id	sid	tid	token	lemma	pos
cana-2868	1	1	communications	communication	NOUN
cana-2868	1	2	on	on	ADP
cana-2868	1	3	applied	apply	VERB
cana-2868	1	4	nonlinear	nonlinear	ADJ
cana-2868	1	5	analysis	analysis	NOUN
cana-2868	1	6	issn	issn	NOUN
cana-2868	1	7	:	:	PUNCT
cana-2868	1	8	1074	1074	NUM
cana-2868	1	9	-	-	PUNCT
cana-2868	1	10	133x	133x	NUM
cana-2868	1	11	vol	vol	NOUN
cana-2868	1	12	32	32	NUM
cana-2868	1	13	no	no	NOUN
cana-2868	1	14	.	.	PUNCT
cana-2868	2	1	4s	4s	NUM
cana-2868	2	2	(	(	PUNCT
cana-2868	2	3	2025	2025	NUM
cana-2868	2	4	)	)	PUNCT
cana-2868	2	5	513	513	NUM
cana-2868	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2868	2	7	fixed	fix	VERB
cana-2868	2	8	points	point	NOUN
cana-2868	2	9	of	of	ADP
cana-2868	2	10	generalized	generalized	ADJ
cana-2868	2	11	geraghty	geraghty	ADJ
cana-2868	2	12	ciric	ciric	ADJ
cana-2868	2	13	-rational	-rational	ADJ
cana-2868	2	14	type	type	NOUN
cana-2868	2	15	contraction	contraction	NOUN
cana-2868	2	16	in	in	ADP
cana-2868	2	17	bmetric	bmetric	ADJ
cana-2868	2	18	spaces	space	NOUN
cana-2868	2	19	dr	dr	PROPN
cana-2868	2	20	.	.	PROPN
cana-2868	2	21	p.harikrishna1	p.harikrishna1	PROPN
cana-2868	2	22	,	,	PUNCT
cana-2868	2	23	dr	dr	PROPN
cana-2868	2	24	.	.	PROPN
cana-2868	2	25	kusuma	kusuma	PROPN
cana-2868	2	26	tummala2	tummala2	PROPN
cana-2868	2	27	,	,	PUNCT
cana-2868	2	28	dr	dr	PROPN
cana-2868	2	29	.	.	PROPN
cana-2868	2	30	v.sree	v.sree	PROPN
cana-2868	2	31	ramani3	ramani3	PROPN
cana-2868	2	32	,	,	PUNCT
cana-2868	2	33	dr	dr	PROPN
cana-2868	2	34	.	.	PROPN
cana-2868	2	35	y.jayababu4	y.jayababu4	PROPN
cana-2868	2	36	,	,	PUNCT
cana-2868	3	1	dr	dr	PROPN
cana-2868	3	2	.	.	PROPN
cana-2868	3	3	t.	t.	PROPN
cana-2868	3	4	nageswara	nageswara	PROPN
cana-2868	3	5	rao5	rao5	PROPN
cana-2868	3	6	1	1	NUM
cana-2868	3	7	associate	associate	NOUN
cana-2868	3	8	professor	professor	NOUN
cana-2868	3	9	,	,	PUNCT
cana-2868	3	10	vignan	vignan	NOUN
cana-2868	3	11	’s	’s	PART
cana-2868	3	12	institute	institute	PROPN
cana-2868	3	13	of	of	ADP
cana-2868	3	14	information	information	NOUN
cana-2868	3	15	technology	technology	PROPN
cana-2868	3	16	,	,	PUNCT
cana-2868	3	17	visakhapatnam	visakhapatnam	PROPN
cana-2868	3	18	,	,	PUNCT
cana-2868	3	19	,	,	PUNCT
cana-2868	3	20	andhrapradesh	andhrapradesh	PROPN
cana-2868	3	21	,	,	PUNCT
cana-2868	3	22	india	india	PROPN
cana-2868	3	23	.	.	PUNCT
cana-2868	3	24	email	email	NOUN
cana-2868	3	25	:	:	PUNCT
cana-2868	3	26	phk.2003@gmail.com	phk.2003@gmail.com	X
cana-2868	4	1	2assistant	2assistant	NUM
cana-2868	4	2	professor	professor	NOUN
cana-2868	4	3	,	,	PUNCT
cana-2868	4	4	department	department	NOUN
cana-2868	4	5	of	of	ADP
cana-2868	4	6	humanities	humanity	NOUN
cana-2868	4	7	and	and	CCONJ
cana-2868	4	8	sciences	science	NOUN
cana-2868	4	9	,	,	PUNCT
cana-2868	4	10	vnr	vnr	PROPN
cana-2868	4	11	vignana	vignana	PROPN
cana-2868	4	12	jyothi	jyothi	PROPN
cana-2868	4	13	institute	institute	PROPN
cana-2868	4	14	of	of	ADP
cana-2868	4	15	engineering	engineering	PROPN
cana-2868	4	16	and	and	CCONJ
cana-2868	4	17	technology	technology	NOUN
cana-2868	4	18	,	,	PUNCT
cana-2868	4	19	bachupally	bachupally	ADV
cana-2868	4	20	,	,	PUNCT
cana-2868	4	21	kukatpally	kukatpally	ADV
cana-2868	4	22	,	,	PUNCT
cana-2868	4	23	hyderabad-500090	hyderabad-500090	NOUN
cana-2868	4	24	,	,	PUNCT
cana-2868	4	25	telangana	telangana	PROPN
cana-2868	4	26	state	state	PROPN
cana-2868	4	27	,	,	PUNCT
cana-2868	4	28	india	india	PROPN
cana-2868	4	29	.	.	PUNCT
cana-2868	4	30	email	email	NOUN
cana-2868	4	31	:	:	PUNCT
cana-2868	4	32	kusumatummala9@gmail.com	kusumatummala9@gmail.com	X
cana-2868	4	33	3	3	NUM
cana-2868	4	34	assistant	assistant	NOUN
cana-2868	4	35	professor	professor	NOUN
cana-2868	4	36	,	,	PUNCT
cana-2868	4	37	department	department	NOUN
cana-2868	4	38	of	of	ADP
cana-2868	4	39	mathematics	mathematics	PROPN
cana-2868	4	40	,	,	PUNCT
cana-2868	4	41	chaitanya	chaitanya	PROPN
cana-2868	4	42	bharathi	bharathi	PROPN
cana-2868	4	43	institute	institute	PROPN
cana-2868	4	44	of	of	ADP
cana-2868	4	45	technology	technology	PROPN
cana-2868	4	46	,	,	PUNCT
cana-2868	4	47	gandipet	gandipet	NOUN
cana-2868	4	48	,	,	PUNCT
cana-2868	4	49	hyderabad500075	hyderabad500075	PROPN
cana-2868	4	50	,	,	PUNCT
cana-2868	4	51	telangana	telangana	PROPN
cana-2868	4	52	state	state	PROPN
cana-2868	4	53	,	,	PUNCT
cana-2868	4	54	india	india	PROPN
cana-2868	4	55	.	.	PUNCT
cana-2868	4	56	email	email	NOUN
cana-2868	4	57	:	:	PUNCT
cana-2868	4	58	sreeramani_maths@cbit.ac.in	sreeramani_maths@cbit.ac.in	NUM
cana-2868	4	59	4professor	4professor	NUM
cana-2868	4	60	,	,	PUNCT
cana-2868	4	61	department	department	NOUN
cana-2868	4	62	of	of	ADP
cana-2868	4	63	cse	cse	PROPN
cana-2868	4	64	,	,	PUNCT
cana-2868	4	65	pragathi	pragathi	PROPN
cana-2868	4	66	enginnering	enginnering	PROPN
cana-2868	4	67	college	college	PROPN
cana-2868	4	68	,	,	PUNCT
cana-2868	4	69	surampalem	surampalem	NOUN
cana-2868	4	70	.	.	PUNCT
cana-2868	5	1	kakinada	kakinada	PROPN
cana-2868	5	2	.	.	PUNCT
cana-2868	6	1	india	india	PROPN
cana-2868	6	2	,	,	PUNCT
cana-2868	6	3	email	email	NOUN
cana-2868	6	4	:	:	PUNCT
cana-2868	6	5	yjbabu4166@gmail.com	yjbabu4166@gmail.com	X
cana-2868	7	1	5associate	5associate	NUM
cana-2868	7	2	professor	professor	NOUN
cana-2868	7	3	,	,	PUNCT
cana-2868	7	4	department	department	NOUN
