id	sid	tid	token	lemma	pos
ejpam-6243	1	1	european	european	PROPN
ejpam-6243	1	2	journal	journal	PROPN
ejpam-6243	1	3	of	of	ADP
ejpam-6243	1	4	pure	pure	ADJ
ejpam-6243	1	5	and	and	CCONJ
ejpam-6243	1	6	applied	applied	ADJ
ejpam-6243	1	7	mathematics	mathematic	NOUN
ejpam-6243	1	8	2025	2025	NUM
ejpam-6243	1	9	,	,	PUNCT
ejpam-6243	1	10	vol	vol	NOUN
ejpam-6243	1	11	.	.	PROPN
ejpam-6243	1	12	18	18	NUM
ejpam-6243	1	13	,	,	PUNCT
ejpam-6243	1	14	issue	issue	NOUN
ejpam-6243	1	15	3	3	NUM
ejpam-6243	1	16	,	,	PUNCT
ejpam-6243	1	17	article	article	NOUN
ejpam-6243	1	18	number	number	NOUN
ejpam-6243	1	19	6243	6243	NUM
ejpam-6243	1	20	issn	issn	PROPN
ejpam-6243	1	21	1307	1307	NUM
ejpam-6243	1	22	-	-	SYM
ejpam-6243	1	23	5543	5543	NUM
ejpam-6243	1	24	–	–	PUNCT
ejpam-6243	1	25	ejpam.com	ejpam.com	X
ejpam-6243	1	26	published	publish	VERB
ejpam-6243	1	27	by	by	ADP
ejpam-6243	1	28	new	new	PROPN
ejpam-6243	1	29	york	york	PROPN
ejpam-6243	1	30	business	business	PROPN
ejpam-6243	1	31	global	global	PROPN
ejpam-6243	1	32	some	some	DET
ejpam-6243	1	33	fixed	fix	VERB
ejpam-6243	1	34	point	point	NOUN
ejpam-6243	1	35	results	result	NOUN
ejpam-6243	1	36	for	for	ADP
ejpam-6243	1	37	hybrid	hybrid	ADJ
ejpam-6243	1	38	contraction	contraction	NOUN
ejpam-6243	1	39	in	in	ADP
ejpam-6243	1	40	metric	metric	ADJ
ejpam-6243	1	41	spaces	space	NOUN
ejpam-6243	1	42	and	and	CCONJ
ejpam-6243	1	43	ulam	ulam	NOUN
ejpam-6243	1	44	-	-	PUNCT
ejpam-6243	1	45	hyers	hyer	NOUN
ejpam-6243	1	46	stability	stability	NOUN
ejpam-6243	1	47	rajagopalan	rajagopalan	VERB
ejpam-6243	1	48	ramaswamy1∗	ramaswamy1∗	NUM
ejpam-6243	1	49	,	,	PUNCT
ejpam-6243	1	50	manoj	manoj	PROPN
ejpam-6243	1	51	kumar2	kumar2	PROPN
ejpam-6243	1	52	,	,	PUNCT
ejpam-6243	1	53	prem	prem	PROPN
ejpam-6243	1	54	lata3	lata3	PROPN
ejpam-6243	1	55	,	,	PUNCT
ejpam-6243	1	56	rayan	rayan	PROPN
ejpam-6243	1	57	abdulrahman	abdulrahman	PROPN
ejpam-6243	1	58	alkhowaiter1	alkhowaiter1	PROPN
ejpam-6243	1	59	,	,	PUNCT
ejpam-6243	1	60	hossam	hossam	NOUN
ejpam-6243	1	61	a	a	DET
ejpam-6243	1	62	nabway1	nabway1	PROPN
ejpam-6243	1	63	,	,	PUNCT
ejpam-6243	1	64	ola	ola	PROPN
ejpam-6243	1	65	ashour	ashour	VERB
ejpam-6243	1	66	a	a	DET
ejpam-6243	1	67	abdelnaby1,4	abdelnaby1,4	PROPN
ejpam-6243	1	68	,	,	PUNCT
ejpam-6243	1	69	gunaseelan	gunaseelan	ADJ
ejpam-6243	1	70	mani5	mani5	NOUN
ejpam-6243	2	1	1department	1department	NUM
ejpam-6243	2	2	of	of	ADP
ejpam-6243	2	3	mathematics	mathematic	NOUN
ejpam-6243	2	4	,	,	PUNCT
ejpam-6243	2	5	college	college	NOUN
ejpam-6243	2	6	of	of	ADP
ejpam-6243	2	7	science	science	NOUN
ejpam-6243	2	8	and	and	CCONJ
ejpam-6243	2	9	humanities	humanity	NOUN
ejpam-6243	2	10	in	in	ADP
ejpam-6243	2	11	alkharj	alkharj	NOUN
ejpam-6243	2	12	,	,	PUNCT
ejpam-6243	2	13	prince	prince	PROPN
ejpam-6243	2	14	sattam	sattam	PROPN
ejpam-6243	2	15	bin	bin	PROPN
ejpam-6243	2	16	abdulaziz	abdulaziz	PROPN
ejpam-6243	2	17	university	university	PROPN
ejpam-6243	2	18	,	,	PUNCT
ejpam-6243	2	19	alkharj	alkharj	VERB
ejpam-6243	2	20	11942	11942	NUM
ejpam-6243	2	21	,	,	PUNCT
ejpam-6243	2	22	saudi	saudi	PROPN
ejpam-6243	2	23	arabia	arabia	PROPN
ejpam-6243	2	24	2department	2department	NUM
ejpam-6243	2	25	of	of	ADP
ejpam-6243	2	26	mathematics	mathematics	PROPN
ejpam-6243	2	27	,	,	PUNCT
ejpam-6243	2	28	maharishi	maharishi	PROPN
ejpam-6243	2	29	markandeshwar	markandeshwar	PROPN
ejpam-6243	2	30	(	(	PUNCT
ejpam-6243	2	31	deemed	deem	VERB
ejpam-6243	2	32	to	to	PART
ejpam-6243	2	33	be	be	AUX
ejpam-6243	2	34	university	university	NOUN
ejpam-6243	2	35	)	)	PUNCT
ejpam-6243	2	36	,	,	PUNCT
ejpam-6243	2	37	mullana	mullana	PROPN
ejpam-6243	2	38	,	,	PUNCT
ejpam-6243	2	39	ambala-133203	ambala-133203	ADJ
ejpam-6243	2	40	,	,	PUNCT
ejpam-6243	2	41	india	india	PROPN
ejpam-6243	2	42	3department	3department	PROPN
ejpam-6243	2	43	of	of	ADP
ejpam-6243	2	44	mathematics	mathematic	NOUN
ejpam-6243	2	45	,	,	PUNCT
ejpam-6243	2	46	baba	baba	PROPN
ejpam-6243	2	47	masthnath	masthnath	PROPN
ejpam-6243	2	48	university	university	PROPN
ejpam-6243	2	49	,	,	PUNCT
ejpam-6243	2	50	asthal	asthal	NOUN
ejpam-6243	2	51	bohar	bohar	NOUN
ejpam-6243	2	52	,	,	PUNCT
ejpam-6243	2	53	rohtak	rohtak	NOUN
ejpam-6243	2	54	4department	4department	NUM
ejpam-6243	2	55	of	of	ADP
ejpam-6243	2	56	mathematics	mathematic	NOUN
ejpam-6243	2	57	,	,	PUNCT
ejpam-6243	2	58	cairo	cairo	PROPN
ejpam-6243	2	59	university	university	PROPN
ejpam-6243	2	60	,	,	PUNCT
ejpam-6243	2	61	cairo	cairo	PROPN
ejpam-6243	2	62	,	,	PUNCT
ejpam-6243	2	63	egypt	egypt	PROPN
ejpam-6243	2	64	5department	5department	NUM
ejpam-6243	2	65	of	of	ADP
ejpam-6243	2	66	mathematics	mathematic	NOUN
ejpam-6243	2	67	,	,	PUNCT
ejpam-6243	2	68	saveetha	saveetha	PROPN
ejpam-6243	2	69	school	school	PROPN
ejpam-6243	2	70	of	of	ADP
ejpam-6243	2	71	engineering	engineering	PROPN
ejpam-6243	2	72	,	,	PUNCT
ejpam-6243	2	73	saveetha	saveetha	PROPN
ejpam-6243	2	74	institute	institute	PROPN
ejpam-6243	2	75	of	of	ADP
ejpam-6243	2	76	medical	medical	ADJ
ejpam-6243	2	77	and	and	CCONJ
ejpam-6243	2	78	technical	technical	ADJ
ejpam-6243	2	79	sciences	science	NOUN
ejpam-6243	2	80	,	,	PUNCT
ejpam-6243	2	81	chennai	chennai	NOUN
ejpam-6243	2	82	602105	602105	NUM
ejpam-6243	2	83	,	,	PUNCT
ejpam-6243	2	84	india	india	PROPN
ejpam-6243	2	85	abstract	abstract	NOUN
ejpam-6243	2	86	.	.	PUNCT
ejpam-6243	3	1	in	in	ADP
ejpam-6243	3	2	the	the	DET
ejpam-6243	3	3	present	present	ADJ
ejpam-6243	3	4	manuscript	manuscript	NOUN
ejpam-6243	3	5	,	,	PUNCT
ejpam-6243	3	6	we	we	PRON
ejpam-6243	3	7	introduce	introduce	VERB
ejpam-6243	3	8	a	a	DET
ejpam-6243	3	9	new	new	ADJ
ejpam-6243	3	10	notion	notion	NOUN
ejpam-6243	3	11	of	of	ADP
ejpam-6243	3	12	(	(	PUNCT
ejpam-6243	3	13	β	β	X
ejpam-6243	3	14	,	,	PUNCT
ejpam-6243	3	15	ϕ)−	ϕ)−	PROPN
ejpam-6243	3	16	admissible	admissible	ADJ
ejpam-6243	3	17	hybrid	hybrid	ADJ
ejpam-6243	3	18	contractions	contraction	NOUN
ejpam-6243	3	19	in	in	ADP
ejpam-6243	3	20	metric	metric	ADJ
ejpam-6243	3	21	spaces	space	NOUN
ejpam-6243	3	22	and	and	CCONJ
ejpam-6243	3	23	establish	establish	VERB
ejpam-6243	3	24	fixed	fix	VERB
ejpam-6243	3	25	point	point	NOUN
ejpam-6243	3	26	results	result	NOUN
ejpam-6243	3	27	in	in	ADP
ejpam-6243	3	28	the	the	DET
ejpam-6243	3	29	setting	setting	NOUN
ejpam-6243	3	30	of	of	ADP
ejpam-6243	3	31	these	these	DET
ejpam-6243	3	32	spaces	space	NOUN
ejpam-6243	3	33	.	.	PUNCT
ejpam-6243	4	1	the	the	DET
ejpam-6243	4	2	derived	derive	VERB
ejpam-6243	4	3	results	result	NOUN
ejpam-6243	4	4	extend	extend	VERB
ejpam-6243	4	5	the	the	DET
ejpam-6243	4	6	reported	report	VERB
ejpam-6243	4	7	findings	finding	NOUN
ejpam-6243	4	8	of	of	ADP
ejpam-6243	4	9	the	the	DET
ejpam-6243	4	10	past	past	NOUN
ejpam-6243	4	11	.	.	PUNCT
ejpam-6243	5	1	the	the	DET
ejpam-6243	5	2	derived	derived	ADJ
ejpam-6243	5	3	result	result	NOUN
ejpam-6243	5	4	is	be	AUX
ejpam-6243	5	5	supplemented	supplement	VERB
ejpam-6243	5	6	with	with	ADP
ejpam-6243	5	7	a	a	DET
ejpam-6243	5	8	non	non	ADJ
ejpam-6243	5	9	-	-	ADJ
ejpam-6243	5	10	trivial	trivial	ADJ
ejpam-6243	5	11	example	example	NOUN
ejpam-6243	5	12	.	.	PUNCT
ejpam-6243	6	1	we	we	PRON
ejpam-6243	6	2	have	have	AUX
ejpam-6243	6	3	also	also	ADV
ejpam-6243	6	4	analyzed	analyze	VERB
ejpam-6243	6	5	the	the	DET
ejpam-6243	6	6	ulam	ulam	NOUN
ejpam-6243	6	7	-	-	PUNCT
ejpam-6243	6	8	hyers	hyer	NOUN
ejpam-6243	6	9	stability	stability	NOUN
ejpam-6243	6	10	and	and	CCONJ
ejpam-6243	6	11	well	well	ADV
ejpam-6243	6	12	-	-	PUNCT
ejpam-6243	6	13	poisedness	poisedness	NOUN
ejpam-6243	6	14	as	as	ADP
ejpam-6243	6	15	an	an	DET
ejpam-6243	6	16	application	application	NOUN
ejpam-6243	6	17	to	to	ADP
ejpam-6243	6	18	the	the	DET
ejpam-6243	6	19	derived	derive	VERB
ejpam-6243	6	20	results	result	NOUN
ejpam-6243	6	21	.	.	PUNCT
ejpam-6243	7	1	2020	2020	NUM
ejpam-6243	7	2	mathematics	mathematic	NOUN
ejpam-6243	7	3	subject	subject	NOUN
ejpam-6243	7	4	classifications	classification	NOUN
ejpam-6243	7	5	:	:	PUNCT
ejpam-6243	7	6	47h10	47h10	NUM
ejpam-6243	7	7	,	,	PUNCT
ejpam-6243	7	8	54h25	54h25	NUM
ejpam-6243	7	9	key	key	ADJ
ejpam-6243	7	10	words	word	NOUN
ejpam-6243	7	11	and	and	CCONJ
ejpam-6243	7	12	phrases	phrase	NOUN
ejpam-6243	7	13	:	:	PUNCT
ejpam-6243	7	14	(	(	PUNCT
ejpam-6243	7	15	β	β	X
ejpam-6243	7	16	,	,	PUNCT
ejpam-6243	7	17	ϕ)−admissible	ϕ)−admissible	ADJ
ejpam-6243	7	18	hybrid	hybrid	ADJ
ejpam-6243	7	19	contraction	contraction	NOUN
ejpam-6243	7	20	,	,	PUNCT
ejpam-6243	7	21	fixed	fix	VERB
ejpam-6243	7	22	point	point	NOUN
ejpam-6243	7	23	,	,	PUNCT
ejpam-6243	7	24	ulam	ulam	PROPN
ejpam-6243	7	25	hyers	hyer	VERB
ejpam-6243	7	26	stability	stability	NOUN
ejpam-6243	7	27	,	,	PUNCT
ejpam-6243	7	28	metric	metric	ADJ
ejpam-6243	7	29	space	space	NOUN
ejpam-6243	7	30	1	1	NUM
ejpam-6243	7	31	.	.	PUNCT
ejpam-6243	8	1	introduction	introduction	NOUN
ejpam-6243	8	2	fixed	fix	VERB
ejpam-6243	8	3	point	point	NOUN
ejpam-6243	8	4	theory	theory	NOUN
ejpam-6243	8	5	simply	simply	ADV
ejpam-6243	8	6	deals	deal	VERB
ejpam-6243	8	7	with	with	ADP
ejpam-6243	8	8	the	the	DET
ejpam-6243	8	9	solution	solution	NOUN
ejpam-6243	8	10	of	of	ADP
ejpam-6243	8	11	the	the	DET
ejpam-6243	8	12	equation	equation	NOUN
ejpam-6243	8	13	tx	tx	NOUN
ejpam-6243	9	1	=	=	PUNCT
ejpam-6243	10	1	x	x	INTJ
ejpam-6243	10	2	where	where	SCONJ
ejpam-6243	10	3	t	t	PROPN
ejpam-6243	10	4	is	be	AUX
ejpam-6243	10	5	a	a	DET
ejpam-6243	10	6	self	self	NOUN
ejpam-6243	10	7	-	-	PUNCT
ejpam-6243	10	8	map	map	NOUN
ejpam-6243	10	9	on	on	ADP
ejpam-6243	10	10	a	a	DET
ejpam-6243	10	11	non	non	ADJ
ejpam-6243	10	12	-	-	ADJ
ejpam-6243	10	13	empty	empty	ADJ
ejpam-6243	10	14	set	set	NOUN
ejpam-6243	10	15	x.	x.	NOUN
ejpam-6243	11	1	the	the	DET
ejpam-6243	11	2	fixed	fix	VERB
ejpam-6243	11	3	point	point	NOUN
ejpam-6243	11	4	problem	problem	NOUN
ejpam-6243	11	5	first	first	ADV
ejpam-6243	11	6	appeared	appear	VERB
ejpam-6243	11	7	in	in	ADP
ejpam-6243	11	8	the	the	DET
ejpam-6243	11	9	solution	solution	NOUN
ejpam-6243	11	10	of	of	ADP
ejpam-6243	11	11	an	an	DET
ejpam-6243	11	12	initial	initial	ADJ
ejpam-6243	11	13	value	value	NOUN
ejpam-6243	11	14	problem	problem	NOUN
ejpam-6243	11	15	.	.	PUNCT
ejpam-6243	12	1	liouville	liouville	PROPN
ejpam-6243	13	1	[	[	X
ejpam-6243	13	2	1	1	X
ejpam-6243	13	3	]	]	PUNCT
ejpam-6243	13	4	in	in	ADP
ejpam-6243	13	5	1837	1837	NUM
ejpam-6243	13	6	and	and	CCONJ
ejpam-6243	13	7	picard	picard	NOUN
ejpam-6243	13	8	[	[	X
ejpam-6243	13	9	2	2	NUM
ejpam-6243	13	10	]	]	PUNCT
ejpam-6243	13	11	in	in	ADP
ejpam-6243	13	12	1890	1890	NUM
ejpam-6243	13	13	solved	solve	VERB
ejpam-6243	13	14	the	the	DET
ejpam-6243	13	15	problem	problem	NOUN
ejpam-6243	13	16	using	use	VERB
ejpam-6243	13	17	the	the	DET
ejpam-6243	13	18	successive	successive	ADJ
ejpam-6243	13	19	approximation	approximation	NOUN
ejpam-6243	13	20	method	method	NOUN
ejpam-6243	13	21	that	that	PRON
ejpam-6243	13	22	also	also	ADV
ejpam-6243	13	23	provided	provide	VERB
ejpam-6243	13	24	the	the	DET
ejpam-6243	13	25	solution	solution	NOUN
ejpam-6243	13	26	of	of	ADP
ejpam-6243	13	27	the	the	DET
ejpam-6243	13	28	fixed	fix	VERB
ejpam-6243	13	29	point	point	NOUN
ejpam-6243	13	30	equation	equation	NOUN
ejpam-6243	13	31	.	.	PUNCT
ejpam-6243	14	1	before	before	ADP
ejpam-6243	14	2	1922	1922	NUM
ejpam-6243	14	3	,	,	PUNCT
ejpam-6243	14	4	there	there	PRON
ejpam-6243	14	5	was	be	VERB
ejpam-6243	14	6	no	no	DET
ejpam-6243	14	7	any	any	DET
ejpam-6243	14	8	direct	direct	ADJ
ejpam-6243	14	9	method	method	NOUN
ejpam-6243	14	10	to	to	PART
ejpam-6243	14	11	evaluate	evaluate	VERB
ejpam-6243	14	12	the	the	DET
ejpam-6243	14	13	fixed	fix	VERB
ejpam-6243	14	14	point	point	NOUN
ejpam-6243	14	15	of	of	ADP
ejpam-6243	14	16	∗corresponding	∗corresponde	VERB
ejpam-6243	14	17	author	author	NOUN
ejpam-6243	14	18	.	.	PUNCT
ejpam-6243	15	1	doi	doi	NOUN
ejpam-6243	15	2	:	:	PUNCT
ejpam-6243	15	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6243	https://doi.org/10.29020/nybg.ejpam.v18i3.6243	ADJ
ejpam-6243	15	4	email	email	NOUN
ejpam-6243	15	5	addresses	address	NOUN
ejpam-6243	15	6	:	:	PUNCT
ejpam-6243	15	7	r.gopalan@psau.edu.sa	r.gopalan@psau.edu.sa	NOUN
ejpam-6243	15	8	(	(	PUNCT
ejpam-6243	15	9	rajagopalan	rajagopalan	PROPN
ejpam-6243	15	10	r	r	NOUN
ejpam-6243	15	11	)	)	PUNCT
ejpam-6243	15	12	,	,	PUNCT
ejpam-6243	15	13	manojkumar@mmumullana.org	manojkumar@mmumullana.org	PROPN
ejpam-6243	15	14	(	(	PUNCT
ejpam-6243	15	15	m.	m.	PROPN
ejpam-6243	15	16	kumar	kumar	PROPN
ejpam-6243	15	17	)	)	PUNCT
ejpam-6243	15	18	,	,	PUNCT
ejpam-6243	15	19	latasharma0701@gmail.com	latasharma0701@gmail.com	X
ejpam-6243	16	1	(	(	PUNCT
ejpam-6243	16	2	p.	p.	NOUN
ejpam-6243	16	3	lata	lata	PROPN
ejpam-6243	16	4	)	)	PUNCT
ejpam-6243	16	5	,	,	PUNCT
ejpam-6243	16	6	rayanalkhowaiter1@gmail.com	rayanalkhowaiter1@gmail.com	X
ejpam-6243	16	7	(	(	PUNCT
ejpam-6243	16	8	r.	r.	NOUN
ejpam-6243	16	9	a	a	DET
ejpam-6243	16	10	alkhowaiter	alkhowaiter	NOUN
ejpam-6243	16	11	)	)	PUNCT
ejpam-6243	16	12	,	,	PUNCT
ejpam-6243	17	1	eng	eng	PROPN
ejpam-6243	17	2	hossam21@yahoo.com	hossam21@yahoo.com	X
ejpam-6243	17	3	(	(	PUNCT
ejpam-6243	17	4	h.	h.	PROPN
ejpam-6243	17	5	a.	a.	PROPN
ejpam-6243	17	6	nabway	nabway	PROPN
ejpam-6243	17	7	)	)	PUNCT
ejpam-6243	17	8	,	,	PUNCT
ejpam-6243	17	9	o.abdelnaby@psau.edu.sa	o.abdelnaby@psau.edu.sa	PROPN
ejpam-6243	17	10	(	(	PUNCT
ejpam-6243	17	11	o.	o.	NOUN
ejpam-6243	17	12	a.	a.	PROPN
ejpam-6243	17	13	a.	a.	PROPN
ejpam-6243	17	14	abdelnaby	abdelnaby	PROPN
ejpam-6243	17	15	)	)	PUNCT
ejpam-6243	17	16	,	,	PUNCT
ejpam-6243	17	17	mathsguna@yahoo.com	mathsguna@yahoo.com	X
ejpam-6243	17	18	(	(	PUNCT
ejpam-6243	17	19	g.	g.	PROPN
ejpam-6243	17	20	mani	mani	PROPN
ejpam-6243	17	21	)	)	PUNCT
ejpam-6243	17	22	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6243	18	1	1	1	NUM
ejpam-6243	18	2	copyright	copyright	NOUN
ejpam-6243	18	3	:	:	PUNCT
ejpam-6243	18	4	©	©	PROPN
ejpam-6243	18	5	2025	2025	NUM
ejpam-6243	18	6	the	the	DET
ejpam-6243	18	7	author(s	author(s	NOUN
ejpam-6243	18	8	)	)	PUNCT
ejpam-6243	18	9	.	.	PUNCT
ejpam-6243	19	1	(	(	PUNCT
ejpam-6243	19	2	cc	cc	NOUN
ejpam-6243	19	3	by	by	ADP
ejpam-6243	19	4	-	-	PUNCT
ejpam-6243	19	5	nc	nc	PROPN
ejpam-6243	19	6	4.0	4.0	NUM
ejpam-6243	19	7	)	)	PUNCT
ejpam-6243	19	8	r	r	NOUN
ejpam-6243	19	9	ramaswamy	ramaswamy	NOUN
ejpam-6243	19	10	et	et	PROPN
ejpam-6243	19	11	al	al	PROPN
ejpam-6243	19	12	.	.	PUNCT
ejpam-6243	19	13	/	/	SYM
ejpam-6243	19	14	eur	eur	PROPN
ejpam-6243	19	15	.	.	PUNCT
ejpam-6243	20	1	j.	j.	PROPN
ejpam-6243	20	2	pure	pure	PROPN
ejpam-6243	20	3	appl	appl	PROPN
ejpam-6243	20	4	.	.	PROPN
ejpam-6243	20	5	math	math	PROPN
ejpam-6243	20	6	,	,	PUNCT
ejpam-6243	20	7	18	18	NUM
ejpam-6243	20	8	(	(	PUNCT
ejpam-6243	20	9	3	3	NUM
ejpam-6243	20	10	)	)	PUNCT
ejpam-6243	20	11	(	(	PUNCT
ejpam-6243	20	12	2025	2025	NUM
ejpam-6243	20	13	)	)	PUNCT
ejpam-6243	20	14	,	,	PUNCT
ejpam-6243	20	15	6243	6243	NUM
ejpam-6243	20	16	2	2	NUM
ejpam-6243	20	17	of	of	ADP
ejpam-6243	20	18	14	14	NUM
ejpam-6243	20	19	a	a	DET
ejpam-6243	20	20	map	map	NOUN
ejpam-6243	20	21	.	.	PUNCT
ejpam-6243	21	1	in	in	ADP
ejpam-6243	21	2	1922	1922	NUM
ejpam-6243	21	3	,	,	PUNCT
ejpam-6243	21	4	stephen	stephen	NOUN
ejpam-6243	21	5	banach	banach	NOUN
ejpam-6243	22	1	[	[	X
ejpam-6243	22	2	3	3	X
ejpam-6243	22	3	]	]	PUNCT
ejpam-6243	22	4	was	be	AUX
ejpam-6243	22	5	the	the	DET
ejpam-6243	22	6	first	first	ADJ
ejpam-6243	22	7	to	to	PART
ejpam-6243	22	8	introduce	introduce	VERB
ejpam-6243	22	9	the	the	DET
ejpam-6243	22	10	contraction	contraction	NOUN
ejpam-6243	22	11	principle	principle	NOUN
ejpam-6243	22	12	to	to	PART
ejpam-6243	22	13	evaluate	evaluate	VERB
ejpam-6243	22	14	the	the	DET
ejpam-6243	22	15	fixed	fix	VERB
ejpam-6243	22	16	point	point	NOUN
ejpam-6243	22	17	in	in	ADP
ejpam-6243	22	18	the	the	DET
ejpam-6243	22	19	setting	setting	NOUN
ejpam-6243	22	20	of	of	ADP
ejpam-6243	22	21	complete	complete	ADJ
ejpam-6243	22	22	metric	metric	ADJ
ejpam-6243	22	23	spaces	space	NOUN
ejpam-6243	22	24	.	.	PUNCT
ejpam-6243	23	1	metric	metric	ADJ
ejpam-6243	23	2	fixed	fix	VERB
ejpam-6243	23	3	point	point	NOUN
ejpam-6243	23	4	theory	theory	NOUN
ejpam-6243	23	5	has	have	AUX
ejpam-6243	23	6	been	be	AUX
ejpam-6243	23	7	investigated	investigate	VERB
ejpam-6243	23	8	since	since	SCONJ
ejpam-6243	23	9	then	then	ADV
ejpam-6243	23	10	,	,	PUNCT
ejpam-6243	23	11	by	by	ADP
ejpam-6243	23	12	many	many	ADJ
ejpam-6243	23	13	researchers	researcher	NOUN
ejpam-6243	23	14	,	,	PUNCT
ejpam-6243	23	15	as	as	SCONJ
ejpam-6243	23	16	it	it	PRON
ejpam-6243	23	17	is	be	AUX
ejpam-6243	23	18	the	the	DET
ejpam-6243	23	19	natural	natural	ADJ
ejpam-6243	23	20	and	and	CCONJ
ejpam-6243	23	21	strong	strong	ADJ
ejpam-6243	23	22	connection	connection	NOUN
ejpam-6243	23	23	of	of	ADP
ejpam-6243	23	24	the	the	DET
ejpam-6243	23	25	theoretical	theoretical	ADJ
ejpam-6243	23	26	results	result	NOUN
ejpam-6243	23	27	in	in	ADP
ejpam-6243	23	28	non	non	ADJ
ejpam-6243	23	29	-	-	ADJ
ejpam-6243	23	30	linear	linear	ADJ
ejpam-6243	23	31	functional	functional	ADJ
ejpam-6243	23	32	analysis	analysis	NOUN
ejpam-6243	23	33	with	with	ADP
ejpam-6243	23	34	applied	apply	VERB
ejpam-6243	23	35	sciences	science	NOUN
ejpam-6243	23	36	.	.	PUNCT
ejpam-6243	24	1	later	later	ADV
ejpam-6243	24	2	a	a	DET
ejpam-6243	24	3	lot	lot	NOUN
ejpam-6243	24	4	of	of	ADP
ejpam-6243	24	5	generalization	generalization	NOUN
ejpam-6243	24	6	of	of	ADP
ejpam-6243	24	7	banach	banach	NOUN
ejpam-6243	24	8	contraction	contraction	NOUN
ejpam-6243	24	9	principle	principle	NOUN
ejpam-6243	24	10	was	be	AUX
ejpam-6243	24	11	done	do	VERB
ejpam-6243	24	12	by	by	ADP
ejpam-6243	24	13	extending	extend	VERB
ejpam-6243	24	14	and	and	CCONJ
ejpam-6243	24	15	generalizing	generalize	VERB
ejpam-6243	24	16	the	the	DET
ejpam-6243	24	17	topological	topological	ADJ
ejpam-6243	24	18	spaces	space	NOUN
ejpam-6243	24	19	as	as	ADV
ejpam-6243	24	20	well	well	ADV
ejpam-6243	24	21	as	as	ADP
ejpam-6243	24	22	the	the	DET
ejpam-6243	24	23	contraction	contraction	NOUN
ejpam-6243	24	24	conditions	condition	NOUN
ejpam-6243	24	25	.	.	PUNCT
ejpam-6243	25	1	in	in	ADP
ejpam-6243	25	2	the	the	DET
ejpam-6243	25	3	recent	recent	ADJ
ejpam-6243	25	4	past	past	NOUN
ejpam-6243	25	5	,	,	PUNCT
ejpam-6243	25	6	interpolative	interpolative	ADJ
ejpam-6243	25	7	contractive	contractive	ADJ
ejpam-6243	25	8	conditions	condition	NOUN
ejpam-6243	25	9	were	be	AUX
ejpam-6243	25	10	reported	report	VERB
ejpam-6243	25	11	by	by	ADP
ejpam-6243	25	12	some	some	DET
ejpam-6243	25	13	researchers	researcher	NOUN
ejpam-6243	25	14	and	and	CCONJ
ejpam-6243	25	15	fixed	fix	VERB
ejpam-6243	25	16	point	point	NOUN
ejpam-6243	25	17	results	result	NOUN
ejpam-6243	25	18	please	please	INTJ
ejpam-6243	25	19	see	see	VERB
ejpam-6243	25	20	[	[	X
ejpam-6243	25	21	4–12	4–12	NOUN
ejpam-6243	25	22	]	]	X
ejpam-6243	25	23	.	.	PUNCT
ejpam-6243	26	1	the	the	DET
ejpam-6243	26	2	notion	notion	NOUN
ejpam-6243	26	3	of	of	ADP
ejpam-6243	26	4	ulam	ulam	PROPN
ejpam-6243	26	5	stability	stability	PROPN
ejpam-6243	26	6	was	be	AUX
ejpam-6243	26	7	proposed	propose	VERB
ejpam-6243	26	8	by	by	ADP
ejpam-6243	26	9	by	by	ADP
ejpam-6243	26	10	ulam	ulam	PROPN
ejpam-6243	27	1	[	[	X
ejpam-6243	27	2	13	13	NUM
ejpam-6243	27	3	]	]	PUNCT
ejpam-6243	27	4	and	and	CCONJ
ejpam-6243	27	5	developed	develop	VERB
ejpam-6243	27	6	by	by	ADP
ejpam-6243	27	7	hyers	hyer	NOUN
ejpam-6243	27	8	[	[	X
ejpam-6243	27	9	14	14	NUM
ejpam-6243	27	10	]	]	PUNCT
ejpam-6243	27	11	,	,	PUNCT
ejpam-6243	27	12	ulam	ulam	X
ejpam-6243	28	1	[	[	X
ejpam-6243	28	2	15	15	NUM
ejpam-6243	28	3	]	]	PUNCT
ejpam-6243	28	4	,	,	PUNCT
ejpam-6243	28	5	rassias	rassia	VERB
ejpam-6243	29	1	[	[	X
ejpam-6243	29	2	16	16	NUM
ejpam-6243	29	3	]	]	PUNCT
ejpam-6243	29	4	,	,	PUNCT
ejpam-6243	29	5	etc	etc	X
ejpam-6243	29	6	.	.	X
ejpam-6243	30	1	in	in	ADP
ejpam-6243	30	2	2023	2023	NUM
ejpam-6243	30	3	,	,	PUNCT
ejpam-6243	30	4	manoj	manoj	PROPN
ejpam-6243	30	5	et	et	PROPN
ejpam-6243	30	6	al	al	PROPN
ejpam-6243	31	1	[	[	X
ejpam-6243	31	2	17	17	NUM
ejpam-6243	31	3	]	]	PUNCT
ejpam-6243	31	4	reported	report	VERB
ejpam-6243	31	5	ulam	ulam	PROPN
ejpam-6243	31	6	-	-	PUNCT
ejpam-6243	31	7	hyer	hyer	NOUN
ejpam-6243	31	8	’s	’s	PART
ejpam-6243	31	9	stability	stability	NOUN
ejpam-6243	31	10	and	and	CCONJ
ejpam-6243	31	11	well	well	ADV
ejpam-6243	31	12	-	-	PUNCT
ejpam-6243	31	13	posedness	posedness	NOUN
ejpam-6243	31	14	of	of	ADP
ejpam-6243	31	15	fixed	fix	VERB
ejpam-6243	31	16	point	point	NOUN
ejpam-6243	31	17	problems	problem	NOUN
ejpam-6243	31	18	in	in	ADP
ejpam-6243	31	19	the	the	DET
ejpam-6243	31	20	setting	setting	NOUN
ejpam-6243	31	21	of	of	ADP
ejpam-6243	31	22	c∗	c∗	PROPN
ejpam-6243	31	23	algebra	algebra	PROPN
ejpam-6243	31	24	valued	value	VERB
ejpam-6243	31	25	bipolar	bipolar	ADJ
ejpam-6243	31	26	metric	metric	ADJ
ejpam-6243	31	27	spaces	space	NOUN
ejpam-6243	31	28	.	.	PUNCT
ejpam-6243	32	1	inspired	inspire	VERB
ejpam-6243	32	2	,	,	PUNCT
ejpam-6243	32	3	in	in	ADP
ejpam-6243	32	4	this	this	DET
ejpam-6243	32	5	paper	paper	NOUN
ejpam-6243	32	6	,	,	PUNCT
ejpam-6243	32	7	we	we	PRON
ejpam-6243	32	8	introduce	introduce	VERB
ejpam-6243	32	9	the	the	DET
ejpam-6243	32	10	notion	notion	NOUN
ejpam-6243	32	11	of	of	ADP
ejpam-6243	32	12	”	"	PUNCT
ejpam-6243	32	13	(	(	PUNCT
ejpam-6243	32	14	β	β	X
ejpam-6243	32	15	,	,	PUNCT
ejpam-6243	32	16	ϕ	ϕ	NOUN
ejpam-6243	32	17	)	)	PUNCT
ejpam-6243	32	18	admissible	admissible	ADJ
ejpam-6243	32	19	hybrid	hybrid	ADJ
ejpam-6243	32	20	contraction	contraction	NOUN
ejpam-6243	32	21	”	"	PUNCT
ejpam-6243	32	22	that	that	PRON
ejpam-6243	32	23	combines	combine	VERB
ejpam-6243	32	24	and	and	CCONJ
ejpam-6243	32	25	unifies	unify	VERB
ejpam-6243	32	26	several	several	ADJ
ejpam-6243	32	27	existing	exist	VERB
ejpam-6243	32	28	linear	linear	NOUN
ejpam-6243	32	29	and	and	CCONJ
ejpam-6243	32	30	nonlinear	nonlinear	ADJ
ejpam-6243	32	31	contractions	contraction	NOUN
ejpam-6243	32	32	and	and	CCONJ
ejpam-6243	32	33	also	also	ADV
ejpam-6243	32	34	extends	extend	VERB
ejpam-6243	32	35	fixed	fix	VERB
ejpam-6243	32	36	point	point	NOUN
ejpam-6243	32	37	results	result	NOUN
ejpam-6243	32	38	of	of	ADP
ejpam-6243	32	39	such	such	ADJ
ejpam-6243	32	40	contraction	contraction	NOUN
ejpam-6243	32	41	conditions	condition	NOUN
ejpam-6243	32	42	.	.	PUNCT
ejpam-6243	33	1	we	we	PRON
ejpam-6243	33	2	also	also	ADV
ejpam-6243	33	3	analyze	analyze	VERB
ejpam-6243	33	4	the	the	DET
ejpam-6243	33	5	ulamhyer	ulamhyer	NOUN
ejpam-6243	33	6	’s	’s	PART
ejpam-6243	33	7	stability	stability	NOUN
ejpam-6243	33	8	and	and	CCONJ
ejpam-6243	33	9	well	well	ADV
ejpam-6243	33	10	-	-	PUNCT
ejpam-6243	33	11	posedness	posedness	NOUN
ejpam-6243	33	12	of	of	ADP
ejpam-6243	33	13	fixed	fix	VERB
ejpam-6243	33	14	point	point	NOUN
ejpam-6243	33	15	problems	problem	NOUN
ejpam-6243	33	16	by	by	ADP
ejpam-6243	33	17	applying	apply	VERB
ejpam-6243	33	18	the	the	DET
ejpam-6243	33	19	derived	derive	VERB
ejpam-6243	33	20	results	result	NOUN
ejpam-6243	33	21	.	.	PUNCT
ejpam-6243	34	1	accordingly	accordingly	ADV
ejpam-6243	34	2	,	,	PUNCT
ejpam-6243	34	3	the	the	DET
ejpam-6243	34	4	rest	rest	NOUN
ejpam-6243	34	5	of	of	ADP
ejpam-6243	34	6	the	the	DET
ejpam-6243	34	7	paper	paper	NOUN
ejpam-6243	34	8	is	be	AUX
ejpam-6243	34	9	organised	organise	VERB
