id	sid	tid	token	lemma	pos
ejpam-6256	1	1	european	european	PROPN
ejpam-6256	1	2	journal	journal	PROPN
ejpam-6256	1	3	of	of	ADP
ejpam-6256	1	4	pure	pure	ADJ
ejpam-6256	1	5	and	and	CCONJ
ejpam-6256	1	6	applied	applied	ADJ
ejpam-6256	1	7	mathematics	mathematic	NOUN
ejpam-6256	1	8	2025	2025	NUM
ejpam-6256	1	9	,	,	PUNCT
ejpam-6256	1	10	vol	vol	NOUN
ejpam-6256	1	11	.	.	PROPN
ejpam-6256	1	12	18	18	NUM
ejpam-6256	1	13	,	,	PUNCT
ejpam-6256	1	14	issue	issue	NOUN
ejpam-6256	1	15	3	3	NUM
ejpam-6256	1	16	,	,	PUNCT
ejpam-6256	1	17	article	article	NOUN
ejpam-6256	1	18	number	number	NOUN
ejpam-6256	1	19	6256	6256	NUM
ejpam-6256	1	20	issn	issn	VERB
ejpam-6256	1	21	1307	1307	NUM
ejpam-6256	1	22	-	-	SYM
ejpam-6256	1	23	5543	5543	NUM
ejpam-6256	1	24	–	–	PUNCT
ejpam-6256	2	1	ejpam.com	ejpam.com	X
ejpam-6256	2	2	published	publish	VERB
ejpam-6256	2	3	by	by	ADP
ejpam-6256	2	4	new	new	PROPN
ejpam-6256	2	5	york	york	PROPN
ejpam-6256	2	6	business	business	PROPN
ejpam-6256	2	7	global	global	ADJ
ejpam-6256	2	8	iterative	iterative	NOUN
ejpam-6256	2	9	approaches	approach	NOUN
ejpam-6256	2	10	to	to	ADP
ejpam-6256	2	11	multiple	multiple	ADJ
ejpam-6256	2	12	fixed	fix	VERB
ejpam-6256	2	13	points	point	NOUN
ejpam-6256	2	14	in	in	ADP
ejpam-6256	2	15	generalized	generalized	ADJ
ejpam-6256	2	16	metric	metric	ADJ
ejpam-6256	2	17	spaces	space	NOUN
ejpam-6256	2	18	yassin	yassin	PROPN
ejpam-6256	2	19	alzubaidi	alzubaidi	VERB
ejpam-6256	2	20	mathematics	mathematics	PROPN
ejpam-6256	2	21	department	department	PROPN
ejpam-6256	2	22	,	,	PUNCT
ejpam-6256	2	23	umm	umm	INTJ
ejpam-6256	2	24	al	al	PROPN
ejpam-6256	2	25	-	-	PUNCT
ejpam-6256	2	26	qura	qura	PROPN
ejpam-6256	2	27	university	university	PROPN
ejpam-6256	2	28	,	,	PUNCT
ejpam-6256	2	29	makkah	makkah	PROPN
ejpam-6256	2	30	,	,	PUNCT
ejpam-6256	2	31	saudi	saudi	PROPN
ejpam-6256	2	32	arabia	arabia	PROPN
ejpam-6256	2	33	abstract	abstract	NOUN
ejpam-6256	2	34	.	.	PUNCT
ejpam-6256	3	1	we	we	PRON
ejpam-6256	3	2	combine	combine	VERB
ejpam-6256	3	3	the	the	DET
ejpam-6256	3	4	concept	concept	NOUN
ejpam-6256	3	5	of	of	ADP
ejpam-6256	3	6	having	have	VERB
ejpam-6256	3	7	multiple	multiple	ADJ
ejpam-6256	3	8	fixed	fix	VERB
ejpam-6256	3	9	points	point	NOUN
ejpam-6256	3	10	for	for	ADP
ejpam-6256	3	11	a	a	DET
ejpam-6256	3	12	mapping	mapping	NOUN
ejpam-6256	3	13	with	with	ADP
ejpam-6256	3	14	the	the	DET
ejpam-6256	3	15	approach	approach	NOUN
ejpam-6256	3	16	of	of	ADP
ejpam-6256	3	17	obtaining	obtain	VERB
ejpam-6256	3	18	these	these	DET
ejpam-6256	3	19	points	point	NOUN
ejpam-6256	3	20	through	through	ADP
ejpam-6256	3	21	iterative	iterative	NOUN
ejpam-6256	3	22	methods	method	NOUN
ejpam-6256	3	23	.	.	PUNCT
ejpam-6256	4	1	as	as	SCONJ
ejpam-6256	4	2	is	be	AUX
ejpam-6256	4	3	well	well	ADV
ejpam-6256	4	4	known	know	VERB
ejpam-6256	4	5	,	,	PUNCT
ejpam-6256	4	6	contraction	contraction	NOUN
ejpam-6256	4	7	selfmappings	selfmapping	NOUN
ejpam-6256	4	8	in	in	ADP
ejpam-6256	4	9	standard	standard	ADJ
ejpam-6256	4	10	metric	metric	ADJ
ejpam-6256	4	11	spaces	space	NOUN
ejpam-6256	4	12	yield	yield	VERB
ejpam-6256	4	13	unique	unique	ADJ
ejpam-6256	4	14	fixed	fix	VERB
ejpam-6256	4	15	points	point	NOUN
ejpam-6256	4	16	that	that	PRON
ejpam-6256	4	17	can	can	AUX
ejpam-6256	4	18	be	be	AUX
ejpam-6256	4	19	obtained	obtain	VERB
ejpam-6256	4	20	iteratively	iteratively	ADV
ejpam-6256	4	21	.	.	PUNCT
ejpam-6256	5	1	to	to	PART
ejpam-6256	5	2	overcome	overcome	VERB
ejpam-6256	5	3	this	this	DET
ejpam-6256	5	4	limitation	limitation	NOUN
ejpam-6256	5	5	,	,	PUNCT
ejpam-6256	5	6	we	we	PRON
ejpam-6256	5	7	conduct	conduct	VERB
ejpam-6256	5	8	our	our	PRON
ejpam-6256	5	9	study	study	NOUN
ejpam-6256	5	10	within	within	ADP
ejpam-6256	5	11	the	the	DET
ejpam-6256	5	12	framework	framework	NOUN
ejpam-6256	5	13	of	of	ADP
ejpam-6256	5	14	generalized	generalized	ADJ
ejpam-6256	5	15	mp	mp	NOUN
ejpam-6256	5	16	-	-	PUNCT
ejpam-6256	5	17	metric	metric	ADJ
ejpam-6256	5	18	spaces	space	NOUN
ejpam-6256	5	19	,	,	PUNCT
ejpam-6256	5	20	utilizing	utilize	VERB
ejpam-6256	5	21	their	their	PRON
ejpam-6256	5	22	properties	property	NOUN
ejpam-6256	5	23	and	and	CCONJ
ejpam-6256	5	24	the	the	DET
ejpam-6256	5	25	broader	broad	ADJ
ejpam-6256	5	26	concept	concept	NOUN
ejpam-6256	5	27	of	of	ADP
ejpam-6256	5	28	limits	limit	NOUN
ejpam-6256	5	29	.	.	PUNCT
ejpam-6256	6	1	this	this	PRON
ejpam-6256	6	2	enables	enable	VERB
ejpam-6256	6	3	us	we	PRON
ejpam-6256	6	4	to	to	PART
ejpam-6256	6	5	establish	establish	VERB
ejpam-6256	6	6	our	our	PRON
ejpam-6256	6	7	main	main	ADJ
ejpam-6256	6	8	result	result	NOUN
ejpam-6256	6	9	:	:	PUNCT
ejpam-6256	6	10	the	the	DET
ejpam-6256	6	11	existence	existence	NOUN
ejpam-6256	6	12	of	of	ADP
ejpam-6256	6	13	multiple	multiple	ADJ
ejpam-6256	6	14	fixed	fix	VERB
ejpam-6256	6	15	points	point	NOUN
ejpam-6256	6	16	that	that	PRON
ejpam-6256	6	17	can	can	AUX
ejpam-6256	6	18	be	be	AUX
ejpam-6256	6	19	iteratively	iteratively	ADV
ejpam-6256	6	20	obtained	obtain	VERB
ejpam-6256	6	21	for	for	ADP
ejpam-6256	6	22	generalized	generalized	ADJ
ejpam-6256	6	23	contraction	contraction	NOUN
ejpam-6256	6	24	mappings	mapping	NOUN
ejpam-6256	6	25	satisfying	satisfy	VERB
ejpam-6256	6	26	specific	specific	ADJ
ejpam-6256	6	27	conditions	condition	NOUN
ejpam-6256	6	28	.	.	PUNCT
ejpam-6256	7	1	2020	2020	NUM
ejpam-6256	7	2	mathematics	mathematic	NOUN
ejpam-6256	7	3	subject	subject	NOUN
ejpam-6256	7	4	classifications	classification	NOUN
ejpam-6256	7	5	:	:	PUNCT
ejpam-6256	7	6	47h10	47h10	NUM
ejpam-6256	7	7	,	,	PUNCT
ejpam-6256	7	8	54e35	54e35	NUM
ejpam-6256	7	9	key	key	ADJ
ejpam-6256	7	10	words	word	NOUN
ejpam-6256	7	11	and	and	CCONJ
ejpam-6256	7	12	phrases	phrase	NOUN
ejpam-6256	7	13	:	:	PUNCT
ejpam-6256	7	14	fixed	fixed	ADJ
ejpam-6256	7	15	point	point	NOUN
ejpam-6256	7	16	,	,	PUNCT
ejpam-6256	7	17	iterative	iterative	NOUN
ejpam-6256	7	18	methods	method	NOUN
ejpam-6256	7	19	,	,	PUNCT
ejpam-6256	7	20	generalized	generalized	ADJ
ejpam-6256	7	21	contractor	contractor	NOUN
ejpam-6256	7	22	,	,	PUNCT
ejpam-6256	7	23	multiple	multiple	ADJ
ejpam-6256	7	24	fixed	fix	VERB
ejpam-6256	7	25	points	point	NOUN
ejpam-6256	7	26	,	,	PUNCT
ejpam-6256	7	27	generalized	generalized	ADJ
ejpam-6256	7	28	metric	metric	NOUN
ejpam-6256	7	29	,	,	PUNCT
ejpam-6256	7	30	mp	mp	NOUN
ejpam-6256	7	31	-	-	ADJ
ejpam-6256	7	32	metric	metric	ADJ
ejpam-6256	7	33	1	1	NUM
ejpam-6256	7	34	.	.	PUNCT
ejpam-6256	8	1	introduction	introduction	NOUN
ejpam-6256	8	2	fixed	fix	VERB
ejpam-6256	8	3	-	-	PUNCT
ejpam-6256	8	4	point	point	NOUN
ejpam-6256	8	5	theory	theory	NOUN
ejpam-6256	8	6	in	in	ADP
ejpam-6256	8	7	metric	metric	ADJ
ejpam-6256	8	8	spaces	space	NOUN
ejpam-6256	8	9	is	be	AUX
ejpam-6256	8	10	one	one	NUM
ejpam-6256	8	11	of	of	ADP
ejpam-6256	8	12	the	the	DET
ejpam-6256	8	13	most	most	ADV
ejpam-6256	8	14	active	active	ADJ
ejpam-6256	8	15	research	research	NOUN
ejpam-6256	8	16	areas	area	NOUN
ejpam-6256	8	17	in	in	ADP
ejpam-6256	8	18	mathematics	mathematic	NOUN
ejpam-6256	8	19	due	due	ADP
ejpam-6256	8	20	to	to	ADP
ejpam-6256	8	21	its	its	PRON
ejpam-6256	8	22	broad	broad	ADJ
ejpam-6256	8	23	applications	application	NOUN
ejpam-6256	8	24	in	in	ADP
ejpam-6256	8	25	both	both	CCONJ
ejpam-6256	8	26	pure	pure	ADJ
ejpam-6256	8	27	and	and	CCONJ
ejpam-6256	8	28	applied	applied	ADJ
ejpam-6256	8	29	fields	field	NOUN
ejpam-6256	8	30	.	.	PUNCT
ejpam-6256	9	1	it	it	PRON
ejpam-6256	9	2	plays	play	VERB
ejpam-6256	9	3	a	a	DET
ejpam-6256	9	4	crucial	crucial	ADJ
ejpam-6256	9	5	role	role	NOUN
ejpam-6256	9	6	in	in	ADP
ejpam-6256	9	7	theoretical	theoretical	ADJ
ejpam-6256	9	8	research	research	NOUN
ejpam-6256	9	9	and	and	CCONJ
ejpam-6256	9	10	practical	practical	ADJ
ejpam-6256	9	11	disciplines	discipline	NOUN
ejpam-6256	9	12	such	such	ADJ
ejpam-6256	9	13	as	as	ADP
ejpam-6256	9	14	physics	physics	NOUN
ejpam-6256	9	15	,	,	PUNCT
ejpam-6256	9	16	computing	computing	NOUN
ejpam-6256	9	17	,	,	PUNCT
ejpam-6256	9	18	and	and	CCONJ
ejpam-6256	9	19	engineering	engineering	NOUN
ejpam-6256	9	20	[	[	X
ejpam-6256	9	21	1	1	NUM
ejpam-6256	9	22	]	]	PUNCT
ejpam-6256	9	23	,	,	PUNCT
ejpam-6256	9	24	[	[	X
ejpam-6256	9	25	2	2	NUM
ejpam-6256	9	26	]	]	PUNCT
ejpam-6256	9	27	.	.	PUNCT
ejpam-6256	10	1	the	the	DET
ejpam-6256	10	2	origins	origin	NOUN
ejpam-6256	10	3	of	of	ADP
ejpam-6256	10	4	fixed	fix	VERB
ejpam-6256	10	5	-	-	PUNCT
ejpam-6256	10	6	point	point	NOUN
ejpam-6256	10	7	theory	theory	NOUN
ejpam-6256	10	8	trace	trace	NOUN
ejpam-6256	10	9	back	back	ADV
ejpam-6256	10	10	to	to	ADP
ejpam-6256	10	11	the	the	DET
ejpam-6256	10	12	late	late	ADJ
ejpam-6256	10	13	19th	19th	NOUN
ejpam-6256	10	14	and	and	CCONJ
ejpam-6256	10	15	early	early	ADJ
ejpam-6256	10	16	20th	20th	ADJ
ejpam-6256	10	17	centuries	century	NOUN
ejpam-6256	10	18	,	,	PUNCT
ejpam-6256	10	19	with	with	ADP
ejpam-6256	10	20	pioneering	pioneer	VERB
ejpam-6256	10	21	contributions	contribution	NOUN
ejpam-6256	10	22	from	from	ADP
ejpam-6256	10	23	scholars	scholar	NOUN
ejpam-6256	10	24	such	such	ADJ
ejpam-6256	10	25	as	as	ADP
ejpam-6256	10	26	poincaré	poincaré	ADJ
ejpam-6256	10	27	,	,	PUNCT
ejpam-6256	10	28	lefschetz	lefschetz	ADJ
ejpam-6256	10	29	–	–	PUNCT
ejpam-6256	10	30	hopf	hopf	ADJ
ejpam-6256	10	31	,	,	PUNCT
ejpam-6256	10	32	and	and	CCONJ
ejpam-6256	10	33	leray	leray	ADJ
ejpam-6256	10	34	–	–	PUNCT
ejpam-6256	10	35	schauder	schauder	NOUN
ejpam-6256	10	36	[	[	X
ejpam-6256	10	37	3	3	NUM
ejpam-6256	10	38	]	]	PUNCT
ejpam-6256	10	39	.	.	PUNCT
ejpam-6256	11	1	however	however	ADV
ejpam-6256	11	2	,	,	PUNCT
ejpam-6256	11	3	the	the	DET
ejpam-6256	11	4	foundation	foundation	NOUN
ejpam-6256	11	5	of	of	ADP
ejpam-6256	11	6	fixed	fix	VERB
ejpam-6256	11	7	-	-	PUNCT
ejpam-6256	11	8	point	point	NOUN
ejpam-6256	11	9	theory	theory	NOUN
ejpam-6256	11	10	in	in	ADP
ejpam-6256	11	11	metric	metric	ADJ
ejpam-6256	11	12	spaces	space	NOUN
ejpam-6256	11	13	was	be	AUX
ejpam-6256	11	14	largely	largely	ADV
ejpam-6256	11	15	established	establish	VERB
ejpam-6256	11	16	in	in	ADP
ejpam-6256	11	17	1922	1922	NUM
ejpam-6256	11	18	when	when	SCONJ
ejpam-6256	11	19	banach	banach	NOUN
ejpam-6256	11	20	published	publish	VERB
ejpam-6256	11	21	his	his	PRON
ejpam-6256	11	22	seminal	seminal	ADJ
ejpam-6256	11	23	paper	paper	NOUN
ejpam-6256	12	1	[	[	X
ejpam-6256	12	2	4	4	NUM
ejpam-6256	12	3	]	]	PUNCT
ejpam-6256	12	4	,	,	PUNCT
ejpam-6256	12	5	proving	prove	VERB
ejpam-6256	12	6	the	the	DET
ejpam-6256	12	7	existence	existence	NOUN
ejpam-6256	12	8	and	and	CCONJ
ejpam-6256	12	9	uniqueness	uniqueness	NOUN
ejpam-6256	12	10	of	of	ADP
ejpam-6256	12	11	fixed	fix	VERB
ejpam-6256	12	12	points	point	NOUN
ejpam-6256	12	13	for	for	ADP
ejpam-6256	12	14	a	a	DET
ejpam-6256	12	15	special	special	ADJ
ejpam-6256	12	16	class	class	NOUN
ejpam-6256	12	17	of	of	ADP
ejpam-6256	12	18	functions	function	NOUN
ejpam-6256	12	19	called	call	VERB
ejpam-6256	12	20	contraction	contraction	NOUN
ejpam-6256	12	21	mappings	mapping	NOUN
ejpam-6256	12	22	.	.	PUNCT
ejpam-6256	13	1	banach	banach	NOUN
ejpam-6256	13	2	’s	’s	PART
ejpam-6256	13	3	proof	proof	NOUN
ejpam-6256	13	4	was	be	AUX
ejpam-6256	13	5	more	more	ADJ
ejpam-6256	13	6	than	than	ADP
ejpam-6256	13	7	just	just	ADV
ejpam-6256	13	8	a	a	DET
ejpam-6256	13	9	demonstration	demonstration	NOUN
ejpam-6256	13	10	of	of	ADP
ejpam-6256	13	11	the	the	DET
ejpam-6256	13	12	theorem	theorem	NOUN
ejpam-6256	13	13	;	;	PUNCT
ejpam-6256	13	14	it	it	PRON
ejpam-6256	13	15	implicitly	implicitly	ADV
ejpam-6256	13	16	introduced	introduce	VERB
ejpam-6256	13	17	an	an	DET
ejpam-6256	13	18	iterative	iterative	NOUN
ejpam-6256	13	19	method	method	NOUN
ejpam-6256	13	20	—	—	PUNCT
ejpam-6256	13	21	specifically	specifically	ADV
ejpam-6256	13	22	,	,	PUNCT
ejpam-6256	13	23	picard	picard	NOUN
ejpam-6256	13	24	iteration	iteration	NOUN
ejpam-6256	13	25	—	—	PUNCT
ejpam-6256	13	26	to	to	PART
ejpam-6256	13	27	obtain	obtain	VERB
ejpam-6256	13	28	the	the	DET
ejpam-6256	13	29	fixed	fix	VERB
ejpam-6256	13	30	point	point	NOUN
ejpam-6256	13	31	.	.	PUNCT
ejpam-6256	14	1	his	his	PRON
ejpam-6256	14	2	groundbreaking	groundbreake	VERB
ejpam-6256	14	3	result	result	NOUN
ejpam-6256	14	4	inspired	inspire	VERB
ejpam-6256	14	5	numerous	numerous	ADJ
ejpam-6256	14	6	researchers	researcher	NOUN
ejpam-6256	14	7	,	,	PUNCT
ejpam-6256	14	8	leading	lead	VERB
ejpam-6256	14	9	to	to	ADP
ejpam-6256	14	10	extensive	extensive	ADJ
ejpam-6256	14	11	generalizations	generalization	NOUN
ejpam-6256	14	12	and	and	CCONJ
ejpam-6256	14	13	applications	application	NOUN
ejpam-6256	14	14	of	of	ADP
ejpam-6256	14	15	banach	banach	NOUN
ejpam-6256	14	16	’s	’s	PART
ejpam-6256	14	17	theorem	theorem	ADJ
ejpam-6256	14	18	.	.	PROPN
ejpam-6256	15	1	among	among	ADP
ejpam-6256	15	2	the	the	DET
ejpam-6256	15	3	major	major	ADJ
ejpam-6256	15	4	directions	direction	NOUN
ejpam-6256	15	5	of	of	ADP
ejpam-6256	15	6	research	research	NOUN
ejpam-6256	15	7	in	in	ADP
ejpam-6256	15	8	fixed	fix	VERB
ejpam-6256	15	9	-	-	PUNCT
ejpam-6256	15	10	point	point	NOUN
ejpam-6256	15	11	theory	theory	NOUN
ejpam-6256	15	12	,	,	PUNCT
ejpam-6256	15	13	two	two	NUM
ejpam-6256	15	14	prominent	prominent	ADJ
ejpam-6256	15	15	areas	area	NOUN
ejpam-6256	15	16	stand	stand	VERB
ejpam-6256	15	17	out	out	ADP
ejpam-6256	15	18	:	:	PUNCT
ejpam-6256	15	19	the	the	DET
ejpam-6256	15	20	first	first	ADJ
ejpam-6256	15	21	is	be	AUX
ejpam-6256	15	22	the	the	DET
ejpam-6256	15	23	study	study	NOUN
ejpam-6256	15	24	of	of	ADP
ejpam-6256	15	25	iterative	iterative	ADJ
ejpam-6256	15	26	methods	method	NOUN
ejpam-6256	15	27	to	to	PART
ejpam-6256	15	28	obtain	obtain	VERB
ejpam-6256	15	29	fixed	fixed	ADJ
ejpam-6256	15	30	points	point	NOUN
ejpam-6256	15	31	,	,	PUNCT
ejpam-6256	15	32	and	and	CCONJ
ejpam-6256	15	33	the	the	DET
ejpam-6256	15	34	second	second	NOUN
ejpam-6256	15	35	is	be	AUX
ejpam-6256	15	36	the	the	DET
ejpam-6256	15	37	quest	quest	NOUN
ejpam-6256	15	38	for	for	ADP
ejpam-6256	15	39	the	the	DET
ejpam-6256	15	40	existence	existence	NOUN
ejpam-6256	15	41	of	of	ADP
ejpam-6256	15	42	more	more	ADJ
ejpam-6256	15	43	than	than	ADP
ejpam-6256	15	44	one	one	NUM
ejpam-6256	15	45	fixed	fix	VERB
ejpam-6256	15	46	point	point	NOUN
ejpam-6256	15	47	.	.	PUNCT
ejpam-6256	16	1	in	in	ADP
ejpam-6256	16	2	this	this	DET
ejpam-6256	16	3	paper	paper	NOUN
ejpam-6256	16	4	,	,	PUNCT
ejpam-6256	16	5	we	we	PRON
ejpam-6256	16	6	aim	aim	VERB
ejpam-6256	16	7	to	to	PART
ejpam-6256	16	8	combine	combine	VERB
ejpam-6256	16	9	these	these	DET
ejpam-6256	16	10	two	two	NUM
ejpam-6256	16	11	directions	direction	NOUN
ejpam-6256	16	12	and	and	CCONJ
ejpam-6256	16	13	integrate	integrate	VERB
ejpam-6256	16	14	them	they	PRON
ejpam-6256	16	15	.	.	PUNCT
ejpam-6256	17	1	doi	doi	NOUN
ejpam-6256	17	2	:	:	PUNCT
ejpam-6256	17	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6256	https://doi.org/10.29020/nybg.ejpam.v18i3.6256	PRON
ejpam-6256	17	4	email	email	NOUN
ejpam-6256	17	5	address	address	NOUN
ejpam-6256	17	6	:	:	PUNCT
ejpam-6256	17	7	yazubaidi@uqu.edu.sa	yazubaidi@uqu.edu.sa	PROPN
ejpam-6256	17	8	(	(	PUNCT
ejpam-6256	17	9	y.	y.	PROPN
ejpam-6256	17	10	alzubaidi	alzubaidi	PROPN
ejpam-6256	17	11	)	)	PUNCT
ejpam-6256	17	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6256	18	1	1	1	NUM
ejpam-6256	18	2	copyright	copyright	NOUN
ejpam-6256	18	3	:	:	PUNCT
ejpam-6256	18	4	©	©	PROPN
ejpam-6256	18	5	2025	2025	NUM
ejpam-6256	18	6	the	the	DET
ejpam-6256	18	7	author(s	author(s	NOUN
ejpam-6256	18	8	)	)	PUNCT
ejpam-6256	18	9	.	.	PUNCT
ejpam-6256	19	1	(	(	PUNCT
ejpam-6256	19	2	cc	cc	NOUN
ejpam-6256	19	3	by	by	ADP
ejpam-6256	19	4	-	-	PUNCT
ejpam-6256	19	5	nc	nc	PROPN
ejpam-6256	19	6	4.0	4.0	NUM
ejpam-6256	19	7	)	)	PUNCT
ejpam-6256	19	8	y.	y.	NOUN
ejpam-6256	19	9	alzubaidi	alzubaidi	PROPN
ejpam-6256	19	10	/	/	SYM
ejpam-6256	19	11	eur	eur	PROPN
ejpam-6256	19	12	.	.	PUNCT
ejpam-6256	20	1	j.	j.	PROPN
ejpam-6256	20	2	pure	pure	PROPN
ejpam-6256	20	3	appl	appl	PROPN
ejpam-6256	20	4	.	.	PROPN
ejpam-6256	20	5	math	math	PROPN
ejpam-6256	20	6	,	,	PUNCT
ejpam-6256	20	7	18	18	NUM
ejpam-6256	20	8	(	(	PUNCT
ejpam-6256	20	9	3	3	NUM
ejpam-6256	20	10	)	)	PUNCT
ejpam-6256	20	11	(	(	PUNCT
ejpam-6256	20	12	2025	2025	NUM
ejpam-6256	20	13	)	)	PUNCT
ejpam-6256	20	14	,	,	PUNCT
ejpam-6256	20	15	6256	6256	NUM
ejpam-6256	20	16	2	2	NUM
ejpam-6256	20	17	of	of	ADP
ejpam-6256	20	18	11	11	NUM
ejpam-6256	20	19	in	in	ADP
ejpam-6256	20	20	this	this	DET
ejpam-6256	20	21	article	article	NOUN
ejpam-6256	20	22	,	,	PUNCT
ejpam-6256	20	23	the	the	DET
ejpam-6256	20	24	notion	notion	NOUN
ejpam-6256	20	25	of	of	ADP
ejpam-6256	20	26	an	an	DET
ejpam-6256	20	27	iterative	iterative	NOUN
ejpam-6256	20	28	fixed	fix	VERB
ejpam-6256	20	29	point	point	NOUN
ejpam-6256	20	30	refers	refer	VERB
ejpam-6256	20	31	to	to	ADP
ejpam-6256	20	32	a	a	DET
ejpam-6256	20	33	fixed	fix	VERB
ejpam-6256	20	34	point	point	NOUN
ejpam-6256	20	35	that	that	PRON
ejpam-6256	20	36	can	can	AUX
ejpam-6256	20	37	be	be	AUX
ejpam-6256	20	38	obtained	obtain	VERB
ejpam-6256	20	39	through	through	ADP
ejpam-6256	20	40	iterative	iterative	ADJ
ejpam-6256	20	41	methods	method	NOUN
ejpam-6256	20	42	.	.	PUNCT
ejpam-6256	21	1	an	an	DET
ejpam-6256	21	2	iterative	iterative	NOUN
ejpam-6256	21	3	fixed	fix	VERB
ejpam-6256	21	4	point	point	NOUN
ejpam-6256	21	5	is	be	AUX
ejpam-6256	21	6	of	of	ADP
ejpam-6256	21	7	dual	dual	ADJ
ejpam-6256	21	8	importance	importance	NOUN
ejpam-6256	21	9	;	;	PUNCT
ejpam-6256	21	10	not	not	PART
ejpam-6256	21	11	only	only	ADV
ejpam-6256	21	12	is	be	AUX
ejpam-6256	21	13	it	it	PRON
ejpam-6256	21	14	a	a	DET
ejpam-6256	21	15	fixed	fix	VERB
ejpam-6256	21	16	point	point	NOUN
ejpam-6256	21	17	,	,	PUNCT
ejpam-6256	21	18	but	but	CCONJ
ejpam-6256	21	19	we	we	PRON
ejpam-6256	21	20	also	also	ADV
ejpam-6256	21	21	have	have	AUX
ejpam-6256	21	22	established	establish	VERB
ejpam-6256	21	23	methods	method	NOUN
ejpam-6256	21	24	for	for	ADP
ejpam-6256	21	25	locating	locate	VERB
ejpam-6256	21	26	or	or	CCONJ
ejpam-6256	21	27	approximating	approximate	VERB
ejpam-6256	21	28	it	it	PRON
ejpam-6256	21	29	,	,	PUNCT
ejpam-6256	21	30	making	make	VERB
ejpam-6256	21	31	it	it	PRON
ejpam-6256	21	32	particularly	particularly	ADV
ejpam-6256	21	33	useful	useful	ADJ
ejpam-6256	21	34	in	in	ADP
ejpam-6256	21	35	practical	practical	ADJ
ejpam-6256	21	36	applications	application	NOUN
ejpam-6256	21	37	.	.	PUNCT
ejpam-6256	22	1	consequently	consequently	ADV
ejpam-6256	22	2	,	,	PUNCT
ejpam-6256	22	3	it	it	PRON
ejpam-6256	22	4	is	be	AUX
ejpam-6256	22	5	unsurprising	unsurprising	ADJ
ejpam-6256	22	6	that	that	SCONJ
ejpam-6256	22	7	significant	significant	ADJ
ejpam-6256	22	8	attention	attention	NOUN
ejpam-6256	22	9	has	have	AUX
ejpam-6256	22	10	been	be	AUX
ejpam-6256	22	11	devoted	devote	VERB
ejpam-6256	22	12	to	to	ADP
ejpam-6256	22	13	iterative	iterative	ADJ
ejpam-6256	22	14	methods	method	NOUN
ejpam-6256	22	15	in	in	ADP
ejpam-6256	22	16	fixed	fix	VERB
ejpam-6256	22	17	point	point	NOUN
ejpam-6256	22	18	theory	theory	NOUN
ejpam-6256	22	19	,	,	PUNCT
ejpam-6256	22	20	with	with	ADP
ejpam-6256	22	21	various	various	ADJ
ejpam-6256	22	22	approaches	approach	NOUN
ejpam-6256	22	23	being	be	AUX
ejpam-6256	22	24	implemented	implement	VERB
ejpam-6256	22	25	within	within	ADP
ejpam-6256	22	26	this	this	DET
ejpam-6256	22	27	framework	framework	NOUN
ejpam-6256	22	28	.	.	PUNCT
ejpam-6256	23	1	in	in	ADP
ejpam-6256	23	2	addition	addition	NOUN
ejpam-6256	23	3	to	to	ADP
ejpam-6256	23	4	the	the	DET
ejpam-6256	23	5	straightforward	straightforward	ADJ
ejpam-6256	23	6	picard	picard	NOUN
ejpam-6256	23	7	iterative	iterative	NOUN
ejpam-6256	23	8	method	method	NOUN
ejpam-6256	23	9	,	,	PUNCT
ejpam-6256	23	10	more	more	ADV
ejpam-6256	23	11	complex	complex	ADJ
ejpam-6256	23	12	iterative	iterative	NOUN
ejpam-6256	23	13	techniques	technique	NOUN
ejpam-6256	23	14	—	—	PUNCT
ejpam-6256	23	15	such	such	ADJ
ejpam-6256	23	16	as	as	ADP
ejpam-6256	23	17	the	the	DET
ejpam-6256	23	18	krasnoselskij	krasnoselskij	NOUN
ejpam-6256	23	19	,	,	PUNCT
ejpam-6256	23	20	mann	mann	PROPN
ejpam-6256	23	21	,	,	PUNCT
ejpam-6256	23	22	and	and	CCONJ
ejpam-6256	23	23	ishikawa	ishikawa	PROPN
ejpam-6256	23	24	methods	method	NOUN
ejpam-6256	23	25	—	—	PUNCT
ejpam-6256	23	26	have	have	AUX
ejpam-6256	23	27	been	be	AUX
ejpam-6256	23	28	developed	develop	VERB
ejpam-6256	23	29	and	and	CCONJ
ejpam-6256	23	30	successfully	successfully	ADV
ejpam-6256	23	31	employed	employ	VERB
ejpam-6256	23	32	to	to	PART
ejpam-6256	23	33	obtain	obtain	VERB
ejpam-6256	23	34	fixed	fixed	ADJ
ejpam-6256	23	35	points	point	NOUN
ejpam-6256	23	36	.	.	PUNCT
ejpam-6256	24	1	(	(	PUNCT
ejpam-6256	24	2	see	see	VERB
ejpam-6256	24	3	,	,	PUNCT
ejpam-6256	24	4	for	for	ADP
ejpam-6256	24	5	example	example	NOUN
ejpam-6256	24	6	,	,	PUNCT
ejpam-6256	24	7	[	[	X
ejpam-6256	24	8	5	5	NUM
ejpam-6256	24	9	]	]	PUNCT
ejpam-6256	24	10	,	,	PUNCT
ejpam-6256	24	11	[	[	X
ejpam-6256	24	12	6	6	NUM
ejpam-6256	24	13	]	]	PUNCT
ejpam-6256	24	14	,	,	PUNCT
ejpam-6256	24	15	[	[	X
ejpam-6256	24	16	7	7	NUM
ejpam-6256	24	17	]	]	PUNCT
ejpam-6256	24	18	,	,	PUNCT
ejpam-6256	24	19	[	[	X
ejpam-6256	24	20	8	8	NUM
ejpam-6256	24	21	]	]	PUNCT
ejpam-6256	24	22	and	and	CCONJ
ejpam-6256	24	23	[	[	X
ejpam-6256	24	24	9	9	NUM
ejpam-6256	24	25	]	]	PUNCT
ejpam-6256	24	26	)	)	PUNCT
ejpam-6256	24	27	.	.	PUNCT
ejpam-6256	25	1	on	on	ADP
ejpam-6256	25	2	the	the	DET
ejpam-6256	25	3	other	other	ADJ
ejpam-6256	25	4	hand	hand	NOUN
ejpam-6256	25	5	,	,	PUNCT
ejpam-6256	25	6	the	the	DET
ejpam-6256	25	7	study	study	NOUN
ejpam-6256	25	8	of	of	ADP
ejpam-6256	25	9	the	the	DET
ejpam-6256	25	10	existence	existence	NOUN
ejpam-6256	25	11	of	of	ADP
ejpam-6256	25	12	multiple	multiple	ADJ
ejpam-6256	25	13	fixed	fix	VERB
ejpam-6256	25	14	points	point	NOUN
ejpam-6256	25	15	began	begin	VERB
ejpam-6256	25	16	in	in	ADP
ejpam-6256	25	17	the	the	DET
ejpam-6256	25	18	1960s	1960	NOUN
ejpam-6256	25	19	with	with	ADP
ejpam-6256	25	20	the	the	DET
ejpam-6256	25	21	works	work	NOUN
ejpam-6256	25	22	of	of	ADP
ejpam-6256	25	23	browder	browder	NOUN
ejpam-6256	25	24	and	and	CCONJ
ejpam-6256	25	25	kirk	kirk	PROPN
ejpam-6256	25	26	(	(	PUNCT
ejpam-6256	26	1	[	[	X
ejpam-6256	26	2	10	10	NUM
ejpam-6256	26	3	]	]	PUNCT
ejpam-6256	26	4	,	,	PUNCT
ejpam-6256	27	1	[	[	X
ejpam-6256	27	2	11	11	NUM
ejpam-6256	27	3	]	]	PUNCT
ejpam-6256	27	4	,	,	PUNCT
ejpam-6256	27	5	[	[	X
ejpam-6256	27	6	12	12	NUM
ejpam-6256	27	7	]	]	PUNCT
ejpam-6256	27	8	,	,	PUNCT
ejpam-6256	27	9	and	and	CCONJ
ejpam-6256	27	10	[	[	X
ejpam-6256	27	11	13	13	NUM
ejpam-6256	27	12	]	]	NUM
ejpam-6256	27	13	)	)	PUNCT
ejpam-6256	27	14	.	.	PUNCT
ejpam-6256	28	1	their	their	PRON
ejpam-6256	28	2	research	research	NOUN
ejpam-6256	28	3	demonstrated	demonstrate	VERB
ejpam-6256	28	4	that	that	SCONJ
ejpam-6256	28	5	non	non	ADJ
ejpam-6256	28	6	-	-	ADJ
ejpam-6256	28	7	expansive	expansive	ADJ
ejpam-6256	28	8	mappings	mapping	NOUN
ejpam-6256	28	9	can	can	AUX
ejpam-6256	28	10	possess	possess	VERB
ejpam-6256	28	11	more	more	ADJ
ejpam-6256	28	12	than	than	ADP
ejpam-6256	28	13	one	one	NUM
ejpam-6256	28	14	fixed	fix	VERB
ejpam-6256	28	15	point	point	NOUN
ejpam-6256	28	16	,	,	PUNCT
ejpam-6256	28	17	thus	thus	ADV
ejpam-6256	28	18	establishing	establish	VERB
ejpam-6256	28	19	a	a	DET
ejpam-6256	28	20	foundation	foundation	NOUN
ejpam-6256	28	21	for	for	ADP
ejpam-6256	28	22	examining	examine	VERB
ejpam-6256	28	23	the	the	DET
ejpam-6256	28	24	potential	potential	NOUN
ejpam-6256	28	25	of	of	ADP
ejpam-6256	28	26	multiple	multiple	ADJ
ejpam-6256	28	27	fixed	fix	VERB
ejpam-6256	28	28	points	point	NOUN
ejpam-6256	28	29	in	in	ADP
ejpam-6256	28	30	specific	specific	ADJ
ejpam-6256	28	31	contexts	contexts	NOUN
ejpam-6256	28	32	.	.	PUNCT
ejpam-6256	29	1	subsequently	subsequently	ADV
ejpam-6256	29	2	,	,	PUNCT
ejpam-6256	29	3	the	the	DET
ejpam-6256	29	4	explicit	explicit	ADJ
ejpam-6256	29	5	investigation	investigation	NOUN
ejpam-6256	29	6	of	of	ADP
ejpam-6256	29	7	the	the	DET
ejpam-6256	29	8	existence	existence	NOUN
ejpam-6256	29	9	of	of	ADP
ejpam-6256	29	10	multiple	multiple	ADJ
ejpam-6256	29	11	fixed	fix	VERB
ejpam-6256	29	12	points	point	NOUN
ejpam-6256	29	13	gained	gain	VERB
ejpam-6256	29	14	traction	traction	NOUN
ejpam-6256	29	15	.	.	PUNCT
ejpam-6256	30	1	for	for	ADP
ejpam-6256	30	2	instance	instance	NOUN
ejpam-6256	30	3	,	,	PUNCT
ejpam-6256	30	4	results	result	NOUN
ejpam-6256	30	5	concerning	concern	VERB
ejpam-6256	30	6	multiple	multiple	ADJ
ejpam-6256	30	7	fixed	fix	VERB
ejpam-6256	30	8	points	point	NOUN
ejpam-6256	30	9	for	for	ADP
ejpam-6256	30	10	monotone	monotone	ADJ
ejpam-6256	30	11	and	and	CCONJ
ejpam-6256	30	12	positive	positive	ADJ
ejpam-6256	30	13	mappings	mapping	NOUN
ejpam-6256	30	14	are	be	AUX
ejpam-6256	30	15	available	available	ADJ
ejpam-6256	30	16	in	in	ADP
ejpam-6256	30	17	[	[	X
ejpam-6256	30	18	14	14	NUM
ejpam-6256	30	19	]	]	PUNCT
ejpam-6256	30	20	and	and	CCONJ
ejpam-6256	30	21	[	[	X
ejpam-6256	30	22	15	15	NUM
ejpam-6256	30	23	]	]	PUNCT
ejpam-6256	30	24	.	.	PUNCT
ejpam-6256	31	1	moreover	moreover	ADV
ejpam-6256	31	2	,	,	PUNCT
ejpam-6256	31	3	further	further	ADJ
ejpam-6256	31	4	applications	application	NOUN
ejpam-6256	31	5	have	have	AUX
ejpam-6256	31	6	illustrated	illustrate	VERB
ejpam-6256	31	7	that	that	SCONJ
ejpam-6256	31	8	the	the	DET
ejpam-6256	31	9	significance	significance	NOUN
ejpam-6256	31	10	of	of	ADP
ejpam-6256	31	11	multiple	multiple	ADJ
ejpam-6256	31	12	fixed	fix	VERB
ejpam-6256	31	13	points	point	NOUN
ejpam-6256	31	14	is	be	AUX
ejpam-6256	31	15	partially	partially	ADV
ejpam-6256	31	16	derived	derive	VERB
ejpam-6256	31	17	from	from	ADP
ejpam-6256	31	18	the	the	DET
ejpam-6256	31	19	fact	fact	NOUN
ejpam-6256	31	20	that	that	SCONJ
ejpam-6256	31	21	,	,	PUNCT
ejpam-6256	31	22	in	in	ADP
ejpam-6256	31	23	many	many	ADJ
ejpam-6256	31	24	practical	practical	ADJ