cana-2868	7	5	of	of	ADP
cana-2868	7	6	mathematics	mathematic	NOUN
cana-2868	7	7	,	,	PUNCT
cana-2868	7	8	koneru	koneru	AUX
cana-2868	7	9	lakshmaih	lakshmaih	VERB
cana-2868	7	10	education	education	NOUN
cana-2868	7	11	foundation	foundation	PROPN
cana-2868	7	12	(	(	PUNCT
cana-2868	7	13	klef	klef	PROPN
cana-2868	7	14	)	)	PUNCT
cana-2868	7	15	,	,	PUNCT
cana-2868	7	16	guntur	guntur	PROPN
cana-2868	7	17	,	,	PUNCT
cana-2868	7	18	andhrapradesh	andhrapradesh	PROPN
cana-2868	7	19	,	,	PUNCT
cana-2868	7	20	india	india	PROPN
cana-2868	7	21	,	,	PUNCT
cana-2868	7	22	e	e	X
cana-2868	7	23	mail	mail	NOUN
cana-2868	7	24	:	:	PUNCT
cana-2868	7	25	tnraothota@kluniversity.in	tnraothota@kluniversity.in	PROPN
cana-2868	7	26	.	.	PUNCT
cana-2868	8	1	corresponding	correspond	VERB
cana-2868	8	2	author	author	NOUN
cana-2868	8	3	:	:	PUNCT
cana-2868	8	4	email	email	NOUN
cana-2868	8	5	:	:	PUNCT
cana-2868	8	6	phk.2003@gmail.com1	phk.2003@gmail.com1	NOUN
cana-2868	8	7	article	article	NOUN
cana-2868	8	8	history	history	NOUN
cana-2868	8	9	:	:	PUNCT
cana-2868	8	10	received	receive	VERB
cana-2868	8	11	:	:	PUNCT
cana-2868	8	12	30	30	NUM
cana-2868	8	13	-	-	SYM
cana-2868	8	14	09	09	NUM
cana-2868	8	15	-	-	PUNCT
cana-2868	8	16	2024	2024	NUM
cana-2868	8	17	revised	revise	VERB
cana-2868	8	18	:	:	PUNCT
cana-2868	8	19	28	28	NUM
cana-2868	8	20	-	-	SYM
cana-2868	8	21	11	11	NUM
cana-2868	8	22	-	-	PUNCT
cana-2868	8	23	2024	2024	NUM
cana-2868	8	24	accepted	accept	VERB
cana-2868	8	25	:	:	PUNCT
cana-2868	8	26	09	09	NUM
cana-2868	8	27	-	-	SYM
cana-2868	8	28	12	12	NUM
cana-2868	8	29	-	-	PUNCT
cana-2868	8	30	2024	2024	NUM
cana-2868	8	31	abstract	abstract	NOUN
cana-2868	8	32	:	:	PUNCT
cana-2868	8	33	in	in	ADP
cana-2868	8	34	this	this	DET
cana-2868	8	35	paper	paper	NOUN
cana-2868	8	36	we	we	PRON
cana-2868	8	37	prove	prove	VERB
cana-2868	8	38	the	the	DET
cana-2868	8	39	existence	existence	NOUN
cana-2868	8	40	and	and	CCONJ
cana-2868	8	41	uniqueness	uniqueness	NOUN
cana-2868	8	42	of	of	ADP
cana-2868	8	43	the	the	DET
cana-2868	8	44	fixed	fix	VERB
cana-2868	8	45	points	point	NOUN
cana-2868	8	46	generalized	generalize	VERB
cana-2868	8	47	ciric	ciric	ADJ
cana-2868	8	48	type	type	NOUN
cana-2868	8	49	geraghty	geraghty	VERB
cana-2868	8	50	rational	rational	ADJ
cana-2868	8	51	contractions	contraction	NOUN
cana-2868	8	52	in	in	ADP
cana-2868	8	53	b	b	NOUN
cana-2868	8	54	-	-	ADJ
cana-2868	8	55	metric	metric	ADJ
cana-2868	8	56	spaces	space	NOUN
cana-2868	8	57	,	,	PUNCT
cana-2868	8	58	our	our	PRON
cana-2868	8	59	results	result	NOUN
cana-2868	8	60	extend	extend	VERB
cana-2868	8	61	some	some	PRON
cana-2868	8	62	of	of	ADP
cana-2868	8	63	the	the	DET
cana-2868	8	64	known	know	VERB
cana-2868	8	65	theorems	theorem	NOUN
cana-2868	8	66	.	.	PUNCT
cana-2868	9	1	keywords	keyword	NOUN
cana-2868	9	2	:	:	PUNCT
cana-2868	9	3	fixed	fix	VERB
cana-2868	9	4	point	point	NOUN
cana-2868	9	5	;	;	PUNCT
cana-2868	9	6	b	b	X
cana-2868	9	7	-	-	PUNCT
cana-2868	9	8	metric	metric	ADJ
cana-2868	9	9	space	space	NOUN
cana-2868	9	10	;	;	PUNCT
cana-2868	9	11	geraghty	geraghty	PROPN
cana-2868	9	12	–	–	PUNCT
cana-2868	9	13	ciric	ciric	ADJ
cana-2868	9	14	type	type	NOUN
cana-2868	9	15	contraction	contraction	NOUN
cana-2868	9	16	.	.	PUNCT
cana-2868	10	1	ams(2010	ams(2010	NOUN
cana-2868	10	2	)	)	PUNCT
cana-2868	11	1	mathematics	mathematics	PROPN
cana-2868	11	2	subject	subject	ADJ
cana-2868	11	3	classification	classification	NOUN
cana-2868	11	4	:	:	PUNCT
cana-2868	11	5	54h25	54h25	NUM
cana-2868	11	6	,	,	PUNCT
cana-2868	11	7	74h10	74h10	NOUN
cana-2868	11	8	.	.	PUNCT
cana-2868	12	1	introduction	introduction	NOUN
cana-2868	12	2	one	one	NUM
cana-2868	12	3	of	of	ADP
cana-2868	12	4	the	the	DET
cana-2868	12	5	most	most	ADV
cana-2868	12	6	important	important	ADJ
cana-2868	12	7	development	development	NOUN
cana-2868	12	8	of	of	ADP
cana-2868	12	9	nonlinear	nonlinear	ADJ
cana-2868	12	10	analysis	analysis	NOUN
cana-2868	12	11	is	be	AUX
cana-2868	12	12	fixed	fix	VERB
cana-2868	12	13	point	point	NOUN
cana-2868	12	14	theory	theory	NOUN
cana-2868	12	15	.	.	PUNCT
cana-2868	13	1	this	this	DET
cana-2868	13	2	idea	idea	NOUN
cana-2868	13	3	is	be	AUX
cana-2868	13	4	useful	useful	ADJ
cana-2868	13	5	in	in	ADP
cana-2868	13	6	science	science	NOUN
cana-2868	13	7	and	and	CCONJ
cana-2868	13	8	engineering	engineering	NOUN
cana-2868	13	9	fields	field	NOUN
cana-2868	13	10	.	.	PUNCT
cana-2868	14	1	banach	banach	NOUN
cana-2868	15	1	[	[	X
cana-2868	15	2	2	2	NUM
cana-2868	15	3	]	]	PUNCT
cana-2868	15	4	first	first	ADV
cana-2868	15	5	proposed	propose	VERB
cana-2868	15	6	the	the	DET
cana-2868	15	7	principle	principle	NOUN
cana-2868	15	8	,	,	PUNCT
cana-2868	15	9	one	one	NUM
cana-2868	15	10	of	of	ADP
cana-2868	15	11	the	the	DET
cana-2868	15	12	fundamental	fundamental	ADJ
cana-2868	15	13	conclusions	conclusion	NOUN
cana-2868	15	14	of	of	ADP
cana-2868	15	15	conventional	conventional	ADJ
cana-2868	15	16	functional	functional	ADJ
cana-2868	15	17	analysis	analysis	NOUN
cana-2868	15	18	,	,	PUNCT
cana-2868	15	19	in	in	ADP
cana-2868	15	20	1922	1922	NUM
cana-2868	15	21	.	.	PUNCT
cana-2868	16	1	one	one	NUM
cana-2868	16	2	well	well	ADV
cana-2868	16	3	-	-	PUNCT
cana-2868	16	4	known	know	VERB
cana-2868	16	5	and	and	CCONJ
cana-2868	16	6	generally	generally	ADV
