id	sid	tid	token	lemma	pos
cana-5953	1	1	communications	communication	NOUN
cana-5953	1	2	on	on	ADP
cana-5953	1	3	applied	apply	VERB
cana-5953	1	4	nonlinear	nonlinear	ADJ
cana-5953	1	5	analysis	analysis	NOUN
cana-5953	1	6	issn	issn	NOUN
cana-5953	1	7	:	:	PUNCT
cana-5953	1	8	1074	1074	NUM
cana-5953	1	9	-	-	PUNCT
cana-5953	1	10	133x	133x	NUM
cana-5953	1	11	vol	vol	VERB
cana-5953	1	12	32	32	NUM
cana-5953	1	13	no	no	NOUN
cana-5953	1	14	.	.	PUNCT
cana-5953	2	1	10s	10	NOUN
cana-5953	2	2	(	(	PUNCT
cana-5953	2	3	2025	2025	NUM
cana-5953	2	4	)	)	PUNCT
cana-5953	2	5	3160	3160	NUM
cana-5953	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	3	1	some	some	DET
cana-5953	3	2	common	common	ADJ
cana-5953	3	3	fixed	fix	VERB
cana-5953	3	4	point	point	NOUN
cana-5953	3	5	theorems	theorem	NOUN
cana-5953	3	6	for	for	ADP
cana-5953	3	7	weakly	weakly	ADJ
cana-5953	3	8	compatible	compatible	ADJ
cana-5953	3	9	mappings	mapping	NOUN
cana-5953	3	10	in	in	ADP
cana-5953	3	11	rectangular	rectangular	ADJ
cana-5953	3	12	s	s	ADJ
cana-5953	3	13	-	-	ADJ
cana-5953	3	14	metric	metric	ADJ
cana-5953	3	15	spaces	space	NOUN
cana-5953	3	16	t.	t.	PROPN
cana-5953	3	17	r.	r.	PROPN
cana-5953	3	18	shimpi1	shimpi1	PROPN
cana-5953	3	19	and	and	CCONJ
cana-5953	3	20	s.	s.	PROPN
cana-5953	3	21	g.	g.	PROPN
cana-5953	3	22	dapke2	dapke2	PROPN
cana-5953	4	1	1department	1department	NUM
cana-5953	4	2	of	of	ADP
cana-5953	4	3	mathematics	mathematic	NOUN
cana-5953	4	4	,	,	PUNCT
cana-5953	4	5	gf	gf	PROPN
cana-5953	4	6	’s	’s	PART
cana-5953	4	7	godavari	godavari	PROPN
cana-5953	4	8	college	college	PROPN
cana-5953	4	9	of	of	ADP
cana-5953	4	10	engineering	engineering	PROPN
cana-5953	4	11	,	,	PUNCT
cana-5953	4	12	jalgaon	jalgaon	PROPN
cana-5953	4	13	,	,	PUNCT
cana-5953	4	14	india	india	PROPN
cana-5953	4	15	.	.	PUNCT
cana-5953	4	16	email	email	NOUN
cana-5953	4	17	.	.	PUNCT
cana-5953	5	1	trushalishimpi@gmail.com	trushalishimpi@gmail.com	X
cana-5953	6	1	2department	2department	NUM
cana-5953	6	2	of	of	ADP
cana-5953	6	3	mathematics	mathematic	NOUN
cana-5953	6	4	,	,	PUNCT
cana-5953	6	5	iqra	iqra	NOUN
cana-5953	6	6	’s	’s	PART
cana-5953	6	7	h.	h.	PROPN
cana-5953	6	8	j.	j.	PROPN
cana-5953	6	9	thim	thim	PROPN
cana-5953	6	10	college	college	PROPN
cana-5953	6	11	of	of	ADP
cana-5953	6	12	arts	art	NOUN
cana-5953	6	13	and	and	CCONJ
cana-5953	6	14	science	science	NOUN
cana-5953	6	15	,	,	PUNCT
cana-5953	6	16	mehrun	mehrun	NOUN
cana-5953	6	17	,	,	PUNCT
cana-5953	6	18	jalgaon	jalgaon	PROPN
cana-5953	6	19	,	,	PUNCT
cana-5953	6	20	india	india	PROPN
cana-5953	6	21	.	.	PUNCT
cana-5953	6	22	email	email	NOUN
cana-5953	6	23	.	.	PUNCT
cana-5953	7	1	sadashivgdapke@gmail.com	sadashivgdapke@gmail.com	X
cana-5953	8	1	article	article	NOUN
cana-5953	8	2	history	history	NOUN
cana-5953	8	3	:	:	PUNCT
cana-5953	8	4	received	receive	VERB
cana-5953	8	5	:	:	PUNCT
cana-5953	8	6	02	02	NUM
cana-5953	8	7	-	-	PUNCT
cana-5953	8	8	01	01	NUM
cana-5953	8	9	-	-	PUNCT
cana-5953	8	10	2025	2025	NUM
cana-5953	8	11	revised	revise	VERB
cana-5953	8	12	:	:	PUNCT
cana-5953	8	13	25	25	NUM
cana-5953	8	14	-	-	PUNCT
cana-5953	8	15	02	02	NUM
cana-5953	8	16	-	-	PUNCT
cana-5953	8	17	2025	2025	NUM
cana-5953	8	18	accepted	accept	VERB
cana-5953	8	19	:	:	PUNCT
cana-5953	8	20	20	20	NUM
cana-5953	8	21	-	-	SYM
cana-5953	8	22	03	03	NUM
cana-5953	8	23	-	-	PUNCT
cana-5953	8	24	2025	2025	NUM
cana-5953	8	25	abstract	abstract	NOUN
cana-5953	8	26	:	:	PUNCT
cana-5953	8	27	this	this	DET
cana-5953	8	28	paper	paper	NOUN
cana-5953	8	29	explores	explore	VERB
cana-5953	8	30	the	the	DET
cana-5953	8	31	existence	existence	NOUN
cana-5953	8	32	of	of	ADP
cana-5953	8	33	common	common	ADJ
cana-5953	8	34	fixed	fix	VERB
cana-5953	8	35	points	point	NOUN
cana-5953	8	36	for	for	ADP
cana-5953	8	37	weakly	weakly	ADJ
cana-5953	8	38	compatible	compatible	ADJ
cana-5953	8	39	mappings	mapping	NOUN
cana-5953	8	40	in	in	ADP
cana-5953	8	41	the	the	DET
cana-5953	8	42	framework	framework	NOUN
cana-5953	8	43	of	of	ADP
cana-5953	8	44	rectangular	rectangular	ADJ
cana-5953	8	45	s	s	ADJ
cana-5953	8	46	-	-	ADJ
cana-5953	8	47	metric	metric	ADJ
cana-5953	8	48	spaces	space	NOUN
cana-5953	8	49	,	,	PUNCT
cana-5953	8	50	a	a	DET
cana-5953	8	51	generalization	generalization	NOUN
cana-5953	8	52	of	of	ADP
cana-5953	8	53	traditional	traditional	ADJ
cana-5953	8	54	s	s	NOUN
cana-5953	8	55	-	-	ADJ
cana-5953	8	56	metric	metric	ADJ
cana-5953	8	57	spaces	space	NOUN
cana-5953	8	58	.	.	PUNCT
cana-5953	9	1	by	by	ADP
cana-5953	9	2	introducing	introduce	VERB
cana-5953	9	3	suitable	suitable	ADJ
cana-5953	9	4	contractive	contractive	ADJ
cana-5953	9	5	conditions	condition	NOUN
cana-5953	9	6	,	,	PUNCT
cana-5953	9	7	we	we	PRON
cana-5953	9	8	establish	establish	VERB
cana-5953	9	9	new	new	ADJ
cana-5953	9	10	common	common	ADJ
cana-5953	9	11	fixed	fix	VERB
cana-5953	9	12	point	point	NOUN
cana-5953	9	13	theorems	theorem	NOUN
cana-5953	9	14	that	that	PRON
cana-5953	9	15	unify	unify	VERB
cana-5953	9	16	and	and	CCONJ
cana-5953	9	17	extend	extend	VERB
cana-5953	9	18	various	various	ADJ
cana-5953	9	19	known	know	VERB
cana-5953	9	20	results	result	NOUN
cana-5953	9	21	in	in	ADP
cana-5953	9	22	the	the	DET
cana-5953	9	23	field	field	NOUN
cana-5953	9	24	of	of	ADP
cana-5953	9	25	fixed	fix	VERB
cana-5953	9	26	point	point	NOUN
cana-5953	9	27	theory	theory	NOUN
cana-5953	9	28	.	.	PUNCT
cana-5953	10	1	these	these	DET
cana-5953	10	2	theorems	theorem	NOUN
cana-5953	10	3	provide	provide	VERB
cana-5953	10	4	broader	broad	ADJ
cana-5953	10	5	applicability	applicability	NOUN
cana-5953	10	6	and	and	CCONJ
cana-5953	10	7	deeper	deep	ADJ
cana-5953	10	8	insights	insight	NOUN
cana-5953	10	9	into	into	ADP
cana-5953	10	10	the	the	DET
cana-5953	10	11	behaviour	behaviour	NOUN
cana-5953	10	12	of	of	ADP
cana-5953	10	13	such	such	ADJ
cana-5953	10	14	mappings	mapping	NOUN
cana-5953	10	15	under	under	ADP
cana-5953	10	16	generalized	generalized	ADJ
cana-5953	10	17	metric	metric	ADJ
cana-5953	10	18	conditions	condition	NOUN
cana-5953	10	19	.	.	PUNCT
cana-5953	11	1	the	the	DET
cana-5953	11	2	developed	develop	VERB
cana-5953	11	3	results	result	NOUN
cana-5953	11	4	not	not	PART
cana-5953	11	5	only	only	ADV
cana-5953	11	6	contribute	contribute	VERB
cana-5953	11	7	to	to	ADP
cana-5953	11	8	the	the	DET
cana-5953	11	9	theoretical	theoretical	ADJ
cana-5953	11	10	understanding	understanding	NOUN
cana-5953	11	11	but	but	CCONJ
cana-5953	11	12	also	also	ADV
cana-5953	11	13	enhance	enhance	VERB
cana-5953	11	14	the	the	DET
cana-5953	11	15	scope	scope	NOUN
cana-5953	11	16	of	of	ADP
cana-5953	11	17	fixed	fix	VERB
cana-5953	11	18	point	point	NOUN
cana-5953	11	19	analysis	analysis	NOUN
cana-5953	11	20	in	in	ADP
cana-5953	11	21	more	more	ADJ
cana-5953	11	22	generalized	generalized	ADJ
cana-5953	11	23	settings	setting	NOUN
cana-5953	11	24	.	.	PUNCT
cana-5953	12	1	to	to	PART
cana-5953	12	2	illustrate	illustrate	VERB
cana-5953	12	3	the	the	DET
cana-5953	12	4	validity	validity	NOUN
cana-5953	12	5	and	and	CCONJ
cana-5953	12	6	applicability	applicability	NOUN
cana-5953	12	7	of	of	ADP
cana-5953	12	8	the	the	DET
cana-5953	12	9	main	main	ADJ
cana-5953	12	10	results	result	NOUN
cana-5953	12	11	,	,	PUNCT
cana-5953	12	12	we	we	PRON
cana-5953	12	13	provide	provide	VERB
cana-5953	12	14	several	several	ADJ
cana-5953	12	15	relevant	relevant	ADJ
cana-5953	12	16	examples	example	NOUN
cana-5953	12	17	.	.	PUNCT
cana-5953	13	1	these	these	DET
cana-5953	13	2	examples	example	NOUN
cana-5953	13	3	serve	serve	VERB
cana-5953	13	4	to	to	PART
cana-5953	13	5	clarify	clarify	VERB
cana-5953	13	6	the	the	DET
cana-5953	13	7	theoretical	theoretical	ADJ
cana-5953	13	8	findings	finding	NOUN
cana-5953	13	9	and	and	CCONJ
cana-5953	13	10	demonstrate	demonstrate	VERB
cana-5953	13	11	how	how	SCONJ
cana-5953	13	12	the	the	DET
cana-5953	13	13	established	establish	VERB
cana-5953	13	14	theorems	theorem	NOUN
cana-5953	13	15	can	can	AUX
cana-5953	13	16	be	be	AUX
cana-5953	13	17	effectively	effectively	ADV
cana-5953	13	18	applied	apply	VERB
cana-5953	13	19	in	in	ADP
cana-5953	13	20	practice	practice	NOUN
cana-5953	13	21	.	.	PUNCT
cana-5953	14	1	overall	overall	ADV
cana-5953	14	2	,	,	PUNCT
cana-5953	14	3	the	the	DET
cana-5953	14	4	study	study	NOUN
cana-5953	14	5	offers	offer	VERB
cana-5953	14	6	a	a	DET
cana-5953	14	7	meaningful	meaningful	ADJ
cana-5953	14	8	advancement	advancement	NOUN
cana-5953	14	9	in	in	ADP
cana-5953	14	10	the	the	DET
cana-5953	14	11	general	general	ADJ
cana-5953	14	12	theory	theory	NOUN
cana-5953	14	13	of	of	ADP
cana-5953	14	14	fixed	fix	VERB
cana-5953	14	15	points	point	NOUN
cana-5953	14	16	within	within	ADP
cana-5953	14	17	the	the	DET
cana-5953	14	18	context	context	NOUN
cana-5953	14	19	of	of	ADP
cana-5953	14	20	rectangular	rectangular	ADJ
cana-5953	14	21	s	s	ADJ
cana-5953	14	22	-	-	ADJ
cana-5953	14	23	metric	metric	ADJ
cana-5953	14	24	spaces	space	NOUN
cana-5953	14	25	.	.	PUNCT
cana-5953	15	1	keywords	keyword	NOUN
cana-5953	15	2	:	:	PUNCT
cana-5953	15	3	s	s	X
cana-5953	15	4	-	-	ADJ
cana-5953	15	5	metric	metric	ADJ
cana-5953	15	6	space	space	NOUN
cana-5953	15	7	;	;	PUNCT
cana-5953	15	8	rectangular	rectangular	ADJ
cana-5953	15	9	s	s	ADJ
cana-5953	15	10	-	-	ADJ
cana-5953	15	11	metric	metric	ADJ
cana-5953	15	12	space	space	NOUN
cana-5953	15	13	;	;	PUNCT
cana-5953	15	14	common	common	ADJ
cana-5953	15	15	fixed	fix	VERB
cana-5953	15	16	point	point	NOUN
cana-5953	15	17	.	.	PUNCT
cana-5953	16	1	1	1	X
cana-5953	16	2	.	.	X
cana-5953	16	3	introduction	introduction	NOUN
cana-5953	16	4	fixed	fix	VERB
cana-5953	16	5	-	-	PUNCT
cana-5953	16	6	point	point	NOUN
cana-5953	16	7	theory	theory	NOUN
cana-5953	16	8	finds	find	VERB
cana-5953	16	9	applications	application	NOUN
cana-5953	16	10	in	in	ADP
cana-5953	16	11	numerous	numerous	ADJ
cana-5953	16	12	disciplines	discipline	NOUN
cana-5953	16	13	.	.	PUNCT
cana-5953	17	1	within	within	ADP
cana-5953	17	2	this	this	DET
cana-5953	17	3	context	context	NOUN
cana-5953	17	4	,	,	PUNCT
cana-5953	17	5	coincidence	coincidence	NOUN
cana-5953	17	6	points	point	NOUN
cana-5953	17	7	,	,	PUNCT
cana-5953	17	8	which	which	PRON
cana-5953	17	9	generalize	generalize	VERB
cana-5953	17	10	fixed	fix	VERB
cana-5953	17	11	points	point	NOUN
cana-5953	17	12	,	,	PUNCT
cana-5953	17	13	hold	hold	VERB
cana-5953	17	14	significant	significant	ADJ
cana-5953	17	15	importance	importance	NOUN
cana-5953	17	16	.	.	PUNCT
cana-5953	18	1	jungck	jungck	PROPN
cana-5953	19	1	[	[	X
cana-5953	19	2	5	5	NUM
cana-5953	19	3	]	]	PUNCT
cana-5953	19	4	was	be	AUX
cana-5953	19	5	the	the	DET
cana-5953	19	6	first	first	ADJ
cana-5953	19	7	to	to	PART
cana-5953	19	8	introduce	introduce	VERB
cana-5953	19	9	the	the	DET
cana-5953	19	10	concept	concept	NOUN
cana-5953	19	11	of	of	ADP
cana-5953	19	12	commuting	commute	VERB
cana-5953	19	13	mappings	mapping	NOUN
cana-5953	19	14	and	and	CCONJ
cana-5953	19	15	to	to	PART
cana-5953	19	16	define	define	VERB
cana-5953	19	17	common	common	ADJ
cana-5953	19	18	fixed	fix	VERB
cana-5953	19	19	points	point	NOUN
cana-5953	19	20	in	in	ADP
cana-5953	19	21	metric	metric	ADJ
cana-5953	19	22	spaces	space	NOUN
cana-5953	19	23	.	.	PUNCT
cana-5953	20	1	although	although	SCONJ
cana-5953	20	2	commuting	commute	VERB
cana-5953	20	3	mappings	mapping	NOUN
cana-5953	20	4	possess	possess	VERB
cana-5953	20	5	strong	strong	ADJ
cana-5953	20	6	structural	structural	ADJ
cana-5953	20	7	properties	property	NOUN
cana-5953	20	8	,	,	PUNCT
cana-5953	20	9	their	their	PRON
cana-5953	20	10	stringent	stringent	ADJ
cana-5953	20	11	conditions	condition	NOUN
cana-5953	20	12	restrict	restrict	VERB
cana-5953	20	13	their	their	PRON
cana-5953	20	14	applicability	applicability	NOUN
cana-5953	20	15	.	.	PUNCT
cana-5953	21	1	to	to	PART
cana-5953	21	2	overcome	overcome	VERB
cana-5953	21	3	this	this	DET
cana-5953	21	4	limitation	limitation	NOUN
cana-5953	21	5	,	,	PUNCT
cana-5953	21	6	sessa	sessa	NOUN
cana-5953	22	1	[	[	X
cana-5953	22	2	15	15	NUM
cana-5953	22	3	]	]	PUNCT
cana-5953	22	4	proposed	propose	VERB
cana-5953	22	5	the	the	DET
cana-5953	22	6	concept	concept	NOUN
cana-5953	22	7	of	of	ADP
cana-5953	22	8	weakly	weakly	ADJ
cana-5953	22	9	commuting	commuting	NOUN
cana-5953	22	10	mappings	mapping	NOUN
cana-5953	22	11	,	,	PUNCT
cana-5953	22	12	offering	offer	VERB
cana-5953	22	13	a	a	DET
cana-5953	22	14	more	more	ADV
cana-5953	22	15	flexible	flexible	ADJ
cana-5953	22	16	alternative	alternative	NOUN
cana-5953	22	17	.	.	PUNCT
cana-5953	23	1	expanding	expand	VERB
cana-5953	23	2	on	on	ADP
cana-5953	23	3	this	this	DET
cana-5953	23	4	notion	notion	NOUN
cana-5953	23	5	,	,	PUNCT
cana-5953	23	6	jungck	jungck	NOUN
cana-5953	23	7	[	[	X
cana-5953	23	8	6	6	NUM
cana-5953	23	9	]	]	PUNCT
cana-5953	23	10	introduced	introduce	VERB
cana-5953	23	11	compatible	compatible	ADJ
cana-5953	23	12	mappings	mapping	NOUN
cana-5953	23	13	,	,	PUNCT
cana-5953	23	14	which	which	PRON
cana-5953	23	15	further	far	ADV
cana-5953	23	16	generalized	generalize	VERB
cana-5953	23	17	weak	weak	ADJ
cana-5953	23	18	commutatively	commutatively	ADV
cana-5953	23	19	.	.	PUNCT
cana-5953	24	1	he	he	PRON
cana-5953	24	2	showed	show	VERB
cana-5953	24	3	that	that	SCONJ
cana-5953	24	4	every	every	DET
cana-5953	24	5	weakly	weakly	ADJ
cana-5953	24	6	commuting	commuting	NOUN
cana-5953	24	7	pair	pair	NOUN
cana-5953	24	8	is	be	AUX
cana-5953	24	9	compatible	compatible	ADJ
cana-5953	24	10	,	,	PUNCT
cana-5953	24	11	although	although	SCONJ
cana-5953	24	12	the	the	DET
cana-5953	24	13	converse	converse	NOUN
cana-5953	24	14	is	be	AUX
cana-5953	24	15	not	not	PART
cana-5953	24	16	necessarily	necessarily	ADV
cana-5953	24	17	true	true	ADJ
cana-5953	24	18	.	.	PUNCT
cana-5953	25	1	later	later	ADV
cana-5953	25	2	,	,	PUNCT
cana-5953	25	3	jungck	jungck	PROPN
cana-5953	26	1	[	[	X
cana-5953	26	2	7	7	NUM
cana-5953	26	3	,	,	PUNCT
cana-5953	26	4	8	8	NUM
cana-5953	26	5	]	]	PUNCT
cana-5953	26	6	introduced	introduce	VERB
cana-5953	26	7	the	the	DET
cana-5953	26	8	concept	concept	NOUN
cana-5953	26	9	of	of	ADP
cana-5953	26	10	weak	weak	ADJ
cana-5953	26	11	compatibility	compatibility	NOUN
cana-5953	26	12	,	,	PUNCT
cana-5953	26	13	defining	define	VERB
cana-5953	26	14	a	a	DET
cana-5953	26	15	pair	pair	NOUN
cana-5953	26	16	of	of	ADP
cana-5953	26	17	self	self	NOUN
cana-5953	26	18	-	-	PUNCT
cana-5953	26	19	maps	map	NOUN
cana-5953	26	20	as	as	ADV
cana-5953	26	21	weakly	weakly	ADV
cana-5953	26	22	compatible	compatible	ADJ
cana-5953	26	23	if	if	SCONJ
cana-5953	26	24	they	they	PRON
cana-5953	26	25	commute	commute	VERB
cana-5953	26	26	at	at	ADP
cana-5953	26	27	their	their	PRON
cana-5953	26	28	coincidence	coincidence	NOUN
cana-5953	26	29	points	point	NOUN
cana-5953	26	30	.	.	PUNCT
cana-5953	27	1	building	build	VERB
cana-5953	27	2	upon	upon	SCONJ
cana-5953	27	3	this	this	DET
cana-5953	27	4	foundational	foundational	ADJ
cana-5953	27	5	work	work	NOUN
cana-5953	27	6	,	,	PUNCT
cana-5953	27	7	several	several	ADJ
cana-5953	27	8	researchers	researcher	NOUN
cana-5953	27	9	have	have	AUX
cana-5953	27	10	investigated	investigate	VERB
cana-5953	27	11	coincidence	coincidence	NOUN
cana-5953	27	12	points	point	NOUN
cana-5953	27	13	for	for	ADP
cana-5953	27	14	various	various	ADJ
cana-5953	27	15	types	type	NOUN
cana-5953	27	16	of	of	ADP
cana-5953	27	17	mappings	mapping	NOUN
cana-5953	27	18	in	in	ADP
cana-5953	27	19	metric	metric	ADJ
cana-5953	27	20	spaces	space	NOUN
cana-5953	27	21	,	,	PUNCT
cana-5953	27	22	where	where	SCONJ
cana-5953	27	23	distances	distance	NOUN
cana-5953	27	24	are	be	AUX
cana-5953	27	25	traditionally	traditionally	ADV
cana-5953	27	26	measured	measure	VERB
cana-5953	27	27	between	between	ADP
cana-5953	27	28	pairs	pair	NOUN
cana-5953	27	29	of	of	ADP
cana-5953	27	30	points	point	NOUN
cana-5953	27	31	.	.	PUNCT
cana-5953	28	1	while	while	SCONJ
cana-5953	28	2	effective	effective	ADJ
cana-5953	28	3	in	in	ADP
cana-5953	28	4	many	many	ADJ
cana-5953	28	5	settings	setting	NOUN
cana-5953	28	6	,	,	PUNCT
cana-5953	28	7	this	this	DET
cana-5953	28	8	pairwise	pairwise	NOUN
cana-5953	28	9	approach	approach	NOUN
cana-5953	28	10	can	can	AUX
cana-5953	28	11	be	be	AUX
cana-5953	28	12	limiting	limit	VERB
cana-5953	28	13	in	in	ADP
cana-5953	28	14	more	more	ADJ
cana-5953	28	15	complex	complex	ADJ
cana-5953	28	16	scenarios	scenario	NOUN
cana-5953	28	17	.	.	PUNCT
cana-5953	29	1	to	to	PART
cana-5953	29	2	address	address	VERB
cana-5953	29	3	such	such	ADJ
cana-5953	29	4	limitations	limitation	NOUN
cana-5953	29	5	,	,	PUNCT
cana-5953	29	6	sedghi	sedghi	VERB
cana-5953	29	7	et	et	PROPN
cana-5953	29	8	al	al	PROPN
cana-5953	29	9	.	.	PUNCT
cana-5953	30	1	[	[	X
cana-5953	30	2	13	13	NUM
cana-5953	30	3	]	]	PUNCT
cana-5953	30	4	introduced	introduce	VERB
cana-5953	30	5	the	the	DET
cana-5953	30	6	concept	concept	NOUN
cana-5953	30	7	of	of	ADP
cana-5953	30	8	s	s	NOUN
cana-5953	30	9	-	-	ADJ
cana-5953	30	10	metric	metric	ADJ
cana-5953	30	11	spaces	space	NOUN
cana-5953	30	12	,	,	PUNCT
cana-5953	30	13	where	where	SCONJ
cana-5953	30	14	distances	distance	NOUN
cana-5953	30	15	are	be	AUX
cana-5953	30	16	defined	define	VERB
cana-5953	30	17	over	over	ADP
cana-5953	30	18	triplets	triplet	NOUN
cana-5953	30	19	of	of	ADP
cana-5953	30	20	points	point	NOUN
cana-5953	30	21	,	,	PUNCT
cana-5953	30	22	thereby	thereby	ADV
cana-5953	30	23	broadening	broaden	VERB
cana-5953	30	24	the	the	DET
cana-5953	30	25	scope	scope	NOUN
cana-5953	30	26	of	of	ADP
cana-5953	30	27	fixed	fix	VERB
cana-5953	30	28	-	-	PUNCT
cana-5953	30	29	point	point	NOUN
cana-5953	30	30	theory	theory	NOUN
cana-5953	30	31	.	.	PUNCT
cana-5953	31	1	abbas	abbas	PROPN
cana-5953	31	2	and	and	CCONJ
cana-5953	31	3	jungck	jungck	NOUN
cana-5953	32	1	[	[	X
cana-5953	32	2	2	2	X
cana-5953	32	3	]	]	PUNCT
cana-5953	32	4	established	establish	VERB
cana-5953	32	5	results	result	NOUN
cana-5953	32	6	on	on	ADP
cana-5953	32	7	coincidence	coincidence	NOUN
cana-5953	32	8	and	and	CCONJ
cana-5953	32	9	common	common	ADJ
cana-5953	32	10	fixed	fix	VERB
cana-5953	32	11	points	point	NOUN
cana-5953	32	12	under	under	ADP
cana-5953	32	13	contractive	contractive	ADJ
cana-5953	32	14	mailto:trushalishimpi@gmail.com	mailto:trushalishimpi@gmail.com	X
cana-5953	32	15	mailto:sadashivgdapke@gmail.com	mailto:sadashivgdapke@gmail.com	X
cana-5953	32	16	communications	communication	NOUN
cana-5953	32	17	on	on	ADP
cana-5953	32	18	applied	apply	VERB
cana-5953	32	19	nonlinear	nonlinear	ADJ
cana-5953	32	20	analysis	analysis	NOUN
cana-5953	32	21	issn	issn	NOUN
cana-5953	32	22	:	:	PUNCT
cana-5953	32	23	1074	1074	NUM
cana-5953	32	24	-	-	PUNCT
cana-5953	32	25	133x	133x	NUM
cana-5953	32	26	vol	vol	VERB
cana-5953	32	27	32	32	NUM
cana-5953	32	28	no	no	NOUN
cana-5953	32	29	.	.	PUNCT
cana-5953	33	1	10s	10	NOUN
cana-5953	33	2	(	(	PUNCT
cana-5953	33	3	2025	2025	NUM
cana-5953	33	4	)	)	PUNCT
cana-5953	33	5	3161	3161	NUM
cana-5953	33	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	33	7	conditions	condition	NOUN
cana-5953	33	8	in	in	ADP
cana-5953	33	9	cone	cone	NOUN
cana-5953	33	10	metric	metric	ADJ
cana-5953	33	11	spaces	space	NOUN
cana-5953	33	12	,	,	PUNCT
cana-5953	33	13	while	while	SCONJ
cana-5953	33	14	abbas	abbas	PROPN
cana-5953	33	15	and	and	CCONJ
cana-5953	33	16	rhoades	rhoade	NOUN
cana-5953	33	17	[	[	X
cana-5953	33	18	3	3	NUM
cana-5953	33	19	]	]	X
cana-5953	33	20	further	far	ADV
cana-5953	33	21	generalized	generalize	VERB
cana-5953	33	22	fixed	fix	VERB
cana-5953	33	23	-	-	PUNCT
cana-5953	33	24	point	point	NOUN
cana-5953	33	25	theorems	theorem	NOUN
cana-5953	33	26	by	by	ADP
cana-5953	33	27	eliminating	eliminate	VERB
cana-5953	33	28	the	the	DET
cana-5953	33	29	requirement	requirement	NOUN
cana-5953	33	30	of	of	ADP
cana-5953	33	31	commutatively	commutatively	ADV
cana-5953	33	32	.	.	PUNCT
cana-5953	34	1	saluja	saluja	PROPN
cana-5953	35	1	[	[	X
cana-5953	35	2	12	12	NUM
cana-5953	35	3	]	]	PUNCT
cana-5953	35	4	extended	extend	VERB
cana-5953	35	5	fixed	fix	VERB
cana-5953	35	6	-	-	PUNCT
cana-5953	35	7	point	point	NOUN
cana-5953	35	8	results	result	NOUN
cana-5953	35	9	for	for	ADP
cana-5953	35	10	weak	weak	ADJ
cana-5953	35	11	contractions	contraction	NOUN
cana-5953	35	12	within	within	ADP
cana-5953	35	13	complete	complete	ADJ
cana-5953	35	14	s	s	NOUN
cana-5953	35	15	-	-	ADJ
cana-5953	35	16	metric	metric	ADJ
cana-5953	35	17	spaces	space	NOUN
cana-5953	35	18	.	.	PUNCT
cana-5953	36	1	in	in	ADP
cana-5953	36	2	this	this	DET
cana-5953	36	3	paper	paper	NOUN
cana-5953	36	4	,	,	PUNCT
cana-5953	36	5	we	we	PRON
cana-5953	36	6	investigate	investigate	VERB
cana-5953	36	7	coincidence	coincidence	NOUN
cana-5953	36	8	and	and	CCONJ
cana-5953	36	9	common	common	ADJ
cana-5953	36	10	fixed	fix	VERB
cana-5953	36	11	points	point	NOUN
cana-5953	36	12	for	for	ADP
cana-5953	36	13	weakly	weakly	ADJ
cana-5953	36	14	compatible	compatible	ADJ
cana-5953	36	15	mappings	mapping	NOUN
cana-5953	36	16	satisfying	satisfy	VERB
cana-5953	36	17	weak	weak	ADJ
cana-5953	36	18	contraction	contraction	NOUN
cana-5953	36	19	conditions	condition	NOUN
cana-5953	36	20	in	in	ADP
cana-5953	36	21	complete	complete	ADJ
cana-5953	36	22	s	s	NOUN
cana-5953	36	23	-	-	ADJ
cana-5953	36	24	metric	metric	ADJ
cana-5953	36	25	spaces	space	NOUN
cana-5953	36	26	.	.	PUNCT
cana-5953	37	1	the	the	DET
cana-5953	37	2	theoretical	theoretical	ADJ
cana-5953	37	3	results	result	NOUN
cana-5953	37	4	are	be	AUX
cana-5953	37	5	illustrated	illustrate	VERB
cana-5953	37	6	with	with	ADP
cana-5953	37	7	relevant	relevant	ADJ
cana-5953	37	8	examples	example	NOUN
cana-5953	37	9	.	.	PUNCT
cana-5953	38	1	2	2	X
cana-5953	38	2	.	.	X
cana-5953	38	3	methodology	methodology	NOUN
cana-5953	38	4	we	we	PRON
cana-5953	38	5	begin	begin	VERB
cana-5953	38	6	by	by	ADP
cana-5953	38	7	introducing	introduce	VERB
cana-5953	38	8	the	the	DET
cana-5953	38	9	necessary	necessary	ADJ
cana-5953	38	10	definitions	definition	NOUN
cana-5953	38	11	and	and	CCONJ
cana-5953	38	12	properties	property	NOUN
cana-5953	38	13	of	of	ADP
cana-5953	38	14	rectangular	rectangular	ADJ
cana-5953	38	15	s	s	ADJ
cana-5953	38	16	-	-	ADJ
cana-5953	38	17	metric	metric	ADJ
cana-5953	38	18	spaces	space	NOUN
cana-5953	38	19	and	and	CCONJ
cana-5953	38	20	weakly	weakly	ADJ
cana-5953	38	21	compatible	compatible	ADJ
cana-5953	38	22	mappings	mapping	NOUN
cana-5953	38	23	.	.	PUNCT
cana-5953	39	1	using	use	VERB
cana-5953	39	2	suitable	suitable	ADJ
cana-5953	39	3	contractive	contractive	ADJ
cana-5953	39	4	conditions	condition	NOUN
cana-5953	39	5	,	,	PUNCT
cana-5953	39	6	we	we	PRON
cana-5953	39	7	formulate	formulate	VERB
cana-5953	39	8	and	and	CCONJ
cana-5953	39	9	prove	prove	VERB
cana-5953	39	10	new	new	ADJ
cana-5953	39	11	common	common	ADJ
cana-5953	39	12	fixed	fix	VERB
cana-5953	39	13	point	point	NOUN
cana-5953	39	14	theorems	theorem	NOUN
cana-5953	39	15	within	within	ADP
cana-5953	39	16	this	this	DET
cana-5953	39	17	generalized	generalized	ADJ
cana-5953	39	18	framework	framework	NOUN
cana-5953	39	19	.	.	PUNCT
cana-5953	40	1	the	the	DET
cana-5953	40	2	results	result	NOUN
cana-5953	40	3	are	be	AUX
cana-5953	40	4	supported	support	VERB
cana-5953	40	5	by	by	ADP
cana-5953	40	6	illustrative	illustrative	ADJ
cana-5953	40	7	examples	example	NOUN
cana-5953	40	8	that	that	PRON
cana-5953	40	9	verify	verify	VERB
cana-5953	40	10	the	the	DET
cana-5953	40	11	applicability	applicability	NOUN
cana-5953	40	12	of	of	ADP
cana-5953	40	13	the	the	DET
cana-5953	40	14	proposed	propose	VERB
cana-5953	40	15	theorems	theorem	NOUN
cana-5953	40	16	.	.	PUNCT
cana-5953	41	1	this	this	DET
cana-5953	41	2	approach	approach	NOUN
cana-5953	41	3	extends	extend	VERB
cana-5953	41	4	existing	exist	VERB
cana-5953	41	5	results	result	NOUN
cana-5953	41	6	in	in	ADP
cana-5953	41	7	fixed	fix	VERB
cana-5953	41	8	point	point	NOUN
cana-5953	41	9	theory	theory	NOUN
cana-5953	41	10	to	to	ADP
cana-5953	41	11	a	a	DET
cana-5953	41	12	broader	broad	ADJ
cana-5953	41	13	class	class	NOUN
cana-5953	41	14	of	of	ADP
cana-5953	41	15	metric	metric	ADJ
cana-5953	41	16	spaces	space	NOUN
cana-5953	41	17	.	.	PUNCT
cana-5953	42	1	3	3	X
cana-5953	42	2	.	.	X
cana-5953	42	3	preliminaries	preliminary	NOUN
cana-5953	42	4	in	in	ADP
cana-5953	42	5	this	this	DET
cana-5953	42	6	section	section	NOUN
cana-5953	42	7	,	,	PUNCT
cana-5953	42	8	we	we	PRON
cana-5953	42	9	present	present	VERB
cana-5953	42	10	the	the	DET
cana-5953	42	11	basic	basic	ADJ
cana-5953	42	12	definitions	definition	NOUN
cana-5953	42	13	and	and	CCONJ
cana-5953	42	14	foundational	foundational	ADJ
cana-5953	42	15	concepts	concept	NOUN
cana-5953	42	16	necessary	necessary	ADJ
cana-5953	42	17	for	for	ADP
cana-5953	42	18	the	the	DET
cana-5953	42	19	development	development	NOUN
cana-5953	42	20	of	of	ADP
cana-5953	42	21	our	our	PRON
cana-5953	42	22	main	main	ADJ
cana-5953	42	23	results	result	NOUN
cana-5953	42	24	.	.	PUNCT
cana-5953	43	1	definition	definition	NOUN
cana-5953	43	2	3.1	3.1	NUM
cana-5953	43	3	[	[	X
cana-5953	43	4	14	14	NUM
cana-5953	43	5	]	]	PUNCT
cana-5953	43	6	let	let	VERB
cana-5953	43	7	𝑋	𝑋	NOUN
cana-5953	43	8	be	be	AUX
cana-5953	43	9	a	a	DET
cana-5953	43	10	non	non	X
cana-5953	43	11	empty	empty	ADJ
cana-5953	43	12	set	set	NOUN
cana-5953	43	13	and	and	CCONJ
cana-5953	43	14	s	s	NOUN
cana-5953	43	15	:	:	PUNCT
cana-5953	43	16	x3	x3	ADJ
cana-5953	43	17	→	→	SYM
cana-5953	43	18	ℝ+	ℝ+	PUNCT
cana-5953	43	19	be	be	AUX
cana-5953	43	20	a	a	DET
cana-5953	43	21	function	function	NOUN
cana-5953	43	22	satisfying	satisfy	VERB
cana-5953	43	23	the	the	DET
cana-5953	43	24	properties	property	NOUN
cana-5953	43	25	given	give	VERB
cana-5953	43	26	below	below	ADV
cana-5953	43	27	;	;	PUNCT
cana-5953	43	28	•	•	NUM
cana-5953	43	29	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	43	30	,	,	PUNCT
cana-5953	43	31	𝑦	𝑦	NOUN
cana-5953	43	32	,	,	PUNCT
cana-5953	43	33	𝑧	𝑧	NOUN
cana-5953	43	34	)	)	PUNCT
cana-5953	43	35	=	=	SYM
cana-5953	43	36	0	0	PUNCT
cana-5953	44	1	if	if	SCONJ
cana-5953	44	2	and	and	CCONJ
cana-5953	44	3	only	only	ADV
cana-5953	44	4	if	if	SCONJ
cana-5953	44	5	𝑥	𝑥	PRON
cana-5953	44	6	=	=	SYM
cana-5953	44	7	𝑦	𝑦	SYM
cana-5953	44	8	=	=	SYM
cana-5953	44	9	𝑧	𝑧	ADJ
cana-5953	44	10	•	•	NUM
cana-5953	44	11	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	44	12	,	,	PUNCT
cana-5953	44	13	𝑦	𝑦	NOUN
cana-5953	44	14	,	,	PUNCT
cana-5953	44	15	𝑧	𝑧	NOUN
cana-5953	44	16	)	)	PUNCT
cana-5953	44	17	≤	≤	NOUN
cana-5953	44	18	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	44	19	,	,	PUNCT
cana-5953	44	20	𝑥	𝑥	NOUN
cana-5953	44	21	,	,	PUNCT
cana-5953	44	22	𝑎	𝑎	NOUN
cana-5953	44	23	)	)	PUNCT
cana-5953	44	24	+	+	CCONJ
cana-5953	44	25	𝑆(𝑦	𝑆(𝑦	PROPN
cana-5953	44	26	,	,	PUNCT
cana-5953	44	27	𝑦	𝑦	NOUN
cana-5953	44	28	,	,	PUNCT
cana-5953	44	29	𝑎	𝑎	NOUN
cana-5953	44	30	)	)	PUNCT
cana-5953	44	31	+	+	CCONJ
cana-5953	44	32	𝑆(𝑧	𝑆(𝑧	PROPN
cana-5953	44	33	,	,	PUNCT
cana-5953	44	34	𝑧	𝑧	PROPN
cana-5953	44	35	,	,	PUNCT
cana-5953	44	36	𝑎	𝑎	NOUN
cana-5953	44	37	)	)	PUNCT
cana-5953	44	38	for	for	ADP
cana-5953	44	39	all	all	DET
cana-5953	44	40	𝑎	𝑎	PROPN
cana-5953	44	41	,	,	PUNCT
cana-5953	44	42	𝑥	𝑥	NOUN
cana-5953	44	43	,	,	PUNCT
cana-5953	44	44	𝑦	𝑦	NOUN
cana-5953	44	45	,	,	PUNCT
cana-5953	44	46	𝑧	𝑧	DET
cana-5953	44	47	∈	∈	PROPN
cana-5953	44	48	𝑋	𝑋	NOUN
cana-5953	44	49	(	(	PUNCT
cana-5953	44	50	termed	term	VERB
cana-5953	44	51	as	as	ADP
cana-5953	44	52	rectangle	rectangle	ADJ
cana-5953	44	53	inequality	inequality	NOUN
cana-5953	44	54	)	)	PUNCT
cana-5953	44	55	.	.	PUNCT
cana-5953	45	1	then	then	ADV
cana-5953	45	2	,	,	PUNCT
cana-5953	45	3	the	the	DET
cana-5953	45	4	pair	pair	NOUN
cana-5953	45	5	(	(	PUNCT
cana-5953	45	6	𝑋	𝑋	PROPN
cana-5953	45	7	,	,	PUNCT
cana-5953	45	8	𝑆	𝑆	PROPN
cana-5953	45	9	)	)	PUNCT
cana-5953	45	10	is	be	AUX
cana-5953	45	11	named	name	VERB