ejpam-6256	31	25	scenarios	scenario	NOUN
ejpam-6256	31	26	,	,	PUNCT
ejpam-6256	31	27	the	the	DET
ejpam-6256	31	28	solution	solution	NOUN
ejpam-6256	31	29	to	to	ADP
ejpam-6256	31	30	a	a	DET
ejpam-6256	31	31	system	system	NOUN
ejpam-6256	31	32	corresponds	correspond	VERB
ejpam-6256	31	33	to	to	ADP
ejpam-6256	31	34	a	a	DET
ejpam-6256	31	35	fixed	fix	VERB
ejpam-6256	31	36	point	point	NOUN
ejpam-6256	31	37	of	of	ADP
ejpam-6256	31	38	a	a	DET
ejpam-6256	31	39	particular	particular	ADJ
ejpam-6256	31	40	mapping	mapping	NOUN
ejpam-6256	31	41	.	.	PUNCT
ejpam-6256	32	1	this	this	PRON
ejpam-6256	32	2	underscores	underscore	VERB
ejpam-6256	32	3	a	a	DET
ejpam-6256	32	4	side	side	NOUN
ejpam-6256	32	5	of	of	ADP
ejpam-6256	32	6	the	the	DET
ejpam-6256	32	7	importance	importance	NOUN
ejpam-6256	32	8	of	of	ADP
ejpam-6256	32	9	studying	study	VERB
ejpam-6256	32	10	the	the	DET
ejpam-6256	32	11	existence	existence	NOUN
ejpam-6256	32	12	of	of	ADP
ejpam-6256	32	13	more	more	ADJ
ejpam-6256	32	14	than	than	ADP
ejpam-6256	32	15	one	one	NUM
ejpam-6256	32	16	fixed	fix	VERB
ejpam-6256	32	17	point	point	NOUN
ejpam-6256	32	18	.	.	PUNCT
ejpam-6256	33	1	however	however	ADV
ejpam-6256	33	2	,	,	PUNCT
ejpam-6256	33	3	in	in	ADP
ejpam-6256	33	4	the	the	DET
ejpam-6256	33	5	standard	standard	ADJ
ejpam-6256	33	6	metric	metric	ADJ
ejpam-6256	33	7	space	space	NOUN
ejpam-6256	33	8	,	,	PUNCT
ejpam-6256	33	9	the	the	DET
ejpam-6256	33	10	existence	existence	NOUN
ejpam-6256	33	11	of	of	ADP
ejpam-6256	33	12	an	an	DET
ejpam-6256	33	13	iterative	iterative	NOUN
ejpam-6256	33	14	fixed	fix	VERB
ejpam-6256	33	15	point	point	NOUN
ejpam-6256	33	16	often	often	ADV
ejpam-6256	33	17	implies	imply	VERB
ejpam-6256	33	18	its	its	PRON
ejpam-6256	33	19	uniqueness	uniqueness	NOUN
ejpam-6256	33	20	.	.	PUNCT
ejpam-6256	34	1	it	it	PRON
ejpam-6256	34	2	is	be	AUX
ejpam-6256	34	3	common	common	ADJ
ejpam-6256	34	4	to	to	PART
ejpam-6256	34	5	combine	combine	VERB
ejpam-6256	34	6	contraction	contraction	NOUN
ejpam-6256	34	7	conditions	condition	NOUN
ejpam-6256	34	8	and	and	CCONJ
ejpam-6256	34	9	iteration	iteration	NOUN
ejpam-6256	34	10	methods	method	NOUN
ejpam-6256	34	11	to	to	PART
ejpam-6256	34	12	construct	construct	VERB
ejpam-6256	34	13	a	a	DET
ejpam-6256	34	14	cauchy	cauchy	ADJ
ejpam-6256	34	15	sequence	sequence	NOUN
ejpam-6256	34	16	,	,	PUNCT
ejpam-6256	34	17	ensuring	ensure	VERB
ejpam-6256	34	18	its	its	PRON
ejpam-6256	34	19	convergence	convergence	NOUN
ejpam-6256	34	20	through	through	ADP
ejpam-6256	34	21	the	the	DET
ejpam-6256	34	22	space	space	NOUN
ejpam-6256	34	23	’s	’s	PART
ejpam-6256	34	24	completeness	completeness	NOUN
ejpam-6256	34	25	,	,	PUNCT
ejpam-6256	34	26	with	with	ADP
ejpam-6256	34	27	the	the	DET
ejpam-6256	34	28	limit	limit	NOUN
ejpam-6256	34	29	serving	serve	VERB
ejpam-6256	34	30	as	as	ADP
ejpam-6256	34	31	the	the	DET
ejpam-6256	34	32	desired	desire	VERB
ejpam-6256	34	33	fixed	fix	VERB
ejpam-6256	34	34	point	point	NOUN
ejpam-6256	34	35	.	.	PUNCT
ejpam-6256	35	1	therefore	therefore	ADV
ejpam-6256	35	2	,	,	PUNCT
ejpam-6256	35	3	we	we	PRON
ejpam-6256	35	4	can	can	AUX
ejpam-6256	35	5	not	not	PART
ejpam-6256	35	6	get	get	VERB
ejpam-6256	35	7	multiple	multiple	ADJ
ejpam-6256	35	8	fixed	fix	VERB
ejpam-6256	35	9	points	point	NOUN
ejpam-6256	35	10	in	in	ADP
ejpam-6256	35	11	such	such	DET
ejpam-6256	35	12	a	a	DET
ejpam-6256	35	13	setting	setting	NOUN
ejpam-6256	35	14	,	,	PUNCT
ejpam-6256	35	15	as	as	SCONJ
ejpam-6256	35	16	the	the	DET
ejpam-6256	35	17	uniqueness	uniqueness	NOUN
ejpam-6256	35	18	of	of	ADP
ejpam-6256	35	19	the	the	DET
ejpam-6256	35	20	fixed	fix	VERB
ejpam-6256	35	21	point	point	NOUN
ejpam-6256	35	22	primarily	primarily	ADV
ejpam-6256	35	23	stems	stem	VERB
ejpam-6256	35	24	from	from	ADP
ejpam-6256	35	25	the	the	DET
ejpam-6256	35	26	contraction	contraction	NOUN
ejpam-6256	35	27	conditions	condition	NOUN
ejpam-6256	35	28	and	and	CCONJ
ejpam-6256	35	29	implicitly	implicitly	ADV
ejpam-6256	35	30	from	from	ADP
ejpam-6256	35	31	the	the	DET
ejpam-6256	35	32	uniqueness	uniqueness	NOUN
ejpam-6256	35	33	of	of	ADP
ejpam-6256	35	34	the	the	DET
ejpam-6256	35	35	limit	limit	NOUN
ejpam-6256	35	36	,	,	PUNCT
ejpam-6256	35	37	a	a	DET
ejpam-6256	35	38	single	single	ADJ
ejpam-6256	35	39	point	point	NOUN
ejpam-6256	35	40	in	in	ADP
ejpam-6256	35	41	ordinary	ordinary	ADJ
ejpam-6256	35	42	metric	metric	ADJ
ejpam-6256	35	43	spaces	space	NOUN
ejpam-6256	35	44	.	.	PUNCT
ejpam-6256	36	1	we	we	PRON
ejpam-6256	36	2	address	address	VERB
ejpam-6256	36	3	this	this	DET
ejpam-6256	36	4	issue	issue	NOUN
ejpam-6256	36	5	by	by	ADP
ejpam-6256	36	6	conducting	conduct	VERB
ejpam-6256	36	7	our	our	PRON
ejpam-6256	36	8	study	study	NOUN
ejpam-6256	36	9	on	on	ADP
ejpam-6256	36	10	a	a	DET
ejpam-6256	36	11	generalized	generalize	VERB
ejpam-6256	36	12	metric	metric	ADJ
ejpam-6256	36	13	space	space	NOUN
ejpam-6256	36	14	,	,	PUNCT
ejpam-6256	36	15	where	where	SCONJ
ejpam-6256	36	16	we	we	PRON
ejpam-6256	36	17	consider	consider	VERB
ejpam-6256	36	18	the	the	DET
ejpam-6256	36	19	mp	mp	NOUN
ejpam-6256	36	20	-	-	PUNCT
ejpam-6256	36	21	metric	metric	ADJ
ejpam-6256	36	22	space	space	NOUN
ejpam-6256	36	23	and	and	CCONJ
ejpam-6256	36	24	exploit	exploit	VERB
ejpam-6256	36	25	its	its	PRON
ejpam-6256	36	26	properties	property	NOUN
ejpam-6256	36	27	to	to	PART
ejpam-6256	36	28	show	show	VERB
ejpam-6256	36	29	the	the	DET
ejpam-6256	36	30	existence	existence	NOUN
ejpam-6256	36	31	of	of	ADP
ejpam-6256	36	32	multiple	multiple	ADJ
ejpam-6256	36	33	iterative	iterative	NOUN
ejpam-6256	36	34	fixed	fix	VERB
ejpam-6256	36	35	points	point	NOUN
ejpam-6256	36	36	under	under	ADP
ejpam-6256	36	37	certain	certain	ADJ
ejpam-6256	36	38	conditions	condition	NOUN
ejpam-6256	36	39	.	.	PUNCT
ejpam-6256	37	1	compared	compare	VERB
ejpam-6256	37	2	to	to	ADP
ejpam-6256	37	3	other	other	ADJ
ejpam-6256	37	4	results	result	NOUN
ejpam-6256	37	5	in	in	ADP
ejpam-6256	37	6	the	the	DET
ejpam-6256	37	7	field	field	NOUN
ejpam-6256	37	8	,	,	PUNCT
ejpam-6256	37	9	we	we	PRON
ejpam-6256	37	10	observe	observe	VERB
ejpam-6256	37	11	that	that	SCONJ
ejpam-6256	37	12	the	the	DET
ejpam-6256	37	13	finding	finding	NOUN
ejpam-6256	37	14	presented	present	VERB
ejpam-6256	37	15	in	in	ADP
ejpam-6256	37	16	this	this	DET
ejpam-6256	37	17	work	work	NOUN
ejpam-6256	37	18	offers	offer	VERB
ejpam-6256	37	19	a	a	DET
ejpam-6256	37	20	distinct	distinct	ADJ
ejpam-6256	37	21	advantage	advantage	NOUN
ejpam-6256	37	22	:	:	PUNCT
ejpam-6256	37	23	it	it	PRON
ejpam-6256	37	24	establishes	establish	VERB
ejpam-6256	37	25	the	the	DET
ejpam-6256	37	26	existence	existence	NOUN
ejpam-6256	37	27	of	of	ADP
ejpam-6256	37	28	multiple	multiple	ADJ
ejpam-6256	37	29	fixed	fix	VERB
ejpam-6256	37	30	points	point	NOUN
ejpam-6256	37	31	for	for	ADP
ejpam-6256	37	32	a	a	DET
ejpam-6256	37	33	given	give	VERB
ejpam-6256	37	34	function	function	NOUN
ejpam-6256	37	35	while	while	SCONJ
ejpam-6256	37	36	leveraging	leverage	VERB
ejpam-6256	37	37	iterative	iterative	NOUN
ejpam-6256	37	38	methods	method	NOUN
ejpam-6256	37	39	and	and	CCONJ
ejpam-6256	37	40	their	their	PRON
ejpam-6256	37	41	notable	notable	ADJ
ejpam-6256	37	42	practical	practical	ADJ
ejpam-6256	37	43	benefits	benefit	NOUN
ejpam-6256	37	44	.	.	PUNCT
ejpam-6256	38	1	the	the	DET
ejpam-6256	38	2	concept	concept	NOUN
ejpam-6256	38	3	of	of	ADP
ejpam-6256	38	4	multiple	multiple	ADJ
ejpam-6256	38	5	iterative	iterative	ADJ
ejpam-6256	38	6	fixed	fix	VERB
ejpam-6256	38	7	points	point	NOUN
ejpam-6256	38	8	is	be	AUX
ejpam-6256	38	9	a	a	DET
ejpam-6256	38	10	potentially	potentially	ADV
ejpam-6256	38	11	valuable	valuable	ADJ
ejpam-6256	38	12	tool	tool	NOUN
ejpam-6256	38	13	in	in	ADP
ejpam-6256	38	14	both	both	CCONJ
ejpam-6256	38	15	pure	pure	ADJ
ejpam-6256	38	16	and	and	CCONJ
ejpam-6256	38	17	applied	applied	ADJ
ejpam-6256	38	18	mathematics	mathematic	NOUN
ejpam-6256	38	19	,	,	PUNCT
ejpam-6256	38	20	as	as	SCONJ
ejpam-6256	38	21	it	it	PRON
ejpam-6256	38	22	facilitates	facilitate	VERB
ejpam-6256	38	23	the	the	DET
ejpam-6256	38	24	analysis	analysis	NOUN
ejpam-6256	38	25	of	of	ADP
ejpam-6256	38	26	systems	system	NOUN
ejpam-6256	38	27	where	where	SCONJ
ejpam-6256	38	28	uniqueness	uniqueness	NOUN
ejpam-6256	38	29	is	be	AUX
ejpam-6256	38	30	either	either	CCONJ
ejpam-6256	38	31	undesirable	undesirable	ADJ
ejpam-6256	38	32	or	or	CCONJ
ejpam-6256	38	33	can	can	AUX
ejpam-6256	38	34	not	not	PART
ejpam-6256	38	35	be	be	AUX
ejpam-6256	38	36	guaranteed	guarantee	VERB
ejpam-6256	38	37	.	.	PUNCT
ejpam-6256	39	1	2	2	X
ejpam-6256	39	2	.	.	X
ejpam-6256	39	3	preliminaries	preliminary	NOUN
ejpam-6256	39	4	numerous	numerous	ADJ
ejpam-6256	39	5	studies	study	NOUN
ejpam-6256	39	6	have	have	AUX
ejpam-6256	39	7	highlighted	highlight	VERB
ejpam-6256	39	8	the	the	DET
ejpam-6256	39	9	advantages	advantage	NOUN
ejpam-6256	39	10	of	of	ADP
ejpam-6256	39	11	studying	study	VERB
ejpam-6256	39	12	fixed	fix	VERB
ejpam-6256	39	13	points	point	NOUN
ejpam-6256	39	14	in	in	ADP
ejpam-6256	39	15	generalized	generalized	ADJ
ejpam-6256	39	16	metric	metric	ADJ
ejpam-6256	39	17	spaces	space	NOUN
ejpam-6256	39	18	,	,	PUNCT
ejpam-6256	39	19	which	which	PRON
ejpam-6256	39	20	help	help	VERB
ejpam-6256	39	21	overcome	overcome	VERB
ejpam-6256	39	22	some	some	PRON
ejpam-6256	39	23	of	of	ADP
ejpam-6256	39	24	the	the	DET
ejpam-6256	39	25	limitations	limitation	NOUN
ejpam-6256	39	26	of	of	ADP
ejpam-6256	39	27	ordinary	ordinary	ADJ
ejpam-6256	39	28	metrics	metric	NOUN
ejpam-6256	39	29	and	and	CCONJ
ejpam-6256	39	30	provide	provide	VERB
ejpam-6256	39	31	a	a	DET
ejpam-6256	39	32	more	more	ADV
ejpam-6256	39	33	suitable	suitable	ADJ
ejpam-6256	39	34	framework	framework	NOUN
ejpam-6256	39	35	for	for	ADP
ejpam-6256	39	36	achieving	achieve	VERB
ejpam-6256	39	37	desired	desire	VERB
ejpam-6256	39	38	results	result	NOUN
ejpam-6256	39	39	.	.	PUNCT
ejpam-6256	40	1	for	for	ADP
ejpam-6256	40	2	example	example	NOUN
ejpam-6256	40	3	,	,	PUNCT
ejpam-6256	40	4	one	one	PRON
ejpam-6256	40	5	can	can	AUX
ejpam-6256	40	6	refer	refer	VERB
ejpam-6256	40	7	to	to	ADP
ejpam-6256	40	8	[	[	X
ejpam-6256	40	9	16],[17	16],[17	PROPN
ejpam-6256	40	10	]	]	PUNCT
ejpam-6256	40	11	,	,	PUNCT
ejpam-6256	40	12	[	[	X
ejpam-6256	40	13	18	18	NUM
ejpam-6256	40	14	]	]	PUNCT
ejpam-6256	40	15	,	,	PUNCT
ejpam-6256	40	16	[	[	X
ejpam-6256	40	17	19	19	NUM
ejpam-6256	40	18	]	]	PUNCT
ejpam-6256	40	19	,	,	PUNCT
ejpam-6256	40	20	and	and	CCONJ
ejpam-6256	40	21	[	[	X
ejpam-6256	40	22	20	20	NUM
ejpam-6256	40	23	]	]	PUNCT
ejpam-6256	40	24	for	for	ADP
ejpam-6256	40	25	results	result	NOUN
ejpam-6256	40	26	on	on	ADP
ejpam-6256	40	27	fixed	fix	VERB
ejpam-6256	40	28	-	-	PUNCT
ejpam-6256	40	29	point	point	NOUN
ejpam-6256	40	30	theory	theory	NOUN
ejpam-6256	40	31	in	in	ADP
ejpam-6256	40	32	generalized	generalized	ADJ
ejpam-6256	40	33	metric	metric	ADJ
ejpam-6256	40	34	spaces	space	NOUN
ejpam-6256	40	35	,	,	PUNCT
ejpam-6256	40	36	such	such	ADJ
ejpam-6256	40	37	as	as	ADP
ejpam-6256	40	38	g	g	NOUN
ejpam-6256	40	39	-	-	PUNCT
ejpam-6256	40	40	metric	metric	ADJ
ejpam-6256	40	41	and	and	CCONJ
ejpam-6256	40	42	b	b	NOUN
ejpam-6256	40	43	-	-	PUNCT
ejpam-6256	40	44	metric	metric	ADJ
ejpam-6256	40	45	spaces	space	NOUN
ejpam-6256	40	46	.	.	PUNCT
ejpam-6256	41	1	y.	y.	PROPN
ejpam-6256	41	2	alzubaidi	alzubaidi	PROPN
ejpam-6256	41	3	/	/	SYM
ejpam-6256	41	4	eur	eur	PROPN
ejpam-6256	41	5	.	.	PUNCT
ejpam-6256	42	1	j.	j.	PROPN
ejpam-6256	42	2	pure	pure	PROPN
ejpam-6256	42	3	appl	appl	PROPN
ejpam-6256	42	4	.	.	PROPN
ejpam-6256	42	5	math	math	PROPN
ejpam-6256	42	6	,	,	PUNCT
ejpam-6256	42	7	18	18	NUM
ejpam-6256	42	8	(	(	PUNCT
ejpam-6256	42	9	3	3	NUM
ejpam-6256	42	10	)	)	PUNCT
ejpam-6256	42	11	(	(	PUNCT
ejpam-6256	42	12	2025	2025	NUM
ejpam-6256	42	13	)	)	PUNCT
ejpam-6256	42	14	,	,	PUNCT
ejpam-6256	42	15	6256	6256	NUM
ejpam-6256	42	16	3	3	NUM
ejpam-6256	42	17	of	of	ADP
ejpam-6256	42	18	11	11	NUM
ejpam-6256	42	19	in	in	ADP
ejpam-6256	42	20	this	this	DET
ejpam-6256	42	21	section	section	NOUN
ejpam-6256	42	22	,	,	PUNCT
ejpam-6256	42	23	we	we	PRON
ejpam-6256	42	24	review	review	VERB
ejpam-6256	42	25	the	the	DET
ejpam-6256	42	26	generalized	generalize	VERB
ejpam-6256	42	27	mp	mp	NOUN
ejpam-6256	42	28	-	-	PUNCT
ejpam-6256	42	29	metric	metric	ADJ
ejpam-6256	42	30	space	space	NOUN
ejpam-6256	42	31	,	,	PUNCT
ejpam-6256	42	32	highlight	highlight	VERB
ejpam-6256	42	33	its	its	PRON
ejpam-6256	42	34	key	key	ADJ
ejpam-6256	42	35	properties	property	NOUN
ejpam-6256	42	36	,	,	PUNCT
ejpam-6256	42	37	and	and	CCONJ
ejpam-6256	42	38	present	present	VERB
ejpam-6256	42	39	essential	essential	ADJ
ejpam-6256	42	40	theorems	theorem	NOUN
ejpam-6256	42	41	relevant	relevant	ADJ
ejpam-6256	42	42	to	to	ADP
ejpam-6256	42	43	this	this	DET
ejpam-6256	42	44	study	study	NOUN
ejpam-6256	42	45	.	.	PUNCT
ejpam-6256	43	1	definition	definition	NOUN
ejpam-6256	43	2	1	1	NUM
ejpam-6256	43	3	(	(	PUNCT
ejpam-6256	43	4	[	[	X
ejpam-6256	43	5	21	21	NUM
ejpam-6256	43	6	]	]	PUNCT
ejpam-6256	43	7	)	)	PUNCT
ejpam-6256	43	8	.	.	PUNCT
ejpam-6256	44	1	let	let	VERB
ejpam-6256	44	2	p	p	NOUN
ejpam-6256	44	3	∗(x	∗(x	PROPN
ejpam-6256	44	4	)	)	PUNCT
ejpam-6256	44	5	denote	denote	VERB
ejpam-6256	44	6	the	the	DET
ejpam-6256	44	7	set	set	NOUN
ejpam-6256	44	8	of	of	ADP
ejpam-6256	44	9	all	all	DET
ejpam-6256	44	10	non	non	ADJ
ejpam-6256	44	11	-	-	ADJ
ejpam-6256	44	12	empty	empty	ADJ
ejpam-6256	44	13	finite	finite	ADJ
ejpam-6256	44	14	subsets	subset	NOUN
ejpam-6256	44	15	of	of	ADP
ejpam-6256	44	16	x.	x.	NOUN
ejpam-6256	44	17	then	then	ADV
ejpam-6256	44	18	,	,	PUNCT
ejpam-6256	44	19	d	d	X
ejpam-6256	44	20	:	:	PUNCT
ejpam-6256	44	21	p	p	NOUN
ejpam-6256	44	22	∗(x	∗(x	PROPN
ejpam-6256	44	23	)	)	PUNCT
ejpam-6256	44	24	−→	−→	NOUN
ejpam-6256	44	25	[	[	X
ejpam-6256	44	26	0,∞	0,∞	NOUN
ejpam-6256	44	27	)	)	PUNCT
ejpam-6256	44	28	is	be	AUX
ejpam-6256	44	29	called	call	VERB
ejpam-6256	44	30	an	an	DET
ejpam-6256	44	31	mp	mp	NOUN
ejpam-6256	44	32	-	-	PUNCT
ejpam-6256	44	33	metric	metric	NOUN
ejpam-6256	44	34	(	(	PUNCT
ejpam-6256	44	35	a	a	DET
ejpam-6256	44	36	multiple	multiple	ADJ
ejpam-6256	44	37	point	point	NOUN
ejpam-6256	44	38	-	-	PUNCT
ejpam-6256	44	39	metric	metric	ADJ
ejpam-6256	44	40	)	)	PUNCT
ejpam-6256	44	41	if	if	SCONJ
ejpam-6256	44	42	for	for	ADP
ejpam-6256	44	43	all	all	DET
ejpam-6256	44	44	a	a	DET
ejpam-6256	44	45	,	,	PUNCT
ejpam-6256	44	46	b	b	X
ejpam-6256	44	47	∈	∈	PROPN
ejpam-6256	44	48	p	p	NOUN
ejpam-6256	44	49	∗(x	∗(x	PROPN
ejpam-6256	44	50	)	)	PUNCT
ejpam-6256	44	51	,	,	PUNCT
ejpam-6256	44	52	the	the	DET
ejpam-6256	44	53	following	follow	VERB
ejpam-6256	44	54	hold	hold	NOUN
ejpam-6256	44	55	:	:	PUNCT
ejpam-6256	44	56	•	•	NUM
ejpam-6256	44	57	(	(	PUNCT
ejpam-6256	44	58	a1	a1	NOUN
ejpam-6256	44	59	)	)	PUNCT
ejpam-6256	44	60	d(a	d(a	PROPN
ejpam-6256	44	61	)	)	PUNCT
ejpam-6256	45	1	=	=	SYM
ejpam-6256	45	2	0	0	NUM
ejpam-6256	45	3	⇐	⇐	ADJ
ejpam-6256	45	4	⇒	⇒	PROPN
ejpam-6256	45	5	|a|	|a|	PROPN
ejpam-6256	45	6	=	=	SYM
ejpam-6256	45	7	1	1	NUM
ejpam-6256	45	8	,	,	PUNCT
ejpam-6256	45	9	•	•	PRON
ejpam-6256	45	10	(	(	PUNCT
ejpam-6256	45	11	a2	a2	PROPN
ejpam-6256	45	12	)	)	PUNCT
ejpam-6256	45	13	a	a	PRON
ejpam-6256	45	14	⊆	⊆	NUM
ejpam-6256	45	15	b	b	NOUN
ejpam-6256	45	16	=	=	NOUN
ejpam-6256	45	17	⇒	⇒	VERB
ejpam-6256	45	18	d(a	d(a	PROPN
ejpam-6256	45	19	)	)	PUNCT
ejpam-6256	45	20	≤	≤	NOUN
ejpam-6256	45	21	d(b	d(b	PROPN
ejpam-6256	45	22	)	)	PUNCT
ejpam-6256	45	23	,	,	PUNCT
ejpam-6256	45	24	•	•	PRON
ejpam-6256	45	25	(	(	PUNCT
ejpam-6256	45	26	a3	a3	NOUN
ejpam-6256	45	27	)	)	PUNCT
ejpam-6256	45	28	a	a	DET
ejpam-6256	45	29	∩b	∩b	NOUN
ejpam-6256	45	30	̸=	̸=	PROPN
ejpam-6256	45	31	ϕ	ϕ	NOUN
ejpam-6256	45	32	=	=	NOUN
ejpam-6256	45	33	⇒	⇒	VERB
ejpam-6256	45	34	d(a	d(a	PROPN
ejpam-6256	45	35	∪b	∪b	NOUN
ejpam-6256	45	36	)	)	PUNCT
ejpam-6256	45	37	≤	≤	NUM
ejpam-6256	46	1	d(a	d(a	PROPN
ejpam-6256	46	2	)	)	PUNCT
ejpam-6256	47	1	+	+	CCONJ
ejpam-6256	47	2	d(b	d(b	PROPN
ejpam-6256	47	3	)	)	PUNCT
ejpam-6256	47	4	,	,	PUNCT
ejpam-6256	47	5	•	•	X
ejpam-6256	47	6	(	(	PUNCT
ejpam-6256	47	7	a4	a4	NOUN
ejpam-6256	47	8	)	)	PUNCT
ejpam-6256	48	1	d(a	d(a	PROPN
ejpam-6256	48	2	∪	∪	X
ejpam-6256	48	3	{	{	PUNCT
ejpam-6256	48	4	b	b	NOUN
ejpam-6256	48	5	}	}	PUNCT
ejpam-6256	48	6	)	)	PUNCT
ejpam-6256	48	7	=	=	SYM
ejpam-6256	48	8	d(a	d(a	PROPN
ejpam-6256	48	9	)	)	PUNCT
ejpam-6256	48	10	for	for	ADP
ejpam-6256	48	11	all	all	DET
ejpam-6256	48	12	b	b	NOUN
ejpam-6256	48	13	∈	∈	ADP
ejpam-6256	48	14	b	b	NOUN
ejpam-6256	49	1	=	=	NOUN
ejpam-6256	49	2	⇒	⇒	VERB
ejpam-6256	49	3	d(a	d(a	PROPN
ejpam-6256	49	4	∪b	∪b	X
ejpam-6256	49	5	)	)	PUNCT
ejpam-6256	49	6	=	=	SYM
ejpam-6256	49	7	d(a	d(a	PROPN
ejpam-6256	49	8	)	)	PUNCT
ejpam-6256	49	9	,	,	PUNCT
ejpam-6256	49	10	where	where	SCONJ
ejpam-6256	49	11	|a|	|a|	NOUN
ejpam-6256	49	12	represents	represent	VERB
ejpam-6256	49	13	the	the	DET
ejpam-6256	49	14	cardinality	cardinality	NOUN
ejpam-6256	49	15	of	of	ADP
ejpam-6256	49	16	the	the	DET
ejpam-6256	49	17	set	set	NOUN
ejpam-6256	49	18	a.	a.	NOUN
ejpam-6256	49	19	furthermore	furthermore	ADV
ejpam-6256	49	20	,	,	PUNCT
ejpam-6256	49	21	(	(	PUNCT
ejpam-6256	49	22	x	x	X
ejpam-6256	49	23	,	,	PUNCT
ejpam-6256	49	24	d	d	NOUN
ejpam-6256	49	25	)	)	PUNCT
ejpam-6256	49	26	is	be	AUX
ejpam-6256	49	27	called	call	VERB
ejpam-6256	49	28	an	an	DET
ejpam-6256	49	29	mpmetric	mpmetric	ADJ
ejpam-6256	49	30	space	space	NOUN
ejpam-6256	49	31	.	.	PUNCT
ejpam-6256	50	1	to	to	PART
ejpam-6256	50	2	simplify	simplify	VERB
ejpam-6256	50	3	the	the	DET
ejpam-6256	50	4	presentation	presentation	NOUN
ejpam-6256	50	5	below	below	ADV
ejpam-6256	50	6	,	,	PUNCT
ejpam-6256	50	7	it	it	PRON
ejpam-6256	50	8	is	be	AUX
ejpam-6256	50	9	assumed	assume	VERB
ejpam-6256	50	10	that	that	SCONJ
ejpam-6256	50	11	d(a1	d(a1	NOUN
ejpam-6256	50	12	,	,	PUNCT
ejpam-6256	50	13	...	...	PUNCT
ejpam-6256	50	14	,	,	PUNCT
ejpam-6256	50	15	ak	ak	PROPN
ejpam-6256	50	16	)	)	PUNCT
ejpam-6256	50	17	,	,	PUNCT
ejpam-6256	50	18	d({a1	d({a1	PROPN
ejpam-6256	50	19	,	,	PUNCT
ejpam-6256	50	20	...	...	PUNCT
ejpam-6256	50	21	,	,	PUNCT
ejpam-6256	50	22	ak	ak	PROPN
ejpam-6256	50	23	}	}	PUNCT
ejpam-6256	50	24	)	)	PUNCT
ejpam-6256	50	25	,	,	PUNCT
ejpam-6256	50	26	and	and	CCONJ
ejpam-6256	50	27	d(a	d(a	PROPN
ejpam-6256	50	28	)	)	PUNCT
ejpam-6256	50	29	for	for	ADP
ejpam-6256	50	30	a	a	DET
ejpam-6256	50	31	=	=	X
ejpam-6256	50	32	{	{	PUNCT
ejpam-6256	50	33	a1	a1	PROPN
ejpam-6256	50	34	,	,	PUNCT
ejpam-6256	50	35	...	...	PUNCT
ejpam-6256	50	36	,	,	PUNCT
ejpam-6256	50	37	ak	ak	PROPN
ejpam-6256	50	38	}	}	PUNCT
ejpam-6256	50	39	are	be	AUX
ejpam-6256	50	40	identical	identical	ADJ
ejpam-6256	50	41	.	.	PUNCT
ejpam-6256	51	1	if	if	SCONJ
ejpam-6256	51	2	we	we	PRON
ejpam-6256	51	3	let	let	VERB
ejpam-6256	51	4	d(2)(a	d(2)(a	NOUN
ejpam-6256	51	5	,	,	PUNCT
ejpam-6256	51	6	b	b	NOUN
ejpam-6256	51	7	)	)	PUNCT
ejpam-6256	51	8	=	=	SYM
ejpam-6256	51	9	d({a	d({a	PROPN
ejpam-6256	51	10	,	,	PUNCT
ejpam-6256	51	11	b	b	NOUN
ejpam-6256	51	12	}	}	PUNCT
ejpam-6256	51	13	)	)	PUNCT
ejpam-6256	51	14	,	,	PUNCT
ejpam-6256	51	15	we	we	PRON
ejpam-6256	51	16	can	can	AUX
ejpam-6256	51	17	check	check	VERB
ejpam-6256	51	18	that	that	DET
ejpam-6256	51	19	d(2	d(2	PROPN
ejpam-6256	51	20	)	)	PUNCT
ejpam-6256	51	21	is	be	AUX
ejpam-6256	51	22	a	a	DET
ejpam-6256	51	23	standard	standard	ADJ
ejpam-6256	51	24	metric	metric	NOUN
ejpam-6256	51	25	that	that	PRON
ejpam-6256	51	26	is	be	AUX
ejpam-6256	51	27	referred	refer	VERB
ejpam-6256	51	28	to	to	ADP
ejpam-6256	51	29	as	as	ADP
ejpam-6256	51	30	the	the	DET
ejpam-6256	51	31	associated	associated	ADJ
ejpam-6256	51	32	metric	metric	NOUN
ejpam-6256	51	33	.	.	PUNCT
ejpam-6256	52	1	the	the	DET
ejpam-6256	52	2	mp	mp	NOUN
ejpam-6256	52	3	-	-	PUNCT
ejpam-6256	52	4	metric	metric	ADJ
ejpam-6256	52	5	concept	concept	NOUN
ejpam-6256	52	6	can	can	AUX
ejpam-6256	52	7	be	be	AUX
ejpam-6256	52	8	considered	consider	VERB
ejpam-6256	52	9	an	an	DET
ejpam-6256	52	10	extension	extension	NOUN
ejpam-6256	52	11	to	to	ADP
ejpam-6256	52	12	some	some	DET
ejpam-6256	52	13	other	other	ADJ
ejpam-6256	52	14	types	type	NOUN
ejpam-6256	52	15	of	of	ADP
ejpam-6256	52	16	metrics	metric	NOUN
ejpam-6256	52	17	,	,	PUNCT
ejpam-6256	52	18	including	include	VERB
ejpam-6256	52	19	the	the	DET
ejpam-6256	52	20	standard	standard	ADJ
ejpam-6256	52	21	metric	metric	NOUN
ejpam-6256	52	22	(	(	PUNCT
ejpam-6256	52	23	see	see	VERB
ejpam-6256	52	24	[	[	X
ejpam-6256	52	25	21	21	NUM
ejpam-6256	52	26	]	]	PUNCT
ejpam-6256	52	27	)	)	PUNCT
ejpam-6256	52	28	.	.	PUNCT
ejpam-6256	53	1	example	example	NOUN
ejpam-6256	54	1	1	1	NUM
ejpam-6256	54	2	(	(	PUNCT
ejpam-6256	54	3	[	[	X
ejpam-6256	54	4	21	21	NUM
ejpam-6256	54	5	]	]	PUNCT
ejpam-6256	54	6	)	)	PUNCT
ejpam-6256	54	7	.	.	PUNCT
ejpam-6256	55	1	a	a	DET
ejpam-6256	55	2	natural	natural	ADJ
ejpam-6256	55	3	mp	mp	NOUN
ejpam-6256	55	4	-	-	PUNCT
ejpam-6256	55	5	metric	metric	ADJ
ejpam-6256	55	6	d	d	NOUN
ejpam-6256	55	7	:	:	PUNCT
ejpam-6256	55	8	p	p	PROPN
ejpam-6256	55	9	∗(r	∗(r	PROPN
ejpam-6256	55	10	)	)	PUNCT
ejpam-6256	55	11	−→	−→	NOUN
ejpam-6256	55	12	[	[	X
ejpam-6256	55	13	0,∞	0,∞	NOUN
ejpam-6256	55	14	)	)	PUNCT
ejpam-6256	55	15	on	on	ADP
ejpam-6256	55	16	r	r	NOUN
ejpam-6256	55	17	can	can	AUX
ejpam-6256	55	18	be	be	AUX
ejpam-6256	55	19	defined	define	VERB
ejpam-6256	55	20	in	in	ADP
ejpam-6256	55	21	the	the	DET
ejpam-6256	55	22	following	following	ADJ
ejpam-6256	55	23	way	way	NOUN
ejpam-6256	55	24	:	:	PUNCT
ejpam-6256	55	25	d(a	d(a	PROPN
ejpam-6256	55	26	)	)	PUNCT
ejpam-6256	56	1	=	=	SYM
ejpam-6256	56	2	max(a	max(a	PROPN
ejpam-6256	56	3	)	)	PUNCT
ejpam-6256	56	4	−	−	PROPN
ejpam-6256	56	5	min(a	min(a	PROPN
ejpam-6256	56	6	)	)	PUNCT
ejpam-6256	56	7	.	.	PUNCT
ejpam-6256	57	1	the	the	DET
ejpam-6256	57	2	restriction	restriction	NOUN
ejpam-6256	57	3	of	of	ADP
ejpam-6256	57	4	this	this	DET
ejpam-6256	57	5	metric	metric	NOUN
ejpam-6256	57	6	to	to	ADP
ejpam-6256	57	7	two	two	NUM
ejpam-6256	57	8	points	point	NOUN
ejpam-6256	57	9	gives	give	VERB
ejpam-6256	57	10	the	the	DET
ejpam-6256	57	11	usual	usual	ADJ
ejpam-6256	57	12	distance	distance	NOUN
ejpam-6256	57	13	metric	metric	NOUN
ejpam-6256	57	14	on	on	ADP
ejpam-6256	57	15	r	r	NOUN
ejpam-6256	57	16	,	,	PUNCT
ejpam-6256	57	17	where	where	SCONJ
ejpam-6256	57	18	d(2)(a	d(2)(a	NOUN
ejpam-6256	57	19	,	,	PUNCT
ejpam-6256	57	20	b	b	NOUN
ejpam-6256	57	21	)	)	PUNCT
ejpam-6256	57	22	=	=	SYM
ejpam-6256	57	23	d({a	d({a	PROPN
ejpam-6256	57	24	,	,	PUNCT
ejpam-6256	57	25	b	b	NOUN
ejpam-6256	57	26	}	}	PUNCT
ejpam-6256	57	27	)	)	PUNCT
ejpam-6256	57	28	=	=	SYM
ejpam-6256	57	29	max({a	max({a	PROPN
ejpam-6256	57	30	,	,	PUNCT
ejpam-6256	57	31	b})−	b})−	PROPN
ejpam-6256	57	32	min({a	min({a	PROPN
ejpam-6256	57	33	,	,	PUNCT
ejpam-6256	57	34	b	b	NOUN
ejpam-6256	57	35	}	}	PUNCT
ejpam-6256	57	36	)	)	PUNCT
ejpam-6256	58	1	=	=	SYM
ejpam-6256	58	2	|a−	|a−	NOUN
ejpam-6256	58	3	b|	b|	PROPN
ejpam-6256	58	4	.	.	PUNCT
ejpam-6256	59	1	next	next	ADV
ejpam-6256	59	2	,	,	PUNCT
ejpam-6256	59	3	we	we	PRON
ejpam-6256	59	4	present	present	VERB
ejpam-6256	59	5	the	the	DET
ejpam-6256	59	6	concept	concept	NOUN
ejpam-6256	59	7	of	of	ADP
ejpam-6256	59	8	convergence	convergence	NOUN
ejpam-6256	59	9	in	in	ADP
ejpam-6256	59	10	mp	mp	NOUN
ejpam-6256	59	11	-	-	PUNCT
ejpam-6256	59	12	metric	metric	ADJ
ejpam-6256	59	13	spaces	space	NOUN
ejpam-6256	59	14	.	.	PUNCT
ejpam-6256	60	1	it	it	PRON
ejpam-6256	60	2	allows	allow	VERB
ejpam-6256	60	3	us	we	PRON
ejpam-6256	60	4	to	to	PART
ejpam-6256	60	5	directly	directly	ADV
ejpam-6256	60	6	obtain	obtain	VERB
ejpam-6256	60	7	a	a	DET
ejpam-6256	60	8	generalized	generalized	ADJ
ejpam-6256	60	9	form	form	NOUN
ejpam-6256	60	10	of	of	ADP
ejpam-6256	60	11	a	a	DET
ejpam-6256	60	12	sequence	sequence	NOUN
ejpam-6256	60	13	’s	’s	PART
ejpam-6256	60	14	limit	limit	NOUN
ejpam-6256	60	15	,	,	PUNCT
ejpam-6256	60	16	where	where	SCONJ
ejpam-6256	60	17	the	the	DET
ejpam-6256	60	18	limit	limit	NOUN
ejpam-6256	60	19	can	can	AUX
ejpam-6256	60	20	be	be	AUX
ejpam-6256	60	21	a	a	DET
ejpam-6256	60	22	set	set	NOUN
ejpam-6256	60	23	(	(	PUNCT
ejpam-6256	60	24	a	a	DET
ejpam-6256	60	25	compact	compact	ADJ
ejpam-6256	60	26	set	set	NOUN
ejpam-6256	60	27	)	)	PUNCT
ejpam-6256	60	28	rather	rather	ADV
ejpam-6256	60	29	than	than	ADP
ejpam-6256	60	30	a	a	DET
ejpam-6256	60	31	single	single	ADJ
ejpam-6256	60	32	point	point	NOUN
ejpam-6256	60	33	.	.	PUNCT
ejpam-6256	61	1	such	such	DET
ejpam-6256	61	2	a	a	DET
ejpam-6256	61	3	way	way	NOUN
ejpam-6256	61	4	of	of	ADP
ejpam-6256	61	5	recognizing	recognize	VERB
ejpam-6256	61	6	the	the	DET
ejpam-6256	61	7	limit	limit	NOUN
ejpam-6256	61	8	will	will	AUX
ejpam-6256	61	9	be	be	AUX
ejpam-6256	61	10	crucial	crucial	ADJ
ejpam-6256	61	11	in	in	ADP
ejpam-6256	61	12	achieving	achieve	VERB
ejpam-6256	61	13	our	our	PRON
ejpam-6256	61	14	main	main	ADJ
ejpam-6256	61	15	result	result	NOUN
ejpam-6256	61	16	.	.	PUNCT
ejpam-6256	62	1	definition	definition	NOUN
ejpam-6256	62	2	2	2	NUM
ejpam-6256	62	3	(	(	PUNCT
ejpam-6256	62	4	[	[	X
ejpam-6256	62	5	21	21	NUM
ejpam-6256	62	6	]	]	PUNCT
ejpam-6256	62	7	)	)	PUNCT
ejpam-6256	62	8	.	.	PUNCT
ejpam-6256	63	1	let	let	VERB
ejpam-6256	63	2	(	(	PUNCT
ejpam-6256	63	3	x	x	NOUN
ejpam-6256	63	4	,	,	PUNCT
ejpam-6256	63	5	d	d	NOUN
ejpam-6256	63	6	)	)	PUNCT
ejpam-6256	63	7	be	be	AUX
ejpam-6256	63	8	an	an	DET
ejpam-6256	63	9	mp	mp	NOUN
ejpam-6256	63	10	-	-	PUNCT
ejpam-6256	63	11	metric	metric	ADJ
ejpam-6256	63	12	space	space	NOUN
ejpam-6256	63	13	and	and	CCONJ
ejpam-6256	63	14	a	a	DET
ejpam-6256	63	15	∈	∈	NOUN
ejpam-6256	63	16	p	p	NOUN
ejpam-6256	63	17	∗(x	∗(x	PROPN
ejpam-6256	63	18	)	)	PUNCT
ejpam-6256	63	19	.	.	PUNCT
ejpam-6256	64	1	then	then	ADV
ejpam-6256	64	2	d(a	d(a	PROPN
ejpam-6256	64	3	)	)	PUNCT
ejpam-6256	64	4	=	=	PRON
