	176	2	implies	imply	VERB
ejpam-5152	176	3	for	for	ADP
ejpam-5152	176	4	any	any	DET
ejpam-5152	176	5	positive	positive	ADJ
ejpam-5152	176	6	real	real	ADJ
ejpam-5152	176	7	numbers	number	NOUN
ejpam-5152	176	8	p	p	X
ejpam-5152	176	9	,	,	PUNCT
ejpam-5152	176	10	q	q	ADJ
ejpam-5152	176	11	,	,	PUNCT
ejpam-5152	176	12	r	r	NOUN
ejpam-5152	176	13	,	,	PUNCT
ejpam-5152	176	14	s	s	PART
ejpam-5152	176	15	and	and	CCONJ
ejpam-5152	176	16	t	t	NOUN
ejpam-5152	176	17	we	we	PRON
ejpam-5152	176	18	have	have	VERB
ejpam-5152	176	19	max	max	PROPN
ejpam-5152	176	20	{	{	PUNCT
ejpam-5152	176	21	p	p	X
ejpam-5152	176	22	,	,	PUNCT
ejpam-5152	176	23	q	q	ADJ
ejpam-5152	176	24	,	,	PUNCT
ejpam-5152	176	25	r	r	NOUN
ejpam-5152	176	26	,	,	PUNCT
ejpam-5152	176	27	1	1	NUM
ejpam-5152	176	28	2	2	NUM
ejpam-5152	176	29	(	(	PUNCT
ejpam-5152	176	30	s+	s+	NOUN
ejpam-5152	176	31	t	t	PROPN
ejpam-5152	176	32	)	)	PUNCT
ejpam-5152	176	33	}	}	PUNCT
ejpam-5152	176	34	≤	≤	NOUN
ejpam-5152	176	35	max{p	max{p	NOUN
ejpam-5152	176	36	,	,	PUNCT
ejpam-5152	176	37	q	q	NOUN
ejpam-5152	176	38	,	,	PUNCT
ejpam-5152	176	39	r	r	NOUN
ejpam-5152	176	40	,	,	PUNCT
ejpam-5152	176	41	s	s	PROPN
ejpam-5152	176	42	,	,	PUNCT
ejpam-5152	176	43	t	t	PROPN
ejpam-5152	176	44	}	}	PUNCT
ejpam-5152	176	45	.	.	PUNCT
ejpam-5152	177	1	(	(	PUNCT
ejpam-5152	177	2	5	5	NUM
ejpam-5152	177	3	)	)	PUNCT
ejpam-5152	177	4	therefore	therefore	ADV
ejpam-5152	177	5	,	,	PUNCT
ejpam-5152	177	6	we	we	PRON
ejpam-5152	177	7	can	can	AUX
ejpam-5152	177	8	derive	derive	VERB
ejpam-5152	177	9	a	a	DET
ejpam-5152	177	10	generalization	generalization	NOUN
ejpam-5152	177	11	of	of	ADP
ejpam-5152	177	12	contractions	contraction	NOUN
ejpam-5152	177	13	that	that	PRON
ejpam-5152	177	14	the	the	DET
ejpam-5152	177	15	theorem	theorem	NOUN
ejpam-5152	177	16	uses	use	VERB
ejpam-5152	177	17	as	as	ADV
ejpam-5152	177	18	well	well	ADV
ejpam-5152	177	19	as	as	ADP
ejpam-5152	177	20	the	the	DET
ejpam-5152	177	21	corollary	corollary	NOUN
ejpam-5152	177	22	on	on	ADP
ejpam-5152	177	23	the	the	DET
ejpam-5152	177	24	previous	previous	ADJ
ejpam-5152	177	25	discussion	discussion	NOUN
ejpam-5152	177	26	.	.	PUNCT
ejpam-5152	178	1	in	in	ADP
ejpam-5152	178	2	corollary	corollary	ADJ
ejpam-5152	178	3	1	1	NUM
ejpam-5152	178	4	,	,	PUNCT
ejpam-5152	178	5	which	which	PRON
ejpam-5152	178	6	indicates	indicate	VERB
ejpam-5152	178	7	the	the	DET
ejpam-5152	178	8	existence	existence	NOUN
ejpam-5152	178	9	of	of	ADP
ejpam-5152	178	10	a	a	DET
ejpam-5152	178	11	common	common	ADJ
ejpam-5152	178	12	fixed	fix	VERB
ejpam-5152	178	13	point	point	NOUN
ejpam-5152	178	14	of	of	ADP
ejpam-5152	178	15	set	set	NOUN
ejpam-5152	178	16	-	-	PUNCT
ejpam-5152	178	17	valued	value	VERB
ejpam-5152	178	18	mapping	mapping	NOUN
ejpam-5152	178	19	,	,	PUNCT
ejpam-5152	178	20	by	by	ADP
ejpam-5152	178	21	utilizing	utilize	VERB
ejpam-5152	178	22	inequality	inequality	NOUN
ejpam-5152	178	23	(	(	PUNCT
ejpam-5152	178	24	5	5	NUM
ejpam-5152	178	25	)	)	PUNCT
ejpam-5152	178	26	we	we	PRON
ejpam-5152	178	27	can	can	AUX
ejpam-5152	178	28	obtain	obtain	VERB
ejpam-5152	178	29	a	a	DET
ejpam-5152	178	30	generalization	generalization	NOUN
ejpam-5152	178	31	of	of	ADP
ejpam-5152	178	32	contractions	contraction	NOUN
ejpam-5152	178	33	(	(	PUNCT
ejpam-5152	178	34	1	1	X
ejpam-5152	178	35	)	)	PUNCT
ejpam-5152	178	36	as	as	SCONJ
ejpam-5152	178	37	follows	follow	VERB
ejpam-5152	178	38	hp(f	hp(f	PUNCT
ejpam-5152	178	39	(	(	PUNCT
ejpam-5152	178	40	x	x	NOUN
ejpam-5152	178	41	)	)	PUNCT
ejpam-5152	178	42	,	,	PUNCT
ejpam-5152	178	43	g(y	g(y	PROPN
ejpam-5152	178	44	)	)	PUNCT
ejpam-5152	178	45	)	)	PUNCT
ejpam-5152	179	1	≤	≤	NUM
ejpam-5152	179	2	κmax	κmax	VERB
ejpam-5152	179	3	{	{	PUNCT
ejpam-5152	179	4	p(x	p(x	PROPN
ejpam-5152	179	5	,	,	PUNCT
ejpam-5152	179	6	y	y	NOUN
ejpam-5152	179	7	)	)	PUNCT
ejpam-5152	179	8	,	,	PUNCT
ejpam-5152	179	9	p(x	p(x	PROPN
ejpam-5152	179	10	,	,	PUNCT
ejpam-5152	179	11	f	f	PROPN
ejpam-5152	179	12	(	(	PUNCT
ejpam-5152	179	13	x	x	NOUN
ejpam-5152	179	14	)	)	PUNCT
ejpam-5152	179	15	)	)	PUNCT
ejpam-5152	179	16	,	,	PUNCT
ejpam-5152	179	17	p(y	p(y	PROPN
ejpam-5152	179	18	,	,	PUNCT
ejpam-5152	179	19	g(y	g(y	NOUN
ejpam-5152	179	20	)	)	PUNCT
ejpam-5152	179	21	)	)	PUNCT
ejpam-5152	179	22	,	,	PUNCT
ejpam-5152	179	23	p(x	p(x	NOUN
ejpam-5152	179	24	,	,	PUNCT
ejpam-5152	179	25	g(y	g(y	NOUN
ejpam-5152	179	26	)	)	PUNCT
ejpam-5152	179	27	)	)	PUNCT
ejpam-5152	179	28	,	,	PUNCT
ejpam-5152	179	29	p(y	p(y	PROPN
ejpam-5152	179	30	,	,	PUNCT
ejpam-5152	179	31	f	f	PROPN
ejpam-5152	179	32	(	(	PUNCT
ejpam-5152	179	33	x	x	NOUN
ejpam-5152	179	34	)	)	PUNCT
ejpam-5152	179	35	)	)	PUNCT
ejpam-5152	179	36	}	}	PUNCT
ejpam-5152	179	37	.	.	PUNCT
ejpam-5152	180	1	(	(	PUNCT
ejpam-5152	180	2	6	6	NUM
ejpam-5152	180	3	)	)	PUNCT
ejpam-5152	180	4	on	on	ADP
ejpam-5152	180	5	the	the	DET
ejpam-5152	180	6	other	other	ADJ
ejpam-5152	180	7	sides	side	NOUN
ejpam-5152	180	8	,	,	PUNCT
ejpam-5152	180	9	singh	singh	PROPN
ejpam-5152	180	10	has	have	AUX
ejpam-5152	180	11	given	give	VERB
ejpam-5152	180	12	a	a	DET
ejpam-5152	180	13	definition	definition	NOUN
ejpam-5152	180	14	of	of	ADP
ejpam-5152	180	15	a	a	DET
ejpam-5152	180	16	function	function	NOUN
ejpam-5152	180	17	in	in	ADP
ejpam-5152	180	18	generalizing	generalize	VERB
ejpam-5152	180	19	the	the	DET
ejpam-5152	180	20	principle	principle	NOUN
ejpam-5152	180	21	of	of	ADP
ejpam-5152	180	22	contraction	contraction	NOUN
ejpam-5152	180	23	of	of	ADP
ejpam-5152	180	24	several	several	ADJ
ejpam-5152	180	25	references	reference	NOUN
ejpam-5152	180	26	therein	therein	ADV
ejpam-5152	180	27	(	(	PUNCT
ejpam-5152	180	28	definition	definition	NOUN
ejpam-5152	180	29	2.1	2.1	NUM
ejpam-5152	180	30	in	in	ADP
ejpam-5152	180	31	[	[	X
ejpam-5152	180	32	19	19	NUM
ejpam-5152	180	33	]	]	PUNCT
ejpam-5152	180	34	)	)	PUNCT
ejpam-5152	180	35	as	as	SCONJ
ejpam-5152	180	36	follows	follow	VERB
ejpam-5152	180	37	.	.	PUNCT
ejpam-5152	181	1	definition	definition	NOUN
ejpam-5152	181	2	4	4	NUM
ejpam-5152	181	3	.	.	PUNCT
ejpam-5152	181	4	suppose	suppose	VERB
ejpam-5152	181	5	that	that	SCONJ
ejpam-5152	181	6	ϕ	ϕ	NOUN
ejpam-5152	181	7	:	:	PUNCT
ejpam-5152	182	1	[	[	X
ejpam-5152	182	2	0,∞	0,∞	NOUN
ejpam-5152	182	3	)	)	PUNCT
ejpam-5152	182	4	→	→	PUNCT
ejpam-5152	183	1	[	[	X
ejpam-5152	183	2	0,∞	0,∞	NOUN
ejpam-5152	183	3	)	)	PUNCT
ejpam-5152	183	4	a	a	DET
ejpam-5152	183	5	function	function	NOUN
ejpam-5152	183	6	that	that	PRON
ejpam-5152	183	7	satisfy	satisfy	VERB
ejpam-5152	183	8	the	the	DET
ejpam-5152	183	9	following	follow	VERB
ejpam-5152	183	10	conditions	condition	NOUN
ejpam-5152	183	11	:	:	PUNCT
ejpam-5152	183	12	(	(	PUNCT
ejpam-5152	183	13	i	i	NOUN
ejpam-5152	183	14	)	)	PUNCT
ejpam-5152	183	15	ϕ	ϕ	PROPN
ejpam-5152	183	16	is	be	AUX
ejpam-5152	183	17	non	non	ADJ
ejpam-5152	183	18	-	-	ADJ
ejpam-5152	183	19	decreasing	decrease	VERB
ejpam-5152	183	20	upper	upper	ADJ
ejpam-5152	183	21	semi	semi	ADJ
ejpam-5152	183	22	-	-	ADJ
ejpam-5152	183	23	continuous	continuous	ADJ
ejpam-5152	183	24	,	,	PUNCT
ejpam-5152	183	25	(	(	PUNCT
ejpam-5152	183	26	ii	ii	NOUN
ejpam-5152	183	27	)	)	PUNCT
ejpam-5152	183	28	ϕ(2u	ϕ(2u	PROPN
ejpam-5152	183	29	)	)	PUNCT
ejpam-5152	183	30	<	<	X
ejpam-5152	183	31	u	u	NOUN
ejpam-5152	183	32	for	for	ADP
ejpam-5152	183	33	each	each	DET
ejpam-5152	183	34	u	u	NOUN
ejpam-5152	183	35	>	>	X
ejpam-5152	183	36	0	0	NUM
ejpam-5152	183	37	.	.	PUNCT
ejpam-5152	184	1	using	use	VERB
ejpam-5152	184	2	this	this	DET
ejpam-5152	184	3	definition	definition	NOUN
ejpam-5152	184	4	,	,	PUNCT
ejpam-5152	184	5	singh	singh	PROPN
ejpam-5152	184	6	established	establish	VERB
ejpam-5152	184	7	the	the	DET
ejpam-5152	184	8	existence	existence	NOUN
ejpam-5152	184	9	of	of	ADP
ejpam-5152	184	10	common	common	ADJ
ejpam-5152	184	11	fixed	fix	VERB
ejpam-5152	184	12	points	point	NOUN
ejpam-5152	184	13	of	of	ADP
ejpam-5152	184	14	setvalued	setvalue	VERB
ejpam-5152	184	15	mappings	mapping	NOUN
ejpam-5152	184	16	(	(	PUNCT
ejpam-5152	184	17	theorem	theorem	VERB
ejpam-5152	184	18	2.2	2.2	NUM
ejpam-5152	184	19	in	in	ADP
ejpam-5152	184	20	[	[	X
ejpam-5152	184	21	19	19	NUM
ejpam-5152	184	22	]	]	NUM
ejpam-5152	184	23	)	)	PUNCT
ejpam-5152	184	24	.	.	PUNCT
ejpam-5152	185	1	referring	refer	VERB
ejpam-5152	185	2	to	to	ADP
ejpam-5152	185	3	these	these	DET
ejpam-5152	185	4	results	result	NOUN
ejpam-5152	185	5	,	,	PUNCT
ejpam-5152	185	6	we	we	PRON
ejpam-5152	185	7	will	will	AUX
ejpam-5152	185	8	generalize	generalize	VERB
ejpam-5152	185	9	the	the	DET
ejpam-5152	185	10	theorem	theorem	NOUN
ejpam-5152	185	11	to	to	ADP
ejpam-5152	185	12	a	a	DET
ejpam-5152	185	13	more	more	ADV
ejpam-5152	185	14	general	general	ADJ
ejpam-5152	185	15	metric	metric	ADJ
ejpam-5152	185	16	space	space	NOUN
ejpam-5152	185	17	,	,	PUNCT
ejpam-5152	185	18	which	which	PRON
ejpam-5152	185	19	is	be	AUX
ejpam-5152	185	20	a	a	DET
ejpam-5152	185	21	partial	partial	ADJ
ejpam-5152	185	22	metric	metric	ADJ
ejpam-5152	185	23	space	space	NOUN
ejpam-5152	185	24	.	.	PUNCT
ejpam-5152	186	1	a.	a.	PROPN
ejpam-5152	186	2	ekayanti	ekayanti	PROPN
ejpam-5152	186	3	et	et	PROPN
ejpam-5152	186	4	al	al	PROPN
ejpam-5152	186	5	.	.	PUNCT
ejpam-5152	186	6	/	/	SYM
ejpam-5152	186	7	eur	eur	PROPN
ejpam-5152	186	8	.	.	PUNCT
ejpam-5152	187	1	j.	j.	PROPN
ejpam-5152	187	2	pure	pure	PROPN
ejpam-5152	187	3	appl	appl	PROPN
ejpam-5152	187	4	.	.	PROPN
ejpam-5152	187	5	math	math	PROPN
ejpam-5152	187	6	,	,	PUNCT
ejpam-5152	187	7	17	17	NUM
ejpam-5152	187	8	(	(	PUNCT
ejpam-5152	187	9	2	2	NUM
ejpam-5152	187	10	)	)	PUNCT
ejpam-5152	187	11	(	(	PUNCT
ejpam-5152	187	12	2024	2024	NUM
ejpam-5152	187	13	)	)	PUNCT
ejpam-5152	187	14	,	,	PUNCT
ejpam-5152	187	15	996	996	NUM
ejpam-5152	187	16	-	-	SYM
ejpam-5152	187	17	1008	1008	NUM
ejpam-5152	187	18	1003	1003	NUM
ejpam-5152	187	19	theorem	theorem	NOUN
ejpam-5152	187	20	5	5	NUM
ejpam-5152	187	21	.	.	PUNCT
ejpam-5152	188	1	let	let	AUX
ejpam-5152	188	2	(	(	PUNCT
ejpam-5152	188	3	cbp(x	cbp(x	PROPN
ejpam-5152	188	4	)	)	PUNCT
ejpam-5152	188	5	,	,	PUNCT
ejpam-5152	188	6	hp	hp	PROPN
ejpam-5152	188	7	)	)	PUNCT
ejpam-5152	188	8	be	be	VERB
ejpam-5152	188	9	a	a	DET
ejpam-5152	188	10	p	p	ADJ
ejpam-5152	188	11	-	-	PUNCT
ejpam-5152	188	12	pompeiu	pompeiu	NOUN
ejpam-5152	188	13	-	-	PUNCT
ejpam-5152	188	14	hausdorff	hausdorff	NOUN
ejpam-5152	188	15	metric	metric	ADJ
ejpam-5152	188	16	spaces	space	NOUN
ejpam-5152	188	17	.	.	PUNCT
ejpam-5152	189	1	suppose	suppose	VERB
ejpam-5152	189	2	that	that	SCONJ
ejpam-5152	189	3	f	f	X
ejpam-5152	189	4	,	,	PUNCT
ejpam-5152	189	5	g	g	NOUN
ejpam-5152	189	6	:	:	PUNCT
ejpam-5152	189	7	x	x	SYM
ejpam-5152	189	8	→	→	SYM
ejpam-5152	189	9	cbp(x	cbp(x	PROPN
ejpam-5152	189	10	)	)	PUNCT
ejpam-5152	189	11	be	be	VERB
ejpam-5152	189	12	set	set	VERB
ejpam-5152	189	13	-	-	PUNCT
ejpam-5152	189	14	valued	value	VERB
ejpam-5152	189	15	mappings	mapping	NOUN
ejpam-5152	189	16	that	that	PRON
ejpam-5152	189	17	satisfy	satisfy	VERB
ejpam-5152	189	18	the	the	DET
ejpam-5152	189	19	following	follow	VERB
ejpam-5152	189	20	conditions	condition	NOUN
ejpam-5152	189	21	hp(f	hp(f	INTJ
ejpam-5152	189	22	(	(	PUNCT
ejpam-5152	189	23	x	x	NOUN
ejpam-5152	189	24	)	)	PUNCT
ejpam-5152	189	25	,	,	PUNCT
ejpam-5152	189	26	g(y	g(y	PROPN
ejpam-5152	189	27	)	)	PUNCT
ejpam-5152	189	28	)	)	PUNCT
ejpam-5152	189	29	≤	≤	PROPN
ejpam-5152	190	1	ϕ(max{p(x	ϕ(max{p(x	PROPN
ejpam-5152	190	2	,	,	PUNCT
ejpam-5152	190	3	y	y	PROPN
ejpam-5152	190	4	)	)	PUNCT
ejpam-5152	190	5	,	,	PUNCT
ejpam-5152	190	6	p(x	p(x	PROPN
ejpam-5152	190	7	,	,	PUNCT
ejpam-5152	190	8	f	f	PROPN
ejpam-5152	190	9	(	(	PUNCT
ejpam-5152	190	10	x	x	NOUN
ejpam-5152	190	11	)	)	PUNCT
ejpam-5152	190	12	)	)	PUNCT
ejpam-5152	190	13	,	,	PUNCT
ejpam-5152	190	14	p(y	p(y	PROPN
ejpam-5152	190	15	,	,	PUNCT
ejpam-5152	190	16	g(y	g(y	NOUN
ejpam-5152	190	17	)	)	PUNCT
ejpam-5152	190	18	)	)	PUNCT
ejpam-5152	190	19	,	,	PUNCT
ejpam-5152	190	20	p(x	p(x	NOUN
ejpam-5152	190	21	,	,	PUNCT
ejpam-5152	190	22	g(y	g(y	NOUN
ejpam-5152	190	23	)	)	PUNCT
ejpam-5152	190	24	)	)	PUNCT
ejpam-5152	190	25	,	,	PUNCT
ejpam-5152	190	26	p(y	p(y	PROPN
ejpam-5152	190	27	,	,	PUNCT
ejpam-5152	190	28	f	f	PROPN
ejpam-5152	190	29	(	(	PUNCT
ejpam-5152	190	30	x	x	NOUN
ejpam-5152	190	31	)	)	PUNCT
ejpam-5152	190	32	)	)	PUNCT
ejpam-5152	190	33	}	}	PUNCT
ejpam-5152	190	34	)	)	PUNCT
ejpam-5152	190	35	,	,	PUNCT
ejpam-5152	190	36	(	(	PUNCT
ejpam-5152	190	37	7	7	X
ejpam-5152	190	38	)	)	PUNCT
ejpam-5152	190	39	for	for	ADP
ejpam-5152	190	40	each	each	DET
ejpam-5152	190	41	x	x	NOUN
ejpam-5152	190	42	,	,	PUNCT
ejpam-5152	190	43	y	y	PROPN