cana-5953	45	12	as	as	ADP
cana-5953	45	13	a	a	DET
cana-5953	45	14	s	s	NOUN
cana-5953	45	15	-	-	ADJ
cana-5953	45	16	metric	metric	ADJ
cana-5953	45	17	space	space	NOUN
cana-5953	45	18	.	.	PUNCT
cana-5953	46	1	definition	definition	NOUN
cana-5953	46	2	3.2	3.2	NUM
cana-5953	46	3	[	[	X
cana-5953	46	4	1	1	NUM
cana-5953	46	5	]	]	PUNCT
cana-5953	46	6	let	let	VERB
cana-5953	46	7	x	x	PRON
cana-5953	46	8	be	be	AUX
cana-5953	46	9	a	a	DET
cana-5953	46	10	non	non	ADJ
cana-5953	46	11	void	void	NOUN
cana-5953	46	12	set	set	NOUN
cana-5953	46	13	and	and	CCONJ
cana-5953	46	14	s	s	NOUN
cana-5953	46	15	:	:	PUNCT
cana-5953	46	16	x3	x3	ADJ
cana-5953	46	17	→	→	SYM
cana-5953	46	18	ℝ+	ℝ+	PUNCT
cana-5953	46	19	be	be	AUX
cana-5953	46	20	a	a	DET
cana-5953	46	21	function	function	NOUN
cana-5953	46	22	satisfying	satisfy	VERB
cana-5953	46	23	the	the	DET
cana-5953	46	24	conditions	condition	NOUN
cana-5953	46	25	listed	list	VERB
cana-5953	46	26	below	below	ADP
cana-5953	46	27	:	:	PUNCT
cana-5953	46	28	•	•	NUM
cana-5953	46	29	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	46	30	,	,	PUNCT
cana-5953	46	31	𝑦	𝑦	NOUN
cana-5953	46	32	,	,	PUNCT
cana-5953	46	33	𝑧	𝑧	NOUN
cana-5953	46	34	)	)	PUNCT
cana-5953	46	35	=	=	SYM
cana-5953	46	36	0	0	PUNCT
cana-5953	47	1	if	if	SCONJ
cana-5953	47	2	and	and	CCONJ
cana-5953	47	3	only	only	ADV
cana-5953	47	4	if	if	SCONJ
cana-5953	47	5	𝑥	𝑥	PRON
cana-5953	47	6	=	=	SYM
cana-5953	47	7	𝑦	𝑦	SYM
cana-5953	47	8	=	=	SYM
cana-5953	47	9	𝑧	𝑧	ADJ
cana-5953	47	10	•	•	NUM
cana-5953	47	11	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	47	12	,	,	PUNCT
cana-5953	47	13	𝑦	𝑦	NOUN
cana-5953	47	14	,	,	PUNCT
cana-5953	47	15	𝑧	𝑧	NOUN
cana-5953	47	16	)	)	PUNCT
cana-5953	47	17	≤	≤	NOUN
cana-5953	47	18	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	47	19	,	,	PUNCT
cana-5953	47	20	𝑥	𝑥	NOUN
cana-5953	47	21	,	,	PUNCT
cana-5953	47	22	𝑎	𝑎	NOUN
cana-5953	47	23	)	)	PUNCT
cana-5953	47	24	+	+	CCONJ
cana-5953	47	25	𝑆(𝑦	𝑆(𝑦	PROPN
cana-5953	47	26	,	,	PUNCT
cana-5953	47	27	𝑦	𝑦	NOUN
cana-5953	47	28	,	,	PUNCT
cana-5953	47	29	𝑎	𝑎	NOUN
cana-5953	47	30	)	)	PUNCT
cana-5953	47	31	+	+	CCONJ
cana-5953	47	32	𝑆(𝑧	𝑆(𝑧	PROPN
cana-5953	47	33	,	,	PUNCT
cana-5953	47	34	𝑧	𝑧	PROPN
cana-5953	47	35	,	,	PUNCT
cana-5953	47	36	𝑎	𝑎	NOUN
cana-5953	47	37	)	)	PUNCT
cana-5953	47	38	for	for	ADP
cana-5953	47	39	all	all	DET
cana-5953	47	40	𝑥	𝑥	PROPN
cana-5953	47	41	,	,	PUNCT
cana-5953	47	42	𝑦	𝑦	NOUN
cana-5953	47	43	,	,	PUNCT
cana-5953	47	44	𝑧	𝑧	DET
cana-5953	47	45	∈	∈	PROPN
cana-5953	47	46	𝑋	𝑋	NOUN
cana-5953	47	47	and	and	CCONJ
cana-5953	47	48	all	all	DET
cana-5953	47	49	distinct	distinct	ADJ
cana-5953	47	50	points	point	NOUN
cana-5953	47	51	𝑎	𝑎	PRON
cana-5953	47	52	∈	∈	NOUN
cana-5953	47	53	𝑋	𝑋	NOUN
cana-5953	47	54	−	−	PROPN
cana-5953	47	55	{	{	PUNCT
cana-5953	47	56	𝑥	𝑥	PROPN
cana-5953	47	57	,	,	PUNCT
cana-5953	47	58	𝑦	𝑦	NOUN
cana-5953	47	59	,	,	PUNCT
cana-5953	47	60	𝑧	𝑧	NOUN
cana-5953	47	61	}	}	PUNCT
cana-5953	47	62	.	.	PUNCT
cana-5953	48	1	then	then	ADV
cana-5953	48	2	,	,	PUNCT
cana-5953	48	3	the	the	DET
cana-5953	48	4	pair	pair	NOUN
cana-5953	48	5	(	(	PUNCT
cana-5953	48	6	𝑋	𝑋	PROPN
cana-5953	48	7	,	,	PUNCT
cana-5953	48	8	𝑆	𝑆	PROPN
cana-5953	48	9	)	)	PUNCT
cana-5953	48	10	is	be	AUX
cana-5953	48	11	referred	refer	VERB
cana-5953	48	12	as	as	ADP
cana-5953	48	13	a	a	DET
cana-5953	48	14	rectangular	rectangular	ADJ
cana-5953	48	15	s	s	ADJ
cana-5953	48	16	-	-	ADJ
cana-5953	48	17	metric	metric	ADJ
cana-5953	48	18	space	space	NOUN
cana-5953	48	19	.	.	PUNCT
cana-5953	49	1	definition	definition	NOUN
cana-5953	49	2	3.3	3.3	NUM
cana-5953	50	1	[	[	X
cana-5953	50	2	14	14	NUM
cana-5953	50	3	]	]	X
cana-5953	50	4	let	let	VERB
cana-5953	50	5	(	(	PUNCT
cana-5953	50	6	𝑋	𝑋	PROPN
cana-5953	50	7	,	,	PUNCT
cana-5953	50	8	𝑆	𝑆	PROPN
cana-5953	50	9	)	)	PUNCT
cana-5953	50	10	be	be	VERB
cana-5953	50	11	an	an	DET
cana-5953	50	12	s	s	NOUN
cana-5953	50	13	-	-	ADJ
cana-5953	50	14	metric	metric	ADJ
cana-5953	50	15	space	space	NOUN
cana-5953	50	16	and	and	CCONJ
cana-5953	50	17	𝐴	𝐴	PROPN
cana-5953	50	18	⊂	⊂	PROPN
cana-5953	50	19	𝑋.	𝑋.	PROPN
cana-5953	50	20	•	•	ADP
cana-5953	50	21	a	a	DET
cana-5953	50	22	sequence	sequence	NOUN
cana-5953	50	23	{	{	PUNCT
cana-5953	50	24	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	50	25	}	}	PUNCT
cana-5953	50	26	in	in	ADP
cana-5953	50	27	𝑋	𝑋	PROPN
cana-5953	50	28	is	be	AUX
cana-5953	50	29	converge	converge	ADJ
cana-5953	50	30	to	to	ADP
cana-5953	50	31	𝑥	𝑥	PRON
cana-5953	50	32	if	if	SCONJ
cana-5953	50	33	𝑆(𝑥𝑛	𝑆(𝑥𝑛	PROPN
cana-5953	50	34	,	,	PUNCT
cana-5953	50	35	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	50	36	,	,	PUNCT
cana-5953	50	37	𝑥	𝑥	NOUN
cana-5953	50	38	)	)	PUNCT
cana-5953	50	39	→	→	SYM
cana-5953	50	40	0	0	NUM
cana-5953	50	41	𝑎𝑠	𝑎𝑠	PROPN
cana-5953	50	42	𝑛	𝑛	PROPN
cana-5953	50	43	→	→	PUNCT
cana-5953	50	44	∞.	∞.	PROPN
cana-5953	50	45	in	in	ADP
cana-5953	50	46	other	other	ADJ
cana-5953	50	47	words	word	NOUN
cana-5953	50	48	,	,	PUNCT
cana-5953	50	49	for	for	ADP
cana-5953	50	50	every	every	DET
cana-5953	50	51	𝜖	𝜖	X
cana-5953	50	52	>	>	X
cana-5953	50	53	0	0	PUNCT
cana-5953	50	54	there	there	PRON
cana-5953	50	55	exists	exist	VERB
cana-5953	50	56	𝑛0	𝑛0	VERB
cana-5953	50	57	∈	∈	NOUN
cana-5953	50	58	𝑁	𝑁	PROPN
cana-5953	50	59	such	such	ADJ
cana-5953	50	60	that	that	PRON
cana-5953	50	61	for	for	SCONJ
cana-5953	50	62	all	all	DET
cana-5953	50	63	𝑛	𝑛	DET
cana-5953	50	64	≥	≥	NOUN
cana-5953	50	65	𝑛0	𝑛0	VERB
cana-5953	50	66	,	,	PUNCT
cana-5953	50	67	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	50	68	,	,	PUNCT
cana-5953	50	69	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	50	70	,	,	PUNCT
cana-5953	50	71	𝑥	𝑥	NOUN
cana-5953	50	72	)	)	PUNCT
cana-5953	50	73	<	<	X
cana-5953	50	74	𝜖.	𝜖.	NOUN
cana-5953	50	75	in	in	ADP
cana-5953	50	76	this	this	DET
cana-5953	50	77	case	case	NOUN
cana-5953	50	78	,	,	PUNCT
cana-5953	50	79	we	we	PRON
cana-5953	50	80	write	write	VERB
cana-5953	50	81	𝑙𝑖𝑚𝑛→∞	𝑙𝑖𝑚𝑛→∞	PROPN
cana-5953	50	82	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	50	83	=	=	SYM
cana-5953	50	84	𝑥	𝑥	PROPN
cana-5953	51	1	and	and	CCONJ
cana-5953	51	2	we	we	PRON
cana-5953	51	3	say	say	VERB
cana-5953	51	4	that	that	SCONJ
cana-5953	51	5	𝑥	𝑥	PROPN
cana-5953	51	6	is	be	AUX
cana-5953	51	7	the	the	DET
cana-5953	51	8	limit	limit	NOUN
cana-5953	51	9	of	of	ADP
cana-5953	51	10	{	{	PUNCT
cana-5953	51	11	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	51	12	}	}	PUNCT
cana-5953	51	13	in	in	ADP
cana-5953	51	14	𝑋.	𝑋.	PROPN
cana-5953	51	15	•	•	ADP
cana-5953	51	16	a	a	DET
cana-5953	51	17	sequence	sequence	NOUN
cana-5953	51	18	{	{	PUNCT
cana-5953	51	19	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	51	20	}	}	PUNCT
cana-5953	51	21	in	in	ADP
cana-5953	51	22	𝑋	𝑋	PROPN
cana-5953	51	23	is	be	AUX
cana-5953	51	24	reffered	reffere	VERB
cana-5953	51	25	as	as	ADP
cana-5953	51	26	cauchy	cauchy	NOUN
cana-5953	51	27	sequence	sequence	NOUN
cana-5953	51	28	if	if	SCONJ
cana-5953	51	29	for	for	ADP
cana-5953	51	30	each	each	DET
cana-5953	51	31	𝜖	𝜖	X
cana-5953	51	32	>	>	X
cana-5953	51	33	0	0	NUM
cana-5953	51	34	,	,	PUNCT
cana-5953	51	35	there	there	PRON
cana-5953	51	36	exists	exist	VERB
cana-5953	51	37	𝑛0	𝑛0	VERB
cana-5953	51	38	∈	∈	NOUN
cana-5953	51	39	𝑁	𝑁	PROPN
cana-5953	51	40	such	such	ADJ
cana-5953	51	41	that	that	PRON
cana-5953	51	42	for	for	ADP
cana-5953	51	43	each	each	DET
cana-5953	51	44	𝑛	𝑛	NOUN
cana-5953	51	45	,	,	PUNCT
cana-5953	51	46	𝑚	𝑚	X
cana-5953	51	47	≥	≥	NOUN
cana-5953	51	48	𝑛0	𝑛0	VERB
cana-5953	51	49	,	,	PUNCT
cana-5953	51	50	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	51	51	,	,	PUNCT
cana-5953	51	52	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	51	53	,	,	PUNCT
cana-5953	51	54	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	51	55	)	)	PUNCT
cana-5953	51	56	<	<	X
cana-5953	51	57	𝜖.	𝜖.	NOUN
cana-5953	51	58	•	•	ADP
cana-5953	51	59	the	the	DET
cana-5953	51	60	𝑆-metric	𝑆-metric	ADJ
cana-5953	51	61	space	space	NOUN
cana-5953	51	62	(	(	PUNCT
cana-5953	51	63	𝑋	𝑋	PROPN
cana-5953	51	64	,	,	PUNCT
cana-5953	51	65	𝑆	𝑆	PROPN
cana-5953	51	66	)	)	PUNCT
cana-5953	51	67	is	be	AUX
cana-5953	51	68	said	say	VERB
cana-5953	51	69	to	to	PART
cana-5953	51	70	be	be	AUX
cana-5953	51	71	complete	complete	ADJ
cana-5953	51	72	if	if	SCONJ
cana-5953	51	73	every	every	DET
cana-5953	51	74	cauchy	cauchy	ADJ
cana-5953	51	75	sequence	sequence	NOUN
cana-5953	51	76	in	in	ADP
cana-5953	51	77	𝑋	𝑋	PROPN
cana-5953	51	78	converges	converge	NOUN
cana-5953	51	79	to	to	ADP
cana-5953	51	80	a	a	DET
cana-5953	51	81	limit	limit	NOUN
cana-5953	51	82	in	in	ADP
cana-5953	51	83	𝑋.	𝑋.	PROPN
cana-5953	51	84	communications	communication	NOUN
cana-5953	51	85	on	on	ADP
cana-5953	51	86	applied	apply	VERB
cana-5953	51	87	nonlinear	nonlinear	ADJ
cana-5953	51	88	analysis	analysis	NOUN
cana-5953	51	89	issn	issn	NOUN
cana-5953	51	90	:	:	PUNCT
cana-5953	51	91	1074	1074	NUM
cana-5953	51	92	-	-	PUNCT
cana-5953	51	93	133x	133x	NUM
cana-5953	51	94	vol	vol	VERB
cana-5953	51	95	32	32	NUM
cana-5953	51	96	no	no	NOUN
cana-5953	51	97	.	.	PUNCT
cana-5953	52	1	10s	10	NOUN
cana-5953	52	2	(	(	PUNCT
cana-5953	52	3	2025	2025	NUM
cana-5953	52	4	)	)	PUNCT
cana-5953	52	5	3162	3162	NUM
cana-5953	52	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	52	7	definition	definition	NOUN
cana-5953	52	8	3.4	3.4	NUM
cana-5953	52	9	[	[	X
cana-5953	52	10	12	12	NUM
cana-5953	52	11	]	]	X
cana-5953	52	12	let	let	NOUN
cana-5953	52	13	(	(	PUNCT
cana-5953	52	14	𝑋	𝑋	PROPN
cana-5953	52	15	,	,	PUNCT
cana-5953	52	16	𝑆	𝑆	PROPN
cana-5953	52	17	)	)	PUNCT
cana-5953	52	18	be	be	VERB
cana-5953	52	19	a	a	DET
cana-5953	52	20	𝑆-metric	𝑆-metric	ADJ
cana-5953	52	21	space	space	NOUN
cana-5953	52	22	.	.	PUNCT
cana-5953	53	1	a	a	DET
cana-5953	53	2	mapping	mapping	NOUN
cana-5953	53	3	𝑇	𝑇	NOUN
cana-5953	53	4	:	:	PUNCT
cana-5953	53	5	𝑋	𝑋	PROPN
cana-5953	53	6	→	→	SYM
cana-5953	53	7	𝑋	𝑋	PROPN
cana-5953	53	8	is	be	AUX
cana-5953	53	9	said	say	VERB
cana-5953	53	10	to	to	PART
cana-5953	53	11	be	be	AUX
cana-5953	53	12	a	a	DET
cana-5953	53	13	weak	weak	ADJ
cana-5953	53	14	contraction	contraction	NOUN
cana-5953	53	15	on	on	ADP
cana-5953	53	16	𝑋	𝑋	PROPN
cana-5953	53	17	if	if	SCONJ
cana-5953	53	18	there	there	PRON
cana-5953	53	19	exists	exist	VERB
cana-5953	53	20	a	a	DET
cana-5953	53	21	function	function	NOUN
cana-5953	53	22	𝜓	𝜓	NOUN
cana-5953	53	23	:	:	PUNCT
cana-5953	53	24	[	[	X
cana-5953	53	25	0	0	NUM
cana-5953	53	26	,	,	PUNCT
cana-5953	53	27	∞	∞	PROPN
cana-5953	53	28	)	)	PUNCT
cana-5953	53	29	→	→	PUNCT
cana-5953	54	1	[	[	X
cana-5953	54	2	0	0	NUM
cana-5953	54	3	,	,	PUNCT
cana-5953	54	4	∞	∞	PROPN
cana-5953	54	5	)	)	PUNCT
cana-5953	54	6	with	with	ADP
cana-5953	54	7	𝜓(𝑡	𝜓(𝑡	PROPN
cana-5953	54	8	)	)	PUNCT
cana-5953	54	9	=	=	SYM
cana-5953	54	10	0	0	PUNCT
cana-5953	55	1	if	if	SCONJ
cana-5953	55	2	and	and	CCONJ
cana-5953	55	3	only	only	ADV
cana-5953	55	4	if	if	SCONJ
cana-5953	55	5	𝑡	𝑡	X
cana-5953	55	6	=	=	VERB
cana-5953	55	7	0	0	NUM
cana-5953	55	8	that	that	PRON
cana-5953	55	9	satisfies	satisfy	VERB
cana-5953	55	10	𝑆(𝑇𝑥	𝑆(𝑇𝑥	NOUN
cana-5953	55	11	,	,	PUNCT
cana-5953	55	12	𝑇𝑥	𝑇𝑥	NOUN
cana-5953	55	13	,	,	PUNCT
cana-5953	55	14	𝑇𝑦	𝑇𝑦	PROPN
cana-5953	55	15	)	)	PUNCT
cana-5953	55	16	≤	≤	NOUN
cana-5953	55	17	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	55	18	,	,	PUNCT
cana-5953	55	19	𝑥	𝑥	PROPN
cana-5953	55	20	,	,	PUNCT
cana-5953	55	21	𝑦	𝑦	NOUN
cana-5953	55	22	)	)	PUNCT
cana-5953	55	23	−	−	PROPN
cana-5953	55	24	𝛿𝜓(𝑆(𝑥	𝛿𝜓(𝑆(𝑥	PROPN
cana-5953	55	25	,	,	PUNCT
cana-5953	55	26	𝑥	𝑥	PROPN
cana-5953	55	27	,	,	PUNCT
cana-5953	55	28	𝑦	𝑦	NOUN
cana-5953	55	29	)	)	PUNCT
cana-5953	55	30	)	)	PUNCT
cana-5953	55	31	for	for	ADP
cana-5953	55	32	all	all	PRON
cana-5953	55	33	𝑥	𝑥	PROPN
cana-5953	55	34	,	,	PUNCT
cana-5953	55	35	𝑦	𝑦	NOUN
cana-5953	55	36	∈	∈	PROPN
cana-5953	55	37	𝑋	𝑋	PROPN
cana-5953	55	38	,	,	PUNCT
cana-5953	55	39	where	where	SCONJ
cana-5953	55	40	0	0	NUM
cana-5953	55	41	≤	≤	NOUN
cana-5953	55	42	𝛿	𝛿	PRON
cana-5953	55	43	<	<	X
cana-5953	55	44	1	1	NUM
cana-5953	55	45	.	.	PUNCT
cana-5953	55	46	example	example	NOUN
cana-5953	55	47	3.5	3.5	NUM
cana-5953	55	48	let	let	VERB
cana-5953	55	49	x	x	PUNCT
cana-5953	55	50	=	=	SYM
cana-5953	55	51	r	r	NOUN
cana-5953	55	52	and	and	CCONJ
cana-5953	55	53	𝑆	𝑆	PROPN
cana-5953	55	54	:	:	PUNCT
cana-5953	55	55	𝑋	𝑋	PROPN
cana-5953	55	56	×	×	NOUN
cana-5953	55	57	𝑋	𝑋	PROPN
cana-5953	55	58	×	×	NOUN
cana-5953	55	59	𝑋	𝑋	PROPN
cana-5953	55	60	→	→	SYM
cana-5953	55	61	𝑅+	𝑅+	NUM
cana-5953	55	62	be	be	AUX
cana-5953	55	63	a	a	DET
cana-5953	55	64	function	function	NOUN
cana-5953	55	65	such	such	ADJ
cana-5953	55	66	that	that	SCONJ
cana-5953	55	67	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	55	68	,	,	PUNCT
cana-5953	55	69	𝑥	𝑥	NOUN
cana-5953	55	70	,	,	PUNCT
cana-5953	55	71	𝑦	𝑦	NOUN
cana-5953	55	72	)	)	PUNCT
cana-5953	55	73	=	=	PRON
cana-5953	55	74	{	{	PUNCT
cana-5953	56	1	9𝑥2	9𝑥2	NUM
cana-5953	57	1	+	+	NUM
cana-5953	57	2	𝑦2	𝑦2	NOUN
cana-5953	57	3	if𝑥	if𝑥	PROPN
cana-5953	57	4	≠	≠	PROPN
cana-5953	57	5	𝑦	𝑦	PROPN
cana-5953	57	6	,	,	PUNCT
cana-5953	57	7	0	0	NUM
cana-5953	57	8	if𝑥	if𝑥	PROPN
cana-5953	57	9	=	=	SYM
cana-5953	57	10	𝑦	𝑦	PROPN
cana-5953	57	11	,	,	PUNCT
cana-5953	57	12	(	(	PUNCT
cana-5953	57	13	3.1	3.1	NUM
cana-5953	57	14	)	)	PUNCT
cana-5953	57	15	for	for	ADP
cana-5953	57	16	all	all	PRON
cana-5953	57	17	𝑥	𝑥	PROPN
cana-5953	57	18	,	,	PUNCT
cana-5953	57	19	𝑦	𝑦	PROPN
cana-5953	57	20	∈	∈	PROPN
cana-5953	57	21	𝑋.	𝑋.	PROPN
cana-5953	57	22	then	then	ADV
cana-5953	57	23	(	(	PUNCT
cana-5953	57	24	𝑋	𝑋	PROPN
cana-5953	57	25	,	,	PUNCT
cana-5953	57	26	𝑆	𝑆	PROPN
cana-5953	57	27	)	)	PUNCT
cana-5953	57	28	becomes	become	VERB
cana-5953	57	29	a	a	DET
cana-5953	57	30	𝑆-metric	𝑆-metric	ADJ
cana-5953	57	31	space	space	NOUN
cana-5953	57	32	.	.	PUNCT
cana-5953	58	1	let	let	VERB
cana-5953	58	2	𝑓	𝑓	X
cana-5953	58	3	:	:	PUNCT
cana-5953	58	4	𝑋	𝑋	PROPN
cana-5953	58	5	→	→	SYM
cana-5953	58	6	𝑋	𝑋	PROPN
cana-5953	58	7	is	be	AUX
cana-5953	58	8	defined	define	VERB
cana-5953	58	9	as	as	ADP
cana-5953	58	10	𝑓(𝑥	𝑓(𝑥	NOUN
cana-5953	58	11	)	)	PUNCT
cana-5953	58	12	=	=	SYM
cana-5953	59	1	𝑥	𝑥	PROPN
cana-5953	59	2	9	9	NUM
cana-5953	59	3	and	and	CCONJ
cana-5953	59	4	𝜓(𝑡	𝜓(𝑡	PROPN
cana-5953	59	5	)	)	PUNCT
cana-5953	59	6	=	=	NOUN
cana-5953	59	7	80𝑡	80𝑡	NOUN
cana-5953	59	8	for	for	ADP
cana-5953	59	9	all	all	DET
cana-5953	59	10	𝑡	𝑡	PROPN
cana-5953	59	11	≥	≥	NOUN
cana-5953	59	12	0	0	NUM
cana-5953	59	13	,	,	PUNCT
cana-5953	59	14	where	where	SCONJ
cana-5953	59	15	𝜓	𝜓	X
cana-5953	59	16	:	:	PUNCT
cana-5953	59	17	[	[	X
cana-5953	59	18	0	0	NUM
cana-5953	59	19	,	,	PUNCT
cana-5953	59	20	∞	∞	PROPN
cana-5953	59	21	)	)	PUNCT
cana-5953	59	22	→	→	PUNCT
cana-5953	60	1	[	[	X
cana-5953	60	2	0	0	NUM
cana-5953	60	3	,	,	PUNCT
cana-5953	60	4	∞	∞	NUM
cana-5953	60	5	)	)	PUNCT
cana-5953	60	6	is	be	AUX
cana-5953	60	7	non	non	ADJ
cana-5953	60	8	decreasing	decrease	VERB
cana-5953	60	9	continuous	continuous	ADJ
cana-5953	60	10	function	function	NOUN
cana-5953	60	11	.	.	PUNCT
cana-5953	61	1	then	then	ADV
cana-5953	61	2	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	61	3	,	,	PUNCT
cana-5953	61	4	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	61	5	,	,	PUNCT
cana-5953	61	6	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	61	7	)	)	PUNCT
cana-5953	61	8	=	=	SYM
cana-5953	61	9	𝑆	𝑆	PROPN
cana-5953	61	10	(	(	PUNCT
cana-5953	61	11	𝑥	𝑥	PROPN
cana-5953	61	12	9	9	NUM
cana-5953	61	13	,	,	PUNCT
cana-5953	61	14	𝑥	𝑥	PROPN
cana-5953	61	15	9	9	NUM
cana-5953	61	16	,	,	PUNCT
cana-5953	61	17	𝑦	𝑦	NOUN
cana-5953	61	18	9	9	NUM
cana-5953	61	19	)	)	PUNCT
cana-5953	61	20	=	=	SYM
cana-5953	61	21	9	9	NUM
cana-5953	61	22	(	(	PUNCT
cana-5953	61	23	𝑥	𝑥	PROPN
cana-5953	61	24	9	9	NUM
cana-5953	61	25	)	)	PUNCT
cana-5953	61	26	2	2	NUM
cana-5953	61	27	+	+	CCONJ
cana-5953	61	28	(	(	PUNCT
cana-5953	61	29	𝑦	𝑦	NOUN
cana-5953	61	30	9	9	NUM
cana-5953	61	31	)	)	PUNCT
cana-5953	61	32	2	2	NUM
cana-5953	61	33	=	=	SYM
cana-5953	61	34	9	9	NUM
cana-5953	61	35	𝑥2	𝑥2	NOUN
cana-5953	61	36	81	81	NUM
cana-5953	61	37	+	+	CCONJ
cana-5953	61	38	𝑦2	𝑦2	PROPN
cana-5953	61	39	81	81	NUM
cana-5953	61	40	=	=	SYM
cana-5953	61	41	9𝑥2	9𝑥2	NUM
cana-5953	62	1	+	+	NUM
cana-5953	62	2	𝑦2	𝑦2	PROPN
cana-5953	62	3	−	−	PROPN
cana-5953	62	4	80	80	NUM
cana-5953	62	5	81	81	NUM
cana-5953	62	6	(	(	PUNCT
cana-5953	62	7	9𝑥2	9𝑥2	NUM
cana-5953	62	8	+	+	NUM
cana-5953	62	9	𝑦2	𝑦2	NOUN
cana-5953	62	10	)	)	PUNCT
cana-5953	62	11	=	=	SYM
cana-5953	62	12	𝑆(𝑥	𝑆(𝑥	PROPN
cana-5953	62	13	,	,	PUNCT
cana-5953	62	14	𝑥	𝑥	PROPN
cana-5953	62	15	,	,	PUNCT
cana-5953	62	16	𝑦	𝑦	NOUN
cana-5953	62	17	)	)	PUNCT
cana-5953	62	18	−	−	PROPN
cana-5953	62	19	1	1	NUM
cana-5953	62	20	81	81	NUM
cana-5953	62	21	𝜓(𝑆(𝑥	𝜓(𝑆(𝑥	NOUN
cana-5953	62	22	,	,	PUNCT
cana-5953	62	23	𝑥	𝑥	NOUN
cana-5953	62	24	,	,	PUNCT
cana-5953	62	25	𝑦	𝑦	NOUN
cana-5953	62	26	)	)	PUNCT
cana-5953	62	27	)	)	PUNCT
cana-5953	62	28	.	.	PUNCT
cana-5953	63	1	thus	thus	ADV
cana-5953	63	2	,	,	PUNCT
cana-5953	63	3	we	we	PRON
cana-5953	63	4	found	find	VERB
cana-5953	63	5	that	that	SCONJ
cana-5953	63	6	𝑓	𝑓	PRON
cana-5953	63	7	is	be	AUX
cana-5953	63	8	a	a	DET
cana-5953	63	9	weak	weak	ADJ
cana-5953	63	10	contraction	contraction	NOUN
cana-5953	63	11	on	on	ADP
cana-5953	63	12	𝑋.	𝑋.	PROPN
cana-5953	63	13	since	since	ADV
cana-5953	63	14	,	,	PUNCT
cana-5953	63	15	it	it	PRON
cana-5953	63	16	fulfills	fulfill	VERB
cana-5953	63	17	the	the	DET
cana-5953	63	18	requirements	requirement	NOUN
cana-5953	63	19	of	of	ADP
cana-5953	63	20	definition	definition	NOUN
cana-5953	63	21	(	(	PUNCT
cana-5953	63	22	3.4	3.4	NUM
cana-5953	63	23	)	)	PUNCT
cana-5953	63	24	for	for	ADP
cana-5953	63	25	𝛿	𝛿	ADJ
cana-5953	63	26	=	=	SYM
cana-5953	63	27	1	1	NUM
cana-5953	63	28	81	81	NUM
cana-5953	63	29	.	.	PUNCT
cana-5953	64	1	proposition	proposition	NOUN
cana-5953	64	2	3.6	3.6	NUM
cana-5953	64	3	[	[	X
cana-5953	64	4	2	2	X
cana-5953	64	5	]	]	PUNCT
cana-5953	64	6	if	if	SCONJ
cana-5953	64	7	weakly	weakly	ADJ
cana-5953	64	8	compatible	compatible	ADJ
cana-5953	64	9	functions	function	NOUN
cana-5953	64	10	𝑓	𝑓	PRON
cana-5953	64	11	and	and	CCONJ
cana-5953	64	12	𝑔	𝑔	PROPN
cana-5953	64	13	have	have	VERB
cana-5953	64	14	exactly	exactly	ADV
cana-5953	64	15	only	only	ADV
cana-5953	64	16	one	one	NUM
cana-5953	64	17	point	point	NOUN
cana-5953	64	18	of	of	ADP
cana-5953	64	19	coincidence	coincidence	NOUN
cana-5953	64	20	𝑤	𝑤	ADP
cana-5953	64	21	=	=	PUNCT
cana-5953	64	22	𝑓𝑥	𝑓𝑥	PROPN
cana-5953	64	23	=	=	PUNCT
cana-5953	64	24	𝑔𝑥	𝑔𝑥	PROPN
cana-5953	64	25	on	on	ADP
cana-5953	64	26	𝑋	𝑋	PROPN
cana-5953	64	27	,	,	PUNCT
cana-5953	64	28	then	then	ADV
cana-5953	64	29	𝑤	𝑤	ADP
cana-5953	64	30	is	be	AUX
cana-5953	64	31	the	the	DET
cana-5953	64	32	unique	unique	ADJ
cana-5953	64	33	common	common	ADJ
cana-5953	64	34	fixed	fix	VERB
cana-5953	64	35	point	point	NOUN
cana-5953	64	36	of	of	ADP
cana-5953	64	37	𝑓	𝑓	PRON
cana-5953	64	38	and	and	CCONJ
cana-5953	64	39	𝑔.	𝑔.	PROPN
cana-5953	64	40	lemma	lemma	PROPN
cana-5953	64	41	3.7	3.7	NUM
cana-5953	65	1	[	[	X
cana-5953	65	2	14	14	NUM
cana-5953	65	3	]	]	PUNCT
cana-5953	65	4	a	a	DET
cana-5953	65	5	condition	condition	NOUN
cana-5953	65	6	𝑆(𝑥	𝑆(𝑥	NOUN
cana-5953	65	7	,	,	PUNCT
cana-5953	65	8	𝑥	𝑥	NOUN
cana-5953	65	9	,	,	PUNCT
cana-5953	65	10	𝑦	𝑦	NOUN
cana-5953	65	11	)	)	PUNCT
cana-5953	65	12	=	=	SYM
cana-5953	65	13	𝑆(𝑦	𝑆(𝑦	PROPN
cana-5953	65	14	,	,	PUNCT
cana-5953	65	15	𝑦	𝑦	NOUN
cana-5953	65	16	,	,	PUNCT
cana-5953	65	17	𝑥	𝑥	NOUN
cana-5953	65	18	)	)	PUNCT
cana-5953	65	19	holds	hold	VERB
cana-5953	65	20	∀𝑥	∀𝑥	NOUN
cana-5953	65	21	,	,	PUNCT
cana-5953	65	22	𝑦	𝑦	NOUN
cana-5953	65	23	∈	∈	NOUN
cana-5953	65	24	𝑋	𝑋	NOUN
cana-5953	65	25	in	in	ADP
cana-5953	65	26	𝑆-metric	𝑆-metric	ADJ
cana-5953	65	27	space	space	NOUN
cana-5953	65	28	(	(	PUNCT
cana-5953	65	29	𝑋	𝑋	PROPN
cana-5953	65	30	,	,	PUNCT
cana-5953	65	31	𝑆	𝑆	PROPN
cana-5953	65	32	)	)	PUNCT
cana-5953	65	33	.	.	PUNCT
cana-5953	66	1	lemma	lemma	PROPN
cana-5953	66	2	3.8	3.8	NUM
cana-5953	67	1	[	[	X
cana-5953	67	2	14	14	NUM
cana-5953	67	3	]	]	X
cana-5953	67	4	let	let	NOUN
cana-5953	67	5	(	(	PUNCT
cana-5953	67	6	𝑋	𝑋	PROPN
cana-5953	67	7	,	,	PUNCT
cana-5953	67	8	𝑆	𝑆	PROPN
cana-5953	67	9	)	)	PUNCT
cana-5953	67	10	be	be	VERB
cana-5953	67	11	an	an	DET
cana-5953	67	12	𝑆-metric	𝑆-metric	ADJ
cana-5953	67	13	space	space	NOUN
cana-5953	67	14	.	.	PUNCT
cana-5953	68	1	if	if	SCONJ
cana-5953	68	2	{	{	PUNCT
cana-5953	68	3	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	68	4	}	}	PUNCT
cana-5953	68	5	and	and	CCONJ
cana-5953	68	6	{	{	PUNCT
cana-5953	68	7	𝑦𝑛	𝑦𝑛	AUX
cana-5953	68	8	}	}	PUNCT
cana-5953	68	9	are	be	AUX
cana-5953	68	10	sequences	sequence	NOUN
cana-5953	68	11	in	in	ADP
cana-5953	68	12	𝑋	𝑋	NOUN
cana-5953	68	13	converging	converge	VERB
cana-5953	68	14	to	to	ADP
cana-5953	68	15	𝑥	𝑥	PROPN
cana-5953	68	16	and	and	CCONJ
cana-5953	68	17	𝑦	𝑦	NOUN
cana-5953	68	18	respectively	respectively	ADV
cana-5953	68	19	,	,	PUNCT
cana-5953	68	20	that	that	ADV
cana-5953	68	21	is	is	ADV
cana-5953	68	22	,	,	PUNCT
cana-5953	68	23	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	68	24	→	→	SYM
cana-5953	68	25	𝑥	𝑥	X
cana-5953	68	26	and	and	CCONJ
cana-5953	68	27	𝑦𝑛	𝑦𝑛	VERB
cana-5953	68	28	→	→	SYM
cana-5953	68	29	𝑦	𝑦	NOUN
cana-5953	68	30	as	as	ADP
cana-5953	68	31	𝑛	𝑛	PROPN
cana-5953	68	32	→	→	SYM
cana-5953	68	33	∞	∞	PROPN
cana-5953	68	34	,	,	PUNCT
cana-5953	68	35	then	then	ADV
cana-5953	68	36	𝑆(𝑥𝑛	𝑆(𝑥𝑛	PROPN
cana-5953	68	37	,	,	PUNCT
cana-5953	68	38	𝑥𝑛	𝑥𝑛	PRON
cana-5953	68	39	,	,	PUNCT
cana-5953	68	40	𝑦𝑛	𝑦𝑛	NOUN
cana-5953	68	41	)	)	PUNCT
cana-5953	68	42	→	→	SYM
cana-5953	68	43	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	68	44	,	,	PUNCT
cana-5953	68	45	𝑥	𝑥	NOUN
cana-5953	68	46	,	,	PUNCT
cana-5953	68	47	𝑦	𝑦	NOUN
cana-5953	68	48	)	)	PUNCT
cana-5953	68	49	as	as	ADP
cana-5953	68	50	𝑛	𝑛	PROPN
cana-5953	68	51	→	→	SYM
cana-5953	68	52	∞.	∞.	PROPN
cana-5953	68	53	lemma	lemma	PROPN
cana-5953	68	54	3.9	3.9	NUM
cana-5953	69	1	[	[	X
cana-5953	69	2	14	14	NUM
cana-5953	69	3	]	]	X
cana-5953	69	4	let	let	NOUN
cana-5953	69	5	(	(	PUNCT
cana-5953	69	6	𝑋	𝑋	PROPN
cana-5953	69	7	,	,	PUNCT
cana-5953	69	8	𝑆	𝑆	PROPN
cana-5953	69	9	)	)	PUNCT
cana-5953	69	10	be	be	VERB
cana-5953	69	11	an	an	DET
cana-5953	69	12	𝑆-metric	𝑆-metric	ADJ
cana-5953	69	13	space	space	NOUN
cana-5953	69	14	.	.	PUNCT
cana-5953	70	1	if	if	SCONJ
cana-5953	70	2	the	the	DET
cana-5953	70	3	sequence	sequence	NOUN
cana-5953	70	4	{	{	PUNCT
cana-5953	70	5	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	70	6	}	}	PUNCT
cana-5953	70	7	in	in	ADP
cana-5953	70	8	𝑋	𝑋	PROPN
cana-5953	70	9	converges	converge	NOUN
cana-5953	70	10	to	to	ADP
cana-5953	70	11	𝑥	𝑥	PRON
cana-5953	70	12	,	,	PUNCT
cana-5953	70	13	then	then	ADV
cana-5953	70	14	the	the	DET
cana-5953	70	15	limit	limit	NOUN
cana-5953	70	16	𝑥	𝑥	NOUN
cana-5953	70	17	is	be	AUX
cana-5953	70	18	unique	unique	ADJ
cana-5953	70	19	.	.	PUNCT
cana-5953	71	1	lemma	lemma	PROPN
cana-5953	71	2	3.10	3.10	NUM
cana-5953	71	3	[	[	X
cana-5953	71	4	14	14	NUM
cana-5953	71	5	]	]	X
cana-5953	71	6	let	let	NOUN
cana-5953	71	7	(	(	PUNCT
cana-5953	71	8	𝑋	𝑋	PROPN
cana-5953	71	9	,	,	PUNCT
cana-5953	71	10	𝑆	𝑆	PROPN
cana-5953	71	11	)	)	PUNCT
cana-5953	71	12	be	be	VERB
cana-5953	71	13	an	an	DET
cana-5953	71	14	𝑆-metric	𝑆-metric	ADJ
cana-5953	71	15	space	space	NOUN
cana-5953	71	16	.	.	PUNCT
cana-5953	72	1	if	if	SCONJ
cana-5953	72	2	the	the	DET
cana-5953	72	3	sequence	sequence	NOUN
cana-5953	72	4	{	{	PUNCT
cana-5953	72	5	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	72	6	}	}	PUNCT
cana-5953	72	7	in	in	ADP
cana-5953	72	8	𝑋	𝑋	PROPN
cana-5953	72	9	converges	converge	NOUN
cana-5953	72	10	to	to	ADP
cana-5953	72	11	𝑥	𝑥	PRON
cana-5953	72	12	,	,	PUNCT
cana-5953	72	13	then	then	ADV
cana-5953	72	14	{	{	PUNCT
cana-5953	72	15	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	72	16	}	}	PUNCT
cana-5953	72	17	is	be	AUX
cana-5953	72	18	a	a	DET
cana-5953	72	19	cauchy	cauchy	ADJ
cana-5953	72	20	sequence	sequence	NOUN
cana-5953	72	21	.	.	PUNCT
cana-5953	73	1	4	4	X
cana-5953	73	2	.	.	X
cana-5953	73	3	main	main	ADJ
cana-5953	73	4	results	result	NOUN
cana-5953	73	5	in	in	ADP
cana-5953	73	6	this	this	DET
cana-5953	73	7	section	section	NOUN
cana-5953	73	8	,	,	PUNCT
cana-5953	73	9	we	we	PRON
cana-5953	73	10	present	present	VERB
cana-5953	73	11	new	new	ADJ
cana-5953	73	12	common	common	ADJ
cana-5953	73	13	fixed	fix	VERB
cana-5953	73	14	point	point	NOUN
cana-5953	73	15	theorems	theorem	NOUN
cana-5953	73	16	for	for	ADP
cana-5953	73	17	weakly	weakly	ADJ
cana-5953	73	18	compatible	compatible	ADJ
cana-5953	73	19	mappings	mapping	NOUN
cana-5953	73	20	defined	define	VERB
cana-5953	73	21	on	on	ADP
cana-5953	73	22	complete	complete	ADJ
cana-5953	73	23	rectangular	rectangular	ADJ
cana-5953	73	24	𝑆-metric	𝑆-metric	ADJ
cana-5953	73	25	spaces	space	NOUN
cana-5953	73	26	.	.	PUNCT
cana-5953	74	1	these	these	DET
cana-5953	74	2	results	result	NOUN
cana-5953	74	3	are	be	AUX
cana-5953	74	4	obtained	obtain	VERB
cana-5953	74	5	by	by	ADP
cana-5953	74	6	imposing	impose	VERB
cana-5953	74	7	specific	specific	ADJ
cana-5953	74	8	contractive	contractive	ADJ
cana-5953	74	9	conditions	condition	NOUN
cana-5953	74	10	that	that	PRON
cana-5953	74	11	ensure	ensure	VERB
cana-5953	74	12	the	the	DET
cana-5953	74	13	existence	existence	NOUN
cana-5953	74	14	and	and	CCONJ
cana-5953	74	15	uniqueness	uniqueness	NOUN
cana-5953	74	16	of	of	ADP
cana-5953	74	17	common	common	ADJ
cana-5953	74	18	fixed	fix	VERB
cana-5953	74	19	points	point	NOUN
cana-5953	74	20	.	.	PUNCT
cana-5953	75	1	communications	communication	NOUN
cana-5953	75	2	on	on	ADP
cana-5953	75	3	applied	apply	VERB
cana-5953	75	4	nonlinear	nonlinear	ADJ
cana-5953	75	5	analysis	analysis	NOUN
cana-5953	75	6	issn	issn	NOUN
cana-5953	75	7	:	:	PUNCT
cana-5953	75	8	1074	1074	NUM
cana-5953	75	9	-	-	PUNCT
cana-5953	75	10	133x	133x	NUM
cana-5953	75	11	vol	vol	VERB
cana-5953	75	12	32	32	NUM
cana-5953	75	13	no	no	NOUN
cana-5953	75	14	.	.	PUNCT
cana-5953	76	1	10s	10	NOUN
cana-5953	76	2	(	(	PUNCT
cana-5953	76	3	2025	2025	NUM
cana-5953	76	4	)	)	PUNCT
cana-5953	76	5	3163	3163	NUM
cana-5953	76	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	76	7	theorem	theorem	VERB
cana-5953	76	8	4.1	4.1	NUM
cana-5953	76	9	let	let	VERB
cana-5953	76	10	𝑓	𝑓	PRON
cana-5953	76	11	and	and	CCONJ
cana-5953	76	12	𝑔	𝑔	AUX
cana-5953	76	13	be	be	AUX
cana-5953	76	14	a	a	DET
cana-5953	76	15	self	self	NOUN
cana-5953	76	16	mapping	mapping	NOUN
cana-5953	76	17	on	on	ADP
cana-5953	76	18	complete	complete	ADJ
cana-5953	76	19	𝑆-metric	𝑆-metric	ADJ
cana-5953	76	20	space	space	NOUN
cana-5953	76	21	(	(	PUNCT
cana-5953	76	22	𝑋	𝑋	PROPN
cana-5953	76	23	,	,	PUNCT
cana-5953	76	24	𝑆	𝑆	PROPN
cana-5953	76	25	)	)	PUNCT
cana-5953	76	26	.	.	PUNCT
cana-5953	77	1	assume	assume	VERB
cana-5953	77	2	that	that	SCONJ
cana-5953	77	3	𝑓	𝑓	PROPN
cana-5953	77	4	and	and	CCONJ
cana-5953	77	5	𝑔	𝑔	PROPN
cana-5953	77	6	satisfies	satisfie	NOUN
cana-5953	77	7	the	the	DET
cana-5953	77	8	following	follow	VERB
cana-5953	77	9	restrictions	restriction	NOUN
cana-5953	77	10	,	,	PUNCT
cana-5953	77	11	𝜓(𝑆(𝑓𝑥	𝜓(𝑆(𝑓𝑥	PROPN
cana-5953	77	12	,	,	PUNCT
cana-5953	77	13	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	77	14	,	,	PUNCT
cana-5953	77	15	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	77	16	)	)	PUNCT
cana-5953	77	17	,	,	PUNCT
cana-5953	77	18	𝑆(𝑓𝑦	𝑆(𝑓𝑦	VERB
cana-5953	77	19	,	,	PUNCT
cana-5953	77	20	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	77	21	,	,	PUNCT
cana-5953	77	22	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	77	23	)	)	PUNCT
cana-5953	77	24	)	)	PUNCT
cana-5953	78	1	≤	≤	PUNCT
cana-5953	79	1	𝑞𝜓(𝑆(𝑔𝑥	𝑞𝜓(𝑆(𝑔𝑥	PROPN
cana-5953	79	2	,	,	PUNCT