ejpam-6256	64	5	{	{	PUNCT
ejpam-6256	64	6	x	x	PUNCT
ejpam-6256	64	7	∈	∈	NOUN
ejpam-6256	64	8	x	x	X
ejpam-6256	64	9	:	:	PUNCT
ejpam-6256	64	10	d(a	d(a	PROPN
ejpam-6256	64	11	∪	∪	X
ejpam-6256	64	12	{	{	PUNCT
ejpam-6256	64	13	x	x	NOUN
ejpam-6256	64	14	}	}	PUNCT
ejpam-6256	64	15	)	)	PUNCT
ejpam-6256	64	16	=	=	SYM
ejpam-6256	64	17	d(a	d(a	PROPN
ejpam-6256	64	18	)	)	PUNCT
ejpam-6256	64	19	}	}	PUNCT
ejpam-6256	64	20	.	.	PUNCT
ejpam-6256	64	21	is	be	AUX
ejpam-6256	64	22	the	the	DET
ejpam-6256	64	23	set	set	NOUN
ejpam-6256	64	24	of	of	ADP
ejpam-6256	64	25	all	all	DET
ejpam-6256	64	26	d	d	ADJ
ejpam-6256	64	27	-	-	ADJ
ejpam-6256	64	28	dependent	dependent	ADJ
ejpam-6256	64	29	points	point	NOUN
ejpam-6256	64	30	on	on	ADP
ejpam-6256	64	31	a.	a.	NOUN
ejpam-6256	64	32	example	example	NOUN
ejpam-6256	64	33	2	2	NUM
ejpam-6256	64	34	(	(	PUNCT
ejpam-6256	64	35	[	[	X
ejpam-6256	64	36	21	21	NUM
ejpam-6256	64	37	]	]	PUNCT
ejpam-6256	64	38	)	)	PUNCT
ejpam-6256	64	39	.	.	PUNCT
ejpam-6256	65	1	(	(	PUNCT
ejpam-6256	65	2	i)let	i)let	NOUN
ejpam-6256	65	3	(	(	PUNCT
ejpam-6256	65	4	x	x	NOUN
ejpam-6256	65	5	,	,	PUNCT
ejpam-6256	65	6	d	d	NOUN
ejpam-6256	65	7	)	)	PUNCT
ejpam-6256	65	8	be	be	AUX
ejpam-6256	65	9	the	the	DET
ejpam-6256	65	10	mp	mp	NOUN
ejpam-6256	65	11	-	-	PUNCT
ejpam-6256	65	12	metric	metric	ADJ
ejpam-6256	65	13	space	space	NOUN
ejpam-6256	65	14	introduced	introduce	VERB
ejpam-6256	65	15	in	in	ADP
ejpam-6256	65	16	example	example	NOUN
ejpam-6256	65	17	1	1	NUM
ejpam-6256	65	18	and	and	CCONJ
ejpam-6256	65	19	a	a	DET
ejpam-6256	65	20	≤	≤	PROPN
ejpam-6256	65	21	b.	b.	NOUN
ejpam-6256	66	1	then	then	ADV
ejpam-6256	66	2	,	,	PUNCT
ejpam-6256	66	3	d({a	d({a	PROPN
ejpam-6256	66	4	,	,	PUNCT
ejpam-6256	66	5	b	b	NOUN
ejpam-6256	66	6	}	}	PUNCT
ejpam-6256	66	7	)	)	PUNCT
ejpam-6256	66	8	=	=	PRON
ejpam-6256	66	9	{	{	PUNCT
ejpam-6256	66	10	x	x	PUNCT
ejpam-6256	66	11	∈	∈	NOUN
ejpam-6256	66	12	x	x	X
ejpam-6256	66	13	:	:	PUNCT
ejpam-6256	66	14	d({a	d({a	ADJ
ejpam-6256	66	15	,	,	PUNCT
ejpam-6256	66	16	b	b	PROPN
ejpam-6256	66	17	,	,	PUNCT
ejpam-6256	66	18	x	x	NOUN
ejpam-6256	66	19	}	}	PUNCT
ejpam-6256	66	20	)	)	PUNCT
ejpam-6256	66	21	=	=	SYM
ejpam-6256	67	1	d({a	d({a	PROPN
ejpam-6256	67	2	,	,	PUNCT
ejpam-6256	67	3	b	b	NOUN
ejpam-6256	67	4	}	}	PUNCT
ejpam-6256	67	5	)	)	PUNCT
ejpam-6256	67	6	}	}	PUNCT
ejpam-6256	68	1	=	=	PUNCT
ejpam-6256	69	1	[	[	X
ejpam-6256	69	2	a	a	X
ejpam-6256	69	3	,	,	PUNCT
ejpam-6256	69	4	b	b	NOUN
ejpam-6256	69	5	]	]	X
ejpam-6256	69	6	.	.	PUNCT
ejpam-6256	70	1	(	(	PUNCT
ejpam-6256	70	2	ii)in	ii)in	X
ejpam-6256	70	3	any	any	DET
ejpam-6256	70	4	mp	mp	NOUN
ejpam-6256	70	5	-	-	PUNCT
ejpam-6256	70	6	metric	metric	ADJ
ejpam-6256	70	7	space	space	NOUN
ejpam-6256	70	8	and	and	CCONJ
ejpam-6256	70	9	for	for	ADP
ejpam-6256	70	10	any	any	DET
ejpam-6256	70	11	singleton	singleton	NOUN
ejpam-6256	70	12	set	set	NOUN
ejpam-6256	70	13	,	,	PUNCT
ejpam-6256	70	14	it	it	PRON
ejpam-6256	70	15	follows	follow	VERB
ejpam-6256	70	16	from	from	ADP
ejpam-6256	70	17	(	(	PUNCT
ejpam-6256	70	18	a1	a1	NOUN
ejpam-6256	70	19	)	)	PUNCT
ejpam-6256	70	20	that	that	SCONJ
ejpam-6256	70	21	d({a	d({a	PROPN
ejpam-6256	70	22	}	}	PUNCT
ejpam-6256	70	23	)	)	PUNCT
ejpam-6256	71	1	=	=	PRON
ejpam-6256	71	2	{	{	PUNCT
ejpam-6256	71	3	a	a	NOUN
ejpam-6256	71	4	}	}	PUNCT
ejpam-6256	71	5	,	,	PUNCT
ejpam-6256	71	6	since	since	SCONJ
ejpam-6256	71	7	d({a	d({a	PROPN
ejpam-6256	71	8	}	}	PUNCT
ejpam-6256	71	9	)	)	PUNCT
ejpam-6256	72	1	=	=	PRON
ejpam-6256	72	2	{	{	PUNCT
ejpam-6256	72	3	x	x	PUNCT
ejpam-6256	72	4	∈	∈	NOUN
ejpam-6256	72	5	x	x	X
ejpam-6256	72	6	:	:	PUNCT
ejpam-6256	72	7	d({a	d({a	ADJ
ejpam-6256	72	8	,	,	PUNCT
ejpam-6256	72	9	x	x	NOUN
ejpam-6256	72	10	}	}	PUNCT
ejpam-6256	72	11	)	)	PUNCT
ejpam-6256	72	12	=	=	SYM
ejpam-6256	72	13	d({a	d({a	PROPN
ejpam-6256	72	14	}	}	PUNCT
ejpam-6256	72	15	)	)	PUNCT
ejpam-6256	72	16	=	=	PUNCT
ejpam-6256	72	17	0	0	X
ejpam-6256	72	18	}	}	PUNCT
ejpam-6256	72	19	=	=	SYM
ejpam-6256	72	20	{	{	PUNCT
ejpam-6256	72	21	a	a	X
ejpam-6256	72	22	}	}	PUNCT
ejpam-6256	72	23	.	.	PUNCT
ejpam-6256	73	1	y.	y.	PROPN
ejpam-6256	73	2	alzubaidi	alzubaidi	PROPN
ejpam-6256	73	3	/	/	SYM
ejpam-6256	73	4	eur	eur	PROPN
ejpam-6256	73	5	.	.	PUNCT
ejpam-6256	74	1	j.	j.	PROPN
ejpam-6256	74	2	pure	pure	PROPN
ejpam-6256	74	3	appl	appl	PROPN
ejpam-6256	74	4	.	.	PROPN
ejpam-6256	74	5	math	math	PROPN
ejpam-6256	74	6	,	,	PUNCT
ejpam-6256	74	7	18	18	NUM
ejpam-6256	74	8	(	(	PUNCT
ejpam-6256	74	9	3	3	NUM
ejpam-6256	74	10	)	)	PUNCT
ejpam-6256	74	11	(	(	PUNCT
ejpam-6256	74	12	2025	2025	NUM
ejpam-6256	74	13	)	)	PUNCT
ejpam-6256	74	14	,	,	PUNCT
ejpam-6256	74	15	6256	6256	NUM
ejpam-6256	74	16	4	4	NUM
ejpam-6256	74	17	of	of	ADP
ejpam-6256	74	18	11	11	NUM
ejpam-6256	74	19	definition	definition	NOUN
ejpam-6256	74	20	3	3	NUM
ejpam-6256	74	21	(	(	PUNCT
ejpam-6256	74	22	[	[	X
ejpam-6256	74	23	21	21	NUM
ejpam-6256	74	24	]	]	PUNCT
ejpam-6256	74	25	)	)	PUNCT
ejpam-6256	74	26	.	.	PUNCT
ejpam-6256	75	1	let	let	VERB
ejpam-6256	75	2	(	(	PUNCT
ejpam-6256	75	3	x	x	NOUN
ejpam-6256	75	4	,	,	PUNCT
ejpam-6256	75	5	d	d	NOUN
ejpam-6256	75	6	)	)	PUNCT
ejpam-6256	75	7	be	be	AUX
ejpam-6256	75	8	an	an	DET
ejpam-6256	75	9	mp	mp	NOUN
ejpam-6256	75	10	-metric	-metric	ADJ
ejpam-6256	75	11	space	space	NOUN
ejpam-6256	75	12	and	and	CCONJ
ejpam-6256	75	13	d	d	NOUN
ejpam-6256	75	14	⊆	⊆	NUM
ejpam-6256	75	15	x.	x.	NOUN
ejpam-6256	75	16	then	then	ADV
ejpam-6256	75	17	,	,	PUNCT
ejpam-6256	75	18	we	we	PRON
ejpam-6256	75	19	say	say	VERB
ejpam-6256	75	20	that	that	SCONJ
ejpam-6256	75	21	a	a	DET
ejpam-6256	75	22	sequence	sequence	NOUN
ejpam-6256	75	23	xn	xn	PROPN
ejpam-6256	75	24	is	be	AUX
ejpam-6256	75	25	d	d	NOUN
ejpam-6256	75	26	-	-	NOUN
ejpam-6256	75	27	convergent	convergent	ADJ
ejpam-6256	75	28	to	to	ADP
ejpam-6256	75	29	d	d	PROPN
ejpam-6256	75	30	and	and	CCONJ
ejpam-6256	75	31	write	write	VERB
ejpam-6256	75	32	d	d	PROPN
ejpam-6256	75	33	lim	lim	PROPN
ejpam-6256	75	34	n→∞	n→∞	X
ejpam-6256	75	35	xn	xn	PUNCT
ejpam-6256	76	1	=	=	PUNCT
ejpam-6256	76	2	d	d	NOUN
ejpam-6256	76	3	if	if	SCONJ
ejpam-6256	76	4	there	there	PRON
ejpam-6256	76	5	is	be	VERB
ejpam-6256	76	6	a	a	DET
ejpam-6256	76	7	∈	∈	PROPN
ejpam-6256	76	8	p	p	NOUN
ejpam-6256	76	9	∗(x	∗(x	PROPN
ejpam-6256	76	10	)	)	PUNCT
ejpam-6256	76	11	such	such	ADJ
ejpam-6256	76	12	that	that	SCONJ
ejpam-6256	76	13	d	d	PROPN
ejpam-6256	76	14	=	=	SYM
ejpam-6256	76	15	d(a	d(a	PROPN
ejpam-6256	76	16	)	)	PUNCT
ejpam-6256	76	17	and	and	CCONJ
ejpam-6256	76	18	:	:	PUNCT
ejpam-6256	76	19	(	(	PUNCT
ejpam-6256	76	20	i)lim	i)lim	NOUN
ejpam-6256	76	21	ni→∞	ni→∞	NOUN
ejpam-6256	76	22	d({xn1	d({xn1	NOUN
ejpam-6256	76	23	,	,	PUNCT
ejpam-6256	76	24	xn2	xn2	PROPN
ejpam-6256	76	25	...	...	PUNCT
ejpam-6256	76	26	,	,	PUNCT
ejpam-6256	76	27	xnk	xnk	PROPN
ejpam-6256	76	28	}	}	PUNCT
ejpam-6256	76	29	∪a	∪a	NUM
ejpam-6256	76	30	)	)	PUNCT
ejpam-6256	77	1	=	=	SYM
ejpam-6256	77	2	d(a	d(a	PROPN
ejpam-6256	77	3	)	)	PUNCT
ejpam-6256	77	4	for	for	ADP
ejpam-6256	77	5	all	all	DET
ejpam-6256	77	6	k	k	PROPN
ejpam-6256	77	7	∈	∈	PROPN
ejpam-6256	77	8	n	n	CCONJ
ejpam-6256	77	9	,	,	PUNCT
ejpam-6256	77	10	(	(	PUNCT
ejpam-6256	77	11	ii)lim	ii)lim	PROPN
ejpam-6256	77	12	inf	inf	PROPN
ejpam-6256	77	13	n−→∞	n−→∞	PROPN
ejpam-6256	77	14	d(xn	d(xn	PROPN
ejpam-6256	77	15	,	,	PUNCT
ejpam-6256	77	16	a	a	X
ejpam-6256	77	17	)	)	PUNCT
ejpam-6256	77	18	=	=	SYM
ejpam-6256	77	19	0	0	NUM
ejpam-6256	77	20	for	for	ADP
ejpam-6256	77	21	all	all	DET
ejpam-6256	77	22	a	a	DET
ejpam-6256	77	23	∈	∈	NOUN
ejpam-6256	77	24	a.	a.	NOUN
ejpam-6256	77	25	we	we	PRON
ejpam-6256	77	26	will	will	AUX
ejpam-6256	77	27	say	say	VERB
ejpam-6256	77	28	that	that	SCONJ
ejpam-6256	77	29	xn	xn	PROPN
ejpam-6256	77	30	converges	converge	VERB
ejpam-6256	77	31	tod(a	tod(a	PROPN
ejpam-6256	77	32	)	)	PUNCT
ejpam-6256	77	33	,	,	PUNCT
ejpam-6256	77	34	and	and	CCONJ
ejpam-6256	77	35	we	we	PRON
ejpam-6256	77	36	will	will	AUX
ejpam-6256	77	37	write	write	VERB
ejpam-6256	77	38	limxn	limxn	ADV
ejpam-6256	77	39	=	=	SYM
ejpam-6256	77	40	d(a	d(a	NOUN
ejpam-6256	77	41	)	)	PUNCT
ejpam-6256	77	42	to	to	PART
ejpam-6256	77	43	indicate	indicate	VERB
ejpam-6256	77	44	that	that	SCONJ
ejpam-6256	77	45	a	a	PRON
ejpam-6256	77	46	meets	meet	VERB
ejpam-6256	77	47	the	the	DET
ejpam-6256	77	48	convergence	convergence	NOUN
ejpam-6256	77	49	conditions	condition	NOUN
ejpam-6256	77	50	in	in	ADP
ejpam-6256	77	51	the	the	DET
ejpam-6256	77	52	preceding	precede	VERB
ejpam-6256	77	53	formulation	formulation	NOUN
ejpam-6256	77	54	.	.	PUNCT
ejpam-6256	78	1	the	the	DET
ejpam-6256	78	2	above	above	ADJ
ejpam-6256	78	3	definition	definition	NOUN
ejpam-6256	78	4	preserves	preserve	VERB
ejpam-6256	78	5	the	the	DET
ejpam-6256	78	6	characteristics	characteristic	NOUN
ejpam-6256	78	7	of	of	ADP
ejpam-6256	78	8	convergence	convergence	NOUN
ejpam-6256	78	9	in	in	ADP
ejpam-6256	78	10	ordinary	ordinary	ADJ
ejpam-6256	78	11	metric	metric	ADJ
ejpam-6256	78	12	spaces	space	NOUN
ejpam-6256	78	13	and	and	CCONJ
ejpam-6256	78	14	,	,	PUNCT
ejpam-6256	78	15	at	at	ADP
ejpam-6256	78	16	the	the	DET
ejpam-6256	78	17	same	same	ADJ
ejpam-6256	78	18	time	time	NOUN
ejpam-6256	78	19	,	,	PUNCT
ejpam-6256	78	20	provides	provide	VERB
ejpam-6256	78	21	a	a	DET
ejpam-6256	78	22	natural	natural	ADJ
ejpam-6256	78	23	generalization	generalization	NOUN
ejpam-6256	78	24	to	to	ADP
ejpam-6256	78	25	the	the	DET
ejpam-6256	78	26	concept	concept	NOUN
ejpam-6256	78	27	of	of	ADP
ejpam-6256	78	28	limit	limit	NOUN
ejpam-6256	78	29	.	.	PUNCT
ejpam-6256	79	1	proposition	proposition	NOUN
ejpam-6256	79	2	1	1	NUM
ejpam-6256	79	3	(	(	PUNCT
ejpam-6256	79	4	[	[	X
ejpam-6256	79	5	21	21	NUM
ejpam-6256	79	6	]	]	PUNCT
ejpam-6256	79	7	)	)	PUNCT
ejpam-6256	79	8	.	.	PUNCT
ejpam-6256	80	1	let	let	VERB
ejpam-6256	80	2	(	(	PUNCT
ejpam-6256	80	3	x	x	NOUN
ejpam-6256	80	4	,	,	PUNCT
ejpam-6256	80	5	d	d	NOUN
ejpam-6256	80	6	)	)	PUNCT
ejpam-6256	80	7	be	be	AUX
ejpam-6256	80	8	an	an	DET
ejpam-6256	80	9	mp	mp	NOUN
ejpam-6256	80	10	-	-	PUNCT
ejpam-6256	80	11	metric	metric	ADJ
ejpam-6256	80	12	space	space	NOUN
ejpam-6256	80	13	,	,	PUNCT
ejpam-6256	80	14	and	and	CCONJ
ejpam-6256	80	15	let	let	VERB
ejpam-6256	80	16	(	(	PUNCT
ejpam-6256	80	17	x	x	NOUN
ejpam-6256	80	18	,	,	PUNCT
ejpam-6256	80	19	d(2	d(2	PROPN
ejpam-6256	80	20	)	)	PUNCT
ejpam-6256	80	21	)	)	PUNCT
ejpam-6256	80	22	be	be	VERB
ejpam-6256	80	23	the	the	DET
ejpam-6256	80	24	associated	associated	ADJ
ejpam-6256	80	25	metric	metric	ADJ
ejpam-6256	80	26	space	space	NOUN
ejpam-6256	80	27	.	.	PUNCT
ejpam-6256	81	1	a	a	DET
ejpam-6256	81	2	sequence	sequence	NOUN
ejpam-6256	81	3	xn	xn	PROPN
ejpam-6256	81	4	converges	converge	VERB
ejpam-6256	81	5	to	to	ADP
ejpam-6256	81	6	a	a	DET
ejpam-6256	81	7	in	in	ADP
ejpam-6256	81	8	(	(	PUNCT
ejpam-6256	81	9	x	x	NOUN
ejpam-6256	81	10	,	,	PUNCT
ejpam-6256	81	11	d(2	d(2	PROPN
ejpam-6256	81	12	)	)	PUNCT
ejpam-6256	81	13	)	)	PUNCT
ejpam-6256	82	1	if	if	SCONJ
ejpam-6256	82	2	and	and	CCONJ
ejpam-6256	82	3	only	only	ADV
ejpam-6256	82	4	if	if	SCONJ
ejpam-6256	82	5	it	it	PRON
ejpam-6256	82	6	is	be	AUX
ejpam-6256	82	7	d	d	NOUN
ejpam-6256	82	8	-	-	NOUN
ejpam-6256	82	9	convergent	convergent	ADJ
ejpam-6256	82	10	to	to	ADP
ejpam-6256	82	11	d({a	d({a	PROPN
ejpam-6256	82	12	}	}	PUNCT
ejpam-6256	82	13	)	)	PUNCT
ejpam-6256	83	1	=	=	PRON
ejpam-6256	83	2	{	{	PUNCT
ejpam-6256	83	3	a	a	NOUN
ejpam-6256	83	4	}	}	PUNCT
ejpam-6256	83	5	in	in	ADP
ejpam-6256	83	6	(	(	PUNCT
ejpam-6256	83	7	x	x	NOUN
ejpam-6256	83	8	,	,	PUNCT
ejpam-6256	83	9	d	d	NOUN
ejpam-6256	83	10	)	)	PUNCT
ejpam-6256	83	11	.	.	PUNCT
ejpam-6256	84	1	the	the	DET
ejpam-6256	84	2	mp	mp	NOUN
ejpam-6256	84	3	-	-	PUNCT
ejpam-6256	84	4	metric	metric	ADJ
ejpam-6256	84	5	,	,	PUNCT
ejpam-6256	84	6	like	like	ADP
ejpam-6256	84	7	the	the	DET
ejpam-6256	84	8	standard	standard	NOUN
ejpam-6256	84	9	metric	metric	NOUN
ejpam-6256	84	10	,	,	PUNCT
ejpam-6256	84	11	is	be	AUX
ejpam-6256	84	12	considered	consider	VERB
ejpam-6256	84	13	continuous	continuous	ADJ
ejpam-6256	84	14	with	with	ADP
ejpam-6256	84	15	respect	respect	NOUN
ejpam-6256	84	16	to	to	ADP
ejpam-6256	84	17	all	all	PRON
ejpam-6256	84	18	of	of	ADP
ejpam-6256	84	19	its	its	PRON
ejpam-6256	84	20	components	component	NOUN
ejpam-6256	84	21	.	.	PUNCT
ejpam-6256	85	1	proposition	proposition	NOUN
ejpam-6256	85	2	2	2	NUM
ejpam-6256	85	3	(	(	PUNCT
ejpam-6256	85	4	[	[	X
ejpam-6256	85	5	21	21	NUM
ejpam-6256	85	6	]	]	PUNCT
ejpam-6256	85	7	)	)	PUNCT
ejpam-6256	85	8	.	.	PUNCT
ejpam-6256	86	1	let	let	VERB
ejpam-6256	86	2	(	(	PUNCT
ejpam-6256	86	3	x	x	NOUN
ejpam-6256	86	4	,	,	PUNCT
ejpam-6256	86	5	d	d	NOUN
ejpam-6256	86	6	)	)	PUNCT
ejpam-6256	86	7	be	be	AUX
ejpam-6256	86	8	an	an	DET
ejpam-6256	86	9	mp	mp	NOUN
ejpam-6256	86	10	-	-	PUNCT
ejpam-6256	86	11	metric	metric	ADJ
ejpam-6256	86	12	space	space	NOUN
ejpam-6256	86	13	,	,	PUNCT
ejpam-6256	86	14	and	and	CCONJ
ejpam-6256	86	15	let	let	VERB
ejpam-6256	86	16	(	(	PUNCT
ejpam-6256	86	17	x	x	NOUN
ejpam-6256	86	18	,	,	PUNCT
ejpam-6256	86	19	d(2	d(2	PROPN
ejpam-6256	86	20	)	)	PUNCT
ejpam-6256	86	21	)	)	PUNCT
ejpam-6256	86	22	be	be	VERB
ejpam-6256	86	23	the	the	DET
ejpam-6256	86	24	associated	associated	ADJ
ejpam-6256	86	25	metric	metric	ADJ
ejpam-6256	86	26	space	space	NOUN
ejpam-6256	86	27	.	.	PUNCT
ejpam-6256	87	1	(	(	PUNCT
ejpam-6256	87	2	i)if	i)if	NOUN
ejpam-6256	87	3	xn	xn	PROPN
ejpam-6256	87	4	converges	converge	NOUN
ejpam-6256	87	5	to	to	ADP
ejpam-6256	87	6	a	a	PRON
ejpam-6256	87	7	,	,	PUNCT
ejpam-6256	87	8	then	then	ADV
ejpam-6256	87	9	lim	lim	PROPN
ejpam-6256	87	10	n−→∞	n−→∞	PROPN
ejpam-6256	87	11	d({xn	d({xn	PROPN
ejpam-6256	87	12	,	,	PUNCT
ejpam-6256	87	13	a1	a1	NOUN
ejpam-6256	87	14	...	...	PUNCT
ejpam-6256	87	15	,	,	PUNCT
ejpam-6256	87	16	ak	ak	PROPN
ejpam-6256	87	17	}	}	PUNCT
ejpam-6256	87	18	)	)	PUNCT
ejpam-6256	88	1	=	=	SYM
ejpam-6256	88	2	d({a	d({a	PROPN
ejpam-6256	88	3	,	,	PUNCT
ejpam-6256	88	4	a1	a1	PROPN
ejpam-6256	88	5	...	...	PUNCT
ejpam-6256	88	6	,	,	PUNCT
ejpam-6256	88	7	ak	ak	PROPN
ejpam-6256	88	8	}	}	PUNCT
ejpam-6256	88	9	)	)	PUNCT
ejpam-6256	88	10	.	.	PUNCT
ejpam-6256	89	1	(	(	PUNCT
ejpam-6256	89	2	ii)if	ii)if	AUX
ejpam-6256	89	3	xni	xni	PROPN
ejpam-6256	89	4	converges	converge	VERB
ejpam-6256	89	5	to	to	PART
ejpam-6256	89	6	ai	ai	VERB
ejpam-6256	89	7	,	,	PUNCT
ejpam-6256	89	8	∀i	∀i	NOUN
ejpam-6256	89	9	=	=	SYM
ejpam-6256	89	10	1	1	NUM
ejpam-6256	89	11	,	,	PUNCT
ejpam-6256	89	12	...	...	PUNCT
ejpam-6256	89	13	,	,	PUNCT
ejpam-6256	89	14	k	k	NOUN
ejpam-6256	89	15	,	,	PUNCT
ejpam-6256	89	16	then	then	ADV
ejpam-6256	89	17	lim	lim	PROPN
ejpam-6256	89	18	ni−→∞	ni−→∞	VERB
ejpam-6256	89	19	d({xn1	d({xn1	PROPN
ejpam-6256	89	20	,	,	PUNCT
ejpam-6256	89	21	...	...	PUNCT
ejpam-6256	89	22	,	,	PUNCT
ejpam-6256	89	23	xnk	xnk	PROPN
ejpam-6256	89	24	}	}	PUNCT
ejpam-6256	89	25	)	)	PUNCT
ejpam-6256	89	26	=	=	SYM
ejpam-6256	90	1	d({a1	d({a1	ADJ
ejpam-6256	90	2	...	...	PUNCT
ejpam-6256	90	3	,	,	PUNCT
ejpam-6256	90	4	ak	ak	PROPN
ejpam-6256	90	5	}	}	PUNCT
ejpam-6256	90	6	)	)	PUNCT
ejpam-6256	90	7	.	.	PUNCT
ejpam-6256	91	1	the	the	DET
ejpam-6256	91	2	cauchy	cauchy	PROPN
ejpam-6256	91	3	convergence	convergence	NOUN
ejpam-6256	91	4	and	and	CCONJ
ejpam-6256	91	5	completeness	completeness	NOUN
ejpam-6256	91	6	concepts	concept	NOUN
ejpam-6256	91	7	are	be	AUX
ejpam-6256	91	8	fundamental	fundamental	ADJ
ejpam-6256	91	9	tools	tool	NOUN
ejpam-6256	91	10	in	in	ADP
ejpam-6256	91	11	any	any	DET
ejpam-6256	91	12	type	type	NOUN
ejpam-6256	91	13	of	of	ADP
ejpam-6256	91	14	metric	metric	ADJ
ejpam-6256	91	15	space	space	NOUN
ejpam-6256	91	16	.	.	PUNCT
ejpam-6256	92	1	the	the	DET
ejpam-6256	92	2	next	next	ADJ
ejpam-6256	92	3	definition	definition	NOUN
ejpam-6256	92	4	provides	provide	VERB
ejpam-6256	92	5	a	a	DET
ejpam-6256	92	6	generalization	generalization	NOUN
ejpam-6256	92	7	of	of	ADP
ejpam-6256	92	8	the	the	DET
ejpam-6256	92	9	notion	notion	NOUN
ejpam-6256	92	10	of	of	ADP
ejpam-6256	92	11	cauchy	cauchy	ADJ
ejpam-6256	92	12	sequences	sequence	NOUN
ejpam-6256	92	13	that	that	PRON
ejpam-6256	92	14	aligns	align	VERB
ejpam-6256	92	15	with	with	ADP
ejpam-6256	92	16	the	the	DET
ejpam-6256	92	17	concept	concept	NOUN
ejpam-6256	92	18	of	of	ADP
ejpam-6256	92	19	convergence	convergence	NOUN
ejpam-6256	92	20	in	in	ADP
ejpam-6256	92	21	the	the	DET
ejpam-6256	92	22	generalized	generalized	ADJ
ejpam-6256	92	23	mp	mp	NOUN
ejpam-6256	92	24	-	-	PUNCT
ejpam-6256	92	25	metric	metric	ADJ
ejpam-6256	92	26	spaces	space	NOUN
ejpam-6256	92	27	.	.	PUNCT
ejpam-6256	93	1	definition	definition	NOUN
ejpam-6256	93	2	4	4	NUM
ejpam-6256	93	3	(	(	PUNCT
ejpam-6256	93	4	[	[	X
ejpam-6256	93	5	21	21	NUM
ejpam-6256	93	6	]	]	PUNCT
ejpam-6256	93	7	)	)	PUNCT
ejpam-6256	93	8	.	.	PUNCT
ejpam-6256	94	1	assume	assume	VERB
ejpam-6256	94	2	that	that	SCONJ
ejpam-6256	94	3	(	(	PUNCT
ejpam-6256	94	4	x	x	X
ejpam-6256	94	5	,	,	PUNCT
ejpam-6256	94	6	d	d	NOUN
ejpam-6256	94	7	)	)	PUNCT
ejpam-6256	94	8	is	be	AUX
ejpam-6256	94	9	an	an	DET
ejpam-6256	94	10	mp	mp	NOUN
ejpam-6256	94	11	-	-	PUNCT
ejpam-6256	94	12	metric	metric	ADJ
ejpam-6256	94	13	space	space	NOUN
ejpam-6256	94	14	.	.	PUNCT
ejpam-6256	95	1	then	then	ADV
ejpam-6256	95	2	,	,	PUNCT
ejpam-6256	95	3	xn	xn	PROPN
ejpam-6256	95	4	is	be	AUX
ejpam-6256	95	5	called	call	VERB
ejpam-6256	95	6	rcauchy	rcauchy	NOUN
ejpam-6256	95	7	for	for	ADP
ejpam-6256	95	8	some	some	DET
ejpam-6256	95	9	r	r	NOUN
ejpam-6256	95	10	≥	≥	NOUN
ejpam-6256	95	11	0	0	NUM
ejpam-6256	95	12	if	if	SCONJ
ejpam-6256	95	13	there	there	PRON
ejpam-6256	95	14	exists	exist	VERB
ejpam-6256	95	15	n	n	PRON
ejpam-6256	95	16	∈	∈	PROPN
ejpam-6256	95	17	n	n	PRON
ejpam-6256	95	18	such	such	ADJ
ejpam-6256	95	19	that	that	SCONJ
ejpam-6256	95	20	lim	lim	PROPN
ejpam-6256	95	21	m−→∞	m−→∞	PROPN
ejpam-6256	95	22	sup	sup	PROPN
ejpam-6256	95	23	ni	ni	PROPN
ejpam-6256	95	24	>	>	PROPN
ejpam-6256	95	25	m	m	PROPN
ejpam-6256	95	26	d(xn1	d(xn1	PROPN
ejpam-6256	95	27	,	,	PUNCT
ejpam-6256	95	28	xn2	xn2	PROPN
ejpam-6256	95	29	,	,	PUNCT
ejpam-6256	95	30	...	...	PUNCT
ejpam-6256	95	31	,	,	PUNCT
ejpam-6256	95	32	xnk	xnk	PROPN
ejpam-6256	95	33	)	)	PUNCT
ejpam-6256	96	1	=	=	SYM
ejpam-6256	96	2	r	r	NOUN
ejpam-6256	96	3	for	for	ADP
ejpam-6256	96	4	all	all	DET
ejpam-6256	96	5	k	k	PROPN
ejpam-6256	96	6	≥	≥	X
ejpam-6256	96	7	n.	n.	NOUN
ejpam-6256	96	8	the	the	DET
ejpam-6256	96	9	convergence	convergence	NOUN
ejpam-6256	96	10	in	in	ADP
ejpam-6256	96	11	mp	mp	NOUN
ejpam-6256	96	12	-	-	PUNCT
ejpam-6256	96	13	metric	metric	ADJ
ejpam-6256	96	14	spaces	space	NOUN
ejpam-6256	96	15	implies	imply	VERB
ejpam-6256	96	16	some	some	DET
ejpam-6256	96	17	essential	essential	ADJ
ejpam-6256	96	18	results	result	NOUN
ejpam-6256	96	19	,	,	PUNCT
ejpam-6256	96	20	such	such	ADJ
ejpam-6256	96	21	as	as	ADP
ejpam-6256	96	22	boundedness	boundedness	NOUN
ejpam-6256	96	23	,	,	PUNCT
ejpam-6256	96	24	r	r	NOUN
ejpam-6256	96	25	-	-	PUNCT
ejpam-6256	96	26	cauchy	cauchy	ADJ
ejpam-6256	96	27	,	,	PUNCT
ejpam-6256	96	28	and	and	CCONJ
ejpam-6256	96	29	other	other	ADJ
ejpam-6256	96	30	findings	finding	NOUN
ejpam-6256	96	31	that	that	PRON
ejpam-6256	96	32	are	be	AUX
ejpam-6256	96	33	analogous	analogous	ADJ
ejpam-6256	96	34	to	to	ADP
ejpam-6256	96	35	those	those	PRON
ejpam-6256	96	36	in	in	ADP
ejpam-6256	96	37	standard	standard	ADJ
ejpam-6256	96	38	metric	metric	ADJ
ejpam-6256	96	39	theory	theory	NOUN
ejpam-6256	96	40	.	.	PUNCT
ejpam-6256	97	1	theorem	theorem	ADJ
ejpam-6256	97	2	1	1	NUM
ejpam-6256	97	3	(	(	PUNCT
ejpam-6256	97	4	[	[	X
ejpam-6256	97	5	21	21	NUM
ejpam-6256	97	6	]	]	PUNCT
ejpam-6256	97	7	)	)	PUNCT
ejpam-6256	97	8	.	.	PUNCT
ejpam-6256	98	1	let	let	VERB
ejpam-6256	98	2	(	(	PUNCT
ejpam-6256	98	3	x	x	NOUN
ejpam-6256	98	4	,	,	PUNCT
ejpam-6256	98	5	d	d	NOUN
ejpam-6256	98	6	)	)	PUNCT
ejpam-6256	98	7	be	be	AUX
ejpam-6256	98	8	an	an	DET
ejpam-6256	98	9	mp	mp	NOUN
ejpam-6256	98	10	-	-	PUNCT
ejpam-6256	98	11	metric	metric	ADJ
ejpam-6256	98	12	space	space	NOUN
ejpam-6256	98	13	.	.	PUNCT
ejpam-6256	99	1	if	if	SCONJ
ejpam-6256	99	2	limxn	limxn	ADV
ejpam-6256	99	3	=	=	SYM
ejpam-6256	99	4	d(a	d(a	PROPN
ejpam-6256	99	5	)	)	PUNCT
ejpam-6256	99	6	for	for	ADP
ejpam-6256	99	7	some	some	DET
ejpam-6256	99	8	a	a	DET
ejpam-6256	99	9	∈	∈	PROPN
ejpam-6256	99	10	p	p	NOUN
ejpam-6256	99	11	∗(x	∗(x	PROPN
ejpam-6256	99	12	)	)	PUNCT
ejpam-6256	99	13	,	,	PUNCT
ejpam-6256	99	14	then	then	ADV
ejpam-6256	99	15	(	(	PUNCT
ejpam-6256	99	16	xn	xn	X
ejpam-6256	99	17	)	)	PUNCT
ejpam-6256	99	18	is	be	AUX
ejpam-6256	99	19	r	r	NOUN
ejpam-6256	99	20	-	-	PUNCT
ejpam-6256	99	21	cauchy	cauchy	ADJ
ejpam-6256	99	22	,	,	PUNCT
ejpam-6256	99	23	and	and	CCONJ
ejpam-6256	99	24	moreover	moreover	ADV
ejpam-6256	99	25	r	r	NOUN
ejpam-6256	99	26	=	=	SYM
ejpam-6256	99	27	d(a	d(a	PROPN
ejpam-6256	99	28	)	)	PUNCT
ejpam-6256	99	29	.	.	PUNCT
ejpam-6256	100	1	proposition	proposition	NOUN
ejpam-6256	100	2	3	3	NUM
ejpam-6256	100	3	(	(	PUNCT
ejpam-6256	100	4	[	[	X
ejpam-6256	100	5	21	21	NUM
ejpam-6256	100	6	]	]	PUNCT
ejpam-6256	100	7	)	)	PUNCT
ejpam-6256	100	8	.	.	PUNCT
ejpam-6256	101	1	let	let	VERB
ejpam-6256	101	2	(	(	PUNCT
ejpam-6256	101	3	x	x	NOUN
ejpam-6256	101	4	,	,	PUNCT
ejpam-6256	101	5	d	d	NOUN
ejpam-6256	101	6	)	)	PUNCT
ejpam-6256	101	7	be	be	AUX
ejpam-6256	101	8	an	an	DET
ejpam-6256	101	9	mp	mp	NOUN
ejpam-6256	101	10	-	-	PUNCT
ejpam-6256	101	11	metric	metric	ADJ
ejpam-6256	101	12	space	space	NOUN
ejpam-6256	101	13	.	.	PUNCT
ejpam-6256	102	1	if	if	SCONJ
ejpam-6256	102	2	xn	xn	PROPN
ejpam-6256	102	3	is	be	AUX
ejpam-6256	102	4	r	r	NOUN
ejpam-6256	102	5	-	-	NOUN
ejpam-6256	102	6	cauchy	cauchy	NOUN
ejpam-6256	102	7	for	for	ADP
ejpam-6256	102	8	some	some	DET
ejpam-6256	102	9	r	r	NOUN
ejpam-6256	102	10	≥	≥	NOUN
ejpam-6256	102	11	0	0	NUM
ejpam-6256	102	12	,	,	PUNCT
ejpam-6256	102	13	then	then	ADV
ejpam-6256	102	14	xn	xn	PROPN
ejpam-6256	102	15	is	be	AUX
ejpam-6256	102	16	bounded	bound	VERB
ejpam-6256	102	17	.	.	PUNCT
ejpam-6256	103	1	in	in	ADP
ejpam-6256	103	2	an	an	DET
ejpam-6256	103	3	obvious	obvious	ADJ
ejpam-6256	103	4	way	way	NOUN
ejpam-6256	103	5	,	,	PUNCT
ejpam-6256	103	6	the	the	DET
ejpam-6256	103	7	completeness	completeness	NOUN
ejpam-6256	103	8	of	of	ADP
ejpam-6256	103	9	the	the	DET
ejpam-6256	103	10	mp	mp	NOUN
ejpam-6256	103	11	-	-	PUNCT
ejpam-6256	103	12	metric	metric	ADJ
ejpam-6256	103	13	spaces	space	NOUN
ejpam-6256	103	14	is	be	AUX
ejpam-6256	103	15	defined	define	VERB
ejpam-6256	103	16	,	,	PUNCT
ejpam-6256	103	17	where	where	SCONJ
ejpam-6256	103	18	we	we	PRON
ejpam-6256	103	19	require	require	VERB
ejpam-6256	103	20	the	the	DET
ejpam-6256	103	21	equivalence	equivalence	NOUN
ejpam-6256	103	22	between	between	ADP
ejpam-6256	103	23	the	the	DET
ejpam-6256	103	24	convergence	convergence	NOUN
ejpam-6256	103	25	in	in	ADP
ejpam-6256	103	26	mp	mp	NOUN
ejpam-6256	103	27	-	-	PUNCT
ejpam-6256	103	28	metric	metric	ADJ
ejpam-6256	103	29	spaces	space	NOUN
ejpam-6256	103	30	and	and	CCONJ
ejpam-6256	103	31	the	the	DET
ejpam-6256	103	32	satisfaction	satisfaction	NOUN
ejpam-6256	103	33	of	of	ADP
ejpam-6256	103	34	the	the	DET
ejpam-6256	103	35	r	r	NOUN
ejpam-6256	103	36	-	-	PUNCT
ejpam-6256	103	37	cauchy	cauchy	ADJ
ejpam-6256	103	38	criterion	criterion	NOUN
ejpam-6256	103	39	.	.	PUNCT
ejpam-6256	104	1	y.	y.	PROPN
ejpam-6256	104	2	alzubaidi	alzubaidi	PROPN
ejpam-6256	104	3	/	/	SYM
ejpam-6256	104	4	eur	eur	PROPN
ejpam-6256	104	5	.	.	PUNCT
ejpam-6256	105	1	j.	j.	PROPN
ejpam-6256	105	2	pure	pure	PROPN
ejpam-6256	105	3	appl	appl	PROPN
ejpam-6256	105	4	.	.	PROPN
ejpam-6256	105	5	math	math	PROPN
ejpam-6256	105	6	,	,	PUNCT
ejpam-6256	105	7	18	18	NUM
ejpam-6256	105	8	(	(	PUNCT
ejpam-6256	105	9	3	3	NUM
ejpam-6256	105	10	)	)	PUNCT
ejpam-6256	105	11	(	(	PUNCT
ejpam-6256	105	12	2025	2025	NUM
ejpam-6256	105	13	)	)	PUNCT
ejpam-6256	105	14	,	,	PUNCT
ejpam-6256	105	15	6256	6256	NUM
ejpam-6256	105	16	5	5	NUM
ejpam-6256	105	17	of	of	ADP
ejpam-6256	105	18	11	11	NUM
ejpam-6256	105	19	definition	definition	NOUN
ejpam-6256	105	20	5	5	NUM
ejpam-6256	105	21	.	.	PUNCT
ejpam-6256	106	1	an	an	DET
ejpam-6256	106	2	mp	mp	NOUN
ejpam-6256	106	3	-	-	PUNCT
ejpam-6256	106	4	metric	metric	ADJ
ejpam-6256	106	5	space	space	NOUN
ejpam-6256	106	6	(	(	PUNCT
ejpam-6256	106	7	x	x	X
ejpam-6256	106	8	,	,	PUNCT
ejpam-6256	106	9	d	d	NOUN
ejpam-6256	106	10	)	)	PUNCT
ejpam-6256	106	11	is	be	AUX
ejpam-6256	106	12	said	say	VERB
ejpam-6256	106	13	to	to	PART
ejpam-6256	106	14	be	be	AUX
ejpam-6256	106	15	complete	complete	ADJ
ejpam-6256	106	16	mp	mp	NOUN
ejpam-6256	106	17	-	-	PUNCT
ejpam-6256	106	18	metric	metric	ADJ
ejpam-6256	106	19	space	space	NOUN
ejpam-6256	106	20	if	if	SCONJ
ejpam-6256	106	21	,	,	PUNCT
ejpam-6256	106	22	for	for	ADP
ejpam-6256	106	23	all	all	DET
ejpam-6256	106	24	r	r	NOUN
ejpam-6256	106	25	-	-	PUNCT
ejpam-6256	106	26	cauchy	cauchy	ADJ
ejpam-6256	106	27	sequence	sequence	NOUN
ejpam-6256	106	28	(	(	PUNCT
ejpam-6256	106	29	xn	xn	PROPN
ejpam-6256	106	30	)	)	PUNCT
ejpam-6256	106	31	in	in	ADP
ejpam-6256	106	32	(	(	PUNCT
ejpam-6256	106	33	x	x	NOUN
ejpam-6256	106	34	,	,	PUNCT
ejpam-6256	106	35	d	d	NOUN
ejpam-6256	106	36	)	)	PUNCT
ejpam-6256	106	37	with	with	ADP
ejpam-6256	106	38	r	r	NOUN
ejpam-6256	106	39	≥	≥	NOUN
ejpam-6256	106	40	0	0	NUM
ejpam-6256	106	41	,	,	PUNCT
ejpam-6256	106	42	there	there	PRON
ejpam-6256	106	43	exists	exist	VERB
ejpam-6256	106	44	a	a	DET
ejpam-6256	106	45	finite	finite	NOUN
ejpam-6256	106	46	nonempty	nonempty	NOUN
ejpam-6256	106	47	subset	subset	VERB
ejpam-6256	106	48	a	a	DET
ejpam-6256	106	49	∈	∈	PROPN
ejpam-6256	106	50	p	p	NOUN
ejpam-6256	106	51	∗(x	∗(x	PROPN
ejpam-6256	106	52	)	)	PUNCT
ejpam-6256	106	53	such	such	ADJ
ejpam-6256	106	54	that	that	SCONJ
ejpam-6256	106	55	d(a	d(a	PROPN
ejpam-6256	106	56	)	)	PUNCT
ejpam-6256	107	1	=	=	SYM