ejpam-6243	34	10	as	as	SCONJ
ejpam-6243	34	11	follows	follow	VERB
ejpam-6243	34	12	:	:	PUNCT
ejpam-6243	34	13	in	in	ADP
ejpam-6243	34	14	section-2	section-2	NUM
ejpam-6243	34	15	,	,	PUNCT
ejpam-6243	34	16	we	we	PRON
ejpam-6243	34	17	review	review	VERB
ejpam-6243	34	18	some	some	DET
ejpam-6243	34	19	preliminaries	preliminary	NOUN
ejpam-6243	34	20	and	and	CCONJ
ejpam-6243	34	21	monograph	monograph	NOUN
ejpam-6243	34	22	which	which	PRON
ejpam-6243	34	23	are	be	AUX
ejpam-6243	34	24	required	require	VERB
ejpam-6243	34	25	in	in	ADP
ejpam-6243	34	26	the	the	DET
ejpam-6243	34	27	sequel	sequel	NOUN
ejpam-6243	34	28	.	.	PUNCT
ejpam-6243	35	1	in	in	ADP
ejpam-6243	35	2	section-3	section-3	PROPN
ejpam-6243	35	3	,	,	PUNCT
ejpam-6243	35	4	we	we	PRON
ejpam-6243	35	5	present	present	VERB
ejpam-6243	35	6	our	our	PRON
ejpam-6243	35	7	main	main	ADJ
ejpam-6243	35	8	results	result	NOUN
ejpam-6243	35	9	and	and	CCONJ
ejpam-6243	35	10	establish	establish	VERB
ejpam-6243	35	11	fixed	fix	VERB
ejpam-6243	35	12	point	point	NOUN
ejpam-6243	35	13	results	result	NOUN
ejpam-6243	35	14	using	use	VERB
ejpam-6243	35	15	the	the	DET
ejpam-6243	35	16	”	"	PUNCT
ejpam-6243	35	17	(	(	PUNCT
ejpam-6243	35	18	β	β	X
ejpam-6243	35	19	,	,	PUNCT
ejpam-6243	35	20	ϕ	ϕ	NOUN
ejpam-6243	35	21	)	)	PUNCT
ejpam-6243	35	22	admissible	admissible	ADJ
ejpam-6243	35	23	hybrid	hybrid	ADJ
ejpam-6243	35	24	contraction	contraction	NOUN
ejpam-6243	35	25	”	"	PUNCT
ejpam-6243	35	26	and	and	CCONJ
ejpam-6243	35	27	supplement	supplement	VERB
ejpam-6243	35	28	the	the	DET
ejpam-6243	35	29	results	result	NOUN
ejpam-6243	35	30	with	with	ADP
ejpam-6243	35	31	non	non	ADJ
ejpam-6243	35	32	-	-	ADJ
ejpam-6243	35	33	trivial	trivial	ADJ
ejpam-6243	35	34	example	example	NOUN
ejpam-6243	35	35	.	.	PUNCT
ejpam-6243	36	1	in	in	ADP
ejpam-6243	36	2	section-4	section-4	NUM
ejpam-6243	36	3	,	,	PUNCT
ejpam-6243	36	4	we	we	PRON
ejpam-6243	36	5	present	present	VERB
ejpam-6243	36	6	an	an	DET
ejpam-6243	36	7	application	application	NOUN
ejpam-6243	36	8	to	to	PART
ejpam-6243	36	9	analyse	analyse	VERB
ejpam-6243	36	10	ulam	ulam	PROPN
ejpam-6243	36	11	-	-	PUNCT
ejpam-6243	36	12	hyer	hyer	PROPN
ejpam-6243	36	13	’s	’s	PART
ejpam-6243	36	14	stability	stability	NOUN
ejpam-6243	36	15	and	and	CCONJ
ejpam-6243	36	16	welll	welll	ADV
ejpam-6243	36	17	posedness	posedness	NOUN
ejpam-6243	36	18	of	of	ADP
ejpam-6243	36	19	fixed	fix	VERB
ejpam-6243	36	20	point	point	NOUN
ejpam-6243	36	21	problems	problem	NOUN
ejpam-6243	36	22	.	.	PUNCT
ejpam-6243	37	1	2	2	X
ejpam-6243	37	2	.	.	NUM
ejpam-6243	37	3	preliminaries	preliminary	NOUN
ejpam-6243	37	4	the	the	DET
ejpam-6243	37	5	following	follow	VERB
ejpam-6243	37	6	are	be	AUX
ejpam-6243	37	7	required	require	VERB
ejpam-6243	37	8	in	in	ADP
ejpam-6243	37	9	the	the	DET
ejpam-6243	37	10	sequel	sequel	NOUN
ejpam-6243	37	11	.	.	PUNCT
ejpam-6243	38	1	definition	definition	NOUN
ejpam-6243	38	2	1	1	NUM
ejpam-6243	38	3	.	.	PUNCT
ejpam-6243	39	1	[	[	X
ejpam-6243	39	2	7	7	NUM
ejpam-6243	39	3	,	,	PUNCT
ejpam-6243	39	4	18	18	NUM
ejpam-6243	39	5	]	]	PUNCT
ejpam-6243	39	6	let	let	VERB
ejpam-6243	39	7	φ	φ	PROPN
ejpam-6243	39	8	be	be	AUX
ejpam-6243	39	9	the	the	DET
ejpam-6243	39	10	set	set	NOUN
ejpam-6243	39	11	of	of	ADP
ejpam-6243	39	12	functions	function	NOUN
ejpam-6243	39	13	ϕ	ϕ	NOUN
ejpam-6243	39	14	:	:	PUNCT
ejpam-6243	40	1	[	[	X
ejpam-6243	40	2	0,+∞	0,+∞	NUM
ejpam-6243	40	3	)	)	PUNCT
ejpam-6243	40	4	→	→	PUNCT
ejpam-6243	41	1	[	[	X
ejpam-6243	41	2	0,+∞	0,+∞	NUM
ejpam-6243	41	3	)	)	PUNCT
ejpam-6243	41	4	such	such	ADJ
ejpam-6243	41	5	that	that	SCONJ
ejpam-6243	41	6	(	(	PUNCT
ejpam-6243	41	7	i	i	NOUN
ejpam-6243	41	8	)	)	PUNCT
ejpam-6243	41	9	ϕ	ϕ	PROPN
ejpam-6243	41	10	is	be	AUX
ejpam-6243	41	11	non	non	ADJ
ejpam-6243	41	12	-	-	ADJ
ejpam-6243	41	13	decreasing	decrease	VERB
ejpam-6243	41	14	;	;	PUNCT
ejpam-6243	41	15	(	(	PUNCT
ejpam-6243	41	16	ii	ii	NOUN
ejpam-6243	41	17	)	)	PUNCT
ejpam-6243	41	18	there	there	PRON
ejpam-6243	41	19	exists	exist	VERB
ejpam-6243	41	20	n0	n0	PROPN
ejpam-6243	41	21	∈	∈	PROPN
ejpam-6243	41	22	n	n	PRON
ejpam-6243	41	23	and	and	CCONJ
ejpam-6243	41	24	δ	δ	PROPN
ejpam-6243	41	25	∈	∈	PROPN
ejpam-6243	41	26	(	(	PUNCT
ejpam-6243	41	27	0	0	NUM
ejpam-6243	41	28	,	,	PUNCT
ejpam-6243	41	29	1	1	NUM
ejpam-6243	41	30	)	)	PUNCT
ejpam-6243	41	31	and	and	CCONJ
ejpam-6243	41	32	a	a	DET
ejpam-6243	41	33	convergent	convergent	NOUN
ejpam-6243	41	34	series	series	NOUN
ejpam-6243	41	35	∑+∞	∑+∞	ADJ
ejpam-6243	41	36	i=0	i=0	PROPN
ejpam-6243	41	37	vi	vi	PROPN
ejpam-6243	41	38	with	with	ADP
ejpam-6243	41	39	vi	vi	PROPN
ejpam-6243	41	40	≥	≥	X
ejpam-6243	41	41	0	0	NUM
ejpam-6243	41	42	such	such	ADJ
ejpam-6243	41	43	that	that	SCONJ
ejpam-6243	41	44	ϕi+1(t	ϕi+1(t	PROPN
ejpam-6243	41	45	)	)	PUNCT
ejpam-6243	41	46	≤	≤	NOUN
ejpam-6243	41	47	δϕ(t	δϕ(t	NOUN
ejpam-6243	41	48	)	)	PUNCT
ejpam-6243	41	49	+	+	X
ejpam-6243	41	50	vi	vi	NOUN
ejpam-6243	41	51	,	,	PUNCT
ejpam-6243	41	52	(	(	PUNCT
ejpam-6243	41	53	1	1	X
ejpam-6243	41	54	)	)	PUNCT
ejpam-6243	41	55	for	for	ADP
ejpam-6243	41	56	i	i	PRON
ejpam-6243	41	57	≥	≥	NOUN
ejpam-6243	41	58	io	io	NOUN
ejpam-6243	41	59	and	and	CCONJ
ejpam-6243	41	60	t	t	PROPN
ejpam-6243	41	61	≥	≥	NUM
ejpam-6243	41	62	0	0	NUM
ejpam-6243	41	63	.	.	PUNCT
ejpam-6243	42	1	each	each	DET
ejpam-6243	42	2	ϕ	ϕ	PROPN
ejpam-6243	42	3	∈	∈	PROPN
ejpam-6243	42	4	φ	φ	PROPN
ejpam-6243	42	5	is	be	AUX
ejpam-6243	42	6	called	call	VERB
ejpam-6243	42	7	a	a	DET
ejpam-6243	42	8	(	(	PUNCT
ejpam-6243	42	9	c)−comparison	c)−comparison	NOUN
ejpam-6243	42	10	function	function	NOUN
ejpam-6243	42	11	.	.	PUNCT
ejpam-6243	43	1	lemma	lemma	PROPN
ejpam-6243	43	2	1	1	NUM
ejpam-6243	43	3	.	.	PUNCT
ejpam-6243	44	1	[	[	X
ejpam-6243	44	2	18	18	NUM
ejpam-6243	44	3	]	]	X
ejpam-6243	44	4	if	if	SCONJ
ejpam-6243	44	5	ϕ	ϕ	PROPN
ejpam-6243	44	6	∈	∈	PROPN
ejpam-6243	44	7	φ	φ	PROPN
ejpam-6243	44	8	,	,	PUNCT
ejpam-6243	44	9	then	then	ADV
ejpam-6243	44	10	r	r	NOUN
ejpam-6243	44	11	ramaswamy	ramaswamy	NOUN
ejpam-6243	44	12	et	et	PROPN
ejpam-6243	44	13	al	al	PROPN
ejpam-6243	44	14	.	.	PUNCT
ejpam-6243	44	15	/	/	SYM
ejpam-6243	44	16	eur	eur	PROPN
ejpam-6243	44	17	.	.	PUNCT
ejpam-6243	45	1	j.	j.	PROPN
ejpam-6243	45	2	pure	pure	PROPN
ejpam-6243	45	3	appl	appl	PROPN
ejpam-6243	45	4	.	.	PROPN
ejpam-6243	45	5	math	math	PROPN
ejpam-6243	45	6	,	,	PUNCT
ejpam-6243	45	7	18	18	NUM
ejpam-6243	45	8	(	(	PUNCT
ejpam-6243	45	9	3	3	NUM
ejpam-6243	45	10	)	)	PUNCT
ejpam-6243	45	11	(	(	PUNCT
ejpam-6243	45	12	2025	2025	NUM
ejpam-6243	45	13	)	)	PUNCT
ejpam-6243	45	14	,	,	PUNCT
ejpam-6243	45	15	6243	6243	NUM
ejpam-6243	45	16	3	3	NUM
ejpam-6243	45	17	of	of	ADP
ejpam-6243	45	18	14	14	NUM
ejpam-6243	45	19	(	(	PUNCT
ejpam-6243	45	20	i	i	NOUN
ejpam-6243	45	21	)	)	PUNCT
ejpam-6243	45	22	(	(	PUNCT
ejpam-6243	45	23	ϕn(t))n∈n	ϕn(t))n∈n	PROPN
ejpam-6243	45	24	converges	converge	VERB
ejpam-6243	45	25	to	to	ADP
ejpam-6243	45	26	0	0	NUM
ejpam-6243	45	27	as	as	ADP
ejpam-6243	45	28	n	n	PROPN
ejpam-6243	45	29	→	→	SYM
ejpam-6243	45	30	+	+	NUM
ejpam-6243	45	31	∞	∞	PROPN
ejpam-6243	45	32	for	for	ADP
ejpam-6243	45	33	t	t	PROPN
ejpam-6243	45	34	≥	≥	PROPN
ejpam-6243	45	35	0	0	NUM
ejpam-6243	45	36	;	;	PUNCT
ejpam-6243	45	37	(	(	PUNCT
ejpam-6243	45	38	ii	ii	NOUN
ejpam-6243	45	39	)	)	PUNCT
ejpam-6243	45	40	ϕ(t	ϕ(t	NUM
ejpam-6243	45	41	)	)	PUNCT
ejpam-6243	45	42	<	<	X
ejpam-6243	45	43	t	t	PROPN
ejpam-6243	45	44	,	,	PUNCT
ejpam-6243	45	45	for	for	ADP
ejpam-6243	45	46	any	any	DET
ejpam-6243	45	47	t	t	NOUN
ejpam-6243	45	48	∈	∈	PROPN
ejpam-6243	45	49	r+	r+	NOUN
ejpam-6243	45	50	;	;	PUNCT
ejpam-6243	45	51	(	(	PUNCT
ejpam-6243	45	52	iii	iii	X
ejpam-6243	45	53	)	)	PUNCT
ejpam-6243	45	54	ϕ	ϕ	NOUN
ejpam-6243	45	55	is	be	AUX
ejpam-6243	45	56	continuous	continuous	ADJ
ejpam-6243	45	57	at	at	ADP
ejpam-6243	45	58	0	0	NUM
ejpam-6243	45	59	;	;	PUNCT
ejpam-6243	45	60	(	(	PUNCT
ejpam-6243	45	61	iv	iv	X
ejpam-6243	45	62	)	)	PUNCT
ejpam-6243	45	63	the	the	DET
ejpam-6243	45	64	series	series	NOUN
ejpam-6243	45	65	∑+∞	∑+∞	AUX
ejpam-6243	45	66	k=0	k=0	PUNCT
ejpam-6243	45	67	ϕ	ϕ	PROPN
ejpam-6243	45	68	k(t	k(t	PROPN
ejpam-6243	45	69	)	)	PUNCT
ejpam-6243	45	70	is	be	AUX
ejpam-6243	45	71	convergent	convergent	ADJ
ejpam-6243	45	72	for	for	ADP
ejpam-6243	45	73	t	t	PROPN
ejpam-6243	45	74	≥	≥	PROPN
ejpam-6243	45	75	0	0	NUM
ejpam-6243	45	76	.	.	PUNCT
ejpam-6243	46	1	lemma	lemma	PROPN
ejpam-6243	46	2	2	2	NUM
ejpam-6243	46	3	.	.	PUNCT
ejpam-6243	47	1	[	[	X
ejpam-6243	47	2	19	19	NUM
ejpam-6243	47	3	]	]	PUNCT
ejpam-6243	47	4	let	let	VERB
ejpam-6243	47	5	α	α	PRON
ejpam-6243	47	6	:	:	PUNCT
ejpam-6243	47	7	x	x	SYM
ejpam-6243	47	8	×	×	NOUN
ejpam-6243	47	9	x	x	INTJ
ejpam-6243	47	10	→	→	X
ejpam-6243	48	1	[	[	X
ejpam-6243	48	2	0,+∞	0,+∞	NUM
ejpam-6243	48	3	)	)	PUNCT
ejpam-6243	48	4	be	be	AUX
ejpam-6243	48	5	a	a	DET
ejpam-6243	48	6	function	function	NOUN
ejpam-6243	48	7	.	.	PUNCT
ejpam-6243	49	1	we	we	PRON
ejpam-6243	49	2	say	say	VERB
ejpam-6243	49	3	that	that	SCONJ
ejpam-6243	49	4	a	a	DET
ejpam-6243	49	5	mapping	mapping	NOUN
ejpam-6243	49	6	t	t	NOUN
ejpam-6243	49	7	:	:	PUNCT
ejpam-6243	49	8	x	x	X
ejpam-6243	49	9	→	→	PUNCT
ejpam-6243	49	10	x	x	X
ejpam-6243	49	11	is	be	AUX
ejpam-6243	49	12	α−orbital	α−orbital	PRON
ejpam-6243	49	13	admissible	admissible	ADJ
ejpam-6243	49	14	if	if	SCONJ
ejpam-6243	49	15	α(x	α(x	PROPN
ejpam-6243	49	16	,	,	PUNCT
ejpam-6243	49	17	tx	tx	PROPN
ejpam-6243	49	18	)	)	PUNCT
ejpam-6243	49	19	≥	≥	NOUN
ejpam-6243	49	20	1	1	NUM
ejpam-6243	49	21	implies	imply	VERB
ejpam-6243	49	22	α(tx	α(tx	PROPN
ejpam-6243	49	23	,	,	PUNCT
ejpam-6243	49	24	t	t	PROPN
ejpam-6243	49	25	2x	2x	NUM
ejpam-6243	49	26	)	)	PUNCT
ejpam-6243	49	27	≥	≥	NOUN
ejpam-6243	49	28	1	1	NUM
ejpam-6243	49	29	,	,	PUNCT
ejpam-6243	49	30	for	for	ADP
ejpam-6243	49	31	all	all	PRON
ejpam-6243	49	32	x	x	SYM
ejpam-6243	49	33	∈	∈	ADJ
ejpam-6243	49	34	x.	x.	NOUN
ejpam-6243	49	35	(	(	PUNCT
ejpam-6243	49	36	2	2	X
ejpam-6243	49	37	)	)	PUNCT
ejpam-6243	49	38	an	an	DET
ejpam-6243	49	39	α−orbital	α−orbital	ADJ
ejpam-6243	49	40	admissible	admissible	ADJ
ejpam-6243	49	41	mapping	mapping	NOUN
ejpam-6243	49	42	f	f	NOUN
ejpam-6243	49	43	is	be	AUX
ejpam-6243	49	44	called	call	VERB
ejpam-6243	49	45	triangular	triangular	NOUN
ejpam-6243	49	46	α−orbital	α−orbital	DET
ejpam-6243	49	47	admissible	admissible	ADJ
ejpam-6243	49	48	if	if	SCONJ
ejpam-6243	49	49	α(x	α(x	PROPN
ejpam-6243	49	50	,	,	PUNCT
ejpam-6243	49	51	y	y	PROPN
ejpam-6243	49	52	)	)	PUNCT
ejpam-6243	49	53	≥	≥	NOUN
ejpam-6243	49	54	1	1	NUM
ejpam-6243	49	55	and	and	CCONJ
ejpam-6243	49	56	α(y	α(y	NOUN
ejpam-6243	49	57	,	,	PUNCT
ejpam-6243	49	58	ty	ty	PRON
ejpam-6243	49	59	)	)	PUNCT
ejpam-6243	49	60	≥	≥	NOUN
ejpam-6243	49	61	1	1	NUM
ejpam-6243	49	62	implies	imply	VERB
ejpam-6243	49	63	α(x	α(x	PROPN
ejpam-6243	49	64	,	,	PUNCT
ejpam-6243	49	65	y	y	PROPN
ejpam-6243	49	66	)	)	PUNCT
ejpam-6243	49	67	≥	≥	NOUN
ejpam-6243	49	68	1	1	NUM
ejpam-6243	49	69	,	,	PUNCT
ejpam-6243	49	70	(	(	PUNCT
ejpam-6243	49	71	3	3	X
ejpam-6243	49	72	)	)	PUNCT
ejpam-6243	49	73	for	for	ADP
ejpam-6243	49	74	every	every	DET
ejpam-6243	49	75	x	x	PROPN
ejpam-6243	49	76	,	,	PUNCT
ejpam-6243	49	77	y	y	PROPN
ejpam-6243	49	78	∈	∈	PROPN
ejpam-6243	49	79	x	x	X
ejpam-6243	49	80	.	.	PUNCT
ejpam-6243	50	1	lemma	lemma	PROPN
ejpam-6243	50	2	3	3	X
ejpam-6243	50	3	.	.	PUNCT
ejpam-6243	51	1	[	[	X
ejpam-6243	51	2	19	19	NUM
ejpam-6243	51	3	]	]	PUNCT
ejpam-6243	51	4	suppose	suppose	VERB
ejpam-6243	51	5	that	that	SCONJ
ejpam-6243	51	6	for	for	ADP
ejpam-6243	51	7	a	a	DET
ejpam-6243	51	8	triangular	triangular	NOUN
ejpam-6243	51	9	α−orbital	α−orbital	DET
ejpam-6243	51	10	admissible	admissible	ADJ
ejpam-6243	51	11	mapping	mapping	NOUN
ejpam-6243	51	12	f	f	NOUN
ejpam-6243	51	13	:	:	PUNCT
ejpam-6243	51	14	x	x	X
ejpam-6243	51	15	→	→	SYM
ejpam-6243	51	16	x	x	SYM
ejpam-6243	51	17	there	there	PRON
ejpam-6243	51	18	exists	exist	VERB
ejpam-6243	51	19	x0	x0	PROPN
ejpam-6243	51	20	∈	∈	PROPN
ejpam-6243	51	21	x	x	PUNCT
ejpam-6243	51	22	such	such	ADJ
ejpam-6243	51	23	that	that	DET
ejpam-6243	51	24	α(x0	α(x0	ADJ
ejpam-6243	51	25	,	,	PUNCT
ejpam-6243	51	26	tx0	tx0	ADJ
ejpam-6243	51	27	)	)	PUNCT
ejpam-6243	51	28	≥	≥	NOUN
ejpam-6243	52	1	1	1	NUM
ejpam-6243	52	2	.	.	PUNCT
ejpam-6243	52	3	then	then	ADV
ejpam-6243	52	4	α(xn	α(xn	NUM
ejpam-6243	52	5	,	,	PUNCT
ejpam-6243	52	6	xm	xm	PROPN
ejpam-6243	52	7	)	)	PUNCT
ejpam-6243	52	8	≥	≥	NOUN
ejpam-6243	52	9	1	1	NUM
ejpam-6243	52	10	,	,	PUNCT
ejpam-6243	52	11	(	(	PUNCT
ejpam-6243	52	12	4	4	NUM
ejpam-6243	52	13	)	)	PUNCT
ejpam-6243	52	14	for	for	ADP
ejpam-6243	52	15	all	all	DET
ejpam-6243	52	16	n	n	CCONJ
ejpam-6243	52	17	,	,	PUNCT
ejpam-6243	52	18	m	m	VERB
ejpam-6243	52	19	∈	∈	NOUN
ejpam-6243	52	20	n	n	NOUN
ejpam-6243	52	21	,	,	PUNCT
ejpam-6243	52	22	where	where	SCONJ
ejpam-6243	52	23	the	the	DET
ejpam-6243	52	24	sequence	sequence	NOUN
ejpam-6243	52	25	{	{	PUNCT
ejpam-6243	52	26	xn	xn	NOUN
ejpam-6243	52	27	}	}	PUNCT
ejpam-6243	52	28	is	be	AUX
ejpam-6243	52	29	defined	define	VERB
ejpam-6243	52	30	by	by	ADP
ejpam-6243	52	31	xn+1	xn+1	PROPN
ejpam-6243	52	32	=	=	SYM
ejpam-6243	52	33	txn	txn	NOUN
ejpam-6243	52	34	,	,	PUNCT
ejpam-6243	52	35	n	n	NOUN
ejpam-6243	52	36	∈	∈	PROPN
ejpam-6243	52	37	n	n	X
ejpam-6243	52	38	.	.	PUNCT
ejpam-6243	53	1	definition	definition	NOUN
ejpam-6243	53	2	2	2	NUM
ejpam-6243	53	3	.	.	PUNCT
ejpam-6243	54	1	let	let	VERB
ejpam-6243	54	2	α	α	PRON
ejpam-6243	54	3	:	:	PUNCT
ejpam-6243	54	4	x	x	SYM
ejpam-6243	54	5	×	×	NOUN
ejpam-6243	54	6	x	x	INTJ
ejpam-6243	54	7	→	→	X
ejpam-6243	55	1	[	[	X
ejpam-6243	55	2	0,+∞	0,+∞	NUM
ejpam-6243	55	3	)	)	PUNCT
ejpam-6243	55	4	be	be	AUX
ejpam-6243	55	5	a	a	DET
ejpam-6243	55	6	mapping	mapping	NOUN
ejpam-6243	55	7	.	.	PUNCT
ejpam-6243	56	1	the	the	DET
ejpam-6243	56	2	set	set	NOUN
ejpam-6243	56	3	x	x	PUNCT
ejpam-6243	56	4	is	be	AUX
ejpam-6243	56	5	called	call	VERB
ejpam-6243	56	6	regular	regular	ADJ
ejpam-6243	56	7	with	with	ADP
ejpam-6243	56	8	respect	respect	NOUN
ejpam-6243	56	9	to	to	ADP
ejpam-6243	56	10	α	α	PRON
ejpam-6243	56	11	if	if	SCONJ
ejpam-6243	56	12	for	for	ADP
ejpam-6243	56	13	a	a	DET
ejpam-6243	56	14	sequence	sequence	NOUN
ejpam-6243	56	15	{	{	PUNCT
ejpam-6243	56	16	xn	xn	NOUN
ejpam-6243	56	17	}	}	PUNCT
ejpam-6243	56	18	in	in	ADP
ejpam-6243	56	19	x	x	SYM
ejpam-6243	56	20	such	such	ADJ
ejpam-6243	56	21	that	that	SCONJ
ejpam-6243	56	22	α(xn	α(xn	NOUN
ejpam-6243	56	23	,	,	PUNCT
ejpam-6243	56	24	xn+1	xn+1	NUM
ejpam-6243	56	25	)	)	PUNCT
ejpam-6243	56	26	≥	≥	NOUN
ejpam-6243	56	27	1	1	NUM
ejpam-6243	56	28	,	,	PUNCT
ejpam-6243	56	29	for	for	ADP
ejpam-6243	56	30	all	all	DET
ejpam-6243	56	31	n	n	NOUN
ejpam-6243	56	32	and	and	CCONJ
ejpam-6243	56	33	xn	xn	PROPN
ejpam-6243	57	1	→	→	SYM
ejpam-6243	57	2	x	x	PUNCT
ejpam-6243	57	3	∈	∈	PROPN
ejpam-6243	57	4	x	x	PUNCT
ejpam-6243	57	5	as	as	ADP
ejpam-6243	57	6	n	n	PROPN
ejpam-6243	57	7	→	→	SYM
ejpam-6243	57	8	+	+	NUM
ejpam-6243	57	9	∞	∞	NOUN
ejpam-6243	57	10	we	we	PRON
ejpam-6243	57	11	have	have	VERB
ejpam-6243	57	12	α(xn	α(xn	NOUN
ejpam-6243	57	13	,	,	PUNCT
ejpam-6243	57	14	x	x	X
ejpam-6243	57	15	)	)	PUNCT
ejpam-6243	57	16	≥	≥	NOUN
ejpam-6243	57	17	1	1	NUM
ejpam-6243	57	18	for	for	ADP
ejpam-6243	57	19	all	all	DET
ejpam-6243	57	20	n.	n.	NOUN
ejpam-6243	57	21	3	3	NUM
ejpam-6243	57	22	.	.	PUNCT
ejpam-6243	57	23	main	main	ADJ
ejpam-6243	57	24	results	result	NOUN
ejpam-6243	57	25	in	in	ADP
ejpam-6243	57	26	this	this	DET
ejpam-6243	57	27	section	section	NOUN
ejpam-6243	57	28	,	,	PUNCT
ejpam-6243	57	29	we	we	PRON
ejpam-6243	57	30	shall	shall	AUX
ejpam-6243	57	31	introduce	introduce	VERB
ejpam-6243	57	32	a	a	DET
ejpam-6243	57	33	new	new	ADJ
ejpam-6243	57	34	notion	notion	NOUN
ejpam-6243	57	35	of	of	ADP
ejpam-6243	57	36	(	(	PUNCT
ejpam-6243	57	37	β	β	X
ejpam-6243	57	38	,	,	PUNCT
ejpam-6243	57	39	ϕ	ϕ	NOUN
ejpam-6243	57	40	)	)	PUNCT
ejpam-6243	57	41	admissible	admissible	ADJ
ejpam-6243	57	42	hybrid	hybrid	ADJ
ejpam-6243	57	43	contraction	contraction	NOUN
ejpam-6243	57	44	and	and	CCONJ
ejpam-6243	57	45	prove	prove	VERB
ejpam-6243	57	46	some	some	DET
ejpam-6243	57	47	fixed	fix	VERB
ejpam-6243	57	48	point	point	NOUN
ejpam-6243	57	49	results	result	NOUN
ejpam-6243	57	50	for	for	ADP
ejpam-6243	57	51	such	such	ADJ
ejpam-6243	57	52	types	type	NOUN
ejpam-6243	57	53	of	of	ADP
ejpam-6243	57	54	contraction	contraction	NOUN
ejpam-6243	57	55	in	in	ADP
ejpam-6243	57	56	metric	metric	ADJ
ejpam-6243	57	57	spaces	space	NOUN
ejpam-6243	57	58	.	.	PUNCT
ejpam-6243	58	1	in	in	ADP
ejpam-6243	58	2	addition	addition	NOUN
ejpam-6243	58	3	to	to	ADP
ejpam-6243	58	4	this	this	PRON
ejpam-6243	58	5	,	,	PUNCT
ejpam-6243	58	6	an	an	DET
ejpam-6243	58	7	example	example	NOUN
ejpam-6243	58	8	is	be	AUX
ejpam-6243	58	9	also	also	ADV
ejpam-6243	58	10	provided	provide	VERB
ejpam-6243	58	11	for	for	ADP
ejpam-6243	58	12	the	the	DET
ejpam-6243	58	13	validity	validity	NOUN
ejpam-6243	58	14	of	of	ADP
ejpam-6243	58	15	our	our	PRON
ejpam-6243	58	16	result	result	NOUN
ejpam-6243	58	17	.	.	PUNCT
ejpam-6243	59	1	definition	definition	NOUN
ejpam-6243	59	2	3	3	X
ejpam-6243	59	3	.	.	PUNCT
ejpam-6243	60	1	let	let	AUX
ejpam-6243	60	2	(	(	PUNCT
ejpam-6243	60	3	x	x	NOUN
ejpam-6243	60	4	,	,	PUNCT
ejpam-6243	60	5	d	d	NOUN
ejpam-6243	60	6	)	)	PUNCT
ejpam-6243	60	7	be	be	AUX
ejpam-6243	60	8	a	a	DET
ejpam-6243	60	9	metric	metric	ADJ
ejpam-6243	60	10	space	space	NOUN
ejpam-6243	60	11	.	.	PUNCT
ejpam-6243	61	1	a	a	DET
ejpam-6243	61	2	mapping	mapping	NOUN
ejpam-6243	61	3	t	t	NOUN
ejpam-6243	61	4	:	:	PUNCT
ejpam-6243	61	5	x	x	X
ejpam-6243	61	6	→	→	PUNCT
ejpam-6243	61	7	x	x	X
ejpam-6243	61	8	is	be	AUX
ejpam-6243	61	9	said	say	VERB
ejpam-6243	61	10	to	to	PART
ejpam-6243	61	11	be	be	AUX
ejpam-6243	61	12	an	an	DET
ejpam-6243	61	13	(	(	PUNCT
ejpam-6243	61	14	β	β	X
ejpam-6243	61	15	,	,	PUNCT
ejpam-6243	61	16	ϕ	ϕ	NOUN
ejpam-6243	61	17	)	)	PUNCT
ejpam-6243	61	18	admissible	admissible	ADJ
ejpam-6243	61	19	hybrid	hybrid	ADJ
ejpam-6243	61	20	contraction	contraction	NOUN
ejpam-6243	61	21	,	,	PUNCT
ejpam-6243	61	22	if	if	SCONJ
ejpam-6243	61	23	there	there	PRON
ejpam-6243	61	24	exists	exist	VERB
ejpam-6243	61	25	ϕ	ϕ	PROPN
ejpam-6243	61	26	∈	∈	PROPN
ejpam-6243	61	27	φ	φ	PROPN
ejpam-6243	61	28	and	and	CCONJ
ejpam-6243	61	29	β	β	X
ejpam-6243	61	30	:	:	PUNCT
ejpam-6243	61	31	x	x	SYM
ejpam-6243	61	32	×x	×x	X
ejpam-6243	61	33	→	→	PUNCT
ejpam-6243	61	34	[	[	X
ejpam-6243	61	35	0,+∞	0,+∞	NUM
ejpam-6243	61	36	)	)	PUNCT
ejpam-6243	61	37	such	such	ADJ
ejpam-6243	61	38	that	that	SCONJ
ejpam-6243	61	39	β(x	β(x	NOUN
ejpam-6243	61	40	,	,	PUNCT
ejpam-6243	61	41	y)d(tx	y)d(tx	NUM
ejpam-6243	61	42	,	,	PUNCT
ejpam-6243	61	43	ty	ty	X
ejpam-6243	61	44	)	)	PUNCT
ejpam-6243	61	45	≤	≤	NUM
ejpam-6243	61	46	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	61	47	s	s	PART
ejpam-6243	61	48	(	(	PUNCT
ejpam-6243	61	49	x	x	NOUN
ejpam-6243	61	50	,	,	PUNCT
ejpam-6243	61	51	y	y	NOUN
ejpam-6243	61	52	)	)	PUNCT
ejpam-6243	61	53	)	)	PUNCT
ejpam-6243	61	54	,	,	PUNCT
ejpam-6243	61	55	(	(	PUNCT
ejpam-6243	61	56	5	5	X
ejpam-6243	61	57	)	)	PUNCT
ejpam-6243	61	58	r	r	NOUN
ejpam-6243	61	59	ramaswamy	ramaswamy	NOUN
ejpam-6243	61	60	et	et	PROPN
ejpam-6243	61	61	al	al	PROPN
ejpam-6243	61	62	.	.	PUNCT
ejpam-6243	61	63	/	/	SYM
ejpam-6243	61	64	eur	eur	PROPN
ejpam-6243	61	65	.	.	PUNCT
ejpam-6243	62	1	j.	j.	PROPN
ejpam-6243	62	2	pure	pure	PROPN
ejpam-6243	62	3	appl	appl	PROPN
ejpam-6243	62	4	.	.	PROPN
ejpam-6243	62	5	math	math	PROPN
ejpam-6243	62	6	,	,	PUNCT
ejpam-6243	62	7	18	18	NUM
ejpam-6243	62	8	(	(	PUNCT
ejpam-6243	62	9	3	3	NUM
ejpam-6243	62	10	)	)	PUNCT
ejpam-6243	62	11	(	(	PUNCT
ejpam-6243	62	12	2025	2025	NUM
ejpam-6243	62	13	)	)	PUNCT
ejpam-6243	62	14	,	,	PUNCT
ejpam-6243	62	15	6243	6243	NUM
ejpam-6243	62	16	4	4	NUM
ejpam-6243	62	17	of	of	ADP
ejpam-6243	62	18	14	14	NUM
ejpam-6243	62	19	for	for	ADP
ejpam-6243	62	20	all	all	DET
ejpam-6243	62	21	distinct	distinct	ADJ
ejpam-6243	62	22	x	x	NOUN
ejpam-6243	62	23	,	,	PUNCT
ejpam-6243	62	24	y	y	PROPN
ejpam-6243	62	25	∈	∈	PROPN
ejpam-6243	62	26	x	x	NOUN
ejpam-6243	62	27	,	,	PUNCT
ejpam-6243	62	28	where	where	SCONJ
ejpam-6243	62	29	s	s	VERB
ejpam-6243	62	30	≥	≥	NOUN
ejpam-6243	62	31	0	0	NUM
ejpam-6243	62	32	and	and	CCONJ
ejpam-6243	62	33	αi	αi	PRON
ejpam-6243	62	34	≥	≥	NOUN
ejpam-6243	62	35	0	0	NUM
ejpam-6243	62	36	for	for	ADP
ejpam-6243	62	37	i	i	PRON
ejpam-6243	62	38	=	=	SYM
ejpam-6243	62	39	1	1	NUM
ejpam-6243	62	40	,	,	PUNCT
ejpam-6243	62	41	2	2	NUM
ejpam-6243	62	42	such	such	ADJ
ejpam-6243	62	43	that	that	DET
ejpam-6243	62	44	α1	α1	PROPN
ejpam-6243	62	45	+	+	CCONJ
ejpam-6243	62	46	α2	α2	ADJ
ejpam-6243	62	47	=	=	SYM
ejpam-6243	62	48	1	1	NUM
ejpam-6243	62	49	and	and	CCONJ
ejpam-6243	62	50	jt	jt	PROPN
ejpam-6243	62	51	s	s	PART
ejpam-6243	62	52	(	(	PUNCT
ejpam-6243	62	53	x	x	NOUN
ejpam-6243	62	54	,	,	PUNCT
ejpam-6243	62	55	y	y	NOUN
ejpam-6243	62	56	)	)	PUNCT
ejpam-6243	62	57	=	=	SYM
ejpam-6243	62	58	{	{	PUNCT
ejpam-6243	62	59	α1	α1	PROPN
ejpam-6243	62	60	(	(	PUNCT
ejpam-6243	62	61	(	(	PUNCT
ejpam-6243	62	62	d(x	d(x	PROPN
ejpam-6243	62	63	,	,	PUNCT
ejpam-6243	62	64	tx)d(y	tx)d(y	NOUN
ejpam-6243	62	65	,	,	PUNCT
ejpam-6243	62	66	ty	ty	INTJ
ejpam-6243	62	67	)	)	PUNCT
ejpam-6243	62	68	d(x	d(x	PROPN
ejpam-6243	62	69	,	,	PUNCT
ejpam-6243	62	70	y	y	NOUN
ejpam-6243	62	71	)	)	PUNCT
ejpam-6243	62	72	)	)	PUNCT
ejpam-6243	62	73	s	s	PART
ejpam-6243	62	74	+	+	X
ejpam-6243	62	75	α2(d(x	α2(d(x	NUM
ejpam-6243	62	76	,	,	PUNCT
ejpam-6243	62	77	y	y	NOUN
ejpam-6243	62	78	)	)	PUNCT
ejpam-6243	62	79	)	)	PUNCT
ejpam-6243	63	1	s	s	X
ejpam-6243	63	2	]	]	X
ejpam-6243	63	3	1	1	NUM
ejpam-6243	63	4	s	s	NOUN
ejpam-6243	63	5	if	if	SCONJ
ejpam-6243	63	6	s	s	X
ejpam-6243	63	7	>	>	X
ejpam-6243	63	8	0	0	PUNCT
ejpam-6243	64	1	(	(	PUNCT
ejpam-6243	64	2	d(x	d(x	PROPN
ejpam-6243	64	3	,	,	PUNCT
ejpam-6243	64	4	tx))α1(d(y	tx))α1(d(y	ADJ
ejpam-6243	64	5	,	,	PUNCT
ejpam-6243	64	6	ty))α2	ty))α2	PROPN
ejpam-6243	64	7	if	if	SCONJ
ejpam-6243	64	8	s	s	PART
ejpam-6243	64	9	=	=	SYM
ejpam-6243	64	10	0	0	NUM
ejpam-6243	64	11	(	(	PUNCT
ejpam-6243	64	12	6	6	NUM
ejpam-6243	64	13	)	)	PUNCT
ejpam-6243	64	14	here	here	ADV
ejpam-6243	64	15	fixt	fixt	ADJ
ejpam-6243	64	16	(	(	PUNCT
ejpam-6243	64	17	x	x	NOUN
ejpam-6243	64	18	)	)	PUNCT
ejpam-6243	64	19	:	:	PUNCT
ejpam-6243	64	20	=	=	SYM
ejpam-6243	64	21	{	{	PUNCT
ejpam-6243	64	22	x	x	PUNCT
ejpam-6243	64	23	∈	∈	PROPN
ejpam-6243	64	24	x	x	X
ejpam-6243	64	25	:	:	PUNCT
ejpam-6243	64	26	tx	tx	PROPN
ejpam-6243	64	27	=	=	PUNCT
ejpam-6243	64	28	x	x	NOUN
ejpam-6243	64	29	}	}	PUNCT
ejpam-6243	64	30	.	.	PUNCT
ejpam-6243	65	1	remark	remark	NOUN
ejpam-6243	65	2	1	1	NUM
ejpam-6243	65	3	.	.	PUNCT
ejpam-6243	66	1	the	the	DET
ejpam-6243	66	2	concept	concept	NOUN
ejpam-6243	66	3	of	of	ADP
ejpam-6243	66	4	”	"	PUNCT
ejpam-6243	66	5	admissible	admissible	ADJ
ejpam-6243	66	6	hybrid	hybrid	ADJ
ejpam-6243	66	7	contraction	contraction	NOUN
ejpam-6243	66	8	”	"	PUNCT
ejpam-6243	66	9	is	be	AUX
ejpam-6243	66	10	inspired	inspire	VERB
ejpam-6243	66	11	from	from	ADP
ejpam-6243	66	12	the	the	DET
ejpam-6243	66	13	notion	notion	NOUN
ejpam-6243	66	14	of	of	ADP
ejpam-6243	66	15	”	"	PUNCT
ejpam-6243	66	16	interpolative	interpolative	ADJ