cana-2868	16	7	accepted	accept	VERB
cana-2868	16	8	outcome	outcome	NOUN
cana-2868	16	9	of	of	ADP
cana-2868	16	10	fixed	fix	VERB
cana-2868	16	11	point	point	NOUN
cana-2868	16	12	theory	theory	NOUN
cana-2868	16	13	is	be	AUX
cana-2868	16	14	this	this	DET
cana-2868	16	15	idea	idea	NOUN
cana-2868	16	16	.	.	PUNCT
cana-2868	17	1	in	in	ADP
cana-2868	17	2	1973	1973	NUM
cana-2868	17	3	,	,	PUNCT
cana-2868	17	4	geraghty	geraghty	PROPN
cana-2868	17	5	[	[	X
cana-2868	17	6	13	13	NUM
cana-2868	17	7	]	]	PUNCT
cana-2868	17	8	demonstrated	demonstrate	VERB
cana-2868	17	9	the	the	DET
cana-2868	17	10	existence	existence	NOUN
cana-2868	17	11	of	of	ADP
cana-2868	17	12	fixed	fix	VERB
cana-2868	17	13	point	point	NOUN
cana-2868	17	14	solutions	solution	NOUN
cana-2868	17	15	in	in	ADP
cana-2868	17	16	the	the	DET
cana-2868	17	17	context	context	NOUN
cana-2868	17	18	of	of	ADP
cana-2868	17	19	full	full	ADJ
cana-2868	17	20	metric	metric	ADJ
cana-2868	17	21	spaces	space	NOUN
cana-2868	17	22	[	[	X
cana-2868	17	23	ms	ms	X
cana-2868	17	24	]	]	X
cana-2868	17	25	and	and	CCONJ
cana-2868	17	26	provided	provide	VERB
cana-2868	17	27	an	an	DET
cana-2868	17	28	important	important	ADJ
cana-2868	17	29	expansion	expansion	NOUN
cana-2868	17	30	of	of	ADP
cana-2868	17	31	the	the	DET
cana-2868	17	32	banach	banach	NOUN
cana-2868	17	33	contraction	contraction	NOUN
cana-2868	17	34	principle	principle	NOUN
cana-2868	17	35	[	[	X
cana-2868	17	36	bcp	bcp	X
cana-2868	17	37	]	]	PUNCT
cana-2868	17	38	by	by	ADP
cana-2868	17	39	substituting	substitute	VERB
cana-2868	17	40	a	a	DET
cana-2868	17	41	function	function	NOUN
cana-2868	17	42	with	with	ADP
cana-2868	17	43	certain	certain	ADJ
cana-2868	17	44	qualities	quality	NOUN
cana-2868	17	45	for	for	ADP
cana-2868	17	46	a	a	DET
cana-2868	17	47	constant	constant	ADJ
cana-2868	17	48	.	.	PUNCT
cana-2868	18	1	as	as	SCONJ
cana-2868	18	2	you	you	PRON
cana-2868	18	3	can	can	AUX
cana-2868	18	4	see	see	VERB
cana-2868	18	5	from	from	ADP
cana-2868	18	6	[	[	X
cana-2868	18	7	8	8	NUM
cana-2868	18	8	,	,	PUNCT
cana-2868	18	9	9	9	NUM
cana-2868	18	10	,	,	PUNCT
cana-2868	18	11	12	12	NUM
cana-2868	18	12	,	,	PUNCT
cana-2868	18	13	13	13	NUM
cana-2868	18	14	]	]	PUNCT
cana-2868	18	15	and	and	CCONJ
cana-2868	18	16	the	the	DET
cana-2868	18	17	references	reference	NOUN
cana-2868	18	18	therein	therein	ADV
cana-2868	18	19	,	,	PUNCT
cana-2868	18	20	numerous	numerous	ADJ
cana-2868	18	21	researchers	researcher	NOUN
cana-2868	18	22	have	have	AUX
cana-2868	18	23	since	since	SCONJ
cana-2868	18	24	expanded	expand	VERB
cana-2868	18	25	and	and	CCONJ
cana-2868	18	26	broadened	broaden	VERB
cana-2868	18	27	the	the	DET
cana-2868	18	28	geraghty	geraghty	PROPN
cana-2868	18	29	conclusion	conclusion	NOUN
cana-2868	18	30	in	in	ADP
cana-2868	18	31	different	different	ADJ
cana-2868	18	32	ways	way	NOUN
cana-2868	18	33	.	.	PUNCT
cana-2868	19	1	in	in	ADP
cana-2868	19	2	metric	metric	ADJ
cana-2868	19	3	spaces	space	NOUN
cana-2868	19	4	,	,	PUNCT
cana-2868	19	5	ćirić	ćirić	NOUN
cana-2868	19	6	[	[	X
cana-2868	19	7	4,5	4,5	NUM
cana-2868	19	8	]	]	PUNCT
cana-2868	19	9	demonstrated	demonstrate	VERB
cana-2868	19	10	the	the	DET
cana-2868	19	11	ćirić	ćirić	NOUN
cana-2868	19	12	-	-	PUNCT
cana-2868	19	13	type	type	NOUN
cana-2868	19	14	fixed	fix	VERB
cana-2868	19	15	point	point	NOUN
cana-2868	19	16	theorem	theorem	VERB
cana-2868	19	17	,	,	PUNCT
cana-2868	19	18	which	which	PRON
cana-2868	19	19	is	be	AUX
cana-2868	19	20	thought	think	VERB
cana-2868	19	21	to	to	PART
cana-2868	19	22	be	be	AUX
cana-2868	19	23	one	one	NUM
cana-2868	19	24	of	of	ADP
cana-2868	19	25	the	the	DET
cana-2868	19	26	most	most	ADV
cana-2868	19	27	important	important	ADJ
cana-2868	19	28	generalizations	generalization	NOUN
cana-2868	19	29	of	of	ADP
cana-2868	19	30	the	the	DET
cana-2868	19	31	bcp	bcp	PROPN
cana-2868	19	32	definition	definition	NOUN
cana-2868	19	33	1.1	1.1	NUM
cana-2868	19	34	[	[	X
cana-2868	19	35	15	15	NUM
cana-2868	19	36	]	]	PUNCT
cana-2868	19	37	.	.	PUNCT
cana-2868	20	1	let	let	VERB
cana-2868	20	2	h	h	PRON
cana-2868	20	3	be	be	AUX
cana-2868	20	4	a	a	DET
cana-2868	20	5	nonempty	nonempty	ADV
cana-2868	20	6	set	set	VERB
cana-2868	20	7	and	and	CCONJ
cana-2868	20	8	let	let	VERB
cana-2868	20	9	t	t	PROPN
cana-2868	20	10	≥1	≥1	VERB
cana-2868	20	11	.	.	PUNCT
cana-2868	21	1	a	a	DET
cana-2868	21	2	mapping	mapping	NOUN
cana-2868	21	3	𝑑	𝑑	NOUN
cana-2868	21	4	:	:	PUNCT
cana-2868	21	5	𝐻x𝐻	𝐻x𝐻	NOUN
cana-2868	21	6	→r	→r	PUNCT
cana-2868	21	7	is	be	AUX
cana-2868	21	8	said	say	VERB
cana-2868	21	9	to	to	PART
cana-2868	21	10	be	be	AUX
cana-2868	21	11	a	a	DET
cana-2868	21	12	b	b	NOUN
cana-2868	21	13	-	-	PUNCT
cana-2868	21	14	metric	metric	ADJ
cana-2868	21	15	space	space	NOUN
cana-2868	21	16	if	if	SCONJ
cana-2868	21	17	∀	∀	NOUN
cana-2868	21	18	a	a	PRON
cana-2868	21	19	,	,	PUNCT
cana-2868	21	20	b.c	b.c	NOUN
cana-2868	21	21	in	in	ADP
cana-2868	21	22	h	h	NOUN
cana-2868	21	23	,	,	PUNCT
cana-2868	21	24	the	the	DET
cana-2868	21	25	following	follow	VERB
cana-2868	21	26	conditions	condition	NOUN
cana-2868	21	27	are	be	AUX
cana-2868	21	28	satisfied	satisfied	ADJ
cana-2868	21	29	.	.	PUNCT
cana-2868	22	1	(	(	PUNCT
cana-2868	22	2	b1	b1	NOUN
cana-2868	22	3	)	)	PUNCT
cana-2868	22	4	𝑑	𝑑	PROPN
cana-2868	22	5	(	(	PUNCT