ejpam-6256	107	2	r	r	NOUN
ejpam-6256	107	3	and	and	CCONJ
ejpam-6256	107	4	the	the	DET
ejpam-6256	107	5	sequence	sequence	NOUN
ejpam-6256	107	6	(	(	PUNCT
ejpam-6256	107	7	xn	xn	X
ejpam-6256	107	8	)	)	PUNCT
ejpam-6256	107	9	is	be	AUX
ejpam-6256	107	10	d	d	NOUN
ejpam-6256	107	11	-	-	NOUN
ejpam-6256	107	12	convergent	convergent	ADJ
ejpam-6256	107	13	to	to	ADP
ejpam-6256	107	14	d(a	d(a	PROPN
ejpam-6256	107	15	)	)	PUNCT
ejpam-6256	107	16	;	;	PUNCT
ejpam-6256	107	17	that	that	PRON
ejpam-6256	107	18	is	be	AUX
ejpam-6256	107	19	,	,	PUNCT
ejpam-6256	107	20	d	d	PROPN
ejpam-6256	107	21	lim	lim	PROPN
ejpam-6256	107	22	n→∞	n→∞	X
ejpam-6256	107	23	xn	xn	PROPN
ejpam-6256	107	24	=	=	SYM
ejpam-6256	107	25	d(a	d(a	PROPN
ejpam-6256	107	26	)	)	PUNCT
ejpam-6256	107	27	.	.	PUNCT
ejpam-6256	108	1	the	the	DET
ejpam-6256	108	2	following	follow	VERB
ejpam-6256	108	3	theorem	theorem	NOUN
ejpam-6256	108	4	provides	provide	VERB
ejpam-6256	108	5	an	an	DET
ejpam-6256	108	6	example	example	NOUN
ejpam-6256	108	7	of	of	ADP
ejpam-6256	108	8	a	a	DET
ejpam-6256	108	9	complete	complete	ADJ
ejpam-6256	108	10	mp	mp	NOUN
ejpam-6256	108	11	-	-	PUNCT
ejpam-6256	108	12	metric	metric	ADJ
ejpam-6256	108	13	space	space	NOUN
ejpam-6256	108	14	.	.	PUNCT
ejpam-6256	109	1	it	it	PRON
ejpam-6256	109	2	shows	show	VERB
ejpam-6256	109	3	that	that	SCONJ
ejpam-6256	109	4	the	the	DET
ejpam-6256	109	5	euclidean	euclidean	ADJ
ejpam-6256	109	6	space	space	NOUN
ejpam-6256	109	7	is	be	AUX
ejpam-6256	109	8	complete	complete	ADJ
ejpam-6256	109	9	with	with	ADP
ejpam-6256	109	10	respect	respect	NOUN
ejpam-6256	109	11	to	to	ADP
ejpam-6256	109	12	the	the	DET
ejpam-6256	109	13	mp	mp	NOUN
ejpam-6256	109	14	-	-	PUNCT
ejpam-6256	109	15	metric	metric	NOUN
ejpam-6256	109	16	introduced	introduce	VERB
ejpam-6256	109	17	in	in	ADP
ejpam-6256	109	18	example	example	NOUN
ejpam-6256	109	19	1	1	NUM
ejpam-6256	109	20	.	.	PUNCT
ejpam-6256	110	1	this	this	DET
ejpam-6256	110	2	result	result	NOUN
ejpam-6256	110	3	is	be	AUX
ejpam-6256	110	4	valid	valid	ADJ
ejpam-6256	110	5	for	for	ADP
ejpam-6256	110	6	higher	high	ADJ
ejpam-6256	110	7	dimensions	dimension	NOUN
ejpam-6256	110	8	when	when	SCONJ
ejpam-6256	110	9	appropriate	appropriate	ADJ
ejpam-6256	110	10	mp	mp	NOUN
ejpam-6256	110	11	-	-	PUNCT
ejpam-6256	110	12	metrics	metric	NOUN
ejpam-6256	110	13	are	be	AUX
ejpam-6256	110	14	implemented	implement	VERB
ejpam-6256	110	15	in	in	ADP
ejpam-6256	110	16	rn	rn	PROPN
ejpam-6256	110	17	(	(	PUNCT
ejpam-6256	110	18	see	see	VERB
ejpam-6256	110	19	[	[	X
ejpam-6256	110	20	21	21	NUM
ejpam-6256	110	21	]	]	PUNCT
ejpam-6256	110	22	)	)	PUNCT
ejpam-6256	110	23	.	.	PUNCT
ejpam-6256	111	1	theorem	theorem	ADJ
ejpam-6256	111	2	2	2	NUM
ejpam-6256	111	3	(	(	PUNCT
ejpam-6256	111	4	[	[	X
ejpam-6256	111	5	21	21	NUM
ejpam-6256	111	6	]	]	PUNCT
ejpam-6256	111	7	)	)	PUNCT
ejpam-6256	111	8	.	.	PUNCT
ejpam-6256	112	1	let	let	AUX
ejpam-6256	112	2	(	(	PUNCT
ejpam-6256	112	3	r	r	NOUN
ejpam-6256	112	4	,	,	PUNCT
ejpam-6256	112	5	d	d	NOUN
ejpam-6256	112	6	)	)	PUNCT
ejpam-6256	112	7	be	be	AUX
ejpam-6256	112	8	the	the	DET
ejpam-6256	112	9	mp	mp	NOUN
ejpam-6256	112	10	-	-	PUNCT
ejpam-6256	112	11	metric	metric	ADJ
ejpam-6256	112	12	space	space	NOUN
ejpam-6256	112	13	introduced	introduce	VERB
ejpam-6256	112	14	in	in	ADP
ejpam-6256	112	15	example	example	NOUN
ejpam-6256	113	1	1	1	NUM
ejpam-6256	113	2	.	.	PUNCT
ejpam-6256	114	1	if	if	SCONJ
ejpam-6256	114	2	xn	xn	PROPN
ejpam-6256	114	3	is	be	AUX
ejpam-6256	114	4	r	r	NOUN
ejpam-6256	114	5	-	-	NOUN
ejpam-6256	114	6	cauchy	cauchy	NOUN
ejpam-6256	114	7	for	for	ADP
ejpam-6256	114	8	some	some	DET
ejpam-6256	114	9	r	r	NOUN
ejpam-6256	114	10	>	>	X
ejpam-6256	114	11	0	0	PUNCT
ejpam-6256	115	1	in	in	ADP
ejpam-6256	115	2	(	(	PUNCT
ejpam-6256	115	3	r	r	NOUN
ejpam-6256	115	4	,	,	PUNCT
ejpam-6256	115	5	d	d	NOUN
ejpam-6256	115	6	)	)	PUNCT
ejpam-6256	115	7	,	,	PUNCT
ejpam-6256	115	8	then	then	ADV
ejpam-6256	115	9	there	there	PRON
ejpam-6256	115	10	are	be	VERB
ejpam-6256	115	11	a	a	DET
ejpam-6256	115	12	,	,	PUNCT
ejpam-6256	115	13	b	b	X
ejpam-6256	115	14	∈	∈	NOUN
ejpam-6256	115	15	r	r	NOUN
ejpam-6256	115	16	such	such	ADJ
ejpam-6256	115	17	that	that	SCONJ
ejpam-6256	115	18	xn	xn	PROPN
ejpam-6256	115	19	is	be	AUX
ejpam-6256	115	20	d	d	NOUN
ejpam-6256	115	21	-	-	NOUN
ejpam-6256	115	22	convergent	convergent	ADJ
ejpam-6256	115	23	to	to	ADP
ejpam-6256	115	24	d({a	d({a	PROPN
ejpam-6256	115	25	,	,	PUNCT
ejpam-6256	115	26	b	b	NOUN
ejpam-6256	115	27	}	}	PUNCT
ejpam-6256	115	28	)	)	PUNCT
ejpam-6256	115	29	.	.	PUNCT
ejpam-6256	116	1	another	another	DET
ejpam-6256	116	2	notion	notion	NOUN
ejpam-6256	116	3	that	that	SCONJ
ejpam-6256	116	4	we	we	PRON
ejpam-6256	116	5	will	will	AUX
ejpam-6256	116	6	need	need	VERB
ejpam-6256	116	7	in	in	ADP
ejpam-6256	116	8	the	the	DET
ejpam-6256	116	9	next	next	ADJ
ejpam-6256	116	10	section	section	NOUN
ejpam-6256	116	11	is	be	AUX
ejpam-6256	116	12	the	the	DET
ejpam-6256	116	13	notion	notion	NOUN
ejpam-6256	116	14	of	of	ADP
ejpam-6256	116	15	the	the	DET
ejpam-6256	116	16	boundedness	boundedness	NOUN
ejpam-6256	116	17	of	of	ADP
ejpam-6256	116	18	an	an	DET
ejpam-6256	116	19	mp	mp	NOUN
ejpam-6256	116	20	-	-	PUNCT
ejpam-6256	116	21	metric	metric	ADJ
ejpam-6256	116	22	space	space	NOUN
ejpam-6256	116	23	.	.	PUNCT
ejpam-6256	117	1	therefore	therefore	ADV
ejpam-6256	117	2	,	,	PUNCT
ejpam-6256	117	3	we	we	PRON
ejpam-6256	117	4	introduce	introduce	VERB
ejpam-6256	117	5	the	the	DET
ejpam-6256	117	6	following	following	ADJ
ejpam-6256	117	7	definition	definition	NOUN
ejpam-6256	117	8	.	.	PUNCT
ejpam-6256	118	1	definition	definition	NOUN
ejpam-6256	118	2	6	6	NUM
ejpam-6256	118	3	.	.	PUNCT
ejpam-6256	119	1	an	an	DET
ejpam-6256	119	2	mp	mp	NOUN
ejpam-6256	119	3	-	-	PUNCT
ejpam-6256	119	4	metric	metric	ADJ
ejpam-6256	119	5	space	space	NOUN
ejpam-6256	119	6	(	(	PUNCT
ejpam-6256	119	7	x	x	X
ejpam-6256	119	8	,	,	PUNCT
ejpam-6256	119	9	d	d	NOUN
ejpam-6256	119	10	)	)	PUNCT
ejpam-6256	119	11	is	be	AUX
ejpam-6256	119	12	called	call	VERB
ejpam-6256	119	13	bounded	bounded	ADJ
ejpam-6256	119	14	mp	mp	PROPN
ejpam-6256	119	15	-	-	PUNCT
ejpam-6256	119	16	metric	metric	ADJ
ejpam-6256	119	17	space	space	NOUN
ejpam-6256	119	18	if	if	SCONJ
ejpam-6256	119	19	there	there	PRON
ejpam-6256	119	20	is	be	VERB
ejpam-6256	119	21	a	a	DET
ejpam-6256	119	22	positive	positive	ADJ
ejpam-6256	119	23	constant	constant	ADJ
ejpam-6256	119	24	c	c	NOUN
ejpam-6256	119	25	such	such	ADJ
ejpam-6256	119	26	that	that	SCONJ
ejpam-6256	119	27	d(a	d(a	PROPN
ejpam-6256	119	28	)	)	PUNCT
ejpam-6256	119	29	<	<	X
ejpam-6256	119	30	c	c	NOUN
ejpam-6256	119	31	for	for	ADP
ejpam-6256	119	32	all	all	DET
ejpam-6256	119	33	a	a	DET
ejpam-6256	119	34	∈	∈	PROPN
ejpam-6256	119	35	p	p	NOUN
ejpam-6256	119	36	∗(x	∗(x	PROPN
ejpam-6256	119	37	)	)	PUNCT
ejpam-6256	119	38	.	.	PUNCT
ejpam-6256	120	1	3	3	X
ejpam-6256	120	2	.	.	X
ejpam-6256	120	3	construction	construction	NOUN
ejpam-6256	120	4	and	and	CCONJ
ejpam-6256	120	5	main	main	ADJ
ejpam-6256	120	6	results	result	NOUN
ejpam-6256	120	7	we	we	PRON
ejpam-6256	120	8	start	start	VERB
ejpam-6256	120	9	this	this	DET
ejpam-6256	120	10	section	section	NOUN
ejpam-6256	120	11	by	by	ADP
ejpam-6256	120	12	introducing	introduce	VERB
ejpam-6256	120	13	some	some	DET
ejpam-6256	120	14	constructive	constructive	ADJ
ejpam-6256	120	15	definitions	definition	NOUN
ejpam-6256	120	16	and	and	CCONJ
ejpam-6256	120	17	propositions	proposition	NOUN
ejpam-6256	120	18	that	that	PRON
ejpam-6256	120	19	will	will	AUX
ejpam-6256	120	20	help	help	VERB
ejpam-6256	120	21	simplify	simplify	VERB
ejpam-6256	120	22	the	the	DET
ejpam-6256	120	23	presentation	presentation	NOUN
ejpam-6256	120	24	of	of	ADP
ejpam-6256	120	25	the	the	DET
ejpam-6256	120	26	main	main	ADJ
ejpam-6256	120	27	result	result	NOUN
ejpam-6256	120	28	.	.	PUNCT
ejpam-6256	121	1	for	for	ADP
ejpam-6256	121	2	a	a	DET
ejpam-6256	121	3	mapping	mapping	NOUN
ejpam-6256	121	4	t	t	NOUN
ejpam-6256	121	5	,	,	PUNCT
ejpam-6256	121	6	we	we	PRON
ejpam-6256	121	7	use	use	VERB
ejpam-6256	121	8	tn	tn	NOUN
ejpam-6256	121	9	to	to	PART
ejpam-6256	121	10	denote	denote	VERB
ejpam-6256	121	11	applying	apply	VERB
ejpam-6256	121	12	the	the	DET
ejpam-6256	121	13	mapping	mapping	NOUN
ejpam-6256	121	14	t	t	PROPN
ejpam-6256	121	15	n	n	PROPN
ejpam-6256	121	16	times	time	NOUN
ejpam-6256	121	17	.	.	PUNCT
ejpam-6256	122	1	also	also	ADV
ejpam-6256	122	2	,	,	PUNCT
ejpam-6256	122	3	we	we	PRON
ejpam-6256	122	4	note	note	VERB
ejpam-6256	122	5	that	that	SCONJ
ejpam-6256	122	6	for	for	ADP
ejpam-6256	122	7	any	any	DET
ejpam-6256	122	8	a	a	DET
ejpam-6256	122	9	∈	∈	PROPN
ejpam-6256	122	10	p	p	NOUN
ejpam-6256	122	11	∗(x	∗(x	PROPN
ejpam-6256	122	12	)	)	PUNCT
ejpam-6256	122	13	,	,	PUNCT
ejpam-6256	122	14	we	we	PRON
ejpam-6256	122	15	can	can	AUX
ejpam-6256	122	16	always	always	ADV
ejpam-6256	122	17	enumerate	enumerate	VERB
ejpam-6256	122	18	its	its	PRON
ejpam-6256	122	19	elements	element	NOUN
ejpam-6256	122	20	and	and	CCONJ
ejpam-6256	122	21	write	write	VERB
ejpam-6256	122	22	a	a	DET
ejpam-6256	122	23	=	=	X
ejpam-6256	122	24	{	{	PUNCT
ejpam-6256	122	25	a1	a1	PROPN
ejpam-6256	122	26	,	,	PUNCT
ejpam-6256	122	27	a2	a2	PROPN
ejpam-6256	122	28	,	,	PUNCT
ejpam-6256	122	29	...	...	PUNCT
ejpam-6256	122	30	,	,	PUNCT
ejpam-6256	122	31	ak	ak	PROPN
ejpam-6256	122	32	}	}	PUNCT
ejpam-6256	122	33	.	.	PUNCT
ejpam-6256	123	1	definition	definition	NOUN
ejpam-6256	123	2	7	7	NUM
ejpam-6256	123	3	.	.	PUNCT
ejpam-6256	124	1	let	let	AUX
ejpam-6256	124	2	(	(	PUNCT
ejpam-6256	124	3	x	x	NOUN
ejpam-6256	124	4	,	,	PUNCT
ejpam-6256	124	5	d	d	NOUN
ejpam-6256	124	6	)	)	PUNCT
ejpam-6256	124	7	be	be	AUX
ejpam-6256	124	8	an	an	DET
ejpam-6256	124	9	mp	mp	NOUN
ejpam-6256	124	10	-	-	PUNCT
ejpam-6256	124	11	metric	metric	ADJ
ejpam-6256	124	12	space	space	NOUN
ejpam-6256	124	13	and	and	CCONJ
ejpam-6256	124	14	t	t	PROPN
ejpam-6256	124	15	is	be	AUX
ejpam-6256	124	16	a	a	DET
ejpam-6256	124	17	self	self	NOUN
ejpam-6256	124	18	mapping	mapping	NOUN
ejpam-6256	124	19	on	on	ADP
ejpam-6256	124	20	x	x	NOUN
ejpam-6256	124	21	,	,	PUNCT
ejpam-6256	124	22	and	and	CCONJ
ejpam-6256	124	23	a	a	DET
ejpam-6256	124	24	=	=	X
ejpam-6256	124	25	{	{	PUNCT
ejpam-6256	124	26	a1	a1	PROPN
ejpam-6256	124	27	,	,	PUNCT
ejpam-6256	124	28	...	...	PUNCT
ejpam-6256	124	29	,	,	PUNCT
ejpam-6256	124	30	ak	ak	PROPN
ejpam-6256	124	31	}	}	PUNCT
ejpam-6256	124	32	∈	∈	PROPN
ejpam-6256	124	33	p	p	NOUN
ejpam-6256	124	34	∗(x	∗(x	PROPN
ejpam-6256	124	35	)	)	PUNCT
ejpam-6256	124	36	.	.	PUNCT
ejpam-6256	125	1	then	then	ADV
ejpam-6256	125	2	for	for	ADP
ejpam-6256	125	3	all	all	DET
ejpam-6256	125	4	j	j	PROPN
ejpam-6256	125	5	∈	∈	PROPN
ejpam-6256	125	6	n	n	PART
ejpam-6256	125	7	∪	∪	X
ejpam-6256	125	8	{	{	PUNCT
ejpam-6256	125	9	0	0	NUM
ejpam-6256	125	10	}	}	PUNCT
ejpam-6256	125	11	,	,	PUNCT
ejpam-6256	125	12	we	we	PRON
ejpam-6256	125	13	define	define	VERB
ejpam-6256	125	14	the	the	DET
ejpam-6256	125	15	set	set	NOUN
ejpam-6256	125	16	sj	sj	PROPN
ejpam-6256	125	17	t	t	PROPN
ejpam-6256	125	18	(	(	PUNCT
ejpam-6256	125	19	a	a	NOUN
ejpam-6256	125	20	)	)	PUNCT
ejpam-6256	125	21	=	=	SYM
ejpam-6256	125	22	∞⋃	∞⋃	PROPN
ejpam-6256	125	23	n	n	CCONJ
ejpam-6256	125	24	=	=	SYM
ejpam-6256	125	25	j	j	NOUN
ejpam-6256	125	26	tn(a	tn(a	NOUN
ejpam-6256	125	27	)	)	PUNCT
ejpam-6256	125	28	,	,	PUNCT
ejpam-6256	125	29	(	(	PUNCT
ejpam-6256	125	30	1	1	X
ejpam-6256	125	31	)	)	PUNCT
ejpam-6256	125	32	and	and	CCONJ
ejpam-6256	125	33	for	for	ADP
ejpam-6256	125	34	j	j	PROPN
ejpam-6256	125	35	>	>	PUNCT
ejpam-6256	125	36	0	0	NUM
ejpam-6256	125	37	and	and	CCONJ
ejpam-6256	125	38	1	1	NUM
ejpam-6256	125	39	≤	≤	NUM
ejpam-6256	125	40	l	l	NOUN
ejpam-6256	125	41	≤	≤	NOUN
ejpam-6256	126	1	k	k	X
ejpam-6256	126	2	,	,	PUNCT
ejpam-6256	126	3	we	we	PRON
ejpam-6256	126	4	define	define	VERB
ejpam-6256	126	5	the	the	DET
ejpam-6256	126	6	set	set	NOUN
ejpam-6256	126	7	sj	sj	NOUN
ejpam-6256	126	8	,	,	PUNCT
ejpam-6256	126	9	l	l	PROPN
ejpam-6256	126	10	t	t	PROPN
ejpam-6256	126	11	(	(	PUNCT
ejpam-6256	126	12	a	a	NOUN
ejpam-6256	126	13	)	)	PUNCT
ejpam-6256	126	14	=	=	SYM
ejpam-6256	126	15	(	(	PUNCT
ejpam-6256	126	16	∞⋃	∞⋃	PROPN
ejpam-6256	126	17	n	n	PROPN
ejpam-6256	126	18	=	=	PROPN
ejpam-6256	126	19	j	j	NOUN
ejpam-6256	126	20	tn(a	tn(a	NUM
ejpam-6256	126	21	)	)	PUNCT
ejpam-6256	126	22	)	)	PUNCT
ejpam-6256	126	23	∪	∪	ADP
ejpam-6256	126	24	t	t	PROPN
ejpam-6256	126	25	j−1({al	j−1({al	PROPN
ejpam-6256	126	26	,	,	PUNCT
ejpam-6256	126	27	...	...	PUNCT
ejpam-6256	126	28	,	,	PUNCT
ejpam-6256	126	29	ak	ak	PROPN
ejpam-6256	126	30	}	}	PUNCT
ejpam-6256	126	31	)	)	PUNCT
ejpam-6256	126	32	(	(	PUNCT
ejpam-6256	126	33	2	2	X
ejpam-6256	126	34	)	)	PUNCT
ejpam-6256	126	35	the	the	DET
ejpam-6256	126	36	sets	set	NOUN
ejpam-6256	126	37	introduced	introduce	VERB
ejpam-6256	126	38	above	above	ADV
ejpam-6256	126	39	have	have	VERB
ejpam-6256	126	40	some	some	DET
ejpam-6256	126	41	properties	property	NOUN
ejpam-6256	126	42	that	that	PRON
ejpam-6256	126	43	are	be	AUX
ejpam-6256	126	44	directly	directly	ADV
ejpam-6256	126	45	derived	derive	VERB
ejpam-6256	126	46	from	from	ADP
ejpam-6256	126	47	their	their	PRON
ejpam-6256	126	48	definitions	definition	NOUN
ejpam-6256	126	49	.	.	PUNCT
ejpam-6256	127	1	the	the	DET
ejpam-6256	127	2	following	follow	VERB
ejpam-6256	127	3	proposition	proposition	NOUN
ejpam-6256	127	4	discusses	discuss	VERB
ejpam-6256	127	5	some	some	PRON
ejpam-6256	127	6	of	of	ADP
ejpam-6256	127	7	these	these	DET
ejpam-6256	127	8	properties	property	NOUN
ejpam-6256	127	9	,	,	PUNCT
ejpam-6256	127	10	including	include	VERB
ejpam-6256	127	11	the	the	DET
ejpam-6256	127	12	effect	effect	NOUN
ejpam-6256	127	13	of	of	ADP
ejpam-6256	127	14	the	the	DET
ejpam-6256	127	15	mapping	mapping	NOUN
ejpam-6256	127	16	t.	t.	NOUN
ejpam-6256	127	17	proposition	proposition	NOUN
ejpam-6256	127	18	4	4	X
ejpam-6256	127	19	.	.	PUNCT
ejpam-6256	128	1	let	let	AUX
ejpam-6256	128	2	(	(	PUNCT
ejpam-6256	128	3	x	x	NOUN
ejpam-6256	128	4	,	,	PUNCT
ejpam-6256	128	5	d	d	NOUN
ejpam-6256	128	6	)	)	PUNCT
ejpam-6256	128	7	be	be	AUX
ejpam-6256	128	8	an	an	DET
ejpam-6256	128	9	mp	mp	NOUN
ejpam-6256	128	10	-	-	PUNCT
ejpam-6256	128	11	metric	metric	ADJ
ejpam-6256	128	12	space	space	NOUN
ejpam-6256	128	13	,	,	PUNCT
ejpam-6256	128	14	and	and	CCONJ
ejpam-6256	128	15	let	let	VERB
ejpam-6256	128	16	t	t	PROPN
ejpam-6256	128	17	be	be	AUX
ejpam-6256	128	18	a	a	DET
ejpam-6256	128	19	self	self	NOUN
ejpam-6256	128	20	mapping	mapping	NOUN
ejpam-6256	128	21	on	on	ADP
ejpam-6256	128	22	x	x	NOUN
ejpam-6256	128	23	,	,	PUNCT
ejpam-6256	128	24	and	and	CCONJ
ejpam-6256	128	25	a	a	DET
ejpam-6256	128	26	=	=	X
ejpam-6256	128	27	{	{	PUNCT
ejpam-6256	128	28	a1	a1	PROPN
ejpam-6256	128	29	,	,	PUNCT
ejpam-6256	128	30	...	...	PUNCT
ejpam-6256	128	31	,	,	PUNCT
ejpam-6256	128	32	ak	ak	PROPN
ejpam-6256	128	33	}	}	PUNCT
ejpam-6256	128	34	∈	∈	PROPN
ejpam-6256	128	35	p	p	NOUN
ejpam-6256	128	36	∗(x	∗(x	PROPN
ejpam-6256	128	37	)	)	PUNCT
ejpam-6256	128	38	.	.	PUNCT
ejpam-6256	129	1	then	then	ADV
ejpam-6256	129	2	(	(	PUNCT
ejpam-6256	129	3	i	i	NOUN
ejpam-6256	129	4	)	)	PUNCT
ejpam-6256	129	5	t	t	PROPN
ejpam-6256	129	6	(	(	PUNCT
ejpam-6256	129	7	sj	sj	PROPN
ejpam-6256	129	8	t	t	PROPN
ejpam-6256	129	9	(	(	PUNCT
ejpam-6256	129	10	a	a	NOUN
ejpam-6256	129	11	)	)	PUNCT
ejpam-6256	129	12	)	)	PUNCT
ejpam-6256	130	1	=	=	SYM
ejpam-6256	130	2	sj+1	sj+1	PRON
ejpam-6256	131	1	t	t	NOUN
ejpam-6256	131	2	(	(	PUNCT
ejpam-6256	131	3	a	a	NOUN
ejpam-6256	131	4	)	)	PUNCT
ejpam-6256	131	5	for	for	ADP
ejpam-6256	131	6	all	all	DET
ejpam-6256	131	7	j	j	PROPN
ejpam-6256	131	8	∈	∈	PROPN
ejpam-6256	131	9	n	n	PART
ejpam-6256	131	10	∪	∪	X
ejpam-6256	131	11	{	{	PUNCT
ejpam-6256	131	12	0	0	NUM
ejpam-6256	131	13	}	}	PUNCT
ejpam-6256	131	14	,	,	PUNCT
ejpam-6256	131	15	y.	y.	PROPN
ejpam-6256	131	16	alzubaidi	alzubaidi	PROPN
ejpam-6256	131	17	/	/	SYM
ejpam-6256	131	18	eur	eur	PROPN
ejpam-6256	131	19	.	.	PUNCT
ejpam-6256	132	1	j.	j.	PROPN
ejpam-6256	132	2	pure	pure	PROPN
ejpam-6256	132	3	appl	appl	PROPN
ejpam-6256	132	4	.	.	PROPN
ejpam-6256	132	5	math	math	PROPN
ejpam-6256	132	6	,	,	PUNCT
ejpam-6256	132	7	18	18	NUM
ejpam-6256	132	8	(	(	PUNCT
ejpam-6256	132	9	3	3	NUM
ejpam-6256	132	10	)	)	PUNCT
ejpam-6256	132	11	(	(	PUNCT
ejpam-6256	132	12	2025	2025	NUM
ejpam-6256	132	13	)	)	PUNCT
ejpam-6256	132	14	,	,	PUNCT
ejpam-6256	132	15	6256	6256	NUM
ejpam-6256	132	16	6	6	NUM
ejpam-6256	132	17	of	of	ADP
ejpam-6256	132	18	11	11	NUM
ejpam-6256	132	19	(	(	PUNCT
ejpam-6256	132	20	ii	ii	NOUN
ejpam-6256	132	21	)	)	PUNCT
ejpam-6256	132	22	t	t	PROPN
ejpam-6256	132	23	(	(	PUNCT
ejpam-6256	132	24	sj	sj	INTJ
ejpam-6256	132	25	,	,	PUNCT
ejpam-6256	132	26	l	l	PROPN
ejpam-6256	132	27	t	t	PROPN
ejpam-6256	132	28	(	(	PUNCT
ejpam-6256	132	29	a	a	NOUN
ejpam-6256	132	30	)	)	PUNCT
ejpam-6256	132	31	)	)	PUNCT
ejpam-6256	133	1	=	=	SYM
ejpam-6256	133	2	sj+1,l	sj+1,l	PROPN
ejpam-6256	133	3	t	t	PROPN
ejpam-6256	133	4	(	(	PUNCT
ejpam-6256	133	5	a	a	NOUN
ejpam-6256	133	6	)	)	PUNCT
ejpam-6256	133	7	for	for	ADP
ejpam-6256	133	8	all	all	DET
ejpam-6256	133	9	j	j	PROPN
ejpam-6256	133	10	∈	∈	PROPN
ejpam-6256	133	11	n	n	PRON
ejpam-6256	133	12	and	and	CCONJ
ejpam-6256	133	13	1	1	NUM
ejpam-6256	133	14	≤	≤	NUM
ejpam-6256	133	15	l	l	NOUN
ejpam-6256	133	16	≤	≤	NOUN
ejpam-6256	134	1	k	k	X
ejpam-6256	134	2	,	,	PUNCT
ejpam-6256	134	3	(	(	PUNCT
ejpam-6256	134	4	iii	iii	X
ejpam-6256	134	5	)	)	PUNCT
ejpam-6256	134	6	sj	sj	PROPN
ejpam-6256	134	7	t	t	PROPN
ejpam-6256	134	8	(	(	PUNCT
ejpam-6256	134	9	a	a	NOUN
ejpam-6256	134	10	)	)	PUNCT
ejpam-6256	135	1	=	=	SYM
ejpam-6256	135	2	sj+1,1	sj+1,1	PROPN
ejpam-6256	135	3	t	t	PROPN
ejpam-6256	135	4	(	(	PUNCT
ejpam-6256	135	5	a	a	NOUN
ejpam-6256	135	6	)	)	PUNCT
ejpam-6256	135	7	for	for	ADP
ejpam-6256	135	8	all	all	DET
ejpam-6256	135	9	j	j	PROPN
ejpam-6256	135	10	∈	∈	PROPN
ejpam-6256	135	11	n	n	CCONJ
ejpam-6256	135	12	,	,	PUNCT
ejpam-6256	135	13	(	(	PUNCT
ejpam-6256	135	14	iv	iv	X
ejpam-6256	135	15	)	)	PUNCT
ejpam-6256	135	16	sj1	sj1	NOUN
ejpam-6256	135	17	t	t	PROPN
ejpam-6256	135	18	(	(	PUNCT
ejpam-6256	135	19	a	a	NOUN
ejpam-6256	135	20	)	)	PUNCT
ejpam-6256	136	1	⊆	⊆	NUM
ejpam-6256	136	2	sj2	sj2	NOUN
ejpam-6256	136	3	t	t	PROPN
ejpam-6256	136	4	(	(	PUNCT
ejpam-6256	136	5	a	a	NOUN
ejpam-6256	136	6	)	)	PUNCT
ejpam-6256	136	7	for	for	ADP
ejpam-6256	136	8	all	all	DET
ejpam-6256	136	9	j1	j1	PROPN
ejpam-6256	136	10	,	,	PUNCT
ejpam-6256	136	11	j2	j2	PROPN
ejpam-6256	136	12	∈	∈	PROPN
ejpam-6256	136	13	n	n	PART
ejpam-6256	136	14	∪	∪	X
ejpam-6256	136	15	{	{	PUNCT
ejpam-6256	136	16	0	0	NUM
ejpam-6256	136	17	}	}	PUNCT
ejpam-6256	136	18	and	and	CCONJ
ejpam-6256	136	19	j2	j2	PROPN
ejpam-6256	136	20	≤	≤	PROPN
ejpam-6256	136	21	j1	j1	PROPN
ejpam-6256	136	22	(	(	PUNCT
ejpam-6256	136	23	v	v	NOUN
ejpam-6256	136	24	)	)	PUNCT
ejpam-6256	136	25	sj	sj	PROPN
ejpam-6256	136	26	,	,	PUNCT
ejpam-6256	136	27	l1	l1	PROPN
ejpam-6256	136	28	t	t	PROPN
ejpam-6256	136	29	(	(	PUNCT
ejpam-6256	136	30	a	a	NOUN
ejpam-6256	136	31	)	)	PUNCT
ejpam-6256	136	32	⊆	⊆	NUM
ejpam-6256	136	33	sj	sj	NOUN
ejpam-6256	136	34	,	,	PUNCT
ejpam-6256	136	35	l2	l2	PROPN
ejpam-6256	136	36	t	t	PROPN
ejpam-6256	136	37	(	(	PUNCT
ejpam-6256	136	38	a	a	NOUN
ejpam-6256	136	39	)	)	PUNCT
ejpam-6256	136	40	for	for	ADP
ejpam-6256	136	41	all	all	DET
ejpam-6256	136	42	j	j	PROPN
ejpam-6256	136	43	∈	∈	PROPN
ejpam-6256	136	44	n	n	CCONJ
ejpam-6256	136	45	,	,	PUNCT
ejpam-6256	136	46	1	1	NUM
ejpam-6256	136	47	≤	≤	NUM
ejpam-6256	136	48	l1	l1	PROPN
ejpam-6256	136	49	,	,	PUNCT
ejpam-6256	136	50	l2	l2	VERB
ejpam-6256	136	51	≤	≤	ADJ
ejpam-6256	136	52	k	k	PROPN
ejpam-6256	136	53	and	and	CCONJ
ejpam-6256	136	54	j2	j2	PROPN
ejpam-6256	136	55	≤	≤	PROPN
ejpam-6256	136	56	j1	j1	PROPN
ejpam-6256	136	57	(	(	PUNCT
ejpam-6256	136	58	vi	vi	NOUN
ejpam-6256	136	59	)	)	PUNCT
ejpam-6256	136	60	sj1,l1	sj1,l1	NOUN
ejpam-6256	136	61	t	t	PROPN
ejpam-6256	136	62	(	(	PUNCT
ejpam-6256	136	63	a	a	NOUN
ejpam-6256	136	64	)	)	PUNCT
ejpam-6256	136	65	⊆	⊆	NUM
ejpam-6256	136	66	sj2,l2	sj2,l2	NOUN
ejpam-6256	136	67	t	t	NOUN
ejpam-6256	136	68	(	(	PUNCT
ejpam-6256	136	69	a	a	NOUN
ejpam-6256	136	70	)	)	PUNCT
ejpam-6256	136	71	for	for	ADP
ejpam-6256	136	72	all	all	DET
ejpam-6256	136	73	j1	j1	PROPN
ejpam-6256	136	74	,	,	PUNCT
ejpam-6256	136	75	j2	j2	PROPN
ejpam-6256	136	76	∈	∈	PROPN
ejpam-6256	136	77	n	n	CCONJ
ejpam-6256	136	78	,	,	PUNCT
ejpam-6256	136	79	1	1	NUM
ejpam-6256	136	80	≤	≤	NUM
ejpam-6256	136	81	l1	l1	PROPN
ejpam-6256	136	82	,	,	PUNCT
ejpam-6256	136	83	l2	l2	VERB
ejpam-6256	136	84	≤	≤	ADJ
ejpam-6256	136	85	k	k	PROPN
ejpam-6256	136	86	,	,	PUNCT
ejpam-6256	136	87	j2	j2	PROPN
ejpam-6256	136	88	≤	≤	PROPN
ejpam-6256	136	89	j1	j1	PROPN
ejpam-6256	136	90	and	and	CCONJ
ejpam-6256	136	91	j2	j2	PROPN
ejpam-6256	136	92	≤	≤	PROPN
ejpam-6256	136	93	j1	j1	PROPN
ejpam-6256	136	94	proof	proof	NOUN
ejpam-6256	136	95	.	.	PUNCT
ejpam-6256	137	1	the	the	DET
ejpam-6256	137	2	results	result	NOUN
ejpam-6256	137	3	of	of	ADP
ejpam-6256	137	4	this	this	DET
ejpam-6256	137	5	proposition	proposition	NOUN
ejpam-6256	137	6	follow	follow	VERB
ejpam-6256	137	7	directly	directly	ADV
ejpam-6256	137	8	from	from	ADP
ejpam-6256	137	9	straightforward	straightforward	ADJ
ejpam-6256	137	10	calculations	calculation	NOUN
ejpam-6256	137	11	using	use	VERB
ejpam-6256	137	12	definition	definition	NOUN
ejpam-6256	137	13	7	7	NUM
ejpam-6256	137	14	.	.	PUNCT
ejpam-6256	138	1	however	however	ADV
ejpam-6256	138	2	,	,	PUNCT
ejpam-6256	138	3	for	for	ADP
ejpam-6256	138	4	the	the	DET
ejpam-6256	138	5	convenience	convenience	NOUN
ejpam-6256	138	6	of	of	ADP
ejpam-6256	138	7	the	the	DET
ejpam-6256	138	8	reader	reader	NOUN
ejpam-6256	138	9	,	,	PUNCT
ejpam-6256	138	10	we	we	PRON
ejpam-6256	138	11	provide	provide	VERB
ejpam-6256	138	12	a	a	DET
ejpam-6256	138	13	proof	proof	NOUN
ejpam-6256	138	14	for	for	ADP
ejpam-6256	138	15	part	part	NOUN
ejpam-6256	138	16	(	(	PUNCT
ejpam-6256	138	17	iii	iii	NOUN
ejpam-6256	138	18	)	)	PUNCT
ejpam-6256	138	19	;	;	PUNCT
ejpam-6256	138	20	the	the	DET
ejpam-6256	138	21	remaining	remain	VERB
ejpam-6256	138	22	parts	part	NOUN
ejpam-6256	138	23	can	can	AUX
ejpam-6256	138	24	be	be	AUX
ejpam-6256	138	25	established	establish	VERB
ejpam-6256	138	26	in	in	ADP
ejpam-6256	138	27	a	a	DET
ejpam-6256	138	28	similar	similar	ADJ
ejpam-6256	138	29	manner	manner	NOUN
ejpam-6256	138	30	.	.	PUNCT
ejpam-6256	139	1	sj	sj	PROPN
ejpam-6256	139	2	t	t	PROPN
ejpam-6256	139	3	(	(	PUNCT
ejpam-6256	139	4	a	a	NOUN
ejpam-6256	139	5	)	)	PUNCT
ejpam-6256	139	6	=	=	SYM
ejpam-6256	139	7	∞⋃	∞⋃	PROPN
ejpam-6256	139	8	n	n	CCONJ
ejpam-6256	139	9	=	=	SYM
ejpam-6256	139	10	j	j	NOUN
ejpam-6256	139	11	tn(a	tn(a	NOUN
ejpam-6256	139	12	)	)	PUNCT
ejpam-6256	139	13	=	=	SYM
ejpam-6256	139	14	(	(	PUNCT
ejpam-6256	139	15	∞⋃	∞⋃	PROPN
ejpam-6256	139	16	n	n	CCONJ
ejpam-6256	139	17	=	=	NOUN
ejpam-6256	139	18	j+1	j+1	NUM
ejpam-6256	139	19	tn(a	tn(a	NOUN
ejpam-6256	139	20	)	)	PUNCT
ejpam-6256	139	21	)	)	PUNCT
ejpam-6256	139	22	∪	∪	ADP
ejpam-6256	139	23	t	t	PROPN
ejpam-6256	139	24	j(a	j(a	PROPN
ejpam-6256	139	25	)	)	PUNCT
ejpam-6256	140	1	=	=	PRON
ejpam-6256	140	2	(	(	PUNCT
ejpam-6256	140	3	∞⋃	∞⋃	PROPN
ejpam-6256	140	4	n	n	CCONJ
ejpam-6256	140	5	=	=	NOUN
ejpam-6256	140	6	j+1	j+1	NUM
ejpam-6256	140	7	tn(a	tn(a	NOUN
ejpam-6256	140	8	)	)	PUNCT
ejpam-6256	140	9	)	)	PUNCT
ejpam-6256	140	10	∪	∪	ADP
ejpam-6256	140	11	t	t	NOUN
ejpam-6256	140	12	j+1−1({a1	j+1−1({a1	NOUN
ejpam-6256	140	13	,	,	PUNCT
ejpam-6256	140	14	...	...	PUNCT
ejpam-6256	140	15	,	,	PUNCT
ejpam-6256	140	16	ak	ak	PROPN
ejpam-6256	140	17	}	}	PUNCT
ejpam-6256	140	18	)	)	PUNCT
ejpam-6256	141	1	=	=	SYM
ejpam-6256	141	2	sj+1,1	sj+1,1	PROPN
ejpam-6256	141	3	t	t	PROPN
ejpam-6256	141	4	(	(	PUNCT
ejpam-6256	141	5	a	a	NOUN
ejpam-6256	141	6	)	)	PUNCT
ejpam-6256	141	7	.	.	PUNCT
ejpam-6256	142	1	note	note	VERB
ejpam-6256	142	2	that	that	SCONJ
ejpam-6256	142	3	the	the	DET
ejpam-6256	142	4	elements	element	NOUN
ejpam-6256	142	5	of	of	ADP
ejpam-6256	142	6	the	the	DET
ejpam-6256	142	7	set	set	NOUN
ejpam-6256	142	8	s0	s0	PROPN
ejpam-6256	142	9	t	t	PROPN
ejpam-6256	142	10	(	(	PUNCT
ejpam-6256	142	11	a	a	X
ejpam-6256	142	12	)	)	PUNCT
ejpam-6256	142	13	can	can	AUX
ejpam-6256	142	14	be	be	AUX
ejpam-6256	142	15	written	write	VERB
ejpam-6256	142	16	as	as	ADP
ejpam-6256	142	17	{	{	PUNCT
ejpam-6256	142	18	a1	a1	NOUN
ejpam-6256	142	19	,	,	PUNCT
ejpam-6256	142	20	...	...	PUNCT
ejpam-6256	142	21	,	,	PUNCT
ejpam-6256	142	22	ak	ak	PROPN
ejpam-6256	142	23	,	,	PUNCT
ejpam-6256	142	24	t	t	PROPN
ejpam-6256	142	25	(	(	PUNCT
ejpam-6256	142	26	a1	a1	PROPN
ejpam-6256	142	27	)	)	PUNCT
ejpam-6256	142	28	,	,	PUNCT
ejpam-6256	142	29	...	...	PUNCT
ejpam-6256	142	30	,	,	PUNCT
ejpam-6256	142	31	t	t	PROPN
ejpam-6256	142	32	(	(	PUNCT
ejpam-6256	142	33	ak	ak	PROPN
ejpam-6256	142	34	)	)	PUNCT
ejpam-6256	142	35	,	,	PUNCT
ejpam-6256	142	36	t	t	PROPN
ejpam-6256	142	37	2(a1	2(a1	NUM
ejpam-6256	142	38	)	)	PUNCT
ejpam-6256	142	39	,	,	PUNCT
ejpam-6256	142	40	...	...	PUNCT
ejpam-6256	142	41	}	}	PUNCT
ejpam-6256	142	42	,	,	PUNCT
ejpam-6256	142	43	suggesting	suggest	VERB
ejpam-6256	142	44	that	that	SCONJ
ejpam-6256	142	45	we	we	PRON
ejpam-6256	142	46	may	may	AUX
ejpam-6256	142	47	think	think	VERB
ejpam-6256	142	48	of	of	ADP
ejpam-6256	142	49	s0	s0	PROPN
ejpam-6256	142	50	as	as	ADP
ejpam-6256	142	51	a	a	DET
ejpam-6256	142	52	sequence	sequence	NOUN
ejpam-6256	142	53	while	while	SCONJ
ejpam-6256	142	54	sj	sj	NOUN
ejpam-6256	142	55	,	,	PUNCT
ejpam-6256	142	56	l	l	PROPN
ejpam-6256	142	57	t	t	PROPN
ejpam-6256	142	58	(	(	PUNCT
ejpam-6256	142	59	a	a	NOUN
ejpam-6256	142	60	)	)	PUNCT
ejpam-6256	142	61	as	as	SCONJ
ejpam-6256	142	62	tail	tail	NOUN