ejpam-5152	190	44	∈	∈	PROPN
ejpam-5152	190	45	x	x	PUNCT
ejpam-5152	190	46	where	where	SCONJ
ejpam-5152	190	47	ϕ	ϕ	NOUN
ejpam-5152	190	48	:	:	PUNCT
ejpam-5152	190	49	r+	r+	NOUN
ejpam-5152	190	50	→	→	SYM
ejpam-5152	190	51	r+	r+	NOUN
ejpam-5152	190	52	such	such	ADJ
ejpam-5152	190	53	that	that	SCONJ
ejpam-5152	190	54	ϕ	ϕ	NOUN
ejpam-5152	190	55	be	be	AUX
ejpam-5152	190	56	a	a	DET
ejpam-5152	190	57	non	non	ADJ
ejpam-5152	190	58	-	-	ADJ
ejpam-5152	190	59	decreasing	decrease	VERB
ejpam-5152	190	60	upper	upper	ADJ
ejpam-5152	190	61	semicontinuous	semicontinuous	NOUN
ejpam-5152	190	62	and	and	CCONJ
ejpam-5152	190	63	ϕ(2u	ϕ(2u	NUM
ejpam-5152	190	64	)	)	PUNCT
ejpam-5152	190	65	<	<	X
ejpam-5152	190	66	u	u	NOUN
ejpam-5152	190	67	for	for	ADP
ejpam-5152	190	68	u	u	PROPN
ejpam-5152	190	69	>	>	X
ejpam-5152	190	70	0	0	PROPN
ejpam-5152	190	71	,	,	PUNCT
ejpam-5152	190	72	then	then	ADV
ejpam-5152	190	73	set	set	VERB
ejpam-5152	190	74	-	-	PUNCT
ejpam-5152	190	75	valued	value	VERB
ejpam-5152	190	76	mappings	mapping	NOUN
ejpam-5152	190	77	f	f	NOUN
ejpam-5152	190	78	and	and	CCONJ
ejpam-5152	190	79	g	g	PROPN
ejpam-5152	190	80	have	have	VERB
ejpam-5152	190	81	a	a	DET
ejpam-5152	190	82	unique	unique	ADJ
ejpam-5152	190	83	common	common	ADJ
ejpam-5152	190	84	fixed	fix	VERB
ejpam-5152	190	85	point	point	NOUN
ejpam-5152	190	86	.	.	PUNCT
ejpam-5152	191	1	proof	proof	NOUN
ejpam-5152	191	2	.	.	PUNCT
ejpam-5152	192	1	take	take	VERB
ejpam-5152	192	2	any	any	DET
ejpam-5152	192	3	x0	x0	PROPN
ejpam-5152	192	4	∈	∈	PROPN
ejpam-5152	192	5	x	x	NOUN
ejpam-5152	192	6	,	,	PUNCT
ejpam-5152	192	7	but	but	CCONJ
ejpam-5152	192	8	fixed	fix	VERB
ejpam-5152	192	9	.	.	PUNCT
ejpam-5152	193	1	let	let	VERB
ejpam-5152	193	2	x0	x0	PROPN
ejpam-5152	193	3	/∈	/∈	PUNCT
ejpam-5152	194	1	f	f	X
ejpam-5152	194	2	(	(	PUNCT
ejpam-5152	194	3	x0	x0	PROPN
ejpam-5152	194	4	)	)	PUNCT
ejpam-5152	195	1	and	and	CCONJ
ejpam-5152	195	2	take	take	VERB
ejpam-5152	195	3	x1	x1	PROPN
ejpam-5152	195	4	∈	∈	PROPN
ejpam-5152	195	5	f	f	X
ejpam-5152	195	6	(	(	PUNCT
ejpam-5152	195	7	x0	x0	PROPN
ejpam-5152	195	8	)	)	PUNCT
ejpam-5152	195	9	,	,	PUNCT
ejpam-5152	195	10	then	then	ADV
ejpam-5152	195	11	from	from	ADP
ejpam-5152	195	12	(	(	PUNCT
ejpam-5152	195	13	7	7	X
ejpam-5152	195	14	)	)	PUNCT
ejpam-5152	195	15	we	we	PRON
ejpam-5152	195	16	obtain	obtain	VERB
ejpam-5152	195	17	p(x1	p(x1	ADJ
ejpam-5152	195	18	,	,	PUNCT
ejpam-5152	195	19	g(x1	g(x1	NOUN
ejpam-5152	195	20	)	)	PUNCT
ejpam-5152	195	21	)	)	PUNCT
ejpam-5152	195	22	≤	≤	NOUN
ejpam-5152	195	23	hp(f	hp(f	PUNCT
ejpam-5152	195	24	(	(	PUNCT
ejpam-5152	195	25	x0	x0	PROPN
ejpam-5152	195	26	)	)	PUNCT
ejpam-5152	195	27	,	,	PUNCT
ejpam-5152	195	28	g(x1	g(x1	NOUN
ejpam-5152	195	29	)	)	PUNCT
ejpam-5152	195	30	)	)	PUNCT
ejpam-5152	195	31	≤	≤	PROPN
ejpam-5152	195	32	ϕ(max{p(x0	ϕ(max{p(x0	NOUN
ejpam-5152	195	33	,	,	PUNCT
ejpam-5152	195	34	x1	x1	PROPN
ejpam-5152	195	35	)	)	PUNCT
ejpam-5152	195	36	,	,	PUNCT
ejpam-5152	195	37	p(x0	p(x0	NOUN
ejpam-5152	195	38	,	,	PUNCT
ejpam-5152	195	39	f	f	PROPN
ejpam-5152	195	40	(	(	PUNCT
ejpam-5152	195	41	x0	x0	PROPN
ejpam-5152	195	42	)	)	PUNCT
ejpam-5152	195	43	)	)	PUNCT
ejpam-5152	195	44	,	,	PUNCT
ejpam-5152	195	45	p(x1	p(x1	NOUN
ejpam-5152	195	46	,	,	PUNCT
ejpam-5152	195	47	g(x1	g(x1	NOUN
ejpam-5152	195	48	)	)	PUNCT
ejpam-5152	195	49	,	,	PUNCT
ejpam-5152	195	50	p(x0	p(x0	NOUN
ejpam-5152	195	51	,	,	PUNCT
ejpam-5152	195	52	g(x1	g(x1	NOUN
ejpam-5152	195	53	)	)	PUNCT
ejpam-5152	195	54	)	)	PUNCT
ejpam-5152	195	55	,	,	PUNCT
ejpam-5152	195	56	p(x1	p(x1	PROPN
ejpam-5152	195	57	,	,	PUNCT
ejpam-5152	195	58	f	f	PROPN
ejpam-5152	195	59	(	(	PUNCT
ejpam-5152	195	60	x0	x0	PROPN
ejpam-5152	195	61	)	)	PUNCT
ejpam-5152	195	62	)	)	PUNCT
ejpam-5152	195	63	}	}	PUNCT
ejpam-5152	195	64	)	)	PUNCT
ejpam-5152	195	65	≤	≤	PROPN
ejpam-5152	195	66	ϕ(max{p(x0	ϕ(max{p(x0	NOUN
ejpam-5152	195	67	,	,	PUNCT
ejpam-5152	195	68	x1	x1	PROPN
ejpam-5152	195	69	)	)	PUNCT
ejpam-5152	195	70	,	,	PUNCT
ejpam-5152	195	71	p(x1	p(x1	NOUN
ejpam-5152	195	72	,	,	PUNCT
ejpam-5152	195	73	g(x1	g(x1	NOUN
ejpam-5152	195	74	)	)	PUNCT
ejpam-5152	195	75	)	)	PUNCT
ejpam-5152	195	76	}	}	PUNCT
ejpam-5152	195	77	)	)	PUNCT
ejpam-5152	196	1	≤	≤	NUM
ejpam-5152	196	2	ϕ(p(x0	ϕ(p(x0	PROPN
ejpam-5152	196	3	,	,	PUNCT
ejpam-5152	196	4	x1	x1	PROPN
ejpam-5152	196	5	)	)	PUNCT
ejpam-5152	196	6	+	+	CCONJ
ejpam-5152	196	7	p(x1	p(x1	ADJ
ejpam-5152	196	8	,	,	PUNCT
ejpam-5152	196	9	g(x1	g(x1	NOUN
ejpam-5152	196	10	)	)	PUNCT
ejpam-5152	196	11	)	)	PUNCT
ejpam-5152	196	12	)	)	PUNCT
ejpam-5152	196	13	.	.	PUNCT
ejpam-5152	197	1	consider	consider	VERB
ejpam-5152	197	2	that	that	PRON
ejpam-5152	197	3	:	:	PUNCT
ejpam-5152	197	4	if	if	SCONJ
ejpam-5152	197	5	p(x0	p(x0	NOUN
ejpam-5152	197	6	,	,	PUNCT
ejpam-5152	197	7	x1	x1	PROPN
ejpam-5152	197	8	)	)	PUNCT
ejpam-5152	197	9	<	<	X
ejpam-5152	197	10	p(x1	p(x1	PROPN
ejpam-5152	197	11	,	,	PUNCT
ejpam-5152	197	12	g(x1	g(x1	NOUN
ejpam-5152	197	13	)	)	PUNCT
ejpam-5152	197	14	)	)	PUNCT
ejpam-5152	198	1	then	then	ADV
ejpam-5152	198	2	p(x1	p(x1	NOUN
ejpam-5152	198	3	,	,	PUNCT
ejpam-5152	198	4	g(x1	g(x1	NOUN
ejpam-5152	198	5	)	)	PUNCT
ejpam-5152	198	6	)	)	PUNCT
ejpam-5152	199	1	≤	≤	NUM
ejpam-5152	199	2	ϕ(p(x0	ϕ(p(x0	PROPN
ejpam-5152	199	3	,	,	PUNCT
ejpam-5152	199	4	x1	x1	PROPN
ejpam-5152	199	5	)	)	PUNCT
ejpam-5152	199	6	+	+	CCONJ
ejpam-5152	199	7	p(x1	p(x1	ADJ
ejpam-5152	199	8	,	,	PUNCT
ejpam-5152	199	9	g(x1	g(x1	NOUN
ejpam-5152	199	10	)	)	PUNCT
ejpam-5152	199	11	)	)	PUNCT
ejpam-5152	199	12	)	)	PUNCT
ejpam-5152	200	1	<	<	X
ejpam-5152	200	2	ϕ(p(x1	ϕ(p(x1	PROPN
ejpam-5152	200	3	,	,	PUNCT
ejpam-5152	200	4	g(x1	g(x1	NOUN
ejpam-5152	200	5	)	)	PUNCT
ejpam-5152	200	6	)	)	PUNCT
ejpam-5152	201	1	+	+	CCONJ
ejpam-5152	201	2	p(x1	p(x1	ADJ
ejpam-5152	201	3	,	,	PUNCT
ejpam-5152	201	4	g(x1	g(x1	NOUN
ejpam-5152	201	5	)	)	PUNCT
ejpam-5152	201	6	)	)	PUNCT
ejpam-5152	201	7	)	)	PUNCT
ejpam-5152	202	1	=	=	PUNCT
ejpam-5152	203	1	ϕ(2p(x1	ϕ(2p(x1	NOUN
ejpam-5152	203	2	,	,	PUNCT
ejpam-5152	203	3	g(x1	g(x1	NOUN
ejpam-5152	203	4	)	)	PUNCT
ejpam-5152	203	5	)	)	PUNCT
ejpam-5152	203	6	)	)	PUNCT
ejpam-5152	204	1	<	<	X
ejpam-5152	204	2	p(x1	p(x1	PROPN
ejpam-5152	204	3	,	,	PUNCT
ejpam-5152	204	4	g(x1	g(x1	NOUN
ejpam-5152	204	5	)	)	PUNCT
ejpam-5152	204	6	.	.	PUNCT
ejpam-5152	205	1	this	this	DET
ejpam-5152	205	2	condition	condition	NOUN
ejpam-5152	205	3	shows	show	VERB
ejpam-5152	205	4	a	a	DET
ejpam-5152	205	5	contradiction	contradiction	NOUN
ejpam-5152	205	6	,	,	PUNCT
ejpam-5152	205	7	then	then	ADV
ejpam-5152	205	8	it	it	PRON
ejpam-5152	205	9	must	must	AUX
ejpam-5152	205	10	be	be	AUX
ejpam-5152	205	11	p(x0	p(x0	NOUN
ejpam-5152	205	12	,	,	PUNCT
ejpam-5152	205	13	x1	x1	PROPN
ejpam-5152	205	14	)	)	PUNCT
ejpam-5152	205	15	≥	≥	NOUN
ejpam-5152	205	16	p(x1	p(x1	NOUN
ejpam-5152	205	17	,	,	PUNCT
ejpam-5152	205	18	g(x1	g(x1	NOUN
ejpam-5152	205	19	)	)	PUNCT
ejpam-5152	205	20	)	)	PUNCT
ejpam-5152	205	21	.	.	PUNCT
ejpam-5152	206	1	therefore	therefore	ADV
ejpam-5152	206	2	,	,	PUNCT
ejpam-5152	206	3	we	we	PRON
ejpam-5152	206	4	have	have	VERB
ejpam-5152	206	5	p(x1	p(x1	ADJ
ejpam-5152	206	6	,	,	PUNCT
ejpam-5152	206	7	g(x1	g(x1	NOUN
ejpam-5152	206	8	)	)	PUNCT
ejpam-5152	206	9	)	)	PUNCT
ejpam-5152	206	10	≤	≤	NUM
ejpam-5152	207	1	ϕ(p(x0	ϕ(p(x0	PROPN
ejpam-5152	207	2	,	,	PUNCT
ejpam-5152	207	3	x1	x1	PROPN
ejpam-5152	207	4	)	)	PUNCT
ejpam-5152	208	1	+	+	NUM
ejpam-5152	208	2	p(x0	p(x0	NOUN
ejpam-5152	208	3	,	,	PUNCT
ejpam-5152	208	4	x1	x1	PROPN
ejpam-5152	208	5	)	)	PUNCT
ejpam-5152	208	6	)	)	PUNCT
ejpam-5152	209	1	=	=	PUNCT
ejpam-5152	210	1	ϕ(2p(x0	ϕ(2p(x0	PROPN
ejpam-5152	210	2	,	,	PUNCT
ejpam-5152	210	3	x1	x1	PROPN
ejpam-5152	210	4	)	)	PUNCT
ejpam-5152	210	5	)	)	PUNCT
ejpam-5152	211	1	<	<	X
ejpam-5152	211	2	p(x0	p(x0	NOUN
ejpam-5152	211	3	,	,	PUNCT
ejpam-5152	211	4	x1	x1	PROPN
ejpam-5152	211	5	)	)	PUNCT
ejpam-5152	211	6	.	.	PUNCT
ejpam-5152	212	1	furthermore	furthermore	ADV
ejpam-5152	212	2	,	,	PUNCT
ejpam-5152	212	3	we	we	PRON
ejpam-5152	212	4	can	can	AUX
ejpam-5152	212	5	take	take	VERB
ejpam-5152	212	6	x2	x2	PROPN
ejpam-5152	212	7	∈	∈	PROPN
ejpam-5152	212	8	g(x1	g(x1	NOUN
ejpam-5152	212	9	)	)	PUNCT
ejpam-5152	212	10	such	such	ADJ
ejpam-5152	212	11	that	that	DET
ejpam-5152	212	12	p(x1	p(x1	NOUN
ejpam-5152	212	13	,	,	PUNCT
ejpam-5152	212	14	x2	x2	NUM
ejpam-5152	212	15	)	)	PUNCT
ejpam-5152	212	16	≤	≤	NOUN
ejpam-5152	212	17	p(x0	p(x0	NOUN
ejpam-5152	212	18	,	,	PUNCT
ejpam-5152	212	19	x1	x1	PROPN
ejpam-5152	212	20	)	)	PUNCT
ejpam-5152	212	21	.	.	PUNCT
ejpam-5152	213	1	thus	thus	ADV
ejpam-5152	213	2	,	,	PUNCT
ejpam-5152	213	3	by	by	ADP
ejpam-5152	213	4	using	use	VERB
ejpam-5152	213	5	inequality	inequality	NOUN
ejpam-5152	213	6	(	(	PUNCT
ejpam-5152	213	7	7	7	X
ejpam-5152	213	8	)	)	PUNCT
ejpam-5152	213	9	we	we	PRON
ejpam-5152	213	10	have	have	VERB
ejpam-5152	213	11	p(x2	p(x2	NOUN
ejpam-5152	213	12	,	,	PUNCT
ejpam-5152	213	13	f	f	PROPN
ejpam-5152	213	14	(	(	PUNCT
ejpam-5152	213	15	x2	x2	PROPN
ejpam-5152	213	16	)	)	PUNCT
ejpam-5152	213	17	)	)	PUNCT
ejpam-5152	213	18	≤	≤	NUM
ejpam-5152	214	1	hp(g(x1	hp(g(x1	PROPN
ejpam-5152	214	2	)	)	PUNCT
ejpam-5152	214	3	,	,	PUNCT
ejpam-5152	214	4	f	f	PROPN
ejpam-5152	214	5	(	(	PUNCT
ejpam-5152	214	6	x2	x2	PROPN
ejpam-5152	214	7	)	)	PUNCT
ejpam-5152	214	8	)	)	PUNCT
ejpam-5152	214	9	≤	≤	NUM
ejpam-5152	214	10	ϕ(max{p(x1	ϕ(max{p(x1	NOUN
ejpam-5152	214	11	,	,	PUNCT
ejpam-5152	214	12	x2	x2	PROPN
ejpam-5152	214	13	)	)	PUNCT
ejpam-5152	214	14	,	,	PUNCT
ejpam-5152	214	15	p(x1	p(x1	NOUN
ejpam-5152	214	16	,	,	PUNCT
ejpam-5152	214	17	g(x1	g(x1	NOUN
ejpam-5152	214	18	)	)	PUNCT
ejpam-5152	214	19	)	)	PUNCT
ejpam-5152	214	20	,	,	PUNCT
ejpam-5152	214	21	p(x2	p(x2	NOUN
ejpam-5152	214	22	,	,	PUNCT
ejpam-5152	214	23	f	f	PROPN
ejpam-5152	214	24	(	(	PUNCT
ejpam-5152	214	25	x2	x2	PROPN
ejpam-5152	214	26	)	)	PUNCT
ejpam-5152	214	27	,	,	PUNCT
ejpam-5152	214	28	p(x1	p(x1	PROPN
ejpam-5152	214	29	,	,	PUNCT
ejpam-5152	214	30	f	f	PROPN
ejpam-5152	214	31	(	(	PUNCT
ejpam-5152	214	32	x2	x2	PROPN
ejpam-5152	214	33	)	)	PUNCT
ejpam-5152	214	34	)	)	PUNCT
ejpam-5152	214	35	,	,	PUNCT
ejpam-5152	214	36	p(x2	p(x2	NOUN
ejpam-5152	214	37	,	,	PUNCT
ejpam-5152	214	38	g(x1	g(x1	NOUN
ejpam-5152	214	39	)	)	PUNCT
ejpam-5152	214	40	)	)	PUNCT
ejpam-5152	214	41	}	}	PUNCT
ejpam-5152	214	42	)	)	PUNCT
ejpam-5152	214	43	≤	≤	NUM
ejpam-5152	214	44	ϕ(max{p(x1	ϕ(max{p(x1	NOUN
ejpam-5152	214	45	,	,	PUNCT
ejpam-5152	214	46	x2	x2	PROPN
ejpam-5152	214	47	)	)	PUNCT
ejpam-5152	214	48	,	,	PUNCT
ejpam-5152	214	49	p(x2	p(x2	NOUN
ejpam-5152	214	50	,	,	PUNCT
ejpam-5152	214	51	f	f	PROPN
ejpam-5152	214	52	(	(	PUNCT
ejpam-5152	214	53	x2	x2	PROPN
ejpam-5152	214	54	)	)	PUNCT
ejpam-5152	214	55	)	)	PUNCT
ejpam-5152	214	56	}	}	PUNCT
ejpam-5152	214	57	)	)	PUNCT
ejpam-5152	214	58	≤	≤	PROPN
ejpam-5152	215	1	ϕ(p(x1	ϕ(p(x1	PROPN
ejpam-5152	215	2	,	,	PUNCT
ejpam-5152	215	3	x2	x2	PROPN
ejpam-5152	215	4	)	)	PUNCT
ejpam-5152	215	5	+	+	NUM
ejpam-5152	215	6	p(x2	p(x2	NOUN
ejpam-5152	215	7	,	,	PUNCT
ejpam-5152	215	8	f	f	PROPN
ejpam-5152	215	9	(	(	PUNCT
ejpam-5152	215	10	x2	x2	PROPN
ejpam-5152	215	11	)	)	PUNCT
ejpam-5152	215	12	)	)	PUNCT
ejpam-5152	215	13	)	)	PUNCT
ejpam-5152	215	14	.	.	PUNCT
ejpam-5152	216	1	let	let	VERB
ejpam-5152	216	2	’s	’s	PRON
ejpam-5152	216	3	observe	observe	VERB
ejpam-5152	216	4	,	,	PUNCT
ejpam-5152	216	5	when	when	SCONJ
ejpam-5152	216	6	p(x1	p(x1	ADJ
ejpam-5152	216	7	,	,	PUNCT
ejpam-5152	216	8	x2	x2	PROPN
ejpam-5152	216	9	)	)	PUNCT
ejpam-5152	216	10	<	<	X
ejpam-5152	216	11	p(x2	p(x2	PROPN
ejpam-5152	216	12	,	,	PUNCT
ejpam-5152	216	13	f	f	PROPN
ejpam-5152	216	14	(	(	PUNCT
ejpam-5152	216	15	x2	x2	PROPN
ejpam-5152	216	16	)	)	PUNCT
ejpam-5152	216	17	)	)	PUNCT
ejpam-5152	216	18	then	then	ADV
ejpam-5152	216	19	we	we	PRON
ejpam-5152	216	20	obtain	obtain	VERB
ejpam-5152	216	21	p(x2	p(x2	NOUN
ejpam-5152	216	22	,	,	PUNCT
ejpam-5152	216	23	f	f	PROPN
ejpam-5152	216	24	(	(	PUNCT
ejpam-5152	216	25	x2	x2	PROPN
ejpam-5152	216	26	)	)	PUNCT
ejpam-5152	216	27	)	)	PUNCT
ejpam-5152	217	1	≤	≤	PROPN
ejpam-5152	218	1	ϕ(p(x1	ϕ(p(x1	PROPN
ejpam-5152	218	2	,	,	PUNCT
ejpam-5152	218	3	x2	x2	PROPN
ejpam-5152	218	4	)	)	PUNCT
ejpam-5152	218	5	+	+	NUM
ejpam-5152	218	6	p(x2	p(x2	NOUN
ejpam-5152	218	7	,	,	PUNCT
ejpam-5152	218	8	f	f	PROPN
ejpam-5152	218	9	(	(	PUNCT
ejpam-5152	218	10	x2	x2	PROPN
ejpam-5152	218	11	)	)	PUNCT
ejpam-5152	218	12	)	)	PUNCT
ejpam-5152	218	13	)	)	PUNCT
ejpam-5152	219	1	<	<	X
ejpam-5152	219	2	ϕ(p(x2	ϕ(p(x2	PROPN
ejpam-5152	219	3	,	,	PUNCT