ejpam-6243	66	17	contractions	contraction	NOUN
ejpam-6243	66	18	”	"	PUNCT
ejpam-6243	66	19	,	,	PUNCT
ejpam-6243	66	20	see	see	VERB
ejpam-6243	66	21	e.g.	e.g.	ADV
ejpam-6243	66	22	[	[	X
ejpam-6243	66	23	1	1	NUM
ejpam-6243	66	24	-	-	SYM
ejpam-6243	66	25	3	3	NUM
ejpam-6243	66	26	,	,	PUNCT
ejpam-6243	66	27	7	7	NUM
ejpam-6243	66	28	-	-	SYM
ejpam-6243	66	29	9	9	NUM
ejpam-6243	66	30	]	]	PUNCT
ejpam-6243	66	31	the	the	DET
ejpam-6243	66	32	main	main	ADJ
ejpam-6243	66	33	results	result	NOUN
ejpam-6243	66	34	of	of	ADP
ejpam-6243	66	35	this	this	DET
ejpam-6243	66	36	manuscript	manuscript	NOUN
ejpam-6243	66	37	is	be	AUX
ejpam-6243	66	38	the	the	DET
ejpam-6243	66	39	following	follow	VERB
ejpam-6243	66	40	theorem	theorem	NOUN
ejpam-6243	66	41	:	:	PUNCT
ejpam-6243	66	42	theorem	theorem	NOUN
ejpam-6243	66	43	1	1	X
ejpam-6243	66	44	.	.	PUNCT
ejpam-6243	67	1	let	let	VERB
ejpam-6243	67	2	(	(	PUNCT
ejpam-6243	67	3	x	x	NOUN
ejpam-6243	67	4	,	,	PUNCT
ejpam-6243	67	5	d	d	X
ejpam-6243	67	6	)	)	PUNCT
ejpam-6243	67	7	be	be	AUX
ejpam-6243	67	8	complete	complete	ADJ
ejpam-6243	67	9	metric	metric	ADJ
ejpam-6243	67	10	space	space	NOUN
ejpam-6243	67	11	and	and	CCONJ
ejpam-6243	67	12	let	let	VERB
ejpam-6243	67	13	t	t	NOUN
ejpam-6243	67	14	:	:	PUNCT
ejpam-6243	67	15	x	x	X
ejpam-6243	67	16	→	→	PUNCT
ejpam-6243	67	17	x	x	X
ejpam-6243	67	18	be	be	AUX
ejpam-6243	67	19	(	(	PUNCT
ejpam-6243	67	20	β	β	X
ejpam-6243	67	21	,	,	PUNCT
ejpam-6243	67	22	ϕ)−	ϕ)−	PROPN
ejpam-6243	67	23	admissible	admissible	ADJ
ejpam-6243	67	24	hybrid	hybrid	ADJ
ejpam-6243	67	25	contraction	contraction	NOUN
ejpam-6243	67	26	satisfying	satisfy	VERB
ejpam-6243	67	27	the	the	DET
ejpam-6243	67	28	followings	following	NOUN
ejpam-6243	67	29	;	;	PUNCT
ejpam-6243	67	30	(	(	PUNCT
ejpam-6243	67	31	i	i	NOUN
ejpam-6243	67	32	)	)	PUNCT
ejpam-6243	67	33	t	t	PROPN
ejpam-6243	67	34	is	be	AUX
ejpam-6243	67	35	triangular	triangular	NOUN
ejpam-6243	67	36	βorbital	βorbital	ADJ
ejpam-6243	67	37	admissible	admissible	ADJ
ejpam-6243	67	38	;	;	PUNCT
ejpam-6243	67	39	(	(	PUNCT
ejpam-6243	67	40	ii	ii	NOUN
ejpam-6243	67	41	)	)	PUNCT
ejpam-6243	67	42	there	there	PRON
ejpam-6243	67	43	exists	exist	VERB
ejpam-6243	67	44	x0	x0	PROPN
ejpam-6243	67	45	∈	∈	PROPN
ejpam-6243	67	46	x	x	X
ejpam-6243	67	47	s.t	s.t	PROPN
ejpam-6243	67	48	.	.	PROPN
ejpam-6243	67	49	β(x0	β(x0	PROPN
ejpam-6243	67	50	,	,	PUNCT
ejpam-6243	67	51	tx0	tx0	PROPN
ejpam-6243	67	52	)	)	PUNCT
ejpam-6243	67	53	≥	≥	NOUN
ejpam-6243	67	54	1	1	NUM
ejpam-6243	67	55	;	;	PUNCT
ejpam-6243	67	56	(	(	PUNCT
ejpam-6243	67	57	iii	iii	NOUN
ejpam-6243	67	58	)	)	PUNCT
ejpam-6243	67	59	either	either	CCONJ
ejpam-6243	67	60	t	t	PROPN
ejpam-6243	67	61	is	be	AUX
ejpam-6243	67	62	continuous	continuous	ADJ
ejpam-6243	67	63	,	,	PUNCT
ejpam-6243	67	64	or	or	CCONJ
ejpam-6243	67	65	(	(	PUNCT
ejpam-6243	67	66	iv	iv	X
ejpam-6243	67	67	)	)	PUNCT
ejpam-6243	67	68	t	t	NOUN
ejpam-6243	67	69	2	2	NUM
ejpam-6243	67	70	is	be	AUX
ejpam-6243	67	71	continuous	continuous	ADJ
ejpam-6243	67	72	and	and	CCONJ
ejpam-6243	67	73	β(tx	β(tx	NUM
ejpam-6243	67	74	,	,	PUNCT
ejpam-6243	67	75	x	x	X
ejpam-6243	67	76	)	)	PUNCT
ejpam-6243	67	77	≥	≥	NOUN
ejpam-6243	67	78	1	1	NUM
ejpam-6243	67	79	for	for	ADP
ejpam-6243	67	80	any	any	DET
ejpam-6243	67	81	x	x	SYM
ejpam-6243	67	82	∈	∈	PROPN
ejpam-6243	67	83	ft	ft	X
ejpam-6243	67	84	(	(	PUNCT
ejpam-6243	67	85	x	x	NOUN
ejpam-6243	67	86	)	)	PUNCT
ejpam-6243	67	87	=	=	SYM
ejpam-6243	67	88	{	{	PUNCT
ejpam-6243	67	89	x	x	PUNCT
ejpam-6243	67	90	∈	∈	PROPN
ejpam-6243	67	91	x	x	X
ejpam-6243	67	92	:	:	PUNCT
ejpam-6243	67	93	tx	tx	PROPN
ejpam-6243	67	94	=	=	PUNCT
ejpam-6243	67	95	x	x	X
ejpam-6243	67	96	}	}	PUNCT
ejpam-6243	67	97	.	.	PUNCT
ejpam-6243	68	1	then	then	ADV
ejpam-6243	68	2	t	t	PROPN
ejpam-6243	68	3	has	have	VERB
ejpam-6243	68	4	a	a	DET
ejpam-6243	68	5	unique	unique	ADJ
ejpam-6243	68	6	fixed	fix	VERB
ejpam-6243	68	7	point	point	NOUN
ejpam-6243	68	8	.	.	PUNCT
ejpam-6243	69	1	proof	proof	NOUN
ejpam-6243	69	2	.	.	PUNCT
ejpam-6243	70	1	we	we	PRON
ejpam-6243	70	2	recursively	recursively	ADV
ejpam-6243	70	3	construct	construct	VERB
ejpam-6243	70	4	up	up	ADP
ejpam-6243	70	5	the	the	DET
ejpam-6243	70	6	sequence	sequence	NOUN
ejpam-6243	70	7	{	{	PUNCT
ejpam-6243	70	8	xn	xn	NUM
ejpam-6243	70	9	}	}	PUNCT
ejpam-6243	70	10	,	,	PUNCT
ejpam-6243	70	11	starting	start	VERB
ejpam-6243	70	12	from	from	ADP
ejpam-6243	70	13	any	any	DET
ejpam-6243	70	14	random	random	ADJ
ejpam-6243	70	15	point	point	NOUN
ejpam-6243	70	16	x0	x0	PROPN
ejpam-6243	70	17	in	in	ADP
ejpam-6243	70	18	x	x	PROPN
ejpam-6243	70	19	,	,	PUNCT
ejpam-6243	70	20	such	such	ADJ
ejpam-6243	70	21	that	that	SCONJ
ejpam-6243	70	22	xn	xn	PROPN
ejpam-6243	71	1	=	=	SYM
ejpam-6243	71	2	tnx0	tnx0	PROPN
ejpam-6243	71	3	for	for	ADP
ejpam-6243	71	4	every	every	DET
ejpam-6243	71	5	n	n	NOUN
ejpam-6243	71	6	∈	∈	NOUN
ejpam-6243	71	7	n	n	NOUN
ejpam-6243	71	8	.	.	PUNCT
ejpam-6243	72	1	assuming	assume	VERB
ejpam-6243	72	2	that	that	SCONJ
ejpam-6243	72	3	there	there	PRON
ejpam-6243	72	4	is	be	VERB
ejpam-6243	72	5	some	some	DET
ejpam-6243	72	6	m	m	NOUN
ejpam-6243	72	7	∈	∈	NOUN
ejpam-6243	72	8	n	n	PRON
ejpam-6243	72	9	such	such	ADJ
ejpam-6243	72	10	that	that	SCONJ
ejpam-6243	72	11	txm	txm	PROPN
ejpam-6243	72	12	=	=	SYM
ejpam-6243	72	13	xm+1	xm+1	PROPN
ejpam-6243	72	14	=	=	SYM
ejpam-6243	72	15	xm	xm	PROPN
ejpam-6243	72	16	,	,	PUNCT
ejpam-6243	72	17	we	we	PRON
ejpam-6243	72	18	conclude	conclude	VERB
ejpam-6243	72	19	the	the	DET
ejpam-6243	72	20	proof	proof	NOUN
ejpam-6243	72	21	by	by	ADP
ejpam-6243	72	22	finding	find	VERB
ejpam-6243	72	23	that	that	SCONJ
ejpam-6243	72	24	xm	xm	PROPN
ejpam-6243	72	25	is	be	AUX
ejpam-6243	72	26	a	a	DET
ejpam-6243	72	27	fixed	fix	VERB
ejpam-6243	72	28	point	point	NOUN
ejpam-6243	72	29	of	of	ADP
ejpam-6243	72	30	t	t	PROPN
ejpam-6243	72	31	.	.	PUNCT
ejpam-6243	73	1	thus	thus	ADV
ejpam-6243	73	2	,	,	PUNCT
ejpam-6243	73	3	for	for	ADP
ejpam-6243	73	4	all	all	DET
ejpam-6243	73	5	n	n	DET
ejpam-6243	73	6	∈	∈	PROPN
ejpam-6243	73	7	n	n	NOUN
ejpam-6243	73	8	,	,	PUNCT
ejpam-6243	73	9	we	we	PRON
ejpam-6243	73	10	can	can	AUX
ejpam-6243	73	11	assume	assume	VERB
ejpam-6243	73	12	going	go	VERB
ejpam-6243	73	13	forward	forward	ADV
ejpam-6243	73	14	that	that	SCONJ
ejpam-6243	73	15	xn	xn	PROPN
ejpam-6243	73	16	̸=	̸=	PROPN
ejpam-6243	73	17	xn−1	xn−1	PROPN
ejpam-6243	73	18	.	.	PUNCT
ejpam-6243	74	1	assuming	assume	VERB
ejpam-6243	74	2	(	(	PUNCT
ejpam-6243	74	3	i	i	NOUN
ejpam-6243	74	4	)	)	PUNCT
ejpam-6243	74	5	that	that	PRON
ejpam-6243	74	6	t	t	PROPN
ejpam-6243	74	7	is	be	AUX
ejpam-6243	74	8	an	an	DET
ejpam-6243	74	9	admissible	admissible	ADJ
ejpam-6243	74	10	hybrid	hybrid	ADJ
ejpam-6243	74	11	contraction	contraction	NOUN
ejpam-6243	74	12	,	,	PUNCT
ejpam-6243	74	13	we	we	PRON
ejpam-6243	74	14	obtain	obtain	VERB
ejpam-6243	74	15	by	by	ADP
ejpam-6243	74	16	replacing	replace	VERB
ejpam-6243	74	17	x	x	PUNCT
ejpam-6243	74	18	by	by	ADP
ejpam-6243	74	19	xn−1	xn−1	PROPN
ejpam-6243	74	20	.	.	PUNCT
ejpam-6243	75	1	and	and	CCONJ
ejpam-6243	75	2	y	y	PROPN
ejpam-6243	75	3	by	by	ADP
ejpam-6243	75	4	xn	xn	PROPN
ejpam-6243	75	5	in	in	ADP
ejpam-6243	75	6	equation	equation	NOUN
ejpam-6243	75	7	(	(	PUNCT
ejpam-6243	75	8	5	5	NUM
ejpam-6243	75	9	)	)	PUNCT
ejpam-6243	75	10	β(xn−1	β(xn−1	NOUN
ejpam-6243	75	11	,	,	PUNCT
ejpam-6243	75	12	xn)d(txn−1	xn)d(txn−1	NUM
ejpam-6243	75	13	,	,	PUNCT
ejpam-6243	75	14	txn	txn	NOUN
ejpam-6243	75	15	)	)	PUNCT
ejpam-6243	75	16	≤	≤	NUM
ejpam-6243	76	1	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	76	2	s	s	PART
ejpam-6243	76	3	(	(	PUNCT
ejpam-6243	76	4	xn−1	xn−1	PROPN
ejpam-6243	76	5	,	,	PUNCT
ejpam-6243	76	6	xn	xn	PROPN
ejpam-6243	76	7	)	)	PUNCT
ejpam-6243	76	8	)	)	PUNCT
ejpam-6243	76	9	.	.	PUNCT
ejpam-6243	77	1	(	(	PUNCT
ejpam-6243	77	2	7	7	X
ejpam-6243	77	3	)	)	PUNCT
ejpam-6243	77	4	considering	consider	VERB
ejpam-6243	77	5	that	that	SCONJ
ejpam-6243	77	6	t	t	PROPN
ejpam-6243	77	7	is	be	AUX
ejpam-6243	77	8	triangular	triangular	ADJ
ejpam-6243	77	9	β−	β−	PUNCT
ejpam-6243	77	10	orbital	orbital	ADJ
ejpam-6243	77	11	admissible	admissible	NOUN
ejpam-6243	77	12	,	,	PUNCT
ejpam-6243	77	13	along	along	ADP
ejpam-6243	77	14	with	with	ADP
ejpam-6243	77	15	(	(	PUNCT
ejpam-6243	77	16	4	4	X
ejpam-6243	77	17	)	)	PUNCT
ejpam-6243	77	18	holding	holding	NOUN
ejpam-6243	77	19	,	,	PUNCT
ejpam-6243	77	20	the	the	DET
ejpam-6243	77	21	above	above	ADJ
ejpam-6243	77	22	inequality	inequality	NOUN
ejpam-6243	77	23	becomes	become	VERB
ejpam-6243	77	24	d(xn	d(xn	PROPN
ejpam-6243	77	25	,	,	PUNCT
ejpam-6243	77	26	xn+1	xn+1	NUM
ejpam-6243	77	27	)	)	PUNCT
ejpam-6243	77	28	≤	≤	NOUN
ejpam-6243	77	29	β(xn−1	β(xn−1	PUNCT
ejpam-6243	77	30	,	,	PUNCT
ejpam-6243	77	31	xn)d(txn−1	xn)d(txn−1	NUM
ejpam-6243	77	32	,	,	PUNCT
ejpam-6243	77	33	txn	txn	NOUN
ejpam-6243	77	34	)	)	PUNCT
ejpam-6243	77	35	.	.	PUNCT
ejpam-6243	78	1	(	(	PUNCT
ejpam-6243	78	2	8)	8)	NUM
ejpam-6243	78	3	<	<	X
ejpam-6243	78	4	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	78	5	s	s	PART
ejpam-6243	78	6	(	(	PUNCT
ejpam-6243	78	7	xn−1	xn−1	PROPN
ejpam-6243	78	8	,	,	PUNCT
ejpam-6243	78	9	xn	xn	PROPN
ejpam-6243	78	10	)	)	PUNCT
ejpam-6243	78	11	)	)	PUNCT
ejpam-6243	78	12	.	.	PUNCT
ejpam-6243	79	1	case	case	NOUN
ejpam-6243	79	2	1	1	NUM
ejpam-6243	79	3	:	:	PUNCT
ejpam-6243	79	4	for	for	SCONJ
ejpam-6243	79	5	the	the	DET
ejpam-6243	79	6	case	case	NOUN
ejpam-6243	79	7	s	s	VERB
ejpam-6243	79	8	>	>	X
ejpam-6243	79	9	0	0	NUM
ejpam-6243	80	1	we	we	PRON
ejpam-6243	80	2	have	have	VERB
ejpam-6243	80	3	jt	jt	PROPN
ejpam-6243	80	4	s	s	PART
ejpam-6243	80	5	(	(	PUNCT
ejpam-6243	80	6	xn−1	xn−1	PROPN
ejpam-6243	80	7	,	,	PUNCT
ejpam-6243	80	8	xn	xn	PRON
ejpam-6243	80	9	)	)	PUNCT
ejpam-6243	80	10	=	=	NOUN
ejpam-6243	81	1	[	[	X
ejpam-6243	81	2	α1	α1	PROPN
ejpam-6243	81	3	(	(	PUNCT
ejpam-6243	81	4	(	(	PUNCT
ejpam-6243	81	5	d(xn−1	d(xn−1	PROPN
ejpam-6243	81	6	,	,	PUNCT
ejpam-6243	81	7	txn−1)d(xn	txn−1)d(xn	NOUN
ejpam-6243	81	8	,	,	PUNCT
ejpam-6243	81	9	txn	txn	NOUN
ejpam-6243	81	10	)	)	PUNCT
ejpam-6243	81	11	d(xn−1	d(xn−1	PROPN
ejpam-6243	81	12	,	,	PUNCT
ejpam-6243	81	13	xn	xn	PROPN
ejpam-6243	81	14	)	)	PUNCT
ejpam-6243	81	15	)	)	PUNCT
ejpam-6243	81	16	)	)	PUNCT
ejpam-6243	81	17	s	s	PART
ejpam-6243	82	1	+	+	X
ejpam-6243	82	2	α2(d(xn−1	α2(d(xn−1	NUM
ejpam-6243	82	3	,	,	PUNCT
ejpam-6243	82	4	xn	xn	PROPN
ejpam-6243	82	5	)	)	PUNCT
ejpam-6243	82	6	)	)	PUNCT
ejpam-6243	83	1	s	s	X
ejpam-6243	83	2	]	]	PUNCT
ejpam-6243	83	3	1	1	NUM
ejpam-6243	83	4	s	s	NOUN
ejpam-6243	83	5	r	r	NOUN
ejpam-6243	83	6	ramaswamy	ramaswamy	NOUN
ejpam-6243	83	7	et	et	PROPN
ejpam-6243	83	8	al	al	PROPN
ejpam-6243	83	9	.	.	PUNCT
ejpam-6243	83	10	/	/	SYM
ejpam-6243	83	11	eur	eur	PROPN
ejpam-6243	83	12	.	.	PUNCT
ejpam-6243	84	1	j.	j.	PROPN
ejpam-6243	84	2	pure	pure	PROPN
ejpam-6243	84	3	appl	appl	PROPN
ejpam-6243	84	4	.	.	PROPN
ejpam-6243	84	5	math	math	PROPN
ejpam-6243	84	6	,	,	PUNCT
ejpam-6243	84	7	18	18	NUM
ejpam-6243	84	8	(	(	PUNCT
ejpam-6243	84	9	3	3	NUM
ejpam-6243	84	10	)	)	PUNCT
ejpam-6243	84	11	(	(	PUNCT
ejpam-6243	84	12	2025	2025	NUM
ejpam-6243	84	13	)	)	PUNCT
ejpam-6243	84	14	,	,	PUNCT
ejpam-6243	84	15	6243	6243	NUM
ejpam-6243	84	16	5	5	NUM
ejpam-6243	84	17	of	of	ADP
ejpam-6243	84	18	14	14	NUM
ejpam-6243	84	19	=	=	SYM
ejpam-6243	85	1	[	[	X
ejpam-6243	85	2	α1	α1	PROPN
ejpam-6243	85	3	(	(	PUNCT
ejpam-6243	85	4	d(xn−1	d(xn−1	PROPN
ejpam-6243	85	5	,	,	PUNCT
ejpam-6243	85	6	xn)d(xn	xn)d(xn	PROPN
ejpam-6243	85	7	,	,	PUNCT
ejpam-6243	85	8	xn+1	xn+1	NUM
ejpam-6243	85	9	)	)	PUNCT
ejpam-6243	85	10	d(xn−1	d(xn−1	NOUN
ejpam-6243	85	11	,	,	PUNCT
ejpam-6243	85	12	xn	xn	PROPN
ejpam-6243	85	13	)	)	PUNCT
ejpam-6243	85	14	)	)	PUNCT
ejpam-6243	85	15	s	s	VERB
ejpam-6243	85	16	+	+	X
ejpam-6243	85	17	α2(d(xn−1	α2(d(xn−1	NUM
ejpam-6243	85	18	,	,	PUNCT
ejpam-6243	85	19	xn	xn	PROPN
ejpam-6243	85	20	)	)	PUNCT
ejpam-6243	85	21	)	)	PUNCT
ejpam-6243	86	1	s	s	X
ejpam-6243	86	2	]	]	X
ejpam-6243	86	3	1	1	NUM
ejpam-6243	86	4	s	s	NOUN
ejpam-6243	86	5	=	=	PUNCT
ejpam-6243	87	1	[	[	X
ejpam-6243	87	2	α1(d(xn	α1(d(xn	X
ejpam-6243	87	3	,	,	PUNCT
ejpam-6243	87	4	xn+1	xn+1	NUM
ejpam-6243	87	5	)	)	PUNCT
ejpam-6243	87	6	)	)	PUNCT
ejpam-6243	88	1	s	s	VERB
ejpam-6243	88	2	+	+	X
ejpam-6243	88	3	α2(d(xn−1	α2(d(xn−1	NUM
ejpam-6243	88	4	,	,	PUNCT
ejpam-6243	88	5	xn	xn	PROPN
ejpam-6243	88	6	)	)	PUNCT
ejpam-6243	88	7	)	)	PUNCT
ejpam-6243	89	1	s	s	X
ejpam-6243	89	2	]	]	PUNCT
ejpam-6243	89	3	1	1	NUM
ejpam-6243	89	4	s	s	NOUN
ejpam-6243	89	5	.	.	PUNCT
ejpam-6243	90	1	and	and	CCONJ
ejpam-6243	90	2	from	from	ADP
ejpam-6243	90	3	(	(	PUNCT
ejpam-6243	90	4	8)	8)	NUM
ejpam-6243	90	5	we	we	PRON
ejpam-6243	90	6	get	get	VERB
ejpam-6243	90	7	d(xn	d(xn	NOUN
ejpam-6243	90	8	,	,	PUNCT
ejpam-6243	90	9	xn+1	xn+1	NUM
ejpam-6243	90	10	)	)	PUNCT
ejpam-6243	90	11	≤	≤	NOUN
ejpam-6243	90	12	β(xn−1	β(xn−1	PUNCT
ejpam-6243	90	13	,	,	PUNCT
ejpam-6243	90	14	xn)d(txn−1	xn)d(txn−1	NUM
ejpam-6243	90	15	,	,	PUNCT
ejpam-6243	90	16	txn	txn	NOUN
ejpam-6243	90	17	)	)	PUNCT
ejpam-6243	91	1	<	<	X
ejpam-6243	91	2	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	91	3	s	s	PART
ejpam-6243	91	4	(	(	PUNCT
ejpam-6243	91	5	xn−1	xn−1	PROPN
ejpam-6243	91	6	,	,	PUNCT
ejpam-6243	91	7	xn	xn	PROPN
ejpam-6243	91	8	)	)	PUNCT
ejpam-6243	91	9	)	)	PUNCT
ejpam-6243	92	1	(	(	PUNCT
ejpam-6243	92	2	9	9	X
ejpam-6243	92	3	)	)	PUNCT
ejpam-6243	92	4	=	=	SYM
ejpam-6243	92	5	ϕ[α1(d(xn	ϕ[α1(d(xn	PROPN
ejpam-6243	92	6	,	,	PUNCT
ejpam-6243	92	7	xn+1	xn+1	NUM
ejpam-6243	92	8	)	)	PUNCT
ejpam-6243	92	9	)	)	PUNCT
ejpam-6243	92	10	s	s	VERB
ejpam-6243	92	11	+	+	X
ejpam-6243	92	12	α2(d(xn−1	α2(d(xn−1	NUM
ejpam-6243	92	13	,	,	PUNCT
ejpam-6243	92	14	xn	xn	PROPN
ejpam-6243	92	15	)	)	PUNCT
ejpam-6243	92	16	)	)	PUNCT
ejpam-6243	93	1	s	s	X
ejpam-6243	93	2	]	]	X
ejpam-6243	93	3	1	1	NUM
ejpam-6243	93	4	s	s	NOUN
ejpam-6243	93	5	.	.	PUNCT
ejpam-6243	94	1	since	since	SCONJ
ejpam-6243	94	2	β	β	PROPN
ejpam-6243	94	3	is	be	AUX
ejpam-6243	94	4	a	a	DET
ejpam-6243	94	5	non	non	ADJ
ejpam-6243	94	6	-	-	ADJ
ejpam-6243	94	7	decreasing	decrease	VERB
ejpam-6243	94	8	function	function	NOUN
ejpam-6243	94	9	,	,	PUNCT
ejpam-6243	94	10	let	let	VERB
ejpam-6243	94	11	us	we	PRON
ejpam-6243	94	12	assume	assume	VERB
ejpam-6243	94	13	that	that	SCONJ
ejpam-6243	94	14	d(xn−1	d(xn−1	PROPN
ejpam-6243	94	15	,	,	PUNCT
ejpam-6243	94	16	xn	xn	PROPN
ejpam-6243	94	17	)	)	PUNCT
ejpam-6243	94	18	≤	≤	PUNCT
ejpam-6243	95	1	d(xn	d(xn	PROPN
ejpam-6243	95	2	,	,	PUNCT
ejpam-6243	95	3	xn+1	xn+1	NUM
ejpam-6243	95	4	)	)	PUNCT
ejpam-6243	95	5	,	,	PUNCT
ejpam-6243	95	6	d(xn	d(xn	PROPN
ejpam-6243	95	7	,	,	PUNCT
ejpam-6243	95	8	xn−1	xn−1	PROPN
ejpam-6243	95	9	)	)	PUNCT
ejpam-6243	95	10	≤	≤	NOUN
ejpam-6243	95	11	β(xn−1	β(xn−1	PUNCT
ejpam-6243	95	12	,	,	PUNCT
ejpam-6243	95	13	xn)d(txn−1	xn)d(txn−1	NUM
ejpam-6243	95	14	,	,	PUNCT
ejpam-6243	95	15	txn	txn	NOUN
ejpam-6243	95	16	)	)	PUNCT
ejpam-6243	95	17	≤	≤	NOUN
ejpam-6243	96	1	ϕ[α1(d(xn	ϕ[α1(d(xn	PROPN
ejpam-6243	96	2	,	,	PUNCT
ejpam-6243	96	3	xn+1	xn+1	NUM
ejpam-6243	96	4	)	)	PUNCT
ejpam-6243	96	5	)	)	PUNCT
ejpam-6243	96	6	s	s	VERB
ejpam-6243	96	7	+	+	X
ejpam-6243	96	8	α2(d(xn−1	α2(d(xn−1	NUM
ejpam-6243	96	9	,	,	PUNCT
ejpam-6243	96	10	xn	xn	PROPN
ejpam-6243	96	11	)	)	PUNCT
ejpam-6243	96	12	)	)	PUNCT
ejpam-6243	97	1	s	s	X
ejpam-6243	97	2	]	]	X
ejpam-6243	97	3	s	s	PART
ejpam-6243	97	4	≤	≤	NUM
ejpam-6243	97	5	ϕ[(α1	ϕ[(α1	NUM
ejpam-6243	97	6	+	+	CCONJ
ejpam-6243	97	7	α2)(d(xn	α2)(d(xn	NOUN
ejpam-6243	97	8	,	,	PUNCT
ejpam-6243	97	9	xn+1	xn+1	NUM
ejpam-6243	97	10	)	)	PUNCT
ejpam-6243	97	11	)	)	PUNCT
ejpam-6243	98	1	s	s	X
ejpam-6243	98	2	]	]	X
ejpam-6243	98	3	1	1	NUM
ejpam-6243	98	4	s	s	PART
ejpam-6243	98	5	(	(	PUNCT
ejpam-6243	98	6	10	10	NUM
ejpam-6243	98	7	)	)	PUNCT
ejpam-6243	98	8	=	=	VERB
ejpam-6243	99	1	ϕ[[(α1	ϕ[[(α1	VERB
ejpam-6243	99	2	+	+	CCONJ
ejpam-6243	99	3	α2	α2	ADJ
ejpam-6243	99	4	)	)	PUNCT
ejpam-6243	99	5	]	]	PUNCT
ejpam-6243	100	1	1	1	NUM
ejpam-6243	100	2	s	s	NOUN
ejpam-6243	100	3	d(xn	d(xn	PROPN
ejpam-6243	100	4	,	,	PUNCT
ejpam-6243	100	5	xn+1	xn+1	NUM
ejpam-6243	100	6	)	)	PUNCT
ejpam-6243	100	7	]	]	PUNCT
ejpam-6243	101	1	<	<	X
ejpam-6243	101	2	(	(	PUNCT
ejpam-6243	101	3	α1	α1	PROPN
ejpam-6243	101	4	+	+	CCONJ
ejpam-6243	101	5	α2	α2	ADJ
ejpam-6243	101	6	)	)	PUNCT
ejpam-6243	101	7	]	]	PUNCT
ejpam-6243	101	8	1	1	NUM
ejpam-6243	101	9	s	s	NOUN
ejpam-6243	101	10	d(xn	d(xn	PROPN
ejpam-6243	101	11	,	,	PUNCT
ejpam-6243	101	12	xn+1	xn+1	NUM
ejpam-6243	101	13	)	)	PUNCT
ejpam-6243	101	14	≤	≤	PUNCT
ejpam-6243	102	1	d(xn	d(xn	PROPN
ejpam-6243	102	2	,	,	PUNCT
ejpam-6243	102	3	xn+1	xn+1	NUM
ejpam-6243	102	4	)	)	PUNCT
ejpam-6243	102	5	,	,	PUNCT
ejpam-6243	102	6	which	which	PRON
ejpam-6243	102	7	contradicts	contradict	VERB
ejpam-6243	102	8	itself	itself	PRON
ejpam-6243	102	9	.	.	PUNCT
ejpam-6243	103	1	consequently	consequently	ADV
ejpam-6243	103	2	,	,	PUNCT
ejpam-6243	103	3	for	for	ADP
ejpam-6243	103	4	any	any	DET
ejpam-6243	103	5	n	n	PRON
ejpam-6243	103	6	∈	∈	PROPN
ejpam-6243	103	7	n	n	NOUN
ejpam-6243	103	8	,	,	PUNCT
ejpam-6243	103	9	we	we	PRON
ejpam-6243	103	10	have	have	VERB
ejpam-6243	103	11	d(xn	d(xn	NOUN
ejpam-6243	103	12	,	,	PUNCT
ejpam-6243	103	13	xn+1	xn+1	NUM
ejpam-6243	103	14	)	)	PUNCT
ejpam-6243	103	15	≤	≤	NOUN
ejpam-6243	103	16	d(xn−1	d(xn−1	NOUN
ejpam-6243	103	17	,	,	PUNCT
ejpam-6243	103	18	xn	xn	PROPN
ejpam-6243	103	19	)	)	PUNCT
ejpam-6243	103	20	,	,	PUNCT
ejpam-6243	103	21	and	and	CCONJ
ejpam-6243	103	22	the	the	DET
ejpam-6243	103	23	inequality	inequality	NOUN
ejpam-6243	103	24	(	(	PUNCT
ejpam-6243	103	25	8)	8)	NUM
ejpam-6243	103	26	yields	yield	NOUN
ejpam-6243	103	27	d(xn	d(xn	NOUN
ejpam-6243	103	28	,	,	PUNCT
ejpam-6243	103	29	xn+1	xn+1	NUM
ejpam-6243	103	30	)	)	PUNCT
ejpam-6243	103	31	≤	≤	NOUN
ejpam-6243	103	32	ϕ[α1(d(xn	ϕ[α1(d(xn	PROPN
ejpam-6243	103	33	,	,	PUNCT
ejpam-6243	103	34	xn+1	xn+1	NUM
ejpam-6243	103	35	)	)	PUNCT
ejpam-6243	103	36	)	)	PUNCT
ejpam-6243	103	37	s	s	VERB
ejpam-6243	104	1	+	+	X
ejpam-6243	104	2	α2(d(xn−1	α2(d(xn−1	NUM
ejpam-6243	104	3	,	,	PUNCT
ejpam-6243	104	4	xn	xn	PROPN
ejpam-6243	104	5	)	)	PUNCT
ejpam-6243	104	6	)	)	PUNCT
ejpam-6243	105	1	s	s	X
ejpam-6243	105	2	]	]	X
ejpam-6243	105	3	1	1	NUM
ejpam-6243	105	4	s	s	PART
ejpam-6243	105	5	≤	≤	NUM
ejpam-6243	105	6	ϕ[(α1	ϕ[(α1	NUM
ejpam-6243	105	7	+	+	CCONJ
ejpam-6243	105	8	α2)(d(xn−1	α2)(d(xn−1	PROPN
ejpam-6243	105	9	,	,	PUNCT
ejpam-6243	105	10	xn	xn	NUM
ejpam-6243	105	11	)	)	PUNCT
ejpam-6243	105	12	)	)	PUNCT
ejpam-6243	106	1	s	s	X
ejpam-6243	106	2	]	]	X
ejpam-6243	106	3	1	1	NUM
ejpam-6243	106	4	s	s	PART
ejpam-6243	106	5	≤	≤	NOUN
ejpam-6243	106	6	ϕ(α1	ϕ(α1	NOUN
ejpam-6243	106	7	+	+	CCONJ
ejpam-6243	106	8	α2	α2	ADJ
ejpam-6243	106	9	)	)	PUNCT
ejpam-6243	106	10	]	]	PUNCT
ejpam-6243	106	11	1	1	NUM
ejpam-6243	106	12	s	s	PART
ejpam-6243	106	13	d(xn−1	d(xn−1	NOUN
ejpam-6243	106	14	,	,	PUNCT
ejpam-6243	106	15	xn	xn	PROPN
ejpam-6243	106	16	)	)	PUNCT
ejpam-6243	106	17	(	(	PUNCT
ejpam-6243	106	18	11	11	NUM
ejpam-6243	106	19	)	)	PUNCT
ejpam-6243	106	20	≤	≤	NOUN
ejpam-6243	106	21	ϕ(d(xn−1	ϕ(d(xn−1	PROPN
ejpam-6243	106	22	,	,	PUNCT
ejpam-6243	106	23	xn	xn	PROPN
ejpam-6243	106	24	)	)	PUNCT
ejpam-6243	106	25	)	)	PUNCT
ejpam-6243	106	26	.	.	PUNCT
ejpam-6243	107	1	≤	≤	NOUN
ejpam-6243	107	2	ϕ2(d(xn−2	ϕ2(d(xn−2	ADV
ejpam-6243	107	3	,	,	PUNCT
ejpam-6243	107	4	xn−1	xn−1	PROPN
ejpam-6243	107	5	)	)	PUNCT
ejpam-6243	107	6	)	)	PUNCT
ejpam-6243	107	7	...	...	PUNCT
ejpam-6243	108	1	<	<	X
ejpam-6243	108	2	ϕn(d(x0	ϕn(d(x0	PROPN
ejpam-6243	108	3	,	,	PUNCT
ejpam-6243	108	4	x1	x1	PROPN
ejpam-6243	108	5	)	)	PUNCT
ejpam-6243	108	6	)	)	PUNCT
ejpam-6243	108	7	.	.	PUNCT
ejpam-6243	109	1	assume	assume	VERB
ejpam-6243	109	2	that	that	SCONJ
ejpam-6243	109	3	p	p	PROPN
ejpam-6243	109	4	>	>	X
ejpam-6243	109	5	m	m	VERB
ejpam-6243	109	6	for	for	ADP
ejpam-6243	109	7	any	any	DET
ejpam-6243	109	8	m	m	NOUN
ejpam-6243	109	9	,	,	PUNCT
ejpam-6243	109	10	p	p	PROPN
ejpam-6243	109	11	∈	∈	PROPN
ejpam-6243	109	12	n	n	ADV
ejpam-6243	109	13	.	.	PUNCT
ejpam-6243	110	1	given	give	VERB
ejpam-6243	110	2	that	that	DET
ejpam-6243	110	3	d(xm	d(xm	PROPN
ejpam-6243	110	4	,	,	PUNCT
ejpam-6243	110	5	xm+1	xm+1	NUM
ejpam-6243	110	6	)	)	PUNCT
ejpam-6243	110	7	<	<	X
ejpam-6243	111	1	ϕ(d(x0	ϕ(d(x0	PROPN
ejpam-6243	111	2	,	,	PUNCT
ejpam-6243	111	3	x1	x1	PROPN
ejpam-6243	111	4	)	)	PUNCT
ejpam-6243	111	5	)	)	PUNCT
ejpam-6243	111	6	for	for	ADP
ejpam-6243	111	7	each	each	DET
ejpam-6243	111	8	x	x	NOUN
ejpam-6243	111	9	,	,	PUNCT
ejpam-6243	111	10	the	the	DET
ejpam-6243	111	11	triangle	triangle	NOUN
ejpam-6243	111	12	inequality	inequality	NOUN
ejpam-6243	111	13	.	.	PUNCT
ejpam-6243	112	1	given	give	VERB
ejpam-6243	112	2	m	m	PRON
ejpam-6243	112	3	∈	∈	NOUN
ejpam-6243	112	4	n	n	NOUN
ejpam-6243	112	5	,	,	PUNCT
ejpam-6243	112	6	we	we	PRON
ejpam-6243	112	7	have	have	VERB
ejpam-6243	112	8	d(xm	d(xm	PROPN
ejpam-6243	112	9	,	,	PUNCT
ejpam-6243	112	10	xp	xp	ADJ
ejpam-6243	112	11	)	)	PUNCT
ejpam-6243	112	12	≤	≤	PROPN
ejpam-6243	113	1	d(xm	d(xm	PROPN
ejpam-6243	113	2	,	,	PUNCT
ejpam-6243	113	3	xm+1	xm+1	NUM
ejpam-6243	113	4	)	)	PUNCT
ejpam-6243	113	5	+	+	CCONJ
ejpam-6243	113	6	d(xm+1	d(xm+1	PROPN
ejpam-6243	113	7	,	,	PUNCT
ejpam-6243	113	8	xm+2	xm+2	PROPN
ejpam-6243	113	9	)	)	PUNCT
ejpam-6243	113	10	+	+	CCONJ
ejpam-6243	113	11	...	...	PUNCT
ejpam-6243	114	1	+	+	CCONJ
ejpam-6243	114	2	d(xp−1	d(xp−1	ADJ
ejpam-6243	114	3	,	,	PUNCT
ejpam-6243	114	4	xp	xp	ADJ
ejpam-6243	114	5	)	)	PUNCT
ejpam-6243	114	6	r	r	NOUN
ejpam-6243	114	7	ramaswamy	ramaswamy	NOUN
ejpam-6243	114	8	et	et	PROPN
ejpam-6243	114	9	al	al	PROPN
ejpam-6243	114	10	.	.	PUNCT
ejpam-6243	114	11	/	/	SYM
ejpam-6243	114	12	eur	eur	PROPN
ejpam-6243	114	13	.	.	PUNCT
ejpam-6243	115	1	j.	j.	PROPN
ejpam-6243	115	2	pure	pure	PROPN
ejpam-6243	115	3	appl	appl	PROPN
ejpam-6243	115	4	.	.	PROPN
ejpam-6243	115	5	math	math	PROPN
ejpam-6243	115	6	,	,	PUNCT
ejpam-6243	115	7	18	18	NUM
ejpam-6243	115	8	(	(	PUNCT
ejpam-6243	115	9	3	3	NUM
ejpam-6243	115	10	)	)	PUNCT
ejpam-6243	115	11	(	(	PUNCT
ejpam-6243	115	12	2025	2025	NUM
ejpam-6243	115	13	)	)	PUNCT
ejpam-6243	115	14	,	,	PUNCT
ejpam-6243	115	15	6243	6243	NUM
ejpam-6243	115	16	6	6	NUM
ejpam-6243	115	17	of	of	ADP
ejpam-6243	115	18	14	14	NUM
ejpam-6243	115	19	=	=	PUNCT
ejpam-6243	115	20	p−1∑	p−1∑	PROPN
ejpam-6243	115	21	j	j	X
ejpam-6243	115	22	=	=	NOUN
ejpam-6243	115	23	m	m	VERB
ejpam-6243	115	24	d(xj	d(xj	ADJ
ejpam-6243	115	25	,	,	PUNCT
ejpam-6243	115	26	xj+1	xj+1	X
ejpam-6243	115	27	)	)	PUNCT
ejpam-6243	115	28	≤	≤	NUM
ejpam-6243	115	29	p−1∑	p−1∑	PROPN
ejpam-6243	115	30	j	j	PROPN
ejpam-6243	116	1	=	=	NOUN
ejpam-6243	116	2	m	m	VERB
ejpam-6243	116	3	ϕj(d(x0	ϕj(d(x0	PROPN
ejpam-6243	116	4	,	,	PUNCT
ejpam-6243	116	5	x1	x1	PROPN
ejpam-6243	116	6	)	)	PUNCT
ejpam-6243	116	7	)	)	PUNCT
ejpam-6243	116	8	.	.	PUNCT
ejpam-6243	117	1	given	give	VERB
ejpam-6243	117	2	that	that	DET
ejpam-6243	117	3	ϕ	ϕ	NOUN
ejpam-6243	117	4	functions	function	NOUN
ejpam-6243	117	5	as	as	ADP
ejpam-6243	117	6	a	a	DET
ejpam-6243	117	7	c−comparison	c−comparison	NOUN
ejpam-6243	117	8	,	,	PUNCT
ejpam-6243	117	9	the	the	DET
ejpam-6243	117	10	series	series	NOUN