ejpam-6256	142	63	subsequences	subsequence	NOUN
ejpam-6256	142	64	for	for	ADP
ejpam-6256	142	65	all	all	DET
ejpam-6256	142	66	j	j	PROPN
ejpam-6256	142	67	∈	∈	PROPN
ejpam-6256	142	68	n.	n.	NOUN
ejpam-6256	142	69	in	in	ADP
ejpam-6256	142	70	particular	particular	ADJ
ejpam-6256	142	71	,	,	PUNCT
ejpam-6256	142	72	we	we	PRON
ejpam-6256	142	73	have	have	VERB
ejpam-6256	142	74	the	the	DET
ejpam-6256	142	75	following	follow	VERB
ejpam-6256	142	76	definition	definition	NOUN
ejpam-6256	142	77	.	.	PUNCT
ejpam-6256	143	1	definition	definition	NOUN
ejpam-6256	143	2	8	8	NUM
ejpam-6256	143	3	.	.	PUNCT
ejpam-6256	144	1	let	let	VERB
ejpam-6256	144	2	(	(	PUNCT
ejpam-6256	144	3	x	x	NOUN
ejpam-6256	144	4	,	,	PUNCT
ejpam-6256	144	5	d	d	NOUN
ejpam-6256	144	6	)	)	PUNCT
ejpam-6256	144	7	be	be	AUX
ejpam-6256	144	8	an	an	DET
ejpam-6256	144	9	mp	mp	NOUN
ejpam-6256	144	10	-	-	PUNCT
ejpam-6256	144	11	metric	metric	ADJ
ejpam-6256	144	12	space	space	NOUN
ejpam-6256	144	13	and	and	CCONJ
ejpam-6256	144	14	t	t	PROPN
ejpam-6256	144	15	is	be	AUX
ejpam-6256	144	16	a	a	DET
ejpam-6256	144	17	self	self	NOUN
ejpam-6256	144	18	mapping	mapping	NOUN
ejpam-6256	144	19	on	on	ADP
ejpam-6256	144	20	x	x	NOUN
ejpam-6256	144	21	,	,	PUNCT
ejpam-6256	144	22	and	and	CCONJ
ejpam-6256	144	23	a	a	DET
ejpam-6256	144	24	=	=	X
ejpam-6256	144	25	{	{	PUNCT
ejpam-6256	144	26	a1	a1	PROPN
ejpam-6256	144	27	,	,	PUNCT
ejpam-6256	144	28	...	...	PUNCT
ejpam-6256	144	29	,	,	PUNCT
ejpam-6256	144	30	ak	ak	PROPN
ejpam-6256	144	31	}	}	PUNCT
ejpam-6256	144	32	∈	∈	PROPN
ejpam-6256	144	33	p	p	NOUN
ejpam-6256	144	34	∗(x	∗(x	PROPN
ejpam-6256	144	35	)	)	PUNCT
ejpam-6256	144	36	.	.	PUNCT
ejpam-6256	145	1	then	then	ADV
ejpam-6256	145	2	for	for	ADP
ejpam-6256	145	3	all	all	DET
ejpam-6256	145	4	j	j	PROPN
ejpam-6256	145	5	∈	∈	PROPN
ejpam-6256	145	6	n	n	PART
ejpam-6256	145	7	∪	∪	X
ejpam-6256	145	8	{	{	PUNCT
ejpam-6256	145	9	0	0	NUM
ejpam-6256	145	10	}	}	PUNCT
ejpam-6256	145	11	,	,	PUNCT
ejpam-6256	145	12	we	we	PRON
ejpam-6256	145	13	define	define	VERB
ejpam-6256	145	14	the	the	DET
ejpam-6256	145	15	sequence	sequence	NOUN
ejpam-6256	145	16	(	(	PUNCT
ejpam-6256	145	17	αn	αn	NOUN
ejpam-6256	145	18	)	)	PUNCT
ejpam-6256	145	19	as	as	ADP
ejpam-6256	145	20	αn	αn	NOUN
ejpam-6256	145	21	=	=	SYM
ejpam-6256	145	22	αn(a	αn(a	PROPN
ejpam-6256	145	23	,	,	PUNCT
ejpam-6256	145	24	t	t	NOUN
ejpam-6256	145	25	)	)	PUNCT
ejpam-6256	146	1	=	=	SYM
ejpam-6256	146	2	t	t	PROPN
ejpam-6256	146	3	j(an−jk	j(an−jk	PROPN
ejpam-6256	146	4	)	)	PUNCT
ejpam-6256	146	5	(	(	PUNCT
ejpam-6256	146	6	3	3	X
ejpam-6256	146	7	)	)	PUNCT
ejpam-6256	147	1	where	where	SCONJ
ejpam-6256	147	2	j	j	PROPN
ejpam-6256	147	3	=	=	PRON
ejpam-6256	147	4	int(n−1	int(n−1	PROPN
ejpam-6256	147	5	k	k	PROPN
ejpam-6256	147	6	)	)	PUNCT
ejpam-6256	147	7	(	(	PUNCT
ejpam-6256	147	8	the	the	DET
ejpam-6256	147	9	function	function	NOUN
ejpam-6256	147	10	of	of	ADP
ejpam-6256	147	11	the	the	DET
ejpam-6256	147	12	greatest	great	ADJ
ejpam-6256	147	13	integer	integer	NOUN
ejpam-6256	147	14	less	less	ADJ
ejpam-6256	147	15	than	than	ADP
ejpam-6256	147	16	or	or	CCONJ
ejpam-6256	147	17	equal	equal	ADJ
ejpam-6256	147	18	)	)	PUNCT
ejpam-6256	147	19	the	the	DET
ejpam-6256	147	20	following	follow	VERB
ejpam-6256	147	21	proposition	proposition	NOUN
ejpam-6256	147	22	describes	describe	VERB
ejpam-6256	147	23	the	the	DET
ejpam-6256	147	24	relationship	relationship	NOUN
ejpam-6256	147	25	between	between	ADP
ejpam-6256	147	26	the	the	DET
ejpam-6256	147	27	sequence	sequence	NOUN
ejpam-6256	147	28	αn(a	αn(a	NUM
ejpam-6256	147	29	,	,	PUNCT
ejpam-6256	147	30	t	t	PROPN
ejpam-6256	147	31	)	)	PUNCT
ejpam-6256	147	32	and	and	CCONJ
ejpam-6256	147	33	the	the	DET
ejpam-6256	147	34	sets	set	NOUN
ejpam-6256	147	35	ss	ss	PROPN
ejpam-6256	147	36	,	,	PUNCT
ejpam-6256	147	37	l	l	PROPN
ejpam-6256	147	38	t	t	PROPN
ejpam-6256	147	39	(	(	PUNCT
ejpam-6256	147	40	a	a	NOUN
ejpam-6256	147	41	)	)	PUNCT
ejpam-6256	147	42	.	.	PUNCT
ejpam-6256	148	1	proposition	proposition	NOUN
ejpam-6256	148	2	5	5	NUM
ejpam-6256	148	3	.	.	PUNCT
ejpam-6256	149	1	let	let	AUX
ejpam-6256	149	2	(	(	PUNCT
ejpam-6256	149	3	x	x	NOUN
ejpam-6256	149	4	,	,	PUNCT
ejpam-6256	149	5	d	d	NOUN
ejpam-6256	149	6	)	)	PUNCT
ejpam-6256	149	7	be	be	AUX
ejpam-6256	149	8	an	an	DET
ejpam-6256	149	9	mp	mp	NOUN
ejpam-6256	149	10	-	-	PUNCT
ejpam-6256	149	11	metric	metric	ADJ
ejpam-6256	149	12	space	space	NOUN
ejpam-6256	149	13	,	,	PUNCT
ejpam-6256	149	14	and	and	CCONJ
ejpam-6256	149	15	let	let	VERB
ejpam-6256	149	16	t	t	PROPN
ejpam-6256	149	17	be	be	AUX
ejpam-6256	149	18	a	a	DET
ejpam-6256	149	19	self	self	NOUN
ejpam-6256	149	20	mapping	mapping	NOUN
ejpam-6256	149	21	on	on	ADP
ejpam-6256	149	22	x	x	NOUN
ejpam-6256	149	23	,	,	PUNCT
ejpam-6256	149	24	and	and	CCONJ
ejpam-6256	149	25	a	a	DET
ejpam-6256	149	26	=	=	X
ejpam-6256	149	27	{	{	PUNCT
ejpam-6256	149	28	a1	a1	PROPN
ejpam-6256	149	29	,	,	PUNCT
ejpam-6256	149	30	...	...	PUNCT
ejpam-6256	149	31	,	,	PUNCT
ejpam-6256	149	32	ak	ak	PROPN
ejpam-6256	149	33	}	}	PUNCT
ejpam-6256	149	34	∈	∈	PROPN
ejpam-6256	149	35	p	p	NOUN
ejpam-6256	149	36	∗(x	∗(x	PROPN
ejpam-6256	149	37	)	)	PUNCT
ejpam-6256	149	38	.	.	PUNCT
ejpam-6256	150	1	then	then	ADV
ejpam-6256	150	2	(	(	PUNCT
ejpam-6256	150	3	i	i	NOUN
ejpam-6256	150	4	)	)	PUNCT
ejpam-6256	150	5	ss	ss	PROPN
ejpam-6256	150	6	,	,	PUNCT
ejpam-6256	150	7	l	l	PROPN
ejpam-6256	150	8	t	t	PROPN
ejpam-6256	150	9	(	(	PUNCT
ejpam-6256	150	10	a	a	X
ejpam-6256	150	11	)	)	PUNCT
ejpam-6256	150	12	=	=	SYM
ejpam-6256	150	13	{	{	PUNCT
ejpam-6256	150	14	αn	αn	NOUN
ejpam-6256	150	15	:	:	PUNCT
ejpam-6256	150	16	n	n	PRON
ejpam-6256	150	17	≥	≥	NOUN
ejpam-6256	150	18	m	m	PROPN
ejpam-6256	150	19	}	}	PUNCT
ejpam-6256	150	20	where	where	SCONJ
ejpam-6256	150	21	m	m	VERB
ejpam-6256	150	22	=	=	SYM
ejpam-6256	150	23	l	l	NOUN
ejpam-6256	151	1	+	+	CCONJ
ejpam-6256	151	2	(	(	PUNCT
ejpam-6256	151	3	s−	s−	PROPN
ejpam-6256	151	4	1)k	1)k	NUM
ejpam-6256	151	5	,	,	PUNCT
ejpam-6256	151	6	(	(	PUNCT
ejpam-6256	151	7	s	s	NOUN
ejpam-6256	151	8	=	=	X
ejpam-6256	151	9	int(m−1	int(m−1	PROPN
ejpam-6256	151	10	k	k	NOUN
ejpam-6256	151	11	)	)	PUNCT
ejpam-6256	152	1	+	+	CCONJ
ejpam-6256	152	2	1	1	X
ejpam-6256	152	3	)	)	PUNCT
ejpam-6256	152	4	(	(	PUNCT
ejpam-6256	152	5	ii	ii	NOUN
ejpam-6256	152	6	)	)	PUNCT
ejpam-6256	152	7	ss	ss	PROPN
ejpam-6256	152	8	t	t	PROPN
ejpam-6256	152	9	(	(	PUNCT
ejpam-6256	152	10	a	a	X
ejpam-6256	152	11	)	)	PUNCT
ejpam-6256	152	12	=	=	SYM
ejpam-6256	152	13	{	{	PUNCT
ejpam-6256	152	14	αn	αn	NOUN
ejpam-6256	152	15	:	:	PUNCT
ejpam-6256	152	16	n	n	CCONJ
ejpam-6256	152	17	>	>	X
ejpam-6256	152	18	sk	sk	PROPN
ejpam-6256	152	19	}	}	PUNCT
ejpam-6256	152	20	,	,	PUNCT
ejpam-6256	152	21	(	(	PUNCT
ejpam-6256	152	22	iii	iii	X
ejpam-6256	152	23	)	)	PUNCT
ejpam-6256	152	24	s0	s0	PROPN
ejpam-6256	152	25	t	t	PROPN
ejpam-6256	152	26	(	(	PUNCT
ejpam-6256	152	27	a	a	X
ejpam-6256	152	28	)	)	PUNCT
ejpam-6256	152	29	=	=	SYM
ejpam-6256	152	30	{	{	PUNCT
ejpam-6256	152	31	αn	αn	NOUN
ejpam-6256	152	32	:	:	PUNCT
ejpam-6256	152	33	n	n	CCONJ
ejpam-6256	152	34	∈	∈	PROPN
ejpam-6256	152	35	n	n	CCONJ
ejpam-6256	152	36	}	}	PUNCT
ejpam-6256	152	37	,	,	PUNCT
ejpam-6256	152	38	proof	proof	NOUN
ejpam-6256	152	39	.	.	PUNCT
ejpam-6256	153	1	to	to	PART
ejpam-6256	153	2	show	show	VERB
ejpam-6256	153	3	(	(	PUNCT
ejpam-6256	153	4	i	i	NOUN
ejpam-6256	153	5	)	)	PUNCT
ejpam-6256	153	6	,	,	PUNCT
ejpam-6256	153	7	let	let	VERB
ejpam-6256	153	8	x	x	PUNCT
ejpam-6256	153	9	∈	∈	PROPN
ejpam-6256	153	10	ss	ss	PROPN
ejpam-6256	153	11	,	,	PUNCT
ejpam-6256	153	12	l	l	PROPN
ejpam-6256	153	13	t	t	PROPN
ejpam-6256	153	14	(	(	PUNCT
ejpam-6256	153	15	a	a	X
ejpam-6256	153	16	)	)	PUNCT
ejpam-6256	153	17	=	=	SYM
ejpam-6256	153	18	ss	ss	PROPN
ejpam-6256	153	19	t	t	PROPN
ejpam-6256	153	20	(	(	PUNCT
ejpam-6256	153	21	a	a	X
ejpam-6256	153	22	)	)	PUNCT
ejpam-6256	153	23	∪	∪	ADP
ejpam-6256	153	24	t	t	PROPN
ejpam-6256	153	25	s−1({al	s−1({al	PROPN
ejpam-6256	153	26	,	,	PUNCT
ejpam-6256	153	27	al+1	al+1	PROPN
ejpam-6256	153	28	...	...	PUNCT
ejpam-6256	153	29	,	,	PUNCT
ejpam-6256	153	30	ak	ak	PROPN
ejpam-6256	153	31	}	}	PUNCT
ejpam-6256	153	32	)	)	PUNCT
ejpam-6256	153	33	.	.	PUNCT
ejpam-6256	154	1	therefore	therefore	ADV
ejpam-6256	154	2	,	,	PUNCT
ejpam-6256	154	3	either	either	CCONJ
ejpam-6256	154	4	x	x	X
ejpam-6256	154	5	=	=	SYM
ejpam-6256	154	6	t	t	NOUN
ejpam-6256	154	7	t(ar	t(ar	PROPN
ejpam-6256	154	8	)	)	PUNCT
ejpam-6256	154	9	for	for	ADP
ejpam-6256	154	10	some	some	DET
ejpam-6256	154	11	t	t	PROPN
ejpam-6256	154	12	≥	≥	NOUN
ejpam-6256	154	13	s	s	NOUN
ejpam-6256	154	14	and	and	CCONJ
ejpam-6256	154	15	1	1	NUM
ejpam-6256	154	16	≤	≤	NOUN
ejpam-6256	154	17	r	r	NOUN
ejpam-6256	154	18	≤	≤	NUM
ejpam-6256	154	19	k	k	NOUN
ejpam-6256	154	20	,	,	PUNCT
ejpam-6256	154	21	or	or	CCONJ
ejpam-6256	154	22	x	x	X
ejpam-6256	154	23	=	=	SYM
ejpam-6256	154	24	t	t	PROPN
ejpam-6256	154	25	s−1(ar	s−1(ar	PROPN
ejpam-6256	154	26	)	)	PUNCT
ejpam-6256	154	27	for	for	ADP
ejpam-6256	154	28	some	some	DET
ejpam-6256	154	29	r	r	NOUN
ejpam-6256	154	30	∈	∈	PROPN
ejpam-6256	154	31	{	{	PUNCT
ejpam-6256	154	32	l	l	NOUN
ejpam-6256	154	33	,	,	PUNCT
ejpam-6256	154	34	...	...	PUNCT
ejpam-6256	154	35	,	,	PUNCT
ejpam-6256	154	36	k	k	NOUN
ejpam-6256	154	37	}	}	PUNCT
ejpam-6256	154	38	.	.	PUNCT
ejpam-6256	155	1	in	in	ADP
ejpam-6256	155	2	the	the	DET
ejpam-6256	155	3	first	first	ADJ
ejpam-6256	155	4	case	case	NOUN
ejpam-6256	155	5	,	,	PUNCT
ejpam-6256	155	6	we	we	PRON
ejpam-6256	155	7	choose	choose	VERB
ejpam-6256	155	8	n	n	NOUN
ejpam-6256	155	9	=	=	SYM
ejpam-6256	156	1	r	r	NOUN
ejpam-6256	156	2	+	+	PROPN
ejpam-6256	156	3	tk	tk	PROPN
ejpam-6256	156	4	≥	≥	PROPN
ejpam-6256	156	5	l	l	NOUN
ejpam-6256	157	1	+	+	CCONJ
ejpam-6256	157	2	(	(	PUNCT
ejpam-6256	157	3	s	s	NOUN
ejpam-6256	157	4	−	−	PROPN
ejpam-6256	157	5	1)k	1)k	NUM
ejpam-6256	157	6	=	=	SYM
ejpam-6256	157	7	m	m	PROPN
ejpam-6256	157	8	,	,	PUNCT
ejpam-6256	157	9	while	while	SCONJ
ejpam-6256	157	10	in	in	ADP
ejpam-6256	157	11	the	the	DET
ejpam-6256	157	12	second	second	ADJ
ejpam-6256	157	13	case	case	NOUN
ejpam-6256	157	14	,	,	PUNCT
ejpam-6256	157	15	we	we	PRON
ejpam-6256	157	16	set	set	VERB
ejpam-6256	157	17	n	n	NOUN
ejpam-6256	157	18	=	=	SYM
ejpam-6256	157	19	r	r	NOUN
ejpam-6256	157	20	+	+	CCONJ
ejpam-6256	157	21	(	(	PUNCT
ejpam-6256	157	22	s	s	NOUN
ejpam-6256	157	23	−	−	PROPN
ejpam-6256	157	24	1)k	1)k	NUM
ejpam-6256	157	25	≥	≥	NOUN
ejpam-6256	157	26	l	l	NOUN
ejpam-6256	158	1	+	+	CCONJ
ejpam-6256	158	2	(	(	PUNCT
ejpam-6256	158	3	s	s	NOUN
ejpam-6256	158	4	−	−	PROPN
ejpam-6256	158	5	1)k	1)k	NUM
ejpam-6256	158	6	=	=	SYM
ejpam-6256	158	7	m.	m.	NOUN
ejpam-6256	158	8	hence	hence	ADV
ejpam-6256	158	9	,	,	PUNCT
ejpam-6256	158	10	we	we	PRON
ejpam-6256	158	11	get	get	VERB
ejpam-6256	158	12	x	x	X
ejpam-6256	158	13	=	=	VERB
ejpam-6256	158	14	αn	αn	NOUN
ejpam-6256	158	15	and	and	CCONJ
ejpam-6256	158	16	n	n	PRON
ejpam-6256	158	17	≥	≥	NOUN
ejpam-6256	159	1	m.	m.	NOUN
ejpam-6256	159	2	y.	y.	PROPN
ejpam-6256	159	3	alzubaidi	alzubaidi	PROPN
ejpam-6256	159	4	/	/	SYM
ejpam-6256	159	5	eur	eur	PROPN
ejpam-6256	159	6	.	.	PUNCT
ejpam-6256	160	1	j.	j.	PROPN
ejpam-6256	160	2	pure	pure	PROPN
ejpam-6256	160	3	appl	appl	PROPN
ejpam-6256	160	4	.	.	PROPN
ejpam-6256	160	5	math	math	PROPN
ejpam-6256	160	6	,	,	PUNCT
ejpam-6256	160	7	18	18	NUM
ejpam-6256	160	8	(	(	PUNCT
ejpam-6256	160	9	3	3	NUM
ejpam-6256	160	10	)	)	PUNCT
ejpam-6256	160	11	(	(	PUNCT
ejpam-6256	160	12	2025	2025	NUM
ejpam-6256	160	13	)	)	PUNCT
ejpam-6256	160	14	,	,	PUNCT
ejpam-6256	160	15	6256	6256	NUM
ejpam-6256	160	16	7	7	NUM
ejpam-6256	160	17	of	of	ADP
ejpam-6256	160	18	11	11	NUM
ejpam-6256	160	19	for	for	ADP
ejpam-6256	160	20	the	the	DET
ejpam-6256	160	21	other	other	ADJ
ejpam-6256	160	22	inclusion	inclusion	NOUN
ejpam-6256	160	23	,	,	PUNCT
ejpam-6256	160	24	let	let	VERB
ejpam-6256	160	25	αn	αn	NOUN
ejpam-6256	160	26	=	=	SYM
ejpam-6256	160	27	t	t	PROPN
ejpam-6256	160	28	t(an−tk	t(an−tk	PROPN
ejpam-6256	160	29	)	)	PUNCT
ejpam-6256	160	30	where	where	SCONJ
ejpam-6256	160	31	t	t	NOUN
ejpam-6256	160	32	=	=	PUNCT
ejpam-6256	160	33	int(n−1	int(n−1	PROPN
ejpam-6256	160	34	k	k	PROPN
ejpam-6256	160	35	)	)	PUNCT
ejpam-6256	160	36	for	for	ADP
ejpam-6256	160	37	some	some	DET
ejpam-6256	160	38	n	n	PRON
ejpam-6256	160	39	≥	≥	NOUN
ejpam-6256	160	40	m.	m.	NOUN
ejpam-6256	160	41	we	we	PRON
ejpam-6256	160	42	set	set	VERB
ejpam-6256	160	43	s	s	X
ejpam-6256	160	44	=	=	SYM
ejpam-6256	160	45	1	1	NUM
ejpam-6256	160	46	+	+	NUM
ejpam-6256	160	47	int(m−1	int(m−1	PROPN
ejpam-6256	160	48	k	k	NOUN
ejpam-6256	160	49	)	)	PUNCT
ejpam-6256	160	50	and	and	CCONJ
ejpam-6256	160	51	l	l	NOUN
ejpam-6256	160	52	=	=	PUNCT
ejpam-6256	161	1	m	m	VERB
ejpam-6256	161	2	−	−	NOUN
ejpam-6256	162	1	(	(	PUNCT
ejpam-6256	162	2	s	s	NOUN
ejpam-6256	162	3	−	−	PROPN
ejpam-6256	162	4	1)k	1)k	NUM
ejpam-6256	162	5	to	to	PART
ejpam-6256	162	6	get	get	VERB
ejpam-6256	162	7	t	t	PROPN
ejpam-6256	162	8	≥	≥	NOUN
ejpam-6256	162	9	s	s	PART
ejpam-6256	162	10	−	−	PROPN
ejpam-6256	162	11	1	1	NUM
ejpam-6256	162	12	.	.	PUNCT
ejpam-6256	163	1	if	if	SCONJ
ejpam-6256	163	2	t	t	PROPN
ejpam-6256	163	3	≥	≥	NOUN
ejpam-6256	163	4	s	s	PART
ejpam-6256	163	5	,	,	PUNCT
ejpam-6256	163	6	then	then	ADV
ejpam-6256	163	7	αn	αn	NOUN
ejpam-6256	163	8	=	=	SYM
ejpam-6256	163	9	t	t	PROPN
ejpam-6256	163	10	t(an−tk	t(an−tk	PROPN
ejpam-6256	163	11	)	)	PUNCT
ejpam-6256	163	12	∈	∈	PROPN
ejpam-6256	163	13	ss	ss	PROPN
ejpam-6256	163	14	t	t	PROPN
ejpam-6256	163	15	(	(	PUNCT
ejpam-6256	163	16	a	a	NOUN
ejpam-6256	163	17	)	)	PUNCT
ejpam-6256	163	18	⊆	⊆	NUM
ejpam-6256	163	19	ss	ss	NOUN
ejpam-6256	163	20	,	,	PUNCT
ejpam-6256	163	21	l	l	PROPN
ejpam-6256	163	22	t	t	PROPN
ejpam-6256	163	23	(	(	PUNCT
ejpam-6256	163	24	a	a	NOUN
ejpam-6256	163	25	)	)	PUNCT
ejpam-6256	163	26	.	.	PUNCT
ejpam-6256	164	1	if	if	SCONJ
ejpam-6256	164	2	t	t	PROPN
ejpam-6256	164	3	=	=	SYM
ejpam-6256	164	4	s	s	PART
ejpam-6256	164	5	−	−	PROPN
ejpam-6256	164	6	1	1	NUM
ejpam-6256	164	7	,	,	PUNCT
ejpam-6256	164	8	then	then	ADV
ejpam-6256	164	9	n	n	CCONJ
ejpam-6256	164	10	−	−	PROPN
ejpam-6256	164	11	tk	tk	PROPN
ejpam-6256	164	12	≥	≥	PROPN
ejpam-6256	164	13	m	m	VERB
ejpam-6256	164	14	−	−	PROPN
ejpam-6256	165	1	(	(	PUNCT
ejpam-6256	165	2	s	s	NOUN
ejpam-6256	165	3	−	−	PROPN
ejpam-6256	165	4	1)k	1)k	NUM
ejpam-6256	165	5	=	=	SYM
ejpam-6256	165	6	l	l	NOUN
ejpam-6256	165	7	,	,	PUNCT
ejpam-6256	165	8	and	and	CCONJ
ejpam-6256	165	9	hence	hence	ADV
ejpam-6256	165	10	αn	αn	NOUN
ejpam-6256	165	11	=	=	SYM
ejpam-6256	165	12	t	t	PROPN
ejpam-6256	165	13	t(an−tk	t(an−tk	PROPN
ejpam-6256	165	14	)	)	PUNCT
ejpam-6256	165	15	∈	∈	PROPN
ejpam-6256	165	16	t	t	PROPN
ejpam-6256	165	17	s−1({al	s−1({al	PROPN
ejpam-6256	165	18	,	,	PUNCT
ejpam-6256	165	19	al+1	al+1	PROPN
ejpam-6256	165	20	...	...	PUNCT
ejpam-6256	165	21	,	,	PUNCT
ejpam-6256	165	22	ak	ak	PROPN
ejpam-6256	165	23	}	}	PUNCT
ejpam-6256	165	24	)	)	PUNCT
ejpam-6256	165	25	⊆	⊆	NUM
ejpam-6256	165	26	ss	ss	NOUN
ejpam-6256	165	27	,	,	PUNCT
ejpam-6256	165	28	l	l	PROPN
ejpam-6256	165	29	t	t	PROPN
ejpam-6256	165	30	(	(	PUNCT
ejpam-6256	165	31	a	a	NOUN
ejpam-6256	165	32	)	)	PUNCT
ejpam-6256	165	33	.	.	PUNCT
ejpam-6256	166	1	(	(	PUNCT
ejpam-6256	166	2	ii	ii	NOUN
ejpam-6256	166	3	)	)	PUNCT
ejpam-6256	166	4	and	and	CCONJ
ejpam-6256	166	5	(	(	PUNCT
ejpam-6256	166	6	iii	iii	X
ejpam-6256	166	7	)	)	PUNCT
ejpam-6256	166	8	follow	follow	VERB
ejpam-6256	166	9	from	from	ADP
ejpam-6256	166	10	(	(	PUNCT
ejpam-6256	166	11	i	i	NOUN
ejpam-6256	166	12	)	)	PUNCT
ejpam-6256	166	13	using	use	VERB
ejpam-6256	166	14	proposition	proposition	NOUN
ejpam-6256	166	15	4-(iii	4-(iii	NUM
ejpam-6256	166	16	)	)	PUNCT
ejpam-6256	166	17	.	.	PUNCT
ejpam-6256	167	1	proposition	proposition	NOUN
ejpam-6256	167	2	6	6	NUM
ejpam-6256	167	3	.	.	PUNCT
ejpam-6256	168	1	let	let	AUX
ejpam-6256	168	2	(	(	PUNCT
ejpam-6256	168	3	x	x	NOUN
ejpam-6256	168	4	,	,	PUNCT
ejpam-6256	168	5	d	d	NOUN
ejpam-6256	168	6	)	)	PUNCT
ejpam-6256	168	7	be	be	AUX
ejpam-6256	168	8	an	an	DET
ejpam-6256	168	9	mp	mp	NOUN
ejpam-6256	168	10	-	-	PUNCT
ejpam-6256	168	11	metric	metric	ADJ
ejpam-6256	168	12	space	space	NOUN
ejpam-6256	168	13	,	,	PUNCT
ejpam-6256	168	14	and	and	CCONJ
ejpam-6256	168	15	let	let	VERB
ejpam-6256	168	16	t	t	PROPN
ejpam-6256	168	17	be	be	AUX
ejpam-6256	168	18	a	a	DET
ejpam-6256	168	19	self	self	NOUN
ejpam-6256	168	20	mapping	mapping	NOUN
ejpam-6256	168	21	on	on	ADP
ejpam-6256	168	22	x	x	NOUN
ejpam-6256	168	23	,	,	PUNCT
ejpam-6256	168	24	and	and	CCONJ
ejpam-6256	168	25	a	a	DET
ejpam-6256	168	26	=	=	X
ejpam-6256	168	27	{	{	PUNCT
ejpam-6256	168	28	a1	a1	PROPN
ejpam-6256	168	29	,	,	PUNCT
ejpam-6256	168	30	...	...	PUNCT
ejpam-6256	168	31	,	,	PUNCT
ejpam-6256	168	32	ak	ak	PROPN
ejpam-6256	168	33	}	}	PUNCT
ejpam-6256	168	34	∈	∈	PROPN
ejpam-6256	168	35	p	p	NOUN
ejpam-6256	168	36	∗(x	∗(x	PROPN
ejpam-6256	168	37	)	)	PUNCT
ejpam-6256	168	38	.	.	PUNCT
ejpam-6256	169	1	if	if	SCONJ
ejpam-6256	169	2	(	(	PUNCT
ejpam-6256	169	3	αnu	αnu	NOUN
ejpam-6256	169	4	)	)	PUNCT
ejpam-6256	169	5	is	be	AUX
ejpam-6256	169	6	a	a	DET
ejpam-6256	169	7	subsequence	subsequence	NOUN
ejpam-6256	169	8	of	of	ADP
ejpam-6256	169	9	(	(	PUNCT
ejpam-6256	169	10	αn(a	αn(a	PROPN
ejpam-6256	169	11	,	,	PUNCT
ejpam-6256	169	12	t	t	NOUN
ejpam-6256	169	13	)	)	PUNCT
ejpam-6256	169	14	)	)	PUNCT
ejpam-6256	169	15	.	.	PUNCT
ejpam-6256	170	1	then	then	ADV
ejpam-6256	170	2	there	there	PRON
ejpam-6256	170	3	are	be	VERB
ejpam-6256	170	4	a	a	DET
ejpam-6256	170	5	∈	∈	PROPN
ejpam-6256	170	6	a	a	PRON
ejpam-6256	170	7	and	and	CCONJ
ejpam-6256	170	8	sv	sv	ADP
ejpam-6256	170	9	∈	∈	PROPN
ejpam-6256	170	10	n	n	PRON
ejpam-6256	170	11	such	such	ADJ
ejpam-6256	170	12	that	that	SCONJ
ejpam-6256	170	13	t	t	PROPN
ejpam-6256	170	14	sv(a	sv(a	PROPN
ejpam-6256	170	15	)	)	PUNCT
ejpam-6256	170	16	is	be	AUX
ejpam-6256	170	17	a	a	DET
ejpam-6256	170	18	subsequence	subsequence	NOUN
ejpam-6256	170	19	of	of	ADP
ejpam-6256	170	20	(	(	PUNCT
ejpam-6256	170	21	αnu	αnu	NOUN
ejpam-6256	170	22	)	)	PUNCT
ejpam-6256	170	23	.	.	PUNCT
ejpam-6256	171	1	proof	proof	NOUN
ejpam-6256	171	2	.	.	PUNCT
ejpam-6256	172	1	this	this	DET
ejpam-6256	172	2	result	result	NOUN
ejpam-6256	172	3	is	be	AUX
ejpam-6256	172	4	a	a	DET
ejpam-6256	172	5	direct	direct	ADJ
ejpam-6256	172	6	consequence	consequence	NOUN
ejpam-6256	172	7	of	of	ADP
ejpam-6256	172	8	the	the	DET
ejpam-6256	172	9	finiteness	finiteness	NOUN
ejpam-6256	172	10	of	of	ADP
ejpam-6256	172	11	a	a	PRON
ejpam-6256	172	12	,	,	PUNCT
ejpam-6256	172	13	as	as	ADP
ejpam-6256	172	14	assuming	assume	VERB
ejpam-6256	172	15	the	the	DET
ejpam-6256	172	16	opposite	opposite	ADJ
ejpam-6256	172	17	leads	lead	VERB
ejpam-6256	172	18	to	to	ADP
ejpam-6256	172	19	a	a	DET
ejpam-6256	172	20	contradiction	contradiction	NOUN
ejpam-6256	172	21	with	with	ADP
ejpam-6256	172	22	(	(	PUNCT
ejpam-6256	172	23	αnu	αnu	NOUN
ejpam-6256	172	24	)	)	PUNCT
ejpam-6256	172	25	being	be	AUX
ejpam-6256	172	26	a	a	DET
ejpam-6256	172	27	subsequence	subsequence	NOUN
ejpam-6256	172	28	.	.	PUNCT
ejpam-6256	173	1	the	the	DET
ejpam-6256	173	2	conclusion	conclusion	NOUN
ejpam-6256	173	3	of	of	ADP
ejpam-6256	173	4	this	this	DET
ejpam-6256	173	5	proposition	proposition	NOUN
ejpam-6256	173	6	is	be	AUX
ejpam-6256	173	7	equivalent	equivalent	ADJ
ejpam-6256	173	8	to	to	ADP
ejpam-6256	173	9	the	the	DET
ejpam-6256	173	10	following	follow	VERB
ejpam-6256	173	11	statement	statement	NOUN
ejpam-6256	173	12	:	:	PUNCT
ejpam-6256	173	13	∃	∃	PROPN
ejpam-6256	173	14	ai	ai	PROPN
ejpam-6256	173	15	∈	∈	PROPN
ejpam-6256	173	16	a	a	DET
ejpam-6256	173	17	s.t	s.t	PROPN
ejpam-6256	173	18	∀	∀	X
ejpam-6256	173	19	m	m	VERB
ejpam-6256	173	20	∈	∈	PROPN
ejpam-6256	173	21	n	n	PRON
ejpam-6256	173	22	∃	∃	PROPN
ejpam-6256	173	23	sv	sv	PROPN
ejpam-6256	173	24	,	,	PUNCT
ejpam-6256	173	25	nu	nu	PROPN
ejpam-6256	173	26	>	>	X
ejpam-6256	173	27	m	m	PROPN
ejpam-6256	173	28	,	,	PUNCT
ejpam-6256	173	29	s.t	s.t	PROPN
ejpam-6256	173	30	.	.	PROPN
ejpam-6256	173	31	t	t	PROPN
ejpam-6256	173	32	sv(ai	sv(ai	PROPN
ejpam-6256	173	33	)	)	PUNCT
ejpam-6256	173	34	=	=	SYM
ejpam-6256	173	35	αnu	αnu	NOUN
ejpam-6256	173	36	.	.	PUNCT
ejpam-6256	174	1	thus	thus	ADV
ejpam-6256	174	2	,	,	PUNCT
ejpam-6256	174	3	its	its	PRON
ejpam-6256	174	4	negation	negation	NOUN
ejpam-6256	174	5	is	be	AUX
ejpam-6256	174	6	:	:	PUNCT
ejpam-6256	174	7	∀	∀	X
ejpam-6256	174	8	ai	ai	VERB
ejpam-6256	174	9	∈	∈	PROPN
ejpam-6256	174	10	a	a	DET
ejpam-6256	174	11	∃	∃	PROPN
ejpam-6256	174	12	mi	mi	PROPN
ejpam-6256	174	13	∈	∈	PROPN
ejpam-6256	174	14	n	n	PRON
ejpam-6256	174	15	s.t	s.t	PROPN
ejpam-6256	174	16	.	.	PROPN
ejpam-6256	174	17	t	t	PROPN
ejpam-6256	174	18	sv(ai	sv(ai	PROPN
ejpam-6256	174	19	)	)	PUNCT
ejpam-6256	174	20	̸=	̸=	PROPN
ejpam-6256	174	21	αnu	αnu	NOUN
ejpam-6256	174	22	.	.	PUNCT
ejpam-6256	174	23	∀	∀	PUNCT
ejpam-6256	175	1	sv	sv	ADP
ejpam-6256	175	2	,	,	PUNCT
ejpam-6256	175	3	nu	nu	PROPN
ejpam-6256	175	4	>	>	X
ejpam-6256	175	5	mi	mi	PROPN
ejpam-6256	175	6	.	.	PROPN
ejpam-6256	175	7	using	use	VERB
ejpam-6256	175	8	the	the	DET
ejpam-6256	175	9	finiteness	finiteness	NOUN
ejpam-6256	175	10	of	of	ADP
ejpam-6256	175	11	a	a	DET
ejpam-6256	175	12	=	=	X
ejpam-6256	175	13	{	{	PUNCT
ejpam-6256	175	14	a1	a1	PROPN
ejpam-6256	175	15	,	,	PUNCT
ejpam-6256	175	16	...	...	PUNCT
ejpam-6256	175	17	,	,	PUNCT
ejpam-6256	175	18	ak	ak	PROPN
ejpam-6256	175	19	}	}	PUNCT
ejpam-6256	175	20	,	,	PUNCT
ejpam-6256	175	21	we	we	PRON
ejpam-6256	175	22	define	define	VERB
ejpam-6256	175	23	m	m	PROPN
ejpam-6256	175	24	=	=	ADJ
ejpam-6256	175	25	max	max	PROPN
ejpam-6256	175	26	i=1,	i=1,	PROPN
ejpam-6256	175	27	...	...	PUNCT
ejpam-6256	175	28	,k	,k	PROPN
ejpam-6256	175	29	mi	mi	PROPN
ejpam-6256	175	30	.	.	PROPN
ejpam-6256	175	31	therefore	therefore	ADV
ejpam-6256	175	32	,	,	PUNCT
ejpam-6256	175	33	t	t	PROPN
ejpam-6256	175	34	sv(ai	sv(ai	PROPN
ejpam-6256	175	35	)	)	PUNCT
ejpam-6256	175	36	̸=	̸=	PROPN
ejpam-6256	175	37	αnu	αnu	NOUN
ejpam-6256	175	38	.	.	PUNCT
ejpam-6256	175	39	∀	∀	PUNCT
ejpam-6256	176	1	sv	sv	ADP
ejpam-6256	176	2	,	,	PUNCT
ejpam-6256	176	3	nu	nu	INTJ
ejpam-6256	176	4	>	>	X
ejpam-6256	176	5	m	m	PROPN
ejpam-6256	176	6	,	,	PUNCT
ejpam-6256	176	7	and	and	CCONJ
ejpam-6256	176	8	∀	∀	NUM
ejpam-6256	176	9	ai	ai	VERB
ejpam-6256	176	10	∈	∈	PROPN
ejpam-6256	176	11	a	a	PRON
ejpam-6256	176	12	,	,	PUNCT
ejpam-6256	176	13	which	which	PRON
ejpam-6256	176	14	contradicts	contradict	VERB
ejpam-6256	176	15	the	the	DET
ejpam-6256	176	16	assumption	assumption	NOUN
ejpam-6256	176	17	that	that	SCONJ
ejpam-6256	176	18	(	(	PUNCT
ejpam-6256	176	19	αnu	αnu	NOUN
ejpam-6256	176	20	)	)	PUNCT
ejpam-6256	176	21	is	be	AUX
ejpam-6256	176	22	a	a	DET
ejpam-6256	176	23	subsequence	subsequence	NOUN
ejpam-6256	176	24	of	of	ADP
ejpam-6256	176	25	(	(	PUNCT
ejpam-6256	176	26	αn(a	αn(a	PROPN
ejpam-6256	176	27	,	,	PUNCT
ejpam-6256	176	28	t	t	NOUN
ejpam-6256	176	29	)	)	PUNCT
ejpam-6256	176	30	)	)	PUNCT
ejpam-6256	177	1	=	=	PRON
ejpam-6256	177	2	(	(	PUNCT
ejpam-6256	177	3	t	t	PROPN
ejpam-6256	177	4	j(an−jk	j(an−jk	PROPN
ejpam-6256	177	5	)	)	PUNCT
ejpam-6256	177	6	)	)	PUNCT
ejpam-6256	177	7	where	where	SCONJ
ejpam-6256	177	8	it	it	PRON
ejpam-6256	177	9	directly	directly	ADV
ejpam-6256	177	10	conflicts	conflict	VERB
ejpam-6256	177	11	with	with	ADP
ejpam-6256	177	12	the	the	DET
ejpam-6256	177	13	construction	construction	NOUN
ejpam-6256	177	14	of	of	ADP
ejpam-6256	177	15	(	(	PUNCT
ejpam-6256	177	16	αnu	αnu	NOUN
ejpam-6256	177	17	)	)	PUNCT
ejpam-6256	177	18	as	as	ADP
ejpam-6256	177	19	a	a	DET
ejpam-6256	177	20	subsequence	subsequence	NOUN
ejpam-6256	177	21	formed	form	VERB
ejpam-6256	177	22	by	by	ADP
ejpam-6256	177	23	applying	apply	VERB
ejpam-6256	177	24	the	the	DET
ejpam-6256	177	25	iteration	iteration	NOUN
ejpam-6256	177	26	of	of	ADP
ejpam-6256	177	27	t	t	PROPN
ejpam-6256	177	28	over	over	ADP
ejpam-6256	177	29	elements	element	NOUN
ejpam-6256	177	30	of	of	ADP
ejpam-6256	177	31	a	a	PRON
ejpam-6256	177	32	(	(	PUNCT
ejpam-6256	177	33	see	see	VERB
ejpam-6256	177	34	definition	definition	NOUN
ejpam-6256	177	35	8	8	NUM
ejpam-6256	177	36	,	,	PUNCT
ejpam-6256	177	37	equation	equation	NOUN
ejpam-6256	177	38	(	(	PUNCT
ejpam-6256	177	39	3	3	NUM
ejpam-6256	177	40	)	)	PUNCT
ejpam-6256	177	41	)	)	PUNCT
ejpam-6256	177	42	.	.	PUNCT
ejpam-6256	178	1	we	we	PRON
ejpam-6256	178	2	now	now	ADV
ejpam-6256	178	3	present	present	VERB
ejpam-6256	178	4	our	our	PRON
ejpam-6256	178	5	main	main	ADJ
ejpam-6256	178	6	result	result	NOUN
ejpam-6256	178	7	,	,	PUNCT
ejpam-6256	178	8	which	which	PRON
ejpam-6256	178	9	provides	provide	VERB
ejpam-6256	178	10	sufficient	sufficient	ADJ
ejpam-6256	178	11	conditions	condition	NOUN
ejpam-6256	178	12	for	for	ADP
ejpam-6256	178	13	a	a	DET
ejpam-6256	178	14	mapping	mapping	NOUN
ejpam-6256	178	15	to	to	PART
ejpam-6256	178	16	attain	attain	VERB
ejpam-6256	178	17	several	several	ADJ
ejpam-6256	178	18	iterative	iterative	ADJ
ejpam-6256	178	19	fixed	fix	VERB
ejpam-6256	178	20	points	point	NOUN
ejpam-6256	178	21	.	.	PUNCT
ejpam-6256	179	1	theorem	theorem	NOUN
ejpam-6256	179	2	3	3	X
ejpam-6256	179	3	.	.	PUNCT
ejpam-6256	180	1	let	let	AUX
ejpam-6256	180	2	(	(	PUNCT
ejpam-6256	180	3	x	x	NOUN
ejpam-6256	180	4	,	,	PUNCT
ejpam-6256	180	5	d	d	NOUN
ejpam-6256	180	6	)	)	PUNCT
ejpam-6256	180	7	be	be	AUX
ejpam-6256	180	8	a	a	DET
ejpam-6256	180	9	bounded	bounded	ADJ
ejpam-6256	180	10	complete	complete	ADJ
ejpam-6256	180	11	mp	mp	NOUN
ejpam-6256	180	12	-	-	PUNCT
ejpam-6256	180	13	metric	metric	ADJ
ejpam-6256	180	14	space	space	NOUN
ejpam-6256	180	15	,	,	PUNCT
ejpam-6256	180	16	and	and	CCONJ
ejpam-6256	180	17	let	let	VERB
ejpam-6256	180	18	t	t	NOUN
ejpam-6256	180	19	:	:	PUNCT
ejpam-6256	180	20	x	x	X