ejpam-5152	219	4	f	f	PROPN
ejpam-5152	219	5	(	(	PUNCT
ejpam-5152	219	6	x2	x2	PROPN
ejpam-5152	219	7	)	)	PUNCT
ejpam-5152	219	8	)	)	PUNCT
ejpam-5152	220	1	+	+	CCONJ
ejpam-5152	220	2	p(x2	p(x2	NOUN
ejpam-5152	220	3	,	,	PUNCT
ejpam-5152	220	4	f	f	PROPN
ejpam-5152	220	5	(	(	PUNCT
ejpam-5152	220	6	x2	x2	PROPN
ejpam-5152	220	7	)	)	PUNCT
ejpam-5152	220	8	)	)	PUNCT
ejpam-5152	220	9	)	)	PUNCT
ejpam-5152	221	1	=	=	SYM
ejpam-5152	221	2	ϕ(2p(x2	ϕ(2p(x2	PROPN
ejpam-5152	221	3	,	,	PUNCT
ejpam-5152	221	4	f	f	PROPN
ejpam-5152	221	5	(	(	PUNCT
ejpam-5152	221	6	x2	x2	PROPN
ejpam-5152	221	7	)	)	PUNCT
ejpam-5152	221	8	)	)	PUNCT
ejpam-5152	221	9	)	)	PUNCT
ejpam-5152	222	1	<	<	X
ejpam-5152	222	2	p(x2	p(x2	PROPN
ejpam-5152	222	3	,	,	PUNCT
ejpam-5152	222	4	f	f	PROPN
ejpam-5152	222	5	(	(	PUNCT
ejpam-5152	222	6	x2	x2	PROPN
ejpam-5152	222	7	)	)	PUNCT
ejpam-5152	222	8	.	.	PUNCT
ejpam-5152	223	1	thus	thus	ADV
ejpam-5152	223	2	,	,	PUNCT
ejpam-5152	223	3	we	we	PRON
ejpam-5152	223	4	found	find	VERB
ejpam-5152	223	5	a	a	DET
ejpam-5152	223	6	contradiction	contradiction	NOUN
ejpam-5152	223	7	.	.	PUNCT
ejpam-5152	224	1	it	it	PRON
ejpam-5152	224	2	must	must	AUX
ejpam-5152	224	3	holds	hold	VERB
ejpam-5152	224	4	p(x1	p(x1	ADJ
ejpam-5152	224	5	,	,	PUNCT
ejpam-5152	224	6	x2	x2	PROPN
ejpam-5152	224	7	)	)	PUNCT
ejpam-5152	224	8	≥	≥	PROPN
ejpam-5152	224	9	p(x2	p(x2	NOUN
ejpam-5152	224	10	,	,	PUNCT
ejpam-5152	224	11	f	f	PROPN
ejpam-5152	224	12	(	(	PUNCT
ejpam-5152	224	13	x2	x2	PROPN
ejpam-5152	224	14	)	)	PUNCT
ejpam-5152	224	15	)	)	PUNCT
ejpam-5152	224	16	.	.	PUNCT
ejpam-5152	225	1	therefore	therefore	ADV
ejpam-5152	225	2	,	,	PUNCT
ejpam-5152	225	3	we	we	PRON
ejpam-5152	225	4	obtain	obtain	VERB
ejpam-5152	225	5	p(x2	p(x2	NOUN
ejpam-5152	225	6	,	,	PUNCT
ejpam-5152	225	7	f	f	PROPN
ejpam-5152	225	8	(	(	PUNCT
ejpam-5152	225	9	x2	x2	PROPN
ejpam-5152	225	10	)	)	PUNCT
ejpam-5152	225	11	)	)	PUNCT
ejpam-5152	225	12	≤	≤	PROPN
ejpam-5152	226	1	ϕ(p(x1	ϕ(p(x1	PROPN
ejpam-5152	226	2	,	,	PUNCT
ejpam-5152	226	3	x2	x2	PROPN
ejpam-5152	226	4	)	)	PUNCT
ejpam-5152	226	5	+	+	CCONJ
ejpam-5152	226	6	p(x1	p(x1	ADJ
ejpam-5152	226	7	,	,	PUNCT
ejpam-5152	226	8	x2	x2	PROPN
ejpam-5152	226	9	)	)	PUNCT
ejpam-5152	226	10	)	)	PUNCT
ejpam-5152	227	1	=	=	PUNCT
ejpam-5152	228	1	ϕ(2p(x1	ϕ(2p(x1	NOUN
ejpam-5152	228	2	,	,	PUNCT
ejpam-5152	228	3	x2	x2	PROPN
ejpam-5152	228	4	)	)	PUNCT
ejpam-5152	228	5	)	)	PUNCT
ejpam-5152	229	1	<	<	X
ejpam-5152	229	2	p(x1	p(x1	PROPN
ejpam-5152	229	3	,	,	PUNCT
ejpam-5152	229	4	x2	x2	PROPN
ejpam-5152	229	5	)	)	PUNCT
ejpam-5152	229	6	.	.	PUNCT
ejpam-5152	230	1	a.	a.	PROPN
ejpam-5152	230	2	ekayanti	ekayanti	PROPN
ejpam-5152	230	3	et	et	PROPN
ejpam-5152	230	4	al	al	PROPN
ejpam-5152	230	5	.	.	PUNCT
ejpam-5152	230	6	/	/	SYM
ejpam-5152	230	7	eur	eur	PROPN
ejpam-5152	230	8	.	.	PUNCT
ejpam-5152	231	1	j.	j.	PROPN
ejpam-5152	231	2	pure	pure	PROPN
ejpam-5152	231	3	appl	appl	PROPN
ejpam-5152	231	4	.	.	PROPN
ejpam-5152	231	5	math	math	PROPN
ejpam-5152	231	6	,	,	PUNCT
ejpam-5152	231	7	17	17	NUM
ejpam-5152	231	8	(	(	PUNCT
ejpam-5152	231	9	2	2	NUM
ejpam-5152	231	10	)	)	PUNCT
ejpam-5152	231	11	(	(	PUNCT
ejpam-5152	231	12	2024	2024	NUM
ejpam-5152	231	13	)	)	PUNCT
ejpam-5152	231	14	,	,	PUNCT
ejpam-5152	231	15	996	996	NUM
ejpam-5152	231	16	-	-	SYM
ejpam-5152	231	17	1008	1008	NUM
ejpam-5152	231	18	1004	1004	NUM
ejpam-5152	231	19	in	in	ADP
ejpam-5152	231	20	the	the	DET
ejpam-5152	231	21	similar	similar	ADJ
ejpam-5152	231	22	line	line	NOUN
ejpam-5152	231	23	,	,	PUNCT
ejpam-5152	231	24	we	we	PRON
ejpam-5152	231	25	can	can	AUX
ejpam-5152	231	26	choose	choose	VERB
ejpam-5152	231	27	x3	x3	PROPN
ejpam-5152	231	28	∈	∈	PROPN
ejpam-5152	231	29	f	f	X
ejpam-5152	231	30	(	(	PUNCT
ejpam-5152	231	31	x2	x2	PROPN
ejpam-5152	231	32	)	)	PUNCT
ejpam-5152	231	33	,	,	PUNCT
ejpam-5152	231	34	then	then	ADV
ejpam-5152	231	35	we	we	PRON
ejpam-5152	231	36	will	will	AUX
ejpam-5152	231	37	have	have	VERB
ejpam-5152	231	38	p(x2	p(x2	NOUN
ejpam-5152	231	39	,	,	PUNCT
ejpam-5152	231	40	x3	x3	ADJ
ejpam-5152	231	41	)	)	PUNCT
ejpam-5152	231	42	≤	≤	NOUN
ejpam-5152	231	43	p(x1	p(x1	NOUN
ejpam-5152	231	44	,	,	PUNCT
ejpam-5152	231	45	x2	x2	PROPN
ejpam-5152	231	46	)	)	PUNCT
ejpam-5152	231	47	.	.	PUNCT
ejpam-5152	232	1	if	if	SCONJ
ejpam-5152	232	2	this	this	DET
ejpam-5152	232	3	process	process	NOUN
ejpam-5152	232	4	is	be	AUX
ejpam-5152	232	5	continued	continue	VERB
ejpam-5152	232	6	then	then	ADV
ejpam-5152	232	7	a	a	DET
ejpam-5152	232	8	sequence	sequence	NOUN
ejpam-5152	232	9	(	(	PUNCT
ejpam-5152	232	10	xn	xn	X
ejpam-5152	232	11	)	)	PUNCT
ejpam-5152	232	12	in	in	ADP
ejpam-5152	232	13	x	x	PROPN
ejpam-5152	232	14	is	be	AUX
ejpam-5152	232	15	obtained	obtain	VERB
ejpam-5152	232	16	with	with	ADP
ejpam-5152	232	17	the	the	DET
ejpam-5152	232	18	form	form	NOUN
ejpam-5152	232	19	as	as	SCONJ
ejpam-5152	232	20	follows	follow	VERB
ejpam-5152	232	21	x2n+1	x2n+1	PROPN
ejpam-5152	232	22	∈	∈	PROPN
ejpam-5152	232	23	f	f	PROPN
ejpam-5152	232	24	(	(	PUNCT
ejpam-5152	232	25	x2n	x2n	PROPN
ejpam-5152	232	26	)	)	PUNCT
ejpam-5152	232	27	,	,	PUNCT
ejpam-5152	232	28	and	and	CCONJ
ejpam-5152	232	29	x2n+2	x2n+2	PUNCT
ejpam-5152	232	30	∈	∈	PROPN
ejpam-5152	232	31	g(x2n+1	g(x2n+1	NOUN
ejpam-5152	232	32	)	)	PUNCT
ejpam-5152	232	33	,	,	PUNCT
ejpam-5152	232	34	and	and	CCONJ
ejpam-5152	232	35	also	also	ADV
ejpam-5152	232	36	p(xn	p(xn	NOUN
ejpam-5152	232	37	,	,	PUNCT
ejpam-5152	232	38	xn+1	xn+1	NUM
ejpam-5152	232	39	)	)	PUNCT
ejpam-5152	232	40	≤	≤	NOUN
ejpam-5152	232	41	p(xn−1	p(xn−1	NOUN
ejpam-5152	232	42	,	,	PUNCT
ejpam-5152	232	43	xn	xn	PROPN
ejpam-5152	232	44	)	)	PUNCT
ejpam-5152	232	45	.	.	PUNCT
ejpam-5152	233	1	(	(	PUNCT
ejpam-5152	233	2	8)	8)	NUM
ejpam-5152	233	3	furthermore	furthermore	ADV
ejpam-5152	233	4	,	,	PUNCT
ejpam-5152	233	5	we	we	PRON
ejpam-5152	233	6	defined	define	VERB
ejpam-5152	233	7	pn	pn	PROPN
ejpam-5152	233	8	=	=	SYM
ejpam-5152	233	9	p(xn	p(xn	NOUN
ejpam-5152	233	10	,	,	PUNCT
ejpam-5152	233	11	xn+1	xn+1	NUM
ejpam-5152	233	12	)	)	PUNCT
ejpam-5152	233	13	.	.	PUNCT
ejpam-5152	234	1	from	from	ADP
ejpam-5152	234	2	inequality	inequality	NOUN
ejpam-5152	234	3	(	(	PUNCT
ejpam-5152	234	4	8)	8)	NUM
ejpam-5152	234	5	then	then	ADV
ejpam-5152	234	6	we	we	PRON
ejpam-5152	234	7	obtain	obtain	VERB
ejpam-5152	234	8	pn	pn	PROPN
ejpam-5152	234	9	≤	≤	PROPN
ejpam-5152	234	10	pn+1	pn+1	PROPN
ejpam-5152	234	11	.	.	PUNCT
ejpam-5152	235	1	this	this	PRON
ejpam-5152	235	2	means	mean	VERB
ejpam-5152	235	3	that	that	SCONJ
ejpam-5152	235	4	pn	pn	PROPN
ejpam-5152	235	5	is	be	AUX
ejpam-5152	235	6	a	a	DET
ejpam-5152	235	7	non	non	ADJ
ejpam-5152	235	8	-	-	ADJ
ejpam-5152	235	9	decreasing	decrease	VERB
ejpam-5152	235	10	sequences	sequence	NOUN
ejpam-5152	235	11	of	of	ADP
ejpam-5152	235	12	real	real	ADJ
ejpam-5152	235	13	numbers	number	NOUN
ejpam-5152	235	14	and	and	CCONJ
ejpam-5152	235	15	is	be	AUX
ejpam-5152	235	16	bounded	bound	VERB
ejpam-5152	235	17	below	below	ADV
ejpam-5152	235	18	by	by	ADP
ejpam-5152	235	19	zero	zero	NUM
ejpam-5152	235	20	.	.	PUNCT
ejpam-5152	236	1	therefore	therefore	ADV
ejpam-5152	236	2	pn	pn	PROPN
ejpam-5152	236	3	is	be	AUX
ejpam-5152	236	4	a	a	DET
ejpam-5152	236	5	convergent	convergent	ADJ
ejpam-5152	236	6	sequences	sequence	NOUN
ejpam-5152	236	7	.	.	PUNCT
ejpam-5152	237	1	suppose	suppose	VERB
ejpam-5152	237	2	that	that	SCONJ
ejpam-5152	237	3	limn→∞pn	limn→∞pn	PROPN
ejpam-5152	237	4	=	=	PUNCT
ejpam-5152	237	5	q.	q.	PROPN
ejpam-5152	237	6	let	let	VERB
ejpam-5152	237	7	q	q	PROPN
ejpam-5152	237	8	>	>	X
ejpam-5152	237	9	0	0	NUM
ejpam-5152	237	10	,	,	PUNCT
ejpam-5152	237	11	consider	consider	VERB
ejpam-5152	237	12	that	that	SCONJ
ejpam-5152	237	13	p(xn	p(xn	NOUN
ejpam-5152	237	14	,	,	PUNCT
ejpam-5152	237	15	xn+1	xn+1	NUM
ejpam-5152	237	16	)	)	PUNCT
ejpam-5152	237	17	≤	≤	NOUN
ejpam-5152	237	18	ϕ(2p(xn−1	ϕ(2p(xn−1	NOUN
ejpam-5152	237	19	,	,	PUNCT
ejpam-5152	237	20	xn	xn	PROPN
ejpam-5152	237	21	)	)	PUNCT
ejpam-5152	237	22	)	)	PUNCT
ejpam-5152	238	1	<	<	X
ejpam-5152	238	2	p(xn−1	p(xn−1	PROPN
ejpam-5152	238	3	,	,	PUNCT
ejpam-5152	238	4	xn	xn	PROPN
ejpam-5152	238	5	)	)	PUNCT
ejpam-5152	238	6	,	,	PUNCT
ejpam-5152	238	7	thus	thus	ADV
ejpam-5152	238	8	p(xn	p(xn	NOUN
ejpam-5152	238	9	)	)	PUNCT
ejpam-5152	238	10	≤	≤	NOUN
ejpam-5152	238	11	ϕ(2pn−1	ϕ(2pn−1	PROPN
ejpam-5152	238	12	)	)	PUNCT
ejpam-5152	238	13	<	<	X
ejpam-5152	239	1	pn−1	pn−1	PROPN
ejpam-5152	239	2	.	.	PUNCT
ejpam-5152	240	1	(	(	PUNCT
ejpam-5152	240	2	9	9	X
ejpam-5152	240	3	)	)	PUNCT
ejpam-5152	240	4	take	take	VERB
ejpam-5152	240	5	n	n	NOUN
ejpam-5152	240	6	→	→	SYM
ejpam-5152	240	7	∞	∞	NUM
ejpam-5152	240	8	on	on	ADP
ejpam-5152	240	9	inequality	inequality	NOUN
ejpam-5152	240	10	(	(	PUNCT
ejpam-5152	240	11	9	9	NUM
ejpam-5152	240	12	)	)	PUNCT
ejpam-5152	240	13	then	then	ADV
ejpam-5152	240	14	we	we	PRON
ejpam-5152	240	15	obtain	obtain	VERB
ejpam-5152	240	16	q	q	PROPN
ejpam-5152	240	17	≤	≤	NUM
ejpam-5152	240	18	ϕ(2q	ϕ(2q	NUM
ejpam-5152	240	19	)	)	PUNCT
ejpam-5152	240	20	<	<	X
ejpam-5152	240	21	q.	q.	PROPN
ejpam-5152	240	22	therefore	therefore	ADV
ejpam-5152	240	23	,	,	PUNCT
ejpam-5152	240	24	we	we	PRON
ejpam-5152	240	25	have	have	VERB
ejpam-5152	240	26	a	a	DET
ejpam-5152	240	27	contradiction	contradiction	NOUN
ejpam-5152	240	28	.	.	PUNCT
ejpam-5152	241	1	hence	hence	ADV
ejpam-5152	241	2	,	,	PUNCT
ejpam-5152	241	3	q	q	X
ejpam-5152	241	4	=	=	SYM
ejpam-5152	241	5	0	0	NUM
ejpam-5152	241	6	,	,	PUNCT
ejpam-5152	241	7	i.e.	i.e.	X
ejpam-5152	241	8	,	,	PUNCT
ejpam-5152	241	9	limn→∞pn	limn→∞pn	ADV
ejpam-5152	241	10	=	=	SYM
ejpam-5152	241	11	limn→∞p(xn	limn→∞p(xn	X
ejpam-5152	241	12	,	,	PUNCT
ejpam-5152	241	13	xn+1	xn+1	X
ejpam-5152	241	14	)	)	PUNCT
ejpam-5152	241	15	=	=	SYM
ejpam-5152	242	1	0	0	X
ejpam-5152	242	2	.	.	PUNCT
ejpam-5152	243	1	furthermore	furthermore	ADV
ejpam-5152	243	2	,	,	PUNCT
ejpam-5152	243	3	we	we	PRON
ejpam-5152	243	4	will	will	AUX
ejpam-5152	243	5	show	show	VERB
ejpam-5152	243	6	that	that	SCONJ
ejpam-5152	243	7	(	(	PUNCT
ejpam-5152	243	8	xn	xn	X
ejpam-5152	243	9	)	)	PUNCT
ejpam-5152	243	10	is	be	AUX
ejpam-5152	243	11	a	a	DET
ejpam-5152	243	12	cauchy	cauchy	ADJ
ejpam-5152	243	13	sequences	sequence	NOUN
ejpam-5152	243	14	.	.	PUNCT
ejpam-5152	244	1	based	base	VERB
ejpam-5152	244	2	on	on	ADP
ejpam-5152	244	3	the	the	DET
ejpam-5152	244	4	construction	construction	NOUN
ejpam-5152	244	5	of	of	ADP
ejpam-5152	244	6	sequence	sequence	NOUN
ejpam-5152	244	7	(	(	PUNCT
ejpam-5152	244	8	xn	xn	PROPN
ejpam-5152	244	9	)	)	PUNCT
ejpam-5152	244	10	,	,	PUNCT
ejpam-5152	244	11	in	in	ADP
ejpam-5152	244	12	showing	show	VERB
ejpam-5152	244	13	that	that	DET
ejpam-5152	244	14	sequence	sequence	NOUN
ejpam-5152	244	15	(	(	PUNCT
ejpam-5152	244	16	xn	xn	X
ejpam-5152	244	17	)	)	PUNCT
ejpam-5152	244	18	is	be	AUX
ejpam-5152	244	19	a	a	DET
ejpam-5152	244	20	cauchy	cauchy	NOUN
ejpam-5152	244	21	can	can	AUX
ejpam-5152	244	22	be	be	AUX
ejpam-5152	244	23	done	do	VERB
ejpam-5152	244	24	by	by	ADP
ejpam-5152	244	25	showing	show	VERB
ejpam-5152	244	26	that	that	SCONJ
ejpam-5152	244	27	(	(	PUNCT
ejpam-5152	244	28	x2n	x2n	NOUN
ejpam-5152	244	29	)	)	PUNCT
ejpam-5152	244	30	is	be	AUX
ejpam-5152	244	31	a	a	DET
ejpam-5152	244	32	cauchy	cauchy	ADJ
ejpam-5152	244	33	sequence	sequence	NOUN
ejpam-5152	244	34	.	.	PUNCT
ejpam-5152	245	1	as	as	ADP
ejpam-5152	245	2	for	for	ADP
ejpam-5152	245	3	the	the	DET
ejpam-5152	245	4	proof	proof	NOUN
ejpam-5152	245	5	using	use	VERB
ejpam-5152	245	6	contradiction	contradiction	NOUN
ejpam-5152	245	7	,	,	PUNCT
ejpam-5152	245	8	that	that	ADV
ejpam-5152	245	9	is	is	ADV
ejpam-5152	245	10	,	,	PUNCT
ejpam-5152	245	11	if	if	SCONJ
ejpam-5152	245	12	(	(	PUNCT
ejpam-5152	245	13	x2n	x2n	ADJ
ejpam-5152	245	14	)	)	PUNCT
ejpam-5152	245	15	is	be	AUX
ejpam-5152	245	16	not	not	PART
ejpam-5152	245	17	a	a	DET
ejpam-5152	245	18	cauchy	cauchy	ADJ
ejpam-5152	245	19	sequence	sequence	NOUN
ejpam-5152	245	20	then	then	ADV
ejpam-5152	245	21	there	there	PRON
ejpam-5152	245	22	exist	exist	VERB
ejpam-5152	245	23	ε	ε	PROPN
ejpam-5152	245	24	>	>	X
ejpam-5152	245	25	0	0	NUM
ejpam-5152	245	26	such	such	ADJ
ejpam-5152	245	27	that	that	PRON
ejpam-5152	245	28	for	for	ADP
ejpam-5152	245	29	every	every	DET
ejpam-5152	245	30	positive	positive	ADJ
ejpam-5152	245	31	integer	integer	NOUN
ejpam-5152	245	32	2	2	NUM
ejpam-5152	245	33	t	t	NOUN
ejpam-5152	245	34	there	there	PRON
ejpam-5152	245	35	is	be	VERB
ejpam-5152	245	36	sequence	sequence	NOUN
ejpam-5152	245	37	(	(	PUNCT
ejpam-5152	245	38	2mt	2mt	ADJ
ejpam-5152	245	39	)	)	PUNCT