ejpam-6243	117	11	is	be	AUX
ejpam-6243	117	12	∑+	∑+	PROPN
ejpam-6243	117	13	j=0∞ϕj(d(x0	j=0∞ϕj(d(x0	PROPN
ejpam-6243	117	14	,	,	PUNCT
ejpam-6243	117	15	x1	x1	PROPN
ejpam-6243	117	16	)	)	PUNCT
ejpam-6243	117	17	)	)	PUNCT
ejpam-6243	117	18	convergent	convergent	NOUN
ejpam-6243	117	19	,	,	PUNCT
ejpam-6243	117	20	sn	sn	PROPN
ejpam-6243	117	21	=	=	SYM
ejpam-6243	118	1	∑n	∑n	PROPN
ejpam-6243	118	2	j=0	j=0	PROPN
ejpam-6243	118	3	ϕ	ϕ	PROPN
ejpam-6243	118	4	j(d(x0	j(d(x0	PROPN
ejpam-6243	118	5	,	,	PUNCT
ejpam-6243	118	6	x1	x1	PROPN
ejpam-6243	118	7	)	)	PUNCT
ejpam-6243	118	8	)	)	PUNCT
ejpam-6243	118	9	transforms	transform	VERB
ejpam-6243	118	10	the	the	DET
ejpam-6243	118	11	inequality	inequality	NOUN
ejpam-6243	118	12	above	above	ADP
ejpam-6243	118	13	into	into	ADP
ejpam-6243	118	14	:	:	PUNCT
ejpam-6243	118	15	d(xm	d(xm	PROPN
ejpam-6243	118	16	,	,	PUNCT
ejpam-6243	118	17	xp	xp	ADJ
ejpam-6243	118	18	)	)	PUNCT
ejpam-6243	118	19	≤	≤	PUNCT
ejpam-6243	118	20	δp−1	δp−1	VERB
ejpam-6243	118	21	−	−	PROPN
ejpam-6243	118	22	δm−1	δm−1	PROPN
ejpam-6243	118	23	,	,	PUNCT
ejpam-6243	118	24	and	and	CCONJ
ejpam-6243	118	25	as	as	ADP
ejpam-6243	118	26	m	m	PROPN
ejpam-6243	118	27	,	,	PUNCT
ejpam-6243	118	28	p	p	X
ejpam-6243	118	29	→	→	PUNCT
ejpam-6243	118	30	+	+	NOUN
ejpam-6243	118	31	∞	∞	NOUN
ejpam-6243	118	32	we	we	PRON
ejpam-6243	118	33	get	get	VERB
ejpam-6243	118	34	d(xm	d(xm	PROPN
ejpam-6243	118	35	,	,	PUNCT
ejpam-6243	118	36	xp	xp	ADJ
ejpam-6243	118	37	)	)	PUNCT
ejpam-6243	118	38	→	→	SYM
ejpam-6243	118	39	0	0	X
ejpam-6243	118	40	.	.	PUNCT
ejpam-6243	119	1	(	(	PUNCT
ejpam-6243	119	2	12	12	NUM
ejpam-6243	119	3	)	)	PUNCT
ejpam-6243	119	4	this	this	PRON
ejpam-6243	119	5	indicates	indicate	VERB
ejpam-6243	119	6	that	that	SCONJ
ejpam-6243	119	7	there	there	PRON
ejpam-6243	119	8	exists	exist	VERB
ejpam-6243	119	9	z	z	NOUN
ejpam-6243	119	10	such	such	ADJ
ejpam-6243	119	11	that	that	SCONJ
ejpam-6243	119	12	{	{	PUNCT
ejpam-6243	119	13	xn	xn	X
ejpam-6243	119	14	}	}	PUNCT
ejpam-6243	119	15	is	be	AUX
ejpam-6243	119	16	a	a	DET
ejpam-6243	119	17	cauchy	cauchy	ADJ
ejpam-6243	119	18	sequence	sequence	NOUN
ejpam-6243	119	19	on	on	ADP
ejpam-6243	119	20	a	a	DET
ejpam-6243	119	21	complete	complete	ADJ
ejpam-6243	119	22	metric	metric	ADJ
ejpam-6243	119	23	space	space	NOUN
ejpam-6243	119	24	lim	lim	PROPN
ejpam-6243	119	25	n→+∞	n→+∞	PROPN
ejpam-6243	119	26	d(xm	d(xm	PROPN
ejpam-6243	119	27	,	,	PUNCT
ejpam-6243	119	28	z	z	NOUN
ejpam-6243	119	29	)	)	PUNCT
ejpam-6243	119	30	=	=	SYM
ejpam-6243	120	1	0	0	X
ejpam-6243	120	2	.	.	PUNCT
ejpam-6243	121	1	(	(	PUNCT
ejpam-6243	121	2	13	13	NUM
ejpam-6243	121	3	)	)	PUNCT
ejpam-6243	121	4	we	we	PRON
ejpam-6243	121	5	’ll	’ll	AUX
ejpam-6243	121	6	demonstrate	demonstrate	VERB
ejpam-6243	121	7	that	that	SCONJ
ejpam-6243	121	8	z	z	PROPN
ejpam-6243	121	9	is	be	AUX
ejpam-6243	121	10	a	a	DET
ejpam-6243	121	11	fixed	fix	VERB
ejpam-6243	121	12	point	point	NOUN
ejpam-6243	121	13	of	of	ADP
ejpam-6243	121	14	t	t	PROPN
ejpam-6243	121	15	at	at	ADP
ejpam-6243	121	16	this	this	DET
ejpam-6243	121	17	point	point	NOUN
ejpam-6243	121	18	.	.	PUNCT
ejpam-6243	122	1	given	give	VERB
ejpam-6243	122	2	assumption	assumption	NOUN
ejpam-6243	122	3	(	(	PUNCT
ejpam-6243	122	4	3	3	NUM
ejpam-6243	122	5	)	)	PUNCT
ejpam-6243	122	6	,	,	PUNCT
ejpam-6243	122	7	if	if	SCONJ
ejpam-6243	122	8	t	t	PROPN
ejpam-6243	122	9	is	be	AUX
ejpam-6243	122	10	continuous	continuous	ADJ
ejpam-6243	122	11	,	,	PUNCT
ejpam-6243	122	12	then	then	ADV
ejpam-6243	122	13	lim	lim	PROPN
ejpam-6243	122	14	n→+∞	n→+∞	PROPN
ejpam-6243	122	15	d(xn+1	d(xn+1	PROPN
ejpam-6243	122	16	,	,	PUNCT
ejpam-6243	122	17	t	t	PROPN
ejpam-6243	122	18	z	z	NOUN
ejpam-6243	122	19	)	)	PUNCT
ejpam-6243	123	1	=	=	VERB
ejpam-6243	123	2	lim	lim	PROPN
ejpam-6243	123	3	n→+∞	n→+∞	VERB
ejpam-6243	123	4	d(xn	d(xn	PROPN
ejpam-6243	123	5	,	,	PUNCT
ejpam-6243	123	6	txn	txn	NOUN
ejpam-6243	123	7	)	)	PUNCT
ejpam-6243	123	8	=	=	SYM
ejpam-6243	124	1	0	0	X
ejpam-6243	124	2	.	.	PUNCT
ejpam-6243	125	1	so	so	ADV
ejpam-6243	125	2	we	we	PRON
ejpam-6243	125	3	get	get	VERB
ejpam-6243	125	4	that	that	PRON
ejpam-6243	125	5	tz	tz	NOUN
ejpam-6243	125	6	=	=	SYM
ejpam-6243	125	7	z	z	PROPN
ejpam-6243	125	8	,	,	PUNCT
ejpam-6243	125	9	that	that	ADV
ejpam-6243	125	10	is	is	ADV
ejpam-6243	125	11	,	,	PUNCT
ejpam-6243	125	12	z	z	NOUN
ejpam-6243	125	13	is	be	AUX
ejpam-6243	125	14	a	a	DET
ejpam-6243	125	15	fixed	fix	VERB
ejpam-6243	125	16	point	point	NOUN
ejpam-6243	125	17	of	of	ADP
ejpam-6243	125	18	t	t	PROPN
ejpam-6243	125	19	.	.	PUNCT
ejpam-6243	126	1	in	in	ADP
ejpam-6243	126	2	the	the	DET
ejpam-6243	126	3	alternative	alternative	ADJ
ejpam-6243	126	4	hypothesis	hypothesis	NOUN
ejpam-6243	126	5	,	,	PUNCT
ejpam-6243	126	6	that	that	SCONJ
ejpam-6243	126	7	t	t	PROPN
ejpam-6243	126	8	2	2	NUM
ejpam-6243	126	9	is	be	AUX
ejpam-6243	126	10	continuous	continuous	ADJ
ejpam-6243	126	11	we	we	PRON
ejpam-6243	126	12	have	have	VERB
ejpam-6243	126	13	t	t	NOUN
ejpam-6243	126	14	2z	2z	NUM
ejpam-6243	126	15	=	=	SYM
ejpam-6243	126	16	limn→+∞t	limn→+∞t	NOUN
ejpam-6243	126	17	2xn	2xn	NOUN
ejpam-6243	127	1	=	=	PUNCT
ejpam-6243	128	1	z	z	NOUN
ejpam-6243	129	1	and	and	CCONJ
ejpam-6243	129	2	we	we	PRON
ejpam-6243	129	3	want	want	VERB
ejpam-6243	129	4	to	to	PART
ejpam-6243	129	5	show	show	VERB
ejpam-6243	129	6	that	that	SCONJ
ejpam-6243	129	7	tz	tz	NOUN
ejpam-6243	129	8	=	=	PUNCT
ejpam-6243	129	9	z.	z.	PROPN
ejpam-6243	129	10	assuming	assume	VERB
ejpam-6243	129	11	that	that	SCONJ
ejpam-6243	129	12	,	,	PUNCT
ejpam-6243	129	13	on	on	ADP
ejpam-6243	129	14	the	the	DET
ejpam-6243	129	15	contrary	contrary	NOUN
ejpam-6243	129	16	,	,	PUNCT
ejpam-6243	129	17	tz	tz	PROPN
ejpam-6243	129	18	̸=	̸=	PROPN
ejpam-6243	129	19	z	z	PROPN
ejpam-6243	129	20	,	,	PUNCT
ejpam-6243	129	21	we	we	PRON
ejpam-6243	129	22	have	have	VERB
ejpam-6243	129	23	from	from	ADP
ejpam-6243	129	24	(	(	PUNCT
ejpam-6243	129	25	5	5	NUM
ejpam-6243	129	26	)	)	PUNCT
ejpam-6243	129	27	d(z	d(z	PROPN
ejpam-6243	129	28	,	,	PUNCT
ejpam-6243	129	29	tz	tz	NOUN
ejpam-6243	129	30	)	)	PUNCT
ejpam-6243	129	31	=	=	SYM
ejpam-6243	130	1	d(t	d(t	PROPN
ejpam-6243	130	2	2z	2z	NUM
ejpam-6243	130	3	,	,	PUNCT
ejpam-6243	130	4	tz	tz	NOUN
ejpam-6243	130	5	)	)	PUNCT
ejpam-6243	130	6	≤	≤	NOUN
ejpam-6243	130	7	β(tz	β(tz	PROPN
ejpam-6243	130	8	,	,	PUNCT
ejpam-6243	130	9	z)d(tz	z)d(tz	NUM
ejpam-6243	130	10	,	,	PUNCT
ejpam-6243	130	11	z	z	NOUN
ejpam-6243	130	12	)	)	PUNCT
ejpam-6243	130	13	≤	≤	NUM
ejpam-6243	130	14	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	130	15	s	s	PART
ejpam-6243	130	16	(	(	PUNCT
ejpam-6243	130	17	tz	tz	PROPN
ejpam-6243	130	18	,	,	PUNCT
ejpam-6243	130	19	z	z	NOUN
ejpam-6243	130	20	)	)	PUNCT
ejpam-6243	130	21	)	)	PUNCT
ejpam-6243	131	1	<	<	X
ejpam-6243	131	2	jt	jt	PROPN
ejpam-6243	131	3	s	s	PROPN
ejpam-6243	131	4	(	(	PUNCT
ejpam-6243	131	5	tz	tz	PROPN
ejpam-6243	131	6	,	,	PUNCT
ejpam-6243	131	7	z	z	NOUN
ejpam-6243	131	8	)	)	PUNCT
ejpam-6243	131	9	=	=	PUNCT
ejpam-6243	132	1	[	[	X
ejpam-6243	132	2	α1	α1	PROPN
ejpam-6243	132	3	(	(	PUNCT
ejpam-6243	132	4	d(tz	d(tz	PROPN
ejpam-6243	132	5	,	,	PUNCT
ejpam-6243	132	6	t	t	PROPN
ejpam-6243	132	7	2z)d(z	2z)d(z	NUM
ejpam-6243	132	8	,	,	PUNCT
ejpam-6243	132	9	tz	tz	NOUN
ejpam-6243	132	10	)	)	PUNCT
ejpam-6243	132	11	d(tz	d(tz	PROPN
ejpam-6243	132	12	,	,	PUNCT
ejpam-6243	132	13	z	z	NOUN
ejpam-6243	132	14	)	)	PUNCT
ejpam-6243	132	15	)	)	PUNCT
ejpam-6243	132	16	s	s	PART
ejpam-6243	133	1	+	+	NUM
ejpam-6243	133	2	α2(d(tz	α2(d(tz	NOUN
ejpam-6243	133	3	,	,	PUNCT
ejpam-6243	133	4	z	z	NOUN
ejpam-6243	133	5	)	)	PUNCT
ejpam-6243	133	6	s	s	PART
ejpam-6243	133	7	]	]	PUNCT
ejpam-6243	133	8	1	1	NUM
ejpam-6243	133	9	s	s	NOUN
ejpam-6243	133	10	.	.	PUNCT
ejpam-6243	134	1	=	=	PUNCT
ejpam-6243	135	1	[	[	X
ejpam-6243	135	2	α1	α1	PROPN
ejpam-6243	135	3	(	(	PUNCT
ejpam-6243	135	4	d(tz	d(tz	PROPN
ejpam-6243	135	5	,	,	PUNCT
ejpam-6243	135	6	z)d(z	z)d(z	NUM
ejpam-6243	135	7	,	,	PUNCT
ejpam-6243	135	8	tz	tz	PROPN
ejpam-6243	135	9	)	)	PUNCT
ejpam-6243	135	10	d(tz	d(tz	PROPN
ejpam-6243	135	11	,	,	PUNCT
ejpam-6243	135	12	z	z	NOUN
ejpam-6243	135	13	)	)	PUNCT
ejpam-6243	135	14	)	)	PUNCT
ejpam-6243	135	15	s	s	PART
ejpam-6243	136	1	+	+	NUM
ejpam-6243	136	2	α2(d(tz	α2(d(tz	NOUN
ejpam-6243	136	3	,	,	PUNCT
ejpam-6243	136	4	z	z	NOUN
ejpam-6243	136	5	)	)	PUNCT
ejpam-6243	136	6	s	s	PART
ejpam-6243	136	7	]	]	X
ejpam-6243	136	8	1	1	NUM
ejpam-6243	136	9	s	s	NOUN
ejpam-6243	136	10	=	=	PUNCT
ejpam-6243	137	1	[	[	X
ejpam-6243	137	2	α1(d(z	α1(d(z	NUM
ejpam-6243	137	3	,	,	PUNCT
ejpam-6243	137	4	tz	tz	PROPN
ejpam-6243	137	5	)	)	PUNCT
ejpam-6243	137	6	)	)	PUNCT
ejpam-6243	138	1	s	s	PART
ejpam-6243	139	1	+	+	NUM
ejpam-6243	139	2	α2(d(tz	α2(d(tz	NOUN
ejpam-6243	139	3	,	,	PUNCT
ejpam-6243	139	4	z	z	NOUN
ejpam-6243	139	5	)	)	PUNCT
ejpam-6243	139	6	s	s	PART
ejpam-6243	139	7	]	]	X
ejpam-6243	139	8	1	1	NUM
ejpam-6243	139	9	s	s	NOUN
ejpam-6243	139	10	=	=	X
ejpam-6243	139	11	[	[	X
ejpam-6243	139	12	(	(	PUNCT
ejpam-6243	139	13	α1	α1	PROPN
ejpam-6243	139	14	+	+	CCONJ
ejpam-6243	139	15	α2)(d(tz	α2)(d(tz	NOUN
ejpam-6243	139	16	,	,	PUNCT
ejpam-6243	139	17	z	z	NOUN
ejpam-6243	139	18	)	)	PUNCT
ejpam-6243	139	19	)	)	PUNCT
ejpam-6243	140	1	s	s	X
ejpam-6243	140	2	]	]	X
ejpam-6243	140	3	1	1	NUM
ejpam-6243	140	4	s	s	NOUN
ejpam-6243	140	5	=	=	PUNCT
ejpam-6243	140	6	(	(	PUNCT
ejpam-6243	140	7	α1	α1	PROPN
ejpam-6243	140	8	+	+	CCONJ
ejpam-6243	140	9	α2	α2	ADJ
ejpam-6243	140	10	)	)	PUNCT
ejpam-6243	140	11	1	1	NUM
ejpam-6243	140	12	s	s	X
ejpam-6243	140	13	(	(	PUNCT
ejpam-6243	140	14	d(tz	d(tz	PROPN
ejpam-6243	140	15	,	,	PUNCT
ejpam-6243	140	16	z	z	NOUN
ejpam-6243	140	17	)	)	PUNCT
ejpam-6243	140	18	)	)	PUNCT
ejpam-6243	140	19	≤	≤	NUM
ejpam-6243	140	20	d(tz	d(tz	NOUN
ejpam-6243	140	21	,	,	PUNCT
ejpam-6243	140	22	z	z	NOUN
ejpam-6243	140	23	)	)	PUNCT
ejpam-6243	140	24	.	.	PUNCT
ejpam-6243	141	1	r	r	NOUN
ejpam-6243	141	2	ramaswamy	ramaswamy	PROPN
ejpam-6243	141	3	et	et	PROPN
ejpam-6243	141	4	al	al	PROPN
ejpam-6243	141	5	.	.	PUNCT
ejpam-6243	141	6	/	/	SYM
ejpam-6243	141	7	eur	eur	PROPN
ejpam-6243	141	8	.	.	PUNCT
ejpam-6243	142	1	j.	j.	PROPN
ejpam-6243	142	2	pure	pure	PROPN
ejpam-6243	142	3	appl	appl	PROPN
ejpam-6243	142	4	.	.	PROPN
ejpam-6243	142	5	math	math	PROPN
ejpam-6243	142	6	,	,	PUNCT
ejpam-6243	142	7	18	18	NUM
ejpam-6243	142	8	(	(	PUNCT
ejpam-6243	142	9	3	3	NUM
ejpam-6243	142	10	)	)	PUNCT
ejpam-6243	142	11	(	(	PUNCT
ejpam-6243	142	12	2025	2025	NUM
ejpam-6243	142	13	)	)	PUNCT
ejpam-6243	142	14	,	,	PUNCT
ejpam-6243	142	15	6243	6243	NUM
ejpam-6243	142	16	7	7	NUM
ejpam-6243	142	17	of	of	ADP
ejpam-6243	142	18	14	14	NUM
ejpam-6243	142	19	this	this	PRON
ejpam-6243	142	20	is	be	AUX
ejpam-6243	142	21	a	a	DET
ejpam-6243	142	22	contradiction	contradiction	NOUN
ejpam-6243	142	23	,	,	PUNCT
ejpam-6243	142	24	so	so	SCONJ
ejpam-6243	142	25	that	that	SCONJ
ejpam-6243	142	26	tz	tz	PROPN
ejpam-6243	142	27	=	=	SYM
ejpam-6243	142	28	z.	z.	PROPN
ejpam-6243	142	29	case	case	NOUN
ejpam-6243	142	30	2	2	NUM
ejpam-6243	142	31	:	:	PUNCT
ejpam-6243	142	32	for	for	ADP
ejpam-6243	142	33	the	the	DET
ejpam-6243	142	34	case	case	NOUN
ejpam-6243	142	35	s	s	PART
ejpam-6243	142	36	=	=	SYM
ejpam-6243	142	37	0	0	NUM
ejpam-6243	142	38	taking	take	VERB
ejpam-6243	142	39	x	x	PUNCT
ejpam-6243	142	40	=	=	PUNCT
ejpam-6243	142	41	xn−1	xn−1	PROPN
ejpam-6243	142	42	and	and	CCONJ
ejpam-6243	142	43	y	y	PROPN
ejpam-6243	143	1	=	=	PUNCT
ejpam-6243	143	2	xn	xn	PROPN
ejpam-6243	144	1	we	we	PRON
ejpam-6243	144	2	have	have	VERB
ejpam-6243	144	3	jt	jt	PROPN
ejpam-6243	144	4	s	s	PART
ejpam-6243	144	5	(	(	PUNCT
ejpam-6243	144	6	xn−1	xn−1	PROPN
ejpam-6243	144	7	,	,	PUNCT
ejpam-6243	144	8	xn	xn	PUNCT
ejpam-6243	144	9	)	)	PUNCT
ejpam-6243	145	1	=	=	PUNCT
ejpam-6243	146	1	[	[	X
ejpam-6243	146	2	(	(	PUNCT
ejpam-6243	146	3	d(xn−1	d(xn−1	PROPN
ejpam-6243	146	4	,	,	PUNCT
ejpam-6243	146	5	txn−1	txn−1	PROPN
ejpam-6243	146	6	)	)	PUNCT
ejpam-6243	146	7	)	)	PUNCT
ejpam-6243	146	8	α1	α1	PROPN
ejpam-6243	146	9	+	+	CCONJ
ejpam-6243	146	10	(	(	PUNCT
ejpam-6243	146	11	d(xn	d(xn	ADJ
ejpam-6243	146	12	,	,	PUNCT
ejpam-6243	146	13	txn	txn	NOUN
ejpam-6243	146	14	)	)	PUNCT
ejpam-6243	146	15	)	)	PUNCT
ejpam-6243	147	1	α2	α2	ADV
ejpam-6243	147	2	]	]	PUNCT
ejpam-6243	148	1	=	=	SYM
ejpam-6243	148	2	(	(	PUNCT
ejpam-6243	148	3	d(xn−1	d(xn−1	PROPN
ejpam-6243	148	4	,	,	PUNCT
ejpam-6243	148	5	xn	xn	NUM
ejpam-6243	148	6	)	)	PUNCT
ejpam-6243	148	7	)	)	PUNCT
ejpam-6243	148	8	α1	α1	PROPN
ejpam-6243	148	9	+	+	CCONJ
ejpam-6243	148	10	(	(	PUNCT
ejpam-6243	148	11	d(xn	d(xn	PROPN
ejpam-6243	148	12	,	,	PUNCT
ejpam-6243	148	13	xn+1	xn+1	NUM
ejpam-6243	148	14	)	)	PUNCT
ejpam-6243	148	15	)	)	PUNCT
ejpam-6243	149	1	α2	α2	ADJ
ejpam-6243	149	2	and	and	CCONJ
ejpam-6243	149	3	from	from	ADP
ejpam-6243	149	4	(	(	PUNCT
ejpam-6243	149	5	5	5	NUM
ejpam-6243	149	6	)	)	PUNCT
ejpam-6243	149	7	d(xn	d(xn	PROPN
ejpam-6243	149	8	,	,	PUNCT
ejpam-6243	149	9	xn+1	xn+1	NUM
ejpam-6243	149	10	)	)	PUNCT
ejpam-6243	149	11	≤	≤	NOUN
ejpam-6243	149	12	β(xn−1	β(xn−1	PUNCT
ejpam-6243	149	13	,	,	PUNCT
ejpam-6243	149	14	xn)d(txn−1	xn)d(txn−1	NUM
ejpam-6243	149	15	,	,	PUNCT
ejpam-6243	149	16	txn	txn	NOUN
ejpam-6243	149	17	)	)	PUNCT
ejpam-6243	149	18	≤	≤	NUM
ejpam-6243	150	1	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	150	2	s	s	PART
ejpam-6243	150	3	(	(	PUNCT
ejpam-6243	150	4	xn−1	xn−1	PROPN
ejpam-6243	150	5	,	,	PUNCT
ejpam-6243	150	6	xn	xn	PROPN
ejpam-6243	150	7	)	)	PUNCT
ejpam-6243	150	8	)	)	PUNCT
ejpam-6243	150	9	.	.	PUNCT
ejpam-6243	151	1	(	(	PUNCT
ejpam-6243	151	2	14	14	NUM
ejpam-6243	151	3	)	)	PUNCT
ejpam-6243	151	4	for	for	ADP
ejpam-6243	151	5	the	the	DET
ejpam-6243	151	6	same	same	ADJ
ejpam-6243	151	7	reason	reason	NOUN
ejpam-6243	151	8	as	as	ADP
ejpam-6243	151	9	the	the	DET
ejpam-6243	151	10	previous	previous	ADJ
ejpam-6243	151	11	case	case	NOUN
ejpam-6243	151	12	,	,	PUNCT
ejpam-6243	151	13	d(xn−1	d(xn−1	PROPN
ejpam-6243	151	14	,	,	PUNCT
ejpam-6243	151	15	xn	xn	PROPN
ejpam-6243	151	16	)	)	PUNCT
ejpam-6243	151	17	>	>	X
ejpam-6243	152	1	d(xn	d(xn	PROPN
ejpam-6243	152	2	,	,	PUNCT
ejpam-6243	152	3	xn+1	xn+1	NUM
ejpam-6243	152	4	)	)	PUNCT
ejpam-6243	152	5	because	because	SCONJ
ejpam-6243	152	6	the	the	DET
ejpam-6243	152	7	other	other	ADJ
ejpam-6243	152	8	case	case	NOUN
ejpam-6243	152	9	contradicts	contradict	VERB
ejpam-6243	152	10	itself	itself	PRON
ejpam-6243	152	11	.	.	PUNCT
ejpam-6243	153	1	furthermore	furthermore	ADV
ejpam-6243	153	2	,	,	PUNCT
ejpam-6243	153	3	if	if	SCONJ
ejpam-6243	153	4	we	we	PRON
ejpam-6243	153	5	assume	assume	VERB
ejpam-6243	153	6	absurdum	absurdum	NOUN
ejpam-6243	153	7	that	that	SCONJ
ejpam-6243	153	8	d(xn−1	d(xn−1	PROPN
ejpam-6243	153	9	,	,	PUNCT
ejpam-6243	153	10	xn	xn	PROPN
ejpam-6243	153	11	)	)	PUNCT
ejpam-6243	153	12	≤	≤	PUNCT
ejpam-6243	154	1	d(xn	d(xn	PROPN
ejpam-6243	154	2	,	,	PUNCT
ejpam-6243	154	3	xn+1	xn+1	NUM
ejpam-6243	154	4	)	)	PUNCT
ejpam-6243	154	5	,	,	PUNCT
ejpam-6243	154	6	we	we	PRON
ejpam-6243	154	7	obtain	obtain	VERB
ejpam-6243	154	8	.	.	PUNCT
ejpam-6243	155	1	d(xn	d(xn	X
ejpam-6243	155	2	,	,	PUNCT
ejpam-6243	155	3	xn+1	xn+1	NUM
ejpam-6243	155	4	)	)	PUNCT
ejpam-6243	155	5	<	<	X
ejpam-6243	155	6	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	155	7	s	s	PART
ejpam-6243	155	8	(	(	PUNCT
ejpam-6243	155	9	xn−1	xn−1	PROPN
ejpam-6243	155	10	,	,	PUNCT
ejpam-6243	155	11	xn	xn	PROPN
ejpam-6243	155	12	)	)	PUNCT
ejpam-6243	155	13	)	)	PUNCT
ejpam-6243	156	1	<	<	X
ejpam-6243	156	2	(	(	PUNCT
ejpam-6243	156	3	d(xn	d(xn	PROPN
ejpam-6243	156	4	,	,	PUNCT
ejpam-6243	156	5	xn+1	xn+1	NUM
ejpam-6243	156	6	)	)	PUNCT
ejpam-6243	156	7	)	)	PUNCT
ejpam-6243	157	1	α1+α2	α1+α2	PROPN
ejpam-6243	157	2	=	=	SYM
ejpam-6243	157	3	d(xn	d(xn	X
ejpam-6243	157	4	,	,	PUNCT
ejpam-6243	157	5	xn+1	xn+1	NUM
ejpam-6243	157	6	)	)	PUNCT
ejpam-6243	157	7	.	.	PUNCT
ejpam-6243	158	1	this	this	PRON
ejpam-6243	158	2	is	be	AUX
ejpam-6243	158	3	a	a	DET
ejpam-6243	158	4	contradiction	contradiction	NOUN
ejpam-6243	158	5	.	.	PUNCT
ejpam-6243	159	1	then	then	ADV
ejpam-6243	159	2	from	from	ADP
ejpam-6243	159	3	(	(	PUNCT
ejpam-6243	159	4	14	14	NUM
ejpam-6243	159	5	)	)	PUNCT
ejpam-6243	159	6	we	we	PRON
ejpam-6243	159	7	obtain	obtain	VERB
ejpam-6243	159	8	the	the	DET
ejpam-6243	159	9	following	following	NOUN
ejpam-6243	159	10	:	:	PUNCT
ejpam-6243	159	11	d(xn	d(xn	X
ejpam-6243	159	12	,	,	PUNCT
ejpam-6243	159	13	xn+1	xn+1	NUM
ejpam-6243	159	14	)	)	PUNCT
ejpam-6243	159	15	≤	≤	NUM
ejpam-6243	159	16	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	159	17	s	s	PART
ejpam-6243	159	18	(	(	PUNCT
ejpam-6243	159	19	xn−1	xn−1	PROPN
ejpam-6243	159	20	,	,	PUNCT
ejpam-6243	159	21	xn	xn	PROPN
ejpam-6243	159	22	)	)	PUNCT
ejpam-6243	159	23	)	)	PUNCT
ejpam-6243	160	1	<	<	X
ejpam-6243	160	2	ϕ(xn−1	ϕ(xn−1	PUNCT
ejpam-6243	160	3	,	,	PUNCT
ejpam-6243	160	4	xn	xn	PROPN
ejpam-6243	160	5	)	)	PUNCT
ejpam-6243	160	6	(	(	PUNCT
ejpam-6243	160	7	15	15	NUM
ejpam-6243	160	8	)	)	PUNCT
ejpam-6243	160	9	and	and	CCONJ
ejpam-6243	160	10	inductively	inductively	ADV
ejpam-6243	160	11	we	we	PRON
ejpam-6243	160	12	get	get	VERB
ejpam-6243	160	13	d(xn	d(xn	NOUN
ejpam-6243	160	14	,	,	PUNCT
ejpam-6243	160	15	xn+1	xn+1	NUM
ejpam-6243	160	16	)	)	PUNCT
ejpam-6243	160	17	≤	≤	NOUN
ejpam-6243	161	1	ϕn(d(xn	ϕn(d(xn	PROPN
ejpam-6243	161	2	,	,	PUNCT
ejpam-6243	161	3	xn+1	xn+1	NUM
ejpam-6243	161	4	)	)	PUNCT
ejpam-6243	161	5	)	)	PUNCT
ejpam-6243	161	6	.	.	PUNCT
ejpam-6243	162	1	we	we	PRON
ejpam-6243	162	2	can	can	AUX
ejpam-6243	162	3	readily	readily	ADV
ejpam-6243	162	4	determine	determine	VERB
ejpam-6243	162	5	that	that	SCONJ
ejpam-6243	162	6	{	{	PUNCT
ejpam-6243	162	7	xn	xn	X
ejpam-6243	162	8	}	}	PUNCT
ejpam-6243	162	9	is	be	AUX
ejpam-6243	162	10	a	a	DET
ejpam-6243	162	11	cauchy	cauchy	ADJ
ejpam-6243	162	12	sequence	sequence	NOUN
ejpam-6243	162	13	in	in	ADP
ejpam-6243	162	14	a	a	DET
ejpam-6243	162	15	complete	complete	ADJ
ejpam-6243	162	16	metric	metric	ADJ
ejpam-6243	162	17	space	space	NOUN
ejpam-6243	162	18	by	by	ADP
ejpam-6243	162	19	applying	apply	VERB
ejpam-6243	162	20	the	the	DET
ejpam-6243	162	21	same	same	ADJ
ejpam-6243	162	22	arguments	argument	NOUN
ejpam-6243	162	23	as	as	ADP
ejpam-6243	162	24	in	in	ADP
ejpam-6243	162	25	the	the	DET
ejpam-6243	162	26	case	case	NOUN
ejpam-6243	162	27	s	s	VERB
ejpam-6243	162	28	>	>	X
ejpam-6243	162	29	0	0	NUM
ejpam-6243	162	30	.	.	PUNCT
ejpam-6243	163	1	consequently	consequently	ADV
ejpam-6243	163	2	,	,	PUNCT
ejpam-6243	163	3	there	there	PRON
ejpam-6243	163	4	exists	exist	VERB
ejpam-6243	163	5	z	z	NOUN
ejpam-6243	163	6	such	such	ADJ
ejpam-6243	163	7	that	that	SCONJ
ejpam-6243	163	8	limn→+∞xn	limn→+∞xn	NUM
ejpam-6243	163	9	=	=	PUNCT
ejpam-6243	163	10	z.	z.	PROPN
ejpam-6243	163	11	we	we	PRON
ejpam-6243	163	12	claim	claim	VERB
ejpam-6243	163	13	that	that	SCONJ
ejpam-6243	163	14	z	z	NOUN
ejpam-6243	163	15	is	be	AUX
ejpam-6243	163	16	a	a	DET
ejpam-6243	163	17	fixed	fix	VERB
ejpam-6243	163	18	point	point	NOUN
ejpam-6243	163	19	of	of	ADP
ejpam-6243	163	20	t	t	PROPN
ejpam-6243	163	21	under	under	ADP
ejpam-6243	163	22	the	the	DET
ejpam-6243	163	23	assumption	assumption	NOUN
ejpam-6243	163	24	that	that	SCONJ
ejpam-6243	163	25	t	t	PROPN
ejpam-6243	163	26	is	be	AUX
ejpam-6243	163	27	continuous	continuous	ADJ
ejpam-6243	163	28	we	we	PRON
ejpam-6243	163	29	have	have	VERB
ejpam-6243	163	30	lim	lim	PROPN
ejpam-6243	163	31	n→+∞	n→+∞	PROPN
ejpam-6243	163	32	d(xn+1	d(xn+1	PROPN
ejpam-6243	163	33	,	,	PUNCT
ejpam-6243	163	34	t	t	PROPN
ejpam-6243	163	35	z	z	NOUN
ejpam-6243	163	36	)	)	PUNCT
ejpam-6243	164	1	=	=	SYM
ejpam-6243	164	2	lim	lim	PROPN
ejpam-6243	164	3	n→+∞	n→+∞	PROPN
ejpam-6243	164	4	d(txn	d(txn	PROPN
ejpam-6243	164	5	,	,	PUNCT
ejpam-6243	164	6	t	t	PROPN
ejpam-6243	164	7	z	z	PROPN
ejpam-6243	164	8	)	)	PUNCT
ejpam-6243	164	9	=	=	SYM
ejpam-6243	164	10	0	0	NUM
ejpam-6243	164	11	,	,	PUNCT
ejpam-6243	164	12	and	and	CCONJ
ejpam-6243	164	13	also	also	ADV
ejpam-6243	164	14	together	together	ADV
ejpam-6243	164	15	with	with	ADP
ejpam-6243	164	16	the	the	DET
ejpam-6243	164	17	uniqueness	uniqueness	NOUN
ejpam-6243	164	18	of	of	ADP
ejpam-6243	164	19	limit	limit	NOUN
ejpam-6243	164	20	,	,	PUNCT
ejpam-6243	164	21	tz	tz	PROPN
ejpam-6243	164	22	=	=	SYM
ejpam-6243	164	23	z.	z.	PROPN
ejpam-6243	164	24	also	also	ADV
ejpam-6243	164	25	,	,	PUNCT
ejpam-6243	164	26	if	if	SCONJ
ejpam-6243	164	27	t	t	PROPN
ejpam-6243	164	28	2	2	NUM
ejpam-6243	164	29	is	be	AUX
ejpam-6243	164	30	continuous	continuous	ADJ
ejpam-6243	164	31	as	as	ADP
ejpam-6243	164	32	in	in	ADP
ejpam-6243	164	33	case	case	NOUN
ejpam-6243	164	34	(	(	PUNCT
ejpam-6243	164	35	1	1	X
ejpam-6243	164	36	)	)	PUNCT
ejpam-6243	164	37	we	we	PRON
ejpam-6243	164	38	have	have	VERB
ejpam-6243	164	39	that	that	PRON
ejpam-6243	164	40	tz	tz	NOUN
ejpam-6243	165	1	=	=	SYM
ejpam-6243	165	2	z	z	PROPN
ejpam-6243	165	3	then	then	ADV
ejpam-6243	165	4	d(z	d(z	PROPN
ejpam-6243	165	5	,	,	PUNCT
ejpam-6243	165	6	tz	tz	NOUN
ejpam-6243	165	7	)	)	PUNCT
ejpam-6243	165	8	=	=	SYM
ejpam-6243	166	1	d(t	d(t	PROPN
ejpam-6243	166	2	2z	2z	NUM
ejpam-6243	166	3	,	,	PUNCT
ejpam-6243	166	4	tz	tz	NOUN
ejpam-6243	166	5	)	)	PUNCT
ejpam-6243	166	6	≤	≤	NOUN
ejpam-6243	166	7	β(tz	β(tz	PROPN
ejpam-6243	166	8	,	,	PUNCT
ejpam-6243	166	9	z)d(t	z)d(t	NOUN
ejpam-6243	166	10	2z	2z	NUM
ejpam-6243	166	11	,	,	PUNCT
ejpam-6243	166	12	tz	tz	NOUN
ejpam-6243	166	13	)	)	PUNCT
ejpam-6243	166	14	≤	≤	NUM
ejpam-6243	166	15	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	166	16	s	s	PART
ejpam-6243	166	17	(	(	PUNCT
ejpam-6243	166	18	t	t	PROPN
ejpam-6243	166	19	2z	2z	NUM
ejpam-6243	166	20	,	,	PUNCT
ejpam-6243	166	21	tz	tz	NOUN
ejpam-6243	166	22	)	)	PUNCT
ejpam-6243	166	23	)	)	PUNCT
ejpam-6243	167	1	r	r	NOUN
ejpam-6243	167	2	ramaswamy	ramaswamy	NOUN
ejpam-6243	167	3	et	et	PROPN
ejpam-6243	167	4	al	al	PROPN
ejpam-6243	167	5	.	.	PUNCT
ejpam-6243	167	6	/	/	SYM
ejpam-6243	167	7	eur	eur	PROPN
ejpam-6243	167	8	.	.	PUNCT
ejpam-6243	168	1	j.	j.	PROPN
ejpam-6243	168	2	pure	pure	PROPN
ejpam-6243	168	3	appl	appl	PROPN
ejpam-6243	168	4	.	.	PROPN
ejpam-6243	168	5	math	math	PROPN
ejpam-6243	168	6	,	,	PUNCT
ejpam-6243	168	7	18	18	NUM
ejpam-6243	168	8	(	(	PUNCT
ejpam-6243	168	9	3	3	NUM
ejpam-6243	168	10	)	)	PUNCT
ejpam-6243	168	11	(	(	PUNCT
ejpam-6243	168	12	2025	2025	NUM
ejpam-6243	168	13	)	)	PUNCT
ejpam-6243	168	14	,	,	PUNCT
ejpam-6243	168	15	6243	6243	NUM
ejpam-6243	168	16	8	8	NUM
ejpam-6243	168	17	of	of	ADP
ejpam-6243	168	18	14	14	NUM
ejpam-6243	168	19	≤	≤	NOUN
ejpam-6243	168	20	ϕ(d(xn	ϕ(d(xn	NOUN
ejpam-6243	168	21	,	,	PUNCT
ejpam-6243	168	22	xn+1	xn+1	NUM
ejpam-6243	168	23	)	)	PUNCT
ejpam-6243	168	24	)	)	PUNCT
ejpam-6243	169	1	α1+α2	α1+α2	PROPN
ejpam-6243	169	2	<	<	X
ejpam-6243	169	3	d(z	d(z	PROPN
ejpam-6243	169	4	,	,	PUNCT
ejpam-6243	169	5	tz	tz	PROPN
ejpam-6243	169	6	)	)	PUNCT
ejpam-6243	169	7	<	<	X
ejpam-6243	169	8	d(z	d(z	PROPN
ejpam-6243	169	9	,	,	PUNCT
ejpam-6243	169	10	tz	tz	PROPN
ejpam-6243	169	11	)	)	PUNCT
ejpam-6243	169	12	.	.	PUNCT
ejpam-6243	170	1	this	this	DET
ejpam-6243	170	2	contradiction	contradiction	NOUN
ejpam-6243	170	3	shows	show	VERB
ejpam-6243	170	4	that	that	SCONJ