ejpam-6256	180	21	→	→	PUNCT
ejpam-6256	180	22	x	x	PUNCT
ejpam-6256	180	23	be	be	AUX
ejpam-6256	180	24	a	a	DET
ejpam-6256	180	25	continuous	continuous	ADJ
ejpam-6256	180	26	mapping	mapping	NOUN
ejpam-6256	180	27	.	.	PUNCT
ejpam-6256	181	1	if	if	SCONJ
ejpam-6256	181	2	there	there	PRON
ejpam-6256	181	3	exist	exist	VERB
ejpam-6256	181	4	positive	positive	ADJ
ejpam-6256	181	5	constants	constant	NOUN
ejpam-6256	181	6	r	r	NOUN
ejpam-6256	181	7	,	,	PUNCT
ejpam-6256	181	8	p	p	X
ejpam-6256	181	9	,	,	PUNCT
ejpam-6256	181	10	and	and	CCONJ
ejpam-6256	181	11	q	q	NOUN
ejpam-6256	181	12	,	,	PUNCT
ejpam-6256	181	13	and	and	CCONJ
ejpam-6256	181	14	n	n	PRON
ejpam-6256	181	15	∈	∈	PROPN
ejpam-6256	181	16	n	n	PRON
ejpam-6256	181	17	such	such	ADJ
ejpam-6256	181	18	that	that	PRON
ejpam-6256	181	19	p+	p+	VERB
ejpam-6256	181	20	q	q	PROPN
ejpam-6256	181	21	≤	≤	NUM
ejpam-6256	181	22	1	1	NUM
ejpam-6256	181	23	and	and	CCONJ
ejpam-6256	181	24	the	the	DET
ejpam-6256	181	25	following	follow	VERB
ejpam-6256	181	26	inequalities	inequality	NOUN
ejpam-6256	181	27	hold	hold	VERB
ejpam-6256	181	28	r	r	NOUN
ejpam-6256	181	29	≤	≤	NUM
ejpam-6256	181	30	sup	sup	NOUN
ejpam-6256	181	31	xi∈ss	xi∈ss	PROPN
ejpam-6256	181	32	,	,	PUNCT
ejpam-6256	181	33	l	l	PROPN
ejpam-6256	181	34	t	t	PROPN
ejpam-6256	181	35	(	(	PUNCT
ejpam-6256	181	36	a	a	NOUN
ejpam-6256	181	37	)	)	PUNCT
ejpam-6256	181	38	d(t	d(t	PROPN
ejpam-6256	181	39	(	(	PUNCT
ejpam-6256	181	40	{	{	PUNCT
ejpam-6256	181	41	x1	x1	PROPN
ejpam-6256	181	42	,	,	PUNCT
ejpam-6256	181	43	...	...	PUNCT
ejpam-6256	181	44	,	,	PUNCT
ejpam-6256	181	45	xu	xu	PROPN
ejpam-6256	181	46	}	}	PUNCT
ejpam-6256	181	47	)	)	PUNCT
ejpam-6256	181	48	)	)	PUNCT
ejpam-6256	181	49	≤	≤	NUM
ejpam-6256	181	50	q	q	NOUN
ejpam-6256	181	51	sup	sup	NOUN
ejpam-6256	181	52	xi∈ss	xi∈ss	PROPN
ejpam-6256	181	53	,	,	PUNCT
ejpam-6256	181	54	l	l	PROPN
ejpam-6256	181	55	t	t	PROPN
ejpam-6256	181	56	(	(	PUNCT
ejpam-6256	181	57	a	a	X
ejpam-6256	181	58	)	)	PUNCT
ejpam-6256	181	59	d({x1	d({x1	PROPN
ejpam-6256	181	60	,	,	PUNCT
ejpam-6256	181	61	...	...	PUNCT
ejpam-6256	181	62	,	,	PUNCT
ejpam-6256	181	63	xu	xu	PROPN
ejpam-6256	181	64	}	}	PUNCT
ejpam-6256	181	65	)	)	PUNCT
ejpam-6256	182	1	+	+	CCONJ
ejpam-6256	182	2	pr	pr	NOUN
ejpam-6256	182	3	for	for	ADP
ejpam-6256	182	4	all	all	DET
ejpam-6256	182	5	u	u	PROPN
ejpam-6256	182	6	≥	≥	PRON
ejpam-6256	182	7	n.	n.	NOUN
ejpam-6256	182	8	(	(	PUNCT
ejpam-6256	182	9	4	4	NUM
ejpam-6256	182	10	)	)	PUNCT
ejpam-6256	182	11	for	for	ADP
ejpam-6256	182	12	some	some	DET
ejpam-6256	182	13	a	a	DET
ejpam-6256	182	14	∈	∈	PROPN
ejpam-6256	182	15	p	p	NOUN
ejpam-6256	182	16	∗(x	∗(x	PROPN
ejpam-6256	182	17	)	)	PUNCT
ejpam-6256	182	18	,	,	PUNCT
ejpam-6256	182	19	then	then	ADV
ejpam-6256	182	20	t	t	PROPN
ejpam-6256	182	21	has	have	VERB
ejpam-6256	182	22	at	at	ADV
ejpam-6256	182	23	least	least	ADJ
ejpam-6256	182	24	two	two	NUM
ejpam-6256	182	25	distinct	distinct	ADJ
ejpam-6256	182	26	iterative	iterative	NOUN
ejpam-6256	182	27	fixed	fix	VERB
ejpam-6256	182	28	points	point	NOUN
ejpam-6256	182	29	.	.	PUNCT
ejpam-6256	183	1	proof	proof	NOUN
ejpam-6256	183	2	.	.	PUNCT
ejpam-6256	184	1	assume	assume	VERB
ejpam-6256	184	2	a	a	DET
ejpam-6256	184	3	=	=	X
ejpam-6256	184	4	{	{	PUNCT
ejpam-6256	184	5	a1	a1	PROPN
ejpam-6256	184	6	,	,	PUNCT
ejpam-6256	184	7	...	...	PUNCT
ejpam-6256	184	8	,	,	PUNCT
ejpam-6256	184	9	ak	ak	PROPN
ejpam-6256	184	10	}	}	PUNCT
ejpam-6256	184	11	satisfying	satisfy	VERB
ejpam-6256	184	12	(	(	PUNCT
ejpam-6256	184	13	4	4	NUM
ejpam-6256	184	14	)	)	PUNCT
ejpam-6256	184	15	and	and	CCONJ
ejpam-6256	184	16	set	set	VERB
ejpam-6256	184	17	αn	αn	NOUN
ejpam-6256	184	18	=	=	SYM
ejpam-6256	184	19	t	t	PROPN
ejpam-6256	184	20	s(an−ks	s(an−ks	PROPN
ejpam-6256	184	21	)	)	PUNCT
ejpam-6256	184	22	with	with	ADP
ejpam-6256	184	23	s	s	NOUN
ejpam-6256	184	24	=	=	PUNCT
ejpam-6256	184	25	int(n−1	int(n−1	PROPN
ejpam-6256	184	26	k	k	PROPN
ejpam-6256	184	27	)	)	PUNCT
ejpam-6256	184	28	as	as	SCONJ
ejpam-6256	184	29	defined	define	VERB
ejpam-6256	184	30	in	in	ADP
ejpam-6256	184	31	(	(	PUNCT
ejpam-6256	184	32	3	3	NUM
ejpam-6256	184	33	)	)	PUNCT
ejpam-6256	184	34	.	.	PUNCT
ejpam-6256	185	1	then	then	ADV
ejpam-6256	185	2	sup	sup	NOUN
ejpam-6256	185	3	ni≥m	ni≥m	PROPN
ejpam-6256	185	4	d(αn1	d(αn1	NOUN
ejpam-6256	185	5	,	,	PUNCT
ejpam-6256	185	6	...	...	PUNCT
ejpam-6256	185	7	,	,	PUNCT
ejpam-6256	185	8	αnu	αnu	NOUN
ejpam-6256	185	9	)	)	PUNCT
ejpam-6256	185	10	=	=	SYM
ejpam-6256	185	11	sup	sup	NOUN
ejpam-6256	185	12	ni≥m	ni≥m	PROPN
ejpam-6256	185	13	d(t	d(t	PROPN
ejpam-6256	185	14	s1(an1−ks1	s1(an1−ks1	PROPN
ejpam-6256	185	15	)	)	PUNCT
ejpam-6256	185	16	,	,	PUNCT
ejpam-6256	185	17	...	...	PUNCT
ejpam-6256	185	18	,	,	PUNCT
ejpam-6256	185	19	t	t	PROPN
ejpam-6256	185	20	su(anu−ksu	su(anu−ksu	NUM
ejpam-6256	185	21	)	)	PUNCT
ejpam-6256	185	22	)	)	PUNCT
ejpam-6256	186	1	=	=	SYM
ejpam-6256	186	2	sup	sup	NOUN
ejpam-6256	186	3	xi∈ss	xi∈ss	PROPN
ejpam-6256	186	4	,	,	PUNCT
ejpam-6256	186	5	l	l	PROPN
ejpam-6256	186	6	t	t	PROPN
ejpam-6256	186	7	(	(	PUNCT
ejpam-6256	186	8	a	a	X
ejpam-6256	186	9	)	)	PUNCT
ejpam-6256	186	10	d({x1	d({x1	PROPN
ejpam-6256	186	11	,	,	PUNCT
ejpam-6256	186	12	...	...	PUNCT
ejpam-6256	186	13	,	,	PUNCT
ejpam-6256	186	14	xu	xu	PROPN
ejpam-6256	186	15	}	}	PUNCT
ejpam-6256	186	16	)	)	PUNCT
ejpam-6256	186	17	(	(	PUNCT
ejpam-6256	186	18	by	by	ADP
ejpam-6256	186	19	proposition	proposition	NOUN
ejpam-6256	186	20	5-(i	5-(i	NUM
ejpam-6256	186	21	)	)	PUNCT
ejpam-6256	186	22	.	.	PUNCT
ejpam-6256	186	23	)	)	PUNCT
ejpam-6256	187	1	=	=	SYM
ejpam-6256	187	2	sup	sup	NOUN
ejpam-6256	187	3	xi∈ss−1,l	xi∈ss−1,l	PROPN
ejpam-6256	187	4	t	t	PROPN
ejpam-6256	187	5	(	(	PUNCT
ejpam-6256	187	6	a	a	NOUN
ejpam-6256	187	7	)	)	PUNCT
ejpam-6256	187	8	d(t	d(t	PROPN
ejpam-6256	187	9	(	(	PUNCT
ejpam-6256	187	10	{	{	PUNCT
ejpam-6256	187	11	x1	x1	PROPN
ejpam-6256	187	12	,	,	PUNCT
ejpam-6256	187	13	...	...	PUNCT
ejpam-6256	187	14	,	,	PUNCT
ejpam-6256	187	15	xu	xu	PROPN
ejpam-6256	187	16	}	}	PUNCT
ejpam-6256	187	17	)	)	PUNCT
ejpam-6256	187	18	)	)	PUNCT
ejpam-6256	187	19	(	(	PUNCT
ejpam-6256	187	20	by	by	ADP
ejpam-6256	187	21	proposition	proposition	NOUN
ejpam-6256	187	22	4-(ii	4-(ii	NUM
ejpam-6256	187	23	)	)	PUNCT
ejpam-6256	187	24	.	.	PUNCT
ejpam-6256	187	25	)	)	PUNCT
ejpam-6256	188	1	(	(	PUNCT
ejpam-6256	188	2	5	5	X
ejpam-6256	188	3	)	)	PUNCT
ejpam-6256	188	4	y.	y.	NOUN
ejpam-6256	188	5	alzubaidi	alzubaidi	PROPN
ejpam-6256	188	6	/	/	SYM
ejpam-6256	188	7	eur	eur	PROPN
ejpam-6256	188	8	.	.	PUNCT
ejpam-6256	189	1	j.	j.	PROPN
ejpam-6256	189	2	pure	pure	PROPN
ejpam-6256	189	3	appl	appl	PROPN
ejpam-6256	189	4	.	.	PROPN
ejpam-6256	189	5	math	math	PROPN
ejpam-6256	189	6	,	,	PUNCT
ejpam-6256	189	7	18	18	NUM
ejpam-6256	189	8	(	(	PUNCT
ejpam-6256	189	9	3	3	NUM
ejpam-6256	189	10	)	)	PUNCT
ejpam-6256	189	11	(	(	PUNCT
ejpam-6256	189	12	2025	2025	NUM
ejpam-6256	189	13	)	)	PUNCT
ejpam-6256	189	14	,	,	PUNCT
ejpam-6256	189	15	6256	6256	NUM
ejpam-6256	189	16	8	8	NUM
ejpam-6256	189	17	of	of	ADP
ejpam-6256	189	18	11	11	NUM
ejpam-6256	189	19	using	use	VERB
ejpam-6256	189	20	(	(	PUNCT
ejpam-6256	189	21	4	4	NUM
ejpam-6256	189	22	)	)	PUNCT
ejpam-6256	189	23	and	and	CCONJ
ejpam-6256	189	24	applying	apply	VERB
ejpam-6256	189	25	the	the	DET
ejpam-6256	189	26	standard	standard	ADJ
ejpam-6256	189	27	iteration	iteration	NOUN
ejpam-6256	189	28	keeping	keeping	NOUN
ejpam-6256	189	29	in	in	ADP
ejpam-6256	189	30	mind	mind	NOUN
ejpam-6256	189	31	the	the	DET
ejpam-6256	189	32	properties	property	NOUN
ejpam-6256	189	33	of	of	ADP
ejpam-6256	189	34	the	the	DET
ejpam-6256	189	35	sets	set	NOUN
ejpam-6256	189	36	ss	ss	PROPN
ejpam-6256	189	37	,	,	PUNCT
ejpam-6256	189	38	l	l	PROPN
ejpam-6256	189	39	t	t	NOUN
ejpam-6256	189	40	,	,	PUNCT
ejpam-6256	189	41	we	we	PRON
ejpam-6256	189	42	get	get	VERB
ejpam-6256	189	43	sup	sup	NOUN
ejpam-6256	189	44	xi∈ss−1,l	xi∈ss−1,l	PROPN
ejpam-6256	190	1	t	t	PROPN
ejpam-6256	190	2	(	(	PUNCT
ejpam-6256	190	3	a	a	NOUN
ejpam-6256	190	4	)	)	PUNCT
ejpam-6256	190	5	d(t	d(t	PROPN
ejpam-6256	190	6	(	(	PUNCT
ejpam-6256	190	7	{	{	PUNCT
ejpam-6256	190	8	x1	x1	PROPN
ejpam-6256	190	9	,	,	PUNCT
ejpam-6256	190	10	...	...	PUNCT
ejpam-6256	190	11	,	,	PUNCT
ejpam-6256	190	12	xu	xu	PROPN
ejpam-6256	190	13	}	}	PUNCT
ejpam-6256	190	14	)	)	PUNCT
ejpam-6256	190	15	)	)	PUNCT
ejpam-6256	190	16	≤	≤	NUM
ejpam-6256	191	1	q	q	NOUN
ejpam-6256	191	2	sup	sup	NOUN
ejpam-6256	191	3	xi∈ss−1,l	xi∈ss−1,l	PROPN
ejpam-6256	191	4	t	t	PROPN
ejpam-6256	191	5	(	(	PUNCT
ejpam-6256	191	6	a	a	X
ejpam-6256	191	7	)	)	PUNCT
ejpam-6256	191	8	d({x1	d({x1	PROPN
ejpam-6256	191	9	,	,	PUNCT
ejpam-6256	191	10	...	...	PUNCT
ejpam-6256	191	11	,	,	PUNCT
ejpam-6256	191	12	xu	xu	PROPN
ejpam-6256	191	13	}	}	PUNCT
ejpam-6256	191	14	)	)	PUNCT
ejpam-6256	192	1	+	+	CCONJ
ejpam-6256	192	2	pr	pr	NOUN
ejpam-6256	192	3	for	for	ADP
ejpam-6256	192	4	all	all	DET
ejpam-6256	192	5	u	u	PROPN
ejpam-6256	192	6	≥	≥	NOUN
ejpam-6256	192	7	n.	n.	NOUN
ejpam-6256	192	8	=	=	PUNCT
ejpam-6256	192	9	q	q	NOUN
ejpam-6256	193	1	sup	sup	NOUN
ejpam-6256	193	2	xi∈ss−2,l	xi∈ss−2,l	PROPN
ejpam-6256	194	1	t	t	PROPN
ejpam-6256	194	2	(	(	PUNCT
ejpam-6256	194	3	a	a	NOUN
ejpam-6256	194	4	)	)	PUNCT
ejpam-6256	194	5	d(t	d(t	PROPN
ejpam-6256	194	6	(	(	PUNCT
ejpam-6256	194	7	{	{	PUNCT
ejpam-6256	194	8	x1	x1	PROPN
ejpam-6256	194	9	,	,	PUNCT
ejpam-6256	194	10	...	...	PUNCT
ejpam-6256	194	11	,	,	PUNCT
ejpam-6256	194	12	xu	xu	PROPN
ejpam-6256	194	13	}	}	PUNCT
ejpam-6256	194	14	)	)	PUNCT
ejpam-6256	194	15	)	)	PUNCT
ejpam-6256	195	1	+	+	CCONJ
ejpam-6256	195	2	pr	pr	X
ejpam-6256	195	3	≤	≤	NUM
ejpam-6256	195	4	q[q	q[q	ADJ
ejpam-6256	195	5	sup	sup	NOUN
ejpam-6256	195	6	xi∈ss−2,l	xi∈ss−2,l	ADP
ejpam-6256	195	7	t	t	PROPN
ejpam-6256	195	8	(	(	PUNCT
ejpam-6256	195	9	a	a	X
ejpam-6256	195	10	)	)	PUNCT
ejpam-6256	195	11	d({x1	d({x1	PROPN
ejpam-6256	195	12	,	,	PUNCT
ejpam-6256	195	13	...	...	PUNCT
ejpam-6256	195	14	,	,	PUNCT
ejpam-6256	195	15	xu	xu	PROPN
ejpam-6256	195	16	}	}	PUNCT
ejpam-6256	195	17	)	)	PUNCT
ejpam-6256	196	1	+	+	CCONJ
ejpam-6256	196	2	pr	pr	X
ejpam-6256	196	3	]	]	X
ejpam-6256	196	4	+	+	CCONJ
ejpam-6256	196	5	pr	pr	NOUN
ejpam-6256	196	6	=	=	SYM
ejpam-6256	196	7	q2	q2	NOUN
ejpam-6256	196	8	sup	sup	NOUN
ejpam-6256	196	9	xi∈ss−2,l	xi∈ss−2,l	PROPN
ejpam-6256	197	1	t	t	PROPN
ejpam-6256	197	2	(	(	PUNCT
ejpam-6256	197	3	a	a	X
ejpam-6256	197	4	)	)	PUNCT
ejpam-6256	197	5	d({x1	d({x1	PROPN
ejpam-6256	197	6	,	,	PUNCT
ejpam-6256	197	7	...	...	PUNCT
ejpam-6256	197	8	,	,	PUNCT
ejpam-6256	197	9	xu	xu	PROPN
ejpam-6256	197	10	}	}	PUNCT
ejpam-6256	197	11	)	)	PUNCT
ejpam-6256	198	1	+	+	CCONJ
ejpam-6256	198	2	qpr	qpr	NOUN
ejpam-6256	198	3	+	+	CCONJ
ejpam-6256	198	4	pr	pr	NOUN
ejpam-6256	198	5	=	=	SYM
ejpam-6256	198	6	q2	q2	NOUN
ejpam-6256	198	7	sup	sup	NOUN
ejpam-6256	198	8	xi∈ss−3,l	xi∈ss−3,l	PROPN
ejpam-6256	199	1	t	t	PROPN
ejpam-6256	199	2	(	(	PUNCT
ejpam-6256	199	3	a	a	NOUN
ejpam-6256	199	4	)	)	PUNCT
ejpam-6256	199	5	d(t	d(t	PROPN
ejpam-6256	199	6	(	(	PUNCT
ejpam-6256	199	7	{	{	PUNCT
ejpam-6256	199	8	x1	x1	PROPN
ejpam-6256	199	9	,	,	PUNCT
ejpam-6256	199	10	...	...	PUNCT
ejpam-6256	199	11	,	,	PUNCT
ejpam-6256	199	12	xu	xu	PROPN
ejpam-6256	199	13	}	}	PUNCT
ejpam-6256	199	14	)	)	PUNCT
ejpam-6256	199	15	)	)	PUNCT
ejpam-6256	200	1	+	+	CCONJ
ejpam-6256	200	2	(	(	PUNCT
ejpam-6256	200	3	q0	q0	PROPN
ejpam-6256	200	4	+	+	NUM
ejpam-6256	200	5	q1)pr	q1)pr	PROPN
ejpam-6256	200	6	.	.	PUNCT
ejpam-6256	201	1	(	(	PUNCT
ejpam-6256	201	2	6	6	X
ejpam-6256	201	3	)	)	PUNCT
ejpam-6256	201	4	repeating	repeat	VERB
ejpam-6256	201	5	the	the	DET
ejpam-6256	201	6	above	above	ADJ
ejpam-6256	201	7	process	process	NOUN
ejpam-6256	201	8	,	,	PUNCT
ejpam-6256	201	9	we	we	PRON
ejpam-6256	201	10	get	get	VERB
ejpam-6256	201	11	sup	sup	NOUN
ejpam-6256	201	12	xi∈ss−1,l	xi∈ss−1,l	PROPN
ejpam-6256	201	13	t	t	PROPN
ejpam-6256	201	14	(	(	PUNCT
ejpam-6256	201	15	a	a	NOUN
ejpam-6256	201	16	)	)	PUNCT
ejpam-6256	201	17	d(t	d(t	PROPN
ejpam-6256	201	18	(	(	PUNCT
ejpam-6256	201	19	{	{	PUNCT
ejpam-6256	201	20	x1	x1	PROPN
ejpam-6256	201	21	,	,	PUNCT
ejpam-6256	201	22	...	...	PUNCT
ejpam-6256	201	23	,	,	PUNCT
ejpam-6256	201	24	xu	xu	PROPN
ejpam-6256	201	25	}	}	PUNCT
ejpam-6256	201	26	)	)	PUNCT
ejpam-6256	201	27	)	)	PUNCT
ejpam-6256	202	1	≤	≤	NUM
ejpam-6256	202	2	qs	qs	ADP
ejpam-6256	202	3	sup	sup	NOUN
ejpam-6256	202	4	xi∈s0	xi∈s0	PROPN
ejpam-6256	203	1	t	t	PROPN
ejpam-6256	203	2	(	(	PUNCT
ejpam-6256	203	3	a	a	X
ejpam-6256	203	4	)	)	PUNCT
ejpam-6256	203	5	d({x1	d({x1	PROPN
ejpam-6256	203	6	,	,	PUNCT
ejpam-6256	203	7	...	...	PUNCT
ejpam-6256	203	8	,	,	PUNCT
ejpam-6256	203	9	xu	xu	PROPN
ejpam-6256	203	10	}	}	PUNCT
ejpam-6256	203	11	)	)	PUNCT
ejpam-6256	204	1	+	+	CCONJ
ejpam-6256	204	2	(	(	PUNCT
ejpam-6256	204	3	q0	q0	PROPN
ejpam-6256	204	4	+	+	NUM
ejpam-6256	204	5	q1	q1	PROPN
ejpam-6256	204	6	+	+	CCONJ
ejpam-6256	204	7	...	...	PUNCT
ejpam-6256	204	8	+	+	CCONJ
ejpam-6256	204	9	qs−1)pr	qs−1)pr	NOUN
ejpam-6256	204	10	≤	≤	NUM
ejpam-6256	204	11	qs	qs	NOUN
ejpam-6256	204	12	sup	sup	NOUN
ejpam-6256	204	13	xi∈s0	xi∈s0	PROPN
ejpam-6256	204	14	t	t	PROPN
ejpam-6256	204	15	(	(	PUNCT
ejpam-6256	204	16	a	a	X
ejpam-6256	204	17	)	)	PUNCT
ejpam-6256	204	18	d({x1	d({x1	PROPN
ejpam-6256	204	19	,	,	PUNCT
ejpam-6256	204	20	...	...	PUNCT
ejpam-6256	204	21	,	,	PUNCT
ejpam-6256	204	22	xu	xu	PROPN
ejpam-6256	204	23	}	}	PUNCT
ejpam-6256	204	24	)	)	PUNCT
ejpam-6256	205	1	+	+	CCONJ
ejpam-6256	205	2	r	r	NOUN
ejpam-6256	205	3	for	for	ADP
ejpam-6256	205	4	all	all	DET
ejpam-6256	205	5	u	u	PROPN
ejpam-6256	205	6	≥	≥	PRON
ejpam-6256	205	7	n.	n.	NOUN
ejpam-6256	205	8	(	(	PUNCT
ejpam-6256	205	9	7	7	NUM
ejpam-6256	205	10	)	)	PUNCT
ejpam-6256	205	11	where	where	SCONJ
ejpam-6256	205	12	in	in	ADP
ejpam-6256	205	13	the	the	DET
ejpam-6256	205	14	last	last	ADJ
ejpam-6256	205	15	step	step	NOUN
ejpam-6256	205	16	we	we	PRON
ejpam-6256	205	17	used	use	VERB
ejpam-6256	205	18	(	(	PUNCT
ejpam-6256	205	19	q0	q0	PROPN
ejpam-6256	205	20	+	+	NUM
ejpam-6256	205	21	q1	q1	PROPN
ejpam-6256	205	22	+	+	CCONJ
ejpam-6256	205	23	...	...	PUNCT
ejpam-6256	205	24	+	+	CCONJ
ejpam-6256	205	25	qs−1)pr	qs−1)pr	NOUN
ejpam-6256	205	26	≤	≤	X
ejpam-6256	205	27	pr	pr	VERB
ejpam-6256	205	28	1−	1−	NUM
ejpam-6256	205	29	q	q	NOUN
ejpam-6256	205	30	≤	≤	ADJ
ejpam-6256	205	31	r	r	NOUN
ejpam-6256	205	32	,	,	PUNCT
ejpam-6256	205	33	since	since	SCONJ
ejpam-6256	205	34	p+	p+	NOUN
ejpam-6256	205	35	q	q	PROPN
ejpam-6256	205	36	≤	≤	NUM
ejpam-6256	205	37	1	1	NUM
ejpam-6256	205	38	⇐	⇐	ADJ
ejpam-6256	205	39	⇒	⇒	NOUN
ejpam-6256	205	40	p	p	X
ejpam-6256	205	41	1−	1−	NUM
ejpam-6256	205	42	q	q	PROPN
ejpam-6256	205	43	≤	≤	NUM
ejpam-6256	205	44	1	1	NUM
ejpam-6256	205	45	.	.	PUNCT
ejpam-6256	206	1	combining	combine	VERB
ejpam-6256	206	2	(	(	PUNCT
ejpam-6256	206	3	4	4	NUM
ejpam-6256	206	4	)	)	PUNCT
ejpam-6256	206	5	,	,	PUNCT
ejpam-6256	206	6	(	(	PUNCT
ejpam-6256	206	7	5	5	NUM
ejpam-6256	206	8	)	)	PUNCT
ejpam-6256	206	9	and	and	CCONJ
ejpam-6256	206	10	(	(	PUNCT
ejpam-6256	206	11	7	7	NUM
ejpam-6256	206	12	)	)	PUNCT
ejpam-6256	206	13	,	,	PUNCT
ejpam-6256	206	14	we	we	PRON
ejpam-6256	206	15	get	get	VERB
ejpam-6256	206	16	r	r	NOUN
ejpam-6256	206	17	≤	≤	NUM
ejpam-6256	206	18	sup	sup	NOUN
ejpam-6256	206	19	ni≥m	ni≥m	PROPN
ejpam-6256	206	20	d(αn1	d(αn1	NOUN
ejpam-6256	206	21	,	,	PUNCT
ejpam-6256	206	22	...	...	PUNCT
ejpam-6256	206	23	,	,	PUNCT
ejpam-6256	206	24	αnu	αnu	NOUN
ejpam-6256	206	25	)	)	PUNCT
ejpam-6256	206	26	≤	≤	NUM
ejpam-6256	206	27	qs	qs	ADP
ejpam-6256	206	28	sup	sup	NOUN
ejpam-6256	206	29	xi∈s0	xi∈s0	PROPN
ejpam-6256	207	1	t	t	PROPN
ejpam-6256	207	2	(	(	PUNCT
ejpam-6256	207	3	a	a	X
ejpam-6256	207	4	)	)	PUNCT
ejpam-6256	207	5	d({x1	d({x1	PROPN
ejpam-6256	207	6	,	,	PUNCT
ejpam-6256	207	7	...	...	PUNCT
ejpam-6256	207	8	,	,	PUNCT
ejpam-6256	207	9	xu	xu	PROPN
ejpam-6256	207	10	}	}	PUNCT
ejpam-6256	207	11	)	)	PUNCT
ejpam-6256	208	1	+	+	CCONJ
ejpam-6256	208	2	r	r	X
ejpam-6256	208	3	,	,	PUNCT
ejpam-6256	208	4	for	for	ADP
ejpam-6256	208	5	all	all	DET
ejpam-6256	208	6	u	u	PROPN
ejpam-6256	208	7	≥	≥	PRON
ejpam-6256	208	8	n.	n.	NOUN
ejpam-6256	208	9	(	(	PUNCT
ejpam-6256	208	10	8)	8)	NUM
ejpam-6256	208	11	using	use	VERB
ejpam-6256	208	12	the	the	DET
ejpam-6256	208	13	boundedness	boundedness	NOUN
ejpam-6256	208	14	of	of	ADP
ejpam-6256	208	15	(	(	PUNCT
ejpam-6256	208	16	x	x	X
ejpam-6256	208	17	,	,	PUNCT
ejpam-6256	208	18	d	d	NOUN
ejpam-6256	208	19	)	)	PUNCT
ejpam-6256	208	20	and	and	CCONJ
ejpam-6256	208	21	the	the	DET
ejpam-6256	208	22	fact	fact	NOUN
ejpam-6256	208	23	m	m	VERB
ejpam-6256	208	24	→	→	SYM
ejpam-6256	208	25	∞	∞	NUM
ejpam-6256	208	26	⇐	⇐	ADJ
ejpam-6256	208	27	⇒	⇒	NOUN
ejpam-6256	208	28	s	s	PART
ejpam-6256	208	29	→	→	SYM
ejpam-6256	208	30	∞	∞	PROPN
ejpam-6256	208	31	,	,	PUNCT
ejpam-6256	208	32	we	we	PRON
ejpam-6256	208	33	get	get	VERB
ejpam-6256	208	34	lim	lim	PROPN
ejpam-6256	208	35	m→∞	m→∞	NUM
ejpam-6256	208	36	sup	sup	NOUN
ejpam-6256	208	37	ni≥m	ni≥m	PROPN
ejpam-6256	208	38	d(αn1	d(αn1	NOUN
ejpam-6256	208	39	,	,	PUNCT
ejpam-6256	208	40	...	...	PUNCT
ejpam-6256	208	41	,	,	PUNCT
ejpam-6256	208	42	αnu	αnu	NOUN
ejpam-6256	208	43	)	)	PUNCT
ejpam-6256	208	44	=	=	SYM
ejpam-6256	209	1	r	r	NOUN
ejpam-6256	209	2	,	,	PUNCT
ejpam-6256	209	3	for	for	ADP
ejpam-6256	209	4	all	all	DET
ejpam-6256	209	5	u	u	PROPN
ejpam-6256	209	6	≥	≥	PRON
ejpam-6256	209	7	n.	n.	NOUN
ejpam-6256	209	8	(	(	PUNCT
ejpam-6256	209	9	9	9	NUM
ejpam-6256	209	10	)	)	PUNCT
ejpam-6256	209	11	therefore	therefore	ADV
ejpam-6256	209	12	,	,	PUNCT
ejpam-6256	209	13	(	(	PUNCT
ejpam-6256	209	14	αn	αn	X
ejpam-6256	209	15	)	)	PUNCT
ejpam-6256	209	16	is	be	AUX
ejpam-6256	209	17	r	r	NOUN
ejpam-6256	209	18	-	-	PUNCT
ejpam-6256	209	19	cauchy	cauchy	ADJ
ejpam-6256	209	20	sequence	sequence	NOUN
ejpam-6256	209	21	and	and	CCONJ
ejpam-6256	209	22	hence	hence	ADV
ejpam-6256	209	23	by	by	ADP
ejpam-6256	209	24	the	the	DET
ejpam-6256	209	25	completeness	completeness	NOUN
ejpam-6256	209	26	of	of	ADP
ejpam-6256	209	27	the	the	DET
ejpam-6256	209	28	mp	mp	NOUN
ejpam-6256	209	29	-	-	PUNCT
ejpam-6256	209	30	metric	metric	ADJ
ejpam-6256	209	31	space	space	NOUN
ejpam-6256	209	32	(	(	PUNCT
ejpam-6256	209	33	x	x	X
ejpam-6256	209	34	,	,	PUNCT
ejpam-6256	209	35	d	d	NOUN
ejpam-6256	209	36	)	)	PUNCT
ejpam-6256	209	37	there	there	PRON
ejpam-6256	209	38	is	be	VERB
ejpam-6256	209	39	b	b	PROPN
ejpam-6256	209	40	∈	∈	PROPN
ejpam-6256	209	41	p	p	NOUN
ejpam-6256	209	42	∗(x	∗(x	PROPN
ejpam-6256	209	43	)	)	PUNCT
ejpam-6256	210	1	such	such	ADJ
ejpam-6256	210	2	that	that	SCONJ
ejpam-6256	210	3	d	d	PROPN
ejpam-6256	210	4	lim	lim	PROPN
ejpam-6256	210	5	n→∞	n→∞	X
ejpam-6256	210	6	αn	αn	NOUN
ejpam-6256	210	7	=	=	SYM
ejpam-6256	210	8	d(b	d(b	X
ejpam-6256	210	9	)	)	PUNCT
ejpam-6256	210	10	(	(	PUNCT
ejpam-6256	210	11	10	10	NUM
ejpam-6256	210	12	)	)	PUNCT
ejpam-6256	210	13	in	in	ADP
ejpam-6256	210	14	particular	particular	ADJ
ejpam-6256	210	15	lim	lim	PROPN
ejpam-6256	210	16	inf	inf	PROPN
ejpam-6256	210	17	n→∞	n→∞	X
ejpam-6256	210	18	d(αn	d(αn	PROPN
ejpam-6256	210	19	,	,	PUNCT
ejpam-6256	210	20	b	b	NOUN
ejpam-6256	210	21	)	)	PUNCT
ejpam-6256	210	22	=	=	SYM
ejpam-6256	210	23	0	0	NUM
ejpam-6256	210	24	for	for	ADP
ejpam-6256	210	25	all	all	DET
ejpam-6256	210	26	b	b	PROPN
ejpam-6256	210	27	∈	∈	ADP
ejpam-6256	210	28	b	b	PROPN
ejpam-6256	210	29	(	(	PUNCT
ejpam-6256	210	30	see	see	VERB
ejpam-6256	210	31	definition	definition	NOUN
ejpam-6256	210	32	3	3	NUM
ejpam-6256	210	33	)	)	PUNCT
ejpam-6256	211	1	.	.	PUNCT
ejpam-6256	212	1	hence	hence	ADV
ejpam-6256	212	2	,	,	PUNCT
ejpam-6256	212	3	there	there	PRON
ejpam-6256	212	4	is	be	VERB
ejpam-6256	212	5	a	a	DET
ejpam-6256	212	6	subsequence	subsequence	NOUN
ejpam-6256	212	7	(	(	PUNCT
ejpam-6256	212	8	αnm	αnm	NOUN
ejpam-6256	212	9	)	)	PUNCT
ejpam-6256	212	10	such	such	ADJ
ejpam-6256	212	11	that	that	SCONJ
ejpam-6256	212	12	lim	lim	PROPN
ejpam-6256	212	13	nm→∞	nm→∞	PROPN
ejpam-6256	212	14	d(αnm	d(αnm	PROPN
ejpam-6256	212	15	,	,	PUNCT
ejpam-6256	212	16	b	b	X
ejpam-6256	212	17	)	)	PUNCT
ejpam-6256	212	18	=	=	SYM
ejpam-6256	212	19	0	0	X
ejpam-6256	212	20	.	.	PUNCT
ejpam-6256	213	1	by	by	ADP
ejpam-6256	213	2	proposition	proposition	NOUN
ejpam-6256	213	3	6	6	NUM
ejpam-6256	213	4	,	,	PUNCT
ejpam-6256	213	5	and	and	CCONJ
ejpam-6256	213	6	since	since	SCONJ
ejpam-6256	213	7	a	a	PRON
ejpam-6256	213	8	is	be	AUX
ejpam-6256	213	9	finite	finite	ADJ
ejpam-6256	213	10	,	,	PUNCT
ejpam-6256	213	11	there	there	PRON
ejpam-6256	213	12	is	be	VERB
ejpam-6256	213	13	a	a	DET
ejpam-6256	213	14	∈	∈	PROPN
ejpam-6256	213	15	a	a	DET
ejpam-6256	213	16	such	such	ADJ
ejpam-6256	213	17	that	that	PRON
ejpam-6256	213	18	(	(	PUNCT
ejpam-6256	213	19	t	t	NOUN
ejpam-6256	213	20	sl(a	sl(a	NUM
ejpam-6256	213	21	)	)	PUNCT
ejpam-6256	213	22	)	)	PUNCT
ejpam-6256	214	1	is	be	AUX
ejpam-6256	214	2	a	a	DET
ejpam-6256	214	3	subsequence	subsequence	NOUN
ejpam-6256	214	4	of	of	ADP
ejpam-6256	214	5	(	(	PUNCT
ejpam-6256	214	6	αnm	αnm	NOUN
ejpam-6256	214	7	)	)	PUNCT
ejpam-6256	214	8	.	.	PUNCT
ejpam-6256	215	1	thus	thus	ADV
ejpam-6256	215	2	,	,	PUNCT
ejpam-6256	215	3	using	use	VERB
ejpam-6256	215	4	the	the	DET
ejpam-6256	215	5	continuity	continuity	NOUN
ejpam-6256	215	6	of	of	ADP
ejpam-6256	215	7	d	d	PROPN
ejpam-6256	215	8	and	and	CCONJ
ejpam-6256	215	9	t	t	PROPN
ejpam-6256	215	10	,	,	PUNCT
ejpam-6256	215	11	we	we	PRON
ejpam-6256	215	12	obtain	obtain	VERB
ejpam-6256	215	13	0	0	NUM
ejpam-6256	216	1	=	=	SYM
ejpam-6256	216	2	lim	lim	PROPN
ejpam-6256	216	3	sl→∞	sl→∞	PROPN
ejpam-6256	216	4	d(t	d(t	PROPN
ejpam-6256	216	5	sl(a	sl(a	NUM
ejpam-6256	216	6	)	)	PUNCT
ejpam-6256	216	7	,	,	PUNCT
ejpam-6256	216	8	b	b	X
ejpam-6256	216	9	)	)	PUNCT
ejpam-6256	217	1	=	=	SYM
ejpam-6256	217	2	lim	lim	PROPN
ejpam-6256	217	3	sl→∞	sl→∞	PROPN
ejpam-6256	217	4	d(t	d(t	PROPN
ejpam-6256	217	5	(	(	PUNCT
ejpam-6256	217	6	t	t	PROPN
ejpam-6256	217	7	sl−1(a	sl−1(a	PROPN
ejpam-6256	217	8	)	)	PUNCT
ejpam-6256	217	9	)	)	PUNCT
ejpam-6256	217	10	,	,	PUNCT
ejpam-6256	217	11	b	b	X
ejpam-6256	217	12	)	)	PUNCT
ejpam-6256	217	13	=	=	SYM
ejpam-6256	217	14	d(t	d(t	PROPN
ejpam-6256	217	15	(	(	PUNCT
ejpam-6256	217	16	lim	lim	PROPN
ejpam-6256	217	17	sl→∞	sl→∞	PROPN
ejpam-6256	217	18	t	t	PROPN
ejpam-6256	217	19	sl−1(a	sl−1(a	PROPN
ejpam-6256	217	20	)	)	PUNCT
ejpam-6256	217	21	)	)	PUNCT
ejpam-6256	217	22	,	,	PUNCT
ejpam-6256	217	23	b	b	X
ejpam-6256	217	24	)	)	PUNCT
ejpam-6256	217	25	=	=	SYM
ejpam-6256	217	26	d(t	d(t	PROPN
ejpam-6256	217	27	(	(	PUNCT
ejpam-6256	217	28	b	b	NOUN
ejpam-6256	217	29	)	)	PUNCT
ejpam-6256	217	30	,	,	PUNCT
ejpam-6256	217	31	b	b	X
ejpam-6256	217	32	)	)	PUNCT
ejpam-6256	217	33	y.	y.	NOUN
ejpam-6256	217	34	alzubaidi	alzubaidi	PROPN
ejpam-6256	217	35	/	/	SYM
ejpam-6256	217	36	eur	eur	PROPN
ejpam-6256	217	37	.	.	PUNCT
ejpam-6256	218	1	j.	j.	PROPN
ejpam-6256	218	2	pure	pure	PROPN
ejpam-6256	218	3	appl	appl	PROPN
ejpam-6256	218	4	.	.	PROPN
ejpam-6256	218	5	math	math	PROPN
ejpam-6256	218	6	,	,	PUNCT
ejpam-6256	218	7	18	18	NUM
ejpam-6256	218	8	(	(	PUNCT
ejpam-6256	218	9	3	3	NUM
ejpam-6256	218	10	)	)	PUNCT
ejpam-6256	218	11	(	(	PUNCT
ejpam-6256	218	12	2025	2025	NUM
ejpam-6256	218	13	)	)	PUNCT
ejpam-6256	218	14	,	,	PUNCT
ejpam-6256	218	15	6256	6256	NUM
ejpam-6256	218	16	9	9	NUM
ejpam-6256	218	17	of	of	ADP
ejpam-6256	218	18	11	11	NUM
ejpam-6256	218	19	that	that	PRON
ejpam-6256	218	20	is	be	AUX
ejpam-6256	218	21	,	,	PUNCT
ejpam-6256	218	22	b	b	PRON
ejpam-6256	218	23	is	be	AUX
ejpam-6256	218	24	a	a	DET
ejpam-6256	218	25	fixed	fix	VERB
ejpam-6256	218	26	point	point	NOUN
ejpam-6256	218	27	for	for	ADP
ejpam-6256	218	28	all	all	DET
ejpam-6256	218	29	b	b	PROPN
ejpam-6256	218	30	∈	∈	PROPN
ejpam-6256	218	31	b.	b.	NOUN
ejpam-6256	219	1	it	it	PRON
ejpam-6256	219	2	remains	remain	VERB
ejpam-6256	219	3	to	to	PART
ejpam-6256	219	4	estimate	estimate	VERB
ejpam-6256	219	5	the	the	DET
ejpam-6256	219	6	number	number	NOUN
ejpam-6256	219	7	of	of	ADP
ejpam-6256	219	8	the	the	DET
ejpam-6256	219	9	fixed	fix	VERB
ejpam-6256	219	10	points	point	NOUN
ejpam-6256	219	11	.	.	PUNCT
ejpam-6256	220	1	using	use	VERB
ejpam-6256	220	2	the	the	DET
ejpam-6256	220	3	properties	property	NOUN
ejpam-6256	220	4	of	of	ADP
ejpam-6256	220	5	the	the	DET
ejpam-6256	220	6	mp	mp	NOUN
ejpam-6256	220	7	-	-	PUNCT
ejpam-6256	220	8	metric	metric	NOUN
ejpam-6256	220	9	,	,	PUNCT
ejpam-6256	220	10	we	we	PRON
ejpam-6256	220	11	have	have	VERB
ejpam-6256	220	12	d(αn1	d(αn1	NOUN
ejpam-6256	220	13	,	,	PUNCT
ejpam-6256	220	14	...	...	PUNCT
ejpam-6256	220	15	,	,	PUNCT
ejpam-6256	220	16	αnu	αnu	NOUN
ejpam-6256	220	17	)	)	PUNCT
ejpam-6256	221	1	≤	≤	NOUN
ejpam-6256	222	1	d({αn1	d({αn1	NOUN
ejpam-6256	222	2	,	,	PUNCT
ejpam-6256	222	3	...	...	PUNCT
ejpam-6256	222	4	,	,	PUNCT
ejpam-6256	222	5	αnu	αnu	NOUN
ejpam-6256	222	6	}	}	PUNCT
ejpam-6256	222	7	∪b	∪b	VERB
ejpam-6256	222	8	)	)	PUNCT
ejpam-6256	222	9	(	(	PUNCT
ejpam-6256	222	10	11	11	NUM
ejpam-6256	222	11	)	)	PUNCT
ejpam-6256	222	12	which	which	PRON
ejpam-6256	222	13	in	in	ADP
ejpam-6256	222	14	turn	turn	NOUN
ejpam-6256	222	15	gives	give	VERB
ejpam-6256	222	16	r	r	NOUN
ejpam-6256	222	17	=	=	SYM
ejpam-6256	222	18	lim	lim	PROPN
ejpam-6256	222	19	m→∞	m→∞	NUM
ejpam-6256	222	20	sup	sup	NOUN
ejpam-6256	222	21	ni≥m	ni≥m	PROPN
ejpam-6256	222	22	d(αn1	d(αn1	NOUN
ejpam-6256	222	23	,	,	PUNCT
ejpam-6256	222	24	...	...	PUNCT
ejpam-6256	222	25	,	,	PUNCT
ejpam-6256	222	26	αnu	αnu	NOUN
ejpam-6256	222	27	)	)	PUNCT
ejpam-6256	222	28	≤	≤	NOUN
ejpam-6256	222	29	lim	lim	PROPN
ejpam-6256	222	30	m.→∞	m.→∞	PROPN
ejpam-6256	222	31	sup	sup	PROPN
ejpam-6256	222	32	ni≥m	ni≥m	PROPN
ejpam-6256	222	33	d({αn1	d({αn1	NOUN
ejpam-6256	222	34	,	,	PUNCT
ejpam-6256	222	35	...	...	PUNCT
ejpam-6256	222	36	,	,	PUNCT