cana-2868	22	6	a	a	PRON
cana-2868	22	7	,	,	PUNCT
cana-2868	23	1	b)=0	b)=0	ADV
cana-2868	23	2	if	if	SCONJ
cana-2868	23	3	and	and	CCONJ
cana-2868	23	4	only	only	ADV
cana-2868	23	5	if	if	SCONJ
cana-2868	23	6	a	a	DET
cana-2868	23	7	=	=	SYM
cana-2868	23	8	b	b	NOUN
cana-2868	23	9	,	,	PUNCT
cana-2868	23	10	(	(	PUNCT
cana-2868	23	11	b2	b2	NOUN
cana-2868	23	12	)	)	PUNCT
cana-2868	23	13	𝑑	𝑑	PROPN
cana-2868	23	14	(	(	PUNCT
cana-2868	23	15	a	a	DET
cana-2868	23	16	,	,	PUNCT
cana-2868	23	17	b	b	NOUN
cana-2868	23	18	)	)	PUNCT
cana-2868	23	19	=	=	SYM
cana-2868	23	20	𝑑	𝑑	PROPN
cana-2868	23	21	(	(	PUNCT
cana-2868	23	22	b	b	NOUN
cana-2868	23	23	,	,	PUNCT
cana-2868	23	24	a	a	PRON
cana-2868	23	25	)	)	PUNCT
cana-2868	23	26	mailto:kusumatummala9@gmail.com	mailto:kusumatummala9@gmail.com	NOUN
cana-2868	23	27	mailto:sreeramani_maths@cbit.ac.in	mailto:sreeramani_maths@cbit.ac.in	PROPN
cana-2868	23	28	mailto:yjbabu4166@gmail.com	mailto:yjbabu4166@gmail.com	PROPN
cana-2868	23	29	mailto:tnraothota@kluniversity.in	mailto:tnraothota@kluniversity.in	PROPN
cana-2868	23	30	mailto:phk.2003@gmail.com	mailto:phk.2003@gmail.com	X
cana-2868	23	31	communications	communication	NOUN
cana-2868	23	32	on	on	ADP
cana-2868	23	33	applied	apply	VERB
cana-2868	23	34	nonlinear	nonlinear	ADJ
cana-2868	23	35	analysis	analysis	NOUN
cana-2868	23	36	issn	issn	NOUN
cana-2868	23	37	:	:	PUNCT
cana-2868	23	38	1074	1074	NUM
cana-2868	23	39	-	-	PUNCT
cana-2868	23	40	133x	133x	NUM
cana-2868	23	41	vol	vol	NOUN
cana-2868	23	42	32	32	NUM
cana-2868	23	43	no	no	NOUN
cana-2868	23	44	.	.	PUNCT
cana-2868	24	1	4s	4s	NUM
cana-2868	24	2	(	(	PUNCT
cana-2868	24	3	2025	2025	NUM
cana-2868	24	4	)	)	PUNCT
cana-2868	24	5	514	514	NUM
cana-2868	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	24	7	(	(	PUNCT
cana-2868	24	8	b3	b3	PROPN
cana-2868	24	9	)	)	PUNCT
cana-2868	24	10	𝑑	𝑑	PROPN
cana-2868	24	11	(	(	PUNCT
cana-2868	24	12	a	a	PRON
cana-2868	24	13	,	,	PUNCT
cana-2868	24	14	c	c	NOUN
cana-2868	24	15	)	)	PUNCT
cana-2868	24	16	≤	≤	NOUN
cana-2868	25	1	t[𝑑	t[𝑑	PRON
cana-2868	25	2	(	(	PUNCT
cana-2868	25	3	a	a	PRON
cana-2868	25	4	,	,	PUNCT
cana-2868	25	5	b)+	b)+	PROPN
cana-2868	25	6	𝑑	𝑑	PROPN
cana-2868	25	7	(	(	PUNCT
cana-2868	25	8	b	b	NOUN
cana-2868	25	9	,	,	PUNCT
cana-2868	25	10	c	c	NOUN
cana-2868	25	11	)	)	PUNCT
cana-2868	25	12	]	]	PUNCT
cana-2868	25	13	.	.	PUNCT
cana-2868	26	1	in	in	ADP
cana-2868	26	2	this	this	DET
cana-2868	26	3	case	case	NOUN
cana-2868	26	4	,	,	PUNCT
cana-2868	26	5	the	the	DET
cana-2868	26	6	pair	pair	NOUN
cana-2868	26	7	(	(	PUNCT
cana-2868	26	8	h	h	NOUN
cana-2868	26	9	,	,	PUNCT
cana-2868	26	10	𝑑	𝑑	NOUN
cana-2868	26	11	)	)	PUNCT
cana-2868	26	12	is	be	AUX
cana-2868	26	13	called	call	VERB
cana-2868	26	14	a	a	DET
cana-2868	26	15	b	b	NOUN
cana-2868	26	16	-	-	PUNCT
cana-2868	26	17	metric	metric	ADJ
cana-2868	26	18	space	space	NOUN
cana-2868	26	19	(	(	PUNCT
cana-2868	26	20	with	with	ADP
cana-2868	26	21	constant	constant	ADJ
cana-2868	26	22	s	s	NOUN
cana-2868	26	23	)	)	PUNCT
cana-2868	26	24	.	.	PUNCT
cana-2868	27	1	note	note	VERB
cana-2868	27	2	that	that	SCONJ
cana-2868	27	3	every	every	DET
cana-2868	27	4	metric	metric	ADJ
cana-2868	27	5	space	space	NOUN
cana-2868	27	6	is	be	AUX
cana-2868	27	7	b	b	NOUN
cana-2868	27	8	-	-	ADJ
cana-2868	27	9	metric	metric	ADJ
cana-2868	27	10	for	for	ADP
cana-2868	27	11	t=1	t=1	PROPN
cana-2868	27	12	,	,	PUNCT
cana-2868	27	13	but	but	CCONJ
cana-2868	27	14	the	the	DET
cana-2868	27	15	converse	converse	NOUN
cana-2868	27	16	is	be	AUX
cana-2868	27	17	not	not	PART
cana-2868	27	18	true	true	ADJ
cana-2868	27	19	.	.	PUNCT
cana-2868	28	1	let	let	VERB
cana-2868	28	2	s	s	PRON
cana-2868	28	3	be	be	AUX
cana-2868	28	4	the	the	DET
cana-2868	28	5	class	class	NOUN
cana-2868	28	6	of	of	ADP
cana-2868	28	7	functions	function	NOUN
cana-2868	28	8	of	of	ADP
cana-2868	28	9	non	non	ADJ
cana-2868	28	10	–	–	PUNCT
cana-2868	28	11	decreasing	decrease	VERB
cana-2868	28	12	functions	function	NOUN
cana-2868	28	13	𝛽	𝛽	NOUN
cana-2868	28	14	:	:	PUNCT
cana-2868	29	1	[	[	X
cana-2868	29	2	0	0	NUM
cana-2868	29	3	,	,	PUNCT
cana-2868	29	4	∞	∞	PROPN
cana-2868	29	5	)	)	PUNCT
cana-2868	29	6	→	→	PUNCT
cana-2868	30	1	[	[	X
cana-2868	30	2	1	1	NUM
cana-2868	30	3	,	,	PUNCT
cana-2868	30	4	1	1	NUM
cana-2868	30	5	t	t	NOUN
cana-2868	30	6	)	)	PUNCT
cana-2868	30	7	which	which	PRON
cana-2868	30	8	satisfy	satisfy	VERB
cana-2868	30	9	the	the	DET
cana-2868	30	10	condition	condition	NOUN
cana-2868	30	11	lim	lim	PROPN
cana-2868	30	12	n→∞	n→∞	X
cana-2868	30	13	𝛽(𝑡𝑛	𝛽(𝑡𝑛	PROPN
cana-2868	30	14	)	)	PUNCT
cana-2868	30	15	=	=	SYM
cana-2868	30	16	1	1	NUM
cana-2868	30	17	𝑡	𝑡	PROPN
cana-2868	30	18	𝑖𝑚𝑝𝑙𝑖𝑒𝑠	𝑖𝑚𝑝𝑙𝑖𝑒𝑠	PROPN
cana-2868	30	19	lim	lim	PROPN
cana-2868	30	20	n→∞	n→∞	NUM
cana-2868	30	21	𝑡𝑛	𝑡𝑛	VERB
cana-2868	30	22	=	=	SYM
cana-2868	30	23	0	0	NUM
cana-2868	30	24	for	for	ADP
cana-2868	30	25	some	some	DET
cana-2868	30	26	𝑡	𝑡	NOUN
cana-2868	30	27	≥	≥	NOUN
cana-2868	30	28	1	1	NUM
cana-2868	30	29	.	.	PUNCT
cana-2868	30	30	geraghty	geraghty	PROPN
cana-2868	31	1	[	[	X
cana-2868	31	2	14	14	NUM
cana-2868	31	3	]	]	PUNCT
cana-2868	31	4	proved	prove	VERB
cana-2868	31	5	the	the	DET
cana-2868	31	6	following	follow	VERB
cana-2868	31	7	theorem	theorem	ADJ
cana-2868	31	8	.	.	PUNCT
cana-2868	31	9	theorem	theorem	VERB