ejpam-6243	170	5	z	z	NOUN
ejpam-6243	170	6	=	=	SYM
ejpam-6243	170	7	tz	tz	PROPN
ejpam-6243	170	8	.	.	PROPN
ejpam-6243	170	9	example	example	NOUN
ejpam-6243	171	1	1	1	NUM
ejpam-6243	171	2	.	.	PUNCT
ejpam-6243	171	3	let	let	VERB
ejpam-6243	171	4	x	x	PUNCT
ejpam-6243	171	5	=	=	PUNCT
ejpam-6243	172	1	[	[	X
ejpam-6243	172	2	0	0	NUM
ejpam-6243	172	3	,	,	PUNCT
ejpam-6243	172	4	2	2	NUM
ejpam-6243	172	5	]	]	PUNCT
ejpam-6243	172	6	,	,	PUNCT
ejpam-6243	172	7	d	d	X
ejpam-6243	172	8	:	:	PUNCT
ejpam-6243	172	9	x	x	PROPN
ejpam-6243	172	10	×x	×x	X
ejpam-6243	172	11	→	→	SYM
ejpam-6243	172	12	[	[	X
ejpam-6243	172	13	0,+∞	0,+∞	NUM
ejpam-6243	172	14	)	)	PUNCT
ejpam-6243	172	15	be	be	VERB
ejpam-6243	172	16	the	the	DET
ejpam-6243	172	17	usual	usual	ADJ
ejpam-6243	172	18	metric	metric	NOUN
ejpam-6243	172	19	,	,	PUNCT
ejpam-6243	172	20	d(x	d(x	PROPN
ejpam-6243	172	21	,	,	PUNCT
ejpam-6243	172	22	y	y	NOUN
ejpam-6243	172	23	)	)	PUNCT
ejpam-6243	172	24	=	=	NOUN
ejpam-6243	172	25	|x	|x	NOUN
ejpam-6243	173	1	−	−	VERB
ejpam-6243	173	2	y|	y|	NOUN
ejpam-6243	173	3	for	for	ADP
ejpam-6243	173	4	all	all	DET
ejpam-6243	173	5	x	x	NOUN
ejpam-6243	173	6	,	,	PUNCT
ejpam-6243	173	7	y	y	PROPN
ejpam-6243	173	8	∈	∈	PROPN
ejpam-6243	173	9	x	x	X
ejpam-6243	173	10	and	and	CCONJ
ejpam-6243	173	11	the	the	DET
ejpam-6243	173	12	mapping	mapping	NOUN
ejpam-6243	173	13	t	t	NOUN
ejpam-6243	173	14	:	:	PUNCT
ejpam-6243	173	15	x	x	X
ejpam-6243	173	16	→	→	PUNCT
ejpam-6243	173	17	x	x	VERB
ejpam-6243	173	18	be	be	AUX
ejpam-6243	173	19	define	define	ADJ
ejpam-6243	173	20	by	by	ADP
ejpam-6243	173	21	t	t	PROPN
ejpam-6243	173	22	(	(	PUNCT
ejpam-6243	173	23	x	x	NOUN
ejpam-6243	173	24	)	)	PUNCT
ejpam-6243	173	25	=	=	PUNCT
ejpam-6243	174	1			PROPN
ejpam-6243	174	2	2	2	NUM
ejpam-6243	174	3	3	3	NUM
ejpam-6243	174	4	,	,	PUNCT
ejpam-6243	174	5	if	if	SCONJ
ejpam-6243	174	6	x	x	PUNCT
ejpam-6243	174	7	∈	∈	PROPN
ejpam-6243	175	1	[	[	X
ejpam-6243	175	2	0	0	NUM
ejpam-6243	175	3	,	,	PUNCT
ejpam-6243	175	4	1	1	NUM
ejpam-6243	175	5	]	]	PUNCT
ejpam-6243	175	6	x	x	SYM
ejpam-6243	175	7	2	2	NUM
ejpam-6243	175	8	,	,	PUNCT
ejpam-6243	175	9	if	if	SCONJ
ejpam-6243	175	10	x	x	SYM
ejpam-6243	175	11	∈	∈	PROPN
ejpam-6243	175	12	(	(	PUNCT
ejpam-6243	175	13	1	1	NUM
ejpam-6243	175	14	,	,	PUNCT
ejpam-6243	175	15	2	2	NUM
ejpam-6243	175	16	]	]	PUNCT
ejpam-6243	175	17	.	.	PUNCT
ejpam-6243	176	1	consider	consider	VERB
ejpam-6243	176	2	also	also	ADV
ejpam-6243	176	3	a	a	DET
ejpam-6243	176	4	function	function	NOUN
ejpam-6243	176	5	β(x	β(x	NOUN
ejpam-6243	176	6	,	,	PUNCT
ejpam-6243	176	7	y	y	NOUN
ejpam-6243	176	8	)	)	PUNCT
ejpam-6243	176	9	=	=	NOUN
ejpam-6243	176	10	{	{	PUNCT
ejpam-6243	177	1	2	2	NUM
ejpam-6243	177	2	,	,	PUNCT
ejpam-6243	177	3	if	if	SCONJ
ejpam-6243	177	4	x	x	NOUN
ejpam-6243	177	5	,	,	PUNCT
ejpam-6243	177	6	y	y	PROPN
ejpam-6243	177	7	∈	∈	PROPN
ejpam-6243	178	1	[	[	X
ejpam-6243	178	2	0	0	NUM
ejpam-6243	178	3	,	,	PUNCT
ejpam-6243	178	4	1	1	NUM
ejpam-6243	178	5	]	]	SYM
ejpam-6243	178	6	1	1	NUM
ejpam-6243	178	7	,	,	PUNCT
ejpam-6243	178	8	if	if	SCONJ
ejpam-6243	178	9	x	x	ADP
ejpam-6243	178	10	=	=	SYM
ejpam-6243	178	11	0	0	NUM
ejpam-6243	178	12	,	,	PUNCT
ejpam-6243	178	13	y	y	PROPN
ejpam-6243	178	14	=	=	SYM
ejpam-6243	178	15	2	2	NUM
ejpam-6243	178	16	and	and	CCONJ
ejpam-6243	178	17	the	the	DET
ejpam-6243	178	18	comparison	comparison	NOUN
ejpam-6243	178	19	function	function	NOUN
ejpam-6243	178	20	ϕ	ϕ	NOUN
ejpam-6243	178	21	:	:	PUNCT
ejpam-6243	179	1	[	[	X
ejpam-6243	179	2	0,+∞	0,+∞	NUM
ejpam-6243	179	3	)	)	PUNCT
ejpam-6243	179	4	→	→	PUNCT
ejpam-6243	180	1	[	[	X
ejpam-6243	180	2	0,+∞	0,+∞	NUM
ejpam-6243	180	3	)	)	PUNCT
ejpam-6243	180	4	,	,	PUNCT
ejpam-6243	180	5	ϕ(t	ϕ(t	NUM
ejpam-6243	180	6	)	)	PUNCT
ejpam-6243	180	7	=	=	SYM
ejpam-6243	180	8	t/5	t/5	NUM
ejpam-6243	180	9	.	.	PUNCT
ejpam-6243	181	1	the	the	DET
ejpam-6243	181	2	assumptions	assumption	NOUN
ejpam-6243	181	3	(	(	PUNCT
ejpam-6243	181	4	1	1	NUM
ejpam-6243	181	5	)	)	PUNCT
ejpam-6243	181	6	and	and	CCONJ
ejpam-6243	181	7	(	(	PUNCT
ejpam-6243	181	8	2	2	X
ejpam-6243	181	9	)	)	PUNCT
ejpam-6243	181	10	are	be	AUX
ejpam-6243	181	11	readily	readily	ADV
ejpam-6243	181	12	shown	show	VERB
ejpam-6243	181	13	to	to	PART
ejpam-6243	181	14	be	be	AUX
ejpam-6243	181	15	true	true	ADJ
ejpam-6243	181	16	,	,	PUNCT
ejpam-6243	181	17	and	and	CCONJ
ejpam-6243	181	18	as	as	ADP
ejpam-6243	181	19	t	t	NOUN
ejpam-6243	181	20	2(x	2(x	NUM
ejpam-6243	181	21	)	)	PUNCT
ejpam-6243	182	1	=	=	SYM
ejpam-6243	182	2	2/3	2/3	NUM
ejpam-6243	182	3	is	be	AUX
ejpam-6243	182	4	continuous	continuous	ADJ
ejpam-6243	182	5	,	,	PUNCT
ejpam-6243	182	6	the	the	DET
ejpam-6243	182	7	assumption	assumption	NOUN
ejpam-6243	182	8	(	(	PUNCT
ejpam-6243	182	9	4	4	X
ejpam-6243	182	10	)	)	PUNCT
ejpam-6243	182	11	is	be	AUX
ejpam-6243	182	12	likewise	likewise	ADV
ejpam-6243	182	13	confirmed	confirm	VERB
ejpam-6243	182	14	.	.	PUNCT
ejpam-6243	183	1	since	since	SCONJ
ejpam-6243	183	2	d(tx	d(tx	PROPN
ejpam-6243	183	3	,	,	PUNCT
ejpam-6243	183	4	ty	ty	PRON
ejpam-6243	183	5	)	)	PUNCT
ejpam-6243	183	6	=	=	SYM
ejpam-6243	183	7	0	0	NUM
ejpam-6243	183	8	holds	hold	VERB
ejpam-6243	183	9	for	for	ADP
ejpam-6243	183	10	every	every	DET
ejpam-6243	183	11	x	x	NOUN
ejpam-6243	183	12	,	,	PUNCT
ejpam-6243	183	13	y	y	PROPN
ejpam-6243	183	14	∈	∈	PROPN
ejpam-6243	184	1	[	[	X
ejpam-6243	184	2	0	0	NUM
ejpam-6243	184	3	,	,	PUNCT
ejpam-6243	184	4	1	1	NUM
ejpam-6243	184	5	]	]	PUNCT
ejpam-6243	184	6	,	,	PUNCT
ejpam-6243	184	7	the	the	DET
ejpam-6243	184	8	inequality	inequality	NOUN
ejpam-6243	184	9	(	(	PUNCT
ejpam-6243	184	10	5	5	NUM
ejpam-6243	184	11	)	)	PUNCT
ejpam-6243	184	12	is	be	AUX
ejpam-6243	184	13	true	true	ADJ
ejpam-6243	184	14	.	.	PUNCT
ejpam-6243	185	1	assuming	assume	VERB
ejpam-6243	185	2	y	y	PROPN
ejpam-6243	185	3	=	=	SYM
ejpam-6243	185	4	2	2	NUM
ejpam-6243	185	5	and	and	CCONJ
ejpam-6243	185	6	x	x	SYM
ejpam-6243	185	7	=	=	SYM
ejpam-6243	185	8	0	0	NUM
ejpam-6243	185	9	,	,	PUNCT
ejpam-6243	185	10	we	we	PRON
ejpam-6243	185	11	get	get	VERB
ejpam-6243	185	12	β(0	β(0	PROPN
ejpam-6243	185	13	,	,	PUNCT
ejpam-6243	185	14	2)d(t0	2)d(t0	NOUN
ejpam-6243	185	15	,	,	PUNCT
ejpam-6243	185	16	t2	t2	NOUN
ejpam-6243	185	17	)	)	PUNCT
ejpam-6243	185	18	=	=	PUNCT
ejpam-6243	186	1	β(0	β(0	PROPN
ejpam-6243	186	2	,	,	PUNCT
ejpam-6243	186	3	2)d	2)d	X
ejpam-6243	186	4	(	(	PUNCT
ejpam-6243	186	5	2	2	NUM
ejpam-6243	186	6	3	3	NUM
ejpam-6243	186	7	,	,	PUNCT
ejpam-6243	186	8	1	1	NUM
ejpam-6243	186	9	)	)	PUNCT
ejpam-6243	186	10	=	=	SYM
ejpam-6243	186	11	1	1	NUM
ejpam-6243	186	12	3	3	NUM
ejpam-6243	186	13	<	<	SYM
ejpam-6243	186	14	1	1	NUM
ejpam-6243	186	15	5	5	NUM
ejpam-6243	186	16	√	√	PROPN
ejpam-6243	186	17	(	(	PUNCT
ejpam-6243	186	18	1	1	NUM
ejpam-6243	186	19	9	9	NUM
ejpam-6243	186	20	+	+	CCONJ
ejpam-6243	186	21	4	4	NUM
ejpam-6243	186	22	)	)	PUNCT
ejpam-6243	186	23	=	=	SYM
ejpam-6243	186	24	(	(	PUNCT
ejpam-6243	186	25	(	(	PUNCT
ejpam-6243	186	26	d(xn	d(xn	X
ejpam-6243	186	27	,	,	PUNCT
ejpam-6243	186	28	txn)d(z	txn)d(z	PROPN
ejpam-6243	186	29	,	,	PUNCT
ejpam-6243	186	30	tz	tz	PROPN
ejpam-6243	186	31	)	)	PUNCT
ejpam-6243	186	32	d(txn	d(txn	PROPN
ejpam-6243	186	33	,	,	PUNCT
ejpam-6243	186	34	z	z	NOUN
ejpam-6243	186	35	)	)	PUNCT
ejpam-6243	186	36	)	)	PUNCT
ejpam-6243	186	37	s	s	X
ejpam-6243	186	38	)	)	PUNCT
ejpam-6243	186	39	1	1	NUM
ejpam-6243	186	40	2	2	NUM
ejpam-6243	186	41	=	=	SYM
ejpam-6243	186	42	(	(	PUNCT
ejpam-6243	186	43	α1	α1	PROPN
ejpam-6243	186	44	(	(	PUNCT
ejpam-6243	186	45	d(xn	d(xn	PROPN
ejpam-6243	186	46	,	,	PUNCT
ejpam-6243	186	47	xn+1)d(z	xn+1)d(z	PROPN
ejpam-6243	186	48	,	,	PUNCT
ejpam-6243	186	49	tz	tz	PROPN
ejpam-6243	186	50	)	)	PUNCT
ejpam-6243	186	51	d(xn	d(xn	PROPN
ejpam-6243	186	52	,	,	PUNCT
ejpam-6243	186	53	z	z	NOUN
ejpam-6243	186	54	)	)	PUNCT
ejpam-6243	186	55	)	)	PUNCT
ejpam-6243	186	56	s	s	PART
ejpam-6243	187	1	+	+	X
ejpam-6243	187	2	α2(d(xn	α2(d(xn	NUM
ejpam-6243	187	3	,	,	PUNCT
ejpam-6243	187	4	z	z	NOUN
ejpam-6243	187	5	)	)	PUNCT
ejpam-6243	187	6	s	s	PART
ejpam-6243	187	7	)	)	PUNCT
ejpam-6243	187	8	1	1	NUM
ejpam-6243	187	9	2	2	NUM
ejpam-6243	187	10	.	.	PUNCT
ejpam-6243	188	1	in	in	ADP
ejpam-6243	188	2	all	all	DET
ejpam-6243	188	3	other	other	ADJ
ejpam-6243	188	4	cases	case	NOUN
ejpam-6243	188	5	,	,	PUNCT
ejpam-6243	188	6	β(x	β(x	NOUN
ejpam-6243	188	7	,	,	PUNCT
ejpam-6243	188	8	y	y	NOUN
ejpam-6243	188	9	)	)	PUNCT
ejpam-6243	188	10	=	=	SYM
ejpam-6243	188	11	0	0	NUM
ejpam-6243	189	1	and	and	CCONJ
ejpam-6243	189	2	(	(	PUNCT
ejpam-6243	189	3	5	5	NUM
ejpam-6243	189	4	)	)	PUNCT
ejpam-6243	189	5	is	be	AUX
ejpam-6243	189	6	obviously	obviously	ADV
ejpam-6243	189	7	satisfied	satisfied	ADJ
ejpam-6243	189	8	.	.	PUNCT
ejpam-6243	190	1	because	because	SCONJ
ejpam-6243	190	2	t	t	PROPN
ejpam-6243	190	3	is	be	AUX
ejpam-6243	190	4	an	an	DET
ejpam-6243	190	5	admissible	admissible	ADJ
ejpam-6243	190	6	hybrid	hybrid	ADJ
ejpam-6243	190	7	contraction	contraction	NOUN
ejpam-6243	190	8	and	and	CCONJ
ejpam-6243	190	9	satisfies	satisfie	NOUN
ejpam-6243	190	10	assumptions	assumption	NOUN
ejpam-6243	190	11	(	(	PUNCT
ejpam-6243	190	12	1	1	NUM
ejpam-6243	190	13	)	)	PUNCT
ejpam-6243	190	14	,	,	PUNCT
ejpam-6243	190	15	(	(	PUNCT
ejpam-6243	190	16	2	2	NUM
ejpam-6243	190	17	)	)	PUNCT
ejpam-6243	190	18	,	,	PUNCT
ejpam-6243	190	19	and	and	CCONJ
ejpam-6243	190	20	(	(	PUNCT
ejpam-6243	190	21	4	4	NUM
ejpam-6243	190	22	)	)	PUNCT
ejpam-6243	190	23	of	of	ADP
ejpam-6243	190	24	theorem	theorem	NOUN
ejpam-6243	190	25	1	1	NUM
ejpam-6243	190	26	,	,	PUNCT
ejpam-6243	190	27	we	we	PRON
ejpam-6243	190	28	may	may	AUX
ejpam-6243	190	29	determine	determine	VERB
ejpam-6243	190	30	that	that	PRON
ejpam-6243	190	31	x	x	NOUN
ejpam-6243	190	32	=	=	SYM
ejpam-6243	190	33	0	0	NUM
ejpam-6243	190	34	is	be	AUX
ejpam-6243	190	35	the	the	DET
ejpam-6243	190	36	fixed	fix	VERB
ejpam-6243	190	37	point	point	NOUN
ejpam-6243	190	38	of	of	ADP
ejpam-6243	190	39	t	t	NOUN
ejpam-6243	190	40	by	by	ADP
ejpam-6243	190	41	letting	let	VERB
ejpam-6243	190	42	β1	β1	PROPN
ejpam-6243	190	43	=	=	SYM
ejpam-6243	190	44	β2	β2	NOUN
ejpam-6243	190	45	=	=	NOUN
ejpam-6243	190	46	1	1	NUM
ejpam-6243	190	47	,	,	PUNCT
ejpam-6243	190	48	and	and	CCONJ
ejpam-6243	190	49	s	s	AUX
ejpam-6243	190	50	=	=	SYM
ejpam-6243	190	51	2	2	X
ejpam-6243	190	52	.	.	PUNCT
ejpam-6243	190	53	theorem	theorem	NOUN
ejpam-6243	190	54	2	2	NUM
ejpam-6243	190	55	.	.	PUNCT
ejpam-6243	191	1	let	let	VERB
ejpam-6243	191	2	(	(	PUNCT
ejpam-6243	191	3	x	x	NOUN
ejpam-6243	191	4	,	,	PUNCT
ejpam-6243	191	5	d	d	X
ejpam-6243	191	6	)	)	PUNCT
ejpam-6243	191	7	be	be	AUX
ejpam-6243	191	8	complete	complete	ADJ
ejpam-6243	191	9	metric	metric	ADJ
ejpam-6243	191	10	space	space	NOUN
ejpam-6243	191	11	and	and	CCONJ
ejpam-6243	191	12	let	let	VERB
ejpam-6243	191	13	t	t	NOUN
ejpam-6243	191	14	:	:	PUNCT
ejpam-6243	191	15	x	x	X
ejpam-6243	191	16	→	→	PUNCT
ejpam-6243	191	17	x	x	X
ejpam-6243	191	18	be	be	AUX
ejpam-6243	191	19	(	(	PUNCT
ejpam-6243	191	20	β	β	X
ejpam-6243	191	21	,	,	PUNCT
ejpam-6243	191	22	ϕ)−	ϕ)−	PROPN
ejpam-6243	191	23	admissible	admissible	ADJ
ejpam-6243	191	24	hybrid	hybrid	ADJ
ejpam-6243	191	25	contraction	contraction	NOUN
ejpam-6243	191	26	satisfying	satisfy	VERB
ejpam-6243	191	27	the	the	DET
ejpam-6243	191	28	followings	following	NOUN
ejpam-6243	191	29	;	;	PUNCT
ejpam-6243	191	30	(	(	PUNCT
ejpam-6243	191	31	i	i	NOUN
ejpam-6243	191	32	)	)	PUNCT
ejpam-6243	191	33	t	t	PROPN
ejpam-6243	191	34	is	be	AUX
ejpam-6243	191	35	triangular	triangular	ADJ
ejpam-6243	191	36	β−	β−	PUNCT
ejpam-6243	191	37	orbital	orbital	ADJ
ejpam-6243	191	38	admissible	admissible	NOUN
ejpam-6243	191	39	;	;	PUNCT
ejpam-6243	191	40	(	(	PUNCT
ejpam-6243	191	41	ii	ii	NOUN
ejpam-6243	191	42	)	)	PUNCT
ejpam-6243	191	43	there	there	PRON
ejpam-6243	191	44	exists	exist	VERB
ejpam-6243	191	45	x0	x0	PROPN
ejpam-6243	191	46	∈	∈	PROPN
ejpam-6243	192	1	x	x	X
ejpam-6243	192	2	s.t	s.t	PROPN
ejpam-6243	192	3	.	.	PROPN
ejpam-6243	192	4	β(x0	β(x0	PROPN
ejpam-6243	192	5	,	,	PUNCT
ejpam-6243	192	6	tx0	tx0	NOUN
ejpam-6243	192	7	)	)	PUNCT
ejpam-6243	192	8	≤	≤	NOUN
ejpam-6243	192	9	1	1	NUM
ejpam-6243	192	10	;	;	PUNCT
ejpam-6243	192	11	r	r	NOUN
ejpam-6243	192	12	ramaswamy	ramaswamy	NOUN
ejpam-6243	192	13	et	et	PROPN
ejpam-6243	192	14	al	al	PROPN
ejpam-6243	192	15	.	.	PUNCT
ejpam-6243	192	16	/	/	SYM
ejpam-6243	192	17	eur	eur	PROPN
ejpam-6243	192	18	.	.	PUNCT
ejpam-6243	193	1	j.	j.	PROPN
ejpam-6243	193	2	pure	pure	PROPN
ejpam-6243	193	3	appl	appl	PROPN
ejpam-6243	193	4	.	.	PROPN
ejpam-6243	193	5	math	math	PROPN
ejpam-6243	193	6	,	,	PUNCT
ejpam-6243	193	7	18	18	NUM
ejpam-6243	193	8	(	(	PUNCT
ejpam-6243	193	9	3	3	NUM
ejpam-6243	193	10	)	)	PUNCT
ejpam-6243	193	11	(	(	PUNCT
ejpam-6243	193	12	2025	2025	NUM
ejpam-6243	193	13	)	)	PUNCT
ejpam-6243	193	14	,	,	PUNCT
ejpam-6243	193	15	6243	6243	NUM
ejpam-6243	193	16	9	9	NUM
ejpam-6243	193	17	of	of	ADP
ejpam-6243	193	18	14	14	NUM
ejpam-6243	193	19	(	(	PUNCT
ejpam-6243	193	20	iii	iii	NOUN
ejpam-6243	193	21	)	)	PUNCT
ejpam-6243	193	22	(	(	PUNCT
ejpam-6243	193	23	x	x	X
ejpam-6243	193	24	,	,	PUNCT
ejpam-6243	193	25	d	d	NOUN
ejpam-6243	193	26	)	)	PUNCT
ejpam-6243	193	27	is	be	AUX
ejpam-6243	193	28	regular	regular	ADJ
ejpam-6243	193	29	with	with	ADP
ejpam-6243	193	30	respect	respect	NOUN
ejpam-6243	193	31	β	β	NOUN
ejpam-6243	193	32	.	.	PUNCT
ejpam-6243	194	1	then	then	ADV
ejpam-6243	194	2	t	t	PROPN
ejpam-6243	194	3	possesses	possess	VERB
ejpam-6243	194	4	a	a	DET
ejpam-6243	194	5	fixed	fix	VERB
ejpam-6243	194	6	point	point	NOUN
ejpam-6243	194	7	.	.	PUNCT
ejpam-6243	195	1	proof	proof	NOUN
ejpam-6243	195	2	.	.	PUNCT
ejpam-6243	196	1	as	as	SCONJ
ejpam-6243	196	2	we	we	PRON
ejpam-6243	196	3	can	can	AUX
ejpam-6243	196	4	see	see	VERB
ejpam-6243	196	5	from	from	ADP
ejpam-6243	196	6	the	the	DET
ejpam-6243	196	7	lines	line	NOUN
ejpam-6243	196	8	in	in	ADP
ejpam-6243	196	9	the	the	DET
ejpam-6243	196	10	proof	proof	NOUN
ejpam-6243	196	11	of	of	ADP
ejpam-6243	196	12	theorem	theorem	NOUN
ejpam-6243	196	13	1	1	NUM
ejpam-6243	196	14	,	,	PUNCT
ejpam-6243	196	15	the	the	DET
ejpam-6243	196	16	sequence	sequence	NOUN
ejpam-6243	196	17	{	{	PUNCT
ejpam-6243	196	18	xn	xn	PUNCT
ejpam-6243	196	19	}	}	PUNCT
ejpam-6243	196	20	is	be	AUX
ejpam-6243	196	21	cauchy	cauchy	ADJ
ejpam-6243	196	22	for	for	ADP
ejpam-6243	196	23	any	any	DET
ejpam-6243	196	24	s	s	X
ejpam-6243	196	25	>	>	X
ejpam-6243	196	26	0	0	NUM
ejpam-6243	196	27	,	,	PUNCT
ejpam-6243	196	28	and	and	CCONJ
ejpam-6243	196	29	there	there	PRON
ejpam-6243	196	30	exists	exist	VERB
ejpam-6243	196	31	a	a	DET
ejpam-6243	196	32	point	point	NOUN
ejpam-6243	196	33	z	z	NOUN
ejpam-6243	196	34	such	such	ADJ
ejpam-6243	196	35	that	that	SCONJ
ejpam-6243	196	36	limn→+∞d(xn	limn→+∞d(xn	NOUN
ejpam-6243	196	37	,	,	PUNCT
ejpam-6243	196	38	z	z	NOUN
ejpam-6243	196	39	)	)	PUNCT
ejpam-6243	196	40	=	=	SYM
ejpam-6243	196	41	0	0	PUNCT
ejpam-6243	196	42	because	because	SCONJ
ejpam-6243	196	43	the	the	DET
ejpam-6243	196	44	metric	metric	ADJ
ejpam-6243	196	45	space	space	NOUN
ejpam-6243	196	46	(	(	PUNCT
ejpam-6243	196	47	x	x	X
ejpam-6243	196	48	,	,	PUNCT
ejpam-6243	196	49	d	d	NOUN
ejpam-6243	196	50	)	)	PUNCT
ejpam-6243	196	51	is	be	AUX
ejpam-6243	196	52	complete	complete	ADJ
ejpam-6243	196	53	.	.	PUNCT
ejpam-6243	197	1	given	give	VERB
ejpam-6243	197	2	that	that	SCONJ
ejpam-6243	197	3	the	the	DET
ejpam-6243	197	4	space	space	NOUN
ejpam-6243	197	5	x	x	PUNCT
ejpam-6243	197	6	is	be	AUX
ejpam-6243	197	7	regular	regular	ADJ
ejpam-6243	197	8	with	with	ADP
ejpam-6243	197	9	regard	regard	NOUN
ejpam-6243	197	10	to	to	ADP
ejpam-6243	197	11	β	β	PRON
ejpam-6243	197	12	,	,	PUNCT
ejpam-6243	197	13	inequality	inequality	NOUN
ejpam-6243	197	14	(	(	PUNCT
ejpam-6243	197	15	5	5	NUM
ejpam-6243	197	16	)	)	PUNCT
ejpam-6243	197	17	together	together	ADV
ejpam-6243	197	18	with	with	ADP
ejpam-6243	197	19	the	the	DET
ejpam-6243	197	20	triangular	triangular	NOUN
ejpam-6243	197	21	inequality	inequality	NOUN
ejpam-6243	197	22	gives	give	VERB
ejpam-6243	197	23	d(z	d(z	PROPN
ejpam-6243	197	24	,	,	PUNCT
ejpam-6243	197	25	tz	tz	NOUN
ejpam-6243	197	26	)	)	PUNCT
ejpam-6243	197	27	≤	≤	NOUN
ejpam-6243	197	28	(	(	PUNCT
ejpam-6243	197	29	d(z	d(z	PROPN
ejpam-6243	197	30	,	,	PUNCT
ejpam-6243	197	31	xn+1	xn+1	NUM
ejpam-6243	197	32	)	)	PUNCT
ejpam-6243	197	33	)	)	PUNCT
ejpam-6243	198	1	+	+	CCONJ
ejpam-6243	198	2	(	(	PUNCT
ejpam-6243	198	3	d(xn+1	d(xn+1	PROPN
ejpam-6243	198	4	,	,	PUNCT
ejpam-6243	198	5	t	t	PROPN
ejpam-6243	198	6	z	z	PROPN
ejpam-6243	198	7	)	)	PUNCT
ejpam-6243	198	8	)	)	PUNCT
ejpam-6243	198	9	≤	≤	NOUN
ejpam-6243	198	10	β(xn	β(xn	NOUN
ejpam-6243	198	11	,	,	PUNCT
ejpam-6243	198	12	z	z	NOUN
ejpam-6243	198	13	)	)	PUNCT
ejpam-6243	199	1	+	+	CCONJ
ejpam-6243	199	2	d(txn	d(txn	PROPN
ejpam-6243	199	3	,	,	PUNCT
ejpam-6243	199	4	t	t	PROPN
ejpam-6243	199	5	z	z	PROPN
ejpam-6243	199	6	)	)	PUNCT
ejpam-6243	200	1	≤	≤	NUM
ejpam-6243	200	2	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	200	3	s	s	PART
ejpam-6243	200	4	(	(	PUNCT
ejpam-6243	200	5	xn	xn	PROPN
ejpam-6243	200	6	,	,	PUNCT
ejpam-6243	200	7	z	z	NOUN
ejpam-6243	200	8	)	)	PUNCT
ejpam-6243	200	9	)	)	PUNCT
ejpam-6243	201	1	≤	≤	PROPN
ejpam-6243	202	1	jt	jt	PROPN
ejpam-6243	202	2	s	s	PROPN
ejpam-6243	202	3	(	(	PUNCT
ejpam-6243	202	4	xn	xn	PROPN
ejpam-6243	202	5	,	,	PUNCT
ejpam-6243	202	6	z	z	NOUN
ejpam-6243	202	7	)	)	PUNCT
ejpam-6243	202	8	.	.	PUNCT
ejpam-6243	203	1	(	(	PUNCT
ejpam-6243	203	2	16	16	NUM
ejpam-6243	203	3	)	)	PUNCT
ejpam-6243	203	4	again	again	ADV
ejpam-6243	203	5	,	,	PUNCT
ejpam-6243	203	6	we	we	PRON
ejpam-6243	203	7	have	have	VERB
ejpam-6243	203	8	to	to	PART
ejpam-6243	203	9	consider	consider	VERB
ejpam-6243	203	10	two	two	NUM
ejpam-6243	203	11	separate	separate	ADJ
ejpam-6243	203	12	cases	case	NOUN
ejpam-6243	203	13	.	.	PUNCT
ejpam-6243	204	1	for	for	ADP
ejpam-6243	204	2	the	the	DET
ejpam-6243	204	3	case	case	NOUN
ejpam-6243	204	4	s	s	VERB
ejpam-6243	204	5	>	>	X
ejpam-6243	204	6	0	0	PROPN
ejpam-6243	204	7	,	,	PUNCT
ejpam-6243	204	8	jt	jt	PROPN
ejpam-6243	204	9	s	s	PART
ejpam-6243	204	10	(	(	PUNCT
ejpam-6243	204	11	xn	xn	PROPN
ejpam-6243	204	12	,	,	PUNCT
ejpam-6243	204	13	z	z	NOUN
ejpam-6243	204	14	)	)	PUNCT
ejpam-6243	204	15	=	=	PUNCT
ejpam-6243	205	1	[	[	X
ejpam-6243	205	2	α1	α1	PROPN
ejpam-6243	205	3	(	(	PUNCT
ejpam-6243	205	4	d(xn	d(xn	PROPN
ejpam-6243	205	5	,	,	PUNCT
ejpam-6243	205	6	txn)d(z	txn)d(z	PROPN
ejpam-6243	205	7	,	,	PUNCT
ejpam-6243	205	8	tz	tz	PROPN
ejpam-6243	205	9	)	)	PUNCT
ejpam-6243	205	10	d(txn	d(txn	PROPN
ejpam-6243	205	11	,	,	PUNCT
ejpam-6243	205	12	z	z	NOUN
ejpam-6243	205	13	)	)	PUNCT
ejpam-6243	205	14	)	)	PUNCT
ejpam-6243	205	15	s	s	PART
ejpam-6243	206	1	+	+	NUM
ejpam-6243	206	2	α2(d(z	α2(d(z	NUM
ejpam-6243	206	3	,	,	PUNCT
ejpam-6243	206	4	tz	tz	PROPN
ejpam-6243	206	5	)	)	PUNCT
ejpam-6243	206	6	s	s	PART
ejpam-6243	206	7	]	]	PUNCT
ejpam-6243	206	8	1	1	NUM
ejpam-6243	206	9	s	s	NOUN
ejpam-6243	206	10	=	=	PUNCT
ejpam-6243	207	1	[	[	X
ejpam-6243	207	2	α1	α1	PROPN
ejpam-6243	207	3	(	(	PUNCT
ejpam-6243	207	4	d(xn	d(xn	PROPN
ejpam-6243	207	5	,	,	PUNCT
ejpam-6243	207	6	xn+1)d(z	xn+1)d(z	PROPN
ejpam-6243	207	7	,	,	PUNCT
ejpam-6243	207	8	tz	tz	PROPN
ejpam-6243	207	9	)	)	PUNCT
ejpam-6243	207	10	d(xn	d(xn	PROPN
ejpam-6243	207	11	,	,	PUNCT
ejpam-6243	207	12	z	z	NOUN
ejpam-6243	207	13	)	)	PUNCT
ejpam-6243	207	14	)	)	PUNCT
ejpam-6243	207	15	s	s	PART
ejpam-6243	208	1	+	+	X
ejpam-6243	208	2	α2(d(xn	α2(d(xn	NUM
ejpam-6243	208	3	,	,	PUNCT
ejpam-6243	208	4	z	z	NOUN
ejpam-6243	208	5	)	)	PUNCT
ejpam-6243	208	6	s	s	PART
ejpam-6243	208	7	]	]	PUNCT
ejpam-6243	208	8	1	1	NUM
ejpam-6243	208	9	s	s	NOUN
ejpam-6243	208	10	.	.	PUNCT
ejpam-6243	209	1	since	since	SCONJ
ejpam-6243	209	2	limn→+∞jt	limn→+∞jt	PROPN
ejpam-6243	209	3	s	s	PART
ejpam-6243	209	4	(	(	PUNCT
ejpam-6243	209	5	xn	xn	PROPN
ejpam-6243	209	6	,	,	PUNCT
ejpam-6243	209	7	z	z	NOUN
ejpam-6243	209	8	)	)	PUNCT
ejpam-6243	209	9	=	=	SYM
ejpam-6243	209	10	(	(	PUNCT
ejpam-6243	209	11	α1(d(z	α1(d(z	PROPN
ejpam-6243	209	12	,	,	PUNCT
ejpam-6243	209	13	tz	tz	NOUN
ejpam-6243	209	14	)	)	PUNCT
ejpam-6243	209	15	)	)	PUNCT
ejpam-6243	209	16	)	)	PUNCT
ejpam-6243	209	17	,	,	PUNCT
ejpam-6243	209	18	consider	consider	VERB
ejpam-6243	209	19	n	n	PRON
ejpam-6243	209	20	→	→	PUNCT
ejpam-6243	209	21	+	+	NOUN
ejpam-6243	209	22	∞	∞	PROPN
ejpam-6243	209	23	in	in	ADP
ejpam-6243	209	24	(	(	PUNCT
ejpam-6243	209	25	16	16	NUM
ejpam-6243	209	26	)	)	PUNCT
ejpam-6243	209	27	we	we	PRON
ejpam-6243	209	28	obtain	obtain	VERB
ejpam-6243	209	29	d(z	d(z	PROPN
ejpam-6243	209	30	,	,	PUNCT
ejpam-6243	209	31	tz	tz	NOUN
ejpam-6243	209	32	)	)	PUNCT
ejpam-6243	209	33	≤	≤	NOUN
ejpam-6243	209	34	d(z	d(z	PROPN
ejpam-6243	209	35	,	,	PUNCT
ejpam-6243	209	36	tz	tz	PROPN
ejpam-6243	209	37	)	)	PUNCT
ejpam-6243	209	38	.	.	PUNCT
ejpam-6243	210	1	which	which	PRON
ejpam-6243	210	2	implies	imply	VERB
ejpam-6243	210	3	that	that	SCONJ
ejpam-6243	210	4	tz	tz	PROPN
ejpam-6243	210	5	=	=	SYM
ejpam-6243	210	6	z	z	NOUN
ejpam-6243	210	7	similarly	similarly	ADV
ejpam-6243	210	8	,	,	PUNCT
ejpam-6243	210	9	for	for	ADP
ejpam-6243	210	10	the	the	DET
ejpam-6243	210	11	case	case	NOUN
ejpam-6243	210	12	s	s	PART
ejpam-6243	210	13	=	=	SYM
ejpam-6243	210	14	0	0	NUM
ejpam-6243	210	15	,	,	PUNCT
ejpam-6243	210	16	we	we	PRON
ejpam-6243	210	17	get	get	VERB
ejpam-6243	210	18	limn→+∞jt	limn→+∞jt	NUM
ejpam-6243	210	19	s	s	PART
ejpam-6243	210	20	(	(	PUNCT
ejpam-6243	210	21	xn	xn	PROPN
ejpam-6243	210	22	,	,	PUNCT
ejpam-6243	210	23	z	z	NOUN
ejpam-6243	210	24	)	)	PUNCT
ejpam-6243	211	1	=	=	SYM
ejpam-6243	211	2	0	0	NUM
ejpam-6243	211	3	then	then	ADV
ejpam-6243	211	4	d(z	d(z	PROPN
ejpam-6243	211	5	,	,	PUNCT
ejpam-6243	211	6	tz	tz	NOUN
ejpam-6243	211	7	)	)	PUNCT
ejpam-6243	211	8	=	=	SYM
ejpam-6243	211	9	0	0	X
ejpam-6243	211	10	.	.	PUNCT
ejpam-6243	211	11	theorem	theorem	NOUN
ejpam-6243	211	12	3	3	NUM
ejpam-6243	211	13	.	.	PUNCT
ejpam-6243	212	1	if	if	SCONJ
ejpam-6243	212	2	in	in	ADP
ejpam-6243	212	3	theorem	theorem	NOUN
ejpam-6243	212	4	1	1	NUM
ejpam-6243	212	5	and	and	CCONJ
ejpam-6243	212	6	2	2	NUM
ejpam-6243	212	7	,	,	PUNCT
ejpam-6243	212	8	in	in	ADP
ejpam-6243	212	9	the	the	DET
ejpam-6243	212	10	case	case	NOUN
ejpam-6243	212	11	s	s	VERB
ejpam-6243	212	12	>	>	X
ejpam-6243	212	13	0	0	NUM
ejpam-6243	212	14	,	,	PUNCT
ejpam-6243	212	15	we	we	PRON
ejpam-6243	212	16	assume	assume	VERB
ejpam-6243	212	17	supplementary	supplementary	ADJ
ejpam-6243	212	18	that	that	SCONJ
ejpam-6243	212	19	β(x	β(x	NOUN
ejpam-6243	212	20	,	,	PUNCT
ejpam-6243	212	21	y	y	PROPN
ejpam-6243	212	22	)	)	PUNCT
ejpam-6243	212	23	≥	≥	NOUN
ejpam-6243	212	24	1	1	NUM
ejpam-6243	212	25	for	for	ADP
ejpam-6243	212	26	any	any	DET
ejpam-6243	212	27	x	x	NOUN
ejpam-6243	212	28	,	,	PUNCT
ejpam-6243	212	29	y	y	PROPN
ejpam-6243	212	30	∈	∈	PROPN
ejpam-6243	212	31	ft	ft	X
ejpam-6243	212	32	(	(	PUNCT
ejpam-6243	212	33	x	x	X
ejpam-6243	212	34	)	)	PUNCT
ejpam-6243	212	35	then	then	ADV
ejpam-6243	212	36	the	the	DET
ejpam-6243	212	37	fixed	fixed	ADJ
ejpam-6243	212	38	point	point	NOUN
ejpam-6243	212	39	of	of	ADP
ejpam-6243	212	40	t	t	PROPN
ejpam-6243	212	41	is	be	AUX
ejpam-6243	212	42	unique	unique	ADJ
ejpam-6243	212	43	.	.	PUNCT