cana-5953	79	3	𝑔𝑥	𝑔𝑥	PROPN
cana-5953	79	4	,	,	PUNCT
cana-5953	79	5	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	79	6	)	)	PUNCT
cana-5953	79	7	,	,	PUNCT
cana-5953	79	8	𝑆(𝑔𝑦	𝑆(𝑔𝑦	PROPN
cana-5953	79	9	,	,	PUNCT
cana-5953	79	10	𝑔𝑦	𝑔𝑦	PROPN
cana-5953	79	11	,	,	PUNCT
cana-5953	79	12	𝑔𝑥	𝑔𝑥	NOUN
cana-5953	79	13	)	)	PUNCT
cana-5953	79	14	)	)	PUNCT
cana-5953	79	15	,	,	PUNCT
cana-5953	79	16	where	where	SCONJ
cana-5953	79	17	0	0	X
cana-5953	79	18	<	<	X
cana-5953	79	19	𝑞	𝑞	X
cana-5953	79	20	<	<	X
cana-5953	79	21	1	1	NUM
cana-5953	79	22	and	and	CCONJ
cana-5953	79	23	𝜓	𝜓	NOUN
cana-5953	79	24	:	:	PUNCT
cana-5953	79	25	[	[	X
cana-5953	79	26	0	0	NUM
cana-5953	79	27	,	,	PUNCT
cana-5953	79	28	∞)2	∞)2	PROPN
cana-5953	79	29	→	→	SYM
cana-5953	79	30	[	[	X
cana-5953	79	31	0	0	NUM
cana-5953	79	32	,	,	PUNCT
cana-5953	79	33	∞)2	∞)2	PROPN
cana-5953	79	34	is	be	AUX
cana-5953	79	35	a	a	DET
cana-5953	79	36	continuous	continuous	ADJ
cana-5953	79	37	function	function	NOUN
cana-5953	79	38	on	on	ADP
cana-5953	79	39	[	[	X
cana-5953	79	40	0	0	NUM
cana-5953	79	41	,	,	PUNCT
cana-5953	79	42	∞)2	∞)2	ADV
cana-5953	79	43	with	with	ADP
cana-5953	79	44	𝜓(𝑎	𝜓(𝑎	NOUN
cana-5953	79	45	,	,	PUNCT
cana-5953	79	46	𝑏	𝑏	NOUN
cana-5953	79	47	)	)	PUNCT
cana-5953	79	48	=	=	SYM
cana-5953	79	49	0	0	PUNCT
cana-5953	80	1	if	if	SCONJ
cana-5953	80	2	and	and	CCONJ
cana-5953	80	3	only	only	ADV
cana-5953	80	4	if	if	SCONJ
cana-5953	80	5	𝑎	𝑎	PROPN
cana-5953	80	6	=	=	SYM
cana-5953	80	7	0	0	PUNCT
cana-5953	81	1	=	=	NOUN
cana-5953	81	2	𝑏.	𝑏.	NOUN
cana-5953	81	3	also	also	ADV
cana-5953	81	4	,	,	PUNCT
cana-5953	81	5	•	•	PRON
cana-5953	81	6	𝑓(𝑋	𝑓(𝑋	NUM
cana-5953	81	7	)	)	PUNCT
cana-5953	81	8	⊆	⊆	NUM
cana-5953	81	9	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	81	10	)	)	PUNCT
cana-5953	81	11	,	,	PUNCT
cana-5953	81	12	•	•	ADP
cana-5953	81	13	if	if	SCONJ
cana-5953	81	14	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	81	15	)	)	PUNCT
cana-5953	81	16	is	be	AUX
cana-5953	81	17	complete	complete	ADJ
cana-5953	81	18	.	.	PUNCT
cana-5953	82	1	then	then	ADV
cana-5953	82	2	𝑓	𝑓	X
cana-5953	82	3	and	and	CCONJ
cana-5953	82	4	𝑔	𝑔	AUX
cana-5953	82	5	have	have	VERB
cana-5953	82	6	a	a	DET
cana-5953	82	7	unique	unique	ADJ
cana-5953	82	8	coincidence	coincidence	NOUN
cana-5953	82	9	point	point	NOUN
cana-5953	82	10	in	in	ADP
cana-5953	82	11	𝑋.	𝑋.	PROPN
cana-5953	82	12	moreover	moreover	ADV
cana-5953	82	13	,	,	PUNCT
cana-5953	82	14	if	if	SCONJ
cana-5953	82	15	𝑓	𝑓	PRON
cana-5953	82	16	and	and	CCONJ
cana-5953	82	17	𝑔	𝑔	PROPN
cana-5953	82	18	are	be	AUX
cana-5953	82	19	weakly	weakly	ADV
cana-5953	82	20	compatible	compatible	ADJ
cana-5953	82	21	,	,	PUNCT
cana-5953	82	22	then	then	ADV
cana-5953	82	23	𝑓	𝑓	X
cana-5953	82	24	and	and	CCONJ
cana-5953	82	25	𝑔	𝑔	AUX
cana-5953	82	26	have	have	VERB
cana-5953	82	27	a	a	DET
cana-5953	82	28	unique	unique	ADJ
cana-5953	82	29	common	common	ADJ
cana-5953	82	30	fixed	fix	VERB
cana-5953	82	31	point	point	NOUN
cana-5953	82	32	in	in	ADP
cana-5953	82	33	𝑋.	𝑋.	PROPN
cana-5953	82	34	proof	proof	NOUN
cana-5953	82	35	.	.	PUNCT
cana-5953	83	1	let	let	VERB
cana-5953	83	2	𝑥0	𝑥0	NOUN
cana-5953	83	3	be	be	AUX
cana-5953	83	4	any	any	DET
cana-5953	83	5	point	point	NOUN
cana-5953	83	6	in	in	ADP
cana-5953	83	7	𝑋.	𝑋.	PROPN
cana-5953	83	8	since	since	SCONJ
cana-5953	83	9	𝑓𝑥0	𝑓𝑥0	PROPN
cana-5953	83	10	∈	∈	PROPN
cana-5953	83	11	𝑓(𝑋	𝑓(𝑋	PROPN
cana-5953	83	12	)	)	PUNCT
cana-5953	83	13	and	and	CCONJ
cana-5953	83	14	𝑓(𝑋	𝑓(𝑋	NUM
cana-5953	83	15	)	)	PUNCT
cana-5953	83	16	⊆	⊆	NUM
cana-5953	83	17	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	83	18	)	)	PUNCT
cana-5953	83	19	,	,	PUNCT
cana-5953	83	20	there	there	PRON
cana-5953	83	21	exists	exist	VERB
cana-5953	83	22	a	a	DET
cana-5953	83	23	point	point	NOUN
cana-5953	83	24	𝑥1	𝑥1	NOUN
cana-5953	83	25	in	in	ADP
cana-5953	83	26	𝑋	𝑋	PROPN
cana-5953	83	27	such	such	ADJ
cana-5953	83	28	that	that	SCONJ
cana-5953	83	29	𝑓𝑥0	𝑓𝑥0	PROPN
cana-5953	83	30	=	=	SYM
cana-5953	83	31	𝑔𝑥1	𝑔𝑥1	PROPN
cana-5953	83	32	.	.	PUNCT
cana-5953	84	1	clearly	clearly	ADV
cana-5953	84	2	𝑥1	𝑥1	NOUN
cana-5953	84	3	∈	∈	PROPN
cana-5953	84	4	𝑋	𝑋	PROPN
cana-5953	84	5	and	and	CCONJ
cana-5953	84	6	again	again	ADV
cana-5953	84	7	,	,	PUNCT
cana-5953	84	8	by	by	ADP
cana-5953	84	9	inclusion	inclusion	NOUN
cana-5953	84	10	𝑓𝑥1	𝑓𝑥1	NOUN
cana-5953	84	11	∈	∈	PROPN
cana-5953	84	12	𝑓(𝑋	𝑓(𝑋	PROPN
cana-5953	84	13	)	)	PUNCT
cana-5953	84	14	there	there	PRON
cana-5953	84	15	exists	exist	VERB
cana-5953	84	16	𝑥2	𝑥2	NOUN
cana-5953	84	17	in	in	ADP
cana-5953	84	18	𝑋	𝑋	PROPN
cana-5953	84	19	such	such	ADJ
cana-5953	84	20	that	that	PRON
cana-5953	84	21	𝑓𝑥1	𝑓𝑥1	NOUN
cana-5953	84	22	=	=	SYM
cana-5953	84	23	𝑔𝑥2	𝑔𝑥2	PROPN
cana-5953	84	24	.	.	PUNCT
cana-5953	85	1	proceeding	proceeding	NOUN
cana-5953	85	2	in	in	ADP
cana-5953	85	3	this	this	DET
cana-5953	85	4	way	way	NOUN
cana-5953	85	5	,	,	PUNCT
cana-5953	85	6	we	we	PRON
cana-5953	85	7	generate	generate	VERB
cana-5953	85	8	a	a	DET
cana-5953	85	9	sequence	sequence	NOUN
cana-5953	85	10	{	{	PUNCT
cana-5953	85	11	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	85	12	}	}	PUNCT
cana-5953	85	13	in	in	ADP
cana-5953	85	14	𝑋	𝑋	PROPN
cana-5953	85	15	,	,	PUNCT
cana-5953	85	16	such	such	ADJ
cana-5953	85	17	that	that	SCONJ
cana-5953	85	18	each	each	DET
cana-5953	85	19	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5953	85	20	∈	∈	PROPN
cana-5953	85	21	𝑋	𝑋	NOUN
cana-5953	85	22	satisfies	satisfy	VERB
cana-5953	85	23	𝑓𝑥𝑛	𝑓𝑥𝑛	VERB
cana-5953	85	24	=	=	SYM
cana-5953	85	25	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	85	26	,	,	PUNCT
cana-5953	85	27	∀𝑛.	∀𝑛.	PROPN
cana-5953	85	28	now	now	ADV
cana-5953	85	29	,	,	PUNCT
cana-5953	85	30	consider	consider	VERB
cana-5953	85	31	the	the	DET
cana-5953	85	32	following	follow	VERB
cana-5953	85	33	expression	expression	NOUN
cana-5953	85	34	:	:	PUNCT
cana-5953	85	35	𝜓(𝑆(𝑔𝑥𝑛+1	𝜓(𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	85	36	,	,	PUNCT
cana-5953	85	37	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	85	38	,	,	PUNCT
cana-5953	85	39	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	85	40	)	)	PUNCT
cana-5953	85	41	,	,	PUNCT
cana-5953	85	42	𝑆(𝑔𝑥𝑛+2	𝑆(𝑔𝑥𝑛+2	PROPN
cana-5953	85	43	,	,	PUNCT
cana-5953	85	44	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	85	45	,	,	PUNCT
cana-5953	85	46	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	85	47	)	)	PUNCT
cana-5953	85	48	)	)	PUNCT
cana-5953	86	1	=	=	PUNCT
cana-5953	86	2	𝜓(𝑆(𝑓𝑥𝑛	𝜓(𝑆(𝑓𝑥𝑛	ADJ
cana-5953	86	3	,	,	PUNCT
cana-5953	86	4	𝑓𝑥𝑛	𝑓𝑥𝑛	ADJ
cana-5953	86	5	,	,	PUNCT
cana-5953	86	6	𝑓𝑥𝑛+1	𝑓𝑥𝑛+1	NOUN
cana-5953	86	7	)	)	PUNCT
cana-5953	86	8	,	,	PUNCT
cana-5953	86	9	𝑆(𝑓𝑥𝑛+1	𝑆(𝑓𝑥𝑛+1	NOUN
cana-5953	86	10	,	,	PUNCT
cana-5953	86	11	𝑓𝑥𝑛+1	𝑓𝑥𝑛+1	NOUN
cana-5953	86	12	,	,	PUNCT
cana-5953	86	13	𝑓𝑥𝑛	𝑓𝑥𝑛	NOUN
cana-5953	86	14	)	)	PUNCT
cana-5953	86	15	)	)	PUNCT
cana-5953	86	16	≤	≤	PUNCT
cana-5953	87	1	𝑞𝜓(𝑆(𝑔𝑥𝑛	𝑞𝜓(𝑆(𝑔𝑥𝑛	PROPN
cana-5953	87	2	,	,	PUNCT
cana-5953	87	3	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	87	4	,	,	PUNCT
cana-5953	87	5	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	87	6	)	)	PUNCT
cana-5953	87	7	,	,	PUNCT
cana-5953	87	8	𝑆(𝑔𝑥𝑛+1	𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	87	9	,	,	PUNCT
cana-5953	87	10	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	87	11	,	,	PUNCT
cana-5953	87	12	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	87	13	)	)	PUNCT
cana-5953	87	14	)	)	PUNCT
cana-5953	87	15	≤	≤	PROPN
cana-5953	87	16	𝑞2𝜓(𝑆(𝑔𝑥𝑛−1	𝑞2𝜓(𝑆(𝑔𝑥𝑛−1	PROPN
cana-5953	87	17	,	,	PUNCT
cana-5953	87	18	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	87	19	,	,	PUNCT
cana-5953	87	20	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	87	21	)	)	PUNCT
cana-5953	87	22	,	,	PUNCT
cana-5953	87	23	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	NOUN
cana-5953	87	24	,	,	PUNCT
cana-5953	87	25	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	87	26	,	,	PUNCT
cana-5953	87	27	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	87	28	)	)	PUNCT
cana-5953	87	29	)	)	PUNCT
cana-5953	87	30	.	.	PUNCT
cana-5953	88	1	by	by	ADP
cana-5953	88	2	continuing	continue	VERB
cana-5953	88	3	this	this	DET
cana-5953	88	4	process	process	NOUN
cana-5953	88	5	inductively	inductively	ADV
cana-5953	88	6	,	,	PUNCT
cana-5953	88	7	we	we	PRON
cana-5953	88	8	obtain	obtain	VERB
cana-5953	88	9	:	:	PUNCT
cana-5953	88	10	𝜓(𝑆(𝑔𝑥𝑛+1	𝜓(𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	88	11	,	,	PUNCT
cana-5953	88	12	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	88	13	,	,	PUNCT
cana-5953	88	14	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	88	15	)	)	PUNCT
cana-5953	88	16	,	,	PUNCT
cana-5953	88	17	𝑆(𝑔𝑥𝑛+2	𝑆(𝑔𝑥𝑛+2	PROPN
cana-5953	88	18	,	,	PUNCT
cana-5953	88	19	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	88	20	,	,	PUNCT
cana-5953	88	21	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	88	22	)	)	PUNCT
cana-5953	88	23	)	)	PUNCT
cana-5953	88	24	≤	≤	PUNCT
cana-5953	89	1	𝜓(𝑆(𝑔𝑥0	𝜓(𝑆(𝑔𝑥0	PROPN
cana-5953	89	2	,	,	PUNCT
cana-5953	89	3	𝑔𝑥0	𝑔𝑥0	PROPN
cana-5953	89	4	,	,	PUNCT
cana-5953	89	5	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	89	6	)	)	PUNCT
cana-5953	89	7	,	,	PUNCT
cana-5953	89	8	𝑆(𝑔𝑥1	𝑆(𝑔𝑥1	NOUN
cana-5953	89	9	,	,	PUNCT
cana-5953	89	10	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	89	11	,	,	PUNCT
cana-5953	89	12	𝑔𝑥0	𝑔𝑥0	NOUN
cana-5953	89	13	)	)	PUNCT
cana-5953	89	14	)	)	PUNCT
cana-5953	89	15	.	.	PUNCT
cana-5953	90	1	since	since	SCONJ
cana-5953	90	2	0	0	NUM
cana-5953	90	3	<	<	X
cana-5953	90	4	𝑞	𝑞	X
cana-5953	90	5	<	<	X
cana-5953	90	6	1	1	NUM
cana-5953	90	7	taking	take	VERB
cana-5953	90	8	the	the	DET
cana-5953	90	9	limit	limit	NOUN
cana-5953	90	10	as	as	ADP
cana-5953	90	11	𝑛	𝑛	PROPN
cana-5953	90	12	→	→	SYM
cana-5953	90	13	∞	∞	PROPN
cana-5953	90	14	,	,	PUNCT
cana-5953	90	15	gives	give	VERB
cana-5953	90	16	𝜓(𝑆(𝑔𝑥𝑛+1	𝜓(𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	90	17	,	,	PUNCT
cana-5953	90	18	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	90	19	,	,	PUNCT
cana-5953	90	20	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	90	21	)	)	PUNCT
cana-5953	90	22	,	,	PUNCT
cana-5953	90	23	𝑆(𝑔𝑥𝑛+2	𝑆(𝑔𝑥𝑛+2	PROPN
cana-5953	90	24	,	,	PUNCT
cana-5953	90	25	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	90	26	,	,	PUNCT
cana-5953	90	27	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	90	28	)	)	PUNCT
cana-5953	90	29	)	)	PUNCT
cana-5953	90	30	→	→	SYM
cana-5953	90	31	0	0	X
cana-5953	90	32	.	.	PUNCT
cana-5953	90	33	due	due	ADP
cana-5953	90	34	to	to	ADP
cana-5953	90	35	continuity	continuity	NOUN
cana-5953	90	36	of	of	ADP
cana-5953	90	37	𝜓	𝜓	NOUN
cana-5953	90	38	,	,	PUNCT
cana-5953	90	39	it	it	PRON
cana-5953	90	40	follows	follow	VERB
cana-5953	90	41	that	that	PRON
cana-5953	90	42	0	0	X
cana-5953	91	1	=	=	SYM
cana-5953	91	2	lim	lim	PROPN
cana-5953	91	3	𝑛→∞	𝑛→∞	PUNCT
cana-5953	91	4	𝜓(𝑆(𝑔𝑥𝑛+1	𝜓(𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	91	5	,	,	PUNCT
cana-5953	91	6	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	91	7	,	,	PUNCT
cana-5953	91	8	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	91	9	)	)	PUNCT
cana-5953	91	10	,	,	PUNCT
cana-5953	91	11	𝑆(𝑔𝑥𝑛+2	𝑆(𝑔𝑥𝑛+2	PROPN
cana-5953	91	12	,	,	PUNCT
cana-5953	91	13	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	91	14	,	,	PUNCT
cana-5953	91	15	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	91	16	)	)	PUNCT
cana-5953	91	17	)	)	PUNCT
cana-5953	91	18	0	0	NUM
cana-5953	92	1	=	=	SYM
cana-5953	92	2	𝜓	𝜓	PROPN
cana-5953	92	3	(	(	PUNCT
cana-5953	92	4	lim	lim	PROPN
cana-5953	92	5	𝑛→∞	𝑛→∞	NUM
cana-5953	92	6	𝑆(𝑔𝑥𝑛+1	𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	92	7	,	,	PUNCT
cana-5953	92	8	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	92	9	,	,	PUNCT
cana-5953	92	10	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	92	11	)	)	PUNCT
cana-5953	92	12	,	,	PUNCT
cana-5953	92	13	lim	lim	PROPN
cana-5953	92	14	𝑛→∞	𝑛→∞	NUM
cana-5953	92	15	𝑆(𝑔𝑥𝑛+2	𝑆(𝑔𝑥𝑛+2	PROPN
cana-5953	92	16	,	,	PUNCT
cana-5953	92	17	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	92	18	,	,	PUNCT
cana-5953	92	19	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	92	20	)	)	PUNCT
cana-5953	92	21	)	)	PUNCT
cana-5953	92	22	.	.	PUNCT
cana-5953	93	1	thus	thus	ADV
cana-5953	93	2	,	,	PUNCT
cana-5953	93	3	by	by	ADP
cana-5953	93	4	using	use	VERB
cana-5953	93	5	the	the	DET
cana-5953	93	6	property	property	NOUN
cana-5953	93	7	of	of	ADP
cana-5953	93	8	𝜓	𝜓	NOUN
cana-5953	93	9	,	,	PUNCT
cana-5953	93	10	we	we	PRON
cana-5953	93	11	conclude	conclude	VERB
cana-5953	93	12	lim	lim	PROPN
cana-5953	93	13	𝑛→∞	𝑛→∞	NUM
cana-5953	93	14	𝑆(𝑔𝑥𝑛+1	𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	93	15	,	,	PUNCT
cana-5953	93	16	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	93	17	,	,	PUNCT
cana-5953	93	18	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	93	19	)	)	PUNCT
cana-5953	93	20	=	=	SYM
cana-5953	93	21	lim	lim	PROPN
cana-5953	93	22	𝑛→∞	𝑛→∞	NUM
cana-5953	93	23	𝑆(𝑔𝑥𝑛+2	𝑆(𝑔𝑥𝑛+2	PROPN
cana-5953	93	24	,	,	PUNCT
cana-5953	93	25	𝑔𝑥𝑛+2	𝑔𝑥𝑛+2	NOUN
cana-5953	93	26	,	,	PUNCT
cana-5953	93	27	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	93	28	)	)	PUNCT
cana-5953	93	29	=	=	SYM
cana-5953	94	1	0	0	X
cana-5953	94	2	.	.	PUNCT
cana-5953	95	1	now	now	ADV
cana-5953	95	2	,	,	PUNCT
cana-5953	95	3	we	we	PRON
cana-5953	95	4	will	will	AUX
cana-5953	95	5	prove	prove	VERB
cana-5953	95	6	that	that	SCONJ
cana-5953	95	7	the	the	DET
cana-5953	95	8	sequence	sequence	NOUN
cana-5953	95	9	{	{	PUNCT
cana-5953	95	10	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	95	11	}	}	PUNCT
cana-5953	95	12	is	be	AUX
cana-5953	95	13	a	a	DET
cana-5953	95	14	cauchy	cauchy	ADJ
cana-5953	95	15	sequence	sequence	NOUN
cana-5953	95	16	.	.	PUNCT
cana-5953	96	1	assume	assume	VERB
cana-5953	96	2	,	,	PUNCT
cana-5953	96	3	for	for	ADP
cana-5953	96	4	contradiction	contradiction	NOUN
cana-5953	96	5	,	,	PUNCT
cana-5953	96	6	that	that	SCONJ
cana-5953	96	7	{	{	PUNCT
cana-5953	96	8	𝑔𝑥_𝑛	𝑔𝑥_𝑛	ADV
cana-5953	96	9	}	}	PUNCT
cana-5953	96	10	is	be	AUX
cana-5953	96	11	not	not	PART
cana-5953	96	12	a	a	DET
cana-5953	96	13	cauchy	cauchy	ADJ
cana-5953	96	14	sequence	sequence	NOUN
cana-5953	96	15	.	.	PUNCT
cana-5953	97	1	then	then	ADV
cana-5953	97	2	there	there	PRON
cana-5953	97	3	is	be	VERB
cana-5953	97	4	𝜖	𝜖	PROPN
cana-5953	97	5	>	>	X
cana-5953	97	6	0	0	PUNCT
cana-5953	97	7	and	and	CCONJ
cana-5953	97	8	subsequences	subsequence	VERB
cana-5953	97	9	{	{	PUNCT
cana-5953	97	10	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	97	11	)	)	PUNCT
cana-5953	97	12	}	}	PUNCT
cana-5953	97	13	and	and	CCONJ
cana-5953	97	14	{	{	PUNCT
cana-5953	97	15	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	97	16	)	)	PUNCT
cana-5953	97	17	}	}	PUNCT
cana-5953	97	18	such	such	ADJ
cana-5953	97	19	that	that	PRON
cana-5953	97	20	,	,	PUNCT
cana-5953	97	21	∀	∀	X
cana-5953	97	22	𝑘	𝑘	ADP
cana-5953	97	23	∈	∈	PROPN
cana-5953	97	24	𝑁.	𝑁.	PROPN
cana-5953	98	1	then	then	ADV
cana-5953	98	2	,	,	PUNCT
cana-5953	98	3	we	we	PRON
cana-5953	98	4	have	have	VERB
cana-5953	98	5	𝑛(𝑘	𝑛(𝑘	NOUN
cana-5953	98	6	)	)	PUNCT
cana-5953	98	7	>	>	X
cana-5953	99	1	𝑚(𝑘	𝑚(𝑘	PROPN
cana-5953	99	2	)	)	PUNCT
cana-5953	99	3	>	>	X
cana-5953	100	1	𝑘	𝑘	X
cana-5953	100	2	,	,	PUNCT
cana-5953	100	3	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NUM
cana-5953	100	4	)	)	PUNCT
cana-5953	100	5	,	,	PUNCT
cana-5953	100	6	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	100	7	)	)	PUNCT
cana-5953	100	8	,	,	PUNCT
cana-5953	100	9	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	100	10	)	)	PUNCT
cana-5953	100	11	)	)	PUNCT
cana-5953	100	12	≥	≥	NOUN
cana-5953	100	13	𝜖	𝜖	X
cana-5953	100	14	and	and	CCONJ
cana-5953	100	15	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NUM
cana-5953	100	16	)	)	PUNCT
cana-5953	100	17	,	,	PUNCT
cana-5953	100	18	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	100	19	)	)	PUNCT
cana-5953	100	20	,	,	PUNCT
cana-5953	100	21	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	100	22	)	)	PUNCT
cana-5953	100	23	)	)	PUNCT
cana-5953	100	24	<	<	X
cana-5953	100	25	𝜖.	𝜖.	NOUN
cana-5953	100	26	communications	communication	NOUN
cana-5953	100	27	on	on	ADP
cana-5953	100	28	applied	apply	VERB
cana-5953	100	29	nonlinear	nonlinear	ADJ
cana-5953	100	30	analysis	analysis	NOUN
cana-5953	100	31	issn	issn	NOUN
cana-5953	100	32	:	:	PUNCT
cana-5953	100	33	1074	1074	NUM
cana-5953	100	34	-	-	PUNCT
cana-5953	100	35	133x	133x	NUM
cana-5953	100	36	vol	vol	VERB
cana-5953	100	37	32	32	NUM
cana-5953	100	38	no	no	NOUN
cana-5953	100	39	.	.	PUNCT
cana-5953	101	1	10s	10	NOUN
cana-5953	101	2	(	(	PUNCT
cana-5953	101	3	2025	2025	NUM
cana-5953	101	4	)	)	PUNCT
cana-5953	101	5	3164	3164	NUM
cana-5953	101	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	101	7	hence	hence	ADV
cana-5953	101	8	,	,	PUNCT
cana-5953	101	9	𝜖	𝜖	X
cana-5953	101	10	≤	≤	ADJ
cana-5953	101	11	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NUM
cana-5953	101	12	)	)	PUNCT
cana-5953	101	13	,	,	PUNCT
cana-5953	101	14	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	101	15	)	)	PUNCT
cana-5953	101	16	,	,	PUNCT
cana-5953	101	17	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	101	18	)	)	PUNCT
cana-5953	101	19	)	)	PUNCT
cana-5953	102	1	=	=	SYM
cana-5953	102	2	𝑆(𝑔𝑥𝑚(𝑘	𝑆(𝑔𝑥𝑚(𝑘	NOUN
cana-5953	102	3	)	)	PUNCT
cana-5953	102	4	,	,	PUNCT
cana-5953	102	5	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	102	6	)	)	PUNCT
cana-5953	102	7	,	,	PUNCT
cana-5953	102	8	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	102	9	)	)	PUNCT
cana-5953	102	10	)	)	PUNCT
cana-5953	102	11	≤	≤	NOUN
cana-5953	103	1	𝑆(𝑔𝑥𝑚(𝑘	𝑆(𝑔𝑥𝑚(𝑘	ADJ
cana-5953	103	2	)	)	PUNCT
cana-5953	103	3	,	,	PUNCT
cana-5953	103	4	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	103	5	)	)	PUNCT
cana-5953	103	6	,	,	PUNCT
cana-5953	103	7	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	103	8	)	)	PUNCT
cana-5953	103	9	)	)	PUNCT
cana-5953	104	1	+	+	CCONJ
cana-5953	104	2	𝑆(𝑔𝑥𝑚(𝑘	𝑆(𝑔𝑥𝑚(𝑘	NOUN
cana-5953	104	3	)	)	PUNCT
cana-5953	104	4	,	,	PUNCT
cana-5953	104	5	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	104	6	)	)	PUNCT
cana-5953	104	7	,	,	PUNCT
cana-5953	104	8	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	104	9	)	)	PUNCT
cana-5953	104	10	)	)	PUNCT
cana-5953	105	1	+	+	VERB
cana-5953	105	2	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NOUN
cana-5953	105	3	)	)	PUNCT
cana-5953	105	4	,	,	PUNCT
cana-5953	105	5	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	105	6	)	)	PUNCT
cana-5953	105	7	,	,	PUNCT
cana-5953	105	8	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	105	9	)	)	PUNCT
cana-5953	105	10	)	)	PUNCT
cana-5953	105	11	<	<	X
cana-5953	105	12	0	0	PUNCT
cana-5953	106	1	+	+	CCONJ
cana-5953	106	2	0	0	NUM
cana-5953	107	1	+	+	CCONJ
cana-5953	107	2	𝜖.	𝜖.	NOUN
cana-5953	107	3	which	which	PRON
cana-5953	107	4	is	be	AUX
cana-5953	107	5	leads	lead	VERB
cana-5953	107	6	to	to	ADP
cana-5953	107	7	a	a	DET
cana-5953	107	8	contradiction	contradiction	NOUN
cana-5953	107	9	.	.	PUNCT
cana-5953	108	1	so	so	ADV
cana-5953	108	2	,	,	PUNCT
cana-5953	108	3	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NUM
cana-5953	108	4	)	)	PUNCT
cana-5953	108	5	,	,	PUNCT
cana-5953	108	6	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	108	7	)	)	PUNCT
cana-5953	108	8	,	,	PUNCT
cana-5953	108	9	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	108	10	)	)	PUNCT
cana-5953	108	11	)	)	PUNCT
cana-5953	108	12	=	=	SYM
cana-5953	108	13	𝜖.	𝜖.	ADV
cana-5953	108	14	additionally	additionally	ADV
cana-5953	108	15	,	,	PUNCT
cana-5953	108	16	𝑆(𝑔𝑥𝑛(𝑘)−1	𝑆(𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	108	17	,	,	PUNCT
cana-5953	108	18	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	108	19	,	,	PUNCT
cana-5953	108	20	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	108	21	)	)	PUNCT
cana-5953	108	22	≤	≤	NOUN
cana-5953	108	23	𝑆(𝑔𝑥𝑛(𝑘)−1	𝑆(𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	108	24	,	,	PUNCT
cana-5953	108	25	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	108	26	,	,	PUNCT
cana-5953	108	27	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	108	28	)	)	PUNCT
cana-5953	108	29	)	)	PUNCT
cana-5953	109	1	+	+	CCONJ
cana-5953	109	2	𝑆(𝑔𝑥𝑛(𝑘)−1	𝑆(𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	109	3	,	,	PUNCT
cana-5953	109	4	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	109	5	,	,	PUNCT
cana-5953	109	6	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	109	7	)	)	PUNCT
cana-5953	109	8	)	)	PUNCT
cana-5953	110	1	+	+	PUNCT
cana-5953	110	2	𝑆(𝑔𝑥𝑚(𝑘)−1	𝑆(𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	3	,	,	PUNCT
cana-5953	110	4	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	5	,	,	PUNCT
cana-5953	110	6	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	110	7	)	)	PUNCT
cana-5953	110	8	)	)	PUNCT
cana-5953	110	9	=	=	SYM
cana-5953	110	10	2𝑆(𝑔𝑥𝑛(𝑘)−1	2𝑆(𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	110	11	,	,	PUNCT
cana-5953	110	12	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	110	13	,	,	PUNCT
cana-5953	110	14	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	110	15	)	)	PUNCT
cana-5953	110	16	)	)	PUNCT
cana-5953	110	17	+	+	CCONJ
cana-5953	110	18	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NUM
cana-5953	110	19	)	)	PUNCT
cana-5953	110	20	,	,	PUNCT
cana-5953	110	21	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	110	22	)	)	PUNCT
cana-5953	110	23	,	,	PUNCT
cana-5953	110	24	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	25	)	)	PUNCT
cana-5953	110	26	<	<	X
cana-5953	110	27	2(0	2(0	NUM
cana-5953	110	28	)	)	PUNCT
cana-5953	110	29	+	+	NUM
cana-5953	110	30	𝜖	𝜖	X
cana-5953	110	31	=	=	X
cana-5953	110	32	𝜖.	𝜖.	NOUN
cana-5953	110	33	thus	thus	ADV
cana-5953	110	34	,	,	PUNCT
cana-5953	110	35	𝑆(𝑔𝑥𝑛(𝑘)−1	𝑆(𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	110	36	,	,	PUNCT
cana-5953	110	37	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	110	38	,	,	PUNCT
cana-5953	110	39	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	40	)	)	PUNCT
cana-5953	110	41	<	<	X
cana-5953	110	42	𝜖.	𝜖.	X
cana-5953	110	43	consequently	consequently	ADV
cana-5953	110	44	,	,	PUNCT
cana-5953	110	45	𝑆(𝑔𝑥𝑚(𝑘)−1	𝑆(𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	46	,	,	PUNCT
cana-5953	110	47	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	48	,	,	PUNCT
cana-5953	110	49	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	110	50	)	)	PUNCT
cana-5953	110	51	=	=	SYM
cana-5953	110	52	𝑆(𝑔𝑥𝑛(𝑘)−1	𝑆(𝑔𝑥𝑛(𝑘)−1	PROPN
cana-5953	110	53	,	,	PUNCT
cana-5953	110	54	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	110	55	,	,	PUNCT
cana-5953	110	56	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	110	57	)	)	PUNCT
cana-5953	110	58	<	<	X
cana-5953	110	59	𝜖.	𝜖.	X
cana-5953	110	60	now	now	ADV
cana-5953	110	61	,	,	PUNCT
cana-5953	110	62	𝜓(𝑆(𝑔𝑥𝑛(𝑘	𝜓(𝑆(𝑔𝑥𝑛(𝑘	ADJ
cana-5953	110	63	)	)	PUNCT
cana-5953	110	64	,	,	PUNCT
cana-5953	110	65	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	PROPN
cana-5953	110	66	)	)	PUNCT
cana-5953	110	67	,	,	PUNCT
cana-5953	110	68	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	110	69	)	)	PUNCT
cana-5953	110	70	)	)	PUNCT
cana-5953	110	71	,	,	PUNCT
cana-5953	110	72	𝑆(𝑔𝑥𝑚(𝑘	𝑆(𝑔𝑥𝑚(𝑘	NOUN
cana-5953	110	73	)	)	PUNCT
cana-5953	110	74	,	,	PUNCT
cana-5953	110	75	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	110	76	)	)	PUNCT
cana-5953	110	77	,	,	PUNCT
cana-5953	110	78	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	NOUN
cana-5953	110	79	)	)	PUNCT
cana-5953	110	80	)	)	PUNCT
cana-5953	110	81	)	)	PUNCT
cana-5953	111	1	=	=	SYM
cana-5953	111	2	𝜓(𝑆(𝑓𝑥𝑛(𝑘)−1	𝜓(𝑆(𝑓𝑥𝑛(𝑘)−1	PROPN
cana-5953	111	3	,	,	PUNCT
cana-5953	111	4	𝑓𝑥𝑛(𝑘)−1	𝑓𝑥𝑛(𝑘)−1	NOUN
cana-5953	111	5	,	,	PUNCT
cana-5953	111	6	𝑓𝑥𝑚(𝑘)−1	𝑓𝑥𝑚(𝑘)−1	NOUN
cana-5953	111	7	)	)	PUNCT
cana-5953	111	8	,	,	PUNCT
cana-5953	111	9	𝑆(𝑓𝑥𝑚(𝑘)−1	𝑆(𝑓𝑥𝑚(𝑘)−1	NOUN
cana-5953	111	10	,	,	PUNCT
cana-5953	111	11	𝑓𝑥𝑚(𝑘)−1	𝑓𝑥𝑚(𝑘)−1	NOUN
cana-5953	111	12	,	,	PUNCT
cana-5953	111	13	𝑓𝑥𝑛(𝑘)−1	𝑓𝑥𝑛(𝑘)−1	NOUN
cana-5953	111	14	)	)	PUNCT
cana-5953	111	15	)	)	PUNCT
cana-5953	111	16	≤	≤	NUM
cana-5953	111	17	𝑞𝜓(𝑆(𝑔𝑥𝑛(𝑘)−1	𝑞𝜓(𝑆(𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	111	18	,	,	PUNCT
cana-5953	111	19	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	111	20	,	,	PUNCT
cana-5953	111	21	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	111	22	)	)	PUNCT
cana-5953	111	23	,	,	PUNCT
cana-5953	111	24	𝑆(𝑔𝑥𝑚(𝑘)−1	𝑆(𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	111	25	,	,	PUNCT
cana-5953	111	26	𝑔𝑥𝑚(𝑘)−1	𝑔𝑥𝑚(𝑘)−1	NOUN
cana-5953	111	27	,	,	PUNCT
cana-5953	111	28	𝑔𝑥𝑛(𝑘)−1	𝑔𝑥𝑛(𝑘)−1	NOUN
cana-5953	111	29	)	)	PUNCT
cana-5953	111	30	)	)	PUNCT
cana-5953	111	31	<	<	X
cana-5953	111	32	𝑞𝜓(𝜖	𝑞𝜓(𝜖	NOUN
cana-5953	111	33	,	,	PUNCT
cana-5953	111	34	𝜖	𝜖	PROPN
cana-5953	111	35	)	)	PUNCT
cana-5953	111	36	,	,	PUNCT
cana-5953	111	37	which	which	PRON
cana-5953	111	38	gives	give	VERB
cana-5953	111	39	𝜓(𝜖	𝜓(𝜖	PROPN
cana-5953	111	40	,	,	PUNCT
cana-5953	111	41	𝜖	𝜖	X
cana-5953	111	42	)	)	PUNCT
cana-5953	111	43	<	<	X
cana-5953	111	44	𝑞𝜓(𝜖	𝑞𝜓(𝜖	NOUN
cana-5953	111	45	,	,	PUNCT
cana-5953	111	46	𝜖	𝜖	PROPN
cana-5953	111	47	)	)	PUNCT
cana-5953	111	48	.	.	PUNCT
cana-5953	112	1	since	since	SCONJ
cana-5953	112	2	0	0	NUM
cana-5953	112	3	<	<	X
cana-5953	112	4	𝑞	𝑞	X
cana-5953	112	5	<	<	X
cana-5953	112	6	1	1	NUM
cana-5953	112	7	,	,	PUNCT
cana-5953	112	8	this	this	DET
cana-5953	112	9	inequality	inequality	NOUN
cana-5953	112	10	is	be	AUX
cana-5953	112	11	only	only	ADV
cana-5953	112	12	possible	possible	ADJ
cana-5953	112	13	if	if	SCONJ
cana-5953	112	14	𝜖	𝜖	PROPN
cana-5953	112	15	=	=	NOUN
cana-5953	112	16	0𝜓(𝜖	0𝜓(𝜖	PROPN
cana-5953	112	17	,	,	PUNCT
cana-5953	112	18	𝜖	𝜖	X
cana-5953	112	19	)	)	PUNCT
cana-5953	112	20	=	=	SYM
cana-5953	112	21	0	0	X
cana-5953	112	22	.	.	PUNCT
cana-5953	113	1	by	by	ADP
cana-5953	113	2	the	the	DET
cana-5953	113	3	property	property	NOUN
cana-5953	113	4	of	of	ADP
cana-5953	113	5	𝜓	𝜓	NOUN
cana-5953	113	6	we	we	PRON
cana-5953	113	7	conclude	conclude	VERB
cana-5953	113	8	𝑆(𝑔𝑥𝑛(𝑘	𝑆(𝑔𝑥𝑛(𝑘	NUM
cana-5953	113	9	)	)	PUNCT
cana-5953	113	10	,	,	PUNCT
cana-5953	113	11	𝑔𝑥𝑛(𝑘	𝑔𝑥𝑛(𝑘	PROPN
cana-5953	113	12	)	)	PUNCT
cana-5953	113	13	,	,	PUNCT
cana-5953	113	14	𝑔𝑥𝑚(𝑘	𝑔𝑥𝑚(𝑘	NOUN
cana-5953	113	15	)	)	PUNCT
cana-5953	113	16	)	)	PUNCT
cana-5953	114	1	=	=	PUNCT
cana-5953	114	2	0	0	X
cana-5953	114	3	.	.	PUNCT
cana-5953	115	1	this	this	PRON
cana-5953	115	2	leads	lead	VERB
cana-5953	115	3	to	to	ADP
cana-5953	115	4	a	a	DET
cana-5953	115	5	contradiction	contradiction	NOUN
cana-5953	115	6	.	.	PUNCT
cana-5953	116	1	thus	thus	ADV
cana-5953	116	2	we	we	PRON
cana-5953	116	3	obtain	obtain	VERB
cana-5953	116	4	,	,	PUNCT
cana-5953	116	5	{	{	PUNCT
cana-5953	116	6	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	116	7	}	}	PUNCT
cana-5953	116	8	is	be	AUX
cana-5953	116	9	a	a	DET
cana-5953	116	10	cauchy	cauchy	ADJ
cana-5953	116	11	sequence	sequence	NOUN
cana-5953	116	12	in	in	ADP
cana-5953	116	13	𝑔(𝑋	𝑔(𝑋	PROPN
cana-5953	116	14	)	)	PUNCT
cana-5953	116	15	.	.	PUNCT
cana-5953	117	1	since	since	SCONJ
cana-5953	117	2	𝑔(𝑋	𝑔(𝑋	PROPN
cana-5953	117	3	)	)	PUNCT
cana-5953	117	4	is	be	AUX
cana-5953	117	5	complete	complete	ADJ
cana-5953	117	6	,	,	PUNCT
cana-5953	117	7	there	there	PRON
cana-5953	117	8	exist	exist	VERB
cana-5953	117	9	a	a	DET
cana-5953	117	10	point	point	NOUN
cana-5953	117	11	𝑞	𝑞	X
cana-5953	117	12	in	in	ADP
cana-5953	117	13	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	117	14	)	)	PUNCT
cana-5953	117	15	such	such	ADJ
cana-5953	117	16	that	that	DET
cana-5953	117	17	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	117	18	→	→	SYM
cana-5953	117	19	𝑞	𝑞	PROPN