ejpam-5152	245	40	and	and	CCONJ
ejpam-5152	245	41	(	(	PUNCT
ejpam-5152	245	42	2nt	2nt	NOUN
ejpam-5152	245	43	)	)	PUNCT
ejpam-5152	245	44	where	where	SCONJ
ejpam-5152	245	45	t	t	NOUN
ejpam-5152	245	46	<	<	X
ejpam-5152	245	47	nt	not	PART
ejpam-5152	245	48	<	<	X
ejpam-5152	245	49	mt	mt	PROPN
ejpam-5152	246	1	and	and	CCONJ
ejpam-5152	246	2	we	we	PRON
ejpam-5152	246	3	have	have	VERB
ejpam-5152	246	4	p(x2nt	p(x2nt	NOUN
ejpam-5152	246	5	,	,	PUNCT
ejpam-5152	246	6	x2mt	x2mt	PROPN
ejpam-5152	246	7	)	)	PUNCT
ejpam-5152	246	8	>	>	X
ejpam-5152	247	1	ε	ε	PROPN
ejpam-5152	247	2	,	,	PUNCT
ejpam-5152	247	3	t	t	NOUN
ejpam-5152	247	4	=	=	SYM
ejpam-5152	247	5	1	1	NUM
ejpam-5152	247	6	,	,	PUNCT
ejpam-5152	247	7	2	2	NUM
ejpam-5152	247	8	,	,	PUNCT
ejpam-5152	247	9	3	3	NUM
ejpam-5152	247	10	,	,	PUNCT
ejpam-5152	247	11	.	.	PUNCT
ejpam-5152	247	12	.	.	PUNCT
ejpam-5152	247	13	.	.	PUNCT
ejpam-5152	248	1	(	(	PUNCT
ejpam-5152	248	2	10	10	NUM
ejpam-5152	248	3	)	)	PUNCT
ejpam-5152	248	4	suppose	suppose	VERB
ejpam-5152	248	5	that	that	SCONJ
ejpam-5152	248	6	2mt	2mt	NOUN
ejpam-5152	248	7	is	be	AUX
ejpam-5152	248	8	the	the	DET
ejpam-5152	248	9	smallest	small	ADJ
ejpam-5152	248	10	integer	integer	NOUN
ejpam-5152	248	11	that	that	SCONJ
ejpam-5152	248	12	greater	great	ADJ
ejpam-5152	248	13	than	than	ADP
ejpam-5152	248	14	2nt	2nt	NOUN
ejpam-5152	248	15	and	and	CCONJ
ejpam-5152	248	16	satisfies	satisfy	VERB
ejpam-5152	248	17	the	the	DET
ejpam-5152	248	18	inequality	inequality	NOUN
ejpam-5152	248	19	(	(	PUNCT
ejpam-5152	248	20	10	10	NUM
ejpam-5152	248	21	)	)	PUNCT
ejpam-5152	248	22	then	then	ADV
ejpam-5152	248	23	we	we	PRON
ejpam-5152	248	24	have	have	VERB
ejpam-5152	248	25	p(x2nt	p(x2nt	NOUN
ejpam-5152	248	26	,	,	PUNCT
ejpam-5152	248	27	x2mt−2	x2mt−2	NUM
ejpam-5152	248	28	)	)	PUNCT
ejpam-5152	248	29	≤	≤	PUNCT
ejpam-5152	249	1	ε	ε	PROPN
ejpam-5152	249	2	.	.	PUNCT
ejpam-5152	249	3	hence	hence	ADV
ejpam-5152	249	4	ε	ε	PROPN
ejpam-5152	249	5	≤	≤	ADJ
ejpam-5152	249	6	p(x2nt	p(x2nt	NOUN
ejpam-5152	249	7	,	,	PUNCT
ejpam-5152	249	8	x2mt	x2mt	PROPN
ejpam-5152	249	9	)	)	PUNCT
ejpam-5152	249	10	≤	≤	NUM
ejpam-5152	249	11	p(x2nt	p(x2nt	NOUN
ejpam-5152	249	12	,	,	PUNCT
ejpam-5152	249	13	x2mt−2	x2mt−2	NUM
ejpam-5152	249	14	)	)	PUNCT
ejpam-5152	249	15	+	+	NUM
ejpam-5152	249	16	p(x2mt−2	p(x2mt−2	NOUN
ejpam-5152	249	17	,	,	PUNCT
ejpam-5152	249	18	x2mt)−	x2mt)−	PROPN
ejpam-5152	249	19	p(x2mt−2	p(x2mt−2	PROPN
ejpam-5152	249	20	,	,	PUNCT
ejpam-5152	249	21	x2mt−2	x2mt−2	NUM
ejpam-5152	249	22	)	)	PUNCT
ejpam-5152	249	23	≤	≤	NUM
ejpam-5152	249	24	p(x2nt	p(x2nt	NOUN
ejpam-5152	249	25	,	,	PUNCT
ejpam-5152	249	26	x2mt−2	x2mt−2	NUM
ejpam-5152	249	27	)	)	PUNCT
ejpam-5152	249	28	+	+	NUM
ejpam-5152	249	29	p(x2mt−2	p(x2mt−2	NOUN
ejpam-5152	249	30	,	,	PUNCT
ejpam-5152	249	31	x2mt	x2mt	PROPN
ejpam-5152	249	32	)	)	PUNCT
ejpam-5152	249	33	≤	≤	NOUN
ejpam-5152	249	34	ε+	ε+	X
ejpam-5152	249	35	p(x2mt−2	p(x2mt−2	NOUN
ejpam-5152	249	36	,	,	PUNCT
ejpam-5152	249	37	x2mt	x2mt	PROPN
ejpam-5152	249	38	)	)	PUNCT
ejpam-5152	249	39	.	.	PUNCT
ejpam-5152	250	1	a.	a.	PROPN
ejpam-5152	250	2	ekayanti	ekayanti	PROPN
ejpam-5152	250	3	et	et	PROPN
ejpam-5152	250	4	al	al	PROPN
ejpam-5152	250	5	.	.	PUNCT
ejpam-5152	250	6	/	/	SYM
ejpam-5152	250	7	eur	eur	PROPN
ejpam-5152	250	8	.	.	PUNCT
ejpam-5152	251	1	j.	j.	PROPN
ejpam-5152	251	2	pure	pure	PROPN
ejpam-5152	251	3	appl	appl	PROPN
ejpam-5152	251	4	.	.	PROPN
ejpam-5152	251	5	math	math	PROPN
ejpam-5152	251	6	,	,	PUNCT
ejpam-5152	251	7	17	17	NUM
ejpam-5152	251	8	(	(	PUNCT
ejpam-5152	251	9	2	2	NUM
ejpam-5152	251	10	)	)	PUNCT
ejpam-5152	251	11	(	(	PUNCT
ejpam-5152	251	12	2024	2024	NUM
ejpam-5152	251	13	)	)	PUNCT
ejpam-5152	251	14	,	,	PUNCT
ejpam-5152	251	15	996	996	NUM
ejpam-5152	251	16	-	-	SYM
ejpam-5152	251	17	1008	1008	NUM
ejpam-5152	251	18	1005	1005	NUM
ejpam-5152	251	19	let	let	VERB
ejpam-5152	251	20	we	we	PRON
ejpam-5152	251	21	consider	consider	VERB
ejpam-5152	251	22	that	that	DET
ejpam-5152	251	23	p(x2mt−2	p(x2mt−2	NOUN
ejpam-5152	251	24	,	,	PUNCT
ejpam-5152	251	25	x2mt	x2mt	PROPN
ejpam-5152	251	26	)	)	PUNCT
ejpam-5152	251	27	→	→	SYM
ejpam-5152	251	28	0	0	PUNCT
ejpam-5152	251	29	as	as	ADP
ejpam-5152	251	30	mt	mt	PROPN
ejpam-5152	251	31	→	→	SYM
ejpam-5152	251	32	∞	∞	PROPN
ejpam-5152	251	33	,	,	PUNCT
ejpam-5152	251	34	then	then	ADV
ejpam-5152	251	35	we	we	PRON
ejpam-5152	251	36	have	have	VERB
ejpam-5152	251	37	ε	ε	PROPN
ejpam-5152	251	38	≤	≤	ADJ
ejpam-5152	251	39	p(x2nt	p(x2nt	NOUN
ejpam-5152	251	40	,	,	PUNCT
ejpam-5152	251	41	x2mt	x2mt	PROPN
ejpam-5152	251	42	)	)	PUNCT
ejpam-5152	251	43	≤	≤	NUM
ejpam-5152	252	1	ε	ε	PROPN
ejpam-5152	252	2	.	.	PUNCT
ejpam-5152	253	1	consequently	consequently	ADV
ejpam-5152	253	2	,	,	PUNCT
ejpam-5152	253	3	limnt	limnt	NOUN
ejpam-5152	253	4	,	,	PUNCT
ejpam-5152	253	5	mt→∞p(x2nt	mt→∞p(x2nt	NOUN
ejpam-5152	253	6	,	,	PUNCT
ejpam-5152	253	7	x2mt	x2mt	PROPN
ejpam-5152	253	8	)	)	PUNCT
ejpam-5152	254	1	=	=	PUNCT
ejpam-5152	254	2	ε	ε	PROPN
ejpam-5152	254	3	.	.	PUNCT
ejpam-5152	255	1	it	it	PRON
ejpam-5152	255	2	is	be	AUX
ejpam-5152	255	3	noted	note	VERB
ejpam-5152	255	4	that	that	SCONJ
ejpam-5152	255	5	p(x2nt+1	p(x2nt+1	PROPN
ejpam-5152	255	6	,	,	PUNCT
ejpam-5152	255	7	x2mt	x2mt	PROPN
ejpam-5152	255	8	)	)	PUNCT
ejpam-5152	255	9	≤	≤	NUM
ejpam-5152	255	10	hp(f	hp(f	PUNCT
ejpam-5152	255	11	(	(	PUNCT
ejpam-5152	255	12	x2nt	x2nt	NOUN
ejpam-5152	255	13	)	)	PUNCT
ejpam-5152	255	14	,	,	PUNCT
ejpam-5152	255	15	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	255	16	)	)	PUNCT
ejpam-5152	255	17	)	)	PUNCT
ejpam-5152	256	1	≤	≤	NOUN
ejpam-5152	256	2	ϕ(max{p(x2nt	ϕ(max{p(x2nt	NOUN
ejpam-5152	256	3	,	,	PUNCT
ejpam-5152	256	4	x2mt−1	x2mt−1	PROPN
ejpam-5152	256	5	)	)	PUNCT
ejpam-5152	256	6	,	,	PUNCT
ejpam-5152	256	7	p(x2nt	p(x2nt	NOUN
ejpam-5152	256	8	,	,	PUNCT
ejpam-5152	256	9	f	f	PROPN
ejpam-5152	256	10	(	(	PUNCT
ejpam-5152	256	11	x2nt	x2nt	NOUN
ejpam-5152	256	12	)	)	PUNCT
ejpam-5152	256	13	)	)	PUNCT
ejpam-5152	256	14	,	,	PUNCT
ejpam-5152	256	15	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	256	16	,	,	PUNCT
ejpam-5152	256	17	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	256	18	)	)	PUNCT
ejpam-5152	256	19	)	)	PUNCT
ejpam-5152	256	20	,	,	PUNCT
ejpam-5152	256	21	p(x2nt	p(x2nt	NOUN
ejpam-5152	256	22	,	,	PUNCT
ejpam-5152	256	23	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	256	24	)	)	PUNCT
ejpam-5152	256	25	)	)	PUNCT
ejpam-5152	256	26	,	,	PUNCT
ejpam-5152	256	27	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	256	28	,	,	PUNCT
ejpam-5152	256	29	f	f	PROPN
ejpam-5152	256	30	(	(	PUNCT
ejpam-5152	256	31	x2nt	x2nt	NOUN
ejpam-5152	256	32	)	)	PUNCT
ejpam-5152	256	33	)	)	PUNCT
ejpam-5152	256	34	}	}	PUNCT
ejpam-5152	256	35	)	)	PUNCT
ejpam-5152	256	36	≤	≤	NOUN
ejpam-5152	257	1	ϕ(max{p(x2nt	ϕ(max{p(x2nt	NOUN
ejpam-5152	257	2	,	,	PUNCT
ejpam-5152	257	3	x2mt−1	x2mt−1	PROPN
ejpam-5152	257	4	)	)	PUNCT
ejpam-5152	257	5	,	,	PUNCT
ejpam-5152	257	6	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	257	7	,	,	PUNCT
ejpam-5152	257	8	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	257	9	)	)	PUNCT
ejpam-5152	257	10	)	)	PUNCT
ejpam-5152	257	11	}	}	PUNCT
ejpam-5152	257	12	)	)	PUNCT
ejpam-5152	258	1	≤	≤	NUM
ejpam-5152	258	2	ϕ(p(x2nt	ϕ(p(x2nt	NUM
ejpam-5152	258	3	,	,	PUNCT
ejpam-5152	258	4	x2mt−1	x2mt−1	PROPN
ejpam-5152	258	5	)	)	PUNCT
ejpam-5152	259	1	+	+	CCONJ
ejpam-5152	259	2	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	259	3	,	,	PUNCT
ejpam-5152	259	4	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	259	5	)	)	PUNCT
ejpam-5152	259	6	)	)	PUNCT
ejpam-5152	259	7	)	)	PUNCT
ejpam-5152	260	1	≤	≤	ADV
ejpam-5152	260	2	ϕ(p(x2nt	ϕ(p(x2nt	NUM
ejpam-5152	260	3	,	,	PUNCT
ejpam-5152	260	4	x2mt−1	x2mt−1	PROPN
ejpam-5152	260	5	)	)	PUNCT
ejpam-5152	261	1	+	+	CCONJ
ejpam-5152	261	2	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	261	3	,	,	PUNCT
ejpam-5152	261	4	x2nt	x2nt	PUNCT
ejpam-5152	261	5	)	)	PUNCT
ejpam-5152	262	1	+	+	NUM
ejpam-5152	262	2	p(x2nt	p(x2nt	NOUN
ejpam-5152	262	3	,	,	PUNCT
ejpam-5152	262	4	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	262	5	)	)	PUNCT
ejpam-5152	262	6	)	)	PUNCT
ejpam-5152	262	7	−p(x2nt	−p(x2nt	NOUN
ejpam-5152	262	8	,	,	PUNCT
ejpam-5152	262	9	x2nt	x2nt	PROPN
ejpam-5152	262	10	)	)	PUNCT
ejpam-5152	262	11	)	)	PUNCT
ejpam-5152	263	1	≤	≤	ADV
ejpam-5152	263	2	ϕ(p(x2nt	ϕ(p(x2nt	NUM
ejpam-5152	263	3	,	,	PUNCT
ejpam-5152	263	4	x2mt−1	x2mt−1	PROPN
ejpam-5152	263	5	)	)	PUNCT
ejpam-5152	264	1	+	+	CCONJ
ejpam-5152	264	2	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	264	3	,	,	PUNCT
ejpam-5152	264	4	x2nt	x2nt	PUNCT
ejpam-5152	264	5	)	)	PUNCT
ejpam-5152	265	1	+	+	NUM
ejpam-5152	265	2	p(x2nt	p(x2nt	NOUN
ejpam-5152	265	3	,	,	PUNCT
ejpam-5152	265	4	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	265	5	)	)	PUNCT
ejpam-5152	265	6	)	)	PUNCT
ejpam-5152	265	7	≤	≤	ADV
ejpam-5152	265	8	ϕ(p(x2nt	ϕ(p(x2nt	NUM
ejpam-5152	265	9	,	,	PUNCT
ejpam-5152	265	10	x2mt−1	x2mt−1	PROPN
ejpam-5152	265	11	)	)	PUNCT
ejpam-5152	266	1	+	+	CCONJ
ejpam-5152	266	2	p(x2mt−1	p(x2mt−1	NOUN
ejpam-5152	266	3	,	,	PUNCT
ejpam-5152	266	4	x2nt	x2nt	PUNCT
ejpam-5152	266	5	)	)	PUNCT
ejpam-5152	267	1	≤	≤	PUNCT
ejpam-5152	268	1	ϕ(2p(x2nt	ϕ(2p(x2nt	PROPN
ejpam-5152	268	2	,	,	PUNCT
ejpam-5152	268	3	x2mt−1	x2mt−1	PROPN
ejpam-5152	268	4	)	)	PUNCT
ejpam-5152	268	5	)	)	PUNCT
ejpam-5152	268	6	≤	≤	NUM
ejpam-5152	268	7	ϕ(2ε	ϕ(2ε	NOUN
ejpam-5152	268	8	)	)	PUNCT
ejpam-5152	268	9	therefore	therefore	ADV
ejpam-5152	268	10	,	,	PUNCT
ejpam-5152	268	11	we	we	PRON
ejpam-5152	268	12	have	have	VERB
ejpam-5152	268	13	p(x2nt	p(x2nt	NOUN
ejpam-5152	268	14	,	,	PUNCT
ejpam-5152	268	15	x2mt	x2mt	PROPN
ejpam-5152	268	16	)	)	PUNCT
ejpam-5152	269	1	≤	≤	NUM
ejpam-5152	269	2	p(x2nt	p(x2nt	NOUN
ejpam-5152	269	3	,	,	PUNCT
ejpam-5152	269	4	x2nt+1	x2nt+1	PROPN
ejpam-5152	269	5	)	)	PUNCT
ejpam-5152	270	1	+	+	CCONJ
ejpam-5152	271	1	p(x2nt+1	p(x2nt+1	INTJ
ejpam-5152	271	2	,	,	PUNCT
ejpam-5152	271	3	x2mt	x2mt	PROPN
ejpam-5152	271	4	)	)	PUNCT
ejpam-5152	271	5	≤	≤	NUM
ejpam-5152	271	6	p(x2nt	p(x2nt	NOUN
ejpam-5152	271	7	,	,	PUNCT
ejpam-5152	271	8	x2nt+1	x2nt+1	PROPN
ejpam-5152	271	9	)	)	PUNCT
ejpam-5152	272	1	+	+	ADV
ejpam-5152	272	2	hp(f	hp(f	X
ejpam-5152	272	3	(	(	PUNCT
ejpam-5152	272	4	x2nt	x2nt	NOUN
ejpam-5152	272	5	)	)	PUNCT
ejpam-5152	272	6	,	,	PUNCT
ejpam-5152	272	7	g(x2mt−1	g(x2mt−1	NOUN
ejpam-5152	272	8	)	)	PUNCT
ejpam-5152	272	9	)	)	PUNCT
ejpam-5152	272	10	≤	≤	NUM
ejpam-5152	272	11	p(x2nt	p(x2nt	NOUN
ejpam-5152	272	12	,	,	PUNCT
ejpam-5152	272	13	x2nt+1	x2nt+1	PROPN
ejpam-5152	272	14	)	)	PUNCT
ejpam-5152	273	1	+	+	CCONJ
ejpam-5152	273	2	ϕ(2ε	ϕ(2ε	NOUN
ejpam-5152	273	3	)	)	PUNCT
ejpam-5152	273	4	thus	thus	ADV
ejpam-5152	273	5	for	for	ADP
ejpam-5152	273	6	nt	not	PART
ejpam-5152	273	7	,	,	PUNCT
ejpam-5152	273	8	mt	mt	PROPN
ejpam-5152	273	9	→	→	SYM
ejpam-5152	273	10	∞	∞	PROPN
ejpam-5152	273	11	we	we	PRON
ejpam-5152	273	12	have	have	VERB
ejpam-5152	273	13	ε	ε	PROPN
ejpam-5152	273	14	≤	≤	NOUN
ejpam-5152	273	15	ϕ(2ε	ϕ(2ε	NUM
ejpam-5152	273	16	)	)	PUNCT
ejpam-5152	273	17	.	.	PUNCT
ejpam-5152	274	1	since	since	SCONJ
ejpam-5152	274	2	ϕ(2ε	ϕ(2ε	NOUN
ejpam-5152	274	3	)	)	PUNCT
ejpam-5152	274	4	<	<	X
ejpam-5152	274	5	ε	ε	PROPN
ejpam-5152	274	6	then	then	ADV
ejpam-5152	274	7	we	we	PRON
ejpam-5152	274	8	have	have	VERB
ejpam-5152	274	9	a	a	DET
ejpam-5152	274	10	contradiction	contradiction	NOUN
ejpam-5152	274	11	.	.	PUNCT
ejpam-5152	275	1	therefore	therefore	ADV
ejpam-5152	275	2	,	,	PUNCT
ejpam-5152	275	3	it	it	PRON
ejpam-5152	275	4	can	can	AUX
ejpam-5152	275	5	be	be	AUX
ejpam-5152	275	6	concluded	conclude	VERB
ejpam-5152	275	7	that	that	SCONJ
ejpam-5152	275	8	(	(	PUNCT
ejpam-5152	275	9	xn	xn	X
ejpam-5152	275	10	)	)	PUNCT
ejpam-5152	275	11	is	be	AUX
ejpam-5152	275	12	cauchy	cauchy	PROPN
ejpam-5152	275	13	sequences	sequence	NOUN
ejpam-5152	275	14	inx	inx	PROPN
ejpam-5152	275	15	.	.	PUNCT
ejpam-5152	276	1	since	since	SCONJ
ejpam-5152	276	2	(	(	PUNCT
ejpam-5152	276	3	x	x	X
ejpam-5152	276	4	,	,	PUNCT
ejpam-5152	276	5	p	p	NOUN
ejpam-5152	276	6	)	)	PUNCT
ejpam-5152	276	7	is	be	AUX
ejpam-5152	276	8	complete	complete	ADJ
ejpam-5152	276	9	partial	partial	ADJ
ejpam-5152	276	10	metric	metric	ADJ
ejpam-5152	276	11	space	space	NOUN
ejpam-5152	276	12	,	,	PUNCT
ejpam-5152	276	13	then	then	ADV
ejpam-5152	276	14	there	there	PRON
ejpam-5152	276	15	exist	exist	VERB
ejpam-5152	276	16	x	x	X
ejpam-5152	276	17	∈	∈	PROPN
ejpam-5152	276	18	x	x	X
ejpam-5152	276	19	such	such	ADJ