ejpam-6243	213	1	proof	proof	NOUN
ejpam-6243	213	2	.	.	PUNCT
ejpam-6243	214	1	let	let	VERB
ejpam-6243	214	2	v	v	X
ejpam-6243	214	3	∈	∈	PROPN
ejpam-6243	214	4	x	x	PUNCT
ejpam-6243	214	5	be	be	AUX
ejpam-6243	214	6	a	a	DET
ejpam-6243	214	7	different	different	ADJ
ejpam-6243	214	8	fixed	fix	VERB
ejpam-6243	214	9	point	point	NOUN
ejpam-6243	214	10	of	of	ADP
ejpam-6243	214	11	t	t	PROPN
ejpam-6243	214	12	from	from	ADP
ejpam-6243	214	13	z.	z.	PROPN
ejpam-6243	214	14	taking	take	VERB
ejpam-6243	214	15	into	into	ADP
ejpam-6243	214	16	consideration	consideration	NOUN
ejpam-6243	214	17	the	the	DET
ejpam-6243	214	18	extra	extra	ADJ
ejpam-6243	214	19	hypotheses	hypothesis	NOUN
ejpam-6243	214	20	and	and	CCONJ
ejpam-6243	214	21	substituting	substitute	VERB
ejpam-6243	214	22	in	in	ADP
ejpam-6243	214	23	(	(	PUNCT
ejpam-6243	214	24	5	5	NUM
ejpam-6243	214	25	)	)	PUNCT
ejpam-6243	214	26	,	,	PUNCT
ejpam-6243	214	27	we	we	PRON
ejpam-6243	214	28	have	have	VERB
ejpam-6243	214	29	d(z	d(z	NOUN
ejpam-6243	214	30	,	,	PUNCT
ejpam-6243	214	31	v	v	NOUN
ejpam-6243	214	32	)	)	PUNCT
ejpam-6243	214	33	≤	≤	NOUN
ejpam-6243	214	34	β(z	β(z	PROPN
ejpam-6243	214	35	,	,	PUNCT
ejpam-6243	214	36	v)(tz	v)(tz	PROPN
ejpam-6243	214	37	,	,	PUNCT
ejpam-6243	214	38	tv	tv	NOUN
ejpam-6243	214	39	)	)	PUNCT
ejpam-6243	214	40	≤	≤	NOUN
ejpam-6243	214	41	β(jt	β(jt	PROPN
ejpam-6243	214	42	s	s	PART
ejpam-6243	214	43	(	(	PUNCT
ejpam-6243	214	44	z	z	NOUN
ejpam-6243	214	45	,	,	PUNCT
ejpam-6243	214	46	v	v	NOUN
ejpam-6243	214	47	)	)	PUNCT
ejpam-6243	214	48	)	)	PUNCT
ejpam-6243	215	1	<	<	X
ejpam-6243	215	2	jt	jt	PROPN
ejpam-6243	215	3	s	s	PROPN
ejpam-6243	215	4	(	(	PUNCT
ejpam-6243	215	5	z	z	NOUN
ejpam-6243	215	6	,	,	PUNCT
ejpam-6243	215	7	v	v	NOUN
ejpam-6243	215	8	)	)	PUNCT
ejpam-6243	215	9	=	=	PUNCT
ejpam-6243	216	1	[	[	X
ejpam-6243	216	2	α1	α1	PROPN
ejpam-6243	216	3	(	(	PUNCT
ejpam-6243	216	4	d(z	d(z	NOUN
ejpam-6243	216	5	,	,	PUNCT
ejpam-6243	216	6	tz)d(v	tz)d(v	NOUN
ejpam-6243	216	7	,	,	PUNCT
ejpam-6243	216	8	tv	tv	NOUN
ejpam-6243	216	9	)	)	PUNCT
ejpam-6243	216	10	d(z	d(z	PROPN
ejpam-6243	216	11	,	,	PUNCT
ejpam-6243	216	12	v	v	NOUN
ejpam-6243	216	13	)	)	PUNCT
ejpam-6243	216	14	)	)	PUNCT
ejpam-6243	216	15	s	s	PART
ejpam-6243	217	1	+	+	NUM
ejpam-6243	217	2	α2(d(z	α2(d(z	NUM
ejpam-6243	217	3	,	,	PUNCT
ejpam-6243	217	4	v	v	NOUN
ejpam-6243	217	5	)	)	PUNCT
ejpam-6243	217	6	)	)	PUNCT
ejpam-6243	218	1	s	s	X
ejpam-6243	218	2	]	]	X
ejpam-6243	218	3	1	1	NUM
ejpam-6243	218	4	s	s	NOUN
ejpam-6243	218	5	=	=	PUNCT
ejpam-6243	219	1	[	[	X
ejpam-6243	219	2	α1	α1	PROPN
ejpam-6243	219	3	(	(	PUNCT
ejpam-6243	219	4	d(z	d(z	PROPN
ejpam-6243	219	5	,	,	PUNCT
ejpam-6243	219	6	z)d(v	z)d(v	NOUN
ejpam-6243	219	7	,	,	PUNCT
ejpam-6243	219	8	v	v	NOUN
ejpam-6243	219	9	)	)	PUNCT
ejpam-6243	219	10	d(z	d(z	PROPN
ejpam-6243	219	11	,	,	PUNCT
ejpam-6243	219	12	v	v	NOUN
ejpam-6243	219	13	)	)	PUNCT
ejpam-6243	219	14	)	)	PUNCT
ejpam-6243	219	15	s	s	PART
ejpam-6243	220	1	+	+	NUM
ejpam-6243	220	2	α2(d(z	α2(d(z	NUM
ejpam-6243	220	3	,	,	PUNCT
ejpam-6243	220	4	v	v	NOUN
ejpam-6243	220	5	)	)	PUNCT
ejpam-6243	220	6	s	s	PART
ejpam-6243	220	7	]	]	PUNCT
ejpam-6243	220	8	1	1	NUM
ejpam-6243	220	9	s	s	NOUN
ejpam-6243	220	10	=	=	NOUN
ejpam-6243	220	11	α	α	PROPN
ejpam-6243	220	12	1	1	NUM
ejpam-6243	220	13	s	s	NUM
ejpam-6243	220	14	2	2	NUM
ejpam-6243	220	15	d(z	d(z	PROPN
ejpam-6243	220	16	,	,	PUNCT
ejpam-6243	220	17	v	v	NOUN
ejpam-6243	220	18	)	)	PUNCT
ejpam-6243	220	19	≤	≤	NOUN
ejpam-6243	220	20	d(z	d(z	PROPN
ejpam-6243	220	21	,	,	PUNCT
ejpam-6243	220	22	v	v	NOUN
ejpam-6243	220	23	)	)	PUNCT
ejpam-6243	220	24	,	,	PUNCT
ejpam-6243	220	25	which	which	PRON
ejpam-6243	220	26	is	be	AUX
ejpam-6243	220	27	a	a	DET
ejpam-6243	220	28	contradiction	contradiction	NOUN
ejpam-6243	220	29	.	.	PUNCT
ejpam-6243	221	1	this	this	PRON
ejpam-6243	221	2	implies	imply	VERB
ejpam-6243	221	3	that	that	SCONJ
ejpam-6243	221	4	t	t	PROPN
ejpam-6243	221	5	has	have	VERB
ejpam-6243	221	6	exactly	exactly	ADV
ejpam-6243	221	7	one	one	NUM
ejpam-6243	221	8	fixed	fix	VERB
ejpam-6243	221	9	point	point	NOUN
ejpam-6243	221	10	.	.	PUNCT
ejpam-6243	222	1	r	r	NOUN
ejpam-6243	222	2	ramaswamy	ramaswamy	PROPN
ejpam-6243	222	3	et	et	PROPN
ejpam-6243	222	4	al	al	PROPN
ejpam-6243	222	5	.	.	PUNCT
ejpam-6243	222	6	/	/	SYM
ejpam-6243	222	7	eur	eur	PROPN
ejpam-6243	222	8	.	.	PUNCT
ejpam-6243	223	1	j.	j.	PROPN
ejpam-6243	223	2	pure	pure	PROPN
ejpam-6243	223	3	appl	appl	PROPN
ejpam-6243	223	4	.	.	PROPN
ejpam-6243	223	5	math	math	PROPN
ejpam-6243	223	6	,	,	PUNCT
ejpam-6243	223	7	18	18	NUM
ejpam-6243	223	8	(	(	PUNCT
ejpam-6243	223	9	3	3	NUM
ejpam-6243	223	10	)	)	PUNCT
ejpam-6243	223	11	(	(	PUNCT
ejpam-6243	223	12	2025	2025	NUM
ejpam-6243	223	13	)	)	PUNCT
ejpam-6243	223	14	,	,	PUNCT
ejpam-6243	223	15	6243	6243	NUM
ejpam-6243	223	16	10	10	NUM
ejpam-6243	223	17	of	of	ADP
ejpam-6243	223	18	14	14	NUM
ejpam-6243	223	19	example	example	NOUN
ejpam-6243	224	1	2	2	NUM
ejpam-6243	224	2	.	.	PUNCT
ejpam-6243	224	3	let	let	VERB
ejpam-6243	224	4	x	x	PUNCT
ejpam-6243	224	5	=	=	PRON
ejpam-6243	224	6	{	{	PUNCT
ejpam-6243	224	7	a	a	PRON
ejpam-6243	224	8	,	,	PUNCT
ejpam-6243	224	9	b	b	NOUN
ejpam-6243	224	10	,	,	PUNCT
ejpam-6243	224	11	c	c	X
ejpam-6243	224	12	,	,	PUNCT
ejpam-6243	224	13	e	e	NOUN
ejpam-6243	224	14	}	}	PUNCT
ejpam-6243	224	15	and	and	CCONJ
ejpam-6243	224	16	d	d	NOUN
ejpam-6243	224	17	:	:	PUNCT
ejpam-6243	224	18	x×x	x×x	PROPN
ejpam-6243	224	19	→	→	PUNCT
ejpam-6243	225	1	[	[	X
ejpam-6243	225	2	0,+∞	0,+∞	NUM
ejpam-6243	225	3	)	)	PUNCT
ejpam-6243	225	4	such	such	ADJ
ejpam-6243	225	5	that	that	SCONJ
ejpam-6243	225	6	d(y	d(y	NOUN
ejpam-6243	225	7	,	,	PUNCT
ejpam-6243	225	8	x	x	NOUN
ejpam-6243	225	9	)	)	PUNCT
ejpam-6243	225	10	=	=	SYM
ejpam-6243	225	11	d(y	d(y	NOUN
ejpam-6243	225	12	,	,	PUNCT
ejpam-6243	225	13	x	x	NOUN
ejpam-6243	225	14	)	)	PUNCT
ejpam-6243	225	15	,	,	PUNCT
ejpam-6243	225	16	d(x	d(x	PROPN
ejpam-6243	225	17	,	,	PUNCT
ejpam-6243	225	18	x	x	NOUN
ejpam-6243	225	19	)	)	PUNCT
ejpam-6243	225	20	=	=	SYM
ejpam-6243	225	21	0	0	NUM
ejpam-6243	225	22	for	for	ADP
ejpam-6243	225	23	any	any	DET
ejpam-6243	225	24	x	x	NOUN
ejpam-6243	225	25	,	,	PUNCT
ejpam-6243	225	26	y	y	PROPN
ejpam-6243	225	27	∈	∈	PROPN
ejpam-6243	225	28	x	x	X
ejpam-6243	225	29	and	and	CCONJ
ejpam-6243	225	30	d(x	d(x	PROPN
ejpam-6243	225	31	,	,	PUNCT
ejpam-6243	225	32	y	y	NOUN
ejpam-6243	225	33	)	)	PUNCT
ejpam-6243	226	1	=	=	SYM
ejpam-6243	226	2			NOUN
ejpam-6243	226	3	1	1	NUM
ejpam-6243	226	4	,	,	PUNCT
ejpam-6243	226	5	if	if	SCONJ
ejpam-6243	226	6	(	(	PUNCT
ejpam-6243	226	7	x	x	NOUN
ejpam-6243	226	8	,	,	PUNCT
ejpam-6243	226	9	y	y	NOUN
ejpam-6243	226	10	)	)	PUNCT
ejpam-6243	226	11	∈	∈	PROPN
ejpam-6243	226	12	(	(	PUNCT
ejpam-6243	226	13	a	a	DET
ejpam-6243	226	14	,	,	PUNCT
ejpam-6243	226	15	b	b	NOUN
ejpam-6243	226	16	)	)	PUNCT
ejpam-6243	226	17	,	,	PUNCT
ejpam-6243	226	18	(	(	PUNCT
ejpam-6243	226	19	b	b	X
ejpam-6243	226	20	,	,	PUNCT
ejpam-6243	226	21	c	c	NOUN
ejpam-6243	226	22	)	)	PUNCT
ejpam-6243	226	23	,	,	PUNCT
ejpam-6243	226	24	(	(	PUNCT
ejpam-6243	226	25	c	c	X
ejpam-6243	226	26	,	,	PUNCT
ejpam-6243	226	27	e	e	NOUN
ejpam-6243	226	28	)	)	PUNCT
ejpam-6243	226	29	2	2	NUM
ejpam-6243	226	30	,	,	PUNCT
ejpam-6243	226	31	if	if	SCONJ
ejpam-6243	226	32	(	(	PUNCT
ejpam-6243	226	33	x	x	NOUN
ejpam-6243	226	34	,	,	PUNCT
ejpam-6243	226	35	y	y	NOUN
ejpam-6243	226	36	)	)	PUNCT
ejpam-6243	226	37	∈	∈	PROPN
ejpam-6243	226	38	(	(	PUNCT
ejpam-6243	226	39	a	a	DET
ejpam-6243	226	40	,	,	PUNCT
ejpam-6243	226	41	c	c	NOUN
ejpam-6243	226	42	)	)	PUNCT
ejpam-6243	226	43	,	,	PUNCT
ejpam-6243	226	44	(	(	PUNCT
ejpam-6243	226	45	b	b	X
ejpam-6243	226	46	,	,	PUNCT
ejpam-6243	226	47	e	e	NOUN
ejpam-6243	226	48	)	)	PUNCT
ejpam-6243	226	49	3	3	NUM
ejpam-6243	226	50	,	,	PUNCT
ejpam-6243	226	51	if	if	SCONJ
ejpam-6243	226	52	(	(	PUNCT
ejpam-6243	226	53	x	x	NOUN
ejpam-6243	226	54	,	,	PUNCT
ejpam-6243	226	55	y	y	NOUN
ejpam-6243	226	56	)	)	PUNCT
ejpam-6243	226	57	∈	∈	PROPN
ejpam-6243	226	58	(	(	PUNCT
ejpam-6243	226	59	a	a	DET
ejpam-6243	226	60	,	,	PUNCT
ejpam-6243	226	61	e	e	NOUN
ejpam-6243	226	62	)	)	PUNCT
ejpam-6243	226	63	let	let	VERB
ejpam-6243	226	64	’s	’s	PRON
ejpam-6243	226	65	define	define	VERB
ejpam-6243	226	66	the	the	DET
ejpam-6243	226	67	self	self	NOUN
ejpam-6243	226	68	-	-	PUNCT
ejpam-6243	226	69	mapping	mapping	NOUN
ejpam-6243	226	70	t	t	NOUN
ejpam-6243	226	71	on	on	ADP
ejpam-6243	226	72	metric	metric	ADJ
ejpam-6243	226	73	space	space	NOUN
ejpam-6243	226	74	(	(	PUNCT
ejpam-6243	226	75	x	x	X
ejpam-6243	226	76	,	,	PUNCT
ejpam-6243	226	77	d	d	NOUN
ejpam-6243	226	78	)	)	PUNCT
ejpam-6243	226	79	as	as	SCONJ
ejpam-6243	226	80	follows	follow	VERB
ejpam-6243	226	81	:	:	PUNCT
ejpam-6243	226	82	t	t	PROPN
ejpam-6243	226	83	(	(	PUNCT
ejpam-6243	226	84	a	a	NOUN
ejpam-6243	226	85	)	)	PUNCT
ejpam-6243	226	86	=	=	SYM
ejpam-6243	226	87	t	t	PROPN
ejpam-6243	226	88	(	(	PUNCT
ejpam-6243	226	89	b	b	NOUN
ejpam-6243	226	90	)	)	PUNCT
ejpam-6243	226	91	=	=	SYM
ejpam-6243	226	92	a	a	PROPN
ejpam-6243	226	93	,	,	PUNCT
ejpam-6243	226	94	t	t	PROPN
ejpam-6243	226	95	(	(	PUNCT
ejpam-6243	226	96	c	c	NOUN
ejpam-6243	226	97	)	)	PUNCT
ejpam-6243	226	98	=	=	SYM
ejpam-6243	226	99	e	e	X
ejpam-6243	226	100	,	,	PUNCT
ejpam-6243	226	101	t	t	PROPN
ejpam-6243	226	102	(	(	PUNCT
ejpam-6243	226	103	e	e	NOUN
ejpam-6243	226	104	)	)	PUNCT
ejpam-6243	226	105	=	=	SYM
ejpam-6243	226	106	b.	b.	PROPN
ejpam-6243	227	1	the	the	DET
ejpam-6243	227	2	function	function	NOUN
ejpam-6243	227	3	β	β	X
ejpam-6243	227	4	:	:	PUNCT
ejpam-6243	227	5	x	x	SYM
ejpam-6243	227	6	×x	×x	X
ejpam-6243	227	7	→	→	SYM
ejpam-6243	227	8	[	[	X
ejpam-6243	227	9	0,+∞	0,+∞	NUM
ejpam-6243	227	10	)	)	PUNCT
ejpam-6243	227	11	is	be	AUX
ejpam-6243	227	12	also	also	ADV
ejpam-6243	227	13	considered	consider	VERB
ejpam-6243	227	14	,	,	PUNCT
ejpam-6243	227	15	along	along	ADP
ejpam-6243	227	16	with	with	ADP
ejpam-6243	227	17	the	the	DET
ejpam-6243	227	18	comparison	comparison	NOUN
ejpam-6243	227	19	function	function	NOUN
ejpam-6243	227	20	ϕ	ϕ	NOUN
ejpam-6243	227	21	:	:	PUNCT
ejpam-6243	227	22	[	[	X
ejpam-6243	227	23	0,+∞	0,+∞	NUM
ejpam-6243	227	24	)	)	PUNCT
ejpam-6243	227	25	→	→	PUNCT
ejpam-6243	228	1	[	[	X
ejpam-6243	228	2	0,+∞	0,+∞	NUM
ejpam-6243	228	3	)	)	PUNCT
ejpam-6243	228	4	,	,	PUNCT
ejpam-6243	228	5	ϕ(t	ϕ(t	NUM
ejpam-6243	228	6	)	)	PUNCT
ejpam-6243	229	1	=	=	SYM
ejpam-6243	229	2	1√	1√	NOUN
ejpam-6243	229	3	2	2	NUM
ejpam-6243	229	4	where	where	SCONJ
ejpam-6243	229	5	β(x	β(x	NOUN
ejpam-6243	229	6	,	,	PUNCT
ejpam-6243	229	7	a	a	PRON
ejpam-6243	229	8	)	)	PUNCT
ejpam-6243	229	9	=	=	SYM
ejpam-6243	230	1	β(a	β(a	PROPN
ejpam-6243	230	2	,	,	PUNCT
ejpam-6243	230	3	x	x	NOUN
ejpam-6243	230	4	)	)	PUNCT
ejpam-6243	230	5	=	=	SYM
ejpam-6243	230	6	3	3	NUM
ejpam-6243	230	7	for	for	ADP
ejpam-6243	230	8	every	every	DET
ejpam-6243	230	9	x	x	SYM
ejpam-6243	230	10	∈	∈	PROPN
ejpam-6243	230	11	x	x	NOUN
ejpam-6243	230	12	,	,	PUNCT
ejpam-6243	230	13	β(b	β(b	ADJ
ejpam-6243	230	14	,	,	PUNCT
ejpam-6243	230	15	e	e	NOUN
ejpam-6243	230	16	)	)	PUNCT
ejpam-6243	230	17	=	=	SYM
ejpam-6243	230	18	1	1	NUM
ejpam-6243	230	19	,	,	PUNCT
ejpam-6243	230	20	and	and	CCONJ
ejpam-6243	230	21	β(x	β(x	PROPN
ejpam-6243	230	22	,	,	PUNCT
ejpam-6243	230	23	y	y	NOUN
ejpam-6243	230	24	)	)	PUNCT
ejpam-6243	230	25	=	=	SYM
ejpam-6243	230	26	0	0	NUM
ejpam-6243	230	27	in	in	ADP
ejpam-6243	230	28	all	all	DET
ejpam-6243	230	29	other	other	ADJ
ejpam-6243	230	30	cases	case	NOUN
ejpam-6243	230	31	.	.	PUNCT
ejpam-6243	231	1	the	the	DET
ejpam-6243	231	2	application	application	NOUN
ejpam-6243	231	3	of	of	ADP
ejpam-6243	231	4	theorem	theorem	NOUN
ejpam-6243	231	5	1	1	NUM
ejpam-6243	231	6	is	be	AUX
ejpam-6243	231	7	not	not	PART
ejpam-6243	231	8	possible	possible	ADJ
ejpam-6243	231	9	since	since	SCONJ
ejpam-6243	231	10	neither	neither	CCONJ
ejpam-6243	231	11	t	t	PROPN
ejpam-6243	231	12	nor	nor	CCONJ
ejpam-6243	231	13	t	t	PROPN
ejpam-6243	231	14	2	2	NUM
ejpam-6243	231	15	are	be	AUX
ejpam-6243	231	16	continuous	continuous	ADJ
ejpam-6243	231	17	.	.	PUNCT
ejpam-6243	232	1	however	however	ADV
ejpam-6243	232	2	,	,	PUNCT
ejpam-6243	232	3	the	the	DET
ejpam-6243	232	4	triangular	triangular	NOUN
ejpam-6243	232	5	β−	β−	PUNCT
ejpam-6243	232	6	orbital	orbital	ADJ
ejpam-6243	232	7	admissibility	admissibility	NOUN
ejpam-6243	232	8	of	of	ADP
ejpam-6243	232	9	t	t	PROPN
ejpam-6243	232	10	is	be	AUX
ejpam-6243	232	11	readily	readily	ADV
ejpam-6243	232	12	apparent	apparent	ADJ
ejpam-6243	232	13	,	,	PUNCT
ejpam-6243	232	14	and	and	CCONJ
ejpam-6243	232	15	the	the	DET
ejpam-6243	232	16	assumptions	assumption	NOUN
ejpam-6243	232	17	(	(	PUNCT
ejpam-6243	232	18	2	2	NUM
ejpam-6243	232	19	)	)	PUNCT
ejpam-6243	232	20	and	and	CCONJ
ejpam-6243	232	21	(	(	PUNCT
ejpam-6243	232	22	3	3	X
ejpam-6243	232	23	)	)	PUNCT
ejpam-6243	232	24	from	from	ADP
ejpam-6243	232	25	theorem	theorem	ADJ
ejpam-6243	232	26	2	2	NUM
ejpam-6243	232	27	are	be	AUX
ejpam-6243	232	28	likewise	likewise	ADV
ejpam-6243	232	29	met	meet	VERB
ejpam-6243	232	30	.	.	PUNCT
ejpam-6243	233	1	considering	consider	VERB
ejpam-6243	233	2	s	s	PART
ejpam-6243	233	3	=	=	SYM
ejpam-6243	233	4	0	0	NUM
ejpam-6243	233	5	,	,	PUNCT
ejpam-6243	233	6	α(1	α(1	PROPN
ejpam-6243	233	7	)	)	PUNCT
ejpam-6243	233	8	=	=	SYM
ejpam-6243	233	9	α(2	α(2	PROPN
ejpam-6243	233	10	)	)	PUNCT
ejpam-6243	233	11	=	=	SYM
ejpam-6243	233	12	1	1	NUM
ejpam-6243	233	13	and	and	CCONJ
ejpam-6243	233	14	taking	take	VERB
ejpam-6243	233	15	into	into	ADP
ejpam-6243	233	16	account	account	NOUN
ejpam-6243	233	17	the	the	DET
ejpam-6243	233	18	definition	definition	NOUN
ejpam-6243	233	19	of	of	ADP
ejpam-6243	233	20	function	function	NOUN
ejpam-6243	233	21	β	β	X
ejpam-6243	233	22	,	,	PUNCT
ejpam-6243	233	23	we	we	PRON
ejpam-6243	233	24	remark	remark	VERB
ejpam-6243	233	25	that	that	SCONJ
ejpam-6243	233	26	the	the	DET
ejpam-6243	233	27	only	only	ADJ
ejpam-6243	233	28	interesting	interesting	ADJ
ejpam-6243	233	29	case	case	NOUN
ejpam-6243	233	30	is	be	AUX
ejpam-6243	233	31	for	for	ADP
ejpam-6243	233	32	x	x	SYM
ejpam-6243	233	33	=	=	SYM
ejpam-6243	233	34	b	b	PROPN
ejpam-6243	233	35	and	and	CCONJ
ejpam-6243	233	36	y	y	PROPN
ejpam-6243	234	1	=	=	PROPN
ejpam-6243	234	2	e.	e.	PROPN
ejpam-6243	234	3	we	we	PRON
ejpam-6243	234	4	have	have	VERB
ejpam-6243	234	5	in	in	ADP
ejpam-6243	234	6	this	this	DET
ejpam-6243	234	7	case	case	NOUN
ejpam-6243	234	8	:	:	PUNCT
ejpam-6243	234	9	β(b	β(b	PROPN
ejpam-6243	234	10	,	,	PUNCT
ejpam-6243	234	11	e)d(b	e)d(b	PROPN
ejpam-6243	234	12	,	,	PUNCT
ejpam-6243	234	13	te	te	PROPN
ejpam-6243	234	14	)	)	PUNCT
ejpam-6243	234	15	=	=	SYM
ejpam-6243	235	1	d(a	d(a	PROPN
ejpam-6243	235	2	,	,	PUNCT
ejpam-6243	235	3	b	b	NOUN
ejpam-6243	235	4	)	)	PUNCT
ejpam-6243	235	5	=	=	SYM
ejpam-6243	235	6	1	1	NUM
ejpam-6243	235	7	<	<	NOUN
ejpam-6243	235	8	√	√	PROPN
ejpam-6243	235	9	2	2	NUM
ejpam-6243	235	10	=	=	SYM
ejpam-6243	235	11	1√	1√	NUM
ejpam-6243	235	12	2	2	NUM
ejpam-6243	235	13	(	(	PUNCT
ejpam-6243	235	14	21	21	NUM
ejpam-6243	235	15	·	·	SYM
ejpam-6243	235	16	11	11	NUM
ejpam-6243	235	17	)	)	PUNCT
ejpam-6243	235	18	=	=	SYM
ejpam-6243	235	19	1√	1√	NUM
ejpam-6243	235	20	2	2	NUM
ejpam-6243	235	21	(	(	PUNCT
ejpam-6243	235	22	d(b	d(b	PROPN
ejpam-6243	235	23	,	,	PUNCT
ejpam-6243	235	24	t	t	NOUN
ejpam-6243	235	25	b))α1(d(e	b))α1(d(e	NOUN
ejpam-6243	235	26	,	,	PUNCT
ejpam-6243	235	27	te))α2	te))α2	PROPN
ejpam-6243	235	28	=	=	SYM
ejpam-6243	235	29	ϕ((d(b	ϕ((d(b	PROPN
ejpam-6243	235	30	,	,	PUNCT
ejpam-6243	235	31	t	t	PROPN
ejpam-6243	235	32	b))α1(d(e	b))α1(d(e	NOUN
ejpam-6243	235	33	,	,	PUNCT
ejpam-6243	235	34	te))α2	te))α2	PROPN
ejpam-6243	235	35	.	.	PUNCT
ejpam-6243	236	1	3.1	3.1	NUM
ejpam-6243	236	2	.	.	PUNCT
ejpam-6243	236	3	application	application	NOUN
ejpam-6243	236	4	3.1.1	3.1.1	NUM
ejpam-6243	236	5	.	.	PUNCT
ejpam-6243	237	1	ulam	ulam	PROPN
ejpam-6243	237	2	type	type	NOUN
ejpam-6243	237	3	stability	stability	NOUN
ejpam-6243	237	4	in	in	ADP
ejpam-6243	237	5	this	this	DET
ejpam-6243	237	6	section	section	NOUN
ejpam-6243	237	7	we	we	PRON
ejpam-6243	237	8	investigate	investigate	VERB
ejpam-6243	237	9	the	the	DET
ejpam-6243	237	10	general	general	ADJ
ejpam-6243	237	11	ulam	ulam	PROPN
ejpam-6243	237	12	type	type	NOUN
ejpam-6243	237	13	stability	stability	NOUN
ejpam-6243	237	14	in	in	ADP
ejpam-6243	237	15	sense	sense	NOUN
ejpam-6243	237	16	of	of	ADP
ejpam-6243	237	17	a	a	DET
ejpam-6243	237	18	fixed	fix	VERB
ejpam-6243	237	19	point	point	NOUN
ejpam-6243	237	20	problem	problem	NOUN
ejpam-6243	237	21	.	.	PUNCT
ejpam-6243	238	1	suppose	suppose	VERB
ejpam-6243	238	2	that	that	SCONJ
ejpam-6243	238	3	t	t	NOUN
ejpam-6243	238	4	:	:	PUNCT
ejpam-6243	238	5	x	x	X
ejpam-6243	238	6	→	→	PUNCT
ejpam-6243	238	7	x	x	X
ejpam-6243	238	8	is	be	AUX
ejpam-6243	238	9	a	a	DET
ejpam-6243	238	10	self	self	NOUN
ejpam-6243	238	11	mapping	mapping	NOUN
ejpam-6243	238	12	on	on	ADP
ejpam-6243	238	13	a	a	DET
ejpam-6243	238	14	metric	metric	ADJ
ejpam-6243	238	15	space	space	NOUN
ejpam-6243	238	16	(	(	PUNCT
ejpam-6243	238	17	x	x	X
ejpam-6243	238	18	,	,	PUNCT
ejpam-6243	238	19	d	d	NOUN
ejpam-6243	238	20	)	)	PUNCT
ejpam-6243	238	21	.	.	PUNCT
ejpam-6243	239	1	the	the	DET
ejpam-6243	239	2	fixed	fix	VERB
ejpam-6243	239	3	point	point	NOUN
ejpam-6243	239	4	problem	problem	NOUN
ejpam-6243	239	5	x	x	PUNCT
ejpam-6243	239	6	=	=	PRON
ejpam-6243	239	7	tx	tx	PROPN
ejpam-6243	239	8	(	(	PUNCT
ejpam-6243	239	9	17	17	NUM
ejpam-6243	239	10	)	)	PUNCT
ejpam-6243	239	11	has	have	VERB
ejpam-6243	239	12	the	the	DET
ejpam-6243	239	13	general	general	ADJ
ejpam-6243	239	14	ulam	ulam	PROPN
ejpam-6243	239	15	type	type	NOUN
ejpam-6243	239	16	stability	stability	NOUN
ejpam-6243	239	17	if	if	SCONJ
ejpam-6243	239	18	and	and	CCONJ
ejpam-6243	239	19	only	only	ADV
ejpam-6243	239	20	if	if	SCONJ
ejpam-6243	239	21	there	there	PRON
ejpam-6243	239	22	exists	exist	VERB
ejpam-6243	239	23	an	an	DET
ejpam-6243	239	24	increasing	increase	VERB
ejpam-6243	239	25	function	function	NOUN
ejpam-6243	239	26	τ	τ	X
ejpam-6243	239	27	:	:	PUNCT
ejpam-6243	240	1	[	[	X
ejpam-6243	240	2	0,+∞	0,+∞	NUM
ejpam-6243	240	3	)	)	PUNCT
ejpam-6243	240	4	↔	↔	PROPN
ejpam-6243	240	5	(	(	PUNCT
ejpam-6243	240	6	0,+∞	0,+∞	NUM
ejpam-6243	240	7	)	)	PUNCT
ejpam-6243	240	8	,	,	PUNCT
ejpam-6243	240	9	continuous	continuous	ADJ
ejpam-6243	240	10	at	at	ADP
ejpam-6243	240	11	0	0	NUM
ejpam-6243	240	12	with	with	ADP
ejpam-6243	240	13	τ(0	τ(0	NOUN
ejpam-6243	240	14	)	)	PUNCT
ejpam-6243	240	15	=	=	SYM
ejpam-6243	240	16	0	0	NUM
ejpam-6243	240	17	such	such	ADJ
ejpam-6243	240	18	that	that	PRON
ejpam-6243	240	19	for	for	ADP
ejpam-6243	240	20	every	every	DET
ejpam-6243	240	21	ϵ	ϵ	X
ejpam-6243	240	22	>	>	X
ejpam-6243	240	23	0	0	PUNCT
ejpam-6243	240	24	and	and	CCONJ
ejpam-6243	240	25	for	for	ADP
ejpam-6243	240	26	each	each	DET
ejpam-6243	240	27	y∗	y∗	ADV
ejpam-6243	240	28	∈	∈	PROPN
ejpam-6243	240	29	x	x	PUNCT
ejpam-6243	240	30	which	which	PRON
ejpam-6243	240	31	satisfies	satisfy	VERB
ejpam-6243	240	32	the	the	DET
ejpam-6243	240	33	inequality	inequality	NOUN
ejpam-6243	240	34	d(y∗	d(y∗	NOUN
ejpam-6243	240	35	,	,	PUNCT
ejpam-6243	240	36	fy∗	fy∗	NOUN
ejpam-6243	240	37	)	)	PUNCT
ejpam-6243	240	38	≤	≤	NOUN
ejpam-6243	240	39	ϵ	ϵ	ADP
ejpam-6243	240	40	,	,	PUNCT
ejpam-6243	240	41	(	(	PUNCT
ejpam-6243	240	42	18	18	NUM
ejpam-6243	240	43	)	)	PUNCT
ejpam-6243	240	44	there	there	PRON
ejpam-6243	240	45	exists	exist	VERB
ejpam-6243	240	46	a	a	DET
ejpam-6243	240	47	solution	solution	NOUN
ejpam-6243	240	48	z	z	NOUN
ejpam-6243	240	49	∈	∈	PROPN
ejpam-6243	240	50	x	x	SYM
ejpam-6243	240	51	of	of	ADP
ejpam-6243	240	52	(	(	PUNCT
ejpam-6243	240	53	17	17	NUM
ejpam-6243	240	54	)	)	PUNCT
ejpam-6243	240	55	such	such	ADJ
ejpam-6243	240	56	that	that	SCONJ
ejpam-6243	240	57	d(z	d(z	PROPN
ejpam-6243	240	58	,	,	PUNCT
ejpam-6243	240	59	y∗	y∗	PROPN
ejpam-6243	240	60	)	)	PUNCT
ejpam-6243	240	61	≤	≤	NUM
ejpam-6243	240	62	τ(ϵ	τ(ϵ	NOUN
ejpam-6243	240	63	)	)	PUNCT
ejpam-6243	240	64	.	.	PUNCT
ejpam-6243	241	1	(	(	PUNCT
ejpam-6243	241	2	19	19	NUM
ejpam-6243	241	3	)	)	PUNCT
ejpam-6243	241	4	in	in	ADP
ejpam-6243	241	5	case	case	NOUN
ejpam-6243	241	6	that	that	SCONJ
ejpam-6243	241	7	for	for	ADP
ejpam-6243	241	8	c	c	PROPN
ejpam-6243	241	9	>	>	X
ejpam-6243	241	10	0	0	NUM
ejpam-6243	241	11	,	,	PUNCT
ejpam-6243	241	12	we	we	PRON
ejpam-6243	241	13	consider	consider	VERB
ejpam-6243	241	14	τ(t	τ(t	NOUN
ejpam-6243	241	15	)	)	PUNCT
ejpam-6243	241	16	=	=	SYM
ejpam-6243	241	17	ct	ct	PROPN
ejpam-6243	241	18	for	for	ADP
ejpam-6243	241	19	all	all	DET
ejpam-6243	241	20	t	t	PROPN
ejpam-6243	241	21	≥	≥	NOUN
ejpam-6243	241	22	0	0	PUNCT
ejpam-6243	241	23	then	then	ADV
ejpam-6243	241	24	the	the	DET
ejpam-6243	241	25	fixed	fixed	ADJ
ejpam-6243	241	26	point	point	NOUN
ejpam-6243	241	27	equation	equation	NOUN
ejpam-6243	241	28	(	(	PUNCT
ejpam-6243	241	29	17	17	NUM
ejpam-6243	241	30	)	)	PUNCT
ejpam-6243	241	31	is	be	AUX
ejpam-6243	241	32	said	say	VERB
ejpam-6243	241	33	to	to	PART
ejpam-6243	241	34	be	be	AUX
ejpam-6243	241	35	ulam	ulam	ADJ
ejpam-6243	241	36	type	type	NOUN
ejpam-6243	241	37	stable	stable	ADJ
ejpam-6243	241	38	.	.	PUNCT
ejpam-6243	242	1	on	on	ADP
ejpam-6243	242	2	a	a	DET
ejpam-6243	242	3	metric	metric	ADJ
ejpam-6243	242	4	space	space	NOUN
ejpam-6243	242	5	(	(	PUNCT
ejpam-6243	242	6	x	x	X
ejpam-6243	242	7	,	,	PUNCT
ejpam-6243	242	8	d	d	PROPN
ejpam-6243	242	9	)	)	PUNCT
ejpam-6243	242	10	,	,	PUNCT
ejpam-6243	242	11	the	the	DET
ejpam-6243	242	12	fixed	fix	VERB
ejpam-6243	242	13	point	point	NOUN
ejpam-6243	242	14	problem	problem	NOUN
ejpam-6243	242	15	(	(	PUNCT
ejpam-6243	242	16	17	17	NUM
ejpam-6243	242	17	)	)	PUNCT
ejpam-6243	242	18	,	,	PUNCT
ejpam-6243	242	19	where	where	SCONJ
ejpam-6243	242	20	t	t	NOUN
ejpam-6243	242	21	:	:	PUNCT
ejpam-6243	242	22	x	x	X
ejpam-6243	242	23	→	→	SYM
ejpam-6243	242	24	x	x	X
ejpam-6243	242	25	,	,	PUNCT
ejpam-6243	242	26	is	be	AUX
ejpam-6243	242	27	said	say	VERB
ejpam-6243	242	28	to	to	PART
ejpam-6243	242	29	be	be	AUX
ejpam-6243	242	30	well	well	ADV
ejpam-6243	242	31	-	-	PUNCT
ejpam-6243	242	32	posed	pose	VERB
ejpam-6243	242	33	if	if	SCONJ
ejpam-6243	242	34	the	the	DET
ejpam-6243	242	35	following	follow	VERB
ejpam-6243	242	36	assumptions	assumption	NOUN
ejpam-6243	242	37	are	be	AUX
ejpam-6243	242	38	satisfy	satisfy	ADJ
ejpam-6243	242	39	:	:	PUNCT
ejpam-6243	242	40	r	r	NOUN
ejpam-6243	242	41	ramaswamy	ramaswamy	NOUN
ejpam-6243	242	42	et	et	PROPN
ejpam-6243	242	43	al	al	PROPN
ejpam-6243	242	44	.	.	PUNCT
ejpam-6243	242	45	/	/	SYM
ejpam-6243	242	46	eur	eur	PROPN
ejpam-6243	242	47	.	.	PUNCT
ejpam-6243	243	1	j.	j.	PROPN
ejpam-6243	243	2	pure	pure	PROPN
ejpam-6243	243	3	appl	appl	PROPN
ejpam-6243	243	4	.	.	PROPN
ejpam-6243	243	5	math	math	PROPN
ejpam-6243	243	6	,	,	PUNCT
ejpam-6243	243	7	18	18	NUM
ejpam-6243	243	8	(	(	PUNCT
ejpam-6243	243	9	3	3	NUM
ejpam-6243	243	10	)	)	PUNCT
ejpam-6243	243	11	(	(	PUNCT
ejpam-6243	243	12	2025	2025	NUM
ejpam-6243	243	13	)	)	PUNCT
ejpam-6243	243	14	,	,	PUNCT
ejpam-6243	243	15	6243	6243	NUM
ejpam-6243	243	16	11	11	NUM
ejpam-6243	243	17	of	of	ADP
ejpam-6243	243	18	14	14	NUM
ejpam-6243	243	19	(	(	PUNCT
ejpam-6243	243	20	i	i	NOUN