cana-5953	117	20	,	,	PUNCT
cana-5953	117	21	as	as	ADP
cana-5953	117	22	𝑛	𝑛	PROPN
cana-5953	117	23	→	→	SYM
cana-5953	117	24	∞.	∞.	PROPN
cana-5953	117	25	thus	thus	ADV
cana-5953	117	26	,	,	PUNCT
cana-5953	117	27	there	there	PRON
cana-5953	117	28	exists	exist	VERB
cana-5953	117	29	a	a	DET
cana-5953	117	30	point	point	NOUN
cana-5953	117	31	𝑝	𝑝	NOUN
cana-5953	117	32	in	in	ADP
cana-5953	117	33	𝑋	𝑋	NOUN
cana-5953	117	34	with	with	ADP
cana-5953	117	35	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	117	36	=	=	PUNCT
cana-5953	117	37	𝑞.	𝑞.	NOUN
cana-5953	117	38	hence	hence	ADV
cana-5953	117	39	,	,	PUNCT
cana-5953	117	40	𝜓(𝑆(𝑔𝑥𝑛+1	𝜓(𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	117	41	,	,	PUNCT
cana-5953	117	42	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	117	43	,	,	PUNCT
cana-5953	117	44	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	117	45	)	)	PUNCT
cana-5953	117	46	,	,	PUNCT
cana-5953	117	47	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	117	48	,	,	PUNCT
cana-5953	117	49	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	117	50	,	,	PUNCT
cana-5953	117	51	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	117	52	)	)	PUNCT
cana-5953	117	53	)	)	PUNCT
cana-5953	118	1	=	=	PUNCT
cana-5953	118	2	𝜓(𝑆(𝑓𝑥𝑛	𝜓(𝑆(𝑓𝑥𝑛	ADJ
cana-5953	118	3	,	,	PUNCT
cana-5953	118	4	𝑓𝑥𝑛	𝑓𝑥𝑛	ADJ
cana-5953	118	5	,	,	PUNCT
cana-5953	118	6	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	118	7	)	)	PUNCT
cana-5953	118	8	,	,	PUNCT
cana-5953	118	9	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	118	10	,	,	PUNCT
cana-5953	118	11	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	118	12	,	,	PUNCT
cana-5953	118	13	𝑓𝑥𝑛	𝑓𝑥𝑛	NOUN
cana-5953	118	14	)	)	PUNCT
cana-5953	118	15	)	)	PUNCT
cana-5953	118	16	≤	≤	PUNCT
cana-5953	119	1	𝑞𝜓(𝑆(𝑔𝑥𝑛	𝑞𝜓(𝑆(𝑔𝑥𝑛	PROPN
cana-5953	119	2	,	,	PUNCT
cana-5953	119	3	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	119	4	,	,	PUNCT
cana-5953	119	5	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	119	6	)	)	PUNCT
cana-5953	119	7	,	,	PUNCT
cana-5953	119	8	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	119	9	,	,	PUNCT
cana-5953	119	10	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	119	11	,	,	PUNCT
cana-5953	119	12	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	119	13	)	)	PUNCT
cana-5953	119	14	)	)	PUNCT
cana-5953	119	15	,	,	PUNCT
cana-5953	119	16	communications	communication	NOUN
cana-5953	119	17	on	on	ADP
cana-5953	119	18	applied	apply	VERB
cana-5953	119	19	nonlinear	nonlinear	ADJ
cana-5953	119	20	analysis	analysis	NOUN
cana-5953	119	21	issn	issn	NOUN
cana-5953	119	22	:	:	PUNCT
cana-5953	119	23	1074	1074	NUM
cana-5953	119	24	-	-	PUNCT
cana-5953	119	25	133x	133x	NUM
cana-5953	119	26	vol	vol	VERB
cana-5953	119	27	32	32	NUM
cana-5953	119	28	no	no	NOUN
cana-5953	119	29	.	.	PUNCT
cana-5953	120	1	10s	10	NOUN
cana-5953	120	2	(	(	PUNCT
cana-5953	120	3	2025	2025	NUM
cana-5953	120	4	)	)	PUNCT
cana-5953	120	5	3165	3165	NUM
cana-5953	121	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	121	2	taking	take	VERB
cana-5953	121	3	the	the	DET
cana-5953	121	4	limit	limit	NOUN
cana-5953	121	5	as	as	ADP
cana-5953	121	6	𝑛	𝑛	PROPN
cana-5953	121	7	→	→	SYM
cana-5953	121	8	∞	∞	PROPN
cana-5953	121	9	,	,	PUNCT
cana-5953	121	10	we	we	PRON
cana-5953	121	11	get	get	VERB
cana-5953	121	12	𝜓(𝑆(𝑞	𝜓(𝑆(𝑞	NOUN
cana-5953	121	13	,	,	PUNCT
cana-5953	121	14	𝑞	𝑞	X
cana-5953	121	15	,	,	PUNCT
cana-5953	121	16	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	121	17	)	)	PUNCT
cana-5953	121	18	,	,	PUNCT
cana-5953	121	19	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	121	20	,	,	PUNCT
cana-5953	121	21	𝑓𝑝	𝑓𝑝	PROPN
cana-5953	121	22	,	,	PUNCT
cana-5953	121	23	𝑞	𝑞	NOUN
cana-5953	121	24	)	)	PUNCT
cana-5953	121	25	)	)	PUNCT
cana-5953	121	26	≤	≤	NUM
cana-5953	121	27	𝑞𝜓(𝑆(𝑞	𝑞𝜓(𝑆(𝑞	NOUN
cana-5953	121	28	,	,	PUNCT
cana-5953	121	29	𝑞	𝑞	X
cana-5953	121	30	,	,	PUNCT
cana-5953	121	31	𝑞	𝑞	NOUN
cana-5953	121	32	)	)	PUNCT
cana-5953	121	33	,	,	PUNCT
cana-5953	121	34	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	121	35	,	,	PUNCT
cana-5953	121	36	𝑞	𝑞	PROPN
cana-5953	121	37	,	,	PUNCT
cana-5953	121	38	𝑞	𝑞	NOUN
cana-5953	121	39	)	)	PUNCT
cana-5953	121	40	)	)	PUNCT
cana-5953	122	1	=	=	SYM
cana-5953	122	2	𝑞𝜓(0,0	𝑞𝜓(0,0	NOUN
cana-5953	122	3	)	)	PUNCT
cana-5953	122	4	.	.	PUNCT
cana-5953	123	1	so	so	ADV
cana-5953	123	2	,	,	PUNCT
cana-5953	123	3	𝜓(𝑆(𝑞	𝜓(𝑆(𝑞	ADV
cana-5953	123	4	,	,	PUNCT
cana-5953	123	5	𝑞	𝑞	X
cana-5953	123	6	,	,	PUNCT
cana-5953	123	7	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	123	8	)	)	PUNCT
cana-5953	123	9	,	,	PUNCT
cana-5953	123	10	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	123	11	,	,	PUNCT
cana-5953	123	12	𝑓𝑝	𝑓𝑝	PROPN
cana-5953	123	13	,	,	PUNCT
cana-5953	123	14	𝑞	𝑞	NOUN
cana-5953	123	15	)	)	PUNCT
cana-5953	123	16	)	)	PUNCT
cana-5953	124	1	=	=	PUNCT
cana-5953	124	2	0𝑓𝑝	0𝑓𝑝	NOUN
cana-5953	125	1	=	=	PUNCT
cana-5953	125	2	𝑞.	𝑞.	NOUN
cana-5953	125	3	thus	thus	ADV
cana-5953	125	4	,	,	PUNCT
cana-5953	125	5	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	125	6	=	=	PUNCT
cana-5953	125	7	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	125	8	=	=	SYM
cana-5953	125	9	𝑞	𝑞	X
cana-5953	125	10	showing	show	VERB
cana-5953	125	11	that	that	SCONJ
cana-5953	125	12	𝑞	𝑞	PROPN
cana-5953	125	13	is	be	AUX
cana-5953	125	14	the	the	DET
cana-5953	125	15	point	point	NOUN
cana-5953	125	16	of	of	ADP
cana-5953	125	17	coincidence	coincidence	NOUN
cana-5953	125	18	of	of	ADP
cana-5953	125	19	𝑓	𝑓	PRON
cana-5953	125	20	and	and	CCONJ
cana-5953	125	21	𝑔.	𝑔.	NOUN
cana-5953	125	22	to	to	PART
cana-5953	125	23	show	show	VERB
cana-5953	125	24	uniqueness	uniqueness	NOUN
cana-5953	125	25	,	,	PUNCT
cana-5953	125	26	suppose	suppose	VERB
cana-5953	125	27	another	another	DET
cana-5953	125	28	point	point	NOUN
cana-5953	125	29	𝑤	𝑤	ADP
cana-5953	125	30	∈	∈	NOUN
cana-5953	125	31	𝑋	𝑋	NOUN
cana-5953	125	32	such	such	ADJ
cana-5953	125	33	that	that	SCONJ
cana-5953	125	34	𝑓𝑤	𝑓𝑤	ADP
cana-5953	125	35	=	=	PUNCT
cana-5953	125	36	𝑔𝑤	𝑔𝑤	NOUN
cana-5953	125	37	=	=	NOUN
cana-5953	125	38	𝑤.	𝑤.	NOUN
cana-5953	125	39	now	now	ADV
cana-5953	125	40	𝜓(𝑆(𝑞	𝜓(𝑆(𝑞	ADV
cana-5953	125	41	,	,	PUNCT
cana-5953	125	42	𝑞	𝑞	X
cana-5953	125	43	,	,	PUNCT
cana-5953	125	44	𝑤	𝑤	ADP
cana-5953	125	45	)	)	PUNCT
cana-5953	125	46	,	,	PUNCT
cana-5953	125	47	𝑆(𝑤	𝑆(𝑤	PRON
cana-5953	125	48	,	,	PUNCT
cana-5953	125	49	𝑤	𝑤	PROPN
cana-5953	125	50	,	,	PUNCT
cana-5953	125	51	𝑞	𝑞	NOUN
cana-5953	125	52	)	)	PUNCT
cana-5953	125	53	)	)	PUNCT
cana-5953	126	1	=	=	SYM
cana-5953	126	2	𝜓(𝑆(𝑓𝑝	𝜓(𝑆(𝑓𝑝	PROPN
cana-5953	126	3	,	,	PUNCT
cana-5953	126	4	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	126	5	,	,	PUNCT
cana-5953	126	6	𝑓𝑤	𝑓𝑤	NOUN
cana-5953	126	7	)	)	PUNCT
cana-5953	126	8	,	,	PUNCT
cana-5953	126	9	𝑆(𝑓𝑤	𝑆(𝑓𝑤	NOUN
cana-5953	126	10	,	,	PUNCT
cana-5953	126	11	𝑓𝑤	𝑓𝑤	ADP
cana-5953	126	12	,	,	PUNCT
cana-5953	126	13	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	126	14	)	)	PUNCT
cana-5953	126	15	)	)	PUNCT
cana-5953	126	16	≤	≤	NUM
cana-5953	126	17	𝑞𝜓(𝑆(𝑔𝑝	𝑞𝜓(𝑆(𝑔𝑝	PROPN
cana-5953	126	18	,	,	PUNCT
cana-5953	126	19	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	126	20	,	,	PUNCT
cana-5953	126	21	𝑔𝑤	𝑔𝑤	PROPN
cana-5953	126	22	)	)	PUNCT
cana-5953	126	23	,	,	PUNCT
cana-5953	126	24	𝑆(𝑔𝑤	𝑆(𝑔𝑤	PROPN
cana-5953	126	25	,	,	PUNCT
cana-5953	126	26	𝑔𝑤	𝑔𝑤	PROPN
cana-5953	126	27	,	,	PUNCT
cana-5953	126	28	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	126	29	)	)	PUNCT
cana-5953	126	30	)	)	PUNCT
cana-5953	127	1	=	=	SYM
cana-5953	127	2	𝑞𝜓(𝑆(𝑞	𝑞𝜓(𝑆(𝑞	NOUN
cana-5953	127	3	,	,	PUNCT
cana-5953	127	4	𝑞	𝑞	X
cana-5953	127	5	,	,	PUNCT
cana-5953	127	6	𝑤	𝑤	ADP
cana-5953	127	7	)	)	PUNCT
cana-5953	127	8	,	,	PUNCT
cana-5953	127	9	𝑆(𝑤	𝑆(𝑤	PRON
cana-5953	127	10	,	,	PUNCT
cana-5953	127	11	𝑤	𝑤	PROPN
cana-5953	127	12	,	,	PUNCT
cana-5953	127	13	𝑞	𝑞	NOUN
cana-5953	127	14	)	)	PUNCT
cana-5953	127	15	)	)	PUNCT
cana-5953	127	16	.	.	PUNCT
cana-5953	128	1	again	again	ADV
cana-5953	128	2	,	,	PUNCT
cana-5953	128	3	since	since	SCONJ
cana-5953	128	4	0	0	NUM
cana-5953	128	5	<	<	X
cana-5953	128	6	𝑞	𝑞	X
cana-5953	128	7	<	<	X
cana-5953	128	8	1	1	NUM
cana-5953	128	9	,	,	PUNCT
cana-5953	128	10	this	this	DET
cana-5953	128	11	yields	yield	NOUN
cana-5953	128	12	;	;	PUNCT
cana-5953	128	13	𝜓(𝑆(𝑞	𝜓(𝑆(𝑞	ADV
cana-5953	128	14	,	,	PUNCT
cana-5953	128	15	𝑞	𝑞	X
cana-5953	128	16	,	,	PUNCT
cana-5953	128	17	𝑤	𝑤	ADP
cana-5953	128	18	)	)	PUNCT
cana-5953	128	19	,	,	PUNCT
cana-5953	128	20	𝑆(𝑤	𝑆(𝑤	PRON
cana-5953	128	21	,	,	PUNCT
cana-5953	128	22	𝑤	𝑤	PROPN
cana-5953	128	23	,	,	PUNCT
cana-5953	128	24	𝑞	𝑞	NOUN
cana-5953	128	25	)	)	PUNCT
cana-5953	128	26	)	)	PUNCT
cana-5953	129	1	=	=	SYM
cana-5953	129	2	0𝑞	0𝑞	NOUN
cana-5953	129	3	=	=	SYM
cana-5953	129	4	𝑤.	𝑤.	NOUN
cana-5953	129	5	therefore	therefore	ADV
cana-5953	129	6	𝑓	𝑓	PROPN
cana-5953	129	7	and	and	CCONJ
cana-5953	129	8	𝑔	𝑔	PROPN
cana-5953	129	9	have	have	VERB
cana-5953	129	10	a	a	DET
cana-5953	129	11	unique	unique	ADJ
cana-5953	129	12	point	point	NOUN
cana-5953	129	13	of	of	ADP
cana-5953	129	14	coincidence	coincidence	NOUN
cana-5953	129	15	.	.	PUNCT
cana-5953	130	1	since	since	SCONJ
cana-5953	130	2	,	,	PUNCT
cana-5953	130	3	𝑓	𝑓	PRON
cana-5953	130	4	and	and	CCONJ
cana-5953	130	5	𝑔	𝑔	PROPN
cana-5953	130	6	are	be	AUX
cana-5953	130	7	weakly	weakly	ADV
cana-5953	130	8	compatible	compatible	ADJ
cana-5953	130	9	,	,	PUNCT
cana-5953	130	10	by	by	ADP
cana-5953	130	11	proposition	proposition	NOUN
cana-5953	130	12	(	(	PUNCT
cana-5953	130	13	3.6	3.6	NUM
cana-5953	130	14	)	)	PUNCT
cana-5953	130	15	,	,	PUNCT
cana-5953	130	16	it	it	PRON
cana-5953	130	17	follows	follow	VERB
cana-5953	130	18	that	that	SCONJ
cana-5953	130	19	they	they	PRON
cana-5953	130	20	have	have	VERB
cana-5953	130	21	a	a	DET
cana-5953	130	22	unique	unique	ADJ
cana-5953	130	23	common	common	ADJ
cana-5953	130	24	fixed	fix	VERB
cana-5953	130	25	point	point	NOUN
cana-5953	130	26	in	in	ADP
cana-5953	130	27	𝑋.	𝑋.	PROPN
cana-5953	130	28	theorem	theorem	VERB
cana-5953	130	29	4.2	4.2	NUM
cana-5953	130	30	let	let	VERB
cana-5953	130	31	(	(	PUNCT
cana-5953	130	32	𝑋	𝑋	PROPN
cana-5953	130	33	,	,	PUNCT
cana-5953	130	34	𝑆	𝑆	PROPN
cana-5953	130	35	)	)	PUNCT
cana-5953	130	36	be	be	AUX
cana-5953	130	37	a	a	DET
cana-5953	130	38	complete	complete	ADJ
cana-5953	130	39	rectangular	rectangular	ADJ
cana-5953	130	40	s	s	ADJ
cana-5953	130	41	-	-	ADJ
cana-5953	130	42	metric	metric	ADJ
cana-5953	130	43	space	space	NOUN
cana-5953	130	44	and	and	CCONJ
cana-5953	130	45	𝑓	𝑓	PRON
cana-5953	130	46	,	,	PUNCT
cana-5953	130	47	𝑔	𝑔	NOUN
cana-5953	130	48	:	:	PUNCT
cana-5953	130	49	𝑋	𝑋	PROPN
cana-5953	130	50	→	→	SYM
cana-5953	130	51	𝑋	𝑋	PROPN
cana-5953	130	52	be	be	VERB
cana-5953	130	53	two	two	NUM
cana-5953	130	54	mappings	mapping	NOUN
cana-5953	130	55	such	such	ADJ
cana-5953	130	56	that	that	PRON
cana-5953	130	57	for	for	ADP
cana-5953	130	58	all	all	PRON
cana-5953	130	59	𝑥	𝑥	PROPN
cana-5953	130	60	,	,	PUNCT
cana-5953	130	61	𝑦	𝑦	NOUN
cana-5953	130	62	∈	∈	PROPN
cana-5953	130	63	𝑋	𝑋	PROPN
cana-5953	130	64	,	,	PUNCT
cana-5953	130	65	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	130	66	,	,	PUNCT
cana-5953	130	67	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	130	68	,	,	PUNCT
cana-5953	130	69	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	130	70	)	)	PUNCT
cana-5953	130	71	≤	≤	NOUN
cana-5953	130	72	𝛼𝑆(𝑔𝑥	𝛼𝑆(𝑔𝑥	PUNCT
cana-5953	130	73	,	,	PUNCT
cana-5953	130	74	𝑔𝑥	𝑔𝑥	INTJ
cana-5953	130	75	,	,	PUNCT
cana-5953	130	76	𝑔𝑦	𝑔𝑦	PROPN
cana-5953	130	77	)	)	PUNCT
cana-5953	130	78	+	+	CCONJ
cana-5953	131	1	𝛽[𝑆(𝑔𝑥	𝛽[𝑆(𝑔𝑥	ADV
cana-5953	131	2	,	,	PUNCT
cana-5953	131	3	𝑔𝑥	𝑔𝑥	INTJ
cana-5953	131	4	,	,	PUNCT
cana-5953	131	5	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	131	6	)	)	PUNCT
cana-5953	131	7	+	+	CCONJ
cana-5953	131	8	𝑆(𝑔𝑦	𝑆(𝑔𝑦	ADJ
cana-5953	131	9	,	,	PUNCT
cana-5953	131	10	𝑔𝑦	𝑔𝑦	PROPN
cana-5953	131	11	,	,	PUNCT
cana-5953	131	12	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	131	13	)	)	PUNCT
cana-5953	131	14	]	]	PUNCT
cana-5953	131	15	,	,	PUNCT
cana-5953	131	16	where	where	SCONJ
cana-5953	131	17	𝛼	𝛼	X
cana-5953	131	18	,	,	PUNCT
cana-5953	131	19	𝛽	𝛽	NOUN
cana-5953	131	20	>	>	X
cana-5953	131	21	0	0	NUM
cana-5953	131	22	and	and	CCONJ
cana-5953	131	23	𝛼	𝛼	PRON
cana-5953	131	24	+	+	NOUN
cana-5953	131	25	2𝛽	2𝛽	NOUN
cana-5953	131	26	<	<	X
cana-5953	131	27	1	1	X
cana-5953	131	28	.	.	PUNCT
cana-5953	131	29	assume	assume	VERB
cana-5953	131	30	the	the	DET
cana-5953	131	31	following	follow	VERB
cana-5953	131	32	conditions	condition	NOUN
cana-5953	131	33	hold	hold	VERB
cana-5953	131	34	;	;	PUNCT
cana-5953	131	35	•	•	NUM
cana-5953	131	36	𝑓(𝑋	𝑓(𝑋	NUM
cana-5953	131	37	)	)	PUNCT
cana-5953	131	38	⊆	⊆	NUM
cana-5953	131	39	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	131	40	)	)	PUNCT
cana-5953	131	41	,	,	PUNCT
cana-5953	131	42	•	•	ADP
cana-5953	131	43	if	if	SCONJ
cana-5953	131	44	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	131	45	)	)	PUNCT
cana-5953	131	46	is	be	AUX
cana-5953	131	47	complete	complete	ADJ
cana-5953	131	48	.	.	PUNCT
cana-5953	132	1	then	then	ADV
cana-5953	132	2	𝑓	𝑓	X
cana-5953	132	3	and	and	CCONJ
cana-5953	132	4	𝑔	𝑔	AUX
cana-5953	132	5	have	have	VERB
cana-5953	132	6	a	a	DET
cana-5953	132	7	unique	unique	ADJ
cana-5953	132	8	coincidence	coincidence	NOUN
cana-5953	132	9	point	point	NOUN
cana-5953	132	10	in	in	ADP
cana-5953	132	11	𝑋	𝑋	PROPN
cana-5953	132	12	(	(	PUNCT
cana-5953	132	13	i.e.	i.e.	X
cana-5953	132	14	there	there	PRON
cana-5953	132	15	exists	exist	VERB
cana-5953	132	16	unique	unique	ADJ
cana-5953	132	17	𝑧	𝑧	DET
cana-5953	132	18	∈	∈	NOUN
cana-5953	132	19	𝑋	𝑋	NOUN
cana-5953	132	20	such	such	ADJ
cana-5953	132	21	that	that	DET
cana-5953	132	22	𝑓𝑧	𝑓𝑧	NOUN
cana-5953	132	23	=	=	SYM
cana-5953	132	24	𝑔𝑧	𝑔𝑧	PROPN
cana-5953	132	25	)	)	PUNCT
cana-5953	132	26	.	.	PUNCT
cana-5953	133	1	moreover	moreover	ADV
cana-5953	133	2	,	,	PUNCT
cana-5953	133	3	if	if	SCONJ
cana-5953	133	4	𝑓	𝑓	PRON
cana-5953	133	5	and	and	CCONJ
cana-5953	133	6	𝑔	𝑔	PROPN
cana-5953	133	7	are	be	AUX
cana-5953	133	8	weakly	weakly	ADV
cana-5953	133	9	compatible	compatible	ADJ
cana-5953	133	10	(	(	PUNCT
cana-5953	133	11	i.e.	i.e.	X
cana-5953	133	12	,	,	PUNCT
cana-5953	133	13	they	they	PRON
cana-5953	133	14	commute	commute	VERB
cana-5953	133	15	at	at	ADP
cana-5953	133	16	their	their	PRON
cana-5953	133	17	coincidence	coincidence	NOUN
cana-5953	133	18	point	point	NOUN
cana-5953	133	19	)	)	PUNCT
cana-5953	133	20	,	,	PUNCT
cana-5953	133	21	then	then	ADV
cana-5953	133	22	𝑓	𝑓	X
cana-5953	133	23	and	and	CCONJ
cana-5953	133	24	𝑔	𝑔	AUX
cana-5953	133	25	have	have	VERB
cana-5953	133	26	a	a	DET
cana-5953	133	27	unique	unique	ADJ
cana-5953	133	28	common	common	ADJ
cana-5953	133	29	fixed	fix	VERB
cana-5953	133	30	point	point	NOUN
cana-5953	133	31	in	in	ADP
cana-5953	133	32	𝑋	𝑋	PROPN
cana-5953	133	33	(	(	PUNCT
cana-5953	133	34	that	that	PRON
cana-5953	133	35	is	is	ADV
cana-5953	133	36	,	,	PUNCT
cana-5953	133	37	there	there	PRON
cana-5953	133	38	exists	exist	VERB
cana-5953	133	39	a	a	DET
cana-5953	133	40	unique	unique	ADJ
cana-5953	133	41	𝑝	𝑝	NOUN
cana-5953	133	42	∈	∈	NOUN
cana-5953	133	43	𝑋	𝑋	NOUN
cana-5953	133	44	such	such	ADJ
cana-5953	133	45	that	that	SCONJ
cana-5953	133	46	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	133	47	=	=	PUNCT
cana-5953	133	48	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	133	49	=	=	PROPN
cana-5953	133	50	𝑝	𝑝	NOUN
cana-5953	133	51	)	)	PUNCT
cana-5953	133	52	.	.	PUNCT
cana-5953	134	1	proof	proof	NOUN
cana-5953	134	2	.	.	PUNCT
cana-5953	135	1	let	let	VERB
cana-5953	135	2	𝑥0	𝑥0	NOUN
cana-5953	135	3	be	be	AUX
cana-5953	135	4	any	any	DET
cana-5953	135	5	point	point	NOUN
cana-5953	135	6	in	in	ADP
cana-5953	135	7	𝑋.	𝑋.	PROPN
cana-5953	135	8	since	since	SCONJ
cana-5953	135	9	𝑓𝑥0	𝑓𝑥0	PROPN
cana-5953	135	10	∈	∈	PROPN
cana-5953	135	11	𝑓(𝑋	𝑓(𝑋	PROPN
cana-5953	135	12	)	)	PUNCT
cana-5953	135	13	and	and	CCONJ
cana-5953	135	14	𝑓(𝑋	𝑓(𝑋	NUM
cana-5953	135	15	)	)	PUNCT
cana-5953	135	16	⊆	⊆	NUM
cana-5953	135	17	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	135	18	)	)	PUNCT
cana-5953	135	19	,	,	PUNCT
cana-5953	135	20	there	there	PRON
cana-5953	135	21	exists	exist	VERB
cana-5953	135	22	a	a	DET
cana-5953	135	23	point	point	NOUN
cana-5953	135	24	𝑥1	𝑥1	NOUN
cana-5953	135	25	in	in	ADP
cana-5953	135	26	𝑋	𝑋	PROPN
cana-5953	135	27	such	such	ADJ
cana-5953	135	28	that	that	SCONJ
cana-5953	135	29	𝑓𝑥0	𝑓𝑥0	PROPN
cana-5953	135	30	=	=	SYM
cana-5953	135	31	𝑔𝑥1	𝑔𝑥1	PROPN
cana-5953	135	32	.	.	PUNCT
cana-5953	136	1	as	as	ADP
cana-5953	136	2	𝑥1	𝑥1	PROPN
cana-5953	136	3	∈	∈	PROPN
cana-5953	136	4	𝑋	𝑋	PROPN
cana-5953	136	5	,	,	PUNCT
cana-5953	136	6	we	we	PRON
cana-5953	136	7	have	have	VERB
cana-5953	136	8	𝑓𝑥1	𝑓𝑥1	NOUN
cana-5953	136	9	∈	∈	PROPN
cana-5953	136	10	𝑓(𝑋	𝑓(𝑋	PROPN
cana-5953	136	11	)	)	PUNCT
cana-5953	136	12	,	,	PUNCT
cana-5953	136	13	so	so	CCONJ
cana-5953	136	14	there	there	PRON
cana-5953	136	15	exists	exist	VERB
cana-5953	136	16	𝑥2	𝑥2	NOUN
cana-5953	136	17	in	in	ADP
cana-5953	136	18	𝑋	𝑋	PROPN
cana-5953	136	19	such	such	ADJ
cana-5953	136	20	that	that	DET
cana-5953	136	21	𝑓𝑥1	𝑓𝑥1	NOUN
cana-5953	136	22	=	=	SYM
cana-5953	136	23	𝑔𝑥2	𝑔𝑥2	PROPN
cana-5953	136	24	.	.	PUNCT
cana-5953	137	1	continuing	continue	VERB
cana-5953	137	2	in	in	ADP
cana-5953	137	3	this	this	DET
cana-5953	137	4	way	way	NOUN
cana-5953	137	5	,	,	PUNCT
cana-5953	137	6	we	we	PRON
cana-5953	137	7	construct	construct	VERB
cana-5953	137	8	a	a	DET
cana-5953	137	9	sequence	sequence	NOUN
cana-5953	137	10	{	{	PUNCT
cana-5953	137	11	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	137	12	}	}	PUNCT
cana-5953	137	13	in	in	ADP
cana-5953	137	14	𝑋	𝑋	PROPN
cana-5953	137	15	such	such	ADJ
cana-5953	137	16	that	that	SCONJ
cana-5953	137	17	,	,	PUNCT
cana-5953	137	18	𝑓𝑥𝑛	𝑓𝑥𝑛	ADJ
cana-5953	137	19	=	=	SYM
cana-5953	137	20	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	137	21	for	for	ADP
cana-5953	137	22	all	all	DET
cana-5953	137	23	n	n	CCONJ
cana-5953	137	24	,	,	PUNCT
cana-5953	137	25	where	where	SCONJ
cana-5953	137	26	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5953	137	27	∈	∈	PROPN
cana-5953	137	28	𝑋	𝑋	PROPN
cana-5953	137	29	now	now	ADV
cana-5953	137	30	,	,	PUNCT
cana-5953	137	31	consider	consider	VERB
cana-5953	137	32	𝑆(𝑔𝑥𝑛+1	𝑆(𝑔𝑥𝑛+1	NOUN
cana-5953	137	33	,	,	PUNCT
cana-5953	137	34	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	137	35	,	,	PUNCT
cana-5953	137	36	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	137	37	)	)	PUNCT
cana-5953	137	38	=	=	SYM
cana-5953	138	1	𝑆(𝑓𝑥𝑛	𝑆(𝑓𝑥𝑛	VERB
cana-5953	138	2	,	,	PUNCT
cana-5953	138	3	𝑓𝑥𝑛	𝑓𝑥𝑛	INTJ
cana-5953	138	4	,	,	PUNCT
cana-5953	138	5	𝑓𝑥𝑛−1	𝑓𝑥𝑛−1	PROPN
cana-5953	138	6	)	)	PUNCT
cana-5953	138	7	≤	≤	NOUN
cana-5953	138	8	𝛼𝑆(𝑔𝑥𝑛	𝛼𝑆(𝑔𝑥𝑛	NOUN
cana-5953	138	9	,	,	PUNCT
cana-5953	138	10	𝑔𝑥𝑛	𝑔𝑥𝑛	INTJ
cana-5953	138	11	,	,	PUNCT
cana-5953	138	12	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	138	13	)	)	PUNCT
cana-5953	139	1	+	+	NUM
cana-5953	139	2	𝛽[𝑆(𝑔𝑥𝑛	𝛽[𝑆(𝑔𝑥𝑛	PROPN
cana-5953	139	3	,	,	PUNCT
cana-5953	139	4	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	139	5	,	,	PUNCT
cana-5953	139	6	𝑓𝑥𝑛	𝑓𝑥𝑛	NOUN
cana-5953	139	7	)	)	PUNCT
cana-5953	140	1	+	+	CCONJ
cana-5953	140	2	𝑆(𝑔𝑥𝑛−1	𝑆(𝑔𝑥𝑛−1	PROPN
cana-5953	140	3	,	,	PUNCT
cana-5953	140	4	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	140	5	,	,	PUNCT
cana-5953	140	6	𝑓𝑥𝑛−1	𝑓𝑥𝑛−1	PROPN
cana-5953	140	7	)	)	PUNCT
cana-5953	140	8	]	]	PUNCT
cana-5953	141	1	=	=	SYM
cana-5953	141	2	𝛼𝑆(𝑔𝑥𝑛	𝛼𝑆(𝑔𝑥𝑛	X
cana-5953	141	3	,	,	PUNCT
cana-5953	141	4	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	141	5	,	,	PUNCT
cana-5953	141	6	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	141	7	)	)	PUNCT
cana-5953	141	8	+	+	X
cana-5953	141	9	𝛽[𝑆(𝑔𝑥𝑛	𝛽[𝑆(𝑔𝑥𝑛	ADJ
cana-5953	141	10	,	,	PUNCT
cana-5953	141	11	𝑔𝑥𝑛	𝑔𝑥𝑛	X
cana-5953	141	12	,	,	PUNCT
cana-5953	141	13	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	141	14	)	)	PUNCT
cana-5953	141	15	+	+	CCONJ
cana-5953	141	16	𝑆(𝑔𝑥𝑛−1	𝑆(𝑔𝑥𝑛−1	PROPN
cana-5953	141	17	,	,	PUNCT
cana-5953	141	18	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	141	19	,	,	PUNCT
cana-5953	141	20	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	141	21	)	)	PUNCT
cana-5953	141	22	]	]	PUNCT
cana-5953	141	23	communications	communication	NOUN
cana-5953	141	24	on	on	ADP
cana-5953	141	25	applied	apply	VERB
cana-5953	141	26	nonlinear	nonlinear	ADJ
cana-5953	141	27	analysis	analysis	NOUN
cana-5953	141	28	issn	issn	NOUN
cana-5953	141	29	:	:	PUNCT
cana-5953	141	30	1074	1074	NUM
cana-5953	141	31	-	-	PUNCT
cana-5953	141	32	133x	133x	NUM
cana-5953	141	33	vol	vol	VERB
cana-5953	141	34	32	32	NUM
cana-5953	141	35	no	no	NOUN
cana-5953	141	36	.	.	PUNCT
cana-5953	142	1	10s	10	NOUN
cana-5953	142	2	(	(	PUNCT
cana-5953	142	3	2025	2025	NUM
cana-5953	142	4	)	)	PUNCT
cana-5953	142	5	3166	3166	NUM
cana-5953	142	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	142	7	=	=	SYM
cana-5953	142	8	𝛼𝑆(𝑔𝑥𝑛	𝛼𝑆(𝑔𝑥𝑛	X
cana-5953	142	9	,	,	PUNCT
cana-5953	142	10	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	142	11	,	,	PUNCT
cana-5953	142	12	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	142	13	)	)	PUNCT
cana-5953	142	14	+	+	CCONJ
cana-5953	142	15	𝛽[𝑆(𝑔𝑥𝑛+1	𝛽[𝑆(𝑔𝑥𝑛+1	NOUN
cana-5953	142	16	,	,	PUNCT
cana-5953	142	17	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	142	18	,	,	PUNCT
cana-5953	142	19	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	142	20	)	)	PUNCT
cana-5953	142	21	+	+	CCONJ
cana-5953	142	22	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	ADJ
cana-5953	142	23	,	,	PUNCT
cana-5953	142	24	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	142	25	,	,	PUNCT
cana-5953	142	26	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	142	27	)	)	PUNCT
cana-5953	142	28	]	]	PUNCT
cana-5953	143	1	=	=	PUNCT
cana-5953	143	2	(	(	PUNCT
cana-5953	143	3	𝛼	𝛼	X
cana-5953	143	4	+	+	X
cana-5953	143	5	𝛽)𝑆(𝑔𝑥𝑛	𝛽)𝑆(𝑔𝑥𝑛	ADJ
cana-5953	143	6	,	,	PUNCT
cana-5953	143	7	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	8	,	,	PUNCT
cana-5953	143	9	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	143	10	)	)	PUNCT
cana-5953	143	11	+	+	CCONJ
cana-5953	143	12	𝛽𝑆(𝑔𝑥𝑛+1	𝛽𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	143	13	,	,	PUNCT
cana-5953	143	14	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	143	15	,	,	PUNCT
cana-5953	143	16	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	17	)	)	PUNCT
cana-5953	143	18	rewriting	rewriting	NOUN
cana-5953	143	19	;	;	PUNCT
cana-5953	143	20	(	(	PUNCT
cana-5953	143	21	1	1	NUM
cana-5953	143	22	−	−	PROPN
cana-5953	143	23	𝛽)𝑆(𝑔𝑥𝑛+1	𝛽)𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	143	24	,	,	PUNCT
cana-5953	143	25	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	143	26	,	,	PUNCT
cana-5953	143	27	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	28	)	)	PUNCT
cana-5953	143	29	≤	≤	NOUN
cana-5953	143	30	(	(	PUNCT
cana-5953	143	31	𝛼	𝛼	NOUN
cana-5953	143	32	+	+	X
cana-5953	143	33	𝛽)𝑆(𝑔𝑥𝑛	𝛽)𝑆(𝑔𝑥𝑛	ADJ
cana-5953	143	34	,	,	PUNCT
cana-5953	143	35	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	36	,	,	PUNCT
cana-5953	143	37	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	143	38	)	)	PUNCT
cana-5953	143	39	this	this	DET
cana-5953	143	40	yields	yield	NOUN
cana-5953	143	41	;	;	PUNCT
cana-5953	143	42	𝑆(𝑔𝑥𝑛+1	𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	143	43	,	,	PUNCT
cana-5953	143	44	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	143	45	,	,	PUNCT
cana-5953	143	46	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	47	)	)	PUNCT
cana-5953	143	48	≤	≤	NOUN
cana-5953	143	49	𝛼+𝛽	𝛼+𝛽	NUM
cana-5953	143	50	1−𝛽	1−𝛽	NUM
cana-5953	143	51	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	ADJ
cana-5953	143	52	,	,	PUNCT
cana-5953	143	53	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	54	,	,	PUNCT
cana-5953	143	55	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	143	56	)	)	PUNCT
cana-5953	143	57	=	=	SYM
cana-5953	143	58	ℎ𝑆(𝑔𝑥𝑛	ℎ𝑆(𝑔𝑥𝑛	PROPN
cana-5953	143	59	,	,	PUNCT
cana-5953	143	60	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	143	61	,	,	PUNCT
cana-5953	143	62	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	143	63	)	)	PUNCT
cana-5953	143	64	≤	≤	NUM
cana-5953	143	65	ℎ𝑛𝑆(𝑔𝑥1	ℎ𝑛𝑆(𝑔𝑥1	NOUN
cana-5953	143	66	,	,	PUNCT
cana-5953	143	67	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	143	68	,	,	PUNCT
cana-5953	143	69	𝑔𝑥0	𝑔𝑥0	PROPN
cana-5953	143	70	)	)	PUNCT
cana-5953	143	71	,	,	PUNCT
cana-5953	143	72	where	where	SCONJ
cana-5953	143	73	,	,	PUNCT
cana-5953	143	74	ℎ	ℎ	PROPN
cana-5953	143	75	=	=	SYM
cana-5953	143	76	𝛼+𝛽	𝛼+𝛽	NUM
cana-5953	143	77	1−𝛽	1−𝛽	NUM
cana-5953	143	78	<	<	SYM
cana-5953	143	79	1	1	NUM
cana-5953	143	80	by	by	ADP
cana-5953	143	81	assumption	assumption	NOUN
cana-5953	143	82	.	.	PUNCT
cana-5953	144	1	by	by	ADP
cana-5953	144	2	recursion	recursion	NOUN
cana-5953	144	3	:	:	PUNCT
cana-5953	144	4	𝑆(𝑔𝑥𝑛+1	𝑆(𝑔𝑥𝑛+1	PROPN
cana-5953	144	5	,	,	PUNCT
cana-5953	144	6	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	144	7	,	,	PUNCT
cana-5953	144	8	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	144	9	)	)	PUNCT
cana-5953	144	10	≤	≤	NUM
cana-5953	144	11	ℎ𝑛𝑆(𝑔𝑥1	ℎ𝑛𝑆(𝑔𝑥1	NOUN
cana-5953	144	12	,	,	PUNCT
cana-5953	144	13	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	144	14	,	,	PUNCT
cana-5953	144	15	𝑔𝑥0	𝑔𝑥0	PROPN
cana-5953	144	16	)	)	PUNCT
cana-5953	144	17	,	,	PUNCT
cana-5953	144	18	let	let	VERB
cana-5953	144	19	𝑧	𝑧	VERB
cana-5953	144	20	=	=	NOUN
cana-5953	144	21	𝑆(𝑔𝑥1	𝑆(𝑔𝑥1	NOUN
cana-5953	144	22	,	,	PUNCT
cana-5953	144	23	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	144	24	,	,	PUNCT
cana-5953	144	25	𝑔𝑥0	𝑔𝑥0	NOUN
cana-5953	144	26	)	)	PUNCT
cana-5953	144	27	.	.	PUNCT
cana-5953	145	1	then	then	ADV
cana-5953	145	2	,	,	PUNCT
cana-5953	145	3	for	for	ADP
cana-5953	145	4	𝑚	𝑚	X
cana-5953	145	5	<	<	X
cana-5953	145	6	𝑛	𝑛	PROPN
cana-5953	145	7	,	,	PUNCT
cana-5953	145	8	we	we	PRON
cana-5953	145	9	have	have	VERB
cana-5953	145	10	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	ADJ
cana-5953	145	11	,	,	PUNCT
cana-5953	145	12	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	145	13	,	,	PUNCT
cana-5953	145	14	𝑔𝑥𝑚	𝑔𝑥𝑚	NOUN
cana-5953	145	15	)	)	PUNCT
cana-5953	145	16	≤	≤	NOUN
cana-5953	145	17	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	NOUN
cana-5953	145	18	,	,	PUNCT
cana-5953	145	19	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	145	20	,	,	PUNCT
cana-5953	145	21	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	145	22	)	)	PUNCT
cana-5953	145	23	+	+	X
cana-5953	145	24	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	ADJ
cana-5953	145	25	,	,	PUNCT
cana-5953	145	26	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	145	27	,	,	PUNCT
cana-5953	145	28	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	145	29	)	)	PUNCT
cana-5953	146	1	+	+	CCONJ
cana-5953	146	2	𝑆(𝑔𝑥𝑚	𝑆(𝑔𝑥𝑚	NUM
cana-5953	146	3	,	,	PUNCT
cana-5953	146	4	𝑔𝑥𝑚	𝑔𝑥𝑚	NOUN
cana-5953	146	5	,	,	PUNCT
cana-5953	146	6	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	146	7	)	)	PUNCT
cana-5953	146	8	≤	≤	NOUN
cana-5953	146	9	ℎ𝑛−1𝑆(𝑔𝑥1	ℎ𝑛−1𝑆(𝑔𝑥1	PROPN
cana-5953	146	10	,	,	PUNCT
cana-5953	146	11	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	146	12	,	,	PUNCT
cana-5953	146	13	𝑔𝑥0	𝑔𝑥0	NUM
cana-5953	146	14	)	)	PUNCT
cana-5953	146	15	+	+	CCONJ
cana-5953	146	16	ℎ𝑛−1𝑆(𝑔𝑥1	ℎ𝑛−1𝑆(𝑔𝑥1	PROPN
cana-5953	146	17	,	,	PUNCT
cana-5953	146	18	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	146	19	,	,	PUNCT
cana-5953	146	20	𝑔𝑥0	𝑔𝑥0	NUM
cana-5953	146	21	)	)	PUNCT
cana-5953	146	22	+	+	CCONJ
cana-5953	146	23	𝑆(𝑔𝑥𝑚	𝑆(𝑔𝑥𝑚	NUM
cana-5953	146	24	,	,	PUNCT
cana-5953	146	25	𝑔𝑥𝑚	𝑔𝑥𝑚	NOUN
cana-5953	146	26	,	,	PUNCT