ejpam-5152	276	20	that	that	DET
ejpam-5152	276	21	limn→∞xn	limn→∞xn	NOUN
ejpam-5152	277	1	=	=	SYM
ejpam-5152	277	2	x.	x.	NOUN
ejpam-5152	277	3	furthermore	furthermore	ADV
ejpam-5152	277	4	,	,	PUNCT
ejpam-5152	277	5	we	we	PRON
ejpam-5152	277	6	will	will	AUX
ejpam-5152	277	7	establish	establish	VERB
ejpam-5152	277	8	that	that	SCONJ
ejpam-5152	277	9	x	x	PRON
ejpam-5152	277	10	is	be	AUX
ejpam-5152	277	11	common	common	ADJ
ejpam-5152	277	12	fixed	fix	VERB
ejpam-5152	277	13	point	point	NOUN
ejpam-5152	277	14	of	of	ADP
ejpam-5152	277	15	f	f	PROPN
ejpam-5152	277	16	and	and	CCONJ
ejpam-5152	277	17	g.	g.	PROPN
ejpam-5152	277	18	let	let	VERB
ejpam-5152	277	19	p(x	p(x	PROPN
ejpam-5152	277	20	,	,	PUNCT
ejpam-5152	277	21	f	f	PROPN
ejpam-5152	277	22	(	(	PUNCT
ejpam-5152	277	23	x	x	NOUN
ejpam-5152	277	24	)	)	PUNCT
ejpam-5152	277	25	)	)	PUNCT
ejpam-5152	277	26	>	>	X
ejpam-5152	278	1	0	0	X
ejpam-5152	278	2	.	.	PUNCT
ejpam-5152	279	1	let	let	VERB
ejpam-5152	279	2	we	we	PRON
ejpam-5152	279	3	consider	consider	VERB
ejpam-5152	279	4	that	that	SCONJ
ejpam-5152	279	5	p(x2n	p(x2n	NOUN
ejpam-5152	279	6	,	,	PUNCT
ejpam-5152	279	7	f	f	PROPN
ejpam-5152	279	8	(	(	PUNCT
ejpam-5152	279	9	x	x	NOUN
ejpam-5152	279	10	)	)	PUNCT
ejpam-5152	279	11	)	)	PUNCT
ejpam-5152	279	12	≤	≤	NOUN
ejpam-5152	279	13	hp(f	hp(f	PUNCT
ejpam-5152	279	14	(	(	PUNCT
ejpam-5152	279	15	x	x	X
ejpam-5152	279	16	)	)	PUNCT
ejpam-5152	279	17	,	,	PUNCT
ejpam-5152	279	18	g(x2n−1	g(x2n−1	PROPN
ejpam-5152	279	19	)	)	PUNCT
ejpam-5152	279	20	)	)	PUNCT
ejpam-5152	279	21	≤	≤	PROPN
ejpam-5152	280	1	ϕ(max{p(x	ϕ(max{p(x	PROPN
ejpam-5152	280	2	,	,	PUNCT
ejpam-5152	280	3	x2n−1	x2n−1	PROPN
ejpam-5152	280	4	)	)	PUNCT
ejpam-5152	280	5	,	,	PUNCT
ejpam-5152	280	6	p(x	p(x	PROPN
ejpam-5152	280	7	,	,	PUNCT
ejpam-5152	280	8	f	f	PROPN
ejpam-5152	280	9	(	(	PUNCT
ejpam-5152	280	10	x	x	NOUN
ejpam-5152	280	11	)	)	PUNCT
ejpam-5152	280	12	)	)	PUNCT
ejpam-5152	280	13	,	,	PUNCT
ejpam-5152	280	14	p(x2n−1	p(x2n−1	PROPN
ejpam-5152	280	15	,	,	PUNCT
ejpam-5152	280	16	g(x2n−1	g(x2n−1	PROPN
ejpam-5152	280	17	)	)	PUNCT
ejpam-5152	280	18	)	)	PUNCT
ejpam-5152	280	19	,	,	PUNCT
ejpam-5152	280	20	p(x	p(x	NOUN
ejpam-5152	280	21	,	,	PUNCT
ejpam-5152	280	22	g(x2n−1	g(x2n−1	PROPN
ejpam-5152	280	23	)	)	PUNCT
ejpam-5152	280	24	)	)	PUNCT
ejpam-5152	280	25	,	,	PUNCT
ejpam-5152	280	26	p(x2n−1	p(x2n−1	PROPN
ejpam-5152	280	27	,	,	PUNCT
ejpam-5152	280	28	f	f	PROPN
ejpam-5152	280	29	(	(	PUNCT
ejpam-5152	280	30	x	x	NOUN
ejpam-5152	280	31	)	)	PUNCT
ejpam-5152	280	32	)	)	PUNCT
ejpam-5152	280	33	}	}	PUNCT
ejpam-5152	280	34	)	)	PUNCT
ejpam-5152	280	35	(	(	PUNCT
ejpam-5152	280	36	11	11	X
ejpam-5152	280	37	)	)	PUNCT
ejpam-5152	280	38	taking	take	VERB
ejpam-5152	280	39	n	n	PRON
ejpam-5152	280	40	→	→	SYM
ejpam-5152	280	41	∞	∞	NUM
ejpam-5152	280	42	on	on	ADP
ejpam-5152	280	43	the	the	DET
ejpam-5152	280	44	inequality	inequality	NOUN
ejpam-5152	280	45	(	(	PUNCT
ejpam-5152	280	46	11	11	NUM
ejpam-5152	280	47	)	)	PUNCT
ejpam-5152	280	48	above	above	ADV
ejpam-5152	280	49	,	,	PUNCT
ejpam-5152	280	50	we	we	PRON
ejpam-5152	280	51	obtain	obtain	VERB
ejpam-5152	280	52	p(x	p(x	PROPN
ejpam-5152	280	53	,	,	PUNCT
ejpam-5152	280	54	f	f	PROPN
ejpam-5152	280	55	(	(	PUNCT
ejpam-5152	280	56	x	x	NOUN
ejpam-5152	280	57	)	)	PUNCT
ejpam-5152	280	58	)	)	PUNCT
ejpam-5152	281	1	≤	≤	PROPN
ejpam-5152	282	1	ϕ(max{p(x	ϕ(max{p(x	PROPN
ejpam-5152	282	2	,	,	PUNCT
ejpam-5152	282	3	f	f	PROPN
ejpam-5152	282	4	(	(	PUNCT
ejpam-5152	282	5	x	x	NOUN
ejpam-5152	282	6	)	)	PUNCT
ejpam-5152	282	7	)	)	PUNCT
ejpam-5152	282	8	,	,	PUNCT
ejpam-5152	282	9	p(x	p(x	PROPN
ejpam-5152	282	10	,	,	PUNCT
ejpam-5152	282	11	f	f	PROPN
ejpam-5152	282	12	(	(	PUNCT
ejpam-5152	282	13	x	x	NOUN
ejpam-5152	282	14	)	)	PUNCT
ejpam-5152	282	15	)	)	PUNCT
ejpam-5152	282	16	}	}	PUNCT
ejpam-5152	282	17	)	)	PUNCT
ejpam-5152	282	18	≤	≤	PUNCT
ejpam-5152	283	1	ϕ(p(x	ϕ(p(x	PROPN
ejpam-5152	283	2	,	,	PUNCT
ejpam-5152	283	3	f	f	PROPN
ejpam-5152	283	4	(	(	PUNCT
ejpam-5152	283	5	x	x	NOUN
ejpam-5152	283	6	)	)	PUNCT
ejpam-5152	283	7	)	)	PUNCT
ejpam-5152	284	1	+	+	CCONJ
ejpam-5152	284	2	p(x	p(x	PROPN
ejpam-5152	284	3	,	,	PUNCT
ejpam-5152	284	4	f	f	PROPN
ejpam-5152	284	5	(	(	PUNCT
ejpam-5152	284	6	x	x	NOUN
ejpam-5152	284	7	)	)	PUNCT
ejpam-5152	284	8	)	)	PUNCT
ejpam-5152	284	9	)	)	PUNCT
ejpam-5152	285	1	=	=	SYM
ejpam-5152	285	2	ϕ(2p(x	ϕ(2p(x	PROPN
ejpam-5152	285	3	,	,	PUNCT
ejpam-5152	285	4	f	f	PROPN
ejpam-5152	285	5	(	(	PUNCT
ejpam-5152	285	6	x	x	NOUN
ejpam-5152	285	7	)	)	PUNCT
ejpam-5152	285	8	)	)	PUNCT
ejpam-5152	285	9	)	)	PUNCT
ejpam-5152	286	1	<	<	X
ejpam-5152	286	2	p(x	p(x	PROPN
ejpam-5152	286	3	,	,	PUNCT
ejpam-5152	286	4	f	f	PROPN
ejpam-5152	286	5	(	(	PUNCT
ejpam-5152	286	6	x	x	NOUN
ejpam-5152	286	7	)	)	PUNCT
ejpam-5152	286	8	)	)	PUNCT
ejpam-5152	286	9	.	.	PUNCT
ejpam-5152	287	1	then	then	ADV
ejpam-5152	287	2	we	we	PRON
ejpam-5152	287	3	have	have	VERB
ejpam-5152	287	4	a	a	DET
ejpam-5152	287	5	contradiction	contradiction	NOUN
ejpam-5152	287	6	.	.	PUNCT
ejpam-5152	288	1	hence	hence	ADV
ejpam-5152	288	2	,	,	PUNCT
ejpam-5152	288	3	p(x	p(x	PROPN
ejpam-5152	288	4	,	,	PUNCT
ejpam-5152	288	5	f	f	PROPN
ejpam-5152	288	6	(	(	PUNCT
ejpam-5152	288	7	x	x	NOUN
ejpam-5152	288	8	)	)	PUNCT
ejpam-5152	288	9	)	)	PUNCT
ejpam-5152	289	1	=	=	SYM
ejpam-5152	289	2	0	0	NUM
ejpam-5152	289	3	,	,	PUNCT
ejpam-5152	289	4	i.e.	i.e.	X
ejpam-5152	289	5	,	,	PUNCT
ejpam-5152	289	6	x	x	SYM
ejpam-5152	289	7	∈	∈	PROPN
ejpam-5152	289	8	f	f	X
ejpam-5152	289	9	(	(	PUNCT
ejpam-5152	289	10	x	x	NOUN
ejpam-5152	289	11	)	)	PUNCT
ejpam-5152	289	12	.	.	PUNCT
ejpam-5152	290	1	in	in	ADP
ejpam-5152	290	2	the	the	DET
ejpam-5152	290	3	similar	similar	ADJ
ejpam-5152	290	4	way	way	NOUN
ejpam-5152	290	5	it	it	PRON
ejpam-5152	290	6	can	can	AUX
ejpam-5152	290	7	be	be	AUX
ejpam-5152	290	8	shown	show	VERB
ejpam-5152	290	9	that	that	SCONJ
ejpam-5152	290	10	p(x	p(x	NOUN
ejpam-5152	290	11	,	,	PUNCT
ejpam-5152	290	12	g(x	g(x	NOUN
ejpam-5152	290	13	)	)	PUNCT
ejpam-5152	290	14	)	)	PUNCT
ejpam-5152	291	1	=	=	SYM
ejpam-5152	291	2	0	0	NUM
ejpam-5152	291	3	,	,	PUNCT
ejpam-5152	291	4	i.e.	i.e.	X
ejpam-5152	291	5	,	,	PUNCT
ejpam-5152	291	6	x	x	SYM
ejpam-5152	291	7	∈	∈	PROPN
ejpam-5152	291	8	g(x	g(x	NOUN
ejpam-5152	291	9	)	)	PUNCT
ejpam-5152	291	10	.	.	PUNCT
ejpam-5152	292	1	it	it	PRON
ejpam-5152	292	2	means	mean	VERB
ejpam-5152	292	3	,	,	PUNCT
ejpam-5152	292	4	x	x	PRON
ejpam-5152	292	5	is	be	AUX
ejpam-5152	292	6	a	a	DET
ejpam-5152	292	7	common	common	ADJ
ejpam-5152	292	8	fixed	fix	VERB
ejpam-5152	292	9	point	point	NOUN
ejpam-5152	292	10	of	of	ADP
ejpam-5152	292	11	set	set	NOUN
ejpam-5152	292	12	-	-	PUNCT
ejpam-5152	292	13	valued	value	VERB
ejpam-5152	292	14	mapping	mapping	NOUN
ejpam-5152	292	15	f	f	PROPN
ejpam-5152	292	16	and	and	CCONJ
ejpam-5152	292	17	g.	g.	PROPN
ejpam-5152	292	18	furthermore	furthermore	ADV
ejpam-5152	292	19	,	,	PUNCT
ejpam-5152	292	20	we	we	PRON
ejpam-5152	292	21	will	will	AUX
ejpam-5152	292	22	show	show	VERB
ejpam-5152	292	23	the	the	DET
ejpam-5152	292	24	uniqueness	uniqueness	NOUN
ejpam-5152	292	25	of	of	ADP
ejpam-5152	292	26	this	this	DET
ejpam-5152	292	27	common	common	ADJ
ejpam-5152	292	28	fixed	fix	VERB
ejpam-5152	292	29	points	point	NOUN
ejpam-5152	292	30	.	.	PUNCT
ejpam-5152	293	1	a.	a.	PROPN
ejpam-5152	293	2	ekayanti	ekayanti	PROPN
ejpam-5152	293	3	et	et	PROPN
ejpam-5152	293	4	al	al	PROPN
ejpam-5152	293	5	.	.	PUNCT
ejpam-5152	293	6	/	/	SYM
ejpam-5152	293	7	eur	eur	PROPN
ejpam-5152	293	8	.	.	PUNCT
ejpam-5152	294	1	j.	j.	PROPN
ejpam-5152	294	2	pure	pure	PROPN
ejpam-5152	294	3	appl	appl	PROPN
ejpam-5152	294	4	.	.	PROPN
ejpam-5152	294	5	math	math	PROPN
ejpam-5152	294	6	,	,	PUNCT
ejpam-5152	294	7	17	17	NUM
ejpam-5152	294	8	(	(	PUNCT
ejpam-5152	294	9	2	2	NUM
ejpam-5152	294	10	)	)	PUNCT
ejpam-5152	294	11	(	(	PUNCT
ejpam-5152	294	12	2024	2024	NUM
ejpam-5152	294	13	)	)	PUNCT
ejpam-5152	294	14	,	,	PUNCT
ejpam-5152	294	15	996	996	NUM
ejpam-5152	294	16	-	-	SYM
ejpam-5152	294	17	1008	1008	NUM
ejpam-5152	294	18	1006	1006	NUM
ejpam-5152	294	19	suppose	suppose	VERB
ejpam-5152	294	20	that	that	SCONJ
ejpam-5152	294	21	v	v	NOUN
ejpam-5152	294	22	is	be	AUX
ejpam-5152	294	23	another	another	DET
ejpam-5152	294	24	common	common	ADJ
ejpam-5152	294	25	fixed	fix	VERB
ejpam-5152	294	26	point	point	NOUN
ejpam-5152	294	27	of	of	ADP
ejpam-5152	294	28	set	set	NOUN
ejpam-5152	294	29	-	-	PUNCT
ejpam-5152	294	30	valued	value	VERB
ejpam-5152	294	31	mappings	mapping	NOUN
ejpam-5152	294	32	f	f	NOUN
ejpam-5152	294	33	and	and	CCONJ
ejpam-5152	294	34	g	g	PROPN
ejpam-5152	294	35	such	such	DET
ejpam-5152	294	36	that	that	DET
ejpam-5152	294	37	v	v	NUM
ejpam-5152	294	38	∈	∈	PROPN
ejpam-5152	294	39	f	f	X
ejpam-5152	294	40	(	(	PUNCT
ejpam-5152	294	41	v	v	NOUN
ejpam-5152	294	42	)	)	PUNCT
ejpam-5152	294	43	and	and	CCONJ
ejpam-5152	294	44	v	v	ADP
ejpam-5152	294	45	∈	∈	PROPN
ejpam-5152	294	46	g(v	g(v	PROPN
ejpam-5152	294	47	)	)	PUNCT
ejpam-5152	294	48	.	.	PUNCT
ejpam-5152	295	1	let	let	VERB
ejpam-5152	295	2	p(x	p(x	NOUN
ejpam-5152	295	3	,	,	PUNCT
ejpam-5152	295	4	v	v	NOUN
ejpam-5152	295	5	)	)	PUNCT
ejpam-5152	295	6	>	>	X
ejpam-5152	295	7	0	0	PUNCT
ejpam-5152	296	1	then	then	ADV
ejpam-5152	296	2	hp(f	hp(f	VERB
ejpam-5152	296	3	(	(	PUNCT
ejpam-5152	296	4	x	x	X
ejpam-5152	296	5	,	,	PUNCT
ejpam-5152	296	6	g(v	g(v	PROPN
ejpam-5152	296	7	)	)	PUNCT
ejpam-5152	296	8	)	)	PUNCT
ejpam-5152	296	9	≤	≤	PROPN
ejpam-5152	297	1	ϕ(max{p(x	ϕ(max{p(x	PROPN
ejpam-5152	297	2	,	,	PUNCT
ejpam-5152	297	3	v	v	NOUN
ejpam-5152	297	4	)	)	PUNCT
ejpam-5152	297	5	,	,	PUNCT
ejpam-5152	297	6	p(x	p(x	PROPN
ejpam-5152	297	7	,	,	PUNCT
ejpam-5152	297	8	f	f	PROPN
ejpam-5152	297	9	(	(	PUNCT
ejpam-5152	297	10	x	x	NOUN
ejpam-5152	297	11	)	)	PUNCT
ejpam-5152	297	12	)	)	PUNCT
ejpam-5152	297	13	,	,	PUNCT
ejpam-5152	297	14	p(v	p(v	NOUN
ejpam-5152	297	15	,	,	PUNCT
ejpam-5152	297	16	g(v	g(v	PROPN
ejpam-5152	297	17	)	)	PUNCT
ejpam-5152	297	18	)	)	PUNCT
ejpam-5152	297	19	,	,	PUNCT
ejpam-5152	297	20	p(x	p(x	PROPN
ejpam-5152	297	21	,	,	PUNCT
ejpam-5152	297	22	g(v	g(v	PROPN
ejpam-5152	297	23	)	)	PUNCT
ejpam-5152	297	24	)	)	PUNCT
ejpam-5152	297	25	,	,	PUNCT
ejpam-5152	297	26	p(v	p(v	PROPN
ejpam-5152	297	27	,	,	PUNCT
ejpam-5152	297	28	f	f	PROPN
ejpam-5152	297	29	(	(	PUNCT
ejpam-5152	297	30	x	x	NOUN
ejpam-5152	297	31	)	)	PUNCT
ejpam-5152	297	32	)	)	PUNCT
ejpam-5152	297	33	}	}	PUNCT
ejpam-5152	297	34	)	)	PUNCT
ejpam-5152	297	35	≤	≤	NUM
ejpam-5152	298	1	ϕ(p(x	ϕ(p(x	PROPN
ejpam-5152	298	2	,	,	PUNCT
ejpam-5152	298	3	v	v	NOUN
ejpam-5152	298	4	)	)	PUNCT
ejpam-5152	298	5	,	,	PUNCT
ejpam-5152	298	6	p(x	p(x	PROPN
ejpam-5152	298	7	,	,	PUNCT
ejpam-5152	298	8	g(v	g(v	PROPN
ejpam-5152	298	9	)	)	PUNCT
ejpam-5152	298	10	)	)	PUNCT
ejpam-5152	298	11	,	,	PUNCT
ejpam-5152	298	12	p(v	p(v	PROPN
ejpam-5152	298	13	,	,	PUNCT
ejpam-5152	298	14	f	f	PROPN
ejpam-5152	298	15	(	(	PUNCT
ejpam-5152	298	16	x	x	NOUN
ejpam-5152	298	17	)	)	PUNCT
ejpam-5152	298	18	)	)	PUNCT
ejpam-5152	298	19	)	)	PUNCT
ejpam-5152	298	20	≤	≤	NUM
ejpam-5152	299	1	ϕ(p(x	ϕ(p(x	PROPN
ejpam-5152	299	2	,	,	PUNCT
ejpam-5152	299	3	v	v	NOUN
ejpam-5152	299	4	)	)	PUNCT
ejpam-5152	299	5	,	,	PUNCT
ejpam-5152	299	6	p(v	p(v	PROPN
ejpam-5152	299	7	,	,	PUNCT
ejpam-5152	299	8	f	f	PROPN
ejpam-5152	299	9	(	(	PUNCT
ejpam-5152	299	10	x	x	NOUN
ejpam-5152	299	11	)	)	PUNCT
ejpam-5152	299	12	)	)	PUNCT
ejpam-5152	299	13	)	)	PUNCT
ejpam-5152	299	14	≤	≤	NUM
ejpam-5152	300	1	ϕ(p(x	ϕ(p(x	PROPN
ejpam-5152	300	2	,	,	PUNCT
ejpam-5152	300	3	v	v	NOUN
ejpam-5152	300	4	)	)	PUNCT
ejpam-5152	300	5	,	,	PUNCT
ejpam-5152	300	6	p(v	p(v	NOUN
ejpam-5152	300	7	,	,	PUNCT
ejpam-5152	300	8	x	x	NOUN
ejpam-5152	300	9	)	)	PUNCT
ejpam-5152	300	10	)	)	PUNCT
ejpam-5152	300	11	≤	≤	PUNCT
ejpam-5152	301	1	ϕ(2p(x	ϕ(2p(x	PROPN
ejpam-5152	301	2	,	,	PUNCT
ejpam-5152	301	3	v	v	NOUN
ejpam-5152	301	4	)	)	PUNCT
ejpam-5152	301	5	)	)	PUNCT
ejpam-5152	301	6	since	since	SCONJ
ejpam-5152	301	7	p(x	p(x	PROPN
ejpam-5152	301	8	,	,	PUNCT
ejpam-5152	301	9	v	v	NOUN
ejpam-5152	301	10	)	)	PUNCT
ejpam-5152	301	11	≤	≤	NOUN
ejpam-5152	301	12	hp(f	hp(f	PUNCT
ejpam-5152	301	13	(	(	PUNCT
ejpam-5152	301	14	x	x	X
ejpam-5152	301	15	,	,	PUNCT
ejpam-5152	301	16	g(v	g(v	PROPN
ejpam-5152	301	17	)	)	PUNCT
ejpam-5152	301	18	)	)	PUNCT
ejpam-5152	301	19	≤	≤	PUNCT
ejpam-5152	302	1	ϕ(2p(x	ϕ(2p(x	PROPN
ejpam-5152	302	2	,	,	PUNCT
ejpam-5152	302	3	v	v	NOUN
ejpam-5152	302	4	)	)	PUNCT
ejpam-5152	302	5	)	)	PUNCT
ejpam-5152	302	6	<	<	X
ejpam-5152	302	7	p(x	p(x	PROPN
ejpam-5152	302	8	,	,	PUNCT