ejpam-6243	243	21	)	)	PUNCT
ejpam-6243	243	22	t	t	PROPN
ejpam-6243	243	23	has	have	VERB
ejpam-6243	243	24	a	a	DET
ejpam-6243	243	25	unique	unique	ADJ
ejpam-6243	243	26	fixed	fix	VERB
ejpam-6243	243	27	point	point	NOUN
ejpam-6243	243	28	z	z	NOUN
ejpam-6243	243	29	in	in	ADP
ejpam-6243	243	30	x	x	SYM
ejpam-6243	243	31	;	;	PUNCT
ejpam-6243	243	32	(	(	PUNCT
ejpam-6243	243	33	ii	ii	NOUN
ejpam-6243	243	34	)	)	PUNCT
ejpam-6243	243	35	d(xn	d(xn	PROPN
ejpam-6243	243	36	,	,	PUNCT
ejpam-6243	243	37	z	z	NOUN
ejpam-6243	243	38	)	)	PUNCT
ejpam-6243	243	39	=	=	SYM
ejpam-6243	243	40	0	0	NUM
ejpam-6243	243	41	for	for	ADP
ejpam-6243	243	42	each	each	DET
ejpam-6243	243	43	sequence	sequence	NOUN
ejpam-6243	243	44	{	{	PUNCT
ejpam-6243	243	45	xn	xn	NOUN
ejpam-6243	243	46	}	}	PUNCT
ejpam-6243	243	47	∈	∈	PROPN
ejpam-6243	243	48	x	x	PUNCT
ejpam-6243	243	49	such	such	ADJ
ejpam-6243	243	50	that	that	SCONJ
ejpam-6243	243	51	limn→+∞(d(xn	limn→+∞(d(xn	NOUN
ejpam-6243	243	52	,	,	PUNCT
ejpam-6243	243	53	txn	txn	NOUN
ejpam-6243	243	54	)	)	PUNCT
ejpam-6243	243	55	)	)	PUNCT
ejpam-6243	244	1	=	=	SYM
ejpam-6243	244	2	0	0	X
ejpam-6243	244	3	.	.	PUNCT
ejpam-6243	244	4	theorem	theorem	NOUN
ejpam-6243	244	5	4	4	NUM
ejpam-6243	244	6	.	.	PUNCT
ejpam-6243	245	1	let	let	AUX
ejpam-6243	245	2	(	(	PUNCT
ejpam-6243	245	3	x	x	NOUN
ejpam-6243	245	4	,	,	PUNCT
ejpam-6243	245	5	d	d	NOUN
ejpam-6243	245	6	)	)	PUNCT
ejpam-6243	245	7	be	be	AUX
ejpam-6243	245	8	a	a	DET
ejpam-6243	245	9	complete	complete	ADJ
ejpam-6243	245	10	metric	metric	ADJ
ejpam-6243	245	11	space	space	NOUN
ejpam-6243	245	12	.	.	PUNCT
ejpam-6243	246	1	if	if	SCONJ
ejpam-6243	246	2	we	we	PRON
ejpam-6243	246	3	add	add	VERB
ejpam-6243	246	4	the	the	DET
ejpam-6243	246	5	condition	condition	NOUN
ejpam-6243	246	6	α2	α2	ADV
ejpam-6243	246	7	<	<	X
ejpam-6243	246	8	1	1	NUM
ejpam-6243	246	9	c(s	c(	NOUN
ejpam-6243	246	10	)	)	PUNCT
ejpam-6243	246	11	,	,	PUNCT
ejpam-6243	246	12	where	where	SCONJ
ejpam-6243	246	13	c(s	c(	VERB
ejpam-6243	246	14	)	)	PUNCT
ejpam-6243	246	15	=	=	SYM
ejpam-6243	246	16	max{1	max{1	NOUN
ejpam-6243	246	17	,	,	PUNCT
ejpam-6243	246	18	√	√	ADV
ejpam-6243	246	19	2s−1	2s−1	NUM
ejpam-6243	246	20	}	}	PUNCT
ejpam-6243	246	21	to	to	ADP
ejpam-6243	246	22	the	the	DET
ejpam-6243	246	23	assumptions	assumption	NOUN
ejpam-6243	246	24	of	of	ADP
ejpam-6243	246	25	theorem	theorem	NOUN
ejpam-6243	246	26	3	3	NUM
ejpam-6243	246	27	,	,	PUNCT
ejpam-6243	246	28	then	then	ADV
ejpam-6243	246	29	the	the	DET
ejpam-6243	246	30	following	follow	VERB
ejpam-6243	246	31	affirmations	affirmation	NOUN
ejpam-6243	246	32	hold	hold	VERB
ejpam-6243	246	33	:	:	PUNCT
ejpam-6243	246	34	(	(	PUNCT
ejpam-6243	246	35	i	i	NOUN
ejpam-6243	246	36	)	)	PUNCT
ejpam-6243	246	37	the	the	DET
ejpam-6243	246	38	fixed	fix	VERB
ejpam-6243	246	39	point	point	NOUN
ejpam-6243	246	40	equation	equation	NOUN
ejpam-6243	246	41	(	(	PUNCT
ejpam-6243	246	42	17	17	NUM
ejpam-6243	246	43	)	)	PUNCT
ejpam-6243	246	44	is	be	AUX
ejpam-6243	246	45	ulam	ulam	PROPN
ejpam-6243	246	46	hyers	hyer	NOUN
ejpam-6243	246	47	stable	stable	ADJ
ejpam-6243	246	48	if	if	SCONJ
ejpam-6243	246	49	β(u	β(u	PROPN
ejpam-6243	246	50	,	,	PUNCT
ejpam-6243	246	51	v	v	NOUN
ejpam-6243	246	52	)	)	PUNCT
ejpam-6243	246	53	≥	≥	NOUN
ejpam-6243	246	54	1	1	NUM
ejpam-6243	246	55	for	for	ADP
ejpam-6243	246	56	any	any	DET
ejpam-6243	246	57	u	u	NOUN
ejpam-6243	246	58	,	,	PUNCT
ejpam-6243	246	59	v	v	PRON
ejpam-6243	246	60	satisfying	satisfy	VERB
ejpam-6243	246	61	the	the	DET
ejpam-6243	246	62	inequality	inequality	NOUN
ejpam-6243	246	63	(	(	PUNCT
ejpam-6243	246	64	18	18	NUM
ejpam-6243	246	65	)	)	PUNCT
ejpam-6243	246	66	;	;	PUNCT
ejpam-6243	246	67	(	(	PUNCT
ejpam-6243	246	68	ii	ii	X
ejpam-6243	246	69	)	)	PUNCT
ejpam-6243	246	70	the	the	DET
ejpam-6243	246	71	fixed	fix	VERB
ejpam-6243	246	72	point	point	NOUN
ejpam-6243	246	73	equation	equation	NOUN
ejpam-6243	246	74	(	(	PUNCT
ejpam-6243	246	75	17	17	NUM
ejpam-6243	246	76	)	)	PUNCT
ejpam-6243	246	77	is	be	AUX
ejpam-6243	246	78	wellposed	wellpose	VERB
ejpam-6243	246	79	if	if	SCONJ
ejpam-6243	246	80	β(xn	β(xn	NOUN
ejpam-6243	246	81	,	,	PUNCT
ejpam-6243	246	82	z	z	NOUN
ejpam-6243	246	83	)	)	PUNCT
ejpam-6243	246	84	≥	≥	NOUN
ejpam-6243	246	85	1	1	NUM
ejpam-6243	246	86	for	for	ADP
ejpam-6243	246	87	any	any	DET
ejpam-6243	246	88	sequence	sequence	NOUN
ejpam-6243	246	89	{	{	PUNCT
ejpam-6243	246	90	xn	xn	NOUN
ejpam-6243	246	91	}	}	PUNCT
ejpam-6243	246	92	∈	∈	PROPN
ejpam-6243	246	93	x	x	NOUN
ejpam-6243	246	94	such	such	ADJ
ejpam-6243	246	95	that	that	SCONJ
ejpam-6243	246	96	limn→+∞d(xn	limn→+∞d(xn	NOUN
ejpam-6243	246	97	,	,	PUNCT
ejpam-6243	246	98	txn	txn	NOUN
ejpam-6243	246	99	)	)	PUNCT
ejpam-6243	246	100	=	=	SYM
ejpam-6243	246	101	0	0	NUM
ejpam-6243	246	102	and	and	CCONJ
ejpam-6243	246	103	fixt	fixt	ADJ
ejpam-6243	246	104	(	(	PUNCT
ejpam-6243	246	105	x	x	NOUN
ejpam-6243	246	106	)	)	PUNCT
ejpam-6243	246	107	=	=	SYM
ejpam-6243	246	108	z.	z.	PROPN
ejpam-6243	246	109	proof	proof	NOUN
ejpam-6243	246	110	.	.	PUNCT
ejpam-6243	247	1	case	case	NOUN
ejpam-6243	247	2	1	1	NUM
ejpam-6243	247	3	:	:	PUNCT
ejpam-6243	247	4	since	since	SCONJ
ejpam-6243	247	5	from	from	ADP
ejpam-6243	247	6	theorem	theorem	NOUN
ejpam-6243	247	7	3	3	NUM
ejpam-6243	247	8	we	we	PRON
ejpam-6243	247	9	know	know	VERB
ejpam-6243	247	10	that	that	SCONJ
ejpam-6243	247	11	there	there	PRON
ejpam-6243	247	12	is	be	VERB
ejpam-6243	247	13	unique	unique	ADJ
ejpam-6243	247	14	z	z	NOUN
ejpam-6243	247	15	∈	∈	PROPN
ejpam-6243	247	16	x	x	PUNCT
ejpam-6243	247	17	such	such	ADJ
ejpam-6243	247	18	that	that	SCONJ
ejpam-6243	247	19	tz	tz	NOUN
ejpam-6243	247	20	=	=	SYM
ejpam-6243	247	21	z	z	NOUN
ejpam-6243	247	22	,	,	PUNCT
ejpam-6243	247	23	let	let	VERB
ejpam-6243	247	24	y∗	y∗	ADV
ejpam-6243	247	25	∈	∈	PROPN
ejpam-6243	247	26	x	x	PUNCT
ejpam-6243	247	27	such	such	ADJ
ejpam-6243	247	28	that	that	SCONJ
ejpam-6243	247	29	d(y∗	d(y∗	NOUN
ejpam-6243	247	30	,	,	PUNCT
ejpam-6243	247	31	ty∗	ty∗	ADJ
ejpam-6243	247	32	)	)	PUNCT
ejpam-6243	247	33	≤	≤	NOUN
ejpam-6243	247	34	ϵ	ϵ	X
ejpam-6243	247	35	for	for	ADP
ejpam-6243	247	36	all	all	PRON
ejpam-6243	247	37	ϵ	ϵ	X
ejpam-6243	247	38	>	>	X
ejpam-6243	247	39	0	0	X
ejpam-6243	247	40	.	.	NOUN
ejpam-6243	247	41	obvious	obvious	ADJ
ejpam-6243	247	42	,	,	PUNCT
ejpam-6243	247	43	z	z	NOUN
ejpam-6243	247	44	verifies	verifie	NOUN
ejpam-6243	247	45	(	(	PUNCT
ejpam-6243	247	46	18	18	NUM
ejpam-6243	247	47	)	)	PUNCT
ejpam-6243	247	48	so	so	SCONJ
ejpam-6243	247	49	we	we	PRON
ejpam-6243	247	50	have	have	VERB
ejpam-6243	247	51	that	that	DET
ejpam-6243	247	52	β(y∗	β(y∗	NOUN
ejpam-6243	247	53	,	,	PUNCT
ejpam-6243	247	54	z	z	NOUN
ejpam-6243	247	55	)	)	PUNCT
ejpam-6243	247	56	≥	≥	NOUN
ejpam-6243	247	57	1	1	NUM
ejpam-6243	247	58	and	and	CCONJ
ejpam-6243	247	59	then	then	ADV
ejpam-6243	247	60	by	by	ADP
ejpam-6243	247	61	using	use	VERB
ejpam-6243	247	62	the	the	DET
ejpam-6243	247	63	triangular	triangular	NOUN
ejpam-6243	247	64	inequality	inequality	NOUN
ejpam-6243	247	65	we	we	PRON
ejpam-6243	247	66	get	get	VERB
ejpam-6243	247	67	d(z	d(z	NOUN
ejpam-6243	247	68	,	,	PUNCT
ejpam-6243	247	69	y∗	y∗	PROPN
ejpam-6243	247	70	)	)	PUNCT
ejpam-6243	247	71	≤	≤	NUM
ejpam-6243	247	72	d(tz	d(tz	NOUN
ejpam-6243	247	73	,	,	PUNCT
ejpam-6243	247	74	ty∗	ty∗	ADJ
ejpam-6243	247	75	)	)	PUNCT
ejpam-6243	247	76	+	+	CCONJ
ejpam-6243	248	1	d(ty∗	d(ty∗	PROPN
ejpam-6243	248	2	,	,	PUNCT
ejpam-6243	248	3	y∗	y∗	PROPN
ejpam-6243	248	4	)	)	PUNCT
ejpam-6243	248	5	≤	≤	NUM
ejpam-6243	248	6	β(y∗	β(y∗	NUM
ejpam-6243	248	7	,	,	PUNCT
ejpam-6243	248	8	z)d(ty∗	z)d(ty∗	PROPN
ejpam-6243	248	9	,	,	PUNCT
ejpam-6243	248	10	t	t	PROPN
ejpam-6243	248	11	z	z	PROPN
ejpam-6243	248	12	)	)	PUNCT
ejpam-6243	249	1	+	+	CCONJ
ejpam-6243	249	2	d(ty∗	d(ty∗	PROPN
ejpam-6243	249	3	,	,	PUNCT
ejpam-6243	249	4	y∗	y∗	PROPN
ejpam-6243	249	5	)	)	PUNCT
ejpam-6243	249	6	≤	≤	NUM
ejpam-6243	249	7	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	249	8	s	s	PART
ejpam-6243	249	9	(	(	PUNCT
ejpam-6243	249	10	y	y	PROPN
ejpam-6243	249	11	∗	∗	NOUN
ejpam-6243	249	12	,	,	PUNCT
ejpam-6243	249	13	z	z	NOUN
ejpam-6243	249	14	)	)	PUNCT
ejpam-6243	249	15	)	)	PUNCT
ejpam-6243	250	1	+	+	CCONJ
ejpam-6243	250	2	d(ty∗	d(ty∗	PROPN
ejpam-6243	250	3	,	,	PUNCT
ejpam-6243	250	4	y∗	y∗	PROPN
ejpam-6243	250	5	)	)	PUNCT
ejpam-6243	250	6	<	<	X
ejpam-6243	250	7	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	250	8	s	s	PART
ejpam-6243	250	9	(	(	PUNCT
ejpam-6243	250	10	y	y	PROPN
ejpam-6243	250	11	∗	∗	NOUN
ejpam-6243	250	12	,	,	PUNCT
ejpam-6243	250	13	z	z	NOUN
ejpam-6243	250	14	)	)	PUNCT
ejpam-6243	250	15	)	)	PUNCT
ejpam-6243	251	1	+	+	CCONJ
ejpam-6243	252	1	d(ty∗	d(ty∗	PROPN
ejpam-6243	252	2	,	,	PUNCT
ejpam-6243	252	3	y∗	y∗	PROPN
ejpam-6243	252	4	)	)	PUNCT
ejpam-6243	252	5	≤	≤	NOUN
ejpam-6243	253	1	[	[	X
ejpam-6243	253	2	α1	α1	PROPN
ejpam-6243	253	3	(	(	PUNCT
ejpam-6243	253	4	(	(	PUNCT
ejpam-6243	253	5	d(y∗	d(y∗	NOUN
ejpam-6243	253	6	,	,	PUNCT
ejpam-6243	253	7	ty∗)d(z	ty∗)d(z	PROPN
ejpam-6243	253	8	,	,	PUNCT
ejpam-6243	253	9	tz	tz	PROPN
ejpam-6243	253	10	)	)	PUNCT
ejpam-6243	253	11	d(z	d(z	PROPN
ejpam-6243	253	12	,	,	PUNCT
ejpam-6243	253	13	y∗	y∗	PROPN
ejpam-6243	253	14	)	)	PUNCT
ejpam-6243	253	15	)	)	PUNCT
ejpam-6243	253	16	s	s	PART
ejpam-6243	254	1	+	+	NUM
ejpam-6243	254	2	α2(d(z	α2(d(z	NOUN
ejpam-6243	254	3	,	,	PUNCT
ejpam-6243	254	4	y	y	PROPN
ejpam-6243	254	5	∗))s	∗))s	X
ejpam-6243	254	6	]	]	X
ejpam-6243	254	7	1	1	NUM
ejpam-6243	254	8	s	s	NOUN
ejpam-6243	254	9	+	+	NUM
ejpam-6243	254	10	d(ty∗	d(ty∗	PROPN
ejpam-6243	254	11	,	,	PUNCT
ejpam-6243	254	12	y∗	y∗	PROPN
ejpam-6243	254	13	)	)	PUNCT
ejpam-6243	254	14	=	=	PUNCT
ejpam-6243	255	1	[	[	X
ejpam-6243	255	2	α2(d(z	α2(d(z	PROPN
ejpam-6243	255	3	,	,	PUNCT
ejpam-6243	255	4	y	y	PROPN
ejpam-6243	255	5	∗))s	∗))s	X
ejpam-6243	255	6	]	]	X
ejpam-6243	255	7	1	1	NUM
ejpam-6243	255	8	s	s	NOUN
ejpam-6243	255	9	+	+	NUM
ejpam-6243	255	10	d(ty∗	d(ty∗	PROPN
ejpam-6243	255	11	,	,	PUNCT
ejpam-6243	255	12	y∗	y∗	PROPN
ejpam-6243	255	13	)	)	PUNCT
ejpam-6243	255	14	≤	≤	PUNCT
ejpam-6243	256	1	[	[	X
ejpam-6243	256	2	α2(d(z	α2(d(z	X
ejpam-6243	256	3	,	,	PUNCT
ejpam-6243	256	4	y	y	PROPN
ejpam-6243	256	5	∗))s	∗))s	X
ejpam-6243	256	6	]	]	X
ejpam-6243	256	7	1	1	NUM
ejpam-6243	256	8	s	s	NOUN
ejpam-6243	256	9	+	+	X
ejpam-6243	256	10	ϵ.	ϵ.	NOUN
ejpam-6243	256	11	therefore	therefore	ADV
ejpam-6243	256	12	,	,	PUNCT
ejpam-6243	256	13	(	(	PUNCT
ejpam-6243	256	14	d(z	d(z	PROPN
ejpam-6243	256	15	,	,	PUNCT
ejpam-6243	256	16	y∗))s	y∗))s	PROPN
ejpam-6243	256	17	≤	≤	PROPN
ejpam-6243	256	18	c(s)[α2(d(z	c(s)[α2(d(z	PROPN
ejpam-6243	256	19	,	,	PUNCT
ejpam-6243	256	20	y	y	PROPN
ejpam-6243	256	21	∗))s	∗))s	X
ejpam-6243	257	1	+	+	CCONJ
ejpam-6243	257	2	ϵs	ϵ	NOUN
ejpam-6243	257	3	]	]	PUNCT
ejpam-6243	257	4	,	,	PUNCT
ejpam-6243	257	5	where	where	SCONJ
ejpam-6243	257	6	c(s	c(	VERB
ejpam-6243	257	7	)	)	PUNCT
ejpam-6243	257	8	=	=	SYM
ejpam-6243	257	9	max{1	max{1	NOUN
ejpam-6243	257	10	,	,	PUNCT
ejpam-6243	257	11	√	√	ADV
ejpam-6243	257	12	2s−1	2s−1	NUM
ejpam-6243	257	13	}	}	PUNCT
ejpam-6243	257	14	by	by	ADP
ejpam-6243	257	15	simple	simple	ADJ
ejpam-6243	257	16	calculation	calculation	NOUN
ejpam-6243	257	17	,	,	PUNCT
ejpam-6243	257	18	from	from	ADP
ejpam-6243	257	19	the	the	DET
ejpam-6243	257	20	above	above	ADJ
ejpam-6243	257	21	inequality	inequality	NOUN
ejpam-6243	257	22	we	we	PRON
ejpam-6243	257	23	have	have	VERB
ejpam-6243	257	24	d(z	d(z	NOUN
ejpam-6243	257	25	,	,	PUNCT
ejpam-6243	257	26	y∗)s	y∗)s	NOUN
ejpam-6243	257	27	≤	≤	NOUN
ejpam-6243	258	1	c(s	c(	NOUN
ejpam-6243	258	2	)	)	PUNCT
ejpam-6243	258	3	(	(	PUNCT
ejpam-6243	258	4	1−	1−	NUM
ejpam-6243	258	5	c(s)α2	c(s)α2	NOUN
ejpam-6243	258	6	)	)	PUNCT
ejpam-6243	258	7	ϵs	ϵ	NOUN
ejpam-6243	258	8	,	,	PUNCT
ejpam-6243	258	9	r	r	NOUN
ejpam-6243	258	10	ramaswamy	ramaswamy	NOUN
ejpam-6243	258	11	et	et	PROPN
ejpam-6243	258	12	al	al	PROPN
ejpam-6243	258	13	.	.	PUNCT
ejpam-6243	258	14	/	/	SYM
ejpam-6243	258	15	eur	eur	PROPN
ejpam-6243	258	16	.	.	PUNCT
ejpam-6243	259	1	j.	j.	PROPN
ejpam-6243	259	2	pure	pure	PROPN
ejpam-6243	259	3	appl	appl	PROPN
ejpam-6243	259	4	.	.	PROPN
ejpam-6243	259	5	math	math	PROPN
ejpam-6243	259	6	,	,	PUNCT
ejpam-6243	259	7	18	18	NUM
ejpam-6243	259	8	(	(	PUNCT
ejpam-6243	259	9	3	3	NUM
ejpam-6243	259	10	)	)	PUNCT
ejpam-6243	259	11	(	(	PUNCT
ejpam-6243	259	12	2025	2025	NUM
ejpam-6243	259	13	)	)	PUNCT
ejpam-6243	259	14	,	,	PUNCT
ejpam-6243	259	15	6243	6243	NUM
ejpam-6243	259	16	12	12	NUM
ejpam-6243	259	17	of	of	ADP
ejpam-6243	259	18	14	14	NUM
ejpam-6243	259	19	which	which	PRON
ejpam-6243	259	20	is	be	AUX
ejpam-6243	259	21	equivalent	equivalent	ADJ
ejpam-6243	259	22	to	to	ADP
ejpam-6243	259	23	d(z	d(z	PROPN
ejpam-6243	259	24	,	,	PUNCT
ejpam-6243	259	25	y∗	y∗	PROPN
ejpam-6243	259	26	)	)	PUNCT
ejpam-6243	259	27	≤	≤	NUM
ejpam-6243	260	1	cϵ.	cϵ.	NOUN
ejpam-6243	260	2	where	where	SCONJ
ejpam-6243	260	3	c	c	AUX
ejpam-6243	260	4	=	=	PUNCT
ejpam-6243	260	5	(	(	PUNCT
ejpam-6243	260	6	c(s	c(s	X
ejpam-6243	260	7	)	)	PUNCT
ejpam-6243	260	8	(	(	PUNCT
ejpam-6243	260	9	1−c(s)α2	1−c(s)α2	NUM
ejpam-6243	260	10	)	)	PUNCT
ejpam-6243	260	11	1	1	NUM
ejpam-6243	260	12	s	s	NOUN
ejpam-6243	260	13	for	for	ADP
ejpam-6243	260	14	any	any	PRON
ejpam-6243	260	15	s	s	X
ejpam-6243	260	16	>	>	X
ejpam-6243	260	17	0	0	PUNCT
ejpam-6243	260	18	and	and	CCONJ
ejpam-6243	260	19	α2	α2	PROPN
ejpam-6243	260	20	∈	∈	PROPN
ejpam-6243	261	1	[	[	X
ejpam-6243	261	2	0	0	NUM
ejpam-6243	261	3	,	,	PUNCT
ejpam-6243	261	4	1	1	NUM
ejpam-6243	261	5	)	)	PUNCT
ejpam-6243	261	6	such	such	ADJ
ejpam-6243	261	7	that	that	DET
ejpam-6243	261	8	α2	α2	NOUN
ejpam-6243	261	9	<	<	X
ejpam-6243	261	10	1	1	NUM
ejpam-6243	261	11	c(s	c(	NOUN
ejpam-6243	261	12	)	)	PUNCT
ejpam-6243	261	13	.	.	PUNCT
ejpam-6243	262	1	case	case	NOUN
ejpam-6243	262	2	2	2	NUM
ejpam-6243	262	3	:	:	PUNCT
ejpam-6243	262	4	taking	take	VERB
ejpam-6243	262	5	into	into	ADP
ejpam-6243	262	6	account	account	NOUN
ejpam-6243	262	7	the	the	DET
ejpam-6243	262	8	supplementary	supplementary	ADJ
ejpam-6243	262	9	condition	condition	NOUN
ejpam-6243	262	10	and	and	CCONJ
ejpam-6243	262	11	since	since	SCONJ
ejpam-6243	262	12	fixt	fixt	ADJ
ejpam-6243	262	13	(	(	PUNCT
ejpam-6243	262	14	x	x	NOUN
ejpam-6243	262	15	)	)	PUNCT
ejpam-6243	262	16	=	=	SYM
ejpam-6243	263	1	z	z	NOUN
ejpam-6243	263	2	we	we	PRON
ejpam-6243	263	3	have	have	VERB
ejpam-6243	263	4	d(xn	d(xn	ADJ
ejpam-6243	263	5	,	,	PUNCT
ejpam-6243	263	6	z	z	NOUN
ejpam-6243	263	7	)	)	PUNCT
ejpam-6243	263	8	≤	≤	NOUN
ejpam-6243	264	1	d(xn	d(xn	ADJ
ejpam-6243	264	2	,	,	PUNCT
ejpam-6243	264	3	txn	txn	NOUN
ejpam-6243	264	4	)	)	PUNCT
ejpam-6243	265	1	+	+	CCONJ
ejpam-6243	265	2	d(txn	d(txn	PROPN
ejpam-6243	265	3	,	,	PUNCT
ejpam-6243	265	4	t	t	PROPN
ejpam-6243	265	5	z	z	PROPN
ejpam-6243	265	6	)	)	PUNCT
ejpam-6243	265	7	≤	≤	NOUN
ejpam-6243	266	1	d(xn	d(xn	ADJ
ejpam-6243	266	2	,	,	PUNCT
ejpam-6243	266	3	txn	txn	NOUN
ejpam-6243	266	4	)	)	PUNCT
ejpam-6243	266	5	+	+	CCONJ
ejpam-6243	266	6	β(xn	β(xn	NOUN
ejpam-6243	266	7	,	,	PUNCT
ejpam-6243	266	8	z)d(txn	z)d(txn	NOUN
ejpam-6243	266	9	,	,	PUNCT
ejpam-6243	266	10	t	t	PROPN
ejpam-6243	266	11	z	z	PROPN
ejpam-6243	266	12	)	)	PUNCT
ejpam-6243	266	13	≤	≤	NOUN
ejpam-6243	267	1	d(xn	d(xn	ADJ
ejpam-6243	267	2	,	,	PUNCT
ejpam-6243	267	3	txn	txn	NOUN
ejpam-6243	267	4	)	)	PUNCT
ejpam-6243	268	1	+	+	CCONJ
ejpam-6243	268	2	ϕ(jt	ϕ(jt	PROPN
ejpam-6243	268	3	s	s	PART
ejpam-6243	268	4	(	(	PUNCT
ejpam-6243	268	5	xn	xn	PROPN
ejpam-6243	268	6	,	,	PUNCT
ejpam-6243	268	7	z	z	NOUN
ejpam-6243	268	8	)	)	PUNCT
ejpam-6243	268	9	)	)	PUNCT
ejpam-6243	269	1	<	<	X
ejpam-6243	269	2	d(xn	d(xn	X
ejpam-6243	269	3	,	,	PUNCT
ejpam-6243	269	4	txn	txn	NOUN
ejpam-6243	269	5	)	)	PUNCT
ejpam-6243	270	1	+	+	CCONJ
ejpam-6243	270	2	jt	jt	PROPN
ejpam-6243	270	3	s	s	PROPN
ejpam-6243	270	4	(	(	PUNCT
ejpam-6243	270	5	xn	xn	PROPN
ejpam-6243	270	6	,	,	PUNCT
ejpam-6243	270	7	z	z	NOUN
ejpam-6243	270	8	)	)	PUNCT
ejpam-6243	270	9	≤	≤	NOUN
ejpam-6243	271	1	[	[	X
ejpam-6243	271	2	α1	α1	PROPN
ejpam-6243	271	3	(	(	PUNCT
ejpam-6243	271	4	d(xn	d(xn	PROPN
ejpam-6243	271	5	,	,	PUNCT
ejpam-6243	271	6	txn)d(z	txn)d(z	PROPN
ejpam-6243	271	7	,	,	PUNCT
ejpam-6243	271	8	tz	tz	NOUN
ejpam-6243	271	9	)	)	PUNCT
ejpam-6243	271	10	)	)	PUNCT
ejpam-6243	272	1	d(xn	d(xn	PROPN
ejpam-6243	272	2	,	,	PUNCT
ejpam-6243	272	3	z	z	NOUN
ejpam-6243	272	4	)	)	PUNCT
ejpam-6243	272	5	)	)	PUNCT
ejpam-6243	272	6	s	s	PART
ejpam-6243	272	7	+	+	X
ejpam-6243	272	8	α2(d(xn	α2(d(xn	NUM
ejpam-6243	272	9	,	,	PUNCT
ejpam-6243	272	10	z	z	NOUN
ejpam-6243	272	11	)	)	PUNCT
ejpam-6243	272	12	)	)	PUNCT
ejpam-6243	273	1	s	s	X
ejpam-6243	273	2	]	]	X
ejpam-6243	273	3	1	1	NUM
ejpam-6243	273	4	s	s	PART
ejpam-6243	273	5	+	+	NOUN
ejpam-6243	273	6	d(xn	d(xn	ADJ
ejpam-6243	273	7	,	,	PUNCT
ejpam-6243	273	8	txn	txn	NOUN
ejpam-6243	273	9	)	)	PUNCT
ejpam-6243	273	10	=	=	PUNCT
ejpam-6243	274	1	[	[	X
ejpam-6243	274	2	α2(d(xn	α2(d(xn	X
ejpam-6243	274	3	,	,	PUNCT
ejpam-6243	274	4	z	z	NOUN
ejpam-6243	274	5	)	)	PUNCT
ejpam-6243	274	6	)	)	PUNCT
ejpam-6243	275	1	s	s	X
ejpam-6243	275	2	]	]	X
ejpam-6243	275	3	1	1	NUM
ejpam-6243	275	4	s	s	PART
ejpam-6243	275	5	+	+	NOUN
ejpam-6243	275	6	d(xn	d(xn	ADJ
ejpam-6243	275	7	,	,	PUNCT
ejpam-6243	275	8	txn	txn	NOUN
ejpam-6243	275	9	)	)	PUNCT
ejpam-6243	275	10	(	(	PUNCT
ejpam-6243	275	11	d(xn	d(xn	PROPN
ejpam-6243	275	12	,	,	PUNCT
ejpam-6243	275	13	z	z	NOUN
ejpam-6243	275	14	)	)	PUNCT
ejpam-6243	275	15	)	)	PUNCT
ejpam-6243	276	1	s	s	VERB
ejpam-6243	276	2	≤	≤	NOUN
ejpam-6243	276	3	α2(d(xn	α2(d(xn	NUM
ejpam-6243	276	4	,	,	PUNCT
ejpam-6243	276	5	z	z	NOUN
ejpam-6243	276	6	)	)	PUNCT
ejpam-6243	276	7	)	)	PUNCT
ejpam-6243	277	1	s	s	VERB
ejpam-6243	278	1	+	+	X
ejpam-6243	278	2	(	(	PUNCT
ejpam-6243	278	3	d(xn	d(xn	ADJ
ejpam-6243	278	4	,	,	PUNCT
ejpam-6243	278	5	txn	txn	NOUN
ejpam-6243	278	6	)	)	PUNCT
ejpam-6243	278	7	)	)	PUNCT
ejpam-6243	279	1	s	s	PART
ejpam-6243	279	2	(	(	PUNCT
ejpam-6243	279	3	d(xn	d(xn	PROPN
ejpam-6243	279	4	,	,	PUNCT
ejpam-6243	279	5	z	z	NOUN
ejpam-6243	279	6	)	)	PUNCT
ejpam-6243	279	7	)	)	PUNCT
ejpam-6243	279	8	s	s	VERB
ejpam-6243	279	9	≤	≤	NOUN
ejpam-6243	279	10	c(s	c(	NOUN
ejpam-6243	279	11	)	)	PUNCT
ejpam-6243	279	12	(	(	PUNCT
ejpam-6243	279	13	1−	1−	NUM
ejpam-6243	279	14	c(s))α2	c(s))α2	NOUN
ejpam-6243	279	15	(	(	PUNCT
ejpam-6243	279	16	d(xn	d(xn	X
ejpam-6243	279	17	,	,	PUNCT
ejpam-6243	279	18	txn	txn	NOUN
ejpam-6243	279	19	)	)	PUNCT
ejpam-6243	279	20	)	)	PUNCT
ejpam-6243	280	1	s.	s.	PROPN
ejpam-6243	280	2	letting	let	VERB
ejpam-6243	280	3	n	n	X
ejpam-6243	280	4	→	→	SYM
ejpam-6243	280	5	+	+	NUM
ejpam-6243	280	6	∞	∞	PROPN
ejpam-6243	280	7	in	in	ADP
ejpam-6243	280	8	the	the	DET
ejpam-6243	280	9	above	above	ADJ
ejpam-6243	280	10	inequality	inequality	NOUN
ejpam-6243	280	11	and	and	CCONJ
ejpam-6243	280	12	keeping	keeping	NOUN
ejpam-6243	280	13	in	in	ADP
ejpam-6243	280	14	mind	mind	NOUN
ejpam-6243	280	15	that	that	SCONJ
ejpam-6243	280	16	lim	lim	PROPN
ejpam-6243	280	17	n→+∞	n→+∞	VERB
ejpam-6243	280	18	d(xn	d(xn	PROPN
ejpam-6243	280	19	,	,	PUNCT
ejpam-6243	280	20	txn	txn	NOUN
ejpam-6243	280	21	)	)	PUNCT
ejpam-6243	280	22	=	=	SYM
ejpam-6243	280	23	0	0	NUM
ejpam-6243	280	24	,	,	PUNCT
ejpam-6243	280	25	we	we	PRON
ejpam-6243	280	26	obtain	obtain	VERB
ejpam-6243	280	27	lim	lim	NOUN
ejpam-6243	280	28	n→+∞	n→+∞	VERB
ejpam-6243	280	29	d(xn	d(xn	PROPN
ejpam-6243	280	30	,	,	PUNCT
ejpam-6243	280	31	z	z	NOUN
ejpam-6243	280	32	)	)	PUNCT
ejpam-6243	280	33	=	=	SYM
ejpam-6243	281	1	0	0	X
ejpam-6243	281	2	.	.	PUNCT
ejpam-6243	282	1	that	that	PRON
ejpam-6243	282	2	is	be	AUX
ejpam-6243	282	3	,	,	PUNCT
ejpam-6243	282	4	the	the	DET
ejpam-6243	282	5	fixed	fix	VERB
ejpam-6243	282	6	point	point	NOUN
ejpam-6243	282	7	equation	equation	NOUN
ejpam-6243	282	8	(	(	PUNCT
ejpam-6243	282	9	17	17	NUM
ejpam-6243	282	10	)	)	PUNCT
ejpam-6243	282	11	is	be	AUX
ejpam-6243	282	12	well	well	ADV
ejpam-6243	282	13	posed	pose	VERB
ejpam-6243	282	14	.	.	PUNCT
ejpam-6243	283	1	4	4	X
ejpam-6243	283	2	.	.	X
ejpam-6243	283	3	conclusions	conclusion	NOUN
ejpam-6243	283	4	in	in	ADP
ejpam-6243	283	5	our	our	PRON
ejpam-6243	283	6	work	work	NOUN
ejpam-6243	283	7	we	we	PRON
ejpam-6243	283	8	introduced	introduce	VERB
ejpam-6243	283	9	the	the	DET
ejpam-6243	283	10	(	(	PUNCT
ejpam-6243	283	11	β	β	X
ejpam-6243	283	12	,	,	PUNCT
ejpam-6243	283	13	ϕ)−	ϕ)−	PROPN
ejpam-6243	283	14	admissible	admissible	ADJ
ejpam-6243	283	15	hybrid	hybrid	ADJ
ejpam-6243	283	16	contractions	contraction	NOUN
ejpam-6243	283	17	in	in	ADP
ejpam-6243	283	18	metric	metric	ADJ
ejpam-6243	283	19	spaces	space	NOUN
ejpam-6243	283	20	and	and	CCONJ
ejpam-6243	283	21	establish	establish	VERB
ejpam-6243	283	22	fixed	fix	VERB
ejpam-6243	283	23	point	point	NOUN
ejpam-6243	283	24	results	result	NOUN
ejpam-6243	283	25	in	in	ADP
ejpam-6243	283	26	the	the	DET
ejpam-6243	283	27	setting	setting	NOUN
ejpam-6243	283	28	of	of	ADP
ejpam-6243	283	29	these	these	DET
ejpam-6243	283	30	spaces	space	NOUN
ejpam-6243	283	31	and	and	CCONJ
ejpam-6243	283	32	the	the	DET
ejpam-6243	283	33	derived	derive	VERB
ejpam-6243	283	34	results	result	NOUN
ejpam-6243	283	35	have	have	AUX
ejpam-6243	283	36	been	be	AUX
ejpam-6243	283	37	supplemented	supplement	VERB
ejpam-6243	283	38	with	with	ADP
ejpam-6243	283	39	suitable	suitable	ADJ
ejpam-6243	283	40	example	example	NOUN
ejpam-6243	283	41	and	and	CCONJ
ejpam-6243	283	42	an	an	DET
ejpam-6243	283	43	application	application	NOUN
ejpam-6243	283	44	to	to	ADP
ejpam-6243	283	45	ulam	ulam	X
ejpam-6243	283	46	-	-	PUNCT
ejpam-6243	283	47	hyers	hyer	NOUN
ejpam-6243	283	48	stability	stability	NOUN
ejpam-6243	283	49	and	and	CCONJ
ejpam-6243	283	50	well	well	ADV
ejpam-6243	283	51	-	-	PUNCT
ejpam-6243	283	52	poisedness	poisedness	NOUN
ejpam-6243	283	53	has	have	AUX
ejpam-6243	283	54	also	also	ADV
ejpam-6243	283	55	been	be	AUX
ejpam-6243	283	56	provided	provide	VERB
ejpam-6243	283	57	.	.	PUNCT
ejpam-6243	284	1	it	it	PRON
ejpam-6243	284	2	is	be	AUX
ejpam-6243	284	3	an	an	DET
ejpam-6243	284	4	open	open	ADJ
ejpam-6243	284	5	problem	problem	NOUN
ejpam-6243	284	6	to	to	PART
ejpam-6243	284	7	extend	extend	VERB
ejpam-6243	284	8	and	and	CCONJ
ejpam-6243	284	9	generalize	generalize	VERB
ejpam-6243	284	10	our	our	PRON
ejpam-6243	284	11	result	result	NOUN
ejpam-6243	284	12	in	in	ADP
ejpam-6243	284	13	the	the	DET
ejpam-6243	284	14	setting	setting	NOUN
ejpam-6243	284	15	of	of	ADP
ejpam-6243	284	16	other	other	ADJ
ejpam-6243	284	17	topological	topological	ADJ
ejpam-6243	284	18	spaces	space	NOUN
ejpam-6243	284	19	and	and	CCONJ
ejpam-6243	284	20	some	some	DET
ejpam-6243	284	21	other	other	ADJ
ejpam-6243	284	22	generalized	generalized	ADJ
ejpam-6243	284	23	contractions	contraction	NOUN
ejpam-6243	284	24	.	.	PUNCT
ejpam-6243	285	1	acknowledgements	acknowledgement	NOUN
ejpam-6243	285	2	(	(	PUNCT
ejpam-6243	285	3	i	i	NOUN
ejpam-6243	285	4	)	)	PUNCT
ejpam-6243	285	5	this	this	DET
ejpam-6243	285	6	study	study	NOUN
ejpam-6243	285	7	is	be	AUX
ejpam-6243	285	8	supported	support	VERB
ejpam-6243	285	9	via	via	ADP
ejpam-6243	285	10	funding	funding	NOUN
ejpam-6243	285	11	from	from	ADP
ejpam-6243	285	12	prince	prince	PROPN