cana-5953	146	27	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	146	28	)	)	PUNCT
cana-5953	146	29	=	=	PUNCT
cana-5953	146	30	2ℎ𝑛−1𝑆(𝑔𝑥1	2ℎ𝑛−1𝑆(𝑔𝑥1	NUM
cana-5953	146	31	,	,	PUNCT
cana-5953	146	32	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	146	33	,	,	PUNCT
cana-5953	146	34	𝑔𝑥0	𝑔𝑥0	NUM
cana-5953	146	35	)	)	PUNCT
cana-5953	146	36	+	+	CCONJ
cana-5953	146	37	𝑆(𝑔𝑥𝑚	𝑆(𝑔𝑥𝑚	NUM
cana-5953	146	38	,	,	PUNCT
cana-5953	146	39	𝑔𝑥𝑚	𝑔𝑥𝑚	NOUN
cana-5953	146	40	,	,	PUNCT
cana-5953	146	41	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	146	42	)	)	PUNCT
cana-5953	146	43	≤	≤	NOUN
cana-5953	147	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	148	1	+	+	CCONJ
cana-5953	148	2	[	[	X
cana-5953	148	3	𝑆(𝑔𝑥𝑚	𝑆(𝑔𝑥𝑚	ADJ
cana-5953	148	4	,	,	PUNCT
cana-5953	148	5	𝑔𝑥𝑚	𝑔𝑥𝑚	NOUN
cana-5953	148	6	,	,	PUNCT
cana-5953	148	7	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	148	8	)	)	PUNCT
cana-5953	149	1	+	+	CCONJ
cana-5953	149	2	𝑆(𝑔𝑥𝑚	𝑆(𝑔𝑥𝑚	ADJ
cana-5953	149	3	,	,	PUNCT
cana-5953	149	4	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	149	5	,	,	PUNCT
cana-5953	149	6	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	149	7	)	)	PUNCT
cana-5953	150	1	+	+	CCONJ
cana-5953	150	2	𝑆(𝑔𝑥𝑛−1	𝑆(𝑔𝑥𝑛−1	PROPN
cana-5953	150	3	,	,	PUNCT
cana-5953	150	4	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	150	5	,	,	PUNCT
cana-5953	150	6	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	150	7	)	)	PUNCT
cana-5953	150	8	]	]	PUNCT
cana-5953	151	1	=	=	PUNCT
cana-5953	152	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	152	2	+	+	CCONJ
cana-5953	153	1	[	[	X
cana-5953	153	2	2𝑆(𝑔𝑥𝑚	2𝑆(𝑔𝑥𝑚	NUM
cana-5953	153	3	,	,	PUNCT
cana-5953	153	4	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	153	5	,	,	PUNCT
cana-5953	153	6	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	153	7	)	)	PUNCT
cana-5953	154	1	+	+	CCONJ
cana-5953	154	2	𝑆(𝑔𝑥𝑛−1	𝑆(𝑔𝑥𝑛−1	PROPN
cana-5953	154	3	,	,	PUNCT
cana-5953	154	4	𝑔𝑥𝑛−1	𝑔𝑥𝑛−1	PROPN
cana-5953	154	5	,	,	PUNCT
cana-5953	154	6	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	154	7	)	)	PUNCT
cana-5953	154	8	]	]	PUNCT
cana-5953	154	9	≤	≤	NOUN
cana-5953	155	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	155	2	+	+	CCONJ
cana-5953	156	1	[	[	X
cana-5953	156	2	2𝑆(𝑔𝑥𝑚	2𝑆(𝑔𝑥𝑚	NUM
cana-5953	156	3	,	,	PUNCT
cana-5953	156	4	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	156	5	,	,	PUNCT
cana-5953	156	6	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	156	7	)	)	PUNCT
cana-5953	156	8	+	+	CCONJ
cana-5953	156	9	ℎ𝑛−2𝑆(𝑔𝑥1	ℎ𝑛−2𝑆(𝑔𝑥1	PROPN
cana-5953	156	10	,	,	PUNCT
cana-5953	156	11	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	156	12	,	,	PUNCT
cana-5953	156	13	𝑔𝑥0	𝑔𝑥0	NOUN
cana-5953	156	14	)	)	PUNCT
cana-5953	156	15	]	]	PUNCT
cana-5953	157	1	=	=	PUNCT
cana-5953	158	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	158	2	+	+	CCONJ
cana-5953	158	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	X
cana-5953	158	4	+	+	CCONJ
cana-5953	158	5	2𝑆(𝑔𝑥𝑚	2𝑆(𝑔𝑥𝑚	NUM
cana-5953	158	6	,	,	PUNCT
cana-5953	158	7	𝑔𝑥𝑚	𝑔𝑥𝑚	NOUN
cana-5953	158	8	,	,	PUNCT
cana-5953	158	9	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	158	10	)	)	PUNCT
cana-5953	158	11	≤	≤	NOUN
cana-5953	159	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	159	2	+	+	CCONJ
cana-5953	159	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	NUM
cana-5953	159	4	+	+	CCONJ
cana-5953	159	5	2[2𝑆(𝑔𝑥𝑚	2[2𝑆(𝑔𝑥𝑚	NUM
cana-5953	159	6	,	,	PUNCT
cana-5953	159	7	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	159	8	,	,	PUNCT
cana-5953	159	9	𝑔𝑥𝑛−3	𝑔𝑥𝑛−3	PROPN
cana-5953	159	10	)	)	PUNCT
cana-5953	159	11	+	+	CCONJ
cana-5953	159	12	𝑆(𝑔𝑥𝑛−2	𝑆(𝑔𝑥𝑛−2	PROPN
cana-5953	159	13	,	,	PUNCT
cana-5953	159	14	𝑔𝑥𝑛−2	𝑔𝑥𝑛−2	PROPN
cana-5953	159	15	,	,	PUNCT
cana-5953	159	16	𝑔𝑥𝑛−3	𝑔𝑥𝑛−3	PROPN
cana-5953	159	17	)	)	PUNCT
cana-5953	159	18	]	]	PUNCT
cana-5953	160	1	≤	≤	NOUN
cana-5953	161	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	161	2	+	+	CCONJ
cana-5953	161	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	X
cana-5953	161	4	+	+	CCONJ
cana-5953	161	5	4𝑆(𝑔𝑥𝑚	4𝑆(𝑔𝑥𝑚	NUM
cana-5953	161	6	,	,	PUNCT
cana-5953	161	7	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	161	8	,	,	PUNCT
cana-5953	161	9	𝑔𝑥𝑛−3	𝑔𝑥𝑛−3	PROPN
cana-5953	161	10	)	)	PUNCT
cana-5953	161	11	+	+	CCONJ
cana-5953	161	12	2ℎ𝑛−3𝑧	2ℎ𝑛−3𝑧	X
cana-5953	161	13	≤	≤	NOUN
cana-5953	162	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	162	2	+	+	CCONJ
cana-5953	162	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	X
cana-5953	162	4	+	+	CCONJ
cana-5953	162	5	2ℎ𝑛−3𝑧	2ℎ𝑛−3𝑧	ADJ
cana-5953	162	6	+	+	NUM
cana-5953	162	7	4[2𝑆(𝑔𝑥𝑚	4[2𝑆(𝑔𝑥𝑚	NUM
cana-5953	162	8	,	,	PUNCT
cana-5953	162	9	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	162	10	,	,	PUNCT
cana-5953	162	11	𝑔𝑥𝑛−4	𝑔𝑥𝑛−4	PROPN
cana-5953	162	12	)	)	PUNCT
cana-5953	163	1	+	+	PUNCT
cana-5953	163	2	𝑆(𝑔𝑥𝑛−3	𝑆(𝑔𝑥𝑛−3	PROPN
cana-5953	163	3	,	,	PUNCT
cana-5953	163	4	𝑔𝑥𝑛−3	𝑔𝑥𝑛−3	PROPN
cana-5953	163	5	,	,	PUNCT
cana-5953	163	6	𝑔𝑥𝑛−4	𝑔𝑥𝑛−4	PROPN
cana-5953	163	7	)	)	PUNCT
cana-5953	163	8	]	]	PUNCT
cana-5953	163	9	≤	≤	NOUN
cana-5953	164	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	164	2	+	+	CCONJ
cana-5953	164	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	X
cana-5953	164	4	+	+	CCONJ
cana-5953	164	5	2ℎ𝑛−3𝑧	2ℎ𝑛−3𝑧	PROPN
cana-5953	164	6	+	+	NUM
cana-5953	164	7	8𝑆(𝑔𝑥𝑚	8𝑆(𝑔𝑥𝑚	NUM
cana-5953	164	8	,	,	PUNCT
cana-5953	164	9	𝑔𝑥𝑚	𝑔𝑥𝑚	PROPN
cana-5953	164	10	,	,	PUNCT
cana-5953	164	11	𝑔𝑥𝑛−4	𝑔𝑥𝑛−4	PROPN
cana-5953	164	12	)	)	PUNCT
cana-5953	165	1	+	+	CCONJ
cana-5953	166	1	4ℎ𝑛−4𝑧	4ℎ𝑛−4𝑧	ADJ
cana-5953	166	2	=	=	PUNCT
cana-5953	167	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	167	2	+	+	CCONJ
cana-5953	167	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	X
cana-5953	167	4	+	+	CCONJ
cana-5953	167	5	2ℎ𝑛−3𝑧	2ℎ𝑛−3𝑧	PROPN
cana-5953	168	1	+	+	CCONJ
cana-5953	168	2	4ℎ𝑛−4𝑧	4ℎ𝑛−4𝑧	NOUN
cana-5953	169	1	+	+	PUNCT
cana-5953	169	2	⋯	⋯	VERB
cana-5953	169	3	=	=	SYM
cana-5953	169	4	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	169	5	+	+	CCONJ
cana-5953	169	6	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	NUM
cana-5953	169	7	(	(	PUNCT
cana-5953	169	8	1	1	NUM
cana-5953	169	9	+	+	SYM
cana-5953	169	10	2	2	NUM
cana-5953	169	11	ℎ	ℎ	NOUN
cana-5953	169	12	+	+	CCONJ
cana-5953	169	13	4	4	NUM
cana-5953	169	14	ℎ2	ℎ2	NOUN
cana-5953	169	15	+	+	CCONJ
cana-5953	169	16	8	8	NUM
cana-5953	169	17	ℎ3	ℎ3	NOUN
cana-5953	169	18	+	+	SYM
cana-5953	169	19	⋯	⋯	NOUN
cana-5953	169	20	)	)	PUNCT
cana-5953	169	21	communications	communication	NOUN
cana-5953	169	22	on	on	ADP
cana-5953	169	23	applied	apply	VERB
cana-5953	169	24	nonlinear	nonlinear	ADJ
cana-5953	169	25	analysis	analysis	NOUN
cana-5953	169	26	issn	issn	NOUN
cana-5953	169	27	:	:	PUNCT
cana-5953	169	28	1074	1074	NUM
cana-5953	169	29	-	-	PUNCT
cana-5953	169	30	133x	133x	NUM
cana-5953	169	31	vol	vol	VERB
cana-5953	169	32	32	32	NUM
cana-5953	169	33	no	no	NOUN
cana-5953	169	34	.	.	PUNCT
cana-5953	170	1	10s	10	NOUN
cana-5953	170	2	(	(	PUNCT
cana-5953	170	3	2025	2025	NUM
cana-5953	170	4	)	)	PUNCT
cana-5953	170	5	3167	3167	NUM
cana-5953	170	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	171	1	=	=	PUNCT
cana-5953	172	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	172	2	+	+	CCONJ
cana-5953	172	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	NUM
cana-5953	172	4	(	(	PUNCT
cana-5953	172	5	1	1	NUM
cana-5953	172	6	+	+	CCONJ
cana-5953	172	7	(	(	PUNCT
cana-5953	172	8	2	2	NUM
cana-5953	172	9	ℎ	ℎ	NOUN
cana-5953	172	10	)	)	PUNCT
cana-5953	173	1	+	+	CCONJ
cana-5953	173	2	(	(	PUNCT
cana-5953	173	3	2	2	NUM
cana-5953	173	4	ℎ	ℎ	NOUN
cana-5953	173	5	)	)	PUNCT
cana-5953	173	6	2	2	NUM
cana-5953	174	1	+	+	CCONJ
cana-5953	174	2	(	(	PUNCT
cana-5953	174	3	2	2	NUM
cana-5953	174	4	ℎ	ℎ	NOUN
cana-5953	174	5	)	)	PUNCT
cana-5953	174	6	3	3	NUM
cana-5953	175	1	+	+	CCONJ
cana-5953	175	2	⋯	⋯	NOUN
cana-5953	175	3	)	)	PUNCT
cana-5953	176	1	=	=	PUNCT
cana-5953	177	1	2ℎ𝑛−1𝑧	2ℎ𝑛−1𝑧	ADJ
cana-5953	177	2	+	+	CCONJ
cana-5953	177	3	ℎ𝑛−2𝑧	ℎ𝑛−2𝑧	NUM
cana-5953	177	4	(	(	PUNCT
cana-5953	177	5	1	1	NUM
cana-5953	177	6	−	−	NUM
cana-5953	177	7	2	2	NUM
cana-5953	177	8	ℎ	ℎ	PROPN
cana-5953	177	9	)	)	PUNCT
cana-5953	177	10	−1	−1	NOUN
cana-5953	177	11	,	,	PUNCT
cana-5953	177	12	since	since	SCONJ
cana-5953	177	13	0	0	NUM
cana-5953	177	14	<	<	X
cana-5953	177	15	ℎ	ℎ	X
cana-5953	177	16	<	<	X
cana-5953	177	17	1	1	NUM
cana-5953	177	18	,	,	PUNCT
cana-5953	177	19	by	by	ADP
cana-5953	177	20	taking	take	VERB
cana-5953	177	21	limit	limit	NOUN
cana-5953	177	22	as	as	ADP
cana-5953	177	23	𝑛	𝑛	PROPN
cana-5953	177	24	→	→	SYM
cana-5953	177	25	∞	∞	PROPN
cana-5953	177	26	,	,	PUNCT
cana-5953	177	27	we	we	PRON
cana-5953	177	28	obtain	obtain	VERB
cana-5953	177	29	,	,	PUNCT
cana-5953	177	30	lim	lim	PROPN
cana-5953	177	31	𝑛→∞	𝑛→∞	NUM
cana-5953	177	32	𝑆(𝑔𝑥𝑛	𝑆(𝑔𝑥𝑛	PROPN
cana-5953	177	33	,	,	PUNCT
cana-5953	177	34	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	177	35	,	,	PUNCT
cana-5953	177	36	𝑔𝑥𝑚	𝑔𝑥𝑚	NUM
cana-5953	177	37	)	)	PUNCT
cana-5953	177	38	=	=	SYM
cana-5953	178	1	0	0	X
cana-5953	178	2	.	.	PUNCT
cana-5953	179	1	hence	hence	ADV
cana-5953	179	2	,	,	PUNCT
cana-5953	179	3	we	we	PRON
cana-5953	179	4	obtain	obtain	VERB
cana-5953	179	5	that	that	SCONJ
cana-5953	179	6	the	the	DET
cana-5953	179	7	sequence	sequence	NOUN
cana-5953	179	8	{	{	PUNCT
cana-5953	179	9	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	179	10	}	}	PUNCT
cana-5953	179	11	is	be	AUX
cana-5953	179	12	a	a	DET
cana-5953	179	13	cauchy	cauchy	ADJ
cana-5953	179	14	sequence	sequence	NOUN
cana-5953	179	15	in	in	ADP
cana-5953	179	16	𝑔(𝑋	𝑔(𝑋	PROPN
cana-5953	179	17	)	)	PUNCT
cana-5953	179	18	.	.	PUNCT
cana-5953	180	1	given	give	VERB
cana-5953	180	2	that	that	PRON
cana-5953	180	3	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	180	4	)	)	PUNCT
cana-5953	180	5	is	be	AUX
cana-5953	180	6	complete	complete	ADJ
cana-5953	180	7	,	,	PUNCT
cana-5953	180	8	there	there	PRON
cana-5953	180	9	exist	exist	VERB
cana-5953	180	10	a	a	DET
cana-5953	180	11	point	point	NOUN
cana-5953	180	12	𝑞	𝑞	X
cana-5953	180	13	in	in	ADP
cana-5953	180	14	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	180	15	)	)	PUNCT
cana-5953	180	16	such	such	ADJ
cana-5953	180	17	that	that	DET
cana-5953	180	18	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	180	19	→	→	SYM
cana-5953	180	20	𝑞	𝑞	PROPN
cana-5953	180	21	,	,	PUNCT
cana-5953	180	22	as	as	ADP
cana-5953	180	23	𝑛	𝑛	PROPN
cana-5953	180	24	→	→	SYM
cana-5953	180	25	∞.	∞.	PROPN
cana-5953	180	26	since	since	ADV
cana-5953	180	27	,	,	PUNCT
cana-5953	180	28	𝑞	𝑞	PROPN
cana-5953	180	29	∈	∈	PROPN
cana-5953	180	30	𝑔(𝑋	𝑔(𝑋	PROPN
cana-5953	180	31	)	)	PUNCT
cana-5953	180	32	there	there	PRON
cana-5953	180	33	exists	exist	VERB
cana-5953	180	34	a	a	DET
cana-5953	180	35	point	point	NOUN
cana-5953	180	36	𝑝	𝑝	NOUN
cana-5953	180	37	in	in	ADP
cana-5953	180	38	𝑋	𝑋	NOUN
cana-5953	180	39	with	with	ADP
cana-5953	180	40	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	180	41	=	=	PUNCT
cana-5953	180	42	𝑞.	𝑞.	NOUN
cana-5953	180	43	now	now	ADV
cana-5953	180	44	,	,	PUNCT
cana-5953	180	45	𝑆(𝑓𝑥𝑛	𝑆(𝑓𝑥𝑛	ADJ
cana-5953	180	46	,	,	PUNCT
cana-5953	180	47	𝑓𝑥𝑛	𝑓𝑥𝑛	ADJ
cana-5953	180	48	,	,	PUNCT
cana-5953	180	49	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	180	50	)	)	PUNCT
cana-5953	180	51	≤	≤	NOUN
cana-5953	180	52	𝛼𝑆(𝑔𝑥𝑛	𝛼𝑆(𝑔𝑥𝑛	NOUN
cana-5953	180	53	,	,	PUNCT
cana-5953	180	54	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	180	55	,	,	PUNCT
cana-5953	180	56	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	180	57	)	)	PUNCT
cana-5953	180	58	+	+	CCONJ
cana-5953	180	59	𝛽[𝑆(𝑔𝑥𝑛	𝛽[𝑆(𝑔𝑥𝑛	ADJ
cana-5953	180	60	,	,	PUNCT
cana-5953	180	61	𝑔𝑥𝑛	𝑔𝑥𝑛	INTJ
cana-5953	180	62	,	,	PUNCT
cana-5953	180	63	𝑓𝑥𝑛	𝑓𝑥𝑛	NOUN
cana-5953	180	64	)	)	PUNCT
cana-5953	180	65	+	+	CCONJ
cana-5953	180	66	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	180	67	,	,	PUNCT
cana-5953	180	68	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	180	69	,	,	PUNCT
cana-5953	180	70	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	180	71	)	)	PUNCT
cana-5953	180	72	]	]	PUNCT
cana-5953	181	1	=	=	SYM
cana-5953	181	2	𝛼𝑆(𝑔𝑥𝑛	𝛼𝑆(𝑔𝑥𝑛	X
cana-5953	181	3	,	,	PUNCT
cana-5953	181	4	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	181	5	,	,	PUNCT
cana-5953	181	6	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	181	7	)	)	PUNCT
cana-5953	181	8	+	+	CCONJ
cana-5953	181	9	𝛽[𝑆(𝑔𝑥𝑛	𝛽[𝑆(𝑔𝑥𝑛	PROPN
cana-5953	181	10	,	,	PUNCT
cana-5953	181	11	𝑔𝑥𝑛	𝑔𝑥𝑛	NOUN
cana-5953	181	12	,	,	PUNCT
cana-5953	181	13	𝑔𝑥𝑛+1	𝑔𝑥𝑛+1	PROPN
cana-5953	181	14	)	)	PUNCT
cana-5953	181	15	+	+	CCONJ
cana-5953	181	16	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	181	17	,	,	PUNCT
cana-5953	181	18	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	181	19	,	,	PUNCT
cana-5953	181	20	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	181	21	)	)	PUNCT
cana-5953	181	22	]	]	PUNCT
cana-5953	181	23	.	.	PUNCT
cana-5953	182	1	taking	take	VERB
cana-5953	182	2	the	the	DET
cana-5953	182	3	limit	limit	NOUN
cana-5953	182	4	as	as	ADP
cana-5953	182	5	𝑛	𝑛	PROPN
cana-5953	182	6	→	→	SYM
cana-5953	182	7	∞	∞	NUM
cana-5953	182	8	and	and	CCONJ
cana-5953	182	9	using	use	VERB
cana-5953	182	10	continuity	continuity	NOUN
cana-5953	182	11	and	and	CCONJ
cana-5953	182	12	convergence	convergence	NOUN
cana-5953	182	13	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	182	14	,	,	PUNCT
cana-5953	182	15	𝑞	𝑞	NOUN
cana-5953	182	16	,	,	PUNCT
cana-5953	182	17	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	182	18	)	)	PUNCT
cana-5953	182	19	≤	≤	NOUN
cana-5953	183	1	𝛼𝑆(𝑞	𝛼𝑆(𝑞	PROPN
cana-5953	183	2	,	,	PUNCT
cana-5953	183	3	𝑞	𝑞	PROPN
cana-5953	183	4	,	,	PUNCT
cana-5953	183	5	𝑞	𝑞	PROPN
cana-5953	183	6	)	)	PUNCT
cana-5953	183	7	+	+	NUM
cana-5953	183	8	𝛽[𝑆(𝑞	𝛽[𝑆(𝑞	NOUN
cana-5953	183	9	,	,	PUNCT
cana-5953	183	10	𝑞	𝑞	X
cana-5953	183	11	,	,	PUNCT
cana-5953	183	12	𝑞	𝑞	PROPN
cana-5953	183	13	)	)	PUNCT
cana-5953	183	14	+	+	CCONJ
cana-5953	183	15	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	183	16	,	,	PUNCT
cana-5953	183	17	𝑞	𝑞	X
cana-5953	183	18	,	,	PUNCT
cana-5953	183	19	𝑓𝑝	𝑓𝑝	PROPN
cana-5953	183	20	)	)	PUNCT
cana-5953	183	21	]	]	PUNCT
cana-5953	183	22	which	which	PRON
cana-5953	183	23	implies	imply	VERB
cana-5953	183	24	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	183	25	,	,	PUNCT
cana-5953	183	26	𝑞	𝑞	X
cana-5953	183	27	,	,	PUNCT
cana-5953	183	28	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	183	29	)	)	PUNCT
cana-5953	183	30	≤	≤	NOUN
cana-5953	184	1	𝛽𝑆(𝑞	𝛽𝑆(𝑞	PROPN
cana-5953	184	2	,	,	PUNCT
cana-5953	184	3	𝑞	𝑞	X
cana-5953	184	4	,	,	PUNCT
cana-5953	184	5	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	184	6	)	)	PUNCT
cana-5953	184	7	.	.	PUNCT
cana-5953	185	1	since	since	SCONJ
cana-5953	185	2	,	,	PUNCT
cana-5953	185	3	0	0	NUM
cana-5953	185	4	<	<	X
cana-5953	185	5	𝛽	𝛽	X
cana-5953	185	6	<	<	X
cana-5953	185	7	1	1	NUM
cana-5953	185	8	,	,	PUNCT
cana-5953	185	9	the	the	DET
cana-5953	185	10	only	only	ADJ
cana-5953	185	11	solution	solution	NOUN
cana-5953	185	12	is	be	AUX
cana-5953	185	13	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	185	14	,	,	PUNCT
cana-5953	185	15	𝑞	𝑞	NOUN
cana-5953	185	16	,	,	PUNCT
cana-5953	185	17	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	185	18	)	)	PUNCT
cana-5953	185	19	=	=	SYM
cana-5953	185	20	0	0	NUM
cana-5953	185	21	,	,	PUNCT
cana-5953	185	22	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	185	23	=	=	SYM
cana-5953	185	24	𝑞.	𝑞.	NOUN
cana-5953	185	25	but	but	CCONJ
cana-5953	185	26	also	also	ADV
cana-5953	185	27	,	,	PUNCT
cana-5953	185	28	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	185	29	=	=	SYM
cana-5953	185	30	𝑞	𝑞	X
cana-5953	185	31	so	so	ADV
cana-5953	185	32	𝑓𝑝	𝑓𝑝	ADV
cana-5953	185	33	=	=	PUNCT
cana-5953	185	34	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	185	35	=	=	SYM
cana-5953	185	36	𝑞	𝑞	X
cana-5953	185	37	showing	show	VERB
cana-5953	185	38	that	that	SCONJ
cana-5953	185	39	,	,	PUNCT
cana-5953	185	40	𝑞	𝑞	X
cana-5953	185	41	is	be	AUX
cana-5953	185	42	the	the	DET
cana-5953	185	43	point	point	NOUN
cana-5953	185	44	of	of	ADP
cana-5953	185	45	coincidence	coincidence	NOUN
cana-5953	185	46	of	of	ADP
cana-5953	185	47	𝑓	𝑓	PRON
cana-5953	185	48	and	and	CCONJ
cana-5953	185	49	𝑔.	𝑔.	NOUN
cana-5953	185	50	to	to	PART
cana-5953	185	51	prove	prove	VERB
cana-5953	185	52	uniqueness	uniqueness	NOUN
cana-5953	185	53	,	,	PUNCT
cana-5953	185	54	suppose	suppose	VERB
cana-5953	185	55	𝑤	𝑤	SCONJ
cana-5953	185	56	∈	∈	PROPN
cana-5953	185	57	𝑋	𝑋	NOUN
cana-5953	185	58	is	be	AUX
cana-5953	185	59	another	another	DET
cana-5953	185	60	coincidence	coincidence	NOUN
cana-5953	185	61	point	point	NOUN
cana-5953	185	62	,	,	PUNCT
cana-5953	185	63	i.e.	i.e.	X
cana-5953	185	64	,	,	PUNCT
cana-5953	185	65	𝑓𝑤	𝑓𝑤	ADP
cana-5953	185	66	=	=	ADJ
cana-5953	185	67	𝑔𝑤	𝑔𝑤	NOUN
cana-5953	185	68	=	=	NOUN
cana-5953	185	69	𝑤.	𝑤.	NOUN
cana-5953	185	70	then	then	ADV
cana-5953	185	71	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	185	72	,	,	PUNCT
cana-5953	185	73	𝑞	𝑞	X
cana-5953	185	74	,	,	PUNCT
cana-5953	185	75	𝑤	𝑤	ADP
cana-5953	185	76	)	)	PUNCT
cana-5953	186	1	=	=	SYM
cana-5953	186	2	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	186	3	,	,	PUNCT
cana-5953	186	4	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	186	5	,	,	PUNCT
cana-5953	186	6	𝑓𝑤	𝑓𝑤	NOUN
cana-5953	186	7	)	)	PUNCT
cana-5953	186	8	≤	≤	NUM
cana-5953	186	9	𝛼𝑆(𝑔𝑝	𝛼𝑆(𝑔𝑝	PROPN
cana-5953	186	10	,	,	PUNCT
cana-5953	186	11	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	186	12	,	,	PUNCT
cana-5953	186	13	𝑔𝑤	𝑔𝑤	PROPN
cana-5953	186	14	)	)	PUNCT
cana-5953	186	15	+	+	CCONJ
cana-5953	186	16	𝛽[𝑆(𝑔𝑝	𝛽[𝑆(𝑔𝑝	PROPN
cana-5953	186	17	,	,	PUNCT
cana-5953	186	18	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	186	19	,	,	PUNCT
cana-5953	186	20	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	186	21	)	)	PUNCT
cana-5953	187	1	+	+	NUM
cana-5953	187	2	𝑆(𝑔𝑤	𝑆(𝑔𝑤	NOUN
cana-5953	187	3	,	,	PUNCT
cana-5953	187	4	𝑔𝑤	𝑔𝑤	INTJ
cana-5953	187	5	,	,	PUNCT
cana-5953	187	6	𝑓𝑤	𝑓𝑤	NOUN
cana-5953	187	7	)	)	PUNCT
cana-5953	187	8	]	]	PUNCT
cana-5953	188	1	=	=	PUNCT
cana-5953	188	2	𝛼𝑆(𝑞	𝛼𝑆(𝑞	PROPN
cana-5953	188	3	,	,	PUNCT
cana-5953	188	4	𝑞	𝑞	PROPN
cana-5953	188	5	,	,	PUNCT
cana-5953	188	6	𝑤	𝑤	ADP
cana-5953	188	7	)	)	PUNCT
cana-5953	188	8	+	+	CCONJ
cana-5953	188	9	𝛽[𝑆(𝑞	𝛽[𝑆(𝑞	NOUN
cana-5953	188	10	,	,	PUNCT
cana-5953	188	11	𝑞	𝑞	X
cana-5953	188	12	,	,	PUNCT
cana-5953	188	13	𝑞	𝑞	PROPN
cana-5953	188	14	)	)	PUNCT
cana-5953	188	15	+	+	CCONJ
cana-5953	188	16	𝑆(𝑤	𝑆(𝑤	PROPN
cana-5953	188	17	,	,	PUNCT
cana-5953	188	18	𝑤	𝑤	ADP
cana-5953	188	19	,	,	PUNCT
cana-5953	188	20	𝑤	𝑤	ADP
cana-5953	188	21	)	)	PUNCT
cana-5953	188	22	]	]	PUNCT
cana-5953	189	1	=	=	PUNCT
cana-5953	189	2	𝛼𝑆(𝑞	𝛼𝑆(𝑞	PROPN
cana-5953	189	3	,	,	PUNCT
cana-5953	189	4	𝑞	𝑞	PROPN
cana-5953	189	5	,	,	PUNCT
cana-5953	189	6	𝑤	𝑤	ADP
cana-5953	189	7	)	)	PUNCT
cana-5953	189	8	.	.	PUNCT
cana-5953	190	1	since	since	SCONJ
cana-5953	190	2	,	,	PUNCT
cana-5953	190	3	0	0	NUM
cana-5953	190	4	<	<	X
cana-5953	190	5	𝛼	𝛼	X
cana-5953	190	6	<	<	X
cana-5953	190	7	1	1	NUM
cana-5953	190	8	,	,	PUNCT
cana-5953	190	9	we	we	PRON
cana-5953	190	10	can	can	AUX
cana-5953	190	11	conclude	conclude	VERB
cana-5953	190	12	that	that	PRON
cana-5953	190	13	𝑆(𝑞	𝑆(𝑞	PROPN
cana-5953	190	14	,	,	PUNCT
cana-5953	190	15	𝑞	𝑞	X
cana-5953	190	16	,	,	PUNCT
cana-5953	190	17	𝑤	𝑤	ADP
cana-5953	190	18	)	)	PUNCT
cana-5953	190	19	=	=	NOUN
cana-5953	191	1	0𝑞	0𝑞	NOUN
cana-5953	191	2	=	=	SYM
cana-5953	191	3	𝑤.	𝑤.	NOUN
cana-5953	191	4	hence	hence	ADV
cana-5953	191	5	,	,	PUNCT
cana-5953	191	6	the	the	DET
cana-5953	191	7	coincidence	coincidence	NOUN
cana-5953	191	8	point	point	NOUN
cana-5953	191	9	is	be	AUX
cana-5953	191	10	unique	unique	ADJ
cana-5953	191	11	.	.	PUNCT
cana-5953	192	1	finally	finally	ADV
cana-5953	192	2	since	since	ADV
cana-5953	192	3	,	,	PUNCT
cana-5953	192	4	𝑓	𝑓	PRON
cana-5953	192	5	and	and	CCONJ
cana-5953	192	6	𝑔	𝑔	PROPN
cana-5953	192	7	are	be	AUX
cana-5953	192	8	weakly	weakly	ADV
cana-5953	192	9	compatible	compatible	ADJ
cana-5953	192	10	,	,	PUNCT
cana-5953	192	11	by	by	ADP
cana-5953	192	12	proposition	proposition	NOUN
cana-5953	192	13	(	(	PUNCT
cana-5953	192	14	3.6	3.6	NUM
cana-5953	192	15	)	)	PUNCT
cana-5953	192	16	,	,	PUNCT
cana-5953	192	17	they	they	PRON
cana-5953	192	18	have	have	VERB
cana-5953	192	19	a	a	DET
cana-5953	192	20	unique	unique	ADJ
cana-5953	192	21	common	common	ADJ
cana-5953	192	22	fixed	fix	VERB
cana-5953	192	23	point	point	NOUN
cana-5953	192	24	in	in	ADP
cana-5953	192	25	𝑋.	𝑋.	PROPN
cana-5953	192	26	theorem	theorem	VERB
cana-5953	192	27	4.3	4.3	NUM
cana-5953	192	28	let	let	VERB
cana-5953	192	29	(	(	PUNCT
cana-5953	192	30	𝑋	𝑋	PROPN
cana-5953	192	31	,	,	PUNCT
cana-5953	192	32	𝑆	𝑆	PROPN
cana-5953	192	33	)	)	PUNCT
cana-5953	192	34	be	be	AUX
cana-5953	192	35	a	a	DET
cana-5953	192	36	complete	complete	ADJ
cana-5953	192	37	rectangular	rectangular	ADJ
cana-5953	192	38	s	s	ADJ
cana-5953	192	39	-	-	ADJ
cana-5953	192	40	metric	metric	ADJ
cana-5953	192	41	space	space	NOUN
cana-5953	192	42	,	,	PUNCT
cana-5953	192	43	and	and	CCONJ
cana-5953	192	44	let	let	VERB
cana-5953	192	45	𝑓	𝑓	PRON
cana-5953	192	46	,	,	PUNCT
cana-5953	192	47	𝑔	𝑔	ADJ
cana-5953	192	48	:	:	PUNCT
cana-5953	192	49	𝑋	𝑋	PROPN
cana-5953	192	50	→	→	SYM
cana-5953	192	51	𝑋	𝑋	PROPN
cana-5953	192	52	be	be	VERB
cana-5953	192	53	two	two	NUM
cana-5953	192	54	selfmaps	selfmap	NOUN
cana-5953	192	55	satisfying	satisfy	VERB
cana-5953	192	56	the	the	DET
cana-5953	192	57	inequality	inequality	NOUN
cana-5953	192	58	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	192	59	,	,	PUNCT
cana-5953	192	60	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	192	61	,	,	PUNCT
cana-5953	192	62	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	192	63	)	)	PUNCT
cana-5953	192	64	≤	≤	NOUN
cana-5953	193	1	𝛼𝑆(𝑥	𝛼𝑆(𝑥	NUM
cana-5953	193	2	,	,	PUNCT
cana-5953	193	3	𝑥	𝑥	PRON
cana-5953	193	4	,	,	PUNCT
cana-5953	193	5	𝑦	𝑦	NOUN
cana-5953	193	6	)	)	PUNCT
cana-5953	193	7	+	+	CCONJ
cana-5953	193	8	𝛽[𝑆(𝑥	𝛽[𝑆(𝑥	PROPN
cana-5953	193	9	,	,	PUNCT
cana-5953	193	10	𝑥	𝑥	NOUN
cana-5953	193	11	,	,	PUNCT
cana-5953	193	12	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	193	13	)	)	PUNCT
cana-5953	193	14	+	+	CCONJ
cana-5953	193	15	𝑆(𝑦	𝑆(𝑦	PROPN
cana-5953	193	16	,	,	PUNCT
cana-5953	193	17	𝑦	𝑦	NOUN
cana-5953	193	18	,	,	PUNCT
cana-5953	193	19	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	193	20	)	)	PUNCT
cana-5953	193	21	]	]	PUNCT
cana-5953	194	1	+	+	CCONJ
cana-5953	194	2	𝛾[𝑆(𝑥	𝛾[𝑆(𝑥	X
cana-5953	194	3	,	,	PUNCT
cana-5953	194	4	𝑥	𝑥	NOUN
cana-5953	194	5	,	,	PUNCT
cana-5953	194	6	𝑔𝑦	𝑔𝑦	ADJ
cana-5953	194	7	)	)	PUNCT
cana-5953	194	8	+	+	CCONJ
cana-5953	194	9	𝑆(𝑦	𝑆(𝑦	PROPN
cana-5953	194	10	,	,	PUNCT
cana-5953	194	11	𝑦	𝑦	NOUN
cana-5953	194	12	,	,	PUNCT
cana-5953	194	13	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	194	14	)	)	PUNCT
cana-5953	194	15	]	]	PUNCT
cana-5953	194	16	for	for	ADP
cana-5953	194	17	all	all	PRON
cana-5953	194	18	𝑥	𝑥	PROPN
cana-5953	194	19	,	,	PUNCT
cana-5953	194	20	𝑦	𝑦	NOUN
cana-5953	194	21	∈	∈	PROPN
cana-5953	194	22	𝑋	𝑋	PROPN
cana-5953	194	23	,	,	PUNCT
cana-5953	194	24	where	where	SCONJ
cana-5953	194	25	𝛼	𝛼	X
cana-5953	194	26	,	,	PUNCT
cana-5953	194	27	𝛽	𝛽	NOUN
cana-5953	194	28	,	,	PUNCT
cana-5953	194	29	𝛾	𝛾	VERB
cana-5953	194	30	>	>	X
cana-5953	194	31	0	0	PUNCT
cana-5953	194	32	and	and	CCONJ
cana-5953	194	33	𝛼	𝛼	PRON
cana-5953	194	34	+	+	CCONJ
cana-5953	194	35	2𝛽	2𝛽	NOUN
cana-5953	195	1	+	+	CCONJ
cana-5953	195	2	3𝛾	3𝛾	NUM
cana-5953	195	3	<	<	X
cana-5953	195	4	1	1	NUM
cana-5953	195	5	.	.	PUNCT
cana-5953	196	1	then	then	ADV
cana-5953	196	2	𝑓	𝑓	X
cana-5953	196	3	and	and	CCONJ
cana-5953	196	4	𝑔	𝑔	AUX
cana-5953	196	5	have	have	VERB
cana-5953	196	6	a	a	DET
cana-5953	196	7	unique	unique	ADJ
cana-5953	196	8	common	common	ADJ
cana-5953	196	9	fixed	fix	VERB
cana-5953	196	10	point	point	NOUN
cana-5953	196	11	in	in	ADP
cana-5953	196	12	𝑋.	𝑋.	PROPN
cana-5953	196	13	moreover	moreover	ADV
cana-5953	196	14	,	,	PUNCT
cana-5953	196	15	every	every	DET
cana-5953	196	16	fixed	fix	VERB
cana-5953	196	17	point	point	NOUN
cana-5953	196	18	of	of	ADP
cana-5953	196	19	𝑓	𝑓	PRON
cana-5953	196	20	is	be	AUX
cana-5953	196	21	also	also	ADV
cana-5953	196	22	a	a	DET
cana-5953	196	23	fixed	fix	VERB
cana-5953	196	24	point	point	NOUN
cana-5953	196	25	of	of	ADP
cana-5953	196	26	𝑔	𝑔	NOUN
cana-5953	196	27	,	,	PUNCT
cana-5953	196	28	and	and	CCONJ
cana-5953	196	29	conversely	conversely	ADV
cana-5953	196	30	.	.	PUNCT
cana-5953	197	1	proof	proof	NOUN
cana-5953	197	2	.	.	PUNCT
cana-5953	198	1	let	let	VERB
cana-5953	198	2	𝑥0	𝑥0	NOUN
cana-5953	198	3	be	be	AUX
cana-5953	198	4	an	an	DET
cana-5953	198	5	arbitrary	arbitrary	ADJ
cana-5953	198	6	starting	starting	NOUN
cana-5953	198	7	point	point	NOUN
cana-5953	198	8	of	of	ADP
cana-5953	198	9	𝑋	𝑋	PROPN
cana-5953	198	10	,	,	PUNCT
cana-5953	198	11	and	and	CCONJ
cana-5953	198	12	define	define	VERB
cana-5953	198	13	the	the	DET
cana-5953	198	14	sequence	sequence	NOUN
cana-5953	198	15	{	{	PUNCT
cana-5953	198	16	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	198	17	}	}	PUNCT
cana-5953	198	18	by	by	ADP
cana-5953	198	19	𝑥2𝑛+1	𝑥2𝑛+1	X
cana-5953	198	20	=	=	SYM
cana-5953	198	21	𝑓𝑥2𝑛	𝑓𝑥2𝑛	PROPN
cana-5953	198	22	,	,	PUNCT
cana-5953	198	23	𝑥2𝑛+2	𝑥2𝑛+2	PROPN
cana-5953	198	24	=	=	PROPN
cana-5953	198	25	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-5953	198	26	,	,	PUNCT
cana-5953	198	27	𝑛	𝑛	NOUN
cana-5953	198	28	=	=	SYM
cana-5953	198	29	0,1,2	0,1,2	NUM
cana-5953	198	30	,	,	PUNCT
cana-5953	198	31	⋯	⋯	VERB
cana-5953	198	32	communications	communication	NOUN
cana-5953	198	33	on	on	ADP
cana-5953	198	34	applied	apply	VERB
cana-5953	198	35	nonlinear	nonlinear	ADJ
cana-5953	198	36	analysis	analysis	NOUN
cana-5953	198	37	issn	issn	NOUN
cana-5953	198	38	:	:	PUNCT
cana-5953	198	39	1074	1074	NUM
cana-5953	198	40	-	-	PUNCT
cana-5953	198	41	133x	133x	NUM
cana-5953	198	42	vol	vol	VERB
cana-5953	198	43	32	32	NUM
cana-5953	198	44	no	no	NOUN
cana-5953	198	45	.	.	PUNCT
cana-5953	199	1	10s	10	NOUN
cana-5953	199	2	(	(	PUNCT
cana-5953	199	3	2025	2025	NUM
cana-5953	199	4	)	)	PUNCT
cana-5953	199	5	3168	3168	NUM
cana-5953	199	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	199	7	now	now	ADV
cana-5953	199	8	,	,	PUNCT
cana-5953	199	9	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	199	10	,	,	PUNCT
cana-5953	199	11	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	199	12	,	,	PUNCT
cana-5953	199	13	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-5953	199	14	)	)	PUNCT
cana-5953	199	15	=	=	SYM
cana-5953	199	16	𝑆(𝑓𝑥2𝑛	𝑆(𝑓𝑥2𝑛	PROPN
cana-5953	199	17	,	,	PUNCT
cana-5953	199	18	𝑓𝑥2𝑛	𝑓𝑥2𝑛	PROPN
cana-5953	199	19	,	,	PUNCT
cana-5953	199	20	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-5953	199	21	)	)	PUNCT
cana-5953	199	22	≤	≤	NOUN
cana-5953	199	23	𝛼𝑆(𝑥2𝑛	𝛼𝑆(𝑥2𝑛	PROPN
cana-5953	199	24	,	,	PUNCT
cana-5953	199	25	𝑥2𝑛	𝑥2𝑛	NOUN
cana-5953	199	26	,	,	PUNCT
cana-5953	199	27	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	199	28	)	)	PUNCT
cana-5953	199	29	+	+	CCONJ
cana-5953	199	30	𝛽[𝑆(𝑥2𝑛	𝛽[𝑆(𝑥2𝑛	NOUN
cana-5953	199	31	,	,	PUNCT
cana-5953	199	32	𝑥2𝑛	𝑥2𝑛	ADV
cana-5953	199	33	,	,	PUNCT
cana-5953	199	34	𝑓𝑥2𝑛	𝑓𝑥2𝑛	ADJ
cana-5953	199	35	)	)	PUNCT
cana-5953	199	36	+	+	CCONJ
cana-5953	200	1	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	200	2	,	,	PUNCT
cana-5953	200	3	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	200	4	,	,	PUNCT
cana-5953	200	5	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-5953	200	6	)	)	PUNCT
cana-5953	200	7	]	]	PUNCT
cana-5953	201	1	+	+	PUNCT
cana-5953	201	2	𝛾[𝑆(𝑥2𝑛	𝛾[𝑆(𝑥2𝑛	NOUN
cana-5953	201	3	,	,	PUNCT