ejpam-5152	302	9	v	v	NOUN
ejpam-5152	302	10	)	)	PUNCT
ejpam-5152	302	11	,	,	PUNCT
ejpam-5152	302	12	thus	thus	ADV
ejpam-5152	302	13	we	we	PRON
ejpam-5152	302	14	have	have	VERB
ejpam-5152	302	15	a	a	DET
ejpam-5152	302	16	contradiction	contradiction	NOUN
ejpam-5152	302	17	.	.	PUNCT
ejpam-5152	303	1	hence	hence	ADV
ejpam-5152	303	2	p(x	p(x	PROPN
ejpam-5152	303	3	,	,	PUNCT
ejpam-5152	303	4	v	v	NOUN
ejpam-5152	303	5	)	)	PUNCT
ejpam-5152	303	6	=	=	SYM
ejpam-5152	303	7	0	0	NUM
ejpam-5152	303	8	,	,	PUNCT
ejpam-5152	303	9	i.e.	i.e.	X
ejpam-5152	303	10	,	,	PUNCT
ejpam-5152	303	11	x	x	SYM
ejpam-5152	303	12	=	=	NOUN
ejpam-5152	304	1	v.	v.	CCONJ
ejpam-5152	304	2	therefore	therefore	ADV
ejpam-5152	304	3	,	,	PUNCT
ejpam-5152	304	4	we	we	PRON
ejpam-5152	304	5	can	can	AUX
ejpam-5152	304	6	conclude	conclude	VERB
ejpam-5152	304	7	that	that	SCONJ
ejpam-5152	304	8	common	common	ADJ
ejpam-5152	304	9	fixed	fix	VERB
ejpam-5152	304	10	point	point	NOUN
ejpam-5152	304	11	x	x	VERB
ejpam-5152	304	12	is	be	AUX
ejpam-5152	304	13	unique	unique	ADJ
ejpam-5152	304	14	.	.	PUNCT
ejpam-5152	305	1	this	this	PRON
ejpam-5152	305	2	complete	complete	VERB
ejpam-5152	305	3	the	the	DET
ejpam-5152	305	4	proof	proof	NOUN
ejpam-5152	305	5	.	.	PUNCT
ejpam-5152	306	1	further	far	ADV
ejpam-5152	306	2	,	,	PUNCT
ejpam-5152	306	3	we	we	PRON
ejpam-5152	306	4	have	have	VERB
ejpam-5152	306	5	corollary	corollary	ADJ
ejpam-5152	306	6	2	2	NUM
ejpam-5152	306	7	.	.	PUNCT
ejpam-5152	307	1	let	let	VERB
ejpam-5152	307	2	(	(	PUNCT
ejpam-5152	307	3	cbp(x	cbp(x	PROPN
ejpam-5152	307	4	)	)	PUNCT
ejpam-5152	307	5	,	,	PUNCT
ejpam-5152	307	6	hp	hp	PROPN
ejpam-5152	307	7	)	)	PUNCT
ejpam-5152	307	8	be	be	VERB
ejpam-5152	307	9	a	a	DET
ejpam-5152	307	10	p	p	ADJ
ejpam-5152	307	11	-	-	PUNCT
ejpam-5152	307	12	pompeiu	pompeiu	NOUN
ejpam-5152	307	13	-	-	PUNCT
ejpam-5152	307	14	hausdorff	hausdorff	NOUN
ejpam-5152	307	15	metric	metric	ADJ
ejpam-5152	307	16	spaces	space	NOUN
ejpam-5152	307	17	.	.	PUNCT
ejpam-5152	308	1	suppose	suppose	VERB
ejpam-5152	308	2	that	that	SCONJ
ejpam-5152	308	3	f	f	X
ejpam-5152	308	4	:	:	PUNCT
ejpam-5152	308	5	x	x	X
ejpam-5152	308	6	→	→	SYM
ejpam-5152	308	7	cbp(x	cbp(x	PROPN
ejpam-5152	308	8	)	)	PUNCT
ejpam-5152	308	9	be	be	VERB
ejpam-5152	308	10	set	set	VERB
ejpam-5152	308	11	-	-	PUNCT
ejpam-5152	308	12	valued	value	VERB
ejpam-5152	308	13	mappings	mapping	NOUN
ejpam-5152	308	14	which	which	PRON
ejpam-5152	308	15	satisfy	satisfy	VERB
ejpam-5152	308	16	hp(f	hp(f	PUNCT
ejpam-5152	308	17	(	(	PUNCT
ejpam-5152	308	18	x	x	X
ejpam-5152	308	19	)	)	PUNCT
ejpam-5152	308	20	,	,	PUNCT
ejpam-5152	308	21	f	f	PROPN
ejpam-5152	308	22	(	(	PUNCT
ejpam-5152	308	23	y	y	NOUN
ejpam-5152	308	24	)	)	PUNCT
ejpam-5152	308	25	)	)	PUNCT
ejpam-5152	308	26	≤	≤	PROPN
ejpam-5152	309	1	ϕ(max{p(x	ϕ(max{p(x	PROPN
ejpam-5152	309	2	,	,	PUNCT
ejpam-5152	309	3	y	y	PROPN
ejpam-5152	309	4	)	)	PUNCT
ejpam-5152	309	5	,	,	PUNCT
ejpam-5152	309	6	p(x	p(x	PROPN
ejpam-5152	309	7	,	,	PUNCT
ejpam-5152	309	8	f	f	PROPN
ejpam-5152	309	9	(	(	PUNCT
ejpam-5152	309	10	x	x	NOUN
ejpam-5152	309	11	)	)	PUNCT
ejpam-5152	309	12	)	)	PUNCT
ejpam-5152	309	13	,	,	PUNCT
ejpam-5152	309	14	p(y	p(y	PROPN
ejpam-5152	309	15	,	,	PUNCT
ejpam-5152	309	16	f	f	PROPN
ejpam-5152	309	17	(	(	PUNCT
ejpam-5152	309	18	y	y	NOUN
ejpam-5152	309	19	)	)	PUNCT
ejpam-5152	309	20	)	)	PUNCT
ejpam-5152	309	21	,	,	PUNCT
ejpam-5152	309	22	p(x	p(x	PROPN
ejpam-5152	309	23	,	,	PUNCT
ejpam-5152	309	24	f	f	PROPN
ejpam-5152	309	25	(	(	PUNCT
ejpam-5152	309	26	y	y	NOUN
ejpam-5152	309	27	)	)	PUNCT
ejpam-5152	309	28	)	)	PUNCT
ejpam-5152	309	29	,	,	PUNCT
ejpam-5152	309	30	p(y	p(y	PROPN
ejpam-5152	309	31	,	,	PUNCT
ejpam-5152	309	32	f	f	PROPN
ejpam-5152	309	33	(	(	PUNCT
ejpam-5152	309	34	x	x	NOUN
ejpam-5152	309	35	)	)	PUNCT
ejpam-5152	309	36	)	)	PUNCT
ejpam-5152	309	37	}	}	PUNCT
ejpam-5152	309	38	)	)	PUNCT
ejpam-5152	309	39	,	,	PUNCT
ejpam-5152	309	40	(	(	PUNCT
ejpam-5152	309	41	12	12	NUM
ejpam-5152	309	42	)	)	PUNCT
ejpam-5152	309	43	for	for	ADP
ejpam-5152	309	44	each	each	DET
ejpam-5152	309	45	x	x	NOUN
ejpam-5152	309	46	,	,	PUNCT
ejpam-5152	309	47	y	y	PROPN
ejpam-5152	309	48	∈	∈	PROPN
ejpam-5152	309	49	x	x	X
ejpam-5152	309	50	and	and	CCONJ
ejpam-5152	309	51	ϕ	ϕ	NOUN
ejpam-5152	309	52	as	as	SCONJ
ejpam-5152	309	53	defined	define	VERB
ejpam-5152	309	54	in	in	ADP
ejpam-5152	309	55	theorem	theorem	NOUN
ejpam-5152	309	56	5	5	NUM
ejpam-5152	309	57	,	,	PUNCT
ejpam-5152	309	58	then	then	ADV
ejpam-5152	309	59	set	set	VERB
ejpam-5152	309	60	-	-	PUNCT
ejpam-5152	309	61	valued	value	VERB
ejpam-5152	309	62	mappings	mapping	NOUN
ejpam-5152	309	63	f	f	X
ejpam-5152	309	64	has	have	VERB
ejpam-5152	309	65	a	a	DET
ejpam-5152	309	66	unique	unique	ADJ
ejpam-5152	309	67	fixed	fix	VERB
ejpam-5152	309	68	point	point	NOUN
ejpam-5152	309	69	.	.	PUNCT
ejpam-5152	310	1	the	the	DET
ejpam-5152	310	2	existence	existence	NOUN
ejpam-5152	310	3	of	of	ADP
ejpam-5152	310	4	a	a	DET
ejpam-5152	310	5	common	common	ADJ
ejpam-5152	310	6	fixed	fix	VERB
ejpam-5152	310	7	point	point	NOUN
ejpam-5152	310	8	of	of	ADP
ejpam-5152	310	9	set	set	NOUN
ejpam-5152	310	10	-	-	PUNCT
ejpam-5152	310	11	valued	value	VERB
ejpam-5152	310	12	mapping	mapping	NOUN
ejpam-5152	310	13	that	that	PRON
ejpam-5152	310	14	satisfies	satisfy	VERB
ejpam-5152	310	15	the	the	DET
ejpam-5152	310	16	contraction	contraction	NOUN
ejpam-5152	310	17	as	as	ADP
ejpam-5152	310	18	in	in	ADP
ejpam-5152	310	19	inequality	inequality	NOUN
ejpam-5152	310	20	(	(	PUNCT
ejpam-5152	310	21	6	6	NUM
ejpam-5152	310	22	)	)	PUNCT
ejpam-5152	310	23	is	be	AUX
ejpam-5152	310	24	the	the	DET
ejpam-5152	310	25	consequence	consequence	NOUN
ejpam-5152	310	26	of	of	ADP
ejpam-5152	310	27	theorem	theorem	NOUN
ejpam-5152	310	28	5	5	NUM
ejpam-5152	310	29	.	.	PUNCT
ejpam-5152	310	30	for	for	ADP
ejpam-5152	310	31	ϕ(βu	ϕ(βu	NOUN
ejpam-5152	310	32	)	)	PUNCT
ejpam-5152	311	1	=	=	PUNCT
ejpam-5152	312	1	βu	βu	X
ejpam-5152	312	2	where	where	SCONJ
ejpam-5152	312	3	β	β	X
ejpam-5152	312	4	∈	∈	PROPN
ejpam-5152	313	1	[	[	X
ejpam-5152	313	2	0	0	NUM
ejpam-5152	313	3	,	,	PUNCT
ejpam-5152	313	4	12	12	NUM
ejpam-5152	313	5	)	)	PUNCT
ejpam-5152	313	6	in	in	ADP
ejpam-5152	313	7	theorem	theorem	NOUN
ejpam-5152	313	8	5	5	NUM
ejpam-5152	313	9	then	then	ADV
ejpam-5152	313	10	we	we	PRON
ejpam-5152	313	11	have	have	VERB
ejpam-5152	313	12	corollary	corollary	ADJ
ejpam-5152	313	13	3	3	NUM
ejpam-5152	313	14	below	below	ADV
ejpam-5152	313	15	.	.	PUNCT
ejpam-5152	314	1	corollary	corollary	ADJ
ejpam-5152	314	2	3	3	X
ejpam-5152	314	3	.	.	PUNCT
ejpam-5152	315	1	let	let	AUX
ejpam-5152	315	2	(	(	PUNCT
ejpam-5152	315	3	cbp(x	cbp(x	PROPN
ejpam-5152	315	4	)	)	PUNCT
ejpam-5152	315	5	,	,	PUNCT
ejpam-5152	315	6	hp	hp	PROPN
ejpam-5152	315	7	)	)	PUNCT
ejpam-5152	315	8	be	be	VERB
ejpam-5152	315	9	a	a	DET
ejpam-5152	315	10	p	p	ADJ
ejpam-5152	315	11	-	-	PUNCT
ejpam-5152	315	12	pompeiu	pompeiu	NOUN
ejpam-5152	315	13	-	-	PUNCT
ejpam-5152	315	14	hausdorff	hausdorff	NOUN
ejpam-5152	315	15	metric	metric	ADJ
ejpam-5152	315	16	spaces	space	NOUN
ejpam-5152	315	17	.	.	PUNCT
ejpam-5152	316	1	suppose	suppose	VERB
ejpam-5152	316	2	that	that	SCONJ
ejpam-5152	316	3	f	f	X
ejpam-5152	316	4	,	,	PUNCT
ejpam-5152	316	5	g	g	NOUN
ejpam-5152	316	6	:	:	PUNCT
ejpam-5152	316	7	x	x	SYM
ejpam-5152	316	8	→	→	SYM
ejpam-5152	316	9	cbp(x	cbp(x	PROPN
ejpam-5152	316	10	)	)	PUNCT
ejpam-5152	316	11	be	be	VERB
ejpam-5152	316	12	set	set	VERB
ejpam-5152	316	13	-	-	PUNCT
ejpam-5152	316	14	valued	value	VERB
ejpam-5152	316	15	mappings	mapping	NOUN
ejpam-5152	316	16	which	which	PRON
ejpam-5152	316	17	satisfy	satisfy	VERB
ejpam-5152	316	18	the	the	DET
ejpam-5152	316	19	contraction	contraction	NOUN
ejpam-5152	316	20	as	as	ADP
ejpam-5152	316	21	in	in	ADP
ejpam-5152	316	22	inequality	inequality	NOUN
ejpam-5152	316	23	(	(	PUNCT
ejpam-5152	316	24	6	6	NUM
ejpam-5152	316	25	)	)	PUNCT
ejpam-5152	316	26	,	,	PUNCT
ejpam-5152	316	27	hp(f	hp(f	X
ejpam-5152	316	28	(	(	PUNCT
ejpam-5152	316	29	x	x	NOUN
ejpam-5152	316	30	)	)	PUNCT
ejpam-5152	316	31	,	,	PUNCT
ejpam-5152	316	32	g(y	g(y	PROPN
ejpam-5152	316	33	)	)	PUNCT
ejpam-5152	316	34	)	)	PUNCT
ejpam-5152	316	35	≤	≤	NUM
ejpam-5152	316	36	κmax	κmax	VERB
ejpam-5152	316	37	{	{	PUNCT
ejpam-5152	316	38	p(x	p(x	PROPN
ejpam-5152	316	39	,	,	PUNCT
ejpam-5152	316	40	y	y	NOUN
ejpam-5152	316	41	)	)	PUNCT
ejpam-5152	316	42	,	,	PUNCT
ejpam-5152	316	43	p(x	p(x	PROPN
ejpam-5152	316	44	,	,	PUNCT
ejpam-5152	316	45	f	f	PROPN
ejpam-5152	316	46	(	(	PUNCT
ejpam-5152	316	47	x	x	NOUN
ejpam-5152	316	48	)	)	PUNCT
ejpam-5152	316	49	)	)	PUNCT
ejpam-5152	316	50	,	,	PUNCT
ejpam-5152	316	51	p(y	p(y	PROPN
ejpam-5152	316	52	,	,	PUNCT
ejpam-5152	316	53	g(y	g(y	NOUN
ejpam-5152	316	54	)	)	PUNCT
ejpam-5152	316	55	)	)	PUNCT
ejpam-5152	316	56	,	,	PUNCT
ejpam-5152	316	57	p(x	p(x	NOUN
ejpam-5152	316	58	,	,	PUNCT
ejpam-5152	316	59	g(y	g(y	NOUN
ejpam-5152	316	60	)	)	PUNCT
ejpam-5152	316	61	)	)	PUNCT
ejpam-5152	316	62	,	,	PUNCT
ejpam-5152	316	63	p(y	p(y	PROPN
ejpam-5152	316	64	,	,	PUNCT
ejpam-5152	316	65	f	f	PROPN
ejpam-5152	316	66	(	(	PUNCT
ejpam-5152	316	67	x	x	NOUN
ejpam-5152	316	68	)	)	PUNCT
ejpam-5152	316	69	)	)	PUNCT
ejpam-5152	316	70	}	}	PUNCT
ejpam-5152	316	71	,	,	PUNCT
ejpam-5152	316	72	for	for	ADP
ejpam-5152	316	73	each	each	DET
ejpam-5152	316	74	x	x	NOUN
ejpam-5152	316	75	,	,	PUNCT
ejpam-5152	316	76	y	y	PROPN
ejpam-5152	316	77	∈	∈	PROPN
ejpam-5152	316	78	x	x	X
ejpam-5152	316	79	and	and	CCONJ
ejpam-5152	316	80	κ	κ	PROPN
ejpam-5152	316	81	∈	∈	PROPN
ejpam-5152	317	1	[	[	X
ejpam-5152	317	2	0	0	NUM
ejpam-5152	317	3	,	,	PUNCT
ejpam-5152	317	4	1	1	NUM
ejpam-5152	317	5	)	)	PUNCT
ejpam-5152	317	6	,	,	PUNCT
ejpam-5152	317	7	then	then	ADV
ejpam-5152	317	8	set	set	VERB
ejpam-5152	317	9	-	-	PUNCT
ejpam-5152	317	10	valued	value	VERB
ejpam-5152	317	11	mappings	mapping	NOUN
ejpam-5152	317	12	f	f	NOUN
ejpam-5152	317	13	and	and	CCONJ
ejpam-5152	317	14	g	g	PROPN
ejpam-5152	317	15	have	have	VERB
ejpam-5152	317	16	a	a	DET
ejpam-5152	317	17	unique	unique	ADJ
ejpam-5152	317	18	common	common	ADJ
ejpam-5152	317	19	fixed	fix	VERB
ejpam-5152	317	20	point	point	NOUN
ejpam-5152	317	21	.	.	PUNCT
ejpam-5152	318	1	4	4	X
ejpam-5152	318	2	.	.	X
ejpam-5152	318	3	conclusion	conclusion	NOUN
ejpam-5152	318	4	in	in	ADP
ejpam-5152	318	5	this	this	DET
ejpam-5152	318	6	manuscript	manuscript	NOUN
ejpam-5152	318	7	,	,	PUNCT
ejpam-5152	318	8	we	we	PRON
ejpam-5152	318	9	have	have	AUX
ejpam-5152	318	10	established	establish	VERB
ejpam-5152	318	11	several	several	ADJ
ejpam-5152	318	12	theorems	theorem	NOUN
ejpam-5152	318	13	concerning	concern	VERB
ejpam-5152	318	14	common	common	ADJ
ejpam-5152	318	15	fixed	fix	VERB
ejpam-5152	318	16	points	point	NOUN
ejpam-5152	318	17	for	for	ADP
ejpam-5152	318	18	set	set	NOUN
ejpam-5152	318	19	-	-	PUNCT
ejpam-5152	318	20	valued	value	VERB
ejpam-5152	318	21	mappings	mapping	NOUN
ejpam-5152	318	22	.	.	PUNCT
ejpam-5152	319	1	these	these	DET
ejpam-5152	319	2	theorems	theorem	NOUN
ejpam-5152	319	3	introduce	introduce	VERB
ejpam-5152	319	4	novel	novel	ADJ
ejpam-5152	319	5	forms	form	NOUN
ejpam-5152	319	6	of	of	ADP
ejpam-5152	319	7	contraction	contraction	NOUN
ejpam-5152	319	8	,	,	PUNCT
ejpam-5152	319	9	which	which	PRON
ejpam-5152	319	10	extend	extend	VERB
ejpam-5152	319	11	the	the	DET
ejpam-5152	319	12	banach	banach	NOUN
ejpam-5152	319	13	contraction	contraction	NOUN
ejpam-5152	319	14	principle	principle	NOUN
ejpam-5152	319	15	to	to	PART
ejpam-5152	319	16	set	set	VERB
ejpam-5152	319	17	-	-	PUNCT
ejpam-5152	319	18	valued	value	VERB
ejpam-5152	319	19	mappings	mapping	NOUN
ejpam-5152	319	20	.	.	PUNCT
ejpam-5152	320	1	among	among	ADP
ejpam-5152	320	2	them	they	PRON
ejpam-5152	320	3	are	be	AUX
ejpam-5152	320	4	contractions	contraction	NOUN
ejpam-5152	320	5	for	for	ADP
ejpam-5152	320	6	sequences	sequence	NOUN
ejpam-5152	320	7	of	of	ADP
ejpam-5152	320	8	set	set	NOUN
ejpam-5152	320	9	-	-	PUNCT
ejpam-5152	320	10	valued	value	VERB
ejpam-5152	320	11	mappings	mapping	NOUN
ejpam-5152	320	12	,	,	PUNCT
ejpam-5152	320	13	indicating	indicate	VERB
ejpam-5152	320	14	the	the	DET
ejpam-5152	320	15	existence	existence	NOUN
ejpam-5152	320	16	of	of	ADP
ejpam-5152	320	17	a	a	DET
ejpam-5152	320	18	common	common	ADJ
ejpam-5152	320	19	fixed	fix	VERB
ejpam-5152	320	20	point	point	NOUN
ejpam-5152	320	21	for	for	ADP
ejpam-5152	320	22	the	the	DET
ejpam-5152	320	23	sequence	sequence	NOUN
ejpam-5152	320	24	.	.	PUNCT
ejpam-5152	321	1	this	this	DET
ejpam-5152	321	2	common	common	ADJ
ejpam-5152	321	3	fixed	fix	VERB