cana-2868	31	10	1.2	1.2	NUM
cana-2868	31	11	.	.	PUNCT
cana-2868	32	1	[	[	X
cana-2868	32	2	14	14	NUM
cana-2868	32	3	]	]	X
cana-2868	32	4	let	let	NOUN
cana-2868	32	5	(	(	PUNCT
cana-2868	32	6	k	k	X
cana-2868	32	7	,	,	PUNCT
cana-2868	32	8	d	d	NOUN
cana-2868	32	9	)	)	PUNCT
cana-2868	32	10	be	be	AUX
cana-2868	32	11	a	a	DET
cana-2868	32	12	cms(complete	cms(complete	ADJ
cana-2868	32	13	metric	metric	ADJ
cana-2868	32	14	space	space	NOUN
cana-2868	32	15	)	)	PUNCT
cana-2868	32	16	.	.	PUNCT
cana-2868	33	1	let	let	VERB
cana-2868	33	2	h	h	NOUN
cana-2868	33	3	:	:	PUNCT
cana-2868	33	4	k	k	X
cana-2868	33	5	→	→	PUNCT
cana-2868	33	6	k	k	X
cana-2868	33	7	be	be	AUX
cana-2868	33	8	a	a	DET
cana-2868	33	9	self	self	NOUN
cana-2868	33	10	map	map	NOUN
cana-2868	33	11	.	.	PUNCT
cana-2868	34	1	if	if	SCONJ
cana-2868	34	2	∃	∃	PROPN
cana-2868	34	3	,	,	PUNCT
cana-2868	34	4	β	β	X
cana-2868	34	5	∈	∈	PROPN
cana-2868	34	6	s	s	VERB
cana-2868	34	7	such	such	ADJ
cana-2868	34	8	that	that	DET
cana-2868	34	9	d(h(u	d(h(u	PROPN
cana-2868	34	10	)	)	PUNCT
cana-2868	34	11	,	,	PUNCT
cana-2868	34	12	h(v	h(v	PROPN
cana-2868	34	13	)	)	PUNCT
cana-2868	34	14	)	)	PUNCT
cana-2868	34	15	≤	≤	PUNCT
cana-2868	35	1	β(d(u	β(d(u	ADV
cana-2868	35	2	,	,	PUNCT
cana-2868	35	3	v))d(u	v))d(u	PROPN
cana-2868	35	4	,	,	PUNCT
cana-2868	35	5	v	v	NOUN
cana-2868	35	6	)	)	PUNCT
cana-2868	35	7	for	for	ADP
cana-2868	35	8	all	all	DET
cana-2868	35	9	u	u	NOUN
cana-2868	35	10	,	,	PUNCT
cana-2868	35	11	v	v	PROPN
cana-2868	35	12	∈	∈	PROPN
cana-2868	35	13	h	h	NOUN
cana-2868	35	14	,	,	PUNCT
cana-2868	35	15	then	then	ADV
cana-2868	35	16	f	f	PROPN
cana-2868	35	17	has	have	VERB
cana-2868	35	18	a	a	DET
cana-2868	35	19	unique	unique	ADJ
cana-2868	35	20	common	common	ADJ
cana-2868	35	21	fixed	fix	VERB
cana-2868	35	22	point	point	NOUN
cana-2868	35	23	in	in	ADP
cana-2868	35	24	h	h	PROPN
cana-2868	35	25	.	.	PUNCT
cana-2868	36	1	definition	definition	NOUN
cana-2868	36	2	1.3	1.3	NUM
cana-2868	36	3	.	.	PUNCT
cana-2868	37	1	[	[	X
cana-2868	37	2	14	14	NUM
cana-2868	37	3	]	]	X
cana-2868	37	4	a	a	DET
cana-2868	37	5	self	self	NOUN
cana-2868	37	6	map	map	NOUN
cana-2868	37	7	h	h	NOUN
cana-2868	37	8	:	:	PUNCT
cana-2868	37	9	k	k	X
cana-2868	37	10	→	→	PUNCT
cana-2868	37	11	k	k	PROPN
cana-2868	37	12	is	be	AUX
cana-2868	37	13	said	say	VERB
cana-2868	37	14	to	to	PART
cana-2868	37	15	be	be	AUX
cana-2868	37	16	a	a	DET
cana-2868	37	17	generalized	generalized	ADJ
cana-2868	37	18	geraghty	geraghty	ADJ
cana-2868	37	19	contraction	contraction	NOUN
cana-2868	37	20	if	if	SCONJ
cana-2868	37	21	∃β	∃β	PROPN
cana-2868	37	22	∈	∈	PROPN
cana-2868	37	23	s	s	VERB
cana-2868	37	24	such	such	ADJ
cana-2868	37	25	that	that	DET
cana-2868	37	26	d(h(r	d(h(r	PROPN
cana-2868	37	27	)	)	PUNCT
cana-2868	37	28	,	,	PUNCT
cana-2868	37	29	h(s	h(s	PROPN
cana-2868	37	30	)	)	PUNCT
cana-2868	37	31	)	)	PUNCT
cana-2868	37	32	≤	≤	NUM
cana-2868	38	1	β	β	X
cana-2868	38	2	(	(	PUNCT
cana-2868	38	3	m	m	PROPN
cana-2868	38	4	(	(	PUNCT
cana-2868	38	5	r	r	NOUN
cana-2868	38	6	,	,	PUNCT
cana-2868	38	7	s))m	s))m	NOUN
cana-2868	38	8	(	(	PUNCT
cana-2868	38	9	r	r	NOUN
cana-2868	38	10	,	,	PUNCT
cana-2868	38	11	s	s	PART
cana-2868	38	12	)	)	PUNCT
cana-2868	38	13	(	(	PUNCT
cana-2868	38	14	1.13.1	1.13.1	NUM
cana-2868	38	15	)	)	PUNCT
cana-2868	38	16	𝑀(𝑟	𝑀(𝑟	NUM
cana-2868	38	17	,	,	PUNCT
cana-2868	38	18	𝑠	𝑠	NOUN
cana-2868	38	19	)	)	PUNCT
cana-2868	38	20	=	=	SYM
cana-2868	38	21	max	max	PROPN
cana-2868	38	22	{	{	PUNCT
cana-2868	38	23	𝑑(𝑟	𝑑(𝑟	PROPN
cana-2868	38	24	,	,	PUNCT
cana-2868	38	25	𝑠	𝑠	NOUN
cana-2868	38	26	)	)	PUNCT
cana-2868	38	27	,	,	PUNCT
cana-2868	38	28	𝑑(𝑟	𝑑(𝑟	PROPN
cana-2868	38	29	,	,	PUNCT
cana-2868	38	30	𝐻𝑟	𝐻𝑟	PROPN
cana-2868	38	31	)	)	PUNCT
cana-2868	38	32	,	,	PUNCT
cana-2868	38	33	𝑑(𝑠	𝑑(𝑠	PROPN
cana-2868	38	34	,	,	PUNCT
cana-2868	38	35	𝐻𝑠	𝐻𝑠	NOUN
cana-2868	38	36	)	)	PUNCT
cana-2868	38	37	,	,	PUNCT
cana-2868	38	38	(	(	PUNCT
cana-2868	38	39	𝑑(r	𝑑(r	PROPN
cana-2868	38	40	,	,	PUNCT
cana-2868	38	41	hs	hs	X
cana-2868	38	42	)	)	PUNCT
cana-2868	38	43	+	+	CCONJ
cana-2868	38	44	𝑑(s	𝑑(	NOUN
cana-2868	38	45	,	,	PUNCT
cana-2868	38	46	hr)/2	hr)/2	NOUN
cana-2868	38	47	}	}	PUNCT
cana-2868	38	48	for	for	ADP
cana-2868	38	49	all	all	DET
cana-2868	38	50	r	r	NOUN
cana-2868	38	51	,	,	PUNCT
cana-2868	38	52	s	s	PROPN
cana-2868	38	53	∈	∈	PROPN
cana-2868	38	54	h.	h.	NOUN
cana-2868	38	55	definition	definition	NOUN
cana-2868	38	56	:	:	PUNCT
cana-2868	38	57	1.4	1.4	NUM
cana-2868	38	58	[	[	X
cana-2868	38	59	1	1	X
cana-2868	38	60	]	]	PUNCT
cana-2868	38	61	a	a	DET
cana-2868	38	62	mapping	mapping	NOUN
cana-2868	38	63	h	h	NOUN
cana-2868	38	64	:	:	PUNCT
cana-2868	38	65	k	k	X
cana-2868	38	66	→	→	PUNCT
cana-2868	38	67	𝐾	𝐾	PROPN
cana-2868	38	68	on	on	ADP
cana-2868	38	69	a	a	DET
cana-2868	38	70	b	b	NOUN
cana-2868	38	71	metric	metric	ADJ
cana-2868	38	72	space	space	NOUN
cana-2868	38	73	(	(	PUNCT
cana-2868	38	74	k	k	X
cana-2868	38	75	,	,	PUNCT
cana-2868	38	76	d	d	PROPN
cana-2868	38	77	,	,	PUNCT
cana-2868	38	78	t	t	PROPN
cana-2868	38	79	)	)	PUNCT
cana-2868	38	80	with	with	ADP
cana-2868	38	81	t	t	PROPN
cana-2868	38	82	≥	≥	NOUN
cana-2868	38	83	1	1	NUM
cana-2868	38	84	is	be	AUX
cana-2868	38	85	called	call	VERB
cana-2868	38	86	a	a	DET
cana-2868	38	87	ciric	ciric	ADJ
cana-2868	38	88	type	type	NOUN