cana-5953	201	4	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	201	5	,	,	PUNCT
cana-5953	201	6	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-5953	201	7	)	)	PUNCT
cana-5953	201	8	+	+	CCONJ
cana-5953	202	1	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	202	2	,	,	PUNCT
cana-5953	202	3	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	202	4	,	,	PUNCT
cana-5953	202	5	𝑓𝑥2𝑛	𝑓𝑥2𝑛	NOUN
cana-5953	202	6	)	)	PUNCT
cana-5953	202	7	]	]	PUNCT
cana-5953	202	8	=	=	SYM
cana-5953	202	9	𝛼𝑆(𝑥2𝑛	𝛼𝑆(𝑥2𝑛	PROPN
cana-5953	202	10	,	,	PUNCT
cana-5953	202	11	𝑥2𝑛	𝑥2𝑛	NOUN
cana-5953	202	12	,	,	PUNCT
cana-5953	202	13	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	202	14	)	)	PUNCT
cana-5953	202	15	+	+	CCONJ
cana-5953	202	16	𝛽[𝑆(𝑥2𝑛	𝛽[𝑆(𝑥2𝑛	NOUN
cana-5953	202	17	,	,	PUNCT
cana-5953	202	18	𝑥2𝑛	𝑥2𝑛	ADV
cana-5953	202	19	,	,	PUNCT
cana-5953	202	20	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	202	21	)	)	PUNCT
cana-5953	202	22	+	+	CCONJ
cana-5953	202	23	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	202	24	,	,	PUNCT
cana-5953	202	25	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	202	26	,	,	PUNCT
cana-5953	202	27	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	202	28	)	)	PUNCT
cana-5953	202	29	]	]	PUNCT
cana-5953	203	1	+	+	PUNCT
cana-5953	203	2	𝛾[𝑆(𝑥2𝑛	𝛾[𝑆(𝑥2𝑛	NOUN
cana-5953	203	3	,	,	PUNCT
cana-5953	203	4	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	203	5	,	,	PUNCT
cana-5953	203	6	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-5953	203	7	)	)	PUNCT
cana-5953	203	8	+	+	CCONJ
cana-5953	203	9	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	203	10	,	,	PUNCT
cana-5953	203	11	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	203	12	,	,	PUNCT
cana-5953	203	13	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	203	14	)	)	PUNCT
cana-5953	203	15	]	]	PUNCT
cana-5953	203	16	=	=	SYM
cana-5953	203	17	𝛼𝑆(𝑥2𝑛	𝛼𝑆(𝑥2𝑛	PROPN
cana-5953	203	18	,	,	PUNCT
cana-5953	203	19	𝑥2𝑛	𝑥2𝑛	NOUN
cana-5953	203	20	,	,	PUNCT
cana-5953	203	21	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	203	22	)	)	PUNCT
cana-5953	203	23	+	+	CCONJ
cana-5953	203	24	𝛽[𝑆(𝑥2𝑛	𝛽[𝑆(𝑥2𝑛	NOUN
cana-5953	203	25	,	,	PUNCT
cana-5953	203	26	𝑥2𝑛	𝑥2𝑛	ADV
cana-5953	203	27	,	,	PUNCT
cana-5953	203	28	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	203	29	)	)	PUNCT
cana-5953	203	30	+	+	CCONJ
cana-5953	203	31	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	203	32	,	,	PUNCT
cana-5953	203	33	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	203	34	,	,	PUNCT
cana-5953	203	35	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	203	36	)	)	PUNCT
cana-5953	203	37	]	]	PUNCT
cana-5953	204	1	+	+	PUNCT
cana-5953	204	2	𝛾[𝑆(𝑥2𝑛	𝛾[𝑆(𝑥2𝑛	NOUN
cana-5953	204	3	,	,	PUNCT
cana-5953	204	4	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	204	5	,	,	PUNCT
cana-5953	204	6	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	204	7	)	)	PUNCT
cana-5953	204	8	+	+	CCONJ
cana-5953	204	9	𝑆(𝑥2𝑛	𝑆(𝑥2𝑛	ADV
cana-5953	204	10	,	,	PUNCT
cana-5953	204	11	𝑥2𝑛	𝑥2𝑛	ADV
cana-5953	204	12	,	,	PUNCT
cana-5953	204	13	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	204	14	)	)	PUNCT
cana-5953	204	15	+	+	CCONJ
cana-5953	204	16	𝑆(𝑥2𝑛+2	𝑆(𝑥2𝑛+2	PROPN
cana-5953	204	17	,	,	PUNCT
cana-5953	204	18	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	204	19	,	,	PUNCT
cana-5953	204	20	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	204	21	)	)	PUNCT
cana-5953	204	22	]	]	PUNCT
cana-5953	204	23	=	=	SYM
cana-5953	204	24	𝛼𝑆(𝑥2𝑛	𝛼𝑆(𝑥2𝑛	PROPN
cana-5953	204	25	,	,	PUNCT
cana-5953	204	26	𝑥2𝑛	𝑥2𝑛	NOUN
cana-5953	204	27	,	,	PUNCT
cana-5953	204	28	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	204	29	)	)	PUNCT
cana-5953	204	30	+	+	CCONJ
cana-5953	204	31	𝛽[𝑆(𝑥2𝑛	𝛽[𝑆(𝑥2𝑛	NOUN
cana-5953	204	32	,	,	PUNCT
cana-5953	204	33	𝑥2𝑛	𝑥2𝑛	ADV
cana-5953	204	34	,	,	PUNCT
cana-5953	204	35	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	204	36	)	)	PUNCT
cana-5953	204	37	+	+	CCONJ
cana-5953	204	38	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	204	39	,	,	PUNCT
cana-5953	204	40	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	204	41	,	,	PUNCT
cana-5953	204	42	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	204	43	)	)	PUNCT
cana-5953	204	44	]	]	PUNCT
cana-5953	205	1	+	+	PUNCT
cana-5953	205	2	𝛾[2𝑆(𝑥2𝑛	𝛾[2𝑆(𝑥2𝑛	PROPN
cana-5953	205	3	,	,	PUNCT
cana-5953	205	4	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	205	5	,	,	PUNCT
cana-5953	205	6	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	205	7	)	)	PUNCT
cana-5953	205	8	+	+	CCONJ
cana-5953	205	9	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	205	10	,	,	PUNCT
cana-5953	205	11	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	205	12	,	,	PUNCT
cana-5953	205	13	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	205	14	)	)	PUNCT
cana-5953	205	15	]	]	PUNCT
cana-5953	206	1	=	=	PUNCT
cana-5953	206	2	(	(	PUNCT
cana-5953	206	3	𝛼	𝛼	X
cana-5953	206	4	+	+	X
cana-5953	206	5	𝛽	𝛽	PROPN
cana-5953	206	6	+	+	NOUN
cana-5953	206	7	2𝛾)𝑆(𝑥2𝑛	2𝛾)𝑆(𝑥2𝑛	NUM
cana-5953	206	8	,	,	PUNCT
cana-5953	206	9	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	206	10	,	,	PUNCT
cana-5953	206	11	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	12	)	)	PUNCT
cana-5953	206	13	+	+	CCONJ
cana-5953	206	14	(	(	PUNCT
cana-5953	206	15	𝛽	𝛽	PROPN
cana-5953	206	16	+	+	PROPN
cana-5953	206	17	𝛾)𝑆(𝑥2𝑛+1	𝛾)𝑆(𝑥2𝑛+1	PROPN
cana-5953	206	18	,	,	PUNCT
cana-5953	206	19	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	20	,	,	PUNCT
cana-5953	206	21	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	206	22	)	)	PUNCT
cana-5953	206	23	rearranging	rearrange	VERB
cana-5953	206	24	gives	give	NOUN
cana-5953	206	25	(	(	PUNCT
cana-5953	206	26	1	1	NUM
cana-5953	206	27	−	−	PROPN
cana-5953	206	28	(	(	PUNCT
cana-5953	206	29	𝛽	𝛽	PROPN
cana-5953	206	30	+	+	X
cana-5953	206	31	𝛾))𝑆(𝑥2𝑛+1	𝛾))𝑆(𝑥2𝑛+1	PROPN
cana-5953	206	32	,	,	PUNCT
cana-5953	206	33	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	34	,	,	PUNCT
cana-5953	206	35	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	206	36	)	)	PUNCT
cana-5953	206	37	≤	≤	NOUN
cana-5953	206	38	(	(	PUNCT
cana-5953	206	39	𝛼	𝛼	X
cana-5953	206	40	+	+	X
cana-5953	206	41	𝛽	𝛽	PROPN
cana-5953	206	42	+	+	NOUN
cana-5953	206	43	2𝛾)𝑆(𝑥2𝑛	2𝛾)𝑆(𝑥2𝑛	NUM
cana-5953	206	44	,	,	PUNCT
cana-5953	206	45	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	206	46	,	,	PUNCT
cana-5953	206	47	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	48	)	)	PUNCT
cana-5953	206	49	≤	≤	NOUN
cana-5953	206	50	(	(	PUNCT
cana-5953	206	51	𝛼+𝛽+2𝛾	𝛼+𝛽+2𝛾	NUM
cana-5953	206	52	)	)	PUNCT
cana-5953	206	53	(	(	PUNCT
cana-5953	206	54	1−(𝛽+𝛾	1−(𝛽+𝛾	NUM
cana-5953	206	55	)	)	PUNCT
cana-5953	206	56	)	)	PUNCT
cana-5953	206	57	𝑆(𝑥2𝑛	𝑆(𝑥2𝑛	ADV
cana-5953	206	58	,	,	PUNCT
cana-5953	206	59	𝑥2𝑛	𝑥2𝑛	ADV
cana-5953	206	60	,	,	PUNCT
cana-5953	206	61	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	62	)	)	PUNCT
cana-5953	206	63	by	by	ADP
cana-5953	206	64	taking	take	VERB
cana-5953	206	65	𝛿	𝛿	PROPN
cana-5953	206	66	=	=	SYM
cana-5953	206	67	(	(	PUNCT
cana-5953	206	68	𝛼+𝛽+2𝛾	𝛼+𝛽+2𝛾	NUM
cana-5953	206	69	)	)	PUNCT
cana-5953	206	70	(	(	PUNCT
cana-5953	206	71	1−(𝛽+𝛾	1−(𝛽+𝛾	NUM
cana-5953	206	72	)	)	PUNCT
cana-5953	206	73	)	)	PUNCT
cana-5953	206	74	<	<	X
cana-5953	206	75	1	1	NUM
cana-5953	206	76	,	,	PUNCT
cana-5953	206	77	we	we	PRON
cana-5953	206	78	get	get	VERB
cana-5953	206	79	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	206	80	,	,	PUNCT
cana-5953	206	81	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	82	,	,	PUNCT
cana-5953	206	83	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	206	84	)	)	PUNCT
cana-5953	206	85	≤	≤	NOUN
cana-5953	206	86	𝛿𝑆(𝑥2𝑛	𝛿𝑆(𝑥2𝑛	PROPN
cana-5953	206	87	,	,	PUNCT
cana-5953	206	88	𝑥2𝑛	𝑥2𝑛	ADJ
cana-5953	206	89	,	,	PUNCT
cana-5953	206	90	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	91	)	)	PUNCT
cana-5953	206	92	applying	apply	VERB
cana-5953	206	93	recursively	recursively	ADV
cana-5953	206	94	,	,	PUNCT
cana-5953	206	95	𝑆(𝑥2𝑛+1	𝑆(𝑥2𝑛+1	PROPN
cana-5953	206	96	,	,	PUNCT
cana-5953	206	97	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-5953	206	98	,	,	PUNCT
cana-5953	206	99	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-5953	206	100	)	)	PUNCT
cana-5953	206	101	≤	≤	NUM
cana-5953	206	102	𝛿2𝑛+1𝑆(𝑥0	𝛿2𝑛+1𝑆(𝑥0	ADJ
cana-5953	206	103	,	,	PUNCT
cana-5953	206	104	𝑥0	𝑥0	NOUN
cana-5953	206	105	,	,	PUNCT
cana-5953	206	106	𝑥1	𝑥1	NOUN
cana-5953	206	107	)	)	PUNCT
cana-5953	206	108	,	,	PUNCT
cana-5953	206	109	for	for	ADP
cana-5953	206	110	𝑚	𝑚	PROPN
cana-5953	206	111	,	,	PUNCT
cana-5953	206	112	𝑛	𝑛	DET
cana-5953	206	113	∈	∈	PROPN
cana-5953	206	114	ℕ	ℕ	PROPN
cana-5953	206	115	with	with	ADP
cana-5953	206	116	𝑚	𝑚	NOUN
cana-5953	206	117	<	<	X
cana-5953	206	118	𝑛	𝑛	PROPN
cana-5953	206	119	and	and	CCONJ
cana-5953	206	120	some	some	DET
cana-5953	206	121	𝑁	𝑁	PROPN
cana-5953	206	122	∈	∈	PROPN
cana-5953	206	123	ℕ	ℕ	PROPN
cana-5953	206	124	,	,	PUNCT
cana-5953	206	125	we	we	PRON
cana-5953	206	126	estimate	estimate	VERB
cana-5953	206	127	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	206	128	,	,	PUNCT
cana-5953	206	129	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	206	130	,	,	PUNCT
cana-5953	206	131	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	206	132	)	)	PUNCT
cana-5953	206	133	≤	≤	NOUN
cana-5953	206	134	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	206	135	,	,	PUNCT
cana-5953	206	136	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	206	137	,	,	PUNCT
cana-5953	206	138	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	206	139	)	)	PUNCT
cana-5953	207	1	+	+	CCONJ
cana-5953	207	2	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	207	3	,	,	PUNCT
cana-5953	207	4	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	207	5	,	,	PUNCT
cana-5953	207	6	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	207	7	)	)	PUNCT
cana-5953	207	8	+	+	CCONJ
cana-5953	207	9	𝑆(𝑥𝑚	𝑆(𝑥𝑚	PROPN
cana-5953	207	10	,	,	PUNCT
cana-5953	207	11	𝑥𝑚	𝑥𝑚	PRON
cana-5953	207	12	,	,	PUNCT
cana-5953	207	13	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5953	207	14	)	)	PUNCT
cana-5953	207	15	=	=	SYM
cana-5953	207	16	2𝑆(𝑥𝑛	2𝑆(𝑥𝑛	NUM
cana-5953	207	17	,	,	PUNCT
cana-5953	207	18	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	207	19	,	,	PUNCT
cana-5953	207	20	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	207	21	)	)	PUNCT
cana-5953	207	22	+	+	NUM
cana-5953	207	23	𝑆(𝑥𝑚	𝑆(𝑥𝑚	NOUN
cana-5953	207	24	,	,	PUNCT
cana-5953	207	25	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	207	26	,	,	PUNCT
cana-5953	207	27	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5953	207	28	)	)	PUNCT
cana-5953	207	29	by	by	ADP
cana-5953	207	30	recursively	recursively	ADV
cana-5953	207	31	applying	apply	VERB
cana-5953	207	32	the	the	DET
cana-5953	207	33	triangle	triangle	NOUN
cana-5953	207	34	-	-	PUNCT
cana-5953	207	35	like	like	ADJ
cana-5953	207	36	inequality	inequality	NOUN
cana-5953	207	37	,	,	PUNCT
cana-5953	207	38	we	we	PRON
cana-5953	207	39	proceed	proceed	VERB
cana-5953	207	40	as	as	ADP
cana-5953	207	41	,	,	PUNCT
cana-5953	207	42	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	207	43	,	,	PUNCT
cana-5953	207	44	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	207	45	,	,	PUNCT
cana-5953	207	46	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	207	47	)	)	PUNCT
cana-5953	207	48	≤	≤	NOUN
cana-5953	207	49	2𝛿𝑛𝑆(𝑥0	2𝛿𝑛𝑆(𝑥0	PROPN
cana-5953	207	50	,	,	PUNCT
cana-5953	207	51	𝑥0	𝑥0	NOUN
cana-5953	207	52	,	,	PUNCT
cana-5953	207	53	𝑥1	𝑥1	NOUN
cana-5953	207	54	)	)	PUNCT
cana-5953	207	55	+	+	CCONJ
cana-5953	208	1	[	[	X
cana-5953	208	2	𝑆(𝑥𝑚	𝑆(𝑥𝑚	X
cana-5953	208	3	,	,	PUNCT
cana-5953	208	4	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	208	5	,	,	PUNCT
cana-5953	208	6	𝑥𝑛+2	𝑥𝑛+2	PUNCT
cana-5953	208	7	)	)	PUNCT
cana-5953	208	8	+	+	CCONJ
cana-5953	208	9	𝑆(𝑥𝑚	𝑆(𝑥𝑚	NOUN
cana-5953	208	10	,	,	PUNCT
cana-5953	208	11	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	208	12	,	,	PUNCT
cana-5953	208	13	𝑥𝑛+2	𝑥𝑛+2	PUNCT
cana-5953	208	14	)	)	PUNCT
cana-5953	208	15	+	+	CCONJ
cana-5953	208	16	𝑆(𝑥𝑛+1	𝑆(𝑥𝑛+1	PROPN
cana-5953	208	17	,	,	PUNCT
cana-5953	208	18	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5953	208	19	,	,	PUNCT
cana-5953	208	20	𝑥𝑛+2	𝑥𝑛+2	NUM
cana-5953	208	21	)	)	PUNCT
cana-5953	208	22	]	]	PUNCT
cana-5953	208	23	by	by	ADP
cana-5953	208	24	taking	take	VERB
cana-5953	208	25	𝑆(𝑔𝑥0	𝑆(𝑔𝑥0	ADJ
cana-5953	208	26	,	,	PUNCT
cana-5953	208	27	𝑔𝑥0	𝑔𝑥0	VERB
cana-5953	208	28	,	,	PUNCT
cana-5953	208	29	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	208	30	)	)	PUNCT
cana-5953	208	31	=	=	SYM
cana-5953	208	32	𝑧	𝑧	X
cana-5953	208	33	,	,	PUNCT
cana-5953	208	34	this	this	DET
cana-5953	208	35	yields	yield	NOUN
cana-5953	208	36	𝑆(𝑥𝑛	𝑆(𝑥𝑛	PROPN
cana-5953	208	37	,	,	PUNCT
cana-5953	208	38	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	208	39	,	,	PUNCT
cana-5953	208	40	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	208	41	)	)	PUNCT
cana-5953	208	42	≤	≤	NOUN
cana-5953	208	43	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	208	44	+	+	PUNCT
cana-5953	209	1	[	[	X
cana-5953	209	2	2𝑆(𝑥𝑚	2𝑆(𝑥𝑚	NUM
cana-5953	209	3	,	,	PUNCT
cana-5953	209	4	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	209	5	,	,	PUNCT
cana-5953	209	6	𝑥𝑛+2	𝑥𝑛+2	PUNCT
cana-5953	209	7	)	)	PUNCT
cana-5953	209	8	+	+	CCONJ
cana-5953	209	9	𝑆(𝑥𝑛+1	𝑆(𝑥𝑛+1	PROPN
cana-5953	209	10	,	,	PUNCT
cana-5953	209	11	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5953	209	12	,	,	PUNCT
cana-5953	209	13	𝑥𝑛+2	𝑥𝑛+2	NUM
cana-5953	209	14	)	)	PUNCT
cana-5953	209	15	]	]	PUNCT
cana-5953	209	16	≤	≤	NOUN
cana-5953	210	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	210	2	+	+	PUNCT
cana-5953	211	1	[	[	X
cana-5953	211	2	2𝑆(𝑥𝑚	2𝑆(𝑥𝑚	NUM
cana-5953	211	3	,	,	PUNCT
cana-5953	211	4	𝑥𝑚	𝑥𝑚	NOUN
cana-5953	211	5	,	,	PUNCT
cana-5953	211	6	𝑥𝑛+2	𝑥𝑛+2	PUNCT
cana-5953	211	7	)	)	PUNCT
cana-5953	211	8	+	+	NUM
cana-5953	211	9	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	211	10	]	]	PUNCT
cana-5953	211	11	≤	≤	NOUN
cana-5953	211	12	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	211	13	+	+	CCONJ
cana-5953	211	14	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	211	15	+	+	CCONJ
cana-5953	211	16	2[𝑆(𝑥𝑚	2[𝑆(𝑥𝑚	NUM
cana-5953	211	17	,	,	PUNCT
cana-5953	211	18	𝑥𝑚	𝑥𝑚	PRON
cana-5953	211	19	,	,	PUNCT
cana-5953	211	20	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	211	21	)	)	PUNCT
cana-5953	212	1	+	+	CCONJ
cana-5953	212	2	𝑆(𝑥𝑚	𝑆(𝑥𝑚	PROPN
cana-5953	212	3	,	,	PUNCT
cana-5953	212	4	𝑥𝑚	𝑥𝑚	PRON
cana-5953	212	5	,	,	PUNCT
cana-5953	212	6	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	212	7	)	)	PUNCT
cana-5953	212	8	+	+	CCONJ
cana-5953	212	9	𝑆(𝑥𝑛+2	𝑆(𝑥𝑛+2	PROPN
cana-5953	212	10	,	,	PUNCT
cana-5953	212	11	𝑥𝑛+2	𝑥𝑛+2	ADV
cana-5953	212	12	,	,	PUNCT
cana-5953	212	13	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	212	14	)	)	PUNCT
cana-5953	212	15	]	]	PUNCT
cana-5953	212	16	≤	≤	NOUN
cana-5953	213	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	213	2	+	+	CCONJ
cana-5953	213	3	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	213	4	+	+	CCONJ
cana-5953	213	5	2[2𝑆(𝑥𝑚	2[2𝑆(𝑥𝑚	NUM
cana-5953	213	6	,	,	PUNCT
cana-5953	213	7	𝑥𝑚	𝑥𝑚	PRON
cana-5953	213	8	,	,	PUNCT
cana-5953	213	9	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	213	10	)	)	PUNCT
cana-5953	214	1	+	+	CCONJ
cana-5953	214	2	𝛿𝑛+2𝑧	𝛿𝑛+2𝑧	NOUN
cana-5953	214	3	]	]	PUNCT
cana-5953	214	4	communications	communication	NOUN
cana-5953	214	5	on	on	ADP
cana-5953	214	6	applied	apply	VERB
cana-5953	214	7	nonlinear	nonlinear	ADJ
cana-5953	214	8	analysis	analysis	NOUN
cana-5953	214	9	issn	issn	NOUN
cana-5953	214	10	:	:	PUNCT
cana-5953	214	11	1074	1074	NUM
cana-5953	214	12	-	-	PUNCT
cana-5953	214	13	133x	133x	NUM
cana-5953	214	14	vol	vol	VERB
cana-5953	214	15	32	32	NUM
cana-5953	214	16	no	no	NOUN
cana-5953	214	17	.	.	PUNCT
cana-5953	215	1	10s	10	NOUN
cana-5953	215	2	(	(	PUNCT
cana-5953	215	3	2025	2025	NUM
cana-5953	215	4	)	)	PUNCT
cana-5953	215	5	3169	3169	NUM
cana-5953	216	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	216	2	≤	≤	PUNCT
cana-5953	217	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	217	2	+	+	CCONJ
cana-5953	217	3	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	217	4	+	+	CCONJ
cana-5953	217	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	217	6	+	+	SYM
cana-5953	217	7	4[𝑆(𝑥𝑚	4[𝑆(𝑥𝑚	NUM
cana-5953	217	8	,	,	PUNCT
cana-5953	217	9	𝑥𝑚	𝑥𝑚	PRON
cana-5953	217	10	,	,	PUNCT
cana-5953	217	11	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	217	12	)	)	PUNCT
cana-5953	217	13	+	+	CCONJ
cana-5953	217	14	𝑆(𝑥𝑚	𝑆(𝑥𝑚	PROPN
cana-5953	217	15	,	,	PUNCT
cana-5953	217	16	𝑥𝑚	𝑥𝑚	PRON
cana-5953	217	17	,	,	PUNCT
cana-5953	217	18	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	217	19	)	)	PUNCT
cana-5953	217	20	≤	≤	NOUN
cana-5953	218	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	218	2	+	+	CCONJ
cana-5953	218	3	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	218	4	+	+	CCONJ
cana-5953	218	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	218	6	+	+	SYM
cana-5953	218	7	4[2𝑆(𝑥𝑚	4[2𝑆(𝑥𝑚	NUM
cana-5953	218	8	,	,	PUNCT
cana-5953	218	9	𝑥𝑚	𝑥𝑚	PRON
cana-5953	218	10	,	,	PUNCT
cana-5953	218	11	𝑥𝑛+4	𝑥𝑛+4	PROPN
cana-5953	218	12	)	)	PUNCT
cana-5953	218	13	+	+	CCONJ
cana-5953	218	14	𝛿𝑛+3𝑧	𝛿𝑛+3𝑧	NOUN
cana-5953	218	15	]	]	PUNCT
cana-5953	219	1	+	+	PUNCT
cana-5953	219	2	𝑆(𝑥𝑛+3	𝑆(𝑥𝑛+3	NOUN
cana-5953	219	3	,	,	PUNCT
cana-5953	219	4	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	219	5	,	,	PUNCT
cana-5953	219	6	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	219	7	)	)	PUNCT
cana-5953	219	8	]	]	PUNCT
cana-5953	219	9	≤	≤	NOUN
cana-5953	220	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	220	2	+	+	CCONJ
cana-5953	220	3	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	220	4	+	+	CCONJ
cana-5953	220	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	220	6	+	+	CCONJ
cana-5953	220	7	4𝛿𝑛+3𝑧	4𝛿𝑛+3𝑧	NUM
cana-5953	220	8	+	+	CCONJ
cana-5953	220	9	8𝛿𝑛+4𝑧	8𝛿𝑛+4𝑧	NUM
cana-5953	221	1	+	+	CCONJ
cana-5953	221	2	⋯	⋯	X
cana-5953	221	3	=	=	SYM
cana-5953	222	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	222	2	+	+	NOUN
cana-5953	222	3	𝛿𝑛+1𝑧(1	𝛿𝑛+1𝑧(1	NOUN
cana-5953	222	4	+	+	CCONJ
cana-5953	222	5	2𝛿	2𝛿	NOUN
cana-5953	222	6	+	+	CCONJ
cana-5953	222	7	4𝛿2	4𝛿2	NUM
cana-5953	222	8	+	+	CCONJ
cana-5953	222	9	8𝛿3	8𝛿3	NUM
cana-5953	222	10	+	+	CCONJ
cana-5953	222	11	⋯	⋯	NOUN
cana-5953	222	12	)	)	PUNCT
cana-5953	222	13	=	=	SYM
cana-5953	223	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	223	2	+	+	NUM
cana-5953	223	3	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	223	4	(	(	PUNCT
cana-5953	223	5	1	1	NUM
cana-5953	223	6	+	+	CCONJ
cana-5953	223	7	(	(	PUNCT
cana-5953	223	8	2𝛿	2𝛿	NOUN
cana-5953	223	9	)	)	PUNCT
cana-5953	223	10	+	+	CCONJ
cana-5953	223	11	(	(	PUNCT
cana-5953	223	12	2𝛿)2	2𝛿)2	NUM
cana-5953	223	13	+	+	CCONJ
cana-5953	223	14	(	(	PUNCT
cana-5953	223	15	2𝛿)3	2𝛿)3	NUM
cana-5953	223	16	+	+	SYM
cana-5953	223	17	⋯	⋯	NOUN
cana-5953	223	18	)	)	PUNCT
cana-5953	223	19	=	=	SYM
cana-5953	224	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	224	2	+	+	NUM
cana-5953	224	3	𝛿𝑛+1𝑧	𝛿𝑛+1𝑧	NOUN
cana-5953	224	4	(	(	PUNCT
cana-5953	224	5	1	1	NUM
cana-5953	224	6	1−2𝛿	1−2𝛿	NUM
cana-5953	224	7	)	)	PUNCT
cana-5953	224	8	,	,	PUNCT
cana-5953	224	9	as	as	ADP
cana-5953	224	10	𝛿	𝛿	PROPN
cana-5953	224	11	<	<	X
cana-5953	224	12	1	1	NUM
cana-5953	224	13	,	,	PUNCT
cana-5953	224	14	it	it	PRON
cana-5953	224	15	follows	follow	VERB
cana-5953	224	16	that	that	SCONJ
cana-5953	224	17	lim	lim	PROPN
cana-5953	224	18	𝑛→∞	𝑛→∞	NUM
cana-5953	224	19	𝑆(𝑥𝑛	𝑆(𝑥𝑛	PROPN
cana-5953	224	20	,	,	PUNCT
cana-5953	224	21	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	224	22	,	,	PUNCT
cana-5953	224	23	𝑥𝑚	𝑥𝑚	ADJ
cana-5953	224	24	)	)	PUNCT
cana-5953	224	25	=	=	SYM
cana-5953	224	26	0	0	X
cana-5953	224	27	.	.	PUNCT
cana-5953	225	1	thus	thus	ADV
cana-5953	225	2	,	,	PUNCT
cana-5953	225	3	the	the	DET
cana-5953	225	4	sequence	sequence	NOUN
cana-5953	225	5	{	{	PUNCT
cana-5953	225	6	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	225	7	}	}	PUNCT
cana-5953	225	8	is	be	AUX
cana-5953	225	9	a	a	DET
cana-5953	225	10	cauchy	cauchy	ADJ
cana-5953	225	11	sequence	sequence	NOUN
cana-5953	225	12	in	in	ADP
cana-5953	225	13	𝑋.	𝑋.	PROPN
cana-5953	225	14	completeness	completeness	NOUN
cana-5953	225	15	of	of	ADP
cana-5953	225	16	𝑋	𝑋	PROPN
cana-5953	225	17	implies	imply	VERB
cana-5953	225	18	that	that	SCONJ
cana-5953	225	19	there	there	PRON
cana-5953	225	20	exists	exist	VERB
cana-5953	225	21	a	a	DET
cana-5953	225	22	point	point	NOUN
cana-5953	225	23	𝑝	𝑝	NOUN
cana-5953	225	24	in	in	ADP
cana-5953	225	25	𝑋	𝑋	PROPN
cana-5953	225	26	such	such	ADJ
cana-5953	225	27	that	that	PRON
cana-5953	225	28	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	225	29	→	→	SYM
cana-5953	225	30	𝑝	𝑝	PROPN
cana-5953	225	31	as	as	ADP
cana-5953	225	32	𝑛	𝑛	PROPN
cana-5953	225	33	→	→	PUNCT
cana-5953	225	34	∞.	∞.	PROPN
cana-5953	225	35	to	to	PART
cana-5953	225	36	show	show	VERB
cana-5953	225	37	that	that	SCONJ
cana-5953	225	38	𝑝	𝑝	PROPN
cana-5953	225	39	is	be	AUX
cana-5953	225	40	a	a	DET
cana-5953	225	41	fixed	fix	VERB
cana-5953	225	42	point	point	NOUN
cana-5953	225	43	of	of	ADP
cana-5953	225	44	𝑔	𝑔	NOUN
cana-5953	225	45	,	,	PUNCT
cana-5953	225	46	we	we	PRON
cana-5953	225	47	consider	consider	VERB
cana-5953	225	48	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	225	49	,	,	PUNCT
cana-5953	225	50	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	225	51	,	,	PUNCT
cana-5953	225	52	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	225	53	)	)	PUNCT
cana-5953	225	54	≤	≤	NOUN
cana-5953	225	55	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	225	56	,	,	PUNCT
cana-5953	225	57	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	225	58	,	,	PUNCT
cana-5953	225	59	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	225	60	)	)	PUNCT
cana-5953	226	1	+	+	CCONJ
cana-5953	226	2	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	226	3	,	,	PUNCT
cana-5953	226	4	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	226	5	,	,	PUNCT
cana-5953	226	6	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	226	7	)	)	PUNCT
cana-5953	226	8	+	+	CCONJ
cana-5953	226	9	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	226	10	,	,	PUNCT
cana-5953	226	11	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	226	12	,	,	PUNCT
cana-5953	226	13	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5953	226	14	)	)	PUNCT
cana-5953	226	15	=	=	SYM
cana-5953	226	16	2𝑆(𝑥𝑛	2𝑆(𝑥𝑛	NUM
cana-5953	226	17	,	,	PUNCT
cana-5953	226	18	𝑥𝑛	𝑥𝑛	NOUN
cana-5953	226	19	,	,	PUNCT
cana-5953	226	20	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	226	21	)	)	PUNCT
cana-5953	226	22	+	+	CCONJ
cana-5953	226	23	𝑆(𝑥𝑛+1	𝑆(𝑥𝑛+1	PROPN
cana-5953	226	24	,	,	PUNCT
cana-5953	226	25	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5953	226	26	,	,	PUNCT
cana-5953	226	27	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	226	28	)	)	PUNCT
cana-5953	226	29	≤	≤	NOUN
cana-5953	227	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	227	2	+	+	PUNCT
cana-5953	228	1	[	[	X
cana-5953	228	2	𝑆(𝑥𝑛+1	𝑆(𝑥𝑛+1	NOUN
cana-5953	228	3	,	,	PUNCT
cana-5953	228	4	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	228	5	,	,	PUNCT
cana-5953	228	6	𝑥𝑛+2	𝑥𝑛+2	NUM
cana-5953	228	7	)	)	PUNCT
cana-5953	228	8	+	+	CCONJ
cana-5953	228	9	𝑆(𝑥𝑛+1	𝑆(𝑥𝑛+1	PROPN
cana-5953	228	10	,	,	PUNCT
cana-5953	228	11	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5953	228	12	,	,	PUNCT
cana-5953	228	13	𝑥𝑛+2	𝑥𝑛+2	PUNCT
cana-5953	228	14	)	)	PUNCT
cana-5953	228	15	+	+	CCONJ
cana-5953	228	16	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	228	17	,	,	PUNCT
cana-5953	228	18	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	228	19	,	,	PUNCT
cana-5953	228	20	𝑥𝑛+2	𝑥𝑛+2	NOUN
cana-5953	228	21	)	)	PUNCT
cana-5953	228	22	]	]	PUNCT
cana-5953	228	23	=	=	PUNCT
cana-5953	229	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	229	2	+	+	PUNCT
cana-5953	230	1	[	[	X
cana-5953	230	2	2𝑆(𝑥𝑛+1	2𝑆(𝑥𝑛+1	ADJ
cana-5953	230	3	,	,	PUNCT
cana-5953	230	4	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5953	230	5	,	,	PUNCT
cana-5953	230	6	𝑥𝑛+2	𝑥𝑛+2	PUNCT
cana-5953	230	7	)	)	PUNCT
cana-5953	230	8	+	+	CCONJ
cana-5953	230	9	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	230	10	,	,	PUNCT
cana-5953	230	11	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	230	12	,	,	PUNCT
cana-5953	230	13	𝑥𝑛+2	𝑥𝑛+2	NUM
cana-5953	230	14	)	)	PUNCT
cana-5953	230	15	]	]	PUNCT
cana-5953	230	16	≤	≤	NOUN
cana-5953	231	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	231	2	+	+	PUNCT
cana-5953	231	3	[	[	X
cana-5953	231	4	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	231	5	+	+	CCONJ
cana-5953	231	6	𝑆(𝑥𝑛+2	𝑆(𝑥𝑛+2	PROPN
cana-5953	231	7	,	,	PUNCT
cana-5953	231	8	𝑥𝑛+2	𝑥𝑛+2	CCONJ
cana-5953	231	9	,	,	PUNCT
cana-5953	231	10	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	231	11	)	)	PUNCT
cana-5953	231	12	]	]	PUNCT
cana-5953	231	13	≤	≤	NOUN
cana-5953	232	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	232	2	+	+	ADJ
cana-5953	232	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	232	4	+	+	SYM
cana-5953	233	1	[	[	X
cana-5953	233	2	𝑆(𝑥𝑛+2	𝑆(𝑥𝑛+2	NUM
cana-5953	233	3	,	,	PUNCT
cana-5953	233	4	𝑥𝑛+2	𝑥𝑛+2	ADV
cana-5953	233	5	,	,	PUNCT
cana-5953	233	6	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	233	7	)	)	PUNCT
cana-5953	233	8	+	+	CCONJ
cana-5953	233	9	𝑆(𝑥𝑛+2	𝑆(𝑥𝑛+2	PROPN
cana-5953	233	10	,	,	PUNCT
cana-5953	233	11	𝑥𝑛+2	𝑥𝑛+2	ADV
cana-5953	233	12	,	,	PUNCT
cana-5953	233	13	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	233	14	)	)	PUNCT
cana-5953	233	15	+	+	CCONJ
cana-5953	233	16	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	233	17	,	,	PUNCT
cana-5953	233	18	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	233	19	,	,	PUNCT
cana-5953	233	20	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	233	21	)	)	PUNCT
cana-5953	233	22	]	]	PUNCT
cana-5953	233	23	=	=	SYM
cana-5953	234	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	234	2	+	+	ADJ
cana-5953	234	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	234	4	+	+	SYM
cana-5953	234	5	[	[	X
cana-5953	234	6	2𝑆(𝑥𝑛+2	2𝑆(𝑥𝑛+2	NUM
cana-5953	234	7	,	,	PUNCT
cana-5953	234	8	𝑥𝑛+2	𝑥𝑛+2	ADV
cana-5953	234	9	,	,	PUNCT
cana-5953	234	10	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	234	11	)	)	PUNCT
cana-5953	234	12	+	+	CCONJ
cana-5953	234	13	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	234	14	,	,	PUNCT
cana-5953	234	15	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	234	16	,	,	PUNCT
cana-5953	234	17	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	234	18	)	)	PUNCT
cana-5953	234	19	]	]	PUNCT
cana-5953	235	1	≤	≤	NOUN
cana-5953	236	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	236	2	+	+	ADJ
cana-5953	236	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	236	4	+	+	SYM
cana-5953	236	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	236	6	+	+	CCONJ
cana-5953	236	7	𝑆(𝑥𝑛+3	𝑆(𝑥𝑛+3	NOUN
cana-5953	236	8	,	,	PUNCT
cana-5953	236	9	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	236	10	,	,	PUNCT
cana-5953	236	11	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	236	12	)	)	PUNCT
cana-5953	236	13	≤	≤	NOUN
cana-5953	237	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	237	2	+	+	ADJ
cana-5953	237	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	237	4	+	+	SYM
cana-5953	237	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	237	6	+	+	CCONJ
cana-5953	238	1	[	[	X
cana-5953	238	2	𝑆(𝑥𝑛+3	𝑆(𝑥𝑛+3	NOUN
cana-5953	238	3	,	,	PUNCT
cana-5953	238	4	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	238	5	,	,	PUNCT