ejpam-5152	321	4	point	point	NOUN
ejpam-5152	321	5	is	be	AUX
ejpam-5152	321	6	then	then	ADV
ejpam-5152	321	7	utilized	utilize	VERB
ejpam-5152	321	8	to	to	PART
ejpam-5152	321	9	infer	infer	VERB
ejpam-5152	321	10	shared	share	VERB
ejpam-5152	321	11	fixed	fix	VERB
ejpam-5152	321	12	points	point	NOUN
ejpam-5152	321	13	of	of	ADP
ejpam-5152	321	14	the	the	DET
ejpam-5152	321	15	set	set	NOUN
ejpam-5152	321	16	-	-	PUNCT
ejpam-5152	321	17	valued	value	VERB
ejpam-5152	321	18	mappings	mapping	NOUN
ejpam-5152	321	19	through	through	ADP
ejpam-5152	321	20	sequence	sequence	NOUN
ejpam-5152	321	21	convergence	convergence	NOUN
ejpam-5152	321	22	.	.	PUNCT
ejpam-5152	322	1	furthermore	furthermore	ADV
ejpam-5152	322	2	,	,	PUNCT
ejpam-5152	322	3	we	we	PRON
ejpam-5152	322	4	present	present	VERB
ejpam-5152	322	5	a	a	DET
ejpam-5152	322	6	new	new	ADJ
ejpam-5152	322	7	,	,	PUNCT
ejpam-5152	322	8	more	more	ADV
ejpam-5152	322	9	general	general	ADJ
ejpam-5152	322	10	contraction	contraction	NOUN
ejpam-5152	322	11	principle	principle	NOUN
ejpam-5152	322	12	.	.	PUNCT
ejpam-5152	323	1	this	this	DET
ejpam-5152	323	2	principle	principle	NOUN
ejpam-5152	323	3	employs	employ	VERB
ejpam-5152	323	4	a	a	DET
ejpam-5152	323	5	non	non	ADJ
ejpam-5152	323	6	-	-	ADJ
ejpam-5152	323	7	decreasing	decrease	VERB
ejpam-5152	323	8	upper	upper	ADJ
ejpam-5152	323	9	semi	semi	ADJ
ejpam-5152	323	10	-	-	ADJ
ejpam-5152	323	11	continuous	continuous	ADJ
ejpam-5152	323	12	ϕ	ϕ	NOUN
ejpam-5152	323	13	function	function	NOUN
ejpam-5152	323	14	to	to	PART
ejpam-5152	323	15	construct	construct	VERB
ejpam-5152	323	16	a	a	DET
ejpam-5152	323	17	contraction	contraction	NOUN
ejpam-5152	323	18	mapping	mapping	NOUN
ejpam-5152	323	19	which	which	PRON
ejpam-5152	323	20	is	be	AUX
ejpam-5152	323	21	then	then	ADV
ejpam-5152	323	22	used	use	VERB
ejpam-5152	323	23	to	to	PART
ejpam-5152	323	24	ensure	ensure	VERB
ejpam-5152	323	25	the	the	DET
ejpam-5152	323	26	existence	existence	NOUN
ejpam-5152	323	27	of	of	ADP
ejpam-5152	323	28	common	common	ADJ
ejpam-5152	323	29	points	point	NOUN
ejpam-5152	323	30	for	for	ADP
ejpam-5152	323	31	the	the	DET
ejpam-5152	323	32	set	set	NOUN
ejpam-5152	323	33	-	-	PUNCT
ejpam-5152	323	34	valued	value	VERB
ejpam-5152	323	35	mappings	mapping	NOUN
ejpam-5152	323	36	.	.	PUNCT
ejpam-5152	324	1	references	reference	NOUN
ejpam-5152	324	2	1007	1007	NUM
ejpam-5152	324	3	acknowledgements	acknowledgement	NOUN
ejpam-5152	324	4	research	research	NOUN
ejpam-5152	324	5	supporting	support	VERB
ejpam-5152	324	6	project	project	NOUN
ejpam-5152	324	7	number	number	NOUN
ejpam-5152	324	8	1130.3	1130.3	NUM
ejpam-5152	324	9	/	/	SYM
ejpam-5152	324	10	un10.c10	un10.c10	ADJ
ejpam-5152	324	11	/	/	SYM
ejpam-5152	325	1	tu/2023	tu/2023	PROPN
ejpam-5152	325	2	dated	date	VERB
ejpam-5152	325	3	19	19	NUM
ejpam-5152	325	4	june	june	PROPN
ejpam-5152	325	5	2023	2023	NUM
ejpam-5152	325	6	directorate	directorate	ADJ
ejpam-5152	325	7	general	general	NOUN
ejpam-5152	325	8	of	of	ADP
ejpam-5152	325	9	research	research	NOUN
ejpam-5152	325	10	and	and	CCONJ
ejpam-5152	325	11	development	development	NOUN
ejpam-5152	325	12	,	,	PUNCT
ejpam-5152	325	13	the	the	DET
ejpam-5152	325	14	ministry	ministry	PROPN
ejpam-5152	325	15	of	of	ADP
ejpam-5152	325	16	education	education	PROPN
ejpam-5152	325	17	,	,	PUNCT
ejpam-5152	325	18	culture	culture	NOUN
ejpam-5152	325	19	,	,	PUNCT
ejpam-5152	325	20	research	research	NOUN
ejpam-5152	325	21	,	,	PUNCT
ejpam-5152	325	22	and	and	CCONJ
ejpam-5152	325	23	technology	technology	NOUN
ejpam-5152	325	24	,	,	PUNCT
ejpam-5152	325	25	indonesia	indonesia	PROPN
ejpam-5152	325	26	.	.	PUNCT
ejpam-5152	326	1	funding	fund	VERB
ejpam-5152	326	2	this	this	DET
ejpam-5152	326	3	research	research	NOUN
ejpam-5152	326	4	is	be	AUX
ejpam-5152	326	5	being	be	AUX
ejpam-5152	326	6	funded	fund	VERB
ejpam-5152	326	7	by	by	ADP
ejpam-5152	326	8	directorate	directorate	ADJ
ejpam-5152	326	9	general	general	NOUN
ejpam-5152	326	10	of	of	ADP
ejpam-5152	326	11	research	research	NOUN
ejpam-5152	326	12	and	and	CCONJ
ejpam-5152	326	13	development	development	NOUN
ejpam-5152	326	14	,	,	PUNCT
ejpam-5152	326	15	the	the	DET
ejpam-5152	326	16	ministry	ministry	PROPN
ejpam-5152	326	17	of	of	ADP
ejpam-5152	326	18	education	education	PROPN
ejpam-5152	326	19	,	,	PUNCT
ejpam-5152	326	20	culture	culture	NOUN
ejpam-5152	326	21	,	,	PUNCT
ejpam-5152	326	22	research	research	NOUN
ejpam-5152	326	23	,	,	PUNCT
ejpam-5152	326	24	and	and	CCONJ
ejpam-5152	326	25	technology	technology	NOUN
ejpam-5152	326	26	via	via	ADP
ejpam-5152	326	27	doctoral	doctoral	ADJ
ejpam-5152	326	28	dissertation	dissertation	NOUN
ejpam-5152	326	29	research	research	NOUN
ejpam-5152	326	30	.	.	PUNCT
ejpam-5152	327	1	references	reference	NOUN
ejpam-5152	327	2	[	[	X
ejpam-5152	327	3	1	1	NUM
ejpam-5152	327	4	]	]	X
ejpam-5152	327	5	m	m	VERB
ejpam-5152	327	6	abbas	abbas	PROPN
ejpam-5152	327	7	,	,	PUNCT
ejpam-5152	327	8	b	b	PROPN
ejpam-5152	327	9	ali	ali	PROPN
ejpam-5152	327	10	,	,	PUNCT
ejpam-5152	327	11	and	and	CCONJ
ejpam-5152	327	12	c	c	PROPN
ejpam-5152	327	13	vetro	vetro	X
ejpam-5152	327	14	.	.	PUNCT
ejpam-5152	328	1	a	a	DET
ejpam-5152	328	2	suzuki	suzuki	NOUN
ejpam-5152	328	3	type	type	NOUN
ejpam-5152	328	4	fixed	fix	VERB
ejpam-5152	328	5	point	point	NOUN
ejpam-5152	328	6	theorem	theorem	NOUN
ejpam-5152	328	7	for	for	ADP
ejpam-5152	328	8	a	a	DET
ejpam-5152	328	9	generalized	generalize	VERB
ejpam-5152	328	10	multivalued	multivalued	ADJ
ejpam-5152	328	11	mapping	mapping	NOUN
ejpam-5152	328	12	on	on	ADP
ejpam-5152	328	13	partial	partial	ADJ
ejpam-5152	328	14	hausdorff	hausdorff	NOUN
ejpam-5152	328	15	metric	metric	ADJ
ejpam-5152	328	16	spaces	space	NOUN
ejpam-5152	328	17	.	.	PUNCT
ejpam-5152	329	1	topol	topol	NOUN
ejpam-5152	329	2	.	.	PUNCT
ejpam-5152	329	3	appl	appl	PROPN
ejpam-5152	329	4	.	.	PROPN
ejpam-5152	329	5	,	,	PUNCT
ejpam-5152	329	6	160:553–563	160:553–563	NUM
ejpam-5152	329	7	,	,	PUNCT
ejpam-5152	329	8	2013	2013	NUM
ejpam-5152	329	9	.	.	PUNCT
ejpam-5152	330	1	[	[	X
ejpam-5152	330	2	2	2	NUM
ejpam-5152	330	3	]	]	PUNCT
ejpam-5152	330	4	t	t	NOUN
ejpam-5152	330	5	abdeljawad	abdeljawad	NOUN
ejpam-5152	330	6	.	.	PUNCT
ejpam-5152	331	1	fixed	fix	VERB
ejpam-5152	331	2	points	point	NOUN
ejpam-5152	331	3	for	for	ADP
ejpam-5152	331	4	generalized	generalized	ADJ
ejpam-5152	331	5	weakly	weakly	ADJ
ejpam-5152	331	6	contractive	contractive	ADJ
ejpam-5152	331	7	mappings	mapping	NOUN
ejpam-5152	331	8	in	in	ADP
ejpam-5152	331	9	partial	partial	ADJ
ejpam-5152	331	10	metric	metric	ADJ
ejpam-5152	331	11	spaces	space	NOUN
ejpam-5152	331	12	.	.	PUNCT
ejpam-5152	332	1	math	math	NOUN
ejpam-5152	332	2	.	.	PUNCT
ejpam-5152	333	1	comput	comput	NOUN
ejpam-5152	333	2	.	.	PUNCT
ejpam-5152	334	1	modelling	modelling	NOUN
ejpam-5152	334	2	,	,	PUNCT
ejpam-5152	334	3	54:2923–2927	54:2923–2927	NUM
ejpam-5152	334	4	,	,	PUNCT
ejpam-5152	334	5	2011	2011	NUM
ejpam-5152	334	6	.	.	PUNCT
ejpam-5152	335	1	[	[	X
ejpam-5152	335	2	3	3	X
ejpam-5152	335	3	]	]	X
ejpam-5152	335	4	j	j	PROPN
ejpam-5152	335	5	ahmad	ahmad	PROPN
ejpam-5152	335	6	,	,	PUNCT
ejpam-5152	335	7	a	a	DET
ejpam-5152	335	8	azam	azam	PROPN
ejpam-5152	335	9	,	,	PUNCT
ejpam-5152	335	10	and	and	CCONJ
ejpam-5152	335	11	m	m	PROPN
ejpam-5152	335	12	arshad	arshad	VERB
ejpam-5152	335	13	.	.	PUNCT
ejpam-5152	336	1	fixed	fix	VERB
ejpam-5152	336	2	points	point	NOUN
ejpam-5152	336	3	of	of	ADP
ejpam-5152	336	4	multivalued	multivalue	VERB
ejpam-5152	336	5	mappings	mapping	NOUN
ejpam-5152	336	6	in	in	ADP
ejpam-5152	336	7	partial	partial	ADJ
ejpam-5152	336	8	metric	metric	ADJ
ejpam-5152	336	9	spaces	space	NOUN
ejpam-5152	336	10	.	.	PUNCT
ejpam-5152	337	1	fixed	fix	VERB
ejpam-5152	337	2	point	point	NOUN
ejpam-5152	337	3	theory	theory	NOUN
ejpam-5152	337	4	and	and	CCONJ
ejpam-5152	337	5	applications	application	NOUN
ejpam-5152	337	6	,	,	PUNCT
ejpam-5152	337	7	316	316	NUM
ejpam-5152	337	8	,	,	PUNCT
ejpam-5152	337	9	2013	2013	NUM
ejpam-5152	337	10	.	.	PUNCT
ejpam-5152	338	1	[	[	X
ejpam-5152	338	2	4	4	NUM
ejpam-5152	338	3	]	]	X
ejpam-5152	338	4	j	j	PROPN
ejpam-5152	338	5	ahmad	ahmad	PROPN
ejpam-5152	338	6	,	,	PUNCT
ejpam-5152	338	7	c	c	NOUN
ejpam-5152	338	8	d	d	NOUN
ejpam-5152	338	9	bari	bari	NOUN
ejpam-5152	338	10	,	,	PUNCT
ejpam-5152	338	11	y	y	PROPN
ejpam-5152	338	12	j	j	PROPN
ejpam-5152	338	13	cho	cho	PROPN
ejpam-5152	338	14	,	,	PUNCT
ejpam-5152	338	15	and	and	CCONJ
ejpam-5152	338	16	m	m	PROPN
ejpam-5152	338	17	arshad	arshad	ADJ
ejpam-5152	338	18	.	.	PUNCT
ejpam-5152	339	1	some	some	DET
ejpam-5152	339	2	fixed	fix	VERB
ejpam-5152	339	3	point	point	NOUN
ejpam-5152	339	4	results	result	NOUN
ejpam-5152	339	5	for	for	ADP
ejpam-5152	339	6	multi	multi	ADJ
ejpam-5152	339	7	-	-	ADJ
ejpam-5152	339	8	valued	value	VERB
ejpam-5152	339	9	mappings	mapping	NOUN
ejpam-5152	339	10	in	in	ADP
ejpam-5152	339	11	partial	partial	ADJ
ejpam-5152	339	12	metric	metric	ADJ
ejpam-5152	339	13	spaces	space	NOUN
ejpam-5152	339	14	.	.	PUNCT
ejpam-5152	340	1	fixed	fix	VERB
ejpam-5152	340	2	point	point	NOUN
ejpam-5152	340	3	theory	theory	NOUN
ejpam-5152	340	4	appl	appl	PROPN
ejpam-5152	340	5	.	.	PROPN
ejpam-5152	340	6	,	,	PUNCT
ejpam-5152	340	7	175	175	NUM
ejpam-5152	340	8	,	,	PUNCT
ejpam-5152	340	9	2013	2013	NUM
ejpam-5152	340	10	.	.	PUNCT
ejpam-5152	341	1	[	[	X
ejpam-5152	341	2	5	5	NUM
ejpam-5152	341	3	]	]	PUNCT
ejpam-5152	341	4	i	i	PRON
ejpam-5152	341	5	altun	altun	NOUN
ejpam-5152	341	6	,	,	PUNCT
ejpam-5152	341	7	f	f	PROPN
ejpam-5152	341	8	sola	sola	PROPN
ejpam-5152	341	9	,	,	PUNCT
ejpam-5152	341	10	and	and	CCONJ
ejpam-5152	341	11	h	h	NOUN
ejpam-5152	341	12	simsek	simsek	NOUN
ejpam-5152	341	13	.	.	PUNCT
ejpam-5152	342	1	generalized	generalized	ADJ
ejpam-5152	342	2	contractions	contraction	NOUN
ejpam-5152	342	3	on	on	ADP
ejpam-5152	342	4	partial	partial	ADJ
ejpam-5152	342	5	metric	metric	ADJ
ejpam-5152	342	6	spaces	space	NOUN
ejpam-5152	342	7	.	.	PUNCT
ejpam-5152	343	1	topology	topology	NOUN
ejpam-5152	343	2	appl	appl	PROPN
ejpam-5152	343	3	.	.	PROPN
ejpam-5152	343	4	,	,	PUNCT
ejpam-5152	343	5	157(18):2778–2785	157(18):2778–2785	NUM
ejpam-5152	343	6	,	,	PUNCT
ejpam-5152	343	7	2010	2010	NUM
ejpam-5152	343	8	.	.	PUNCT
ejpam-5152	344	1	[	[	X
ejpam-5152	344	2	6	6	NUM
ejpam-5152	344	3	]	]	X
ejpam-5152	344	4	h	h	NOUN
ejpam-5152	344	5	aydi	aydi	ADJ
ejpam-5152	344	6	,	,	PUNCT
ejpam-5152	344	7	m	m	NOUN
ejpam-5152	344	8	abbas	abbas	NOUN
ejpam-5152	344	9	,	,	PUNCT
ejpam-5152	344	10	and	and	CCONJ
ejpam-5152	344	11	c	c	PROPN
ejpam-5152	344	12	vetro	vetro	X
ejpam-5152	344	13	.	.	PUNCT
ejpam-5152	345	1	partial	partial	ADJ
ejpam-5152	345	2	hausdorff	hausdorff	PROPN
ejpam-5152	345	3	metric	metric	ADJ
ejpam-5152	345	4	and	and	CCONJ
ejpam-5152	345	5	nadler	nadler	PROPN
ejpam-5152	345	6	’s	’s	PART
ejpam-5152	345	7	fixed	fix	VERB
ejpam-5152	345	8	point	point	NOUN
ejpam-5152	345	9	theorem	theorem	VERB
ejpam-5152	345	10	on	on	ADP
ejpam-5152	345	11	partial	partial	ADJ
ejpam-5152	345	12	metric	metric	ADJ
ejpam-5152	345	13	spaces	space	NOUN
ejpam-5152	345	14	.	.	PUNCT
ejpam-5152	346	1	topol	topol	NOUN
ejpam-5152	346	2	.	.	PUNCT
ejpam-5152	346	3	appl	appl	PROPN
ejpam-5152	346	4	.	.	PROPN
ejpam-5152	346	5	,	,	PUNCT
ejpam-5152	346	6	159:3234–3242	159:3234–3242	PROPN
ejpam-5152	346	7	,	,	PUNCT
ejpam-5152	346	8	2012	2012	NUM
ejpam-5152	346	9	.	.	PUNCT
ejpam-5152	347	1	[	[	X
ejpam-5152	347	2	7	7	X
ejpam-5152	347	3	]	]	X
ejpam-5152	347	4	h	h	NOUN
ejpam-5152	347	5	aydi	aydi	ADJ
ejpam-5152	347	6	,	,	PUNCT
ejpam-5152	347	7	m	m	NOUN
ejpam-5152	347	8	abbas	abbas	NOUN
ejpam-5152	347	9	,	,	PUNCT
ejpam-5152	347	10	and	and	CCONJ
ejpam-5152	347	11	c	c	X
ejpam-5152	347	12	vetro	vetro	X
ejpam-5152	347	13	.	.	PUNCT
ejpam-5152	348	1	common	common	ADJ
ejpam-5152	348	2	fixed	fix	VERB
ejpam-5152	348	3	points	point	NOUN
ejpam-5152	348	4	for	for	ADP
ejpam-5152	348	5	multi	multi	ADJ
ejpam-5152	348	6	-	-	ADJ
ejpam-5152	348	7	valued	value	VERB
ejpam-5152	348	8	generalized	generalized	ADJ
ejpam-5152	348	9	contractions	contraction	NOUN
ejpam-5152	348	10	on	on	ADP
ejpam-5152	348	11	partial	partial	ADJ
ejpam-5152	348	12	metric	metric	ADJ
ejpam-5152	348	13	spaces	space	NOUN
ejpam-5152	348	14	.	.	PUNCT
ejpam-5152	349	1	rev	rev	PROPN
ejpam-5152	349	2	.	.	PROPN
ejpam-5152	349	3	r.	r.	PROPN
ejpam-5152	349	4	acad	acad	PROPN
ejpam-5152	349	5	.	.	PUNCT
ejpam-5152	350	1	cienc	cienc	PROPN
ejpam-5152	350	2	.	.	PUNCT
ejpam-5152	351	1	exactas	exactas	PROPN
ejpam-5152	351	2	f́ıs	f́ıs	PROPN
ejpam-5152	351	3	.	.	PUNCT