ejpam-6243	285	13	sattam	sattam	PROPN
ejpam-6243	285	14	bin	bin	PROPN
ejpam-6243	285	15	abdulaziz	abdulaziz	PROPN
ejpam-6243	285	16	university	university	PROPN
ejpam-6243	285	17	project	project	NOUN
ejpam-6243	285	18	number	number	NOUN
ejpam-6243	285	19	(	(	PUNCT
ejpam-6243	285	20	psau/2025	psau/2025	NOUN
ejpam-6243	285	21	/	/	SYM
ejpam-6243	285	22	r/1446	r/1446	PROPN
ejpam-6243	285	23	)	)	PUNCT
ejpam-6243	285	24	.	.	PUNCT
ejpam-6243	286	1	(	(	PUNCT
ejpam-6243	286	2	ii	ii	X
ejpam-6243	286	3	)	)	PUNCT
ejpam-6243	286	4	the	the	DET
ejpam-6243	286	5	authors	author	NOUN
ejpam-6243	286	6	convey	convey	VERB
ejpam-6243	286	7	their	their	PRON
ejpam-6243	286	8	sincere	sincere	ADJ
ejpam-6243	286	9	appreciation	appreciation	NOUN
ejpam-6243	286	10	to	to	ADP
ejpam-6243	286	11	the	the	DET
ejpam-6243	286	12	anonymous	anonymous	ADJ
ejpam-6243	286	13	reviewers	reviewer	NOUN
ejpam-6243	286	14	for	for	ADP
ejpam-6243	286	15	their	their	PRON
ejpam-6243	286	16	comments	comment	NOUN
ejpam-6243	286	17	,	,	PUNCT
ejpam-6243	286	18	which	which	PRON
ejpam-6243	286	19	helped	help	VERB
ejpam-6243	286	20	to	to	PART
ejpam-6243	286	21	improve	improve	VERB
ejpam-6243	286	22	the	the	DET
ejpam-6243	286	23	manuscript	manuscript	NOUN
ejpam-6243	286	24	to	to	ADP
ejpam-6243	286	25	its	its	PRON
ejpam-6243	286	26	present	present	ADJ
ejpam-6243	286	27	form	form	NOUN
ejpam-6243	286	28	.	.	PUNCT
ejpam-6243	287	1	r	r	NOUN
ejpam-6243	287	2	ramaswamy	ramaswamy	PROPN
ejpam-6243	287	3	et	et	PROPN
ejpam-6243	287	4	al	al	PROPN
ejpam-6243	287	5	.	.	PUNCT
ejpam-6243	287	6	/	/	SYM
ejpam-6243	287	7	eur	eur	PROPN
ejpam-6243	287	8	.	.	PUNCT
ejpam-6243	288	1	j.	j.	PROPN
ejpam-6243	288	2	pure	pure	PROPN
ejpam-6243	288	3	appl	appl	PROPN
ejpam-6243	288	4	.	.	PROPN
ejpam-6243	288	5	math	math	PROPN
ejpam-6243	288	6	,	,	PUNCT
ejpam-6243	288	7	18	18	NUM
ejpam-6243	288	8	(	(	PUNCT
ejpam-6243	288	9	3	3	NUM
ejpam-6243	288	10	)	)	PUNCT
ejpam-6243	288	11	(	(	PUNCT
ejpam-6243	288	12	2025	2025	NUM
ejpam-6243	288	13	)	)	PUNCT
ejpam-6243	288	14	,	,	PUNCT
ejpam-6243	288	15	6243	6243	NUM
ejpam-6243	288	16	13	13	NUM
ejpam-6243	288	17	of	of	ADP
ejpam-6243	288	18	14	14	NUM
ejpam-6243	288	19	conflict	conflict	NOUN
ejpam-6243	288	20	of	of	ADP
ejpam-6243	288	21	interests	interest	NOUN
ejpam-6243	288	22	the	the	DET
ejpam-6243	288	23	authors	author	NOUN
ejpam-6243	288	24	declare	declare	VERB
ejpam-6243	288	25	no	no	DET
ejpam-6243	288	26	conflicts	conflict	NOUN
ejpam-6243	288	27	of	of	ADP
ejpam-6243	288	28	interest	interest	NOUN
ejpam-6243	288	29	.	.	PUNCT
ejpam-6243	289	1	references	reference	NOUN
ejpam-6243	289	2	[	[	X
ejpam-6243	289	3	1	1	NUM
ejpam-6243	289	4	]	]	PUNCT
ejpam-6243	289	5	liouville	liouville	NOUN
ejpam-6243	289	6	j.	j.	PROPN
ejpam-6243	289	7	second	second	PROPN
ejpam-6243	289	8	mmoire	mmoire	PROPN
ejpam-6243	289	9	sur	sur	PROPN
ejpam-6243	289	10	le	le	X
ejpam-6243	289	11	developpement	developpement	PROPN
ejpam-6243	289	12	des	des	PROPN
ejpam-6243	289	13	fonctions	fonctions	PROPN
ejpam-6243	289	14	ou	ou	ADP
ejpam-6243	289	15	parties	party	NOUN
ejpam-6243	289	16	de	de	X
ejpam-6243	289	17	fonctions	fonction	NOUN
ejpam-6243	289	18	en	en	ADP
ejpam-6243	289	19	series	series	NOUN
ejpam-6243	289	20	do	do	AUX
ejpam-6243	289	21	nt	not	PART
ejpam-6243	289	22	divers	diver	NOUN
ejpam-6243	289	23	termes	terme	NOUN
ejpam-6243	289	24	sont	sont	ADV
ejpam-6243	289	25	assujettis	assujettis	NOUN
ejpam-6243	289	26	satisfaire	satisfaire	VERB
ejpam-6243	289	27	a	a	DET
ejpam-6243	289	28	une	une	PROPN
ejpam-6243	289	29	m	m	PROPN
ejpam-6243	289	30	eme	eme	NOUN
ejpam-6243	289	31	equation	equation	NOUN
ejpam-6243	289	32	differentielle	differentielle	VERB
ejpam-6243	289	33	du	du	PROPN
ejpam-6243	289	34	second	second	ADJ
ejpam-6243	289	35	ordre	ordre	PROPN
ejpam-6243	289	36	contenant	contenant	PROPN
ejpam-6243	289	37	un	un	PROPN
ejpam-6243	289	38	parametre	parametre	PROPN
ejpam-6243	289	39	variabl	variabl	PROPN
ejpam-6243	289	40	.	.	PUNCT
ejpam-6243	290	1	j.	j.	PROPN
ejpam-6243	290	2	math	math	PROPN
ejpam-6243	290	3	.	.	PUNCT
ejpam-6243	291	1	pure	pure	ADJ
ejpam-6243	291	2	appl	appl	PROPN
ejpam-6243	291	3	,	,	PUNCT
ejpam-6243	291	4	2:16–35	2:16–35	NUM
ejpam-6243	291	5	,	,	PUNCT
ejpam-6243	291	6	1837	1837	NUM
ejpam-6243	291	7	.	.	PUNCT
ejpam-6243	292	1	[	[	X
ejpam-6243	292	2	2	2	NUM
ejpam-6243	292	3	]	]	X
ejpam-6243	292	4	picard	picard	PROPN
ejpam-6243	292	5	e.	e.	PROPN
ejpam-6243	292	6	emoire	emoire	PROPN
ejpam-6243	292	7	sur	sur	PROPN
ejpam-6243	292	8	la	la	PROPN
ejpam-6243	292	9	theorie	theorie	PROPN
ejpam-6243	292	10	des	des	PROPN
ejpam-6243	292	11	equations	equations	PROPN
ejpam-6243	292	12	aux	aux	PROPN
ejpam-6243	292	13	derivees	derive	VERB
ejpam-6243	292	14	partielles	partielle	NOUN
ejpam-6243	292	15	et	et	PROPN
ejpam-6243	292	16	la	la	PROPN
ejpam-6243	292	17	methode	methode	PROPN
ejpam-6243	292	18	des	des	PROPN
ejpam-6243	292	19	approximations	approximation	NOUN
ejpam-6243	292	20	successive	successive	ADJ
ejpam-6243	292	21	,	,	PUNCT
ejpam-6243	292	22	.	.	PUNCT
ejpam-6243	293	1	,	,	PUNCT
ejpam-6243	293	2	j.	j.	PROPN
ejpam-6243	293	3	math	math	PROPN
ejpam-6243	293	4	.	.	PUNCT
ejpam-6243	294	1	pures	pure	NOUN
ejpam-6243	294	2	appl	appl	PROPN
ejpam-6243	294	3	.	.	PROPN
ejpam-6243	294	4	,	,	PUNCT
ejpam-6243	294	5	6:145–210	6:145–210	PROPN
ejpam-6243	294	6	,	,	PUNCT
ejpam-6243	294	7	1890	1890	NUM
ejpam-6243	294	8	.	.	PUNCT
ejpam-6243	295	1	[	[	X
ejpam-6243	295	2	3	3	X
ejpam-6243	295	3	]	]	X
ejpam-6243	295	4	banach	banach	NOUN
ejpam-6243	295	5	s.	s.	PROPN
ejpam-6243	295	6	sur	sur	PROPN
ejpam-6243	295	7	les	les	PROPN
ejpam-6243	295	8	operations	operation	NOUN
ejpam-6243	295	9	dans	dan	NOUN
ejpam-6243	295	10	les	le	NOUN
ejpam-6243	295	11	ensembles	ensemble	NOUN
ejpam-6243	295	12	abstraits	abstrait	NOUN
ejpam-6243	295	13	et	et	PROPN
ejpam-6243	295	14	leur	leur	X
ejpam-6243	295	15	application	application	PROPN
ejpam-6243	295	16	aux	aux	PROPN
ejpam-6243	295	17	equations	equation	NOUN
ejpam-6243	295	18	integrals	integral	NOUN
ejpam-6243	295	19	,	,	PUNCT
ejpam-6243	295	20	.	.	PUNCT
ejpam-6243	296	1	fundamenta	fundamenta	PROPN
ejpam-6243	296	2	mathematicae	mathematicae	PROPN
ejpam-6243	296	3	.	.	PUNCT
ejpam-6243	297	1	,	,	PUNCT
ejpam-6243	297	2	,	,	PUNCT
ejpam-6243	297	3	3:133–181	3:133–181	NUM
ejpam-6243	297	4	,	,	PUNCT
ejpam-6243	297	5	1922	1922	NUM
ejpam-6243	297	6	.	.	PUNCT
ejpam-6243	298	1	[	[	X
ejpam-6243	298	2	4	4	X
ejpam-6243	298	3	]	]	X
ejpam-6243	298	4	karapinar	karapinar	PROPN
ejpam-6243	298	5	e.	e.	PROPN
ejpam-6243	298	6	agarwal	agarwal	PROPN
ejpam-6243	298	7	r.	r.	PROPN
ejpam-6243	298	8	p.	p.	PROPN
ejpam-6243	299	1	interpolative	interpolative	ADJ
ejpam-6243	299	2	rus	rus	PROPN
ejpam-6243	299	3	-	-	PUNCT
ejpam-6243	299	4	reich	reich	NOUN
ejpam-6243	299	5	-	-	PUNCT
ejpam-6243	299	6	ciric	ciric	ADJ
ejpam-6243	299	7	type	type	NOUN
ejpam-6243	299	8	contractions	contraction	NOUN
ejpam-6243	299	9	via	via	ADP
ejpam-6243	299	10	simulation	simulation	NOUN
ejpam-6243	299	11	functions	function	NOUN
ejpam-6243	299	12	.	.	PUNCT
ejpam-6243	300	1	an	an	DET
ejpam-6243	300	2	.	.	PUNCT
ejpam-6243	300	3	st	st	PROPN
ejpam-6243	300	4	.	.	PROPN
ejpam-6243	300	5	univ	univ	PROPN
ejpam-6243	300	6	.	.	PUNCT
ejpam-6243	301	1	ovidius	ovidius	PROPN
ejpam-6243	301	2	constanta	constanta	PROPN
ejpam-6243	301	3	,	,	PUNCT
ejpam-6243	301	4	,	,	PUNCT
ejpam-6243	301	5	27(3	27(3	NUM
ejpam-6243	301	6	)	)	PUNCT
ejpam-6243	301	7	,	,	PUNCT
ejpam-6243	301	8	2019	2019	NUM
ejpam-6243	301	9	.	.	PUNCT
ejpam-6243	302	1	[	[	X
ejpam-6243	302	2	5	5	X
ejpam-6243	302	3	]	]	X
ejpam-6243	302	4	karapinar	karapinar	PROPN
ejpam-6243	302	5	e.	e.	PROPN
ejpam-6243	302	6	aydi	aydi	PROPN
ejpam-6243	302	7	h.	h.	PROPN
ejpam-6243	302	8	,	,	PUNCT
ejpam-6243	302	9	chen	chen	PROPN
ejpam-6243	302	10	c.	c.	PROPN
ejpam-6243	302	11	m.	m.	PROPN
ejpam-6243	302	12	interpolative	interpolative	ADJ
ejpam-6243	302	13	ciric	ciric	ADJ
ejpam-6243	302	14	-	-	PUNCT
ejpam-6243	302	15	reich	reich	NOUN
ejpam-6243	302	16	-	-	PUNCT
ejpam-6243	302	17	rus	rus	NOUN
ejpam-6243	302	18	type	type	NOUN
ejpam-6243	302	19	contractions	contraction	NOUN
ejpam-6243	302	20	via	via	ADP
ejpam-6243	302	21	the	the	DET
ejpam-6243	302	22	branciari	branciari	ADJ
ejpam-6243	302	23	distance	distance	NOUN
ejpam-6243	302	24	,	,	PUNCT
ejpam-6243	302	25	mathematics	mathematic	NOUN
ejpam-6243	302	26	.	.	PUNCT
ejpam-6243	303	1	mathematics	mathematic	NOUN
ejpam-6243	303	2	,	,	PUNCT
ejpam-6243	303	3	,	,	PUNCT
ejpam-6243	303	4	7(1	7(1	NUM
ejpam-6243	303	5	)	)	PUNCT
ejpam-6243	303	6	,	,	PUNCT
ejpam-6243	303	7	2019	2019	NUM
ejpam-6243	303	8	.	.	PUNCT
ejpam-6243	304	1	[	[	X
ejpam-6243	304	2	6	6	NUM
ejpam-6243	304	3	]	]	PUNCT
ejpam-6243	304	4	roldan	roldan	PROPN
ejpam-6243	304	5	lopez	lopez	PROPN
ejpam-6243	304	6	de	de	PROPN
ejpam-6243	304	7	hierro	hierro	PROPN
ejpam-6243	304	8	a.	a.	PROPN
ejpam-6243	304	9	f.	f.	PROPN
ejpam-6243	304	10	aydi	aydi	PROPN
ejpam-6243	304	11	h.	h.	PROPN
ejpam-6243	304	12	,	,	PUNCT
ejpam-6243	304	13	karapinar	karapinar	PROPN
ejpam-6243	304	14	e.	e.	PROPN
ejpam-6243	304	15	interpolative	interpolative	ADJ
ejpam-6243	304	16	ciric	ciric	PROPN
ejpam-6243	304	17	-	-	PUNCT
ejpam-6243	304	18	reich	reich	NOUN
ejpam-6243	304	19	-	-	PUNCT
ejpam-6243	304	20	rustype	rustype	NOUN
ejpam-6243	304	21	contractions	contraction	NOUN
ejpam-6243	304	22	,	,	PUNCT
ejpam-6243	304	23	.	.	PUNCT
ejpam-6243	305	1	mathematics	mathematic	NOUN
ejpam-6243	305	2	,	,	PUNCT
ejpam-6243	305	3	7(1):57	7(1):57	NUM
ejpam-6243	305	4	,	,	PUNCT
ejpam-6243	305	5	2019	2019	NUM
ejpam-6243	305	6	.	.	PUNCT
ejpam-6243	306	1	[	[	X
ejpam-6243	306	2	7	7	X
ejpam-6243	306	3	]	]	X
ejpam-6243	306	4	grandolfi	grandolfi	NOUN
ejpam-6243	306	5	m.	m.	PROPN
ejpam-6243	306	6	bianchini	bianchini	PROPN
ejpam-6243	306	7	r.	r.	PROPN
ejpam-6243	306	8	m.	m.	PROPN
ejpam-6243	306	9	transformazioni	transformazioni	PROPN
ejpam-6243	306	10	di	di	PROPN
ejpam-6243	306	11	tipo	tipo	PROPN
ejpam-6243	306	12	contracttivo	contracttivo	PROPN
ejpam-6243	306	13	generalizzato	generalizzato	ADJ
ejpam-6243	306	14	in	in	ADP
ejpam-6243	306	15	uno	uno	PROPN
ejpam-6243	306	16	spazio	spazio	PROPN
ejpam-6243	306	17	metric	metric	PROPN
ejpam-6243	306	18	.	.	PUNCT
ejpam-6243	307	1	atti	atti	PROPN
ejpam-6243	307	2	acad	acad	PROPN
ejpam-6243	307	3	.	.	PUNCT
ejpam-6243	308	1	naz	naz	PROPN
ejpam-6243	308	2	.	.	PUNCT
ejpam-6243	309	1	lincei	lincei	NOUN
ejpam-6243	309	2	,	,	PUNCT
ejpam-6243	309	3	vii	vii	PROPN
ejpam-6243	309	4	.	.	PROPN
ejpam-6243	309	5	ser	ser	PROPN
ejpam-6243	309	6	.	.	PUNCT
ejpam-6243	310	1	rend	rend	VERB
ejpam-6243	310	2	.	.	PUNCT
ejpam-6243	311	1	cl	cl	NOUN
ejpam-6243	311	2	.	.	PUNCT
ejpam-6243	312	1	sci	sci	PROPN
ejpam-6243	312	2	.	.	PROPN
ejpam-6243	312	3	fis	fis	PROPN
ejpam-6243	312	4	.	.	PUNCT
ejpam-6243	312	5	mat	mat	PROPN
ejpam-6243	312	6	.	.	PUNCT
ejpam-6243	312	7	natur	natur	PROPN
ejpam-6243	312	8	.	.	PROPN
ejpam-6243	313	1	,	,	PUNCT
ejpam-6243	313	2	,	,	PUNCT
ejpam-6243	313	3	45:212–216	45:212–216	NUM
ejpam-6243	313	4	,	,	PUNCT
ejpam-6243	313	5	1968	1968	NUM
ejpam-6243	313	6	.	.	PUNCT
ejpam-6243	314	1	[	[	X
ejpam-6243	314	2	8	8	NUM
ejpam-6243	314	3	]	]	X
ejpam-6243	314	4	karapinar	karapinar	PROPN
ejpam-6243	314	5	e.	e.	PROPN
ejpam-6243	314	6	revisiting	revisit	VERB
ejpam-6243	314	7	the	the	DET
ejpam-6243	314	8	kannan	kannan	PROPN
ejpam-6243	314	9	type	type	NOUN
ejpam-6243	314	10	contractions	contraction	NOUN
ejpam-6243	314	11	via	via	ADP
ejpam-6243	314	12	interpolation	interpolation	NOUN
ejpam-6243	314	13	.	.	PUNCT
ejpam-6243	315	1	adv.theory	adv.theory	ADJ
ejpam-6243	315	2	nonlinear	nonlinear	ADJ
ejpam-6243	315	3	anal	anal	PROPN
ejpam-6243	315	4	.	.	PUNCT
ejpam-6243	316	1	appl	appl	PROPN
ejpam-6243	316	2	.	.	PROPN
ejpam-6243	316	3	,	,	PUNCT
ejpam-6243	316	4	,	,	PUNCT
ejpam-6243	316	5	2:85–87	2:85–87	NUM
ejpam-6243	316	6	,	,	PUNCT
ejpam-6243	316	7	2018	2018	NUM
ejpam-6243	316	8	.	.	PUNCT
ejpam-6243	317	1	[	[	X
ejpam-6243	317	2	9	9	NUM
ejpam-6243	317	3	]	]	PUNCT
ejpam-6243	317	4	e.	e.	PROPN
ejpam-6243	317	5	karapinar	karapinar	PROPN
ejpam-6243	317	6	.	.	PUNCT
ejpam-6243	318	1	recent	recent	ADJ
ejpam-6243	318	2	advances	advance	NOUN
ejpam-6243	318	3	on	on	ADP
ejpam-6243	318	4	metric	metric	ADJ
ejpam-6243	318	5	fixed	fix	VERB
ejpam-6243	318	6	point	point	NOUN
ejpam-6243	318	7	theory	theory	NOUN
ejpam-6243	318	8	:	:	PUNCT
ejpam-6243	318	9	a	a	DET
ejpam-6243	318	10	review	review	NOUN
ejpam-6243	318	11	,	,	PUNCT
ejpam-6243	318	12	.	.	PUNCT
ejpam-6243	318	13	applied	apply	VERB
ejpam-6243	318	14	and	and	CCONJ
ejpam-6243	318	15	computational	computational	ADJ
ejpam-6243	318	16	mathematics	mathematic	NOUN
ejpam-6243	318	17	,	,	PUNCT
ejpam-6243	318	18	,	,	PUNCT
ejpam-6243	318	19	22(1):3–30	22(1):3–30	NUM
ejpam-6243	318	20	,	,	PUNCT
ejpam-6243	318	21	2023	2023	NUM
ejpam-6243	318	22	.	.	PUNCT
ejpam-6243	319	1	[	[	X
ejpam-6243	319	2	10	10	NUM
ejpam-6243	319	3	]	]	PUNCT
ejpam-6243	319	4	aydi	aydi	VERB
ejpam-6243	319	5	h.	h.	PROPN
ejpam-6243	319	6	karapinar	karapinar	PROPN
ejpam-6243	319	7	e.	e.	PROPN
ejpam-6243	319	8	,	,	PUNCT
ejpam-6243	319	9	agarwal	agarwal	PROPN
ejpam-6243	319	10	r.	r.	PROPN
ejpam-6243	319	11	interpolative	interpolative	PROPN
ejpam-6243	319	12	reich	reich	PROPN
ejpam-6243	319	13	-	-	PUNCT
ejpam-6243	319	14	rus	rus	NOUN
ejpam-6243	319	15	-	-	ADJ
ejpam-6243	319	16	ciric	ciric	ADJ
ejpam-6243	319	17	type	type	NOUN
ejpam-6243	319	18	contractions	contraction	NOUN
ejpam-6243	319	19	on	on	ADP
ejpam-6243	319	20	partial	partial	ADJ
ejpam-6243	319	21	metric	metric	ADJ
ejpam-6243	319	22	spaces	space	NOUN
ejpam-6243	319	23	.	.	PUNCT
ejpam-6243	320	1	mathematics	mathematic	NOUN
ejpam-6243	320	2	,	,	PUNCT
ejpam-6243	320	3	6:256	6:256	NUM
ejpam-6243	320	4	,	,	PUNCT
ejpam-6243	320	5	2018	2018	NUM
ejpam-6243	320	6	.	.	PUNCT
ejpam-6243	321	1	[	[	X
ejpam-6243	321	2	11	11	NUM
ejpam-6243	321	3	]	]	PUNCT
ejpam-6243	321	4	aydi	aydi	VERB
ejpam-6243	321	5	h.	h.	PROPN
ejpam-6243	321	6	o	o	PROPN
ejpam-6243	321	7	karapinar	karapinar	PROPN
ejpam-6243	321	8	e.	e.	PROPN
ejpam-6243	321	9	,	,	PUNCT
ejpam-6243	321	10	alqahtani	alqahtani	PROPN
ejpam-6243	322	1	o.	o.	PROPN
ejpam-6243	322	2	on	on	ADP
ejpam-6243	322	3	interpolative	interpolative	ADJ
ejpam-6243	322	4	hardy	hardy	ADJ
ejpam-6243	322	5	-	-	PUNCT
ejpam-6243	322	6	rogers	rogers	NOUN
ejpam-6243	322	7	type	type	NOUN
ejpam-6243	322	8	contractions	contraction	NOUN
ejpam-6243	322	9	.	.	PUNCT
ejpam-6243	323	1	symmetry	symmetry	NOUN
ejpam-6243	323	2	,	,	PUNCT
ejpam-6243	323	3	11(1):8	11(1):8	PROPN
ejpam-6243	323	4	,	,	PUNCT
ejpam-6243	323	5	2019	2019	NUM
ejpam-6243	323	6	.	.	PUNCT
ejpam-6243	324	1	[	[	X
ejpam-6243	324	2	12	12	NUM
ejpam-6243	324	3	]	]	PUNCT
ejpam-6243	324	4	shukla	shukla	NOUN
ejpam-6243	324	5	s.	s.	PROPN
ejpam-6243	324	6	radenovi’c	radenovi’c	PROPN
ejpam-6243	324	7	s.	s.	PROPN
ejpam-6243	324	8	khojasteh	khojasteh	PROPN
ejpam-6243	324	9	,	,	PUNCT
ejpam-6243	324	10	f.	f.	PROPN
ejpam-6243	324	11	new	new	ADJ
ejpam-6243	324	12	approach	approach	NOUN
ejpam-6243	324	13	to	to	ADP
ejpam-6243	324	14	the	the	DET
ejpam-6243	324	15	study	study	NOUN
ejpam-6243	324	16	of	of	ADP
ejpam-6243	324	17	fixed	fix	VERB
ejpam-6243	324	18	point	point	NOUN
ejpam-6243	324	19	theory	theory	NOUN
ejpam-6243	324	20	for	for	ADP
ejpam-6243	324	21	simulation	simulation	NOUN
ejpam-6243	324	22	functions	function	NOUN
ejpam-6243	324	23	,	,	PUNCT
ejpam-6243	324	24	.	.	PUNCT
ejpam-6243	325	1	filomat	filomat	PROPN
ejpam-6243	325	2	,	,	PUNCT
ejpam-6243	325	3	29(6):1189–1194	29(6):1189–1194	PROPN
ejpam-6243	325	4	,	,	PUNCT
ejpam-6243	325	5	2015	2015	NUM
ejpam-6243	325	6	.	.	PUNCT
ejpam-6243	326	1	[	[	X
ejpam-6243	326	2	13	13	NUM
ejpam-6243	326	3	]	]	X
ejpam-6243	326	4	ulam	ulam	PROPN
ejpam-6243	326	5	s.	s.	PROPN
ejpam-6243	326	6	m.	m.	PROPN
ejpam-6243	326	7	problems	problem	NOUN
ejpam-6243	326	8	in	in	ADP
ejpam-6243	326	9	modern	modern	ADJ
ejpam-6243	326	10	mathematics	mathematic	NOUN
ejpam-6243	326	11	.	.	PUNCT
ejpam-6243	327	1	dover	dover	PROPN
ejpam-6243	327	2	publications	publications	PROPN
ejpam-6243	327	3	,	,	PUNCT
ejpam-6243	327	4	inc	inc	PROPN
ejpam-6243	327	5	.	.	PROPN
ejpam-6243	327	6	,	,	PUNCT
ejpam-6243	327	7	,	,	PUNCT
ejpam-6243	327	8	mineola	mineola	PROPN
ejpam-6243	327	9	,	,	PUNCT
ejpam-6243	327	10	new	new	PROPN
ejpam-6243	327	11	york	york	PROPN
ejpam-6243	327	12	,	,	PUNCT
ejpam-6243	327	13	2004	2004	NUM
ejpam-6243	327	14	.	.	PUNCT
ejpam-6243	328	1	[	[	X
ejpam-6243	328	2	14	14	NUM
ejpam-6243	328	3	]	]	X
ejpam-6243	328	4	hyers	hyer	NOUN
ejpam-6243	328	5	d.	d.	PROPN
ejpam-6243	328	6	h.	h.	PROPN
ejpam-6243	328	7	on	on	ADP
ejpam-6243	328	8	the	the	DET
ejpam-6243	328	9	stability	stability	NOUN
ejpam-6243	328	10	of	of	ADP
ejpam-6243	328	11	linear	linear	ADJ
ejpam-6243	328	12	functional	functional	ADJ
ejpam-6243	328	13	equations	equation	NOUN
ejpam-6243	328	14	.	.	PUNCT
ejpam-6243	329	1	proc	proc	PROPN
ejpam-6243	329	2	.	.	PUNCT
ejpam-6243	330	1	natl	natl	PROPN
ejpam-6243	330	2	.	.	PUNCT
ejpam-6243	331	1	acad	acad	PROPN
ejpam-6243	331	2	.	.	PUNCT
ejpam-6243	332	1	sci	sci	PROPN
ejpam-6243	332	2	.	.	PROPN
ejpam-6243	332	3	,	,	PUNCT
ejpam-6243	332	4	usa	usa	PROPN
ejpam-6243	332	5	,	,	PUNCT
ejpam-6243	332	6	,	,	PUNCT
ejpam-6243	332	7	27:222–224	27:222–224	NUM
ejpam-6243	332	8	,	,	PUNCT
ejpam-6243	332	9	1941	1941	NUM
ejpam-6243	332	10	.	.	PUNCT
ejpam-6243	333	1	[	[	X
ejpam-6243	333	2	15	15	NUM
ejpam-6243	333	3	]	]	X
ejpam-6243	333	4	ulam	ulam	PROPN
ejpam-6243	333	5	s.	s.	PROPN
ejpam-6243	333	6	m.	m.	PROPN
ejpam-6243	333	7	a	a	DET
ejpam-6243	333	8	collection	collection	NOUN
ejpam-6243	333	9	of	of	ADP
ejpam-6243	333	10	mathematical	mathematical	ADJ
ejpam-6243	333	11	problems	problem	NOUN
ejpam-6243	333	12	,	,	PUNCT
ejpam-6243	333	13	.	.	PUNCT
ejpam-6243	334	1	interscience	interscience	NOUN
ejpam-6243	334	2	publishers	publisher	NOUN
ejpam-6243	334	3	:	:	PUNCT
ejpam-6243	334	4	,	,	PUNCT
ejpam-6243	334	5	london	london	PROPN
ejpam-6243	334	6	,	,	PUNCT
ejpam-6243	334	7	1940	1940	NUM
ejpam-6243	334	8	.	.	PUNCT
ejpam-6243	335	1	[	[	X
ejpam-6243	335	2	16	16	NUM
ejpam-6243	335	3	]	]	PUNCT
ejpam-6243	335	4	rassias	rassias	PROPN
ejpam-6243	335	5	t.	t.	PROPN
ejpam-6243	335	6	m.	m.	NOUN
ejpam-6243	335	7	on	on	ADP
ejpam-6243	335	8	the	the	DET
ejpam-6243	335	9	stability	stability	NOUN
ejpam-6243	335	10	of	of	ADP
ejpam-6243	335	11	linear	linear	PROPN
ejpam-6243	335	12	mapping	mapping	NOUN
ejpam-6243	335	13	in	in	ADP
ejpam-6243	335	14	banach	banach	NOUN
ejpam-6243	335	15	spaces	space	NOUN
ejpam-6243	335	16	.	.	PUNCT
ejpam-6243	336	1	proc	proc	NOUN
ejpam-6243	336	2	.	.	PUNCT
ejpam-6243	337	1	am	be	AUX
ejpam-6243	337	2	.	.	PUNCT
ejpam-6243	338	1	math	math	NOUN
ejpam-6243	338	2	.	.	PUNCT
ejpam-6243	339	1	soc	soc	PROPN
ejpam-6243	339	2	.	.	PUNCT
ejpam-6243	340	1	,	,	PUNCT
ejpam-6243	340	2	,	,	PUNCT
ejpam-6243	340	3	72:297–300	72:297–300	PROPN
ejpam-6243	340	4	,	,	PUNCT
ejpam-6243	340	5	1978	1978	NUM
ejpam-6243	340	6	.	.	PUNCT
ejpam-6243	341	1	[	[	X
ejpam-6243	341	2	17	17	NUM
ejpam-6243	341	3	]	]	X
ejpam-6243	341	4	kumar	kumar	PROPN
ejpam-6243	341	5	p.	p.	PROPN
ejpam-6243	341	6	mutlu	mutlu	PROPN
ejpam-6243	341	7	a.	a.	PROPN
ejpam-6243	341	8	ramaswamy	ramaswamy	PROPN
ejpam-6243	341	9	r.	r.	PROPN
ejpam-6243	341	10	o.a.a	o.a.a	PROPN
ejpam-6243	341	11	.	.	PUNCT
ejpam-6243	341	12	radenovic	radenovic	PROPN
ejpam-6243	341	13	s.	s.	PROPN
ejpam-6243	341	14	kumar	kumar	PROPN
ejpam-6243	341	15	,	,	PUNCT
ejpam-6243	341	16	m.	m.	NOUN
ejpam-6243	341	17	ulam	ulam	PROPN
ejpam-6243	341	18	-	-	PUNCT
ejpam-6243	341	19	hyers	hyer	NOUN
ejpam-6243	341	20	r	r	NOUN
ejpam-6243	341	21	ramaswamy	ramaswamy	NOUN
ejpam-6243	341	22	et	et	PROPN
ejpam-6243	341	23	al	al	PROPN
ejpam-6243	341	24	.	.	PUNCT
ejpam-6243	341	25	/	/	SYM
ejpam-6243	341	26	eur	eur	PROPN
ejpam-6243	341	27	.	.	PUNCT
ejpam-6243	342	1	j.	j.	PROPN
ejpam-6243	342	2	pure	pure	PROPN
ejpam-6243	342	3	appl	appl	PROPN
ejpam-6243	342	4	.	.	PROPN
ejpam-6243	342	5	math	math	PROPN
ejpam-6243	342	6	,	,	PUNCT
ejpam-6243	342	7	18	18	NUM
ejpam-6243	342	8	(	(	PUNCT
ejpam-6243	342	9	3	3	NUM
ejpam-6243	342	10	)	)	PUNCT
ejpam-6243	342	11	(	(	PUNCT
ejpam-6243	342	12	2025	2025	NUM
ejpam-6243	342	13	)	)	PUNCT
ejpam-6243	342	14	,	,	PUNCT
ejpam-6243	342	15	6243	6243	NUM
ejpam-6243	342	16	14	14	NUM
ejpam-6243	342	17	of	of	ADP
ejpam-6243	342	18	14	14	NUM
ejpam-6243	342	19	stability	stability	NOUN
ejpam-6243	342	20	and	and	CCONJ
ejpam-6243	342	21	well	well	ADV
ejpam-6243	342	22	-	-	PUNCT
ejpam-6243	342	23	posedness	posedness	NOUN
ejpam-6243	342	24	of	of	ADP
ejpam-6243	342	25	fixed	fix	VERB
ejpam-6243	342	26	point	point	NOUN
ejpam-6243	342	27	problems	problem	NOUN
ejpam-6243	342	28	in	in	ADP
ejpam-6243	342	29	c*-algebra	c*-algebra	PROPN
ejpam-6243	342	30	valued	value	VERB
ejpam-6243	342	31	bipolar	bipolar	ADJ
ejpam-6243	342	32	b	b	NOUN
ejpam-6243	342	33	-	-	PUNCT
ejpam-6243	342	34	metric	metric	ADJ
ejpam-6243	342	35	spaces	space	NOUN
ejpam-6243	342	36	.	.	PUNCT
ejpam-6243	343	1	mathematics	mathematic	NOUN
ejpam-6243	343	2	,	,	PUNCT
ejpam-6243	343	3	11	11	NUM
ejpam-6243	343	4	:	:	PUNCT
ejpam-6243	343	5	doi.org/10.3390	doi.org/10.3390	PROPN
ejpam-6243	343	6	/	/	SYM
ejpam-6243	343	7	math11102323	math11102323	PROPN
ejpam-6243	343	8	,	,	PUNCT
ejpam-6243	343	9	2023	2023	NUM
ejpam-6243	343	10	.	.	PUNCT
ejpam-6243	344	1	[	[	X
ejpam-6243	344	2	18	18	NUM
ejpam-6243	344	3	]	]	PUNCT
ejpam-6243	344	4	rus	rus	PROPN
ejpam-6243	344	5	i.	i.	PROPN
ejpam-6243	344	6	a.	a.	PROPN
ejpam-6243	344	7	generalized	generalize	VERB
ejpam-6243	344	8	contractions	contraction	NOUN
ejpam-6243	344	9	and	and	CCONJ
ejpam-6243	344	10	applications	application	NOUN
ejpam-6243	344	11	,	,	PUNCT
ejpam-6243	344	12	.	.	PUNCT
ejpam-6243	345	1	cluj	cluj	PROPN
ejpam-6243	345	2	univ	univ	PROPN
ejpam-6243	345	3	press	press	PROPN
ejpam-6243	345	4	,	,	PUNCT
ejpam-6243	345	5	clui	clui	NOUN
ejpam-6243	345	6	-	-	PUNCT
ejpam-6243	345	7	napoca	napoca	PROPN
ejpam-6243	345	8	,	,	PUNCT
ejpam-6243	345	9	romania	romania	PROPN
ejpam-6243	345	10	,	,	PUNCT
ejpam-6243	345	11	2001	2001	NUM
ejpam-6243	345	12	.	.	PUNCT
ejpam-6243	346	1	[	[	X
ejpam-6243	346	2	19	19	NUM
ejpam-6243	346	3	]	]	PUNCT
ejpam-6243	346	4	popescu	popescu	NOUN
ejpam-6243	346	5	o.	o.	PROPN
ejpam-6243	347	1	some	some	DET
ejpam-6243	347	2	new	new	ADJ
ejpam-6243	347	3	fixed	fix	VERB
ejpam-6243	347	4	point	point	NOUN
ejpam-6243	347	5	theorems	theorem	NOUN
ejpam-6243	347	6	for	for	ADP
ejpam-6243	347	7	-geraghty	-geraghty	ADJ
ejpam-6243	347	8	contractive	contractive	ADJ
ejpam-6243	347	9	type	type	NOUN
ejpam-6243	347	10	maps	map	NOUN
ejpam-6243	347	11	in	in	ADP
ejpam-6243	347	12	metric	metric	ADJ
ejpam-6243	347	13	spaces	space	NOUN
ejpam-6243	347	14	.	.	PUNCT
ejpam-6243	348	1	fixed	fix	VERB
ejpam-6243	348	2	point	point	NOUN
ejpam-6243	348	3	theory	theory	NOUN
ejpam-6243	348	4	appl	appl	PROPN
ejpam-6243	348	5	.	.	PROPN
ejpam-6243	348	6	,	,	PUNCT
ejpam-6243	348	7	2	2	NUM
ejpam-6243	348	8	,	,	PUNCT
ejpam-6243	348	9	2014:190	2014:190	NUM
ejpam-6243	348	10	,	,	PUNCT
ejpam-6243	348	11	2014	2014	NUM
ejpam-6243	348	12	.	.	PUNCT