cana-2868	38	89	geraghty	geraghty	NOUN
cana-2868	38	90	contraction	contraction	NOUN
cana-2868	38	91	mapping	mapping	NOUN
cana-2868	38	92	if	if	SCONJ
cana-2868	38	93	∃	∃	PROPN
cana-2868	38	94	,	,	PUNCT
cana-2868	38	95	𝛽	𝛽	PROPN
cana-2868	38	96	∈	∈	PROPN
cana-2868	38	97	f	f	NOUN
cana-2868	38	98	such	such	ADJ
cana-2868	38	99	that	that	SCONJ
cana-2868	38	100	𝑑(𝐻𝑢	𝑑(𝐻𝑢	NOUN
cana-2868	38	101	,	,	PUNCT
cana-2868	38	102	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	38	103	)	)	PUNCT
cana-2868	38	104	≤	≤	NOUN
cana-2868	38	105	𝑀(𝑟	𝑀(𝑟	PROPN
cana-2868	38	106	,	,	PUNCT
cana-2868	38	107	𝑠	𝑠	NOUN
cana-2868	38	108	)	)	PUNCT
cana-2868	38	109	,	,	PUNCT
cana-2868	38	110	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2868	38	111	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-2868	38	112	𝑢	𝑢	PROPN
cana-2868	38	113	,	,	PUNCT
cana-2868	38	114	𝑣	𝑣	PRON
cana-2868	38	115	∈	∈	PROPN
cana-2868	38	116	𝐾	𝐾	PROPN
cana-2868	38	117	where	where	SCONJ
cana-2868	38	118	𝑀(𝑟	𝑀(𝑟	NOUN
cana-2868	38	119	,	,	PUNCT
cana-2868	38	120	𝑠	𝑠	NOUN
cana-2868	38	121	)	)	PUNCT
cana-2868	38	122	=	=	PUNCT
cana-2868	38	123	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-2868	38	124	{	{	PUNCT
cana-2868	38	125	𝛽(𝑑(𝑟	𝛽(𝑑(𝑟	PROPN
cana-2868	38	126	,	,	PUNCT
cana-2868	38	127	𝑠))𝑑(𝑟	𝑠))𝑑(𝑟	PROPN
cana-2868	38	128	,	,	PUNCT
cana-2868	38	129	𝑠	𝑠	NOUN
cana-2868	38	130	)	)	PUNCT
cana-2868	38	131	,	,	PUNCT
cana-2868	38	132	𝛽(𝑑(𝑟	𝛽(𝑑(𝑟	PROPN
cana-2868	38	133	,	,	PUNCT
cana-2868	38	134	𝐻𝑟))𝑑(𝑟	𝐻𝑟))𝑑(𝑟	PROPN
cana-2868	38	135	,	,	PUNCT
cana-2868	38	136	𝐻𝑟	𝐻𝑟	PROPN
cana-2868	38	137	)	)	PUNCT
cana-2868	38	138	,	,	PUNCT
cana-2868	38	139	𝛽(𝑑(𝑠	𝛽(𝑑(𝑠	NOUN
cana-2868	38	140	,	,	PUNCT
cana-2868	38	141	𝐻𝑠))𝑑(𝑠	𝐻𝑠))𝑑(𝑠	PROPN
cana-2868	38	142	,	,	PUNCT
cana-2868	38	143	𝐻𝑠	𝐻𝑠	NOUN
cana-2868	38	144	)	)	PUNCT
cana-2868	38	145	,	,	PUNCT
cana-2868	38	146	𝛽(𝑑(𝑟	𝛽(𝑑(𝑟	PROPN
cana-2868	38	147	,	,	PUNCT
cana-2868	38	148	𝐻𝑠))𝑑(𝑟	𝐻𝑠))𝑑(𝑟	ADJ
cana-2868	38	149	,	,	PUNCT
cana-2868	38	150	𝐻𝑠	𝐻𝑠	NOUN
cana-2868	38	151	)	)	PUNCT
cana-2868	38	152	,	,	PUNCT
cana-2868	38	153	𝛽(𝑑(𝑠	𝛽(𝑑(𝑠	NOUN
cana-2868	38	154	,	,	PUNCT
cana-2868	38	155	𝐻𝑟))𝑑(𝑟	𝐻𝑟))𝑑(𝑟	ADJ
cana-2868	38	156	,	,	PUNCT
cana-2868	38	157	𝐻𝑠	𝐻𝑠	NOUN
cana-2868	38	158	)	)	PUNCT
cana-2868	38	159	}	}	PUNCT
cana-2868	38	160	definition1.5	definition1.5	NOUN
cana-2868	38	161	:	:	PUNCT
cana-2868	39	1	[	[	X
cana-2868	39	2	1	1	X
cana-2868	39	3	]	]	PUNCT
cana-2868	39	4	a	a	DET
cana-2868	39	5	mapping	mapping	NOUN
cana-2868	39	6	h	h	NOUN
cana-2868	39	7	:	:	PUNCT
cana-2868	39	8	k	k	X
cana-2868	39	9	→	→	PUNCT
cana-2868	39	10	𝐾	𝐾	PROPN
cana-2868	39	11	on	on	ADP
cana-2868	39	12	a	a	DET
cana-2868	39	13	b	b	NOUN
cana-2868	39	14	metric	metric	ADJ
cana-2868	39	15	space	space	NOUN
cana-2868	39	16	(	(	PUNCT
cana-2868	39	17	k	k	X
cana-2868	39	18	,	,	PUNCT
cana-2868	39	19	d	d	PROPN
cana-2868	39	20	,	,	PUNCT
cana-2868	39	21	t	t	PROPN
cana-2868	39	22	)	)	PUNCT
cana-2868	39	23	with	with	ADP
cana-2868	39	24	t	t	PROPN
cana-2868	39	25	≥	≥	NOUN
cana-2868	39	26	1	1	NUM
cana-2868	39	27	is	be	AUX
cana-2868	39	28	called	call	VERB
cana-2868	39	29	a	a	DET
cana-2868	39	30	ciric	ciric	ADJ
cana-2868	39	31	type	type	NOUN
cana-2868	39	32	geraghty	geraghty	NOUN
cana-2868	39	33	contraction	contraction	NOUN
cana-2868	39	34	mapping	mapping	NOUN
cana-2868	39	35	if	if	SCONJ
cana-2868	39	36	there	there	PRON
cana-2868	39	37	exists	exist	VERB
cana-2868	39	38	𝛽	𝛽	PROPN
cana-2868	39	39	∈	∈	NOUN
cana-2868	39	40	s	s	VERB
cana-2868	39	41	such	such	ADJ
cana-2868	39	42	that	that	SCONJ
cana-2868	39	43	𝑑(𝐻𝑢	𝑑(𝐻𝑢	NOUN
cana-2868	39	44	,	,	PUNCT
cana-2868	39	45	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	39	46	)	)	PUNCT
cana-2868	39	47	≤	≤	NOUN
cana-2868	39	48	𝑀(𝑢	𝑀(𝑢	PROPN
cana-2868	39	49	,	,	PUNCT
cana-2868	39	50	𝑣	𝑣	NOUN
cana-2868	39	51	)	)	PUNCT
cana-2868	39	52	,	,	PUNCT
cana-2868	39	53	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2868	39	54	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-2868	39	55	𝑢	𝑢	PROPN
cana-2868	39	56	,	,	PUNCT
cana-2868	39	57	𝑣	𝑣	PRON
cana-2868	39	58	∈	∈	PROPN
cana-2868	39	59	𝐾	𝐾	PROPN
cana-2868	39	60	where	where	SCONJ
cana-2868	39	61	𝑀(𝑢	𝑀(𝑢	NOUN
cana-2868	39	62	,	,	PUNCT
cana-2868	39	63	𝑣	𝑣	NOUN
cana-2868	39	64	)	)	PUNCT
cana-2868	39	65	=	=	SYM
cana-2868	39	66	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-2868	39	67	{	{	PUNCT
cana-2868	39	68	𝛽(𝑑(𝑢	𝛽(𝑑(𝑢	PROPN
cana-2868	39	69	,	,	PUNCT
cana-2868	39	70	𝑣))𝑑(𝑢	𝑣))𝑑(𝑢	PROPN
cana-2868	39	71	,	,	PUNCT
cana-2868	39	72	𝑣	𝑣	NOUN
cana-2868	39	73	)	)	PUNCT
cana-2868	39	74	,	,	PUNCT
cana-2868	39	75	𝛽(𝑑(𝑢	𝛽(𝑑(𝑢	PROPN
cana-2868	39	76	,	,	PUNCT
cana-2868	39	77	𝐻𝑢))𝑑(𝑢	𝐻𝑢))𝑑(𝑢	ADJ
cana-2868	39	78	,	,	PUNCT
cana-2868	39	79	𝐻𝑢	𝐻𝑢	PROPN
cana-2868	39	80	)	)	PUNCT
cana-2868	39	81	,	,	PUNCT
cana-2868	39	82	𝛽(𝑑(𝑣	𝛽(𝑑(𝑣	PROPN
cana-2868	39	83	,	,	PUNCT
cana-2868	39	84	𝐻𝑣))𝑑(𝑣	𝐻𝑣))𝑑(𝑣	PROPN
cana-2868	39	85	,	,	PUNCT
cana-2868	39	86	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	39	87	)	)	PUNCT
cana-2868	39	88	,	,	PUNCT
cana-2868	39	89	𝛽(𝑑(𝑢	𝛽(𝑑(𝑢	PROPN
cana-2868	39	90	,	,	PUNCT
cana-2868	39	91	𝐻𝑣))𝑑(𝑢	𝐻𝑣))𝑑(𝑢	PROPN
cana-2868	39	92	,	,	PUNCT
cana-2868	39	93	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	39	94	)	)	PUNCT
cana-2868	39	95	,	,	PUNCT
cana-2868	39	96	𝛽(𝑑(𝑣	𝛽(𝑑(𝑣	PROPN