ejpam-6256	222	37	αnu	αnu	NOUN
ejpam-6256	222	38	}	}	PUNCT
ejpam-6256	222	39	∪b	∪b	VERB
ejpam-6256	222	40	)	)	PUNCT
ejpam-6256	222	41	(	(	PUNCT
ejpam-6256	222	42	12	12	NUM
ejpam-6256	222	43	)	)	PUNCT
ejpam-6256	222	44	moreover	moreover	ADV
ejpam-6256	222	45	,	,	PUNCT
ejpam-6256	222	46	in	in	ADP
ejpam-6256	222	47	view	view	NOUN
ejpam-6256	222	48	of	of	ADP
ejpam-6256	222	49	equation	equation	NOUN
ejpam-6256	222	50	(	(	PUNCT
ejpam-6256	222	51	10	10	NUM
ejpam-6256	222	52	)	)	PUNCT
ejpam-6256	222	53	and	and	CCONJ
ejpam-6256	222	54	definition	definition	NOUN
ejpam-6256	222	55	3	3	NUM
ejpam-6256	222	56	,	,	PUNCT
ejpam-6256	222	57	we	we	PRON
ejpam-6256	222	58	observe	observe	VERB
ejpam-6256	222	59	that	that	SCONJ
ejpam-6256	222	60	lim	lim	PROPN
ejpam-6256	222	61	ni→∞	ni→∞	PROPN
ejpam-6256	222	62	d({αn1	d({αn1	NOUN
ejpam-6256	222	63	,	,	PUNCT
ejpam-6256	222	64	...	...	PUNCT
ejpam-6256	222	65	,	,	PUNCT
ejpam-6256	222	66	αnu}∪	αnu}∪	PROPN
ejpam-6256	222	67	b	b	NOUN
ejpam-6256	222	68	)	)	PUNCT
ejpam-6256	222	69	exists	exist	VERB
ejpam-6256	222	70	and	and	CCONJ
ejpam-6256	222	71	is	be	AUX
ejpam-6256	222	72	equal	equal	ADJ
ejpam-6256	222	73	to	to	ADP
ejpam-6256	222	74	d(b	d(b	PROPN
ejpam-6256	222	75	)	)	PUNCT
ejpam-6256	222	76	.	.	PUNCT
ejpam-6256	223	1	hence	hence	ADV
ejpam-6256	223	2	,	,	PUNCT
ejpam-6256	223	3	lim	lim	PROPN
ejpam-6256	223	4	m→∞	m→∞	NOUN
ejpam-6256	223	5	sup	sup	NOUN
ejpam-6256	223	6	ni≥m	ni≥m	PROPN
ejpam-6256	223	7	d({αn1	d({αn1	NOUN
ejpam-6256	223	8	,	,	PUNCT
ejpam-6256	223	9	...	...	PUNCT
ejpam-6256	223	10	,	,	PUNCT
ejpam-6256	223	11	αnu}∪b	αnu}∪b	PROPN
ejpam-6256	223	12	)	)	PUNCT
ejpam-6256	223	13	also	also	ADV
ejpam-6256	223	14	exists	exist	VERB
ejpam-6256	223	15	,	,	PUNCT
ejpam-6256	223	16	and	and	CCONJ
ejpam-6256	223	17	we	we	PRON
ejpam-6256	223	18	have	have	VERB
ejpam-6256	223	19	that	that	DET
ejpam-6256	223	20	lim	lim	PROPN
ejpam-6256	223	21	m→∞	m→∞	NOUN
ejpam-6256	223	22	sup	sup	NOUN
ejpam-6256	223	23	ni≥m	ni≥m	PROPN
ejpam-6256	223	24	d({αn1	d({αn1	NOUN
ejpam-6256	223	25	,	,	PUNCT
ejpam-6256	223	26	...	...	PUNCT
ejpam-6256	223	27	,	,	PUNCT
ejpam-6256	223	28	αnu	αnu	NOUN
ejpam-6256	223	29	}	}	PUNCT
ejpam-6256	223	30	∪b	∪b	NOUN
ejpam-6256	223	31	)	)	PUNCT
ejpam-6256	224	1	=	=	VERB
ejpam-6256	224	2	lim	lim	PROPN
ejpam-6256	224	3	ni→∞	ni→∞	PROPN
ejpam-6256	224	4	d({αn1	d({αn1	NOUN
ejpam-6256	224	5	,	,	PUNCT
ejpam-6256	224	6	...	...	PUNCT
ejpam-6256	224	7	,	,	PUNCT
ejpam-6256	224	8	αnu	αnu	NOUN
ejpam-6256	224	9	}	}	PUNCT
ejpam-6256	224	10	∪b	∪b	NOUN
ejpam-6256	224	11	)	)	PUNCT
ejpam-6256	224	12	=	=	SYM
ejpam-6256	224	13	d(b	d(b	X
ejpam-6256	224	14	)	)	PUNCT
ejpam-6256	224	15	(	(	PUNCT
ejpam-6256	224	16	13	13	X
ejpam-6256	224	17	)	)	PUNCT
ejpam-6256	224	18	combining	combine	VERB
ejpam-6256	224	19	(	(	PUNCT
ejpam-6256	224	20	12	12	NUM
ejpam-6256	224	21	)	)	PUNCT
ejpam-6256	224	22	and	and	CCONJ
ejpam-6256	224	23	(	(	PUNCT
ejpam-6256	224	24	13	13	NUM
ejpam-6256	224	25	)	)	PUNCT
ejpam-6256	224	26	,	,	PUNCT
ejpam-6256	224	27	we	we	PRON
ejpam-6256	224	28	get	get	VERB
ejpam-6256	224	29	r	r	NOUN
ejpam-6256	224	30	≤	≤	NUM
ejpam-6256	224	31	d(b	d(b	PROPN
ejpam-6256	224	32	)	)	PUNCT
ejpam-6256	224	33	)	)	PUNCT
ejpam-6256	225	1	(	(	PUNCT
ejpam-6256	225	2	14	14	NUM
ejpam-6256	225	3	)	)	PUNCT
ejpam-6256	225	4	as	as	ADP
ejpam-6256	225	5	a	a	DET
ejpam-6256	225	6	result	result	NOUN
ejpam-6256	225	7	,	,	PUNCT
ejpam-6256	225	8	d(b	d(b	PROPN
ejpam-6256	225	9	)	)	PUNCT
ejpam-6256	225	10	>	>	X
ejpam-6256	225	11	0	0	PUNCT
ejpam-6256	226	1	(	(	PUNCT
ejpam-6256	226	2	since	since	SCONJ
ejpam-6256	226	3	r	r	NOUN
ejpam-6256	226	4	is	be	AUX
ejpam-6256	226	5	positive	positive	ADJ
ejpam-6256	226	6	)	)	PUNCT
ejpam-6256	226	7	,	,	PUNCT
ejpam-6256	226	8	which	which	PRON
ejpam-6256	226	9	according	accord	VERB
ejpam-6256	226	10	to	to	ADP
ejpam-6256	226	11	axiom	axiom	NOUN
ejpam-6256	226	12	(	(	PUNCT
ejpam-6256	226	13	a1	a1	NOUN
ejpam-6256	226	14	)	)	PUNCT
ejpam-6256	226	15	implies	imply	VERB
ejpam-6256	226	16	that	that	SCONJ
ejpam-6256	226	17	b	b	PROPN
ejpam-6256	226	18	contains	contain	VERB
ejpam-6256	226	19	at	at	ADV
ejpam-6256	226	20	least	least	ADV
ejpam-6256	226	21	two	two	NUM
ejpam-6256	226	22	distinct	distinct	ADJ
ejpam-6256	226	23	elements	element	NOUN
ejpam-6256	226	24	,	,	PUNCT
ejpam-6256	226	25	that	that	PRON
ejpam-6256	226	26	are	be	AUX
ejpam-6256	226	27	fixed	fix	VERB
ejpam-6256	226	28	points	point	NOUN
ejpam-6256	226	29	.	.	PUNCT
ejpam-6256	227	1	this	this	PRON
ejpam-6256	227	2	concludes	conclude	VERB
ejpam-6256	227	3	the	the	DET
ejpam-6256	227	4	proof	proof	NOUN
ejpam-6256	227	5	.	.	PUNCT
ejpam-6256	228	1	example	example	NOUN
ejpam-6256	228	2	3	3	X
ejpam-6256	228	3	.	.	X
ejpam-6256	229	1	consider	consider	VERB
ejpam-6256	229	2	the	the	DET
ejpam-6256	229	3	mp	mp	NOUN
ejpam-6256	229	4	-	-	PUNCT
ejpam-6256	229	5	metric	metric	ADJ
ejpam-6256	229	6	d	d	NOUN
ejpam-6256	229	7	:	:	PUNCT
ejpam-6256	229	8	p	p	NOUN
ejpam-6256	229	9	∗([0	∗([0	NOUN
ejpam-6256	229	10	,	,	PUNCT
ejpam-6256	229	11	1	1	NUM
ejpam-6256	229	12	]	]	PUNCT
ejpam-6256	229	13	)	)	PUNCT
ejpam-6256	230	1	−→	−→	NOUN
ejpam-6256	230	2	[	[	X
ejpam-6256	230	3	0,∞	0,∞	NOUN
ejpam-6256	230	4	)	)	PUNCT
ejpam-6256	230	5	defined	define	VERB
ejpam-6256	230	6	as	as	ADP
ejpam-6256	230	7	d(a	d(a	PROPN
ejpam-6256	230	8	)	)	PUNCT
ejpam-6256	230	9	=	=	SYM
ejpam-6256	230	10	max(a)−	max(a)−	NOUN
ejpam-6256	230	11	min(a	min(a	PROPN
ejpam-6256	230	12	)	)	PUNCT
ejpam-6256	230	13	.	.	PUNCT
ejpam-6256	231	1	the	the	DET
ejpam-6256	231	2	space	space	NOUN
ejpam-6256	231	3	(	(	PUNCT
ejpam-6256	231	4	[	[	X
ejpam-6256	231	5	0	0	NUM
ejpam-6256	231	6	,	,	PUNCT
ejpam-6256	231	7	1	1	NUM
ejpam-6256	231	8	]	]	PUNCT
ejpam-6256	231	9	,	,	PUNCT
ejpam-6256	231	10	d	d	X
ejpam-6256	231	11	)	)	PUNCT
ejpam-6256	231	12	is	be	AUX
ejpam-6256	231	13	a	a	DET
ejpam-6256	231	14	bounded	bounded	ADJ
ejpam-6256	231	15	complete	complete	ADJ
ejpam-6256	231	16	mp	mp	NOUN
ejpam-6256	231	17	-	-	PUNCT
ejpam-6256	231	18	metric	metric	ADJ
ejpam-6256	231	19	space	space	NOUN
ejpam-6256	231	20	.	.	PUNCT
ejpam-6256	232	1	if	if	SCONJ
ejpam-6256	232	2	xn	xn	PROPN
ejpam-6256	232	3	is	be	AUX
ejpam-6256	232	4	r	r	NOUN
ejpam-6256	232	5	-	-	NOUN
ejpam-6256	232	6	cauchy	cauchy	NOUN
ejpam-6256	232	7	for	for	ADP
ejpam-6256	232	8	some	some	DET
ejpam-6256	232	9	r	r	NOUN
ejpam-6256	232	10	>	>	X
ejpam-6256	232	11	0	0	PUNCT
ejpam-6256	233	1	in	in	ADP
ejpam-6256	233	2	(	(	PUNCT
ejpam-6256	233	3	[	[	X
ejpam-6256	233	4	0	0	NUM
ejpam-6256	233	5	,	,	PUNCT
ejpam-6256	233	6	1	1	NUM
ejpam-6256	233	7	]	]	PUNCT
ejpam-6256	233	8	,	,	PUNCT
ejpam-6256	233	9	d	d	NOUN
ejpam-6256	233	10	)	)	PUNCT
ejpam-6256	233	11	,	,	PUNCT
ejpam-6256	233	12	then	then	ADV
ejpam-6256	233	13	there	there	PRON
ejpam-6256	233	14	are	be	VERB
ejpam-6256	233	15	a	a	DET
ejpam-6256	233	16	,	,	PUNCT
ejpam-6256	233	17	b	b	X
ejpam-6256	233	18	∈	∈	PROPN
ejpam-6256	234	1	[	[	X
ejpam-6256	234	2	0	0	NUM
ejpam-6256	234	3	,	,	PUNCT
ejpam-6256	234	4	1	1	NUM
ejpam-6256	234	5	]	]	PUNCT
ejpam-6256	234	6	such	such	ADJ
ejpam-6256	234	7	that	that	SCONJ
ejpam-6256	234	8	xn	xn	PROPN
ejpam-6256	234	9	is	be	AUX
ejpam-6256	234	10	d	d	NOUN
ejpam-6256	234	11	-	-	NOUN
ejpam-6256	234	12	convergent	convergent	ADJ
ejpam-6256	234	13	to	to	ADP
ejpam-6256	234	14	d({a	d({a	PROPN
ejpam-6256	234	15	,	,	PUNCT
ejpam-6256	234	16	b	b	NOUN
ejpam-6256	234	17	}	}	PUNCT
ejpam-6256	234	18	)	)	PUNCT
ejpam-6256	234	19	.	.	PUNCT
ejpam-6256	235	1	let	let	VERB
ejpam-6256	235	2	f	f	NOUN
ejpam-6256	235	3	:	:	PUNCT
ejpam-6256	236	1	[	[	X
ejpam-6256	236	2	0	0	NUM
ejpam-6256	236	3	,	,	PUNCT
ejpam-6256	236	4	1	1	NUM
ejpam-6256	236	5	]	]	X
ejpam-6256	236	6	−→	−→	NOUN
ejpam-6256	236	7	[	[	X
ejpam-6256	236	8	0	0	NUM
ejpam-6256	236	9	,	,	PUNCT
ejpam-6256	236	10	1	1	NUM
ejpam-6256	236	11	]	]	PUNCT
ejpam-6256	236	12	be	be	AUX
ejpam-6256	236	13	defined	define	VERB
ejpam-6256	236	14	as	as	ADP
ejpam-6256	236	15	f	f	PROPN
ejpam-6256	236	16	(	(	PUNCT
ejpam-6256	236	17	x	x	NOUN
ejpam-6256	236	18	)	)	PUNCT
ejpam-6256	236	19	=	=	SYM
ejpam-6256	236	20	xt	xt	X
ejpam-6256	236	21	for	for	ADP
ejpam-6256	236	22	some	some	DET
ejpam-6256	236	23	t	t	PROPN
ejpam-6256	236	24	>	>	X
ejpam-6256	237	1	1	1	X
ejpam-6256	237	2	.	.	PUNCT
ejpam-6256	237	3	then	then	ADV
ejpam-6256	237	4	,	,	PUNCT
ejpam-6256	237	5	choosing	choose	VERB
ejpam-6256	237	6	a	a	DET
ejpam-6256	237	7	=	=	X
ejpam-6256	237	8	{	{	PUNCT
ejpam-6256	237	9	1	1	NUM
ejpam-6256	237	10	2	2	NUM
ejpam-6256	237	11	,	,	PUNCT
ejpam-6256	237	12	1	1	NUM
ejpam-6256	237	13	}	}	PUNCT
ejpam-6256	237	14	,	,	PUNCT
ejpam-6256	237	15	we	we	PRON
ejpam-6256	237	16	have	have	VERB
ejpam-6256	237	17	the	the	DET
ejpam-6256	237	18	sequence	sequence	NOUN
ejpam-6256	237	19	(	(	PUNCT
ejpam-6256	237	20	αn	αn	NOUN
ejpam-6256	237	21	)	)	PUNCT
ejpam-6256	237	22	=	=	PRON
ejpam-6256	237	23	{	{	PUNCT
ejpam-6256	237	24	1	1	NUM
ejpam-6256	237	25	2	2	NUM
ejpam-6256	237	26	,	,	PUNCT
ejpam-6256	237	27	1	1	NUM
ejpam-6256	237	28	,	,	PUNCT
ejpam-6256	237	29	(	(	PUNCT
ejpam-6256	237	30	1	1	NUM
ejpam-6256	237	31	2	2	NUM
ejpam-6256	237	32	)	)	PUNCT
ejpam-6256	237	33	t	t	PROPN
ejpam-6256	237	34	,	,	PUNCT
ejpam-6256	237	35	1	1	NUM
ejpam-6256	237	36	,	,	PUNCT
ejpam-6256	237	37	(	(	PUNCT
ejpam-6256	237	38	12	12	NUM
ejpam-6256	237	39	)	)	PUNCT
ejpam-6256	237	40	t2	t2	NOUN
ejpam-6256	237	41	,	,	PUNCT
ejpam-6256	237	42	1	1	NUM
ejpam-6256	237	43	,	,	PUNCT
ejpam-6256	237	44	...	...	PUNCT
ejpam-6256	237	45	}	}	PUNCT
ejpam-6256	237	46	.	.	PUNCT
ejpam-6256	238	1	therefore	therefore	ADV
ejpam-6256	238	2	,	,	PUNCT
ejpam-6256	238	3	if	if	SCONJ
ejpam-6256	238	4	we	we	PRON
ejpam-6256	238	5	let	let	VERB
ejpam-6256	238	6	r	r	NOUN
ejpam-6256	238	7	=	=	SYM
ejpam-6256	238	8	1	1	NUM
ejpam-6256	238	9	,	,	PUNCT
ejpam-6256	238	10	n	n	NOUN
ejpam-6256	238	11	=	=	SYM
ejpam-6256	238	12	2	2	NUM
ejpam-6256	238	13	and	and	CCONJ
ejpam-6256	238	14	p	p	NOUN
ejpam-6256	238	15	=	=	PROPN
ejpam-6256	238	16	1−	1−	NUM
ejpam-6256	238	17	q	q	NOUN
ejpam-6256	238	18	where	where	SCONJ
ejpam-6256	238	19	q	q	NOUN
ejpam-6256	238	20	can	can	AUX
ejpam-6256	238	21	be	be	AUX
ejpam-6256	238	22	any	any	DET
ejpam-6256	238	23	number	number	NOUN
ejpam-6256	238	24	in	in	ADP
ejpam-6256	238	25	(	(	PUNCT
ejpam-6256	238	26	0	0	NUM
ejpam-6256	238	27	,	,	PUNCT
ejpam-6256	238	28	1	1	NUM
ejpam-6256	238	29	)	)	PUNCT
ejpam-6256	238	30	,	,	PUNCT
ejpam-6256	238	31	we	we	PRON
ejpam-6256	238	32	get	get	VERB
ejpam-6256	238	33	1	1	NUM
ejpam-6256	238	34	≤	≤	NUM
ejpam-6256	238	35	sup	sup	NOUN
ejpam-6256	238	36	xi∈ss	xi∈ss	PROPN
ejpam-6256	238	37	,	,	PUNCT
ejpam-6256	238	38	l	l	PROPN
ejpam-6256	238	39	f	f	X
ejpam-6256	238	40	(	(	PUNCT
ejpam-6256	238	41	a	a	NOUN
ejpam-6256	238	42	)	)	PUNCT
ejpam-6256	238	43	d(f	d(f	NOUN
ejpam-6256	238	44	(	(	PUNCT
ejpam-6256	238	45	{	{	PUNCT
ejpam-6256	238	46	x1	x1	ADJ
ejpam-6256	238	47	,	,	PUNCT
ejpam-6256	238	48	...	...	PUNCT
ejpam-6256	238	49	,	,	PUNCT
ejpam-6256	238	50	xu	xu	PROPN
ejpam-6256	238	51	}	}	PUNCT
ejpam-6256	238	52	)	)	PUNCT
ejpam-6256	238	53	)	)	PUNCT
ejpam-6256	238	54	≤	≤	NUM
ejpam-6256	239	1	q	q	NOUN
ejpam-6256	239	2	sup	sup	NOUN
ejpam-6256	239	3	xi∈ss	xi∈ss	PROPN
ejpam-6256	239	4	,	,	PUNCT
ejpam-6256	239	5	l	l	PROPN
ejpam-6256	239	6	f	f	X
ejpam-6256	239	7	(	(	PUNCT
ejpam-6256	239	8	a	a	X
ejpam-6256	239	9	)	)	PUNCT
ejpam-6256	239	10	d({x1	d({x1	PROPN
ejpam-6256	239	11	,	,	PUNCT
ejpam-6256	239	12	...	...	PUNCT
ejpam-6256	239	13	,	,	PUNCT
ejpam-6256	239	14	xu	xu	PROPN
ejpam-6256	239	15	}	}	PUNCT
ejpam-6256	239	16	)	)	PUNCT
ejpam-6256	240	1	+	+	CCONJ
ejpam-6256	240	2	(	(	PUNCT
ejpam-6256	240	3	1−	1−	NUM
ejpam-6256	240	4	q	q	NOUN
ejpam-6256	240	5	)	)	PUNCT
ejpam-6256	240	6	for	for	ADP
ejpam-6256	240	7	all	all	DET
ejpam-6256	240	8	u	u	NOUN
ejpam-6256	240	9	≥	≥	NOUN
ejpam-6256	240	10	2	2	NUM
ejpam-6256	240	11	.	.	PUNCT
ejpam-6256	241	1	thus	thus	ADV
ejpam-6256	241	2	,	,	PUNCT
ejpam-6256	241	3	f	f	PROPN
ejpam-6256	241	4	and	and	CCONJ
ejpam-6256	241	5	(	(	PUNCT
ejpam-6256	241	6	[	[	X
ejpam-6256	241	7	0	0	NUM
ejpam-6256	241	8	,	,	PUNCT
ejpam-6256	241	9	1	1	NUM
ejpam-6256	241	10	]	]	PUNCT
ejpam-6256	241	11	,	,	PUNCT
ejpam-6256	241	12	d	d	X
ejpam-6256	241	13	)	)	PUNCT
ejpam-6256	241	14	satisfy	satisfy	VERB
ejpam-6256	241	15	the	the	DET
ejpam-6256	241	16	conditions	condition	NOUN
ejpam-6256	241	17	of	of	ADP
ejpam-6256	241	18	theorem	theorem	NOUN
ejpam-6256	241	19	3	3	X
ejpam-6256	241	20	.	.	PUNCT
ejpam-6256	242	1	we	we	PRON
ejpam-6256	242	2	can	can	AUX
ejpam-6256	242	3	check	check	VERB
ejpam-6256	242	4	that	that	PRON
ejpam-6256	242	5	d	d	PROPN
ejpam-6256	242	6	lim	lim	PROPN
ejpam-6256	242	7	n→∞	n→∞	NUM
ejpam-6256	242	8	αn	αn	NOUN
ejpam-6256	242	9	=	=	SYM
ejpam-6256	242	10	d({0	d({0	ADJ
ejpam-6256	242	11	,	,	PUNCT
ejpam-6256	242	12	1	1	NUM
ejpam-6256	242	13	}	}	PUNCT
ejpam-6256	242	14	)	)	PUNCT
ejpam-6256	242	15	.	.	PUNCT
ejpam-6256	243	1	it	it	PRON
ejpam-6256	243	2	is	be	AUX
ejpam-6256	243	3	clear	clear	ADJ
ejpam-6256	243	4	that	that	SCONJ
ejpam-6256	243	5	0	0	NUM
ejpam-6256	243	6	and	and	CCONJ
ejpam-6256	243	7	1	1	NUM
ejpam-6256	243	8	are	be	AUX
ejpam-6256	243	9	fixed	fix	VERB
ejpam-6256	243	10	points	point	NOUN
ejpam-6256	243	11	for	for	ADP
ejpam-6256	243	12	f	f	PROPN
ejpam-6256	243	13	.	.	PUNCT
ejpam-6256	244	1	moreover	moreover	ADV
ejpam-6256	244	2	,	,	PUNCT
ejpam-6256	244	3	it	it	PRON
ejpam-6256	244	4	can	can	AUX
ejpam-6256	244	5	be	be	AUX
ejpam-6256	244	6	observed	observe	VERB
ejpam-6256	244	7	that	that	SCONJ
ejpam-6256	244	8	they	they	PRON
ejpam-6256	244	9	can	can	AUX
ejpam-6256	244	10	be	be	AUX
ejpam-6256	244	11	obtained	obtain	VERB
ejpam-6256	244	12	as	as	ADP
ejpam-6256	244	13	iterative	iterative	ADJ
ejpam-6256	244	14	fixed	fix	VERB
ejpam-6256	244	15	points	point	NOUN
ejpam-6256	244	16	:	:	PUNCT
ejpam-6256	244	17	lim	lim	PROPN
ejpam-6256	244	18	n→∞	n→∞	NUM
ejpam-6256	244	19	fn(12	fn(12	PROPN
ejpam-6256	244	20	)	)	PUNCT
ejpam-6256	244	21	=	=	SYM
ejpam-6256	244	22	0	0	NUM
ejpam-6256	244	23	,	,	PUNCT
ejpam-6256	244	24	and	and	CCONJ
ejpam-6256	244	25	lim	lim	PROPN
ejpam-6256	244	26	n→∞	n→∞	X
ejpam-6256	244	27	fn(1	fn(1	NOUN
ejpam-6256	244	28	)	)	PUNCT
ejpam-6256	244	29	=	=	SYM
ejpam-6256	245	1	1	1	NUM
ejpam-6256	245	2	.	.	NOUN
ejpam-6256	245	3	4	4	NUM
ejpam-6256	245	4	.	.	X
ejpam-6256	245	5	conclusion	conclusion	NOUN
ejpam-6256	245	6	in	in	ADP
ejpam-6256	245	7	this	this	DET
ejpam-6256	245	8	research	research	NOUN
ejpam-6256	245	9	paper	paper	NOUN
ejpam-6256	245	10	,	,	PUNCT
ejpam-6256	245	11	we	we	PRON
ejpam-6256	245	12	examine	examine	VERB
ejpam-6256	245	13	a	a	DET
ejpam-6256	245	14	class	class	NOUN
ejpam-6256	245	15	of	of	ADP
ejpam-6256	245	16	functions	function	NOUN
ejpam-6256	245	17	that	that	PRON
ejpam-6256	245	18	possess	possess	VERB
ejpam-6256	245	19	multiple	multiple	ADJ
ejpam-6256	245	20	fixed	fix	VERB
ejpam-6256	245	21	points	point	NOUN
ejpam-6256	245	22	,	,	PUNCT
ejpam-6256	245	23	which	which	PRON
ejpam-6256	245	24	can	can	AUX
ejpam-6256	245	25	be	be	AUX
ejpam-6256	245	26	obtained	obtain	VERB
ejpam-6256	245	27	through	through	ADP
ejpam-6256	245	28	iterative	iterative	NOUN
ejpam-6256	245	29	methods	method	NOUN
ejpam-6256	245	30	.	.	PUNCT
ejpam-6256	246	1	to	to	PART
ejpam-6256	246	2	achieve	achieve	VERB
ejpam-6256	246	3	this	this	DET
ejpam-6256	246	4	objective	objective	NOUN
ejpam-6256	246	5	,	,	PUNCT
ejpam-6256	246	6	we	we	PRON
ejpam-6256	246	7	conducted	conduct	VERB
ejpam-6256	246	8	our	our	PRON
ejpam-6256	246	9	study	study	NOUN
ejpam-6256	246	10	within	within	ADP
ejpam-6256	246	11	the	the	DET
ejpam-6256	246	12	framework	framework	NOUN
ejpam-6256	246	13	of	of	ADP
ejpam-6256	246	14	generalized	generalized	ADJ
ejpam-6256	246	15	mp	mp	NOUN
ejpam-6256	246	16	-	-	PUNCT
ejpam-6256	246	17	metric	metric	ADJ
ejpam-6256	246	18	spaces	space	NOUN
ejpam-6256	246	19	,	,	PUNCT
ejpam-6256	246	20	leveraging	leverage	VERB
ejpam-6256	246	21	y.	y.	NOUN
ejpam-6256	246	22	alzubaidi	alzubaidi	PROPN
ejpam-6256	246	23	/	/	SYM
ejpam-6256	246	24	eur	eur	PROPN
ejpam-6256	246	25	.	.	PUNCT
ejpam-6256	247	1	j.	j.	PROPN
ejpam-6256	247	2	pure	pure	PROPN
ejpam-6256	247	3	appl	appl	PROPN
ejpam-6256	247	4	.	.	PROPN
ejpam-6256	247	5	math	math	PROPN
ejpam-6256	247	6	,	,	PUNCT
ejpam-6256	247	7	18	18	NUM
ejpam-6256	247	8	(	(	PUNCT
ejpam-6256	247	9	3	3	NUM
ejpam-6256	247	10	)	)	PUNCT
ejpam-6256	247	11	(	(	PUNCT
ejpam-6256	247	12	2025	2025	NUM
ejpam-6256	247	13	)	)	PUNCT
ejpam-6256	247	14	,	,	PUNCT
ejpam-6256	247	15	6256	6256	NUM
ejpam-6256	247	16	10	10	NUM
ejpam-6256	247	17	of	of	ADP
ejpam-6256	247	18	11	11	NUM
ejpam-6256	247	19	their	their	PRON
ejpam-6256	247	20	comprehensive	comprehensive	ADJ
ejpam-6256	247	21	convergence	convergence	NOUN
ejpam-6256	247	22	concepts	concept	NOUN
ejpam-6256	247	23	.	.	PUNCT
ejpam-6256	248	1	furthermore	furthermore	ADV
ejpam-6256	248	2	,	,	PUNCT
ejpam-6256	248	3	by	by	ADP
ejpam-6256	248	4	implementing	implement	VERB
ejpam-6256	248	5	an	an	DET
ejpam-6256	248	6	iterative	iterative	NOUN
ejpam-6256	248	7	process	process	NOUN
ejpam-6256	248	8	,	,	PUNCT
ejpam-6256	248	9	we	we	PRON
ejpam-6256	248	10	established	establish	VERB
ejpam-6256	248	11	the	the	DET
ejpam-6256	248	12	existence	existence	NOUN
ejpam-6256	248	13	of	of	ADP
ejpam-6256	248	14	multiple	multiple	ADJ
ejpam-6256	248	15	fixed	fix	VERB
ejpam-6256	248	16	points	point	NOUN
ejpam-6256	248	17	for	for	ADP
ejpam-6256	248	18	this	this	DET
ejpam-6256	248	19	class	class	NOUN
ejpam-6256	248	20	of	of	ADP
ejpam-6256	248	21	functions	function	NOUN
ejpam-6256	248	22	that	that	PRON
ejpam-6256	248	23	satisfy	satisfy	VERB
ejpam-6256	248	24	general	general	ADJ
ejpam-6256	248	25	contraction	contraction	NOUN
ejpam-6256	248	26	conditions	condition	NOUN
ejpam-6256	248	27	.	.	PUNCT
ejpam-6256	249	1	our	our	PRON
ejpam-6256	249	2	results	result	NOUN
ejpam-6256	249	3	open	open	VERB
ejpam-6256	249	4	several	several	ADJ
ejpam-6256	249	5	avenues	avenue	NOUN
ejpam-6256	249	6	for	for	ADP
ejpam-6256	249	7	further	far	ADV
ejpam-6256	249	8	research.for	research.for	NOUN
ejpam-6256	249	9	instance	instance	NOUN
ejpam-6256	249	10	,	,	PUNCT
ejpam-6256	249	11	potential	potential	ADJ
ejpam-6256	249	12	applications	application	NOUN
ejpam-6256	249	13	can	can	AUX
ejpam-6256	249	14	be	be	AUX
ejpam-6256	249	15	explored	explore	VERB
ejpam-6256	249	16	in	in	ADP
ejpam-6256	249	17	function	function	NOUN
ejpam-6256	249	18	spaces	space	NOUN
ejpam-6256	249	19	where	where	SCONJ
ejpam-6256	249	20	an	an	DET
ejpam-6256	249	21	mp	mp	NOUN
ejpam-6256	249	22	-	-	ADJ
ejpam-6256	249	23	metric	metric	ADJ
ejpam-6256	249	24	d	d	PROPN
ejpam-6256	249	25	on	on	ADP
ejpam-6256	249	26	a	a	DET
ejpam-6256	249	27	set	set	NOUN
ejpam-6256	249	28	x	x	PUNCT
ejpam-6256	249	29	is	be	AUX
ejpam-6256	249	30	extended	extend	VERB
ejpam-6256	249	31	to	to	ADP
ejpam-6256	249	32	an	an	DET
ejpam-6256	249	33	mp	mp	NOUN
ejpam-6256	249	34	-	-	PUNCT
ejpam-6256	249	35	metric	metric	ADJ
ejpam-6256	249	36	δ	δ	PROPN
ejpam-6256	249	37	on	on	ADP
ejpam-6256	249	38	certain	certain	ADJ
ejpam-6256	249	39	spaces	space	NOUN
ejpam-6256	249	40	of	of	ADP
ejpam-6256	249	41	functions	function	NOUN
ejpam-6256	249	42	over	over	ADP
ejpam-6256	249	43	x	x	PUNCT
ejpam-6256	249	44	by	by	ADP
ejpam-6256	249	45	employing	employ	VERB
ejpam-6256	249	46	integration	integration	NOUN
ejpam-6256	249	47	and	and	CCONJ
ejpam-6256	249	48	setting	set	VERB
ejpam-6256	249	49	δ(f1	δ(f1	NOUN
ejpam-6256	249	50	,	,	PUNCT
ejpam-6256	249	51	f2	f2	PROPN
ejpam-6256	249	52	,	,	PUNCT
ejpam-6256	249	53	...	...	PUNCT
ejpam-6256	249	54	,	,	PUNCT
ejpam-6256	249	55	fk	fk	INTJ
ejpam-6256	249	56	)	)	PUNCT
ejpam-6256	249	57	=	=	SYM
ejpam-6256	250	1	∫	∫	PROPN
ejpam-6256	250	2	x	x	PUNCT
ejpam-6256	250	3	d(f1(x	d(f1(x	PROPN
ejpam-6256	250	4	)	)	PUNCT
ejpam-6256	250	5	,	,	PUNCT
ejpam-6256	250	6	f2(x	f2(x	PROPN
ejpam-6256	250	7	)	)	PUNCT
ejpam-6256	250	8	,	,	PUNCT
ejpam-6256	250	9	...	...	PUNCT
ejpam-6256	250	10	,	,	PUNCT
ejpam-6256	250	11	fk(x	fk(x	PROPN
ejpam-6256	250	12	)	)	PUNCT
ejpam-6256	250	13	)	)	PUNCT
ejpam-6256	250	14	.	.	PUNCT
ejpam-6256	251	1	in	in	ADP
ejpam-6256	251	2	this	this	DET
ejpam-6256	251	3	context	context	NOUN
ejpam-6256	251	4	,	,	PUNCT
ejpam-6256	251	5	we	we	PRON
ejpam-6256	251	6	consider	consider	VERB
ejpam-6256	251	7	two	two	NUM
ejpam-6256	251	8	functions	function	NOUN
ejpam-6256	251	9	to	to	PART
ejpam-6256	251	10	be	be	AUX
ejpam-6256	251	11	equal	equal	ADJ
ejpam-6256	251	12	if	if	SCONJ
ejpam-6256	251	13	they	they	PRON
ejpam-6256	251	14	coincide	coincide	VERB
ejpam-6256	251	15	almost	almost	ADV
ejpam-6256	251	16	everywhere	everywhere	ADV
ejpam-6256	251	17	.	.	PUNCT
ejpam-6256	252	1	furthermore	furthermore	ADV
ejpam-6256	252	2	,	,	PUNCT
ejpam-6256	252	3	we	we	PRON
ejpam-6256	252	4	can	can	AUX
ejpam-6256	252	5	verify	verify	VERB
ejpam-6256	252	6	that	that	SCONJ
ejpam-6256	252	7	the	the	DET
ejpam-6256	252	8	space	space	NOUN
ejpam-6256	252	9	of	of	ADP
ejpam-6256	252	10	continuous	continuous	ADJ
ejpam-6256	252	11	functions	function	NOUN
ejpam-6256	252	12	c[a	c[a	NUM
ejpam-6256	252	13	,	,	PUNCT
ejpam-6256	252	14	b	b	NOUN
ejpam-6256	252	15	]	]	X
ejpam-6256	252	16	,	,	PUNCT
ejpam-6256	252	17	equipped	equip	VERB
ejpam-6256	252	18	with	with	ADP
ejpam-6256	252	19	the	the	DET
ejpam-6256	252	20	mp	mp	NOUN
ejpam-6256	252	21	-	-	PUNCT
ejpam-6256	252	22	metric	metric	PROPN
ejpam-6256	252	23	δ	δ	PROPN
ejpam-6256	252	24	,	,	PUNCT
ejpam-6256	252	25	is	be	AUX
ejpam-6256	252	26	not	not	PART
ejpam-6256	252	27	a	a	DET
ejpam-6256	252	28	complete	complete	ADJ
ejpam-6256	252	29	mp	mp	NOUN
ejpam-6256	252	30	-	-	PUNCT
ejpam-6256	252	31	metric	metric	ADJ
ejpam-6256	252	32	space	space	NOUN
ejpam-6256	252	33	.	.	PUNCT
ejpam-6256	253	1	thus	thus	ADV
ejpam-6256	253	2	,	,	PUNCT
ejpam-6256	253	3	a	a	DET
ejpam-6256	253	4	prominent	prominent	ADJ
ejpam-6256	253	5	area	area	NOUN
ejpam-6256	253	6	of	of	ADP
ejpam-6256	253	7	research	research	NOUN
ejpam-6256	253	8	is	be	AUX
ejpam-6256	253	9	the	the	DET
ejpam-6256	253	10	investigation	investigation	NOUN
ejpam-6256	253	11	of	of	ADP
ejpam-6256	253	12	the	the	DET
ejpam-6256	253	13	mp	mp	NOUN
ejpam-6256	253	14	-	-	PUNCT
ejpam-6256	253	15	completion	completion	NOUN
ejpam-6256	253	16	of	of	ADP
ejpam-6256	253	17	the	the	DET
ejpam-6256	253	18	space	space	NOUN
ejpam-6256	253	19	c[a	c[a	NOUN
ejpam-6256	253	20	,	,	PUNCT
ejpam-6256	253	21	b	b	NOUN
ejpam-6256	253	22	]	]	X
ejpam-6256	253	23	.	.	PUNCT
ejpam-6256	254	1	this	this	DET
ejpam-6256	254	2	exploration	exploration	NOUN
ejpam-6256	254	3	will	will	AUX
ejpam-6256	254	4	lay	lay	VERB
ejpam-6256	254	5	a	a	DET
ejpam-6256	254	6	foundation	foundation	NOUN
ejpam-6256	254	7	for	for	ADP
ejpam-6256	254	8	studying	study	VERB
ejpam-6256	254	9	mp	mp	NOUN
ejpam-6256	254	10	-	-	ADJ
ejpam-6256	254	11	metric	metric	ADJ
ejpam-6256	254	12	spaces	space	NOUN
ejpam-6256	254	13	of	of	ADP
ejpam-6256	254	14	functions	function	NOUN
ejpam-6256	254	15	,	,	PUNCT
ejpam-6256	254	16	while	while	SCONJ
ejpam-6256	254	17	also	also	ADV
ejpam-6256	254	18	facilitating	facilitate	VERB
ejpam-6256	254	19	the	the	DET
ejpam-6256	254	20	examination	examination	NOUN
ejpam-6256	254	21	of	of	ADP
ejpam-6256	254	22	potential	potential	ADJ
ejpam-6256	254	23	applications	application	NOUN
ejpam-6256	254	24	for	for	ADP
ejpam-6256	254	25	our	our	PRON
ejpam-6256	254	26	results	result	NOUN
ejpam-6256	254	27	,	,	PUNCT
ejpam-6256	254	28	such	such	ADJ
ejpam-6256	254	29	as	as	ADP
ejpam-6256	254	30	studying	study	VERB
ejpam-6256	254	31	the	the	DET
ejpam-6256	254	32	existence	existence	NOUN
ejpam-6256	254	33	of	of	ADP
ejpam-6256	254	34	multiple	multiple	ADJ
ejpam-6256	254	35	solutions	solution	NOUN
ejpam-6256	254	36	for	for	ADP
ejpam-6256	254	37	certain	certain	ADJ
ejpam-6256	254	38	systems	system	NOUN
ejpam-6256	254	39	.	.	PUNCT
ejpam-6256	255	1	another	another	DET
ejpam-6256	255	2	potential	potential	ADJ
ejpam-6256	255	3	direction	direction	NOUN
ejpam-6256	255	4	for	for	ADP
ejpam-6256	255	5	future	future	ADJ
ejpam-6256	255	6	work	work	NOUN
ejpam-6256	255	7	is	be	AUX
ejpam-6256	255	8	the	the	DET
ejpam-6256	255	9	investigation	investigation	NOUN
ejpam-6256	255	10	of	of	ADP
ejpam-6256	255	11	the	the	DET
ejpam-6256	255	12	geometric	geometric	ADJ
ejpam-6256	255	13	aspects	aspect	NOUN
ejpam-6256	255	14	of	of	ADP
ejpam-6256	255	15	multiple	multiple	ADJ
ejpam-6256	255	16	fixed	fix	VERB
ejpam-6256	255	17	points	point	NOUN
ejpam-6256	255	18	.	.	PUNCT
ejpam-6256	256	1	in	in	ADP
ejpam-6256	256	2	this	this	DET
ejpam-6256	256	3	regard	regard	NOUN
ejpam-6256	256	4	,	,	PUNCT
ejpam-6256	256	5	the	the	DET
ejpam-6256	256	6	concept	concept	NOUN
ejpam-6256	256	7	of	of	ADP
ejpam-6256	256	8	fixed	fix	VERB
ejpam-6256	256	9	discs	disc	NOUN
ejpam-6256	256	10	,	,	PUNCT
ejpam-6256	256	11	where	where	SCONJ
ejpam-6256	256	12	individual	individual	ADJ
ejpam-6256	256	13	points	point	NOUN
ejpam-6256	256	14	are	be	AUX
ejpam-6256	256	15	replaced	replace	VERB
ejpam-6256	256	16	by	by	ADP
ejpam-6256	256	17	discs	disc	NOUN
ejpam-6256	256	18	,	,	PUNCT
ejpam-6256	256	19	becomes	become	VERB
ejpam-6256	256	20	particularly	particularly	ADV
ejpam-6256	256	21	relevant	relevant	ADJ
ejpam-6256	256	22	.	.	PUNCT
ejpam-6256	257	1	this	this	DET
ejpam-6256	257	2	idea	idea	NOUN
ejpam-6256	257	3	has	have	AUX
ejpam-6256	257	4	been	be	AUX
ejpam-6256	257	5	studied	study	VERB
ejpam-6256	257	6	in	in	ADP
ejpam-6256	257	7	both	both	PRON
ejpam-6256	257	8	standard	standard	ADJ
ejpam-6256	257	9	metric	metric	ADJ
ejpam-6256	257	10	spaces	space	NOUN
ejpam-6256	257	11	and	and	CCONJ
ejpam-6256	257	12	some	some	DET
ejpam-6256	257	13	generalized	generalized	ADJ
ejpam-6256	257	14	metric	metric	ADJ
ejpam-6256	257	15	spaces	space	NOUN