cana-5953	238	6	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	238	7	)	)	PUNCT
cana-5953	238	8	+	+	CCONJ
cana-5953	238	9	𝑆(𝑥𝑛+3	𝑆(𝑥𝑛+3	NOUN
cana-5953	238	10	,	,	PUNCT
cana-5953	238	11	𝑥𝑛+3	𝑥𝑛+3	NUM
cana-5953	238	12	,	,	PUNCT
cana-5953	238	13	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	238	14	)	)	PUNCT
cana-5953	238	15	+	+	NOUN
cana-5953	238	16	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	238	17	,	,	PUNCT
cana-5953	238	18	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	238	19	,	,	PUNCT
cana-5953	238	20	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	238	21	)	)	PUNCT
cana-5953	238	22	]	]	PUNCT
cana-5953	238	23	=	=	PUNCT
cana-5953	239	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	239	2	+	+	ADJ
cana-5953	239	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	239	4	+	+	SYM
cana-5953	239	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	239	6	+	+	CCONJ
cana-5953	239	7	[	[	X
cana-5953	239	8	2𝑆(𝑥𝑛+3	2𝑆(𝑥𝑛+3	NUM
cana-5953	239	9	,	,	PUNCT
cana-5953	239	10	𝑥𝑛+3	𝑥𝑛+3	PRON
cana-5953	239	11	,	,	PUNCT
cana-5953	239	12	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	239	13	)	)	PUNCT
cana-5953	239	14	+	+	CCONJ
cana-5953	239	15	𝑆(𝑔𝑝	𝑆(𝑔𝑝	PROPN
cana-5953	239	16	,	,	PUNCT
cana-5953	239	17	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	239	18	,	,	PUNCT
cana-5953	239	19	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	239	20	)	)	PUNCT
cana-5953	239	21	]	]	PUNCT
cana-5953	239	22	≤	≤	NOUN
cana-5953	240	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	240	2	+	+	ADJ
cana-5953	240	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	240	4	+	+	SYM
cana-5953	240	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	240	6	+	+	CCONJ
cana-5953	240	7	2𝛿𝑛+3𝑧	2𝛿𝑛+3𝑧	NUM
cana-5953	240	8	+	+	CCONJ
cana-5953	240	9	𝑆(𝑥𝑛+4	𝑆(𝑥𝑛+4	NOUN
cana-5953	240	10	,	,	PUNCT
cana-5953	240	11	𝑥𝑛+4	𝑥𝑛+4	NUM
cana-5953	240	12	,	,	PUNCT
cana-5953	240	13	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	240	14	)	)	PUNCT
cana-5953	240	15	=	=	SYM
cana-5953	241	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	241	2	+	+	ADJ
cana-5953	241	3	2𝛿𝑛+1𝑧	2𝛿𝑛+1𝑧	NUM
cana-5953	241	4	+	+	SYM
cana-5953	241	5	2𝛿𝑛+2𝑧	2𝛿𝑛+2𝑧	NUM
cana-5953	241	6	+	+	CCONJ
cana-5953	241	7	2𝛿𝑛+3𝑧	2𝛿𝑛+3𝑧	NUM
cana-5953	241	8	+	+	NUM
cana-5953	241	9	⋯	⋯	NOUN
cana-5953	241	10	=	=	SYM
cana-5953	241	11	2𝛿𝑛𝑧(1	2𝛿𝑛𝑧(1	NUM
cana-5953	241	12	+	+	CCONJ
cana-5953	241	13	𝛿	𝛿	ADJ
cana-5953	241	14	+	+	ADJ
cana-5953	241	15	𝛿2	𝛿2	NOUN
cana-5953	241	16	+	+	CCONJ
cana-5953	241	17	𝛿3	𝛿3	NOUN
cana-5953	241	18	+	+	CCONJ
cana-5953	241	19	⋯	⋯	NOUN
cana-5953	241	20	)	)	PUNCT
cana-5953	241	21	therefore	therefore	ADV
cana-5953	241	22	,	,	PUNCT
cana-5953	241	23	𝑆(𝑥𝑛	𝑆(𝑥𝑛	NOUN
cana-5953	241	24	,	,	PUNCT
cana-5953	241	25	𝑥𝑛	𝑥𝑛	PROPN
cana-5953	241	26	,	,	PUNCT
cana-5953	241	27	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	241	28	)	)	PUNCT
cana-5953	241	29	≤	≤	NOUN
cana-5953	242	1	2𝛿𝑛𝑧	2𝛿𝑛𝑧	NUM
cana-5953	242	2	(	(	PUNCT
cana-5953	242	3	1	1	NUM
cana-5953	242	4	1−𝛿	1−𝛿	NUM
cana-5953	242	5	)	)	PUNCT
cana-5953	242	6	,	,	PUNCT
cana-5953	242	7	where	where	SCONJ
cana-5953	242	8	𝑧	𝑧	NOUN
cana-5953	242	9	=	=	SYM
cana-5953	242	10	𝑆(𝑔𝑥0	𝑆(𝑔𝑥0	ADJ
cana-5953	242	11	,	,	PUNCT
cana-5953	242	12	𝑔𝑥0	𝑔𝑥0	VERB
cana-5953	242	13	,	,	PUNCT
cana-5953	242	14	𝑔𝑥1	𝑔𝑥1	NOUN
cana-5953	242	15	)	)	PUNCT
cana-5953	242	16	.	.	PUNCT
cana-5953	243	1	since	since	SCONJ
cana-5953	243	2	𝛿	𝛿	PRON
cana-5953	243	3	<	<	X
cana-5953	243	4	1	1	NUM
cana-5953	243	5	,	,	PUNCT
cana-5953	243	6	as	as	ADP
cana-5953	243	7	𝑛	𝑛	PROPN
cana-5953	243	8	→	→	SYM
cana-5953	243	9	∞.	∞.	PROPN
cana-5953	243	10	thus	thus	ADV
cana-5953	243	11	,	,	PUNCT
cana-5953	243	12	lim𝑛→∞𝑆(𝑝	lim𝑛→∞𝑆(𝑝	PROPN
cana-5953	243	13	,	,	PUNCT
cana-5953	243	14	𝑝	𝑝	NOUN
cana-5953	243	15	,	,	PUNCT
cana-5953	243	16	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	243	17	)	)	PUNCT
cana-5953	243	18	=	=	SYM
cana-5953	243	19	0	0	NUM
cana-5953	243	20	and	and	CCONJ
cana-5953	243	21	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	243	22	=	=	SYM
cana-5953	243	23	𝑝.	𝑝.	NOUN
cana-5953	243	24	also	also	ADV
cana-5953	243	25	,	,	PUNCT
cana-5953	243	26	now	now	ADV
cana-5953	243	27	to	to	PART
cana-5953	243	28	show	show	VERB
cana-5953	243	29	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	243	30	=	=	SYM
cana-5953	243	31	𝑝	𝑝	PROPN
cana-5953	243	32	,	,	PUNCT
cana-5953	243	33	we	we	PRON
cana-5953	243	34	proceed	proceed	VERB
cana-5953	243	35	as	as	ADP
cana-5953	243	36	,	,	PUNCT
cana-5953	243	37	communications	communication	NOUN
cana-5953	243	38	on	on	ADP
cana-5953	243	39	applied	apply	VERB
cana-5953	243	40	nonlinear	nonlinear	ADJ
cana-5953	243	41	analysis	analysis	NOUN
cana-5953	243	42	issn	issn	NOUN
cana-5953	243	43	:	:	PUNCT
cana-5953	243	44	1074	1074	NUM
cana-5953	243	45	-	-	PUNCT
cana-5953	243	46	133x	133x	NUM
cana-5953	243	47	vol	vol	VERB
cana-5953	243	48	32	32	NUM
cana-5953	243	49	no	no	NOUN
cana-5953	243	50	.	.	PUNCT
cana-5953	244	1	10s	10	NOUN
cana-5953	244	2	(	(	PUNCT
cana-5953	244	3	2025	2025	NUM
cana-5953	244	4	)	)	PUNCT
cana-5953	244	5	3170	3170	NUM
cana-5953	244	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	244	7	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	244	8	,	,	PUNCT
cana-5953	244	9	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	244	10	,	,	PUNCT
cana-5953	244	11	𝑝	𝑝	NOUN
cana-5953	244	12	)	)	PUNCT
cana-5953	245	1	=	=	SYM
cana-5953	245	2	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	245	3	,	,	PUNCT
cana-5953	245	4	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	245	5	,	,	PUNCT
cana-5953	245	6	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	245	7	)	)	PUNCT
cana-5953	245	8	≤	≤	NOUN
cana-5953	245	9	𝛼𝑆(𝑝	𝛼𝑆(𝑝	NUM
cana-5953	245	10	,	,	PUNCT
cana-5953	245	11	𝑝	𝑝	NOUN
cana-5953	245	12	,	,	PUNCT
cana-5953	245	13	𝑝	𝑝	NOUN
cana-5953	245	14	)	)	PUNCT
cana-5953	245	15	+	+	CCONJ
cana-5953	245	16	𝛽[𝑆(𝑝	𝛽[𝑆(𝑝	ADJ
cana-5953	245	17	,	,	PUNCT
cana-5953	245	18	𝑝	𝑝	NOUN
cana-5953	245	19	,	,	PUNCT
cana-5953	245	20	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	245	21	)	)	PUNCT
cana-5953	245	22	+	+	CCONJ
cana-5953	245	23	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	245	24	,	,	PUNCT
cana-5953	245	25	𝑝	𝑝	NOUN
cana-5953	245	26	,	,	PUNCT
cana-5953	245	27	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	245	28	)	)	PUNCT
cana-5953	245	29	]	]	PUNCT
cana-5953	246	1	+	+	CCONJ
cana-5953	246	2	𝛾[𝑆(𝑝	𝛾[𝑆(𝑝	NOUN
cana-5953	246	3	,	,	PUNCT
cana-5953	246	4	𝑝	𝑝	NOUN
cana-5953	246	5	,	,	PUNCT
cana-5953	246	6	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	246	7	)	)	PUNCT
cana-5953	246	8	+	+	CCONJ
cana-5953	246	9	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	246	10	,	,	PUNCT
cana-5953	246	11	𝑝	𝑝	NOUN
cana-5953	246	12	,	,	PUNCT
cana-5953	246	13	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	246	14	)	)	PUNCT
cana-5953	246	15	]	]	PUNCT
cana-5953	247	1	=	=	SYM
cana-5953	247	2	𝛽[𝑆(𝑝	𝛽[𝑆(𝑝	NOUN
cana-5953	247	3	,	,	PUNCT
cana-5953	247	4	𝑝	𝑝	NOUN
cana-5953	247	5	,	,	PUNCT
cana-5953	247	6	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	247	7	)	)	PUNCT
cana-5953	248	1	+	+	CCONJ
cana-5953	248	2	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	248	3	,	,	PUNCT
cana-5953	248	4	𝑝	𝑝	NOUN
cana-5953	248	5	,	,	PUNCT
cana-5953	248	6	𝑝	𝑝	NOUN
cana-5953	248	7	)	)	PUNCT
cana-5953	248	8	]	]	PUNCT
cana-5953	249	1	+	+	CCONJ
cana-5953	249	2	𝛾[𝑆(𝑝	𝛾[𝑆(𝑝	NOUN
cana-5953	249	3	,	,	PUNCT
cana-5953	249	4	𝑝	𝑝	NOUN
cana-5953	249	5	,	,	PUNCT
cana-5953	249	6	𝑝	𝑝	NOUN
cana-5953	249	7	)	)	PUNCT
cana-5953	249	8	+	+	CCONJ
cana-5953	249	9	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	249	10	,	,	PUNCT
cana-5953	249	11	𝑝	𝑝	NOUN
cana-5953	249	12	,	,	PUNCT
cana-5953	249	13	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	249	14	)	)	PUNCT
cana-5953	249	15	]	]	PUNCT
cana-5953	250	1	=	=	PUNCT
cana-5953	250	2	(	(	PUNCT
cana-5953	250	3	𝛽	𝛽	PROPN
cana-5953	250	4	+	+	X
cana-5953	250	5	𝛾)𝑆(𝑝	𝛾)𝑆(𝑝	PROPN
cana-5953	250	6	,	,	PUNCT
cana-5953	250	7	𝑝	𝑝	NOUN
cana-5953	250	8	,	,	PUNCT
cana-5953	250	9	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	250	10	)	)	PUNCT
cana-5953	250	11	=	=	SYM
cana-5953	251	1	(	(	PUNCT
cana-5953	251	2	𝛽	𝛽	NOUN
cana-5953	251	3	+	+	CCONJ
cana-5953	251	4	𝛾)𝑆(𝑓𝑝	𝛾)𝑆(𝑓𝑝	PROPN
cana-5953	251	5	,	,	PUNCT
cana-5953	251	6	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	251	7	,	,	PUNCT
cana-5953	251	8	𝑝	𝑝	NOUN
cana-5953	251	9	)	)	PUNCT
cana-5953	251	10	.	.	PUNCT
cana-5953	252	1	since	since	SCONJ
cana-5953	252	2	,	,	PUNCT
cana-5953	252	3	𝛽	𝛽	PROPN
cana-5953	252	4	,	,	PUNCT
cana-5953	252	5	𝛾	𝛾	ADP
cana-5953	252	6	>	>	X
cana-5953	252	7	0	0	X
cana-5953	252	8	.	.	PUNCT
cana-5953	253	1	this	this	PRON
cana-5953	253	2	is	be	AUX
cana-5953	253	3	only	only	ADV
cana-5953	253	4	possible	possible	ADJ
cana-5953	253	5	if	if	SCONJ
cana-5953	253	6	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	253	7	,	,	PUNCT
cana-5953	253	8	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	253	9	,	,	PUNCT
cana-5953	253	10	𝑝	𝑝	NOUN
cana-5953	253	11	)	)	PUNCT
cana-5953	253	12	=	=	SYM
cana-5953	253	13	0	0	NUM
cana-5953	253	14	and	and	CCONJ
cana-5953	253	15	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	253	16	=	=	NOUN
cana-5953	253	17	𝑝.	𝑝.	PROPN
cana-5953	253	18	therefore	therefore	ADV
cana-5953	253	19	,	,	PUNCT
cana-5953	253	20	we	we	PRON
cana-5953	253	21	can	can	AUX
cana-5953	253	22	conclude	conclude	VERB
cana-5953	253	23	that	that	PRON
cana-5953	253	24	,	,	PUNCT
cana-5953	253	25	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	253	26	=	=	PUNCT
cana-5953	253	27	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	253	28	=	=	SYM
cana-5953	253	29	𝑝.	𝑝.	NOUN
cana-5953	253	30	hence	hence	ADV
cana-5953	253	31	,	,	PUNCT
cana-5953	253	32	𝑓	𝑓	PRON
cana-5953	253	33	and	and	CCONJ
cana-5953	253	34	𝑔	𝑔	AUX
cana-5953	253	35	have	have	VERB
cana-5953	253	36	a	a	DET
cana-5953	253	37	common	common	ADJ
cana-5953	253	38	fixed	fix	VERB
cana-5953	253	39	point	point	NOUN
cana-5953	253	40	in	in	ADP
cana-5953	253	41	𝑝	𝑝	PROPN
cana-5953	253	42	∈	∈	PROPN
cana-5953	253	43	𝑋.	𝑋.	PROPN
cana-5953	253	44	uniqueness	uniqueness	NOUN
cana-5953	253	45	:	:	PUNCT
cana-5953	253	46	suppose	suppose	VERB
cana-5953	253	47	another	another	DET
cana-5953	253	48	common	common	ADJ
cana-5953	253	49	fixed	fix	VERB
cana-5953	253	50	point	point	NOUN
cana-5953	253	51	𝑤	𝑤	ADP
cana-5953	253	52	∈	∈	NOUN
cana-5953	253	53	𝑋	𝑋	NOUN
cana-5953	253	54	exists	exist	VERB
cana-5953	253	55	i.e.	i.e.	ADV
cana-5953	253	56	𝑓𝑤	𝑓𝑤	ADP
cana-5953	253	57	=	=	PUNCT
cana-5953	253	58	𝑔𝑤	𝑔𝑤	NOUN
cana-5953	253	59	=	=	SYM
cana-5953	253	60	𝑤	𝑤	X
cana-5953	253	61	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	253	62	,	,	PUNCT
cana-5953	253	63	𝑝	𝑝	NOUN
cana-5953	253	64	,	,	PUNCT
cana-5953	253	65	𝑤	𝑤	ADJ
cana-5953	253	66	)	)	PUNCT
cana-5953	254	1	=	=	SYM
cana-5953	254	2	𝑆(𝑓𝑝	𝑆(𝑓𝑝	NOUN
cana-5953	254	3	,	,	PUNCT
cana-5953	254	4	𝑓𝑝	𝑓𝑝	PROPN
cana-5953	254	5	,	,	PUNCT
cana-5953	254	6	𝑔𝑤	𝑔𝑤	PROPN
cana-5953	254	7	)	)	PUNCT
cana-5953	254	8	≤	≤	NOUN
cana-5953	254	9	𝛼𝑆(𝑝	𝛼𝑆(𝑝	NUM
cana-5953	254	10	,	,	PUNCT
cana-5953	254	11	𝑝	𝑝	NOUN
cana-5953	254	12	,	,	PUNCT
cana-5953	254	13	𝑤	𝑤	ADJ
cana-5953	254	14	)	)	PUNCT
cana-5953	254	15	+	+	CCONJ
cana-5953	254	16	𝛽[𝑆(𝑝	𝛽[𝑆(𝑝	ADJ
cana-5953	254	17	,	,	PUNCT
cana-5953	254	18	𝑝	𝑝	NOUN
cana-5953	254	19	,	,	PUNCT
cana-5953	254	20	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	254	21	)	)	PUNCT
cana-5953	254	22	+	+	CCONJ
cana-5953	255	1	𝑆(𝑤	𝑆(𝑤	PROPN
cana-5953	255	2	,	,	PUNCT
cana-5953	255	3	𝑤	𝑤	ADP
cana-5953	255	4	,	,	PUNCT
cana-5953	255	5	𝑔𝑤	𝑔𝑤	PROPN
cana-5953	255	6	)	)	PUNCT
cana-5953	255	7	]	]	PUNCT
cana-5953	256	1	+	+	CCONJ
cana-5953	256	2	𝛾[𝑆(𝑝	𝛾[𝑆(𝑝	NOUN
cana-5953	256	3	,	,	PUNCT
cana-5953	256	4	𝑝	𝑝	NOUN
cana-5953	256	5	,	,	PUNCT
cana-5953	256	6	𝑔𝑤	𝑔𝑤	PROPN
cana-5953	256	7	)	)	PUNCT
cana-5953	257	1	+	+	CCONJ
cana-5953	257	2	𝑆(𝑤	𝑆(𝑤	X
cana-5953	257	3	,	,	PUNCT
cana-5953	257	4	𝑤	𝑤	ADP
cana-5953	257	5	,	,	PUNCT
cana-5953	257	6	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	257	7	)	)	PUNCT
cana-5953	257	8	]	]	PUNCT
cana-5953	257	9	substituting	substitute	VERB
cana-5953	257	10	𝑓𝑝	𝑓𝑝	NOUN
cana-5953	257	11	=	=	PUNCT
cana-5953	257	12	𝑔𝑝	𝑔𝑝	NOUN
cana-5953	257	13	=	=	PUNCT
cana-5953	257	14	𝑝	𝑝	PROPN
cana-5953	257	15	and	and	CCONJ
cana-5953	257	16	𝑓𝑤	𝑓𝑤	ADP
cana-5953	257	17	=	=	ADJ
cana-5953	257	18	𝑔𝑤	𝑔𝑤	NOUN
cana-5953	257	19	=	=	SYM
cana-5953	257	20	𝑤	𝑤	PROPN
cana-5953	257	21	,	,	PUNCT
cana-5953	257	22	we	we	PRON
cana-5953	257	23	obtain	obtain	VERB
cana-5953	257	24	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	257	25	,	,	PUNCT
cana-5953	257	26	𝑝	𝑝	NOUN
cana-5953	257	27	,	,	PUNCT
cana-5953	257	28	𝑤	𝑤	ADJ
cana-5953	257	29	)	)	PUNCT
cana-5953	257	30	=	=	SYM
cana-5953	257	31	𝛼𝑆(𝑝	𝛼𝑆(𝑝	NUM
cana-5953	257	32	,	,	PUNCT
cana-5953	257	33	𝑝	𝑝	NOUN
cana-5953	257	34	,	,	PUNCT
cana-5953	257	35	𝑤	𝑤	ADJ
cana-5953	257	36	)	)	PUNCT
cana-5953	258	1	+	+	CCONJ
cana-5953	258	2	𝛽[𝑆(𝑝	𝛽[𝑆(𝑝	ADJ
cana-5953	258	3	,	,	PUNCT
cana-5953	258	4	𝑝	𝑝	NOUN
cana-5953	258	5	,	,	PUNCT
cana-5953	258	6	𝑝	𝑝	NOUN
cana-5953	258	7	)	)	PUNCT
cana-5953	259	1	+	+	CCONJ
cana-5953	259	2	𝑆(𝑤	𝑆(𝑤	PROPN
cana-5953	259	3	,	,	PUNCT
cana-5953	259	4	𝑤	𝑤	ADP
cana-5953	259	5	,	,	PUNCT
cana-5953	259	6	𝑤	𝑤	ADP
cana-5953	259	7	)	)	PUNCT
cana-5953	259	8	]	]	PUNCT
cana-5953	260	1	+	+	CCONJ
cana-5953	260	2	𝛾[𝑆(𝑝	𝛾[𝑆(𝑝	NOUN
cana-5953	260	3	,	,	PUNCT
cana-5953	260	4	𝑝	𝑝	NOUN
cana-5953	260	5	,	,	PUNCT
cana-5953	260	6	𝑤	𝑤	ADP
cana-5953	260	7	)	)	PUNCT
cana-5953	260	8	+	+	CCONJ
cana-5953	261	1	𝑆(𝑤	𝑆(𝑤	PROPN
cana-5953	261	2	,	,	PUNCT
cana-5953	261	3	𝑤	𝑤	ADP
cana-5953	261	4	,	,	PUNCT
cana-5953	261	5	𝑝	𝑝	NOUN
cana-5953	261	6	)	)	PUNCT
cana-5953	261	7	]	]	PUNCT
cana-5953	262	1	=	=	SYM
cana-5953	262	2	𝛼𝑆(𝑝	𝛼𝑆(𝑝	NUM
cana-5953	262	3	,	,	PUNCT
cana-5953	262	4	𝑝	𝑝	NOUN
cana-5953	262	5	,	,	PUNCT
cana-5953	262	6	𝑤	𝑤	ADP
cana-5953	262	7	)	)	PUNCT
cana-5953	262	8	+	+	CCONJ
cana-5953	262	9	𝛾[𝑆(𝑝	𝛾[𝑆(𝑝	NOUN
cana-5953	262	10	,	,	PUNCT
cana-5953	262	11	𝑝	𝑝	NOUN
cana-5953	262	12	,	,	PUNCT
cana-5953	262	13	𝑤	𝑤	ADP
cana-5953	262	14	)	)	PUNCT
cana-5953	262	15	+	+	CCONJ
cana-5953	262	16	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	262	17	,	,	PUNCT
cana-5953	262	18	𝑝	𝑝	NOUN
cana-5953	262	19	,	,	PUNCT
cana-5953	262	20	𝑤	𝑤	NOUN
cana-5953	262	21	)	)	PUNCT
cana-5953	262	22	]	]	PUNCT
cana-5953	263	1	=	=	SYM
cana-5953	263	2	𝛼𝑆(𝑝	𝛼𝑆(𝑝	NUM
cana-5953	263	3	,	,	PUNCT
cana-5953	263	4	𝑝	𝑝	NOUN
cana-5953	263	5	,	,	PUNCT
cana-5953	263	6	𝑤	𝑤	ADP
cana-5953	263	7	)	)	PUNCT
cana-5953	263	8	+	+	CCONJ
cana-5953	263	9	2𝛾𝑆(𝑝	2𝛾𝑆(𝑝	NUM
cana-5953	263	10	,	,	PUNCT
cana-5953	263	11	𝑝	𝑝	NOUN
cana-5953	263	12	,	,	PUNCT
cana-5953	263	13	𝑤	𝑤	NOUN
cana-5953	263	14	)	)	PUNCT
cana-5953	263	15	=	=	SYM
cana-5953	263	16	(	(	PUNCT
cana-5953	263	17	𝛼	𝛼	X
cana-5953	263	18	+	+	SYM
cana-5953	263	19	2𝛾)𝑆(𝑝	2𝛾)𝑆(𝑝	NUM
cana-5953	263	20	,	,	PUNCT
cana-5953	263	21	𝑝	𝑝	NOUN
cana-5953	263	22	,	,	PUNCT
cana-5953	263	23	𝑤	𝑤	NOUN
cana-5953	263	24	)	)	PUNCT
cana-5953	263	25	.	.	PUNCT
cana-5953	264	1	since	since	SCONJ
cana-5953	264	2	,	,	PUNCT
cana-5953	264	3	𝛼	𝛼	X
cana-5953	264	4	,	,	PUNCT
cana-5953	264	5	𝛾	𝛾	ADP
cana-5953	264	6	>	>	X
cana-5953	264	7	0	0	NUM
cana-5953	264	8	,	,	PUNCT
cana-5953	264	9	the	the	DET
cana-5953	264	10	only	only	ADJ
cana-5953	264	11	possibility	possibility	NOUN
cana-5953	264	12	is	be	AUX
cana-5953	264	13	𝑆(𝑝	𝑆(𝑝	PROPN
cana-5953	264	14	,	,	PUNCT
cana-5953	264	15	𝑝	𝑝	NOUN
cana-5953	264	16	,	,	PUNCT
cana-5953	264	17	𝑤	𝑤	ADJ
cana-5953	264	18	)	)	PUNCT
cana-5953	264	19	=	=	SYM
cana-5953	265	1	0	0	X
cana-5953	265	2	.	.	PUNCT
cana-5953	266	1	hence	hence	ADV
cana-5953	266	2	,	,	PUNCT
cana-5953	266	3	𝑝	𝑝	NOUN
cana-5953	266	4	=	=	NOUN
cana-5953	266	5	𝑤.	𝑤.	NOUN
cana-5953	266	6	hence	hence	ADV
cana-5953	266	7	,	,	PUNCT
cana-5953	266	8	the	the	DET
cana-5953	266	9	common	common	ADJ
cana-5953	266	10	fixed	fix	VERB
cana-5953	266	11	point	point	NOUN
cana-5953	266	12	is	be	AUX
cana-5953	266	13	unique	unique	ADJ
cana-5953	266	14	,	,	PUNCT
cana-5953	266	15	and	and	CCONJ
cana-5953	266	16	the	the	DET
cana-5953	266	17	proof	proof	NOUN
cana-5953	266	18	is	be	AUX
cana-5953	266	19	complete	complete	ADJ
cana-5953	266	20	.	.	PUNCT
cana-5953	267	1	corollary	corollary	ADJ
cana-5953	267	2	4.4	4.4	NUM
cana-5953	267	3	let	let	VERB
cana-5953	267	4	(	(	PUNCT
cana-5953	267	5	𝑋	𝑋	PROPN
cana-5953	267	6	,	,	PUNCT
cana-5953	267	7	𝑆	𝑆	PROPN
cana-5953	267	8	)	)	PUNCT
cana-5953	267	9	be	be	AUX
cana-5953	267	10	a	a	DET
cana-5953	267	11	complete	complete	ADJ
cana-5953	267	12	rectangular	rectangular	ADJ
cana-5953	267	13	𝑆-metric	𝑆-metric	ADJ
cana-5953	267	14	space	space	NOUN
cana-5953	267	15	and	and	CCONJ
cana-5953	267	16	𝑓	𝑓	NOUN
cana-5953	267	17	,	,	PUNCT
cana-5953	267	18	𝑔	𝑔	NOUN
cana-5953	267	19	:	:	PUNCT
cana-5953	267	20	𝑋	𝑋	PROPN
cana-5953	267	21	→	→	SYM
cana-5953	267	22	𝑋	𝑋	PROPN
cana-5953	267	23	be	be	VERB
cana-5953	267	24	a	a	DET
cana-5953	267	25	self	self	NOUN
cana-5953	267	26	mappings	mapping	NOUN
cana-5953	267	27	such	such	ADJ
cana-5953	267	28	that	that	DET
cana-5953	267	29	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	267	30	,	,	PUNCT
cana-5953	267	31	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	267	32	,	,	PUNCT
cana-5953	267	33	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	267	34	)	)	PUNCT
cana-5953	267	35	≤	≤	NOUN
cana-5953	267	36	ℎ𝑆(𝑔𝑥	ℎ𝑆(𝑔𝑥	PUNCT
cana-5953	267	37	,	,	PUNCT
cana-5953	267	38	𝑔𝑥	𝑔𝑥	PROPN
cana-5953	267	39	,	,	PUNCT
cana-5953	267	40	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	267	41	)	)	PUNCT
cana-5953	267	42	,	,	PUNCT
cana-5953	267	43	where	where	SCONJ
cana-5953	267	44	ℎ	ℎ	X
cana-5953	267	45	∈	∈	PROPN
cana-5953	267	46	[	[	X
cana-5953	267	47	0,1	0,1	NUM
cana-5953	267	48	)	)	PUNCT
cana-5953	267	49	.	.	PUNCT
cana-5953	268	1	also	also	ADV
cana-5953	268	2	,	,	PUNCT
cana-5953	268	3	following	follow	VERB
cana-5953	268	4	conditions	condition	NOUN
cana-5953	268	5	holds	hold	VERB
cana-5953	268	6	•	•	ADP
cana-5953	268	7	𝑓(𝑋	𝑓(𝑋	NUM
cana-5953	268	8	)	)	PUNCT
cana-5953	268	9	⊆	⊆	NUM
cana-5953	268	10	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	268	11	)	)	PUNCT
cana-5953	268	12	,	,	PUNCT
cana-5953	268	13	•	•	ADP
cana-5953	268	14	if	if	SCONJ
cana-5953	268	15	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	268	16	)	)	PUNCT
cana-5953	268	17	is	be	AUX
cana-5953	268	18	complete	complete	ADJ
cana-5953	268	19	.	.	PUNCT
cana-5953	269	1	then	then	ADV
cana-5953	269	2	𝑓	𝑓	X
cana-5953	269	3	and	and	CCONJ
cana-5953	269	4	𝑔	𝑔	AUX
cana-5953	269	5	have	have	VERB
cana-5953	269	6	a	a	DET
cana-5953	269	7	unique	unique	ADJ
cana-5953	269	8	common	common	ADJ
cana-5953	269	9	fixed	fix	VERB
cana-5953	269	10	point	point	NOUN
cana-5953	269	11	in	in	ADP
cana-5953	269	12	𝑋.	𝑋.	PROPN
cana-5953	269	13	proof	proof	NOUN
cana-5953	269	14	.	.	PUNCT
cana-5953	270	1	the	the	DET
cana-5953	270	2	given	give	VERB
cana-5953	270	3	inequality	inequality	NOUN
cana-5953	270	4	is	be	AUX
cana-5953	270	5	a	a	DET
cana-5953	270	6	special	special	ADJ
cana-5953	270	7	case	case	NOUN
cana-5953	270	8	of	of	ADP
cana-5953	270	9	the	the	DET
cana-5953	270	10	condition	condition	NOUN
cana-5953	270	11	in	in	ADP
cana-5953	270	12	theorem	theorem	NOUN
cana-5953	270	13	(	(	PUNCT
cana-5953	270	14	4.2	4.2	NUM
cana-5953	270	15	)	)	PUNCT
cana-5953	270	16	,	,	PUNCT
cana-5953	270	17	obtained	obtain	VERB
cana-5953	270	18	by	by	ADP
cana-5953	270	19	choosing	choose	VERB
cana-5953	270	20	𝛼	𝛼	PRON
cana-5953	270	21	=	=	NOUN
cana-5953	270	22	ℎ	ℎ	NOUN
cana-5953	270	23	and	and	CCONJ
cana-5953	270	24	𝛽	𝛽	NOUN
cana-5953	270	25	=	=	NOUN
cana-5953	270	26	0	0	NUM
cana-5953	270	27	.	.	PUNCT
cana-5953	271	1	hence	hence	ADV
cana-5953	271	2	,	,	PUNCT
cana-5953	271	3	the	the	DET
cana-5953	271	4	result	result	NOUN
cana-5953	271	5	follows	follow	VERB
cana-5953	271	6	directly	directly	ADV
cana-5953	271	7	from	from	ADP
cana-5953	271	8	theorem	theorem	ADJ
cana-5953	271	9	(	(	PUNCT
cana-5953	271	10	4.2	4.2	NUM
cana-5953	271	11	)	)	PUNCT
cana-5953	271	12	.	.	PUNCT
cana-5953	272	1	corollary	corollary	ADJ
cana-5953	272	2	4.5	4.5	NUM
cana-5953	272	3	let	let	VERB
cana-5953	272	4	(	(	PUNCT
cana-5953	272	5	𝑋	𝑋	PROPN
cana-5953	272	6	,	,	PUNCT
cana-5953	272	7	𝑆	𝑆	PROPN
cana-5953	272	8	)	)	PUNCT
cana-5953	272	9	be	be	AUX
cana-5953	272	10	a	a	DET
cana-5953	272	11	complete	complete	ADJ
cana-5953	272	12	rectangular	rectangular	ADJ
cana-5953	272	13	𝑆-metric	𝑆-metric	ADJ
cana-5953	272	14	space	space	NOUN
cana-5953	272	15	and	and	CCONJ
cana-5953	272	16	𝑓	𝑓	NOUN
cana-5953	272	17	,	,	PUNCT
cana-5953	272	18	𝑔	𝑔	NOUN
cana-5953	272	19	:	:	PUNCT
cana-5953	272	20	𝑋	𝑋	PROPN
cana-5953	272	21	→	→	SYM
cana-5953	272	22	𝑋	𝑋	PROPN
cana-5953	272	23	be	be	VERB
cana-5953	272	24	a	a	DET
cana-5953	272	25	self	self	NOUN
cana-5953	272	26	mapping	mapping	NOUN
cana-5953	272	27	satisfies	satisfie	NOUN
cana-5953	272	28	the	the	DET
cana-5953	272	29	condition	condition	NOUN
cana-5953	272	30	,	,	PUNCT
cana-5953	272	31	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	272	32	,	,	PUNCT
cana-5953	272	33	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	272	34	,	,	PUNCT
cana-5953	272	35	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	272	36	)	)	PUNCT
cana-5953	272	37	≤	≤	NOUN
cana-5953	272	38	𝑞[𝑆(𝑔𝑥	𝑞[𝑆(𝑔𝑥	NOUN
cana-5953	272	39	,	,	PUNCT
cana-5953	272	40	𝑔𝑥	𝑔𝑥	INTJ
cana-5953	272	41	,	,	PUNCT
cana-5953	272	42	𝑓𝑦	𝑓𝑦	PROPN
cana-5953	272	43	)	)	PUNCT
cana-5953	272	44	+	+	CCONJ
cana-5953	272	45	𝑆(𝑔𝑦	𝑆(𝑔𝑦	ADJ
cana-5953	272	46	,	,	PUNCT
cana-5953	272	47	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	272	48	,	,	PUNCT
cana-5953	272	49	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	272	50	)	)	PUNCT
cana-5953	272	51	]	]	PUNCT
cana-5953	272	52	,	,	PUNCT
cana-5953	272	53	where	where	SCONJ
cana-5953	272	54	𝑞	𝑞	PROPN
cana-5953	272	55	∈	∈	PROPN
cana-5953	273	1	[	[	X
cana-5953	273	2	0	0	NUM
cana-5953	273	3	,	,	PUNCT
cana-5953	273	4	1	1	NUM
cana-5953	273	5	2	2	NUM
cana-5953	273	6	)	)	PUNCT
cana-5953	273	7	.	.	PUNCT
cana-5953	274	1	assume	assume	VERB
cana-5953	274	2	further	far	ADV
cana-5953	274	3	that	that	SCONJ
cana-5953	274	4	,	,	PUNCT
cana-5953	274	5	communications	communication	NOUN
cana-5953	274	6	on	on	ADP
cana-5953	274	7	applied	apply	VERB
cana-5953	274	8	nonlinear	nonlinear	ADJ
cana-5953	274	9	analysis	analysis	NOUN
cana-5953	274	10	issn	issn	NOUN
cana-5953	274	11	:	:	PUNCT
cana-5953	274	12	1074	1074	NUM
cana-5953	274	13	-	-	PUNCT
cana-5953	274	14	133x	133x	NUM
cana-5953	274	15	vol	vol	VERB
cana-5953	274	16	32	32	NUM
cana-5953	274	17	no	no	NOUN
cana-5953	274	18	.	.	PUNCT
cana-5953	275	1	10s	10	NOUN
cana-5953	275	2	(	(	PUNCT
cana-5953	275	3	2025	2025	NUM
cana-5953	275	4	)	)	PUNCT
cana-5953	275	5	3171	3171	NUM
cana-5953	276	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	276	2	•	•	NUM
cana-5953	276	3	𝑓(𝑋	𝑓(𝑋	PROPN
cana-5953	276	4	)	)	PUNCT
cana-5953	276	5	⊆	⊆	NUM
cana-5953	276	6	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	276	7	)	)	PUNCT
cana-5953	276	8	•	•	ADP
cana-5953	276	9	if	if	SCONJ
cana-5953	276	10	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	276	11	)	)	PUNCT
cana-5953	276	12	is	be	AUX
cana-5953	276	13	complete	complete	ADJ
cana-5953	276	14	subset	subset	NOUN
cana-5953	276	15	of	of	ADP
cana-5953	276	16	𝑋.	𝑋.	PROPN
cana-5953	276	17	then	then	ADV
cana-5953	276	18	there	there	PRON
cana-5953	276	19	exists	exist	VERB
cana-5953	276	20	a	a	DET
cana-5953	276	21	unique	unique	ADJ
cana-5953	276	22	fixed	fix	VERB
cana-5953	276	23	point	point	NOUN
cana-5953	276	24	in	in	ADP
cana-5953	276	25	𝑋	𝑋	PROPN
cana-5953	276	26	for	for	ADP
cana-5953	276	27	𝑓	𝑓	PROPN
cana-5953	276	28	and	and	CCONJ
cana-5953	276	29	𝑔.	𝑔.	PROPN
cana-5953	276	30	corollary	corollary	PROPN
cana-5953	276	31	4.6	4.6	NUM
cana-5953	276	32	let	let	VERB
cana-5953	276	33	(	(	PUNCT
cana-5953	276	34	𝑋	𝑋	PROPN
cana-5953	276	35	,	,	PUNCT
cana-5953	276	36	𝑆	𝑆	PROPN
cana-5953	276	37	)	)	PUNCT
cana-5953	276	38	be	be	AUX
cana-5953	276	39	a	a	DET
cana-5953	276	40	complete	complete	ADJ
cana-5953	276	41	rectangular	rectangular	ADJ
cana-5953	276	42	𝑆-metric	𝑆-metric	ADJ
cana-5953	276	43	space	space	NOUN
cana-5953	276	44	and	and	CCONJ
cana-5953	276	45	𝑓	𝑓	NOUN
cana-5953	276	46	,	,	PUNCT
cana-5953	276	47	𝑔	𝑔	NOUN
cana-5953	276	48	:	:	PUNCT
cana-5953	276	49	𝑋	𝑋	PROPN
cana-5953	276	50	→	→	SYM
cana-5953	276	51	𝑋	𝑋	PROPN
cana-5953	276	52	be	be	VERB
cana-5953	276	53	a	a	DET
cana-5953	276	54	self	self	NOUN
cana-5953	276	55	mapping	mapping	NOUN
cana-5953	276	56	satisfies	satisfie	NOUN
cana-5953	276	57	the	the	DET
cana-5953	276	58	condition	condition	NOUN
cana-5953	276	59	,	,	PUNCT
cana-5953	276	60	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	276	61	,	,	PUNCT
cana-5953	276	62	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	276	63	,	,	PUNCT
cana-5953	276	64	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	276	65	)	)	PUNCT
cana-5953	276	66	≤	≤	NOUN
cana-5953	276	67	𝛽[𝑆(𝑔𝑥	𝛽[𝑆(𝑔𝑥	ADV
cana-5953	276	68	,	,	PUNCT
cana-5953	276	69	𝑔𝑥	𝑔𝑥	INTJ
cana-5953	276	70	,	,	PUNCT
cana-5953	276	71	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	276	72	)	)	PUNCT
cana-5953	276	73	+	+	CCONJ
cana-5953	276	74	𝑆(𝑔𝑦	𝑆(𝑔𝑦	ADJ
cana-5953	276	75	,	,	PUNCT
cana-5953	276	76	𝑔𝑦	𝑔𝑦	PROPN
cana-5953	276	77	,	,	PUNCT
cana-5953	276	78	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	276	79	)	)	PUNCT
cana-5953	276	80	]	]	PUNCT
cana-5953	276	81	,	,	PUNCT
cana-5953	276	82	where	where	SCONJ
cana-5953	276	83	𝑞	𝑞	PROPN
cana-5953	276	84	∈	∈	PROPN
cana-5953	276	85	[	[	X
cana-5953	276	86	0	0	NUM
cana-5953	276	87	,	,	PUNCT
cana-5953	276	88	1	1	NUM
cana-5953	276	89	2	2	NUM
cana-5953	276	90	)	)	PUNCT
cana-5953	276	91	.	.	PUNCT
cana-5953	277	1	suppose	suppose	VERB
cana-5953	277	2	further	far	ADV
cana-5953	277	3	that	that	SCONJ
cana-5953	277	4	,	,	PUNCT
cana-5953	277	5	•	•	NUM
cana-5953	277	6	𝑓(𝑋	𝑓(𝑋	NUM
cana-5953	277	7	)	)	PUNCT
cana-5953	277	8	⊆	⊆	NUM
cana-5953	277	9	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	277	10	)	)	PUNCT
cana-5953	277	11	,	,	PUNCT
cana-5953	277	12	•	•	ADP
cana-5953	277	13	if	if	SCONJ
cana-5953	277	14	𝑔(𝑋	𝑔(𝑋	NOUN
cana-5953	277	15	)	)	PUNCT
cana-5953	277	16	is	be	AUX
cana-5953	277	17	complete	complete	ADJ
cana-5953	277	18	.	.	PUNCT
cana-5953	278	1	then	then	ADV
cana-5953	278	2	𝑓	𝑓	X
cana-5953	278	3	and	and	CCONJ
cana-5953	278	4	𝑔	𝑔	AUX
cana-5953	278	5	have	have	VERB
cana-5953	278	6	a	a	DET
cana-5953	278	7	unique	unique	ADJ
cana-5953	278	8	common	common	ADJ
cana-5953	278	9	fixed	fix	VERB
cana-5953	278	10	point	point	NOUN
cana-5953	278	11	in	in	ADP
cana-5953	278	12	𝑋.	𝑋.	PROPN
cana-5953	278	13	proof	proof	NOUN
cana-5953	278	14	.	.	PUNCT
cana-5953	279	1	this	this	DET
cana-5953	279	2	inequality	inequality	NOUN
cana-5953	279	3	results	result	VERB
cana-5953	279	4	from	from	ADP
cana-5953	279	5	theorem	theorem	ADJ
cana-5953	279	6	(	(	PUNCT
cana-5953	279	7	4.2	4.2	NUM
cana-5953	279	8	)	)	PUNCT
cana-5953	279	9	by	by	ADP
cana-5953	279	10	setting	set	VERB
cana-5953	279	11	𝛼	𝛼	PROPN
cana-5953	279	12	=	=	NOUN
cana-5953	279	13	0	0	NUM
cana-5953	279	14	.	.	PUNCT
cana-5953	280	1	the	the	DET
cana-5953	280	2	conclusion	conclusion	NOUN
cana-5953	280	3	then	then	ADV
cana-5953	280	4	directly	directly	ADV
cana-5953	280	5	follows	follow	VERB
cana-5953	280	6	from	from	ADP
cana-5953	280	7	theorem(4.2	theorem(4.2	NOUN
cana-5953	280	8	)	)	PUNCT
cana-5953	280	9	.	.	PUNCT
cana-5953	281	1	example	example	NOUN
cana-5953	282	1	4.7	4.7	NUM
cana-5953	282	2	let	let	VERB
cana-5953	282	3	𝑋	𝑋	NOUN
cana-5953	282	4	=	=	SYM