cana-2868	39	97	,	,	PUNCT
cana-2868	39	98	𝐻𝑢))𝑑(𝑣	𝐻𝑢))𝑑(𝑣	NOUN
cana-2868	39	99	,	,	PUNCT
cana-2868	39	100	𝐻𝑢	𝐻𝑢	NOUN
cana-2868	39	101	)	)	PUNCT
cana-2868	39	102	}	}	PUNCT
cana-2868	39	103	theorem	theorem	VERB
cana-2868	39	104	1.6	1.6	NUM
cana-2868	39	105	:	:	PUNCT
cana-2868	39	106	[	[	PUNCT
cana-2868	39	107	8,9	8,9	NUM
cana-2868	39	108	]	]	X
cana-2868	39	109	let	let	VERB
cana-2868	39	110	(	(	PUNCT
cana-2868	39	111	m	m	NOUN
cana-2868	39	112	,	,	PUNCT
cana-2868	39	113	d	d	NOUN
cana-2868	39	114	)	)	PUNCT
cana-2868	39	115	be	be	AUX
cana-2868	39	116	a	a	DET
cana-2868	39	117	cms	cms	NOUN
cana-2868	39	118	and	and	CCONJ
cana-2868	39	119	𝑇	𝑇	PROPN
cana-2868	39	120	:	:	PUNCT
cana-2868	39	121	𝑀	𝑀	PROPN
cana-2868	39	122	→	→	SYM
cana-2868	39	123	𝑀	𝑀	PROPN
cana-2868	39	124	be	be	VERB
cana-2868	39	125	a	a	DET
cana-2868	39	126	geraghty	geraghty	ADJ
cana-2868	39	127	–	–	PUNCT
cana-2868	39	128	ciric	ciric	ADJ
cana-2868	39	129	–	–	PUNCT
cana-2868	39	130	contraction	contraction	NOUN
cana-2868	39	131	with	with	ADP
cana-2868	39	132	some	some	DET
cana-2868	39	133	𝛽	𝛽	NOUN
cana-2868	39	134	∈	∈	NOUN
cana-2868	39	135	s	s	PART
cana-2868	39	136	,	,	PUNCT
cana-2868	39	137	then	then	ADV
cana-2868	39	138	t	t	PROPN
cana-2868	39	139	has	have	AUX
cana-2868	39	140	fixed	fix	VERB
cana-2868	39	141	point	point	NOUN
cana-2868	39	142	and	and	CCONJ
cana-2868	39	143	unique	unique	ADJ
cana-2868	39	144	.	.	PUNCT
cana-2868	40	1	in	in	ADP
cana-2868	40	2	2019	2019	NUM
cana-2868	40	3	,	,	PUNCT
cana-2868	40	4	faraji	faraji	NOUN
cana-2868	40	5	et.al	et.al	NOUN
cana-2868	40	6	[	[	PUNCT
cana-2868	40	7	8	8	NUM
cana-2868	40	8	]	]	PUNCT
cana-2868	40	9	proved	prove	VERB
cana-2868	40	10	a	a	DET
cana-2868	40	11	fixed	fix	VERB
cana-2868	40	12	point	point	NOUN
cana-2868	40	13	theorem	theorem	VERB
cana-2868	40	14	with	with	ADP
cana-2868	40	15	geraghty	geraghty	PROPN
cana-2868	40	16	–	–	PUNCT
cana-2868	40	17	type	type	NOUN
cana-2868	40	18	contractive	contractive	ADJ
cana-2868	40	19	maps	map	NOUN
cana-2868	40	20	in	in	ADP
cana-2868	40	21	bmetric	bmetric	ADJ
cana-2868	40	22	spaces	space	NOUN
cana-2868	40	23	.	.	PUNCT
cana-2868	41	1	theorem	theorem	VERB
cana-2868	41	2	1.7	1.7	NUM
cana-2868	41	3	[	[	PUNCT
cana-2868	41	4	8	8	NUM
cana-2868	41	5	]	]	PUNCT
cana-2868	41	6	let	let	VERB
cana-2868	41	7	(	(	PUNCT
cana-2868	41	8	m	m	NOUN
cana-2868	41	9	,	,	PUNCT
cana-2868	41	10	d	d	NOUN
cana-2868	41	11	,	,	PUNCT
cana-2868	41	12	v	v	NOUN
cana-2868	41	13	)	)	PUNCT
cana-2868	41	14	be	be	AUX
cana-2868	41	15	a	a	DET
cana-2868	41	16	completeb	completeb	NOUN
cana-2868	41	17	metric	metric	ADJ
cana-2868	41	18	space	space	NOUN
cana-2868	41	19	with	with	ADP
cana-2868	41	20	𝑣	𝑣	DET
cana-2868	41	21	≥	≥	NOUN
cana-2868	41	22	1	1	NUM
cana-2868	41	23	and	and	CCONJ
cana-2868	41	24	let	let	VERB
cana-2868	41	25	t	t	NOUN
cana-2868	41	26	:	:	PUNCT
cana-2868	41	27	m	m	VERB
cana-2868	41	28	→	→	SYM
cana-2868	41	29	𝑀	𝑀	PROPN
cana-2868	41	30	,	,	PUNCT
cana-2868	41	31	be	be	AUX
cana-2868	41	32	a	a	DET
cana-2868	41	33	self	self	NOUN
cana-2868	41	34	–	–	PUNCT
cana-2868	41	35	mapping	mapping	NOUN
cana-2868	41	36	and	and	CCONJ
cana-2868	41	37	if	if	SCONJ
cana-2868	41	38	there	there	PRON
cana-2868	41	39	exist	exist	VERB
cana-2868	41	40	𝛽	𝛽	PRON
cana-2868	41	41	∈	∈	PROPN
cana-2868	41	42	s	s	NOUN
cana-2868	41	43	,	,	PUNCT
cana-2868	41	44	𝑑(𝑇𝑢	𝑑(𝑇𝑢	NOUN
cana-2868	41	45	,	,	PUNCT
cana-2868	41	46	𝑇𝑣	𝑇𝑣	PROPN
cana-2868	41	47	)	)	PUNCT
cana-2868	41	48	≤	≤	NOUN
cana-2868	41	49	𝛽(𝐿(𝑢	𝛽(𝐿(𝑢	PROPN
cana-2868	41	50	,	,	PUNCT
cana-2868	41	51	𝑣))𝐿(𝑢	𝑣))𝐿(𝑢	PROPN
cana-2868	41	52	,	,	PUNCT
cana-2868	41	53	𝑣	𝑣	NOUN
cana-2868	41	54	)	)	PUNCT
cana-2868	41	55	,	,	PUNCT
cana-2868	41	56	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2868	41	57	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-2868	41	58	𝑢	𝑢	PROPN
cana-2868	41	59	,	,	PUNCT
cana-2868	41	60	𝑣	𝑣	PROPN
cana-2868	41	61	∈	∈	PROPN
cana-2868	41	62	𝑀	𝑀	PROPN
cana-2868	41	63	,	,	PUNCT
cana-2868	41	64	where	where	SCONJ
cana-2868	41	65	𝐿(𝑢	𝐿(𝑢	X
cana-2868	41	66	,	,	PUNCT
cana-2868	41	67	𝑣	𝑣	NOUN
cana-2868	41	68	)	)	PUNCT
cana-2868	41	69	=	=	SYM
cana-2868	41	70	max	max	PROPN
cana-2868	41	71	{	{	PUNCT
cana-2868	41	72	𝑑(𝑢	𝑑(𝑢	PROPN
cana-2868	41	73	,	,	PUNCT
cana-2868	41	74	𝑣	𝑣	NOUN
cana-2868	41	75	)	)	PUNCT
cana-2868	41	76	,	,	PUNCT
cana-2868	41	77	𝑑(𝑢	𝑑(𝑢	NOUN
cana-2868	41	78	,	,	PUNCT
cana-2868	41	79	𝑇𝑢	𝑇𝑢	PROPN
cana-2868	41	80	)	)	PUNCT
cana-2868	41	81	,	,	PUNCT
cana-2868	41	82	𝑑(𝑣	𝑑(𝑣	NOUN
cana-2868	41	83	,	,	PUNCT
cana-2868	41	84	𝑇𝑣	𝑇𝑣	PROPN
cana-2868	41	85	)	)	PUNCT
cana-2868	41	86	,	,	PUNCT
cana-2868	41	87	1	1	NUM
cana-2868	41	88	2𝑣	2𝑣	NOUN
cana-2868	41	89	[	[	X
cana-2868	41	90	𝑑(𝑢	𝑑(𝑢	ADJ
cana-2868	41	91	,	,	PUNCT
cana-2868	41	92	𝑇𝑣	𝑇𝑣	PROPN
cana-2868	41	93	)	)	PUNCT
cana-2868	41	94	+	+	NUM
cana-2868	41	95	𝑑(𝑣	𝑑(𝑣	NOUN
cana-2868	41	96	,	,	PUNCT
cana-2868	41	97	𝑇𝑢	𝑇𝑢	NOUN
cana-2868	41	98	)	)	PUNCT
cana-2868	41	99	]	]	PUNCT
cana-2868	41	100	}	}	PUNCT
cana-2868	41	101	then	then	ADV
cana-2868	41	102	t	t	PROPN
cana-2868	41	103	has	have	VERB
cana-2868	41	104	a	a	DET
cana-2868	41	105	unique	unique	ADJ
cana-2868	41	106	fixed	fix	VERB
cana-2868	41	107	point	point	NOUN
cana-2868	41	108	.	.	PUNCT
cana-2868	42	1	communications	communication	NOUN
cana-2868	42	2	on	on	ADP