ejpam-5152	352	1	nat	nat	PROPN
ejpam-5152	352	2	.	.	PUNCT
ejpam-5152	352	3	,	,	PUNCT
ejpam-5152	352	4	ser	ser	PROPN
ejpam-5152	352	5	.	.	PUNCT
ejpam-5152	353	1	a	a	DET
ejpam-5152	353	2	mat	mat	NOUN
ejpam-5152	353	3	.	.	PROPN
ejpam-5152	353	4	,	,	PUNCT
ejpam-5152	353	5	108:483–501	108:483–501	NUM
ejpam-5152	353	6	,	,	PUNCT
ejpam-5152	353	7	2013	2013	NUM
ejpam-5152	353	8	.	.	PUNCT
ejpam-5152	354	1	[	[	X
ejpam-5152	354	2	8	8	NUM
ejpam-5152	354	3	]	]	PUNCT
ejpam-5152	354	4	m	m	VERB
ejpam-5152	354	5	berinde	berinde	NOUN
ejpam-5152	354	6	and	and	CCONJ
ejpam-5152	354	7	v	v	ADP
ejpam-5152	354	8	berinde	berinde	NOUN
ejpam-5152	354	9	.	.	PUNCT
ejpam-5152	355	1	on	on	ADP
ejpam-5152	355	2	a	a	DET
ejpam-5152	355	3	general	general	ADJ
ejpam-5152	355	4	class	class	NOUN
ejpam-5152	355	5	of	of	ADP
ejpam-5152	355	6	multi	multi	ADJ
ejpam-5152	355	7	-	-	ADJ
ejpam-5152	355	8	valued	value	VERB
ejpam-5152	355	9	weakly	weakly	ADJ
ejpam-5152	355	10	picard	picard	NOUN
ejpam-5152	355	11	mappings	mapping	NOUN
ejpam-5152	355	12	.	.	PUNCT
ejpam-5152	356	1	j.	j.	PROPN
ejpam-5152	356	2	math	math	PROPN
ejpam-5152	356	3	.	.	PUNCT
ejpam-5152	357	1	anal	anal	PROPN
ejpam-5152	357	2	.	.	PUNCT
ejpam-5152	357	3	appl	appl	PROPN
ejpam-5152	357	4	,	,	PUNCT
ejpam-5152	357	5	326:772–782	326:772–782	NUM
ejpam-5152	357	6	,	,	PUNCT
ejpam-5152	357	7	2007	2007	NUM
ejpam-5152	357	8	.	.	PUNCT
ejpam-5152	358	1	[	[	X
ejpam-5152	358	2	9	9	NUM
ejpam-5152	358	3	]	]	SYM
ejpam-5152	358	4	v	v	NOUN
ejpam-5152	358	5	berinde	berinde	NOUN
ejpam-5152	358	6	and	and	CCONJ
ejpam-5152	358	7	m	m	NOUN
ejpam-5152	358	8	păcurar	păcurar	NOUN
ejpam-5152	358	9	.	.	PUNCT
ejpam-5152	359	1	why	why	SCONJ
ejpam-5152	359	2	pompeiu	pompeiu	NOUN
ejpam-5152	359	3	-	-	PUNCT
ejpam-5152	359	4	hausdorff	hausdorff	NOUN
ejpam-5152	359	5	metric	metric	NOUN
ejpam-5152	359	6	instead	instead	ADV
ejpam-5152	359	7	of	of	ADP
ejpam-5152	359	8	hausdorff	hausdorff	PROPN
ejpam-5152	359	9	metric	metric	PROPN
ejpam-5152	359	10	?	?	PUNCT
ejpam-5152	360	1	creat	creat	PROPN
ejpam-5152	360	2	.	.	PUNCT
ejpam-5152	361	1	math	math	PROPN
ejpam-5152	361	2	.	.	PUNCT
ejpam-5152	362	1	inform	inform	NOUN
ejpam-5152	362	2	.	.	PUNCT
ejpam-5152	362	3	,	,	PUNCT
ejpam-5152	362	4	31(1):33–41	31(1):33–41	NUM
ejpam-5152	362	5	,	,	PUNCT
ejpam-5152	362	6	2022	2022	NUM
ejpam-5152	362	7	.	.	PUNCT
ejpam-5152	363	1	[	[	X
ejpam-5152	363	2	10	10	NUM
ejpam-5152	363	3	]	]	X
ejpam-5152	363	4	d	d	X
ejpam-5152	363	5	ilić	ilić	NOUN
ejpam-5152	363	6	,	,	PUNCT
ejpam-5152	363	7	v	v	ADP
ejpam-5152	363	8	pavlović	pavlović	NOUN
ejpam-5152	363	9	,	,	PUNCT
ejpam-5152	363	10	and	and	CCONJ
ejpam-5152	363	11	v	v	ADP
ejpam-5152	363	12	rakocević.	rakocević.	PROPN
ejpam-5152	363	13	some	some	DET
ejpam-5152	363	14	new	new	ADJ
ejpam-5152	363	15	extensions	extension	NOUN
ejpam-5152	363	16	of	of	ADP
ejpam-5152	363	17	banach	banach	NOUN
ejpam-5152	363	18	’s	’s	PART
ejpam-5152	363	19	contraction	contraction	NOUN
ejpam-5152	363	20	principle	principle	NOUN
ejpam-5152	363	21	to	to	ADP
ejpam-5152	363	22	partial	partial	ADJ
ejpam-5152	363	23	metric	metric	ADJ
ejpam-5152	363	24	space	space	NOUN
ejpam-5152	363	25	.	.	PUNCT
ejpam-5152	364	1	appl	appl	PROPN
ejpam-5152	364	2	.	.	PROPN
ejpam-5152	364	3	math	math	PROPN
ejpam-5152	364	4	.	.	PUNCT
ejpam-5152	365	1	lett	lett	PROPN
ejpam-5152	365	2	.	.	PROPN
ejpam-5152	365	3	,	,	PUNCT
ejpam-5152	365	4	24:1326–1330	24:1326–1330	NUM
ejpam-5152	365	5	,	,	PUNCT
ejpam-5152	365	6	2011	2011	NUM
ejpam-5152	365	7	.	.	PUNCT
ejpam-5152	366	1	references	reference	NOUN
ejpam-5152	366	2	1008	1008	PROPN
ejpam-5152	367	1	[	[	X
ejpam-5152	367	2	11	11	NUM
ejpam-5152	367	3	]	]	X
ejpam-5152	367	4	e	e	X
ejpam-5152	367	5	karapınar	karapınar	NOUN
ejpam-5152	367	6	.	.	PUNCT
ejpam-5152	368	1	weak	weak	ADJ
ejpam-5152	368	2	ϕ-contraction	ϕ-contraction	NOUN
ejpam-5152	368	3	on	on	ADP
ejpam-5152	368	4	partial	partial	ADJ
ejpam-5152	368	5	metric	metric	ADJ
ejpam-5152	368	6	spaces	space	NOUN
ejpam-5152	368	7	.	.	PUNCT
ejpam-5152	369	1	j.	j.	PROPN
ejpam-5152	369	2	comput	comput	PROPN
ejpam-5152	369	3	.	.	PUNCT
ejpam-5152	370	1	anal	anal	PROPN
ejpam-5152	370	2	.	.	PUNCT
ejpam-5152	370	3	appl	appl	PROPN
ejpam-5152	370	4	.	.	PROPN
ejpam-5152	370	5	,	,	PUNCT
ejpam-5152	370	6	14	14	NUM
ejpam-5152	370	7	,	,	PUNCT
ejpam-5152	370	8	2011	2011	NUM
ejpam-5152	370	9	.	.	PUNCT
ejpam-5152	371	1	[	[	X
ejpam-5152	371	2	12	12	NUM
ejpam-5152	371	3	]	]	X
ejpam-5152	371	4	t	t	PROPN
ejpam-5152	371	5	kubiak	kubiak	PROPN
ejpam-5152	371	6	.	.	PUNCT
ejpam-5152	371	7	fixed	fix	VERB
ejpam-5152	371	8	point	point	NOUN
ejpam-5152	371	9	theorems	theorem	NOUN
ejpam-5152	371	10	for	for	ADP
ejpam-5152	371	11	contractive	contractive	ADJ
ejpam-5152	371	12	type	type	NOUN
ejpam-5152	371	13	multivalued	multivalue	VERB
ejpam-5152	371	14	mappings	mapping	NOUN
ejpam-5152	371	15	.	.	PUNCT
ejpam-5152	372	1	math	math	PROPN
ejpam-5152	372	2	japonica	japonica	PROPN
ejpam-5152	372	3	,	,	PUNCT
ejpam-5152	372	4	30(1):89–101	30(1):89–101	NUM
ejpam-5152	372	5	,	,	PUNCT
ejpam-5152	372	6	1985	1985	NUM
ejpam-5152	372	7	.	.	PUNCT
ejpam-5152	373	1	[	[	X
ejpam-5152	373	2	13	13	NUM
ejpam-5152	373	3	]	]	SYM
ejpam-5152	373	4	j	j	PROPN
ejpam-5152	373	5	t	t	PROPN
ejpam-5152	373	6	markin	markin	PROPN
ejpam-5152	373	7	.	.	PUNCT
ejpam-5152	374	1	a	a	DET
ejpam-5152	374	2	fixed	fix	VERB
ejpam-5152	374	3	point	point	NOUN
ejpam-5152	374	4	theorem	theorem	NOUN
ejpam-5152	374	5	for	for	ADP
ejpam-5152	374	6	set	set	NOUN
ejpam-5152	374	7	-	-	PUNCT
ejpam-5152	374	8	valued	value	VERB
ejpam-5152	374	9	mappings	mapping	NOUN
ejpam-5152	374	10	.	.	PUNCT
ejpam-5152	375	1	bull	bull	NOUN
ejpam-5152	375	2	.	.	PUNCT
ejpam-5152	376	1	am	be	AUX
ejpam-5152	376	2	.	.	PUNCT
ejpam-5152	377	1	math	math	NOUN
ejpam-5152	377	2	.	.	PUNCT
ejpam-5152	378	1	soc	soc	PROPN
ejpam-5152	378	2	,	,	PUNCT
ejpam-5152	378	3	74:639–640	74:639–640	PROPN
ejpam-5152	378	4	,	,	PUNCT
ejpam-5152	378	5	1968	1968	NUM
ejpam-5152	378	6	.	.	PUNCT
ejpam-5152	379	1	[	[	X
ejpam-5152	379	2	14	14	NUM
ejpam-5152	379	3	]	]	PUNCT
ejpam-5152	379	4	n	n	CCONJ
ejpam-5152	379	5	mizoguchi	mizoguchi	PROPN
ejpam-5152	379	6	and	and	CCONJ
ejpam-5152	379	7	w	w	PROPN
ejpam-5152	379	8	takahashi	takahashi	PROPN
ejpam-5152	379	9	.	.	PUNCT
ejpam-5152	380	1	fixed	fix	VERB
ejpam-5152	380	2	point	point	NOUN
ejpam-5152	380	3	theorems	theorem	NOUN
ejpam-5152	380	4	for	for	ADP
ejpam-5152	380	5	multivalued	multivalued	ADJ
ejpam-5152	380	6	mappings	mapping	NOUN
ejpam-5152	380	7	on	on	ADP
ejpam-5152	380	8	complete	complete	ADJ
ejpam-5152	380	9	metric	metric	ADJ
ejpam-5152	380	10	spaces	space	NOUN
ejpam-5152	380	11	.	.	PUNCT
ejpam-5152	381	1	j.	j.	PROPN
ejpam-5152	381	2	math	math	PROPN
ejpam-5152	381	3	.	.	PUNCT
ejpam-5152	382	1	anal	anal	PROPN
ejpam-5152	382	2	.	.	PUNCT
ejpam-5152	383	1	appl	appl	PROPN
ejpam-5152	383	2	,	,	PUNCT
ejpam-5152	383	3	141:177–188	141:177–188	NUM
ejpam-5152	383	4	,	,	PUNCT
ejpam-5152	383	5	1989	1989	NUM
ejpam-5152	383	6	.	.	PUNCT
ejpam-5152	384	1	[	[	X
ejpam-5152	384	2	15	15	NUM
ejpam-5152	384	3	]	]	X
ejpam-5152	384	4	m	m	NOUN
ejpam-5152	384	5	muslikh	muslikh	NOUN
ejpam-5152	384	6	.	.	PUNCT
ejpam-5152	385	1	partial	partial	ADJ
ejpam-5152	385	2	metric	metric	NOUN
ejpam-5152	385	3	on	on	ADP
ejpam-5152	385	4	space	space	NOUN
ejpam-5152	385	5	of	of	ADP
ejpam-5152	385	6	subsets	subset	NOUN
ejpam-5152	385	7	.	.	PUNCT
ejpam-5152	386	1	global	global	ADJ
ejpam-5152	386	2	journal	journal	PROPN
ejpam-5152	386	3	of	of	ADP
ejpam-5152	386	4	pure	pure	ADJ
ejpam-5152	386	5	and	and	CCONJ
ejpam-5152	386	6	applied	applied	ADJ
ejpam-5152	386	7	mathematics	mathematic	NOUN
ejpam-5152	386	8	,	,	PUNCT
ejpam-5152	386	9	11(5):2719–2734	11(5):2719–2734	NUM
ejpam-5152	386	10	,	,	PUNCT
ejpam-5152	386	11	2015	2015	NUM
ejpam-5152	386	12	.	.	PUNCT
ejpam-5152	387	1	[	[	X
ejpam-5152	387	2	16	16	NUM
ejpam-5152	387	3	]	]	SYM
ejpam-5152	387	4	s	s	PROPN
ejpam-5152	387	5	b	b	PROPN
ejpam-5152	387	6	nadler	nadler	PROPN
ejpam-5152	387	7	.	.	PUNCT
ejpam-5152	388	1	multi	multi	ADJ
ejpam-5152	388	2	-	-	ADJ
ejpam-5152	388	3	valued	value	VERB
ejpam-5152	388	4	contraction	contraction	NOUN
ejpam-5152	388	5	mappings	mapping	NOUN
ejpam-5152	388	6	.	.	PUNCT
ejpam-5152	389	1	pacific	pacific	PROPN
ejpam-5152	389	2	journal	journal	PROPN
ejpam-5152	389	3	of	of	ADP
ejpam-5152	389	4	mathematics	mathematic	NOUN
ejpam-5152	389	5	,	,	PUNCT
ejpam-5152	389	6	30(2	30(2	NUM
ejpam-5152	389	7	)	)	PUNCT
ejpam-5152	389	8	,	,	PUNCT
ejpam-5152	389	9	1969	1969	NUM
ejpam-5152	389	10	.	.	PUNCT
ejpam-5152	390	1	[	[	X
ejpam-5152	390	2	17	17	NUM
ejpam-5152	390	3	]	]	SYM
ejpam-5152	390	4	s	s	VERB
ejpam-5152	390	5	v	v	NOUN
ejpam-5152	390	6	r	r	NOUN
ejpam-5152	390	7	naidu	naidu	PROPN
ejpam-5152	390	8	.	.	PUNCT
ejpam-5152	391	1	fixed	fix	VERB
ejpam-5152	391	2	point	point	NOUN
ejpam-5152	391	3	theorems	theorem	NOUN
ejpam-5152	391	4	for	for	ADP
ejpam-5152	391	5	a	a	DET
ejpam-5152	391	6	broad	broad	ADJ
ejpam-5152	391	7	class	class	NOUN
ejpam-5152	391	8	of	of	ADP
ejpam-5152	391	9	multimaps	multimap	NOUN
ejpam-5152	391	10	.	.	PUNCT
ejpam-5152	392	1	nonlinear	nonlinear	ADJ
ejpam-5152	392	2	anal	anal	PROPN
ejpam-5152	392	3	.	.	PUNCT
ejpam-5152	392	4	,	,	PUNCT
ejpam-5152	392	5	52:961–969	52:961–969	NUM
ejpam-5152	392	6	,	,	PUNCT
ejpam-5152	392	7	2003	2003	NUM
ejpam-5152	392	8	.	.	PUNCT
ejpam-5152	393	1	[	[	X
ejpam-5152	393	2	18	18	NUM
ejpam-5152	393	3	]	]	SYM
ejpam-5152	393	4	b	b	PROPN
ejpam-5152	393	5	e	e	NOUN
ejpam-5152	393	6	rhoades	rhoade	NOUN
ejpam-5152	393	7	.	.	PUNCT
ejpam-5152	394	1	a	a	DET
ejpam-5152	394	2	comparison	comparison	NOUN
ejpam-5152	394	3	of	of	ADP
ejpam-5152	394	4	various	various	ADJ
ejpam-5152	394	5	definitions	definition	NOUN
ejpam-5152	394	6	of	of	ADP
ejpam-5152	394	7	contractive	contractive	ADJ
ejpam-5152	394	8	mappings	mapping	NOUN
ejpam-5152	394	9	.	.	PUNCT
ejpam-5152	395	1	trans	trans	PROPN
ejpam-5152	395	2	,	,	PUNCT
ejpam-5152	395	3	amer	amer	PROPN
ejpam-5152	395	4	,	,	PUNCT
ejpam-5152	395	5	math	math	NOUN
ejpam-5152	395	6	.	.	PUNCT
ejpam-5152	396	1	soc	soc	PROPN
ejpam-5152	396	2	.	.	PROPN
ejpam-5152	396	3	,	,	PUNCT
ejpam-5152	396	4	226:257–290	226:257–290	NUM
ejpam-5152	396	5	,	,	PUNCT
ejpam-5152	396	6	1977	1977	NUM
ejpam-5152	396	7	.	.	PUNCT
ejpam-5152	397	1	[	[	X
ejpam-5152	397	2	19	19	NUM
ejpam-5152	397	3	]	]	X
ejpam-5152	397	4	ph	ph	PROPN
ejpam-5152	397	5	r	r	NOUN
ejpam-5152	397	6	singh	singh	NOUN
ejpam-5152	397	7	.	.	PUNCT
ejpam-5152	398	1	a	a	DET
ejpam-5152	398	2	common	common	ADJ
ejpam-5152	398	3	fixed	fix	VERB
ejpam-5152	398	4	point	point	NOUN
ejpam-5152	398	5	theorem	theorem	NOUN
ejpam-5152	398	6	for	for	ADP
ejpam-5152	398	7	contractive	contractive	ADJ
ejpam-5152	398	8	multi	multi	ADJ
ejpam-5152	398	9	-	-	ADJ
ejpam-5152	398	10	valued	value	VERB
ejpam-5152	398	11	mappings	mapping	NOUN
ejpam-5152	398	12	.	.	PUNCT
ejpam-5152	399	1	int	int	NOUN
ejpam-5152	399	2	.	.	PUNCT
ejpam-5152	400	1	j.	j.	PROPN
ejpam-5152	400	2	contemp	contemp	PROPN
ejpam-5152	400	3	.	.	PUNCT
ejpam-5152	401	1	math	math	NOUN
ejpam-5152	401	2	.	.	PUNCT
ejpam-5152	402	1	sciences	science	NOUN
ejpam-5152	402	2	,	,	PUNCT
ejpam-5152	402	3	9(6):253–256	9(6):253–256	NUM
ejpam-5152	402	4	,	,	PUNCT
ejpam-5152	402	5	2014	2014	NUM
ejpam-5152	402	6	.	.	PUNCT
ejpam-5152	403	1	[	[	X
ejpam-5152	403	2	20	20	NUM
ejpam-5152	403	3	]	]	SYM
ejpam-5152	403	4	c	c	PROPN
ejpam-5152	403	5	k	k	PROPN
ejpam-5152	403	6	zhang	zhang	PROPN
ejpam-5152	403	7	,	,	PUNCT
ejpam-5152	403	8	j	j	PROPN
ejpam-5152	403	9	zhu	zhu	PROPN
ejpam-5152	403	10	,	,	PUNCT
ejpam-5152	403	11	and	and	CCONJ
ejpam-5152	403	12	p	p	NOUN
ejpam-5152	403	13	h	h	NOUN
ejpam-5152	403	14	zhao	zhao	PROPN
ejpam-5152	403	15	.	.	PUNCT
ejpam-5152	404	1	an	an	DET
ejpam-5152	404	2	extension	extension	NOUN
ejpam-5152	404	3	of	of	ADP
ejpam-5152	404	4	multi	multi	ADJ
ejpam-5152	404	5	-	-	ADJ
ejpam-5152	404	6	valued	value	VERB
ejpam-5152	404	7	contraction	contraction	NOUN
ejpam-5152	404	8	mappings	mapping	NOUN
ejpam-5152	404	9	and	and	CCONJ
ejpam-5152	404	10	fixed	fix	VERB
ejpam-5152	404	11	points	point	NOUN
ejpam-5152	404	12	.	.	PUNCT
ejpam-5152	405	1	proc	proc	NOUN
ejpam-5152	405	2	.	.	PUNCT
ejpam-5152	406	1	am	be	AUX
ejpam-5152	406	2	.	.	PUNCT
ejpam-5152	407	1	math	math	NOUN
ejpam-5152	407	2	.	.	PUNCT
ejpam-5152	408	1	soc	soc	PROPN
ejpam-5152	408	2	.	.	PUNCT
ejpam-5152	408	3	,	,	PUNCT
ejpam-5152	408	4	128:2439–2444	128:2439–2444	NUM
ejpam-5152	408	5	,	,	PUNCT
ejpam-5152	408	6	2000	2000	NUM
ejpam-5152	408	7	.	.	PUNCT