ejpam-6256	257	16	(	(	PUNCT
ejpam-6256	257	17	see	see	VERB
ejpam-6256	257	18	,	,	PUNCT
ejpam-6256	257	19	for	for	ADP
ejpam-6256	257	20	example	example	NOUN
ejpam-6256	257	21	,	,	PUNCT
ejpam-6256	257	22	[	[	X
ejpam-6256	257	23	22	22	NUM
ejpam-6256	257	24	]	]	PUNCT
ejpam-6256	257	25	and	and	CCONJ
ejpam-6256	257	26	[	[	X
ejpam-6256	257	27	23	23	NUM
ejpam-6256	257	28	]	]	PUNCT
ejpam-6256	257	29	)	)	PUNCT
ejpam-6256	257	30	.	.	PUNCT
ejpam-6256	258	1	in	in	ADP
ejpam-6256	258	2	connection	connection	NOUN
ejpam-6256	258	3	with	with	ADP
ejpam-6256	258	4	our	our	PRON
ejpam-6256	258	5	work	work	NOUN
ejpam-6256	258	6	,	,	PUNCT
ejpam-6256	258	7	the	the	DET
ejpam-6256	258	8	theory	theory	NOUN
ejpam-6256	258	9	of	of	ADP
ejpam-6256	258	10	fixed	fix	VERB
ejpam-6256	258	11	discs	disc	NOUN
ejpam-6256	258	12	appears	appear	VERB
ejpam-6256	258	13	to	to	PART
ejpam-6256	258	14	be	be	AUX
ejpam-6256	258	15	compatible	compatible	ADJ
ejpam-6256	258	16	with	with	ADP
ejpam-6256	258	17	the	the	DET
ejpam-6256	258	18	framework	framework	NOUN
ejpam-6256	258	19	of	of	ADP
ejpam-6256	258	20	mp	mp	NOUN
ejpam-6256	258	21	-	-	PUNCT
ejpam-6256	258	22	metric	metric	ADJ
ejpam-6256	258	23	spaces	space	NOUN
ejpam-6256	258	24	and	and	CCONJ
ejpam-6256	258	25	aligns	align	VERB
ejpam-6256	258	26	naturally	naturally	ADV
ejpam-6256	258	27	with	with	ADP
ejpam-6256	258	28	the	the	DET
ejpam-6256	258	29	set	set	NOUN
ejpam-6256	258	30	-	-	PUNCT
ejpam-6256	258	31	valued	value	VERB
ejpam-6256	258	32	nature	nature	NOUN
ejpam-6256	258	33	of	of	ADP
ejpam-6256	258	34	limits	limit	NOUN
ejpam-6256	258	35	within	within	ADP
ejpam-6256	258	36	this	this	DET
ejpam-6256	258	37	setting	setting	NOUN
ejpam-6256	258	38	.	.	PUNCT
ejpam-6256	259	1	acknowledgements	acknowledgement	NOUN
ejpam-6256	259	2	the	the	DET
ejpam-6256	259	3	author	author	NOUN
ejpam-6256	259	4	extends	extend	VERB
ejpam-6256	259	5	his	his	PRON
ejpam-6256	259	6	appreciation	appreciation	NOUN
ejpam-6256	259	7	to	to	ADP
ejpam-6256	259	8	umm	umm	INTJ
ejpam-6256	259	9	al	al	PROPN
ejpam-6256	259	10	-	-	PUNCT
ejpam-6256	259	11	qura	qura	PROPN
ejpam-6256	259	12	university	university	PROPN
ejpam-6256	259	13	,	,	PUNCT
ejpam-6256	259	14	saudi	saudi	PROPN
ejpam-6256	259	15	arabia	arabia	PROPN
ejpam-6256	259	16	,	,	PUNCT
ejpam-6256	259	17	for	for	ADP
ejpam-6256	259	18	funding	fund	VERB
ejpam-6256	259	19	this	this	DET
ejpam-6256	259	20	research	research	NOUN
ejpam-6256	259	21	work	work	NOUN
ejpam-6256	259	22	through	through	ADP
ejpam-6256	259	23	grant	grant	NOUN
ejpam-6256	259	24	number	number	NOUN
ejpam-6256	259	25	:	:	PUNCT
ejpam-6256	259	26	25uqu4230209gssr01	25uqu4230209gssr01	NUM
ejpam-6256	259	27	references	reference	NOUN
ejpam-6256	259	28	[	[	X
ejpam-6256	259	29	1	1	NUM
ejpam-6256	259	30	]	]	X
ejpam-6256	259	31	d.	d.	PROPN
ejpam-6256	259	32	panthi	panthi	PROPN
ejpam-6256	259	33	and	and	CCONJ
ejpam-6256	259	34	p.	p.	PROPN
ejpam-6256	259	35	k.	k.	PROPN
ejpam-6256	260	1	jha	jha	PROPN
ejpam-6256	260	2	.	.	PUNCT
ejpam-6256	261	1	a	a	DET
ejpam-6256	261	2	short	short	ADJ
ejpam-6256	261	3	survey	survey	NOUN
ejpam-6256	261	4	on	on	ADP
ejpam-6256	261	5	fixed	fix	VERB
ejpam-6256	261	6	point	point	NOUN
ejpam-6256	261	7	results	result	NOUN
ejpam-6256	261	8	and	and	CCONJ
ejpam-6256	261	9	applications	application	NOUN
ejpam-6256	261	10	.	.	PUNCT
ejpam-6256	262	1	international	international	ADJ
ejpam-6256	262	2	journal	journal	PROPN
ejpam-6256	262	3	of	of	ADP
ejpam-6256	262	4	statistics	statistic	NOUN
ejpam-6256	262	5	and	and	CCONJ
ejpam-6256	262	6	applied	apply	VERB
ejpam-6256	262	7	mathematics	mathematic	NOUN
ejpam-6256	262	8	,	,	PUNCT
ejpam-6256	262	9	2(5):25–34	2(5):25–34	NUM
ejpam-6256	262	10	,	,	PUNCT
ejpam-6256	262	11	2017	2017	NUM
ejpam-6256	262	12	.	.	PUNCT
ejpam-6256	263	1	[	[	X
ejpam-6256	263	2	2	2	NUM
ejpam-6256	263	3	]	]	X
ejpam-6256	263	4	f.	f.	PROPN
ejpam-6256	263	5	kamalov	kamalov	PROPN
ejpam-6256	263	6	and	and	CCONJ
ejpam-6256	263	7	h.	h.	PROPN
ejpam-6256	263	8	h.	h.	PROPN
ejpam-6256	263	9	leung	leung	PROPN
ejpam-6256	263	10	.	.	PUNCT
ejpam-6256	264	1	fixed	fix	VERB
ejpam-6256	264	2	point	point	NOUN
ejpam-6256	264	3	theory	theory	NOUN
ejpam-6256	264	4	:	:	PUNCT
ejpam-6256	264	5	a	a	DET
ejpam-6256	264	6	review	review	NOUN
ejpam-6256	264	7	.	.	PUNCT
ejpam-6256	265	1	arxiv:2309.03226	arxiv:2309.03226	PROPN
ejpam-6256	265	2	,	,	PUNCT
ejpam-6256	265	3	2023	2023	NUM
ejpam-6256	265	4	.	.	PUNCT
ejpam-6256	266	1	[	[	X
ejpam-6256	266	2	3	3	NUM
ejpam-6256	266	3	]	]	PUNCT
ejpam-6256	266	4	a.	a.	NOUN
ejpam-6256	266	5	granas	grana	NOUN
ejpam-6256	266	6	and	and	CCONJ
ejpam-6256	266	7	j.	j.	PROPN
ejpam-6256	266	8	dugundji	dugundji	PROPN
ejpam-6256	266	9	.	.	PUNCT
ejpam-6256	267	1	fixed	fix	VERB
ejpam-6256	267	2	point	point	NOUN
ejpam-6256	267	3	theory	theory	NOUN
ejpam-6256	267	4	.	.	PUNCT
ejpam-6256	268	1	springer	springer	NOUN
ejpam-6256	268	2	,	,	PUNCT
ejpam-6256	268	3	new	new	PROPN
ejpam-6256	268	4	york	york	PROPN
ejpam-6256	268	5	,	,	PUNCT
ejpam-6256	268	6	2003	2003	NUM
ejpam-6256	268	7	.	.	PUNCT
ejpam-6256	269	1	[	[	X
ejpam-6256	269	2	4	4	X
ejpam-6256	269	3	]	]	X
ejpam-6256	269	4	s.	s.	PROPN
ejpam-6256	269	5	banach	banach	PROPN
ejpam-6256	269	6	.	.	PUNCT
ejpam-6256	270	1	sur	sur	PROPN
ejpam-6256	270	2	les	les	X
ejpam-6256	270	3	opérations	opération	NOUN
ejpam-6256	270	4	dans	dan	NOUN
ejpam-6256	270	5	les	les	X
ejpam-6256	270	6	ensembles	ensemble	NOUN
ejpam-6256	270	7	abstraits	abstrait	NOUN
ejpam-6256	270	8	et	et	PROPN
ejpam-6256	270	9	leur	leur	X
ejpam-6256	270	10	application	application	PROPN
ejpam-6256	270	11	aux	aux	PROPN
ejpam-6256	270	12	équations	équations	PROPN
ejpam-6256	270	13	intégrales	intégrale	NOUN
ejpam-6256	270	14	.	.	PUNCT
ejpam-6256	271	1	fundamenta	fundamenta	PROPN
ejpam-6256	271	2	mathematicae	mathematicae	PROPN
ejpam-6256	271	3	,	,	PUNCT
ejpam-6256	271	4	3:133–181	3:133–181	NUM
ejpam-6256	271	5	,	,	PUNCT
ejpam-6256	271	6	1922	1922	NUM
ejpam-6256	271	7	.	.	PUNCT
ejpam-6256	272	1	[	[	X
ejpam-6256	272	2	5	5	NUM
ejpam-6256	272	3	]	]	X
ejpam-6256	272	4	gazal	gazal	PROPN
ejpam-6256	272	5	,	,	PUNCT
ejpam-6256	272	6	s.	s.	PROPN
ejpam-6256	272	7	rathee	rathee	PROPN
ejpam-6256	272	8	,	,	PUNCT
ejpam-6256	272	9	mahima	mahima	PROPN
ejpam-6256	272	10	,	,	PUNCT
ejpam-6256	272	11	a.	a.	PROPN
ejpam-6256	272	12	kadyan	kadyan	PROPN
ejpam-6256	272	13	,	,	PUNCT
ejpam-6256	272	14	minakshi	minakshi	PROPN
ejpam-6256	272	15	,	,	PUNCT
ejpam-6256	272	16	and	and	CCONJ
ejpam-6256	272	17	a.	a.	PROPN
ejpam-6256	272	18	kumar	kumar	PROPN
ejpam-6256	272	19	.	.	PUNCT
ejpam-6256	273	1	existence	existence	PROPN
ejpam-6256	273	2	,	,	PUNCT
ejpam-6256	273	3	approximation	approximation	NOUN
ejpam-6256	273	4	and	and	CCONJ
ejpam-6256	273	5	stability	stability	NOUN
ejpam-6256	273	6	for	for	ADP
ejpam-6256	273	7	fixed	fix	VERB
ejpam-6256	273	8	point	point	NOUN
ejpam-6256	273	9	for	for	ADP
ejpam-6256	273	10	ćirić	ćirić	PROPN
ejpam-6256	273	11	contraction	contraction	NOUN
ejpam-6256	273	12	in	in	ADP
ejpam-6256	273	13	convex	convex	PROPN
ejpam-6256	273	14	b	b	NOUN
ejpam-6256	273	15	-	-	ADJ
ejpam-6256	273	16	metric	metric	ADJ
ejpam-6256	273	17	spaces	space	NOUN
ejpam-6256	273	18	.	.	PUNCT
ejpam-6256	274	1	axioms	axiom	NOUN
ejpam-6256	274	2	,	,	PUNCT
ejpam-6256	274	3	12(7	12(7	NUM
ejpam-6256	274	4	)	)	PUNCT
ejpam-6256	274	5	,	,	PUNCT
ejpam-6256	274	6	2023	2023	NUM
ejpam-6256	274	7	.	.	PUNCT
ejpam-6256	275	1	[	[	X
ejpam-6256	275	2	6	6	NUM
ejpam-6256	275	3	]	]	PUNCT
ejpam-6256	275	4	v.	v.	ADP
ejpam-6256	275	5	berinde	berinde	NOUN
ejpam-6256	275	6	.	.	PUNCT
ejpam-6256	276	1	iterative	iterative	NOUN
ejpam-6256	276	2	approximation	approximation	NOUN
ejpam-6256	276	3	of	of	ADP
ejpam-6256	276	4	fixed	fix	VERB
ejpam-6256	276	5	points	point	NOUN
ejpam-6256	276	6	.	.	PUNCT
ejpam-6256	277	1	springer	springer	NOUN
ejpam-6256	277	2	,	,	PUNCT
ejpam-6256	277	3	london	london	PROPN
ejpam-6256	277	4	,	,	PUNCT
ejpam-6256	277	5	2007	2007	NUM
ejpam-6256	277	6	.	.	PUNCT
ejpam-6256	278	1	y.	y.	PROPN
ejpam-6256	278	2	alzubaidi	alzubaidi	PROPN
ejpam-6256	278	3	/	/	SYM
ejpam-6256	278	4	eur	eur	PROPN
ejpam-6256	278	5	.	.	PUNCT
ejpam-6256	279	1	j.	j.	PROPN
ejpam-6256	279	2	pure	pure	PROPN
ejpam-6256	279	3	appl	appl	PROPN
ejpam-6256	279	4	.	.	PROPN
ejpam-6256	279	5	math	math	PROPN
ejpam-6256	279	6	,	,	PUNCT
ejpam-6256	279	7	18	18	NUM
ejpam-6256	279	8	(	(	PUNCT
ejpam-6256	279	9	3	3	NUM
ejpam-6256	279	10	)	)	PUNCT
ejpam-6256	279	11	(	(	PUNCT
ejpam-6256	279	12	2025	2025	NUM
ejpam-6256	279	13	)	)	PUNCT
ejpam-6256	279	14	,	,	PUNCT
ejpam-6256	279	15	6256	6256	NUM
ejpam-6256	279	16	11	11	NUM
ejpam-6256	279	17	of	of	ADP
ejpam-6256	279	18	11	11	NUM
ejpam-6256	280	1	[	[	X
ejpam-6256	280	2	7	7	NUM
ejpam-6256	280	3	]	]	PUNCT
ejpam-6256	280	4	j.	j.	PROPN
ejpam-6256	280	5	ahmad	ahmad	PROPN
ejpam-6256	280	6	,	,	PUNCT
ejpam-6256	280	7	k.	k.	PROPN
ejpam-6256	280	8	ullah	ullah	PROPN
ejpam-6256	280	9	,	,	PUNCT
ejpam-6256	280	10	m.	m.	NOUN
ejpam-6256	280	11	arshad	arshad	PROPN
ejpam-6256	280	12	,	,	PUNCT
ejpam-6256	280	13	and	and	CCONJ
ejpam-6256	280	14	m.	m.	PROPN
ejpam-6256	280	15	de	de	PROPN
ejpam-6256	280	16	la	la	PROPN
ejpam-6256	280	17	sen	sen	PROPN
ejpam-6256	280	18	.	.	PROPN
ejpam-6256	280	19	iterative	iterative	PROPN
ejpam-6256	280	20	approximation	approximation	NOUN
ejpam-6256	280	21	of	of	ADP
ejpam-6256	280	22	fixed	fix	VERB
ejpam-6256	280	23	points	point	NOUN
ejpam-6256	280	24	by	by	ADP
ejpam-6256	280	25	using	use	VERB
ejpam-6256	280	26	f	f	PROPN
ejpam-6256	280	27	iteration	iteration	NOUN
ejpam-6256	280	28	process	process	NOUN
ejpam-6256	280	29	in	in	ADP
ejpam-6256	280	30	banach	banach	NOUN
ejpam-6256	280	31	spaces	space	NOUN
ejpam-6256	280	32	.	.	PUNCT
ejpam-6256	281	1	journal	journal	NOUN
ejpam-6256	281	2	of	of	ADP
ejpam-6256	281	3	function	function	NOUN
ejpam-6256	281	4	spaces	space	NOUN
ejpam-6256	281	5	,	,	PUNCT
ejpam-6256	281	6	2021(1	2021(1	NUM
ejpam-6256	281	7	)	)	PUNCT
ejpam-6256	281	8	,	,	PUNCT
ejpam-6256	281	9	2021	2021	NUM
ejpam-6256	281	10	.	.	PUNCT
ejpam-6256	282	1	[	[	X
ejpam-6256	282	2	8	8	NUM
ejpam-6256	282	3	]	]	PUNCT
ejpam-6256	282	4	m.	m.	PROPN
ejpam-6256	282	5	r.	r.	PROPN
ejpam-6256	282	6	haddadi	haddadi	PROPN
ejpam-6256	282	7	,	,	PUNCT
ejpam-6256	282	8	v.	v.	CCONJ
ejpam-6256	282	9	parvaneh	parvaneh	NOUN
ejpam-6256	282	10	,	,	PUNCT
ejpam-6256	282	11	and	and	CCONJ
ejpam-6256	282	12	m.	m.	NOUN
ejpam-6256	282	13	bota	bota	NOUN
ejpam-6256	282	14	.	.	PUNCT
ejpam-6256	283	1	further	further	ADJ
ejpam-6256	283	2	generalizations	generalization	NOUN
ejpam-6256	283	3	of	of	ADP
ejpam-6256	283	4	the	the	DET
ejpam-6256	283	5	ishikawa	ishikawa	PROPN
ejpam-6256	283	6	algorithm	algorithm	NOUN
ejpam-6256	283	7	.	.	PUNCT
ejpam-6256	284	1	aims	aim	VERB
ejpam-6256	284	2	mathematics	mathematic	NOUN
ejpam-6256	284	3	,	,	PUNCT
ejpam-6256	284	4	8(5):12185–12194	8(5):12185–12194	PROPN
ejpam-6256	284	5	,	,	PUNCT
ejpam-6256	284	6	2023	2023	NUM
ejpam-6256	284	7	.	.	PUNCT
ejpam-6256	285	1	[	[	X
ejpam-6256	285	2	9	9	NUM
ejpam-6256	285	3	]	]	PUNCT
ejpam-6256	285	4	t.	t.	PROPN
ejpam-6256	285	5	a.	a.	PROPN
ejpam-6256	285	6	burton	burton	PROPN
ejpam-6256	285	7	.	.	PUNCT
ejpam-6256	286	1	a	a	DET
ejpam-6256	286	2	fixed	fix	VERB
ejpam-6256	286	3	-	-	PUNCT
ejpam-6256	286	4	point	point	NOUN
ejpam-6256	286	5	theorem	theorem	NOUN
ejpam-6256	286	6	of	of	ADP
ejpam-6256	286	7	krasnoselskii	krasnoselskii	PROPN
ejpam-6256	286	8	.	.	PUNCT
ejpam-6256	287	1	applied	apply	VERB
ejpam-6256	287	2	mathematics	mathematics	NOUN
ejpam-6256	287	3	letters	letter	NOUN
ejpam-6256	287	4	,	,	PUNCT
ejpam-6256	287	5	11(1):85–88	11(1):85–88	NUM
ejpam-6256	287	6	,	,	PUNCT
ejpam-6256	287	7	1998	1998	NUM
ejpam-6256	287	8	.	.	PUNCT
ejpam-6256	288	1	[	[	X
ejpam-6256	288	2	10	10	NUM
ejpam-6256	288	3	]	]	X
ejpam-6256	288	4	f.	f.	PROPN
ejpam-6256	288	5	e.	e.	PROPN
ejpam-6256	288	6	browder	browder	PROPN
ejpam-6256	288	7	.	.	PUNCT
ejpam-6256	289	1	nonlinear	nonlinear	ADJ
ejpam-6256	289	2	operators	operator	NOUN
ejpam-6256	289	3	and	and	CCONJ
ejpam-6256	289	4	fixed	fix	VERB
ejpam-6256	289	5	points	point	NOUN
ejpam-6256	289	6	.	.	PUNCT
ejpam-6256	290	1	proceedings	proceeding	NOUN
ejpam-6256	290	2	of	of	ADP
ejpam-6256	290	3	the	the	DET
ejpam-6256	290	4	american	american	PROPN
ejpam-6256	290	5	mathematical	mathematical	PROPN
ejpam-6256	290	6	society	society	NOUN
ejpam-6256	290	7	,	,	PUNCT
ejpam-6256	290	8	16:638–640	16:638–640	NUM
ejpam-6256	290	9	,	,	PUNCT
ejpam-6256	290	10	1965	1965	NUM
ejpam-6256	290	11	.	.	PUNCT
ejpam-6256	291	1	[	[	X
ejpam-6256	291	2	11	11	NUM
ejpam-6256	291	3	]	]	X
ejpam-6256	291	4	f.	f.	PROPN
ejpam-6256	291	5	e.	e.	PROPN
ejpam-6256	291	6	browder	browder	PROPN
ejpam-6256	291	7	.	.	PUNCT
ejpam-6256	292	1	fixed	fix	VERB
ejpam-6256	292	2	point	point	NOUN
ejpam-6256	292	3	theorems	theorem	NOUN
ejpam-6256	292	4	for	for	ADP
ejpam-6256	292	5	nonexpansive	nonexpansive	ADJ
ejpam-6256	292	6	mappings	mapping	NOUN
ejpam-6256	292	7	.	.	PUNCT
ejpam-6256	293	1	the	the	DET
ejpam-6256	293	2	journal	journal	NOUN
ejpam-6256	293	3	of	of	ADP
ejpam-6256	293	4	the	the	DET
ejpam-6256	293	5	society	society	NOUN
ejpam-6256	293	6	for	for	ADP
ejpam-6256	293	7	industrial	industrial	ADJ
ejpam-6256	293	8	and	and	CCONJ
ejpam-6256	293	9	applied	apply	VERB
ejpam-6256	293	10	mathematics	mathematic	NOUN
ejpam-6256	293	11	,	,	PUNCT
ejpam-6256	293	12	15(3):517–528	15(3):517–528	NUM
ejpam-6256	293	13	,	,	PUNCT
ejpam-6256	293	14	1967	1967	NUM
ejpam-6256	293	15	.	.	PUNCT
ejpam-6256	294	1	[	[	X
ejpam-6256	294	2	12	12	NUM
ejpam-6256	294	3	]	]	PUNCT
ejpam-6256	294	4	w.	w.	PROPN
ejpam-6256	294	5	a.	a.	PROPN
ejpam-6256	294	6	kirk	kirk	PROPN
ejpam-6256	294	7	.	.	PUNCT
ejpam-6256	295	1	a	a	DET
ejpam-6256	295	2	fixed	fix	VERB
ejpam-6256	295	3	point	point	NOUN
ejpam-6256	295	4	theorem	theorem	NOUN
ejpam-6256	295	5	for	for	ADP
ejpam-6256	295	6	mappings	mapping	NOUN
ejpam-6256	295	7	which	which	PRON
ejpam-6256	295	8	do	do	AUX
ejpam-6256	295	9	not	not	PART
ejpam-6256	295	10	increase	increase	VERB
ejpam-6256	295	11	distances	distance	NOUN
ejpam-6256	295	12	.	.	PUNCT
ejpam-6256	296	1	american	american	PROPN
ejpam-6256	296	2	mathematical	mathematical	PROPN
ejpam-6256	296	3	monthly	monthly	ADV
ejpam-6256	296	4	,	,	PUNCT
ejpam-6256	296	5	72(9):1004–1006	72(9):1004–1006	PROPN
ejpam-6256	296	6	,	,	PUNCT
ejpam-6256	296	7	1965	1965	NUM
ejpam-6256	296	8	.	.	PUNCT
ejpam-6256	297	1	[	[	X
ejpam-6256	297	2	13	13	NUM
ejpam-6256	297	3	]	]	PUNCT
ejpam-6256	297	4	l.	l.	PROPN
ejpam-6256	297	5	p.	p.	PROPN
ejpam-6256	297	6	belluce	belluce	NOUN
ejpam-6256	297	7	and	and	CCONJ
ejpam-6256	297	8	w.	w.	PROPN
ejpam-6256	297	9	a.	a.	PROPN
ejpam-6256	297	10	kirk	kirk	PROPN
ejpam-6256	297	11	.	.	PUNCT
ejpam-6256	298	1	nonexpansive	nonexpansive	ADJ
ejpam-6256	298	2	mappings	mapping	NOUN
ejpam-6256	298	3	and	and	CCONJ
ejpam-6256	298	4	fixed	fix	VERB
ejpam-6256	298	5	-	-	PUNCT
ejpam-6256	298	6	points	point	NOUN
ejpam-6256	298	7	in	in	ADP
ejpam-6256	298	8	banach	banach	NOUN
ejpam-6256	298	9	spaces	space	NOUN
ejpam-6256	298	10	.	.	PUNCT
ejpam-6256	299	1	illinois	illinois	PROPN
ejpam-6256	299	2	journal	journal	PROPN
ejpam-6256	299	3	of	of	ADP
ejpam-6256	299	4	mathematics	mathematic	NOUN
ejpam-6256	299	5	,	,	PUNCT
ejpam-6256	299	6	11(3):474–479	11(3):474–479	PROPN
ejpam-6256	299	7	,	,	PUNCT
ejpam-6256	299	8	1967	1967	NUM
ejpam-6256	299	9	.	.	PUNCT
ejpam-6256	300	1	[	[	X
ejpam-6256	300	2	14	14	NUM
ejpam-6256	300	3	]	]	X
ejpam-6256	300	4	h.	h.	PROPN
ejpam-6256	300	5	persson	persson	PROPN
ejpam-6256	300	6	.	.	PUNCT
ejpam-6256	301	1	a	a	DET
ejpam-6256	301	2	fixed	fix	VERB
ejpam-6256	301	3	point	point	NOUN
ejpam-6256	301	4	theorem	theorem	NOUN
ejpam-6256	301	5	for	for	ADP
ejpam-6256	301	6	monotone	monotone	ADJ
ejpam-6256	301	7	functions	function	NOUN
ejpam-6256	301	8	.	.	PUNCT
ejpam-6256	302	1	applied	apply	VERB
ejpam-6256	302	2	mathematics	mathematics	NOUN
ejpam-6256	302	3	letters	letter	NOUN
ejpam-6256	302	4	,	,	PUNCT
ejpam-6256	302	5	19(11):1207–1209	19(11):1207–1209	NUM
ejpam-6256	302	6	,	,	PUNCT
ejpam-6256	302	7	2006	2006	NUM
ejpam-6256	302	8	.	.	PUNCT
ejpam-6256	303	1	[	[	X
ejpam-6256	303	2	15	15	NUM
ejpam-6256	303	3	]	]	X
ejpam-6256	303	4	e.	e.	PROPN
ejpam-6256	303	5	n.	n.	PROPN
ejpam-6256	303	6	dancer	dancer	PROPN
ejpam-6256	303	7	.	.	PUNCT
ejpam-6256	304	1	multiple	multiple	ADJ
ejpam-6256	304	2	fixed	fix	VERB
ejpam-6256	304	3	points	point	NOUN
ejpam-6256	304	4	of	of	ADP
ejpam-6256	304	5	positive	positive	ADJ
ejpam-6256	304	6	mappings	mapping	NOUN
ejpam-6256	304	7	.	.	PUNCT
ejpam-6256	305	1	journal	journal	PROPN
ejpam-6256	305	2	für	für	AUX
ejpam-6256	305	3	die	die	VERB
ejpam-6256	305	4	reine	reine	PROPN
ejpam-6256	305	5	und	und	PROPN
ejpam-6256	305	6	angewandte	angewandte	PROPN
ejpam-6256	305	7	mathematik	mathematik	PROPN
ejpam-6256	305	8	(	(	PUNCT
ejpam-6256	305	9	crelles	crelle	NOUN
ejpam-6256	305	10	journal	journal	PROPN
ejpam-6256	305	11	)	)	PUNCT
ejpam-6256	305	12	,	,	PUNCT
ejpam-6256	305	13	1986(371):46–66	1986(371):46–66	NUM
ejpam-6256	305	14	,	,	PUNCT
ejpam-6256	305	15	1986	1986	NUM
ejpam-6256	305	16	.	.	PUNCT
ejpam-6256	306	1	[	[	X
ejpam-6256	306	2	16	16	NUM
ejpam-6256	306	3	]	]	PUNCT
ejpam-6256	306	4	r.	r.	PROPN
ejpam-6256	306	5	p.	p.	PROPN
ejpam-6256	306	6	agarwal	agarwal	PROPN
ejpam-6256	306	7	,	,	PUNCT
ejpam-6256	306	8	e.	e.	PROPN
ejpam-6256	306	9	karapınar	karapınar	PROPN
ejpam-6256	306	10	,	,	PUNCT
ejpam-6256	306	11	d.	d.	PROPN
ejpam-6256	306	12	o’regan	o’regan	PROPN
ejpam-6256	306	13	,	,	PUNCT
ejpam-6256	306	14	and	and	CCONJ
ejpam-6256	306	15	a.	a.	PROPN
ejpam-6256	306	16	f.	f.	PROPN
ejpam-6256	306	17	roldán	roldán	PROPN
ejpam-6256	306	18	-	-	PUNCT
ejpam-6256	306	19	lópez	lópez	ADV
ejpam-6256	306	20	-	-	PUNCT
ejpam-6256	306	21	de	de	X
ejpam-6256	306	22	hierro	hierro	PROPN
ejpam-6256	306	23	.	.	PROPN
ejpam-6256	306	24	fixed	fix	VERB
ejpam-6256	306	25	point	point	NOUN
ejpam-6256	306	26	theory	theory	NOUN
ejpam-6256	306	27	in	in	ADP
ejpam-6256	306	28	metric	metric	ADJ
ejpam-6256	306	29	type	type	NOUN
ejpam-6256	306	30	spaces	space	NOUN
ejpam-6256	306	31	.	.	PUNCT
ejpam-6256	307	1	springer	springer	NOUN
ejpam-6256	307	2	,	,	PUNCT
ejpam-6256	307	3	cham	cham	PROPN
ejpam-6256	307	4	,	,	PUNCT
ejpam-6256	307	5	switzerland	switzerland	PROPN
ejpam-6256	307	6	,	,	PUNCT
ejpam-6256	307	7	2015	2015	NUM
ejpam-6256	307	8	.	.	PUNCT
ejpam-6256	308	1	[	[	X
ejpam-6256	308	2	17	17	NUM
ejpam-6256	308	3	]	]	X
ejpam-6256	308	4	d.	d.	PROPN
ejpam-6256	308	5	gopal	gopal	PROPN
ejpam-6256	308	6	,	,	PUNCT
ejpam-6256	308	7	p.	p.	PROPN
ejpam-6256	308	8	agarwal	agarwal	PROPN
ejpam-6256	308	9	,	,	PUNCT
ejpam-6256	308	10	and	and	CCONJ
ejpam-6256	308	11	p.	p.	PROPN
ejpam-6256	308	12	kumam	kumam	PROPN
ejpam-6256	308	13	.	.	PUNCT
ejpam-6256	309	1	metric	metric	ADJ
ejpam-6256	309	2	structures	structure	NOUN
ejpam-6256	309	3	and	and	CCONJ
ejpam-6256	309	4	fixed	fix	VERB
ejpam-6256	309	5	point	point	NOUN
ejpam-6256	309	6	theory	theory	NOUN
ejpam-6256	309	7	.	.	PUNCT
ejpam-6256	310	1	crc	crc	PROPN
ejpam-6256	310	2	press	press	PROPN
ejpam-6256	310	3	,	,	PUNCT
ejpam-6256	310	4	2021	2021	NUM
ejpam-6256	310	5	.	.	PUNCT
ejpam-6256	311	1	[	[	X
ejpam-6256	311	2	18	18	NUM
ejpam-6256	311	3	]	]	X
ejpam-6256	311	4	e.	e.	PROPN
ejpam-6256	311	5	karapınar	karapınar	PROPN
ejpam-6256	311	6	and	and	CCONJ
ejpam-6256	311	7	r.	r.	PROPN
ejpam-6256	311	8	p.	p.	PROPN
ejpam-6256	311	9	agarwal	agarwal	PROPN
ejpam-6256	311	10	.	.	PUNCT
ejpam-6256	312	1	fixed	fix	VERB
ejpam-6256	312	2	point	point	NOUN
ejpam-6256	312	3	theory	theory	NOUN
ejpam-6256	312	4	in	in	ADP
ejpam-6256	312	5	generalized	generalized	ADJ
ejpam-6256	312	6	metric	metric	ADJ
ejpam-6256	312	7	spaces	space	NOUN
ejpam-6256	312	8	.	.	PUNCT
ejpam-6256	313	1	springer	springer	NOUN
ejpam-6256	313	2	,	,	PUNCT
ejpam-6256	313	3	2022	2022	NUM
ejpam-6256	313	4	.	.	PUNCT
ejpam-6256	314	1	[	[	X
ejpam-6256	314	2	19	19	NUM
ejpam-6256	314	3	]	]	X
ejpam-6256	314	4	s.	s.	PROPN
ejpam-6256	314	5	anwar	anwar	PROPN
ejpam-6256	314	6	,	,	PUNCT
ejpam-6256	314	7	m.	m.	NOUN
ejpam-6256	314	8	nazam	nazam	PROPN
ejpam-6256	314	9	,	,	PUNCT
ejpam-6256	314	10	h.	h.	PROPN
ejpam-6256	314	11	h.	h.	PROPN
ejpam-6256	314	12	al	al	PROPN
ejpam-6256	314	13	sulami	sulami	PROPN
ejpam-6256	314	14	,	,	PUNCT
ejpam-6256	314	15	a.	a.	PROPN
ejpam-6256	314	16	hussain	hussain	PROPN
ejpam-6256	314	17	,	,	PUNCT
ejpam-6256	314	18	k.	k.	PROPN
ejpam-6256	314	19	javed	javed	PROPN
ejpam-6256	314	20	,	,	PUNCT
ejpam-6256	314	21	and	and	CCONJ
ejpam-6256	314	22	m.	m.	PROPN
ejpam-6256	314	23	arshad	arshad	PROPN
ejpam-6256	314	24	.	.	PROPN
ejpam-6256	315	1	existence	existence	NOUN
ejpam-6256	315	2	fixed	fix	VERB
ejpam-6256	315	3	-	-	PUNCT
ejpam-6256	315	4	point	point	NOUN
ejpam-6256	315	5	theorems	theorem	NOUN
ejpam-6256	315	6	in	in	ADP
ejpam-6256	315	7	the	the	DET
ejpam-6256	315	8	partial	partial	ADJ
ejpam-6256	315	9	b	b	NOUN
ejpam-6256	315	10	-	-	PUNCT
ejpam-6256	315	11	metric	metric	ADJ
ejpam-6256	315	12	spaces	space	NOUN
ejpam-6256	315	13	and	and	CCONJ
ejpam-6256	315	14	an	an	DET
ejpam-6256	315	15	application	application	NOUN
ejpam-6256	315	16	to	to	ADP
ejpam-6256	315	17	the	the	DET
ejpam-6256	315	18	boundary	boundary	ADJ
ejpam-6256	315	19	value	value	NOUN
ejpam-6256	315	20	problem	problem	NOUN
ejpam-6256	315	21	.	.	PUNCT
ejpam-6256	316	1	aims	aim	VERB
ejpam-6256	316	2	mathematics	mathematic	NOUN
ejpam-6256	316	3	,	,	PUNCT
ejpam-6256	316	4	7(5):8188–8205	7(5):8188–8205	NUM
ejpam-6256	316	5	,	,	PUNCT
ejpam-6256	316	6	2022	2022	NUM
ejpam-6256	316	7	.	.	PUNCT
ejpam-6256	317	1	[	[	X
ejpam-6256	317	2	20	20	NUM
ejpam-6256	317	3	]	]	PUNCT
ejpam-6256	317	4	m.	m.	NOUN
ejpam-6256	317	5	o.	o.	PROPN
ejpam-6256	317	6	olatinwo	olatinwo	PROPN
ejpam-6256	317	7	.	.	PUNCT
ejpam-6256	318	1	some	some	DET
ejpam-6256	318	2	results	result	NOUN
ejpam-6256	318	3	on	on	ADP
ejpam-6256	318	4	multi	multi	ADJ
ejpam-6256	318	5	-	-	ADJ
ejpam-6256	318	6	valued	value	VERB
ejpam-6256	318	7	weakly	weakly	ADJ
ejpam-6256	318	8	jungck	jungck	NOUN
ejpam-6256	318	9	mappings	mapping	NOUN
ejpam-6256	318	10	in	in	ADP
ejpam-6256	318	11	b	b	NOUN
ejpam-6256	318	12	-	-	PUNCT
ejpam-6256	318	13	metric	metric	ADJ
ejpam-6256	318	14	space	space	NOUN
ejpam-6256	318	15	.	.	PUNCT
ejpam-6256	319	1	open	open	ADJ
ejpam-6256	319	2	mathematics	mathematic	NOUN
ejpam-6256	319	3	,	,	PUNCT
ejpam-6256	319	4	6(4):610–621	6(4):610–621	NUM
ejpam-6256	319	5	,	,	PUNCT
ejpam-6256	319	6	2008	2008	NUM
ejpam-6256	319	7	.	.	PUNCT
ejpam-6256	320	1	[	[	X
ejpam-6256	320	2	21	21	NUM
ejpam-6256	320	3	]	]	X
ejpam-6256	320	4	y.	y.	NOUN
ejpam-6256	320	5	alzubaidi	alzubaidi	PROPN
ejpam-6256	320	6	.	.	PUNCT
ejpam-6256	321	1	on	on	ADP
ejpam-6256	321	2	convergence	convergence	NOUN
ejpam-6256	321	3	and	and	CCONJ
ejpam-6256	321	4	completeness	completeness	NOUN
ejpam-6256	321	5	in	in	ADP
ejpam-6256	321	6	generalized	generalized	ADJ
ejpam-6256	321	7	metric	metric	ADJ
ejpam-6256	321	8	spaces	space	NOUN
ejpam-6256	321	9	.	.	PUNCT
ejpam-6256	322	1	asia	asia	PROPN
ejpam-6256	322	2	pacific	pacific	PROPN
ejpam-6256	322	3	journal	journal	PROPN
ejpam-6256	322	4	of	of	ADP
ejpam-6256	322	5	mathematics	mathematic	NOUN
ejpam-6256	322	6	,	,	PUNCT
ejpam-6256	322	7	11(11	11(11	NUM
ejpam-6256	322	8	)	)	PUNCT
ejpam-6256	322	9	,	,	PUNCT
ejpam-6256	322	10	2024	2024	NUM
ejpam-6256	322	11	.	.	PUNCT
ejpam-6256	323	1	[	[	X
ejpam-6256	323	2	22	22	NUM
ejpam-6256	323	3	]	]	PUNCT
ejpam-6256	323	4	nihal	nihal	VERB
ejpam-6256	323	5	taş	taş	NOUN
ejpam-6256	323	6	,	,	PUNCT
ejpam-6256	323	7	nabil	nabil	PROPN
ejpam-6256	323	8	mlaiki	mlaiki	PROPN
ejpam-6256	323	9	,	,	PUNCT
ejpam-6256	323	10	hassen	hassen	PROPN
ejpam-6256	323	11	aydi	aydi	ADV
ejpam-6256	323	12	,	,	PUNCT
ejpam-6256	323	13	and	and	CCONJ
ejpam-6256	323	14	nihal	nihal	ADJ
ejpam-6256	323	15	özgür	özgür	NUM
ejpam-6256	323	16	.	.	PUNCT
ejpam-6256	323	17	fixed	fix	VERB
ejpam-6256	323	18	-	-	PUNCT
ejpam-6256	323	19	disc	disc	NOUN
ejpam-6256	323	20	results	result	NOUN
ejpam-6256	323	21	on	on	ADP
ejpam-6256	323	22	metric	metric	ADJ
ejpam-6256	323	23	spaces	space	NOUN
ejpam-6256	323	24	.	.	PUNCT
ejpam-6256	324	1	filomat	filomat	NOUN
ejpam-6256	324	2	,	,	PUNCT
ejpam-6256	324	3	35(2):447–457	35(2):447–457	PROPN
ejpam-6256	324	4	,	,	PUNCT
ejpam-6256	324	5	2021	2021	NUM
ejpam-6256	324	6	.	.	PUNCT
ejpam-6256	325	1	[	[	X
ejpam-6256	325	2	23	23	NUM
ejpam-6256	325	3	]	]	PUNCT
ejpam-6256	325	4	aftab	aftab	PROPN
ejpam-6256	325	5	hussain	hussain	PROPN
ejpam-6256	325	6	,	,	PUNCT
ejpam-6256	325	7	hamed	hamed	PROPN
ejpam-6256	325	8	al	al	PROPN
ejpam-6256	325	9	-	-	PROPN
ejpam-6256	325	10	sulami	sulami	PROPN
ejpam-6256	325	11	,	,	PUNCT
ejpam-6256	325	12	nawab	nawab	PROPN
ejpam-6256	325	13	hussain	hussain	PROPN
ejpam-6256	325	14	,	,	PUNCT
ejpam-6256	325	15	and	and	CCONJ
ejpam-6256	325	16	hamza	hamza	PROPN
ejpam-6256	325	17	farooq	farooq	PROPN
ejpam-6256	325	18	.	.	PUNCT
ejpam-6256	326	1	newly	newly	ADV
ejpam-6256	326	2	fixed	fix	VERB
ejpam-6256	326	3	disc	disc	NOUN
ejpam-6256	326	4	results	result	NOUN
ejpam-6256	326	5	using	use	VERB
ejpam-6256	326	6	advanced	advanced	ADJ
ejpam-6256	326	7	contractions	contraction	NOUN
ejpam-6256	326	8	on	on	ADP
ejpam-6256	326	9	f	f	ADJ
ejpam-6256	326	10	-	-	PUNCT
ejpam-6256	326	11	metric	metric	ADJ
ejpam-6256	326	12	space	space	NOUN
ejpam-6256	326	13	.	.	PUNCT
ejpam-6256	327	1	journal	journal	PROPN
ejpam-6256	327	2	of	of	ADP
ejpam-6256	327	3	applied	apply	VERB
ejpam-6256	327	4	analysis	analysis	NOUN
ejpam-6256	327	5	and	and	CCONJ
ejpam-6256	327	6	computation	computation	NOUN
ejpam-6256	327	7	,	,	PUNCT
ejpam-6256	327	8	10(6):2313–2322	10(6):2313–2322	NUM
ejpam-6256	327	9	,	,	PUNCT
ejpam-6256	327	10	2020	2020	NUM
ejpam-6256	327	11	.	.	PUNCT