cana-5953	282	5	ℝ	ℝ	PROPN
cana-5953	282	6	and	and	CCONJ
cana-5953	282	7	define	define	VERB
cana-5953	282	8	the	the	DET
cana-5953	282	9	function	function	NOUN
cana-5953	282	10	𝑆	𝑆	PROPN
cana-5953	282	11	:	:	PUNCT
cana-5953	282	12	𝑋	𝑋	PROPN
cana-5953	282	13	×	×	NOUN
cana-5953	282	14	𝑋	𝑋	PROPN
cana-5953	282	15	×	×	NOUN
cana-5953	282	16	𝑋	𝑋	PROPN
cana-5953	282	17	→	→	SYM
cana-5953	282	18	ℝ+	ℝ+	PUNCT
cana-5953	282	19	by	by	ADP
cana-5953	282	20	𝑆(𝑥	𝑆(𝑥	NUM
cana-5953	282	21	,	,	PUNCT
cana-5953	282	22	𝑦	𝑦	NOUN
cana-5953	282	23	,	,	PUNCT
cana-5953	282	24	𝑧	𝑧	NOUN
cana-5953	282	25	)	)	PUNCT
cana-5953	282	26	=	=	SYM
cana-5953	282	27	𝑥𝑦	𝑥𝑦	NOUN
cana-5953	282	28	+	+	NUM
cana-5953	282	29	𝑧2	𝑧2	NOUN
cana-5953	282	30	for	for	ADP
cana-5953	282	31	all	all	DET
cana-5953	282	32	𝑥	𝑥	PROPN
cana-5953	282	33	,	,	PUNCT
cana-5953	282	34	𝑦	𝑦	PRON
cana-5953	282	35	∈	∈	NOUN
cana-5953	282	36	ℝ.	ℝ.	PROPN
cana-5953	282	37	then	then	ADV
cana-5953	282	38	(	(	PUNCT
cana-5953	282	39	𝑋	𝑋	PROPN
cana-5953	282	40	,	,	PUNCT
cana-5953	282	41	𝑆	𝑆	PROPN
cana-5953	282	42	)	)	PUNCT
cana-5953	282	43	forms	form	VERB
cana-5953	282	44	a	a	DET
cana-5953	282	45	𝑆-metric	𝑆-metric	ADJ
cana-5953	282	46	space	space	NOUN
cana-5953	282	47	.	.	PUNCT
cana-5953	283	1	now	now	ADV
cana-5953	283	2	,	,	PUNCT
cana-5953	283	3	define	define	VERB
cana-5953	283	4	the	the	DET
cana-5953	283	5	mappings	mapping	NOUN
cana-5953	283	6	𝑓	𝑓	PRON
cana-5953	283	7	:	:	PUNCT
cana-5953	283	8	𝑋	𝑋	PROPN
cana-5953	283	9	→	→	SYM
cana-5953	283	10	𝑋	𝑋	PROPN
cana-5953	283	11	as	as	SCONJ
cana-5953	283	12	follows	follow	VERB
cana-5953	283	13	𝑓(𝑥	𝑓(𝑥	NOUN
cana-5953	283	14	)	)	PUNCT
cana-5953	283	15	=	=	SYM
cana-5953	283	16	𝑥	𝑥	DET
cana-5953	283	17	4	4	NUM
cana-5953	283	18	and	and	CCONJ
cana-5953	283	19	𝑔(𝑥	𝑔(𝑥	NUM
cana-5953	283	20	)	)	PUNCT
cana-5953	284	1	=	=	SYM
cana-5953	284	2	𝑥	𝑥	DET
cana-5953	284	3	2	2	NUM
cana-5953	284	4	for	for	ADP
cana-5953	284	5	all	all	DET
cana-5953	284	6	𝑥	𝑥	DET
cana-5953	284	7	∈	∈	NOUN
cana-5953	284	8	[	[	X
cana-5953	284	9	0,1	0,1	NUM
cana-5953	284	10	]	]	PUNCT
cana-5953	284	11	.	.	PUNCT
cana-5953	285	1	we	we	PRON
cana-5953	285	2	then	then	ADV
cana-5953	285	3	compute	compute	VERB
cana-5953	285	4	𝑆(𝑓𝑥	𝑆(𝑓𝑥	ADJ
cana-5953	285	5	,	,	PUNCT
cana-5953	285	6	𝑓𝑥	𝑓𝑥	NOUN
cana-5953	285	7	,	,	PUNCT
cana-5953	285	8	𝑓𝑦	𝑓𝑦	NOUN
cana-5953	285	9	)	)	PUNCT
cana-5953	285	10	=	=	SYM
cana-5953	285	11	𝑆	𝑆	PROPN
cana-5953	285	12	(	(	PUNCT
cana-5953	285	13	𝑥	𝑥	PROPN
cana-5953	285	14	4	4	NUM
cana-5953	285	15	,	,	PUNCT
cana-5953	285	16	𝑥	𝑥	PROPN
cana-5953	285	17	4	4	NUM
cana-5953	285	18	,	,	PUNCT
cana-5953	285	19	𝑦	𝑦	NOUN
cana-5953	285	20	4	4	NUM
cana-5953	285	21	)	)	PUNCT
cana-5953	285	22	=	=	SYM
cana-5953	285	23	(	(	PUNCT
cana-5953	285	24	𝑥	𝑥	NOUN
cana-5953	285	25	4	4	NUM
cana-5953	285	26	)	)	PUNCT
cana-5953	285	27	(	(	PUNCT
cana-5953	285	28	𝑥	𝑥	PROPN
cana-5953	285	29	4	4	NUM
cana-5953	285	30	)	)	PUNCT
cana-5953	286	1	+	+	CCONJ
cana-5953	286	2	(	(	PUNCT
cana-5953	286	3	𝑦	𝑦	NOUN
cana-5953	286	4	4	4	NUM
cana-5953	286	5	)	)	SYM
cana-5953	286	6	2	2	NUM
cana-5953	286	7	=	=	SYM
cana-5953	286	8	𝑥2	𝑥2	NOUN
cana-5953	286	9	16	16	NUM
cana-5953	287	1	+	+	CCONJ
cana-5953	287	2	𝑦2	𝑦2	PROPN
cana-5953	287	3	16	16	NUM
cana-5953	287	4	=	=	SYM
cana-5953	287	5	1	1	NUM
cana-5953	287	6	16	16	NUM
cana-5953	287	7	(	(	PUNCT
cana-5953	287	8	𝑥2	𝑥2	NOUN
cana-5953	287	9	+	+	CCONJ
cana-5953	287	10	𝑦2	𝑦2	NOUN
cana-5953	287	11	)	)	PUNCT
cana-5953	287	12	=	=	SYM
cana-5953	287	13	1	1	NUM
cana-5953	287	14	4	4	NUM
cana-5953	287	15	(	(	PUNCT
cana-5953	287	16	1	1	NUM
cana-5953	287	17	4	4	NUM
cana-5953	287	18	(	(	PUNCT
cana-5953	287	19	𝑥2	𝑥2	NOUN
cana-5953	287	20	+	+	CCONJ
cana-5953	287	21	𝑦2	𝑦2	NOUN
cana-5953	287	22	)	)	PUNCT
cana-5953	287	23	)	)	PUNCT
cana-5953	288	1	=	=	SYM
cana-5953	288	2	1	1	NUM
cana-5953	288	3	4	4	NUM
cana-5953	288	4	(	(	PUNCT
cana-5953	288	5	𝑆(𝑔𝑥	𝑆(𝑔𝑥	X
cana-5953	288	6	,	,	PUNCT
cana-5953	288	7	𝑔𝑥	𝑔𝑥	PROPN
cana-5953	288	8	,	,	PUNCT
cana-5953	288	9	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	288	10	)	)	PUNCT
cana-5953	288	11	)	)	PUNCT
cana-5953	289	1	=	=	SYM
cana-5953	289	2	ℎ(𝑆(𝑔𝑥	ℎ(𝑆(𝑔𝑥	PROPN
cana-5953	289	3	,	,	PUNCT
cana-5953	289	4	𝑔𝑥	𝑔𝑥	PROPN
cana-5953	289	5	,	,	PUNCT
cana-5953	289	6	𝑔𝑦	𝑔𝑦	NOUN
cana-5953	289	7	)	)	PUNCT
cana-5953	289	8	)	)	PUNCT
cana-5953	289	9	,	,	PUNCT
cana-5953	289	10	where	where	SCONJ
cana-5953	289	11	ℎ	ℎ	X
cana-5953	289	12	=	=	SYM
cana-5953	289	13	1	1	NUM
cana-5953	289	14	4	4	NUM
cana-5953	289	15	<	<	SYM
cana-5953	289	16	1	1	NUM
cana-5953	289	17	.	.	PUNCT
cana-5953	290	1	thus	thus	ADV
cana-5953	290	2	𝑓	𝑓	DET
cana-5953	290	3	satisfies	satisfie	NOUN
cana-5953	290	4	all	all	DET
cana-5953	290	5	the	the	DET
cana-5953	290	6	conditions	condition	NOUN
cana-5953	290	7	of	of	ADP
cana-5953	290	8	corollary	corollary	ADJ
cana-5953	290	9	(	(	PUNCT
cana-5953	290	10	4.4	4.4	NUM
cana-5953	290	11	)	)	PUNCT
cana-5953	290	12	.	.	PUNCT
cana-5953	291	1	therefore	therefore	ADV
cana-5953	291	2	,	,	PUNCT
cana-5953	291	3	by	by	ADP
cana-5953	291	4	applying	apply	VERB
cana-5953	291	5	corollary(4.4	corollary(4.4	NOUN
cana-5953	291	6	)	)	PUNCT
cana-5953	291	7	.	.	PUNCT
cana-5953	292	1	it	it	PRON
cana-5953	292	2	follows	follow	VERB
cana-5953	292	3	that	that	SCONJ
cana-5953	292	4	𝑓	𝑓	PROPN
cana-5953	292	5	and	and	CCONJ
cana-5953	292	6	𝑔	𝑔	AUX
cana-5953	292	7	have	have	VERB
cana-5953	292	8	a	a	DET
cana-5953	292	9	unique	unique	ADJ
cana-5953	292	10	common	common	ADJ
cana-5953	292	11	fixed	fix	VERB
cana-5953	292	12	point	point	NOUN
cana-5953	292	13	in	in	ADP
cana-5953	292	14	𝑋.	𝑋.	PROPN
cana-5953	292	15	clearly	clearly	ADV
cana-5953	292	16	,	,	PUNCT
cana-5953	292	17	0	0	NUM
cana-5953	292	18	∈	∈	PROPN
cana-5953	292	19	𝑋	𝑋	NOUN
cana-5953	292	20	is	be	AUX
cana-5953	292	21	this	this	DET
cana-5953	292	22	unique	unique	ADJ
cana-5953	292	23	common	common	ADJ
cana-5953	292	24	fixed	fix	VERB
cana-5953	292	25	point	point	NOUN
cana-5953	292	26	.	.	PUNCT
cana-5953	293	1	5	5	X
cana-5953	293	2	.	.	X
cana-5953	293	3	conclusion	conclusion	NOUN
cana-5953	293	4	the	the	DET
cana-5953	293	5	results	result	NOUN
cana-5953	293	6	presented	present	VERB
cana-5953	293	7	in	in	ADP
cana-5953	293	8	this	this	DET
cana-5953	293	9	paper	paper	NOUN
cana-5953	293	10	contribute	contribute	NOUN
cana-5953	293	11	to	to	ADP
cana-5953	293	12	the	the	DET
cana-5953	293	13	ongoing	ongoing	ADJ
cana-5953	293	14	development	development	NOUN
cana-5953	293	15	of	of	ADP
cana-5953	293	16	fixed	fix	VERB
cana-5953	293	17	point	point	NOUN
cana-5953	293	18	theory	theory	NOUN
cana-5953	293	19	by	by	ADP
cana-5953	293	20	extending	extend	VERB
cana-5953	293	21	classical	classical	ADJ
cana-5953	293	22	results	result	NOUN
cana-5953	293	23	to	to	ADP
cana-5953	293	24	the	the	DET
cana-5953	293	25	broader	broad	ADJ
cana-5953	293	26	context	context	NOUN
cana-5953	293	27	of	of	ADP
cana-5953	293	28	rectangular	rectangular	ADJ
cana-5953	293	29	s	s	ADJ
cana-5953	293	30	-	-	ADJ
cana-5953	293	31	metric	metric	ADJ
cana-5953	293	32	spaces	space	NOUN
cana-5953	293	33	.	.	PUNCT
cana-5953	294	1	by	by	ADP
cana-5953	294	2	introducing	introduce	VERB
cana-5953	294	3	new	new	ADJ
cana-5953	294	4	common	common	ADJ
cana-5953	294	5	fixed	fix	VERB
cana-5953	294	6	point	point	NOUN
cana-5953	294	7	theorems	theorem	NOUN
cana-5953	294	8	for	for	ADP
cana-5953	294	9	weakly	weakly	ADJ
cana-5953	294	10	compatible	compatible	ADJ
cana-5953	294	11	mappings	mapping	NOUN
cana-5953	294	12	under	under	ADP
cana-5953	294	13	contractive	contractive	ADJ
cana-5953	294	14	conditions	condition	NOUN
cana-5953	294	15	,	,	PUNCT
cana-5953	294	16	we	we	PRON
cana-5953	294	17	provide	provide	VERB
cana-5953	294	18	a	a	DET
cana-5953	294	19	more	more	ADV
cana-5953	294	20	general	general	ADJ
cana-5953	294	21	framework	framework	NOUN
cana-5953	294	22	that	that	PRON
cana-5953	294	23	encompasses	encompass	VERB
cana-5953	294	24	several	several	ADJ
cana-5953	294	25	existing	exist	VERB
cana-5953	294	26	results	result	NOUN
cana-5953	294	27	as	as	ADP
cana-5953	294	28	special	special	ADJ
cana-5953	294	29	cases	case	NOUN
cana-5953	294	30	.	.	PUNCT
cana-5953	295	1	the	the	DET
cana-5953	295	2	illustrative	illustrative	ADJ
cana-5953	295	3	examples	example	NOUN
cana-5953	295	4	confirm	confirm	VERB
cana-5953	295	5	the	the	DET
cana-5953	295	6	applicability	applicability	NOUN
cana-5953	295	7	of	of	ADP
cana-5953	295	8	our	our	PRON
cana-5953	295	9	theorems	theorem	NOUN
cana-5953	295	10	and	and	CCONJ
cana-5953	295	11	highlight	highlight	VERB
cana-5953	295	12	the	the	DET
cana-5953	295	13	usefulness	usefulness	NOUN
cana-5953	295	14	of	of	ADP
cana-5953	295	15	communications	communication	NOUN
cana-5953	295	16	on	on	ADP
cana-5953	295	17	applied	apply	VERB
cana-5953	295	18	nonlinear	nonlinear	ADJ
cana-5953	295	19	analysis	analysis	NOUN
cana-5953	295	20	issn	issn	NOUN
cana-5953	295	21	:	:	PUNCT
cana-5953	295	22	1074	1074	NUM
cana-5953	295	23	-	-	PUNCT
cana-5953	295	24	133x	133x	NUM
cana-5953	295	25	vol	vol	VERB
cana-5953	295	26	32	32	NUM
cana-5953	295	27	no	no	NOUN
cana-5953	295	28	.	.	PUNCT
cana-5953	296	1	10s	10	NOUN
cana-5953	296	2	(	(	PUNCT
cana-5953	296	3	2025	2025	NUM
cana-5953	296	4	)	)	PUNCT
cana-5953	296	5	3172	3172	NUM
cana-5953	296	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5953	296	7	rectangular	rectangular	ADJ
cana-5953	296	8	s	s	ADJ
cana-5953	296	9	-	-	ADJ
cana-5953	296	10	metric	metric	ADJ
cana-5953	296	11	spaces	space	NOUN
cana-5953	296	12	in	in	ADP
cana-5953	296	13	analysing	analyse	VERB
cana-5953	296	14	complex	complex	ADJ
cana-5953	296	15	mapping	mapping	NOUN
cana-5953	296	16	behaviours	behaviour	NOUN
cana-5953	296	17	.	.	PUNCT
cana-5953	297	1	these	these	DET
cana-5953	297	2	findings	finding	NOUN
cana-5953	297	3	open	open	VERB
cana-5953	297	4	new	new	ADJ
cana-5953	297	5	avenues	avenue	NOUN
cana-5953	297	6	for	for	ADP
cana-5953	297	7	further	further	ADJ
cana-5953	297	8	research	research	NOUN
cana-5953	297	9	in	in	ADP
cana-5953	297	10	generalized	generalized	ADJ
cana-5953	297	11	metric	metric	ADJ
cana-5953	297	12	spaces	space	NOUN
cana-5953	297	13	and	and	CCONJ
cana-5953	297	14	their	their	PRON
cana-5953	297	15	applications	application	NOUN
cana-5953	297	16	.	.	PUNCT
cana-5953	298	1	references	reference	NOUN
cana-5953	298	2	[	[	X
cana-5953	298	3	1	1	X
cana-5953	298	4	]	]	PUNCT
cana-5953	298	5	o.	o.	PROPN
cana-5953	298	6	k.	k.	PROPN
cana-5953	298	7	adewale	adewale	PROPN
cana-5953	298	8	,	,	PUNCT
cana-5953	298	9	c.	c.	PROPN
cana-5953	298	10	iluno	iluno	PROPN
cana-5953	298	11	,	,	PUNCT
cana-5953	298	12	fixed	fix	VERB
cana-5953	298	13	point	point	NOUN
cana-5953	298	14	theorems	theorem	NOUN
cana-5953	298	15	on	on	ADP
cana-5953	298	16	rectangular	rectangular	ADJ
cana-5953	298	17	𝑆-metric	𝑆-metric	ADJ
cana-5953	298	18	spaces	space	NOUN
cana-5953	298	19	,	,	PUNCT
cana-5953	298	20	scientific	scientific	ADJ
cana-5953	298	21	african	african	ADJ
cana-5953	298	22	,	,	PUNCT
cana-5953	298	23	(	(	PUNCT
cana-5953	298	24	2022	2022	NUM
cana-5953	298	25	)	)	PUNCT
cana-5953	298	26	.	.	PUNCT
cana-5953	299	1	[	[	X
cana-5953	299	2	2	2	NUM
cana-5953	299	3	]	]	PUNCT
cana-5953	299	4	m.	m.	NOUN
cana-5953	299	5	abbas	abbas	PROPN
cana-5953	299	6	,	,	PUNCT
cana-5953	299	7	g.	g.	PROPN
cana-5953	299	8	jungck	jungck	PROPN
cana-5953	299	9	,	,	PUNCT
cana-5953	299	10	common	common	ADJ
cana-5953	299	11	fixed	fix	VERB
cana-5953	299	12	point	point	NOUN
cana-5953	299	13	results	result	NOUN
cana-5953	299	14	for	for	ADP
cana-5953	299	15	noncommuting	noncommute	VERB
cana-5953	299	16	mappings	mapping	NOUN
cana-5953	299	17	without	without	ADP
cana-5953	299	18	continuity	continuity	NOUN
cana-5953	299	19	in	in	ADP
cana-5953	299	20	cone	cone	NOUN
cana-5953	299	21	metric	metric	ADJ
cana-5953	299	22	spaces	space	NOUN
cana-5953	299	23	,	,	PUNCT
cana-5953	299	24	j.	j.	PROPN
cana-5953	299	25	math	math	PROPN
cana-5953	299	26	.	.	PUNCT
cana-5953	300	1	anal	anal	PROPN
cana-5953	300	2	.	.	PUNCT
cana-5953	300	3	appl	appl	PROPN
cana-5953	300	4	.	.	PROPN
cana-5953	300	5	,	,	PUNCT
cana-5953	300	6	341	341	NUM
cana-5953	300	7	(	(	PUNCT
cana-5953	300	8	2008	2008	NUM
cana-5953	300	9	)	)	PUNCT
cana-5953	300	10	,	,	PUNCT
cana-5953	300	11	416–420	416–420	NUM
cana-5953	300	12	.	.	PUNCT
cana-5953	301	1	[	[	X
cana-5953	301	2	3	3	NUM
cana-5953	301	3	]	]	PUNCT
cana-5953	301	4	m.	m.	NOUN
cana-5953	301	5	abbas	abbas	PROPN
cana-5953	301	6	,	,	PUNCT
cana-5953	301	7	b.	b.	PROPN
cana-5953	301	8	rhoades	rhoades	PROPN
cana-5953	301	9	,	,	PUNCT
cana-5953	301	10	fixed	fix	VERB
cana-5953	301	11	and	and	CCONJ
cana-5953	301	12	periodic	periodic	ADJ
cana-5953	301	13	point	point	NOUN
cana-5953	301	14	results	result	NOUN
cana-5953	301	15	in	in	ADP
cana-5953	301	16	cone	cone	NOUN
cana-5953	301	17	metric	metric	ADJ
cana-5953	301	18	spaces	space	NOUN
cana-5953	301	19	,	,	PUNCT
cana-5953	301	20	appl	appl	PROPN
cana-5953	301	21	.	.	PROPN
cana-5953	301	22	math	math	PROPN
cana-5953	301	23	.	.	PUNCT
cana-5953	302	1	lett	lett	PROPN
cana-5953	302	2	.	.	PROPN
cana-5953	303	1	,	,	PUNCT
cana-5953	303	2	22	22	NUM
cana-5953	303	3	(	(	PUNCT
cana-5953	303	4	2009	2009	NUM
cana-5953	303	5	)	)	PUNCT
cana-5953	303	6	,	,	PUNCT
cana-5953	303	7	511–515	511–515	NUM
cana-5953	303	8	.	.	PUNCT
cana-5953	304	1	[	[	X
cana-5953	304	2	4	4	X
cana-5953	304	3	]	]	PUNCT
cana-5953	304	4	s.	s.	PROPN
cana-5953	304	5	duraj	duraj	PROPN
cana-5953	304	6	,	,	PUNCT
cana-5953	304	7	s.	s.	PROPN
cana-5953	304	8	liftaj	liftaj	PROPN
cana-5953	304	9	,	,	PUNCT
cana-5953	304	10	a	a	DET
cana-5953	304	11	common	common	ADJ
cana-5953	304	12	fixed	fix	VERB
cana-5953	304	13	-	-	PUNCT
cana-5953	304	14	point	point	NOUN
cana-5953	304	15	theorem	theorem	NOUN
cana-5953	304	16	of	of	ADP
cana-5953	304	17	mappings	mapping	NOUN
cana-5953	304	18	on	on	ADP
cana-5953	304	19	s	s	NOUN
cana-5953	304	20	-	-	ADJ
cana-5953	304	21	metric	metric	ADJ
cana-5953	304	22	spaces	space	NOUN
cana-5953	304	23	,	,	PUNCT
cana-5953	304	24	asian	asian	ADJ
cana-5953	304	25	journal	journal	NOUN
cana-5953	304	26	of	of	ADP
cana-5953	304	27	probability	probability	NOUN
cana-5953	304	28	and	and	CCONJ
cana-5953	304	29	statistics	statistic	NOUN
cana-5953	304	30	,	,	PUNCT
cana-5953	304	31	20	20	NUM
cana-5953	304	32	(	(	PUNCT
cana-5953	304	33	2022	2022	NUM
cana-5953	304	34	)	)	PUNCT
cana-5953	304	35	(	(	PUNCT
cana-5953	304	36	2	2	NUM
cana-5953	304	37	)	)	PUNCT
cana-5953	304	38	,	,	PUNCT
cana-5953	304	39	40	40	NUM
cana-5953	304	40	-	-	SYM
cana-5953	304	41	45	45	NUM
cana-5953	304	42	.	.	PUNCT
cana-5953	305	1	[	[	X
cana-5953	305	2	5	5	X
cana-5953	305	3	]	]	PUNCT
cana-5953	305	4	g.	g.	PROPN
cana-5953	305	5	jungck	jungck	PROPN
cana-5953	305	6	,	,	PUNCT
cana-5953	305	7	commuting	commuting	NOUN
cana-5953	305	8	maps	map	NOUN
cana-5953	305	9	and	and	CCONJ
cana-5953	305	10	fixed	fix	VERB
cana-5953	305	11	points	point	NOUN
cana-5953	305	12	,	,	PUNCT
cana-5953	305	13	amer	amer	PROPN
cana-5953	305	14	.	.	PROPN
cana-5953	305	15	math	math	PROPN
cana-5953	305	16	.	.	PUNCT
cana-5953	306	1	monthly	monthly	ADJ
cana-5953	306	2	,	,	PUNCT
cana-5953	306	3	83	83	NUM
cana-5953	306	4	(	(	PUNCT
cana-5953	306	5	1976	1976	NUM
cana-5953	306	6	)	)	PUNCT
cana-5953	306	7	,	,	PUNCT
cana-5953	306	8	261–263	261–263	NUM
cana-5953	306	9	.	.	PUNCT
cana-5953	307	1	[	[	X
cana-5953	307	2	6	6	NUM
cana-5953	307	3	]	]	PUNCT
cana-5953	307	4	g.	g.	PROPN
cana-5953	307	5	jungck	jungck	PROPN
cana-5953	307	6	,	,	PUNCT
cana-5953	307	7	compatible	compatible	ADJ
cana-5953	307	8	mappings	mapping	NOUN
cana-5953	307	9	and	and	CCONJ
cana-5953	307	10	common	common	ADJ
cana-5953	307	11	fixed	fix	VERB
cana-5953	307	12	points	point	NOUN
cana-5953	307	13	,	,	PUNCT
cana-5953	307	14	internat	internat	PROPN
cana-5953	307	15	.	.	PUNCT
cana-5953	308	1	j.	j.	PROPN
cana-5953	308	2	math	math	PROPN
cana-5953	308	3	.	.	PUNCT
cana-5953	309	1	math	math	NOUN
cana-5953	309	2	.	.	PUNCT
cana-5953	310	1	sci	sci	PROPN
cana-5953	310	2	.	.	PROPN
cana-5953	310	3	,	,	PUNCT
cana-5953	310	4	9(4	9(4	NUM
cana-5953	310	5	)	)	PUNCT
cana-5953	310	6	(	(	PUNCT
cana-5953	310	7	1986	1986	NUM
cana-5953	310	8	)	)	PUNCT
cana-5953	310	9	,	,	PUNCT
cana-5953	310	10	771–779	771–779	NUM
cana-5953	310	11	.	.	PUNCT
cana-5953	311	1	[	[	X
cana-5953	311	2	7	7	X
cana-5953	311	3	]	]	X
cana-5953	311	4	g.	g.	PROPN
cana-5953	311	5	jungck	jungck	PROPN
cana-5953	311	6	,	,	PUNCT
cana-5953	311	7	common	common	ADJ
cana-5953	311	8	fixed	fix	VERB
cana-5953	311	9	points	point	NOUN
cana-5953	311	10	for	for	ADP
cana-5953	311	11	noncontinuous	noncontinuous	ADJ
cana-5953	311	12	nonself	nonself	PROPN
cana-5953	311	13	maps	map	NOUN
cana-5953	311	14	on	on	ADP
cana-5953	311	15	non	non	ADJ
cana-5953	311	16	-	-	ADJ
cana-5953	311	17	metric	metric	ADJ
cana-5953	311	18	spaces	space	NOUN
cana-5953	311	19	,	,	PUNCT
cana-5953	311	20	far	far	PROPN
cana-5953	311	21	east	east	PROPN
cana-5953	311	22	j.	j.	PROPN
cana-5953	311	23	math	math	PROPN
cana-5953	311	24	.	.	PUNCT
cana-5953	312	1	sci	sci	PROPN
cana-5953	312	2	.	.	PUNCT
cana-5953	312	3	(	(	PUNCT
cana-5953	312	4	fjms	fjms	PROPN
cana-5953	312	5	)	)	PUNCT
cana-5953	312	6	,	,	PUNCT
cana-5953	312	7	4	4	NUM
cana-5953	312	8	(	(	PUNCT
cana-5953	312	9	1996	1996	NUM
cana-5953	312	10	)	)	PUNCT
cana-5953	312	11	,	,	PUNCT
cana-5953	312	12	199–215	199–215	NUM
cana-5953	312	13	.	.	PUNCT
cana-5953	313	1	[	[	X
cana-5953	313	2	8	8	NUM
cana-5953	313	3	]	]	X
cana-5953	313	4	g.	g.	PROPN
cana-5953	313	5	jungck	jungck	PROPN
cana-5953	313	6	,	,	PUNCT
cana-5953	313	7	b.e	b.e	PROPN
cana-5953	313	8	.	.	PROPN
cana-5953	313	9	rhoades	rhoade	NOUN
cana-5953	313	10	,	,	PUNCT
cana-5953	313	11	fixed	fix	VERB
cana-5953	313	12	point	point	NOUN
cana-5953	313	13	for	for	ADP
cana-5953	313	14	set	set	NOUN
cana-5953	313	15	-	-	PUNCT
cana-5953	313	16	valued	value	VERB
cana-5953	313	17	functions	function	NOUN
cana-5953	313	18	without	without	ADP
cana-5953	313	19	continuity	continuity	NOUN
cana-5953	313	20	,	,	PUNCT
cana-5953	313	21	indian	indian	ADJ
cana-5953	313	22	j.	j.	PROPN
cana-5953	313	23	pure	pure	PROPN
cana-5953	313	24	appl	appl	PROPN
cana-5953	313	25	.	.	PUNCT
cana-5953	313	26	math	math	PROPN
cana-5953	313	27	.	.	PUNCT
cana-5953	313	28	,	,	PUNCT
cana-5953	313	29	29(3	29(3	NUM
cana-5953	313	30	)	)	PUNCT
cana-5953	313	31	(	(	PUNCT
cana-5953	313	32	1998	1998	NUM
cana-5953	313	33	)	)	PUNCT
cana-5953	313	34	,	,	PUNCT
cana-5953	313	35	227–238	227–238	NUM
cana-5953	313	36	.	.	PUNCT
cana-5953	314	1	[	[	X
cana-5953	314	2	9	9	NUM
cana-5953	314	3	]	]	PUNCT
cana-5953	314	4	v.	v.	X
cana-5953	314	5	ozturk	ozturk	PROPN
cana-5953	314	6	d.	d.	PROPN
cana-5953	314	7	turkoglu	turkoglu	PROPN
cana-5953	314	8	,	,	PUNCT
cana-5953	314	9	common	common	ADJ
cana-5953	314	10	fixed	fix	VERB
cana-5953	314	11	point	point	NOUN
cana-5953	314	12	theorems	theorem	NOUN
cana-5953	314	13	for	for	ADP
cana-5953	314	14	mappings	mapping	NOUN
cana-5953	314	15	satisfying	satisfy	VERB
cana-5953	314	16	(	(	PUNCT
cana-5953	314	17	e.a)-property	e.a)-property	NOUN
cana-5953	314	18	in	in	ADP
cana-5953	314	19	b	b	NOUN
cana-5953	314	20	-	-	ADJ
cana-5953	314	21	metric	metric	ADJ
cana-5953	314	22	spaces	space	NOUN
cana-5953	314	23	,	,	PUNCT
cana-5953	314	24	j.	j.	PROPN
cana-5953	314	25	nonlinear	nonlinear	PROPN
cana-5953	314	26	sci	sci	PROPN
cana-5953	314	27	.	.	PUNCT
cana-5953	314	28	appl	appl	PROPN
cana-5953	314	29	.	.	PROPN
cana-5953	314	30	,	,	PUNCT
cana-5953	314	31	8	8	NUM
cana-5953	314	32	(	(	PUNCT
cana-5953	314	33	2015	2015	NUM
cana-5953	314	34	)	)	PUNCT
cana-5953	314	35	,	,	PUNCT
cana-5953	314	36	1127–1133	1127–1133	NUM
cana-5953	314	37	.	.	PUNCT
cana-5953	315	1	[	[	X
cana-5953	315	2	10	10	NUM
cana-5953	315	3	]	]	X
cana-5953	315	4	p.	p.	PROPN
cana-5953	315	5	prajapati	prajapati	PROPN
cana-5953	315	6	,	,	PUNCT
cana-5953	315	7	common	common	ADJ
cana-5953	315	8	fixed	fix	VERB
cana-5953	315	9	point	point	NOUN
cana-5953	315	10	theorem	theorem	NOUN
cana-5953	315	11	involving	involve	VERB
cana-5953	315	12	contractive	contractive	ADJ
cana-5953	315	13	conditions	condition	NOUN
cana-5953	315	14	of	of	ADP
cana-5953	315	15	rational	rational	ADJ
cana-5953	315	16	type	type	NOUN
cana-5953	315	17	in	in	ADP
cana-5953	315	18	dislocated	dislocated	ADJ
cana-5953	315	19	quasi	quasi	ADJ
cana-5953	315	20	-	-	ADJ
cana-5953	315	21	metric	metric	ADJ
cana-5953	315	22	space	space	NOUN
cana-5953	315	23	,	,	PUNCT
cana-5953	315	24	mathematical	mathematical	ADJ
cana-5953	315	25	analysis	analysis	NOUN
cana-5953	315	26	and	and	CCONJ
cana-5953	315	27	its	its	PRON
cana-5953	315	28	contemporary	contemporary	ADJ
cana-5953	315	29	applications	application	NOUN
cana-5953	315	30	,	,	PUNCT
cana-5953	315	31	6	6	NUM
cana-5953	315	32	(	(	PUNCT
cana-5953	315	33	2024	2024	NUM
cana-5953	315	34	)	)	PUNCT
cana-5953	315	35	(	(	PUNCT
cana-5953	315	36	4	4	NUM
cana-5953	315	37	)	)	PUNCT
cana-5953	315	38	,	,	PUNCT
cana-5953	315	39	01	01	NUM
cana-5953	315	40	-	-	SYM
cana-5953	315	41	21	21	NUM
cana-5953	315	42	.	.	PUNCT
cana-5953	316	1	[	[	X
cana-5953	316	2	11	11	NUM
cana-5953	316	3	]	]	PUNCT
cana-5953	316	4	m.	m.	NOUN
cana-5953	316	5	saadi	saadi	NOUN
cana-5953	316	6	,	,	PUNCT
cana-5953	316	7	t.	t.	PROPN
cana-5953	316	8	hamaizia	hamaizia	PROPN
cana-5953	316	9	,	,	PUNCT
cana-5953	316	10	multivalued	multivalued	ADJ
cana-5953	316	11	common	common	ADJ
cana-5953	316	12	fixed	fix	VERB
cana-5953	316	13	points	point	NOUN
cana-5953	316	14	theorem	theorem	VERB
cana-5953	316	15	in	in	ADP
cana-5953	316	16	complex	complex	ADJ
cana-5953	316	17	b	b	X
cana-5953	316	18	-	-	PUNCT
cana-5953	316	19	metric	metric	ADJ
cana-5953	316	20	spaces	space	NOUN
cana-5953	316	21	,	,	PUNCT
cana-5953	316	22	mathematics	mathematic	NOUN
cana-5953	316	23	,	,	PUNCT
cana-5953	316	24	mdpi	mdpi	PROPN
cana-5953	316	25	,	,	PUNCT
cana-5953	316	26	11	11	NUM
cana-5953	316	27	(	(	PUNCT
cana-5953	316	28	2023	2023	NUM
cana-5953	316	29	)	)	PUNCT
cana-5953	316	30	,	,	PUNCT
cana-5953	316	31	01	01	NUM
cana-5953	316	32	-	-	SYM
cana-5953	316	33	11	11	NUM
cana-5953	316	34	.	.	PUNCT
cana-5953	317	1	[	[	X
cana-5953	317	2	12	12	NUM
cana-5953	317	3	]	]	X
cana-5953	317	4	g.	g.	PROPN
cana-5953	317	5	s.	s.	PROPN
cana-5953	317	6	saluja	saluja	PROPN
cana-5953	317	7	,	,	PUNCT
cana-5953	317	8	some	some	DET
cana-5953	317	9	fixed	fix	VERB
cana-5953	317	10	point	point	NOUN
cana-5953	317	11	theorems	theorem	NOUN
cana-5953	317	12	for	for	ADP
cana-5953	317	13	weak	weak	ADJ
cana-5953	317	14	contraction	contraction	NOUN
cana-5953	317	15	mappings	mapping	NOUN
cana-5953	317	16	in	in	ADP
cana-5953	317	17	𝑆-metric	𝑆-metric	ADJ
cana-5953	317	18	spaces	space	NOUN
cana-5953	317	19	,	,	PUNCT
cana-5953	317	20	jnanabha	jnanabha	PROPN
cana-5953	317	21	,	,	PUNCT
cana-5953	317	22	50(1	50(1	NUM
cana-5953	317	23	)	)	PUNCT
cana-5953	317	24	(	(	PUNCT
cana-5953	317	25	2020	2020	NUM
cana-5953	317	26	)	)	PUNCT
cana-5953	317	27	,	,	PUNCT
cana-5953	317	28	20–26	20–26	NUM
cana-5953	317	29	.	.	PUNCT
cana-5953	318	1	[	[	X
cana-5953	318	2	13	13	NUM
cana-5953	318	3	]	]	X
cana-5953	318	4	s.	s.	PROPN
cana-5953	318	5	sedghi	sedghi	PROPN
cana-5953	318	6	,	,	PUNCT
cana-5953	318	7	n.	n.	PROPN
cana-5953	318	8	shobe	shobe	PROPN
cana-5953	318	9	,	,	PUNCT
cana-5953	318	10	m.	m.	PROPN
cana-5953	318	11	ali	ali	PROPN
cana-5953	318	12	,	,	PUNCT
cana-5953	318	13	a	a	DET
cana-5953	318	14	generalization	generalization	NOUN
cana-5953	318	15	of	of	ADP
cana-5953	318	16	metric	metric	ADJ
cana-5953	318	17	spaces	space	NOUN
cana-5953	318	18	,	,	PUNCT
cana-5953	318	19	mathematical	mathematical	ADJ
cana-5953	318	20	communications	communication	NOUN
cana-5953	318	21	,	,	PUNCT
cana-5953	318	22	17	17	NUM
cana-5953	318	23	(	(	PUNCT
cana-5953	318	24	2012	2012	NUM
cana-5953	318	25	)	)	PUNCT
cana-5953	318	26	,	,	PUNCT
cana-5953	318	27	39–50	39–50	NOUN
cana-5953	318	28	.	.	PUNCT
cana-5953	319	1	[	[	X
cana-5953	319	2	14	14	NUM
cana-5953	319	3	]	]	X
cana-5953	319	4	s.	s.	PROPN
cana-5953	319	5	sedghi	sedghi	PROPN
cana-5953	319	6	,	,	PUNCT
cana-5953	319	7	n.	n.	PROPN
cana-5953	319	8	shobe	shobe	PROPN
cana-5953	319	9	,	,	PUNCT
cana-5953	319	10	a.	a.	PROPN
cana-5953	319	11	aliouche	aliouche	PROPN
cana-5953	319	12	,	,	PUNCT
cana-5953	319	13	a	a	DET
cana-5953	319	14	generalization	generalization	NOUN
cana-5953	319	15	of	of	ADP
cana-5953	319	16	fixed	fix	VERB
cana-5953	319	17	point	point	NOUN
cana-5953	319	18	theorem	theorem	VERB
cana-5953	319	19	in	in	ADP
cana-5953	319	20	𝑆-metric	𝑆-metric	ADJ
cana-5953	319	21	spaces	space	NOUN
cana-5953	319	22	,	,	PUNCT
cana-5953	319	23	mat	mat	PROPN
cana-5953	319	24	.	.	PROPN
cana-5953	319	25	vesnik	vesnik	PROPN
cana-5953	319	26	,	,	PUNCT
cana-5953	319	27	64	64	NUM
cana-5953	319	28	(	(	PUNCT
cana-5953	319	29	2012	2012	NUM
cana-5953	319	30	)	)	PUNCT
cana-5953	319	31	,	,	PUNCT
cana-5953	319	32	258–266	258–266	NUM
cana-5953	319	33	.	.	PUNCT
cana-5953	320	1	[	[	X
cana-5953	320	2	15	15	NUM
cana-5953	320	3	]	]	X
cana-5953	320	4	s.	s.	PROPN
cana-5953	320	5	sessa	sessa	PROPN
cana-5953	320	6	,	,	PUNCT
cana-5953	320	7	on	on	ADP
cana-5953	320	8	a	a	DET
cana-5953	320	9	weak	weak	ADJ
cana-5953	320	10	commutativity	commutativity	NOUN
cana-5953	320	11	condition	condition	NOUN
cana-5953	320	12	of	of	ADP
cana-5953	320	13	mappings	mapping	NOUN
cana-5953	320	14	in	in	ADP
cana-5953	320	15	fixed	fix	VERB
cana-5953	320	16	point	point	NOUN
cana-5953	320	17	consideration	consideration	NOUN
cana-5953	320	18	,	,	PUNCT
cana-5953	320	19	publ	publ	NOUN
cana-5953	320	20	.	.	PUNCT
cana-5953	321	1	inst	inst	PROPN
cana-5953	321	2	.	.	PUNCT
cana-5953	321	3	math	math	NOUN
cana-5953	321	4	.	.	PUNCT
cana-5953	322	1	(	(	PUNCT
cana-5953	322	2	beograd	beograd	PROPN
cana-5953	322	3	)	)	PUNCT
cana-5953	322	4	(	(	PUNCT
cana-5953	322	5	n.s	n.s	PROPN
cana-5953	322	6	.	.	PROPN
cana-5953	322	7	)	)	PUNCT
cana-5953	322	8	,	,	PUNCT
cana-5953	322	9	32	32	NUM
cana-5953	322	10	(	(	PUNCT
cana-5953	322	11	1982	1982	NUM
cana-5953	322	12	)	)	PUNCT
cana-5953	322	13	,	,	PUNCT
cana-5953	322	14	149–153	149–153	NUM
cana-5953	322	15	.	.	PUNCT
cana-5953	323	1	[	[	X
cana-5953	323	2	16	16	NUM
cana-5953	323	3	]	]	X
cana-5953	323	4	l.	l.	PROPN
cana-5953	323	5	wangwe	wangwe	PROPN
cana-5953	323	6	,	,	PUNCT
cana-5953	323	7	s.	s.	PROPN
cana-5953	323	8	kumar	kumar	PROPN
cana-5953	323	9	,	,	PUNCT
cana-5953	323	10	common	common	ADJ
cana-5953	323	11	fixed	fix	VERB
cana-5953	323	12	point	point	NOUN
cana-5953	323	13	theorems	theorem	NOUN
cana-5953	323	14	under	under	ADP
cana-5953	323	15	implicit	implicit	ADJ
cana-5953	323	16	contractive	contractive	ADJ
cana-5953	323	17	condition	condition	NOUN
cana-5953	323	18	using	use	VERB
cana-5953	323	19	e.	e.	PROPN
cana-5953	323	20	a.	a.	NOUN
cana-5953	323	21	property	property	NOUN
cana-5953	323	22	on	on	ADP
cana-5953	323	23	metric	metric	ADJ
cana-5953	323	24	-	-	PUNCT
cana-5953	323	25	like	like	ADJ
cana-5953	323	26	spaces	space	NOUN
cana-5953	323	27	employing	employ	VERB
cana-5953	323	28	an	an	DET
cana-5953	323	29	arbitrary	arbitrary	ADJ
cana-5953	323	30	binary	binary	ADJ
cana-5953	323	31	relation	relation	NOUN
cana-5953	323	32	with	with	ADP
cana-5953	323	33	some	some	DET
cana-5953	323	34	application	application	NOUN
cana-5953	323	35	,	,	PUNCT
cana-5953	323	36	int	int	NOUN
cana-5953	323	37	.	.	PUNCT
cana-5953	324	1	j.	j.	PROPN
cana-5953	324	2	nonlinear	nonlinear	PROPN
cana-5953	324	3	anal	anal	PROPN
cana-5953	324	4	.	.	PUNCT
cana-5953	325	1	appl	appl	PROPN
cana-5953	325	2	.	.	PROPN
cana-5953	325	3	,	,	PUNCT
cana-5953	325	4	13	13	NUM
cana-5953	325	5	(	(	PUNCT
cana-5953	325	6	2022	2022	NUM
cana-5953	325	7	)	)	PUNCT
cana-5953	325	8	2	2	NUM
cana-5953	325	9	,	,	PUNCT
cana-5953	325	10	2325–2346	2325–2346	NUM
cana-5953	325	11	.	.	PUNCT