cana-2868	42	3	applied	apply	VERB
cana-2868	42	4	nonlinear	nonlinear	ADJ
cana-2868	42	5	analysis	analysis	NOUN
cana-2868	42	6	issn	issn	NOUN
cana-2868	42	7	:	:	PUNCT
cana-2868	42	8	1074	1074	NUM
cana-2868	42	9	-	-	PUNCT
cana-2868	42	10	133x	133x	NUM
cana-2868	42	11	vol	vol	NOUN
cana-2868	42	12	32	32	NUM
cana-2868	42	13	no	no	NOUN
cana-2868	42	14	.	.	PUNCT
cana-2868	43	1	4s	4s	NUM
cana-2868	43	2	(	(	PUNCT
cana-2868	43	3	2025	2025	NUM
cana-2868	43	4	)	)	PUNCT
cana-2868	43	5	515	515	NUM
cana-2868	43	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	43	7	in	in	ADP
cana-2868	43	8	1975	1975	NUM
cana-2868	43	9	,	,	PUNCT
cana-2868	43	10	dass	dass	PROPN
cana-2868	43	11	and	and	CCONJ
cana-2868	43	12	gupta	gupta	NOUN
cana-2868	44	1	[	[	X
cana-2868	44	2	6	6	NUM
cana-2868	44	3	]	]	PUNCT
cana-2868	44	4	extended	extend	VERB
cana-2868	44	5	the	the	DET
cana-2868	44	6	bcp	bcp	NOUN
cana-2868	44	7	type	type	NOUN
cana-2868	44	8	of	of	ADP
cana-2868	44	9	rational	rational	ADJ
cana-2868	44	10	terms	term	NOUN
cana-2868	44	11	.	.	PUNCT
cana-2868	45	1	theorem	theorem	ADJ
cana-2868	45	2	1.8	1.8	NUM
cana-2868	45	3	.	.	PUNCT
cana-2868	46	1	(	(	PUNCT
cana-2868	46	2	[	[	X
cana-2868	46	3	3	3	NUM
cana-2868	46	4	]	]	PUNCT
cana-2868	46	5	)	)	PUNCT
cana-2868	46	6	.	.	PUNCT
cana-2868	47	1	let	let	VERB
cana-2868	47	2	(	(	PUNCT
cana-2868	47	3	h	h	NOUN
cana-2868	47	4	,	,	PUNCT
cana-2868	47	5	d	d	NOUN
cana-2868	47	6	)	)	PUNCT
cana-2868	47	7	be	be	AUX
cana-2868	47	8	a	a	DET
cana-2868	47	9	cms	cms	NOUN
cana-2868	47	10	and	and	CCONJ
cana-2868	47	11	h	h	NOUN
cana-2868	47	12	:	:	PUNCT
cana-2868	48	1	k	k	X
cana-2868	48	2	→	→	PUNCT
cana-2868	48	3	k	k	X
cana-2868	48	4	be	be	AUX
cana-2868	48	5	a	a	DET
cana-2868	48	6	mapping	mapping	NOUN
cana-2868	48	7	such	such	ADJ
cana-2868	48	8	that	that	SCONJ
cana-2868	48	9	there	there	PRON
cana-2868	48	10	exist	exist	VERB
cana-2868	48	11	α	α	PRON
cana-2868	48	12	,	,	PUNCT
cana-2868	48	13	β	β	X
cana-2868	48	14	≥	≥	NOUN
cana-2868	48	15	0	0	NUM
cana-2868	48	16	with	with	ADP
cana-2868	48	17	α	α	PROPN
cana-2868	48	18	+	+	X
cana-2868	48	19	β	β	X
cana-2868	48	20	<	<	X
cana-2868	48	21	1	1	NUM
cana-2868	48	22	satisfying	satisfy	VERB
cana-2868	48	23	𝑑(𝐻𝑢	𝑑(𝐻𝑢	NOUN
cana-2868	48	24	,	,	PUNCT
cana-2868	48	25	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	48	26	)	)	PUNCT
cana-2868	48	27	≤	≤	NOUN
cana-2868	48	28	𝛼	𝛼	DET
cana-2868	48	29	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑢,𝐻𝑣	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑢,𝐻𝑣	PROPN
cana-2868	48	30	)	)	PUNCT
cana-2868	48	31	]	]	PUNCT
cana-2868	49	1	1+𝑑(𝑢,𝑣	1+𝑑(𝑢,𝑣	X
cana-2868	49	2	)	)	PUNCT
cana-2868	50	1	+	+	CCONJ
cana-2868	50	2	β𝑑(𝑢	β𝑑(𝑢	NOUN
cana-2868	50	3	,	,	PUNCT
cana-2868	50	4	𝑣	𝑣	NOUN
cana-2868	50	5	)	)	PUNCT
cana-2868	50	6	for	for	ADP
cana-2868	50	7	all	all	DET
cana-2868	50	8	u	u	NOUN
cana-2868	50	9	,	,	PUNCT
cana-2868	50	10	v	v	PROPN
cana-2868	50	11	∈	∈	PROPN
cana-2868	50	12	h.	h.	NOUN
cana-2868	50	13	then	then	ADV
cana-2868	50	14	h	h	PROPN
cana-2868	50	15	has	have	VERB
cana-2868	50	16	a	a	DET
cana-2868	50	17	unique	unique	ADJ
cana-2868	50	18	fixed	fix	VERB
cana-2868	50	19	point	point	NOUN
cana-2868	50	20	.	.	PUNCT
cana-2868	51	1	lemma	lemma	PROPN
cana-2868	51	2	[	[	PUNCT
cana-2868	51	3	9	9	NUM
cana-2868	51	4	]	]	PUNCT
cana-2868	51	5	let	let	VERB
cana-2868	51	6	(	(	PUNCT
cana-2868	51	7	m	m	NOUN
cana-2868	51	8	,	,	PUNCT
cana-2868	51	9	d	d	NOUN
cana-2868	51	10	,	,	PUNCT
cana-2868	51	11	v	v	NOUN
cana-2868	51	12	)	)	PUNCT
cana-2868	51	13	be	be	AUX
cana-2868	51	14	a	a	DET
cana-2868	51	15	metric	metric	ADJ
cana-2868	51	16	space	space	NOUN
cana-2868	51	17	with	with	ADP
cana-2868	51	18	𝑣	𝑣	DET
cana-2868	51	19	≥	≥	NOUN
cana-2868	51	20	1	1	NUM
cana-2868	51	21	and	and	CCONJ
cana-2868	51	22	let	let	VERB
cana-2868	51	23	𝑇	𝑇	PROPN
cana-2868	51	24	:	:	PUNCT
cana-2868	51	25	𝑀	𝑀	PROPN
cana-2868	51	26	→	→	SYM
cana-2868	51	27	𝑀	𝑀	PROPN
cana-2868	51	28	be	be	VERB
cana-2868	51	29	a	a	DET
cana-2868	51	30	self	self	NOUN
cana-2868	51	31	mapping	mapping	NOUN
cana-2868	51	32	.	.	PUNCT
cana-2868	52	1	let	let	VERB
cana-2868	52	2	𝑥0	𝑥0	PROPN
cana-2868	52	3	∈	∈	PROPN
cana-2868	52	4	𝑀	𝑀	PROPN
cana-2868	52	5	be	be	AUX
cana-2868	52	6	given	give	VERB
cana-2868	52	7	and	and	CCONJ
cana-2868	52	8	{	{	PUNCT
cana-2868	52	9	𝑥𝑛	𝑥𝑛	AUX
cana-2868	52	10	}	}	PUNCT
cana-2868	52	11	be	be	AUX
cana-2868	52	12	a	a	DET
cana-2868	52	13	sequence	sequence	NOUN
cana-2868	52	14	in	in	ADP
cana-2868	52	15	m	m	PRON
cana-2868	52	16	such	such	ADJ
cana-2868	52	17	that	that	SCONJ
cana-2868	52	18	𝑥𝑛	𝑥𝑛	PROPN
cana-2868	52	19	=	=	SYM
cana-2868	52	20	𝑇𝑥𝑛−1	𝑇𝑥𝑛−1	PROPN
cana-2868	52	21	for	for	ADP
cana-2868	52	22	all	all	DET
cana-2868	52	23	n	n	NOUN
cana-2868	52	24	in	in	ADP
cana-2868	52	25	n	n	CCONJ
cana-2868	52	26	,	,	PUNCT
cana-2868	52	27	the	the	DET
cana-2868	52	28	sequence	sequence	NOUN
cana-2868	52	29	defined	define	VERB
cana-2868	52	30	by	by	ADP
cana-2868	52	31	an	an	DET
cana-2868	52	32	=	=	PROPN
cana-2868	52	33	max	max	NOUN
cana-2868	52	34	{	{	PUNCT
cana-2868	52	35	𝑑(𝑥𝑝	𝑑(𝑥𝑝	NUM
cana-2868	52	36	,	,	PUNCT
cana-2868	52	37	x𝑞)|0	x𝑞)|0	PROPN
cana-2868	52	38	≤	≤	PROPN
cana-2868	52	39	𝑝	𝑝	PROPN
cana-2868	52	40	,	,	PUNCT
cana-2868	52	41	𝑞	𝑞	PROPN
cana-2868	52	42	≤	≤	NOUN
cana-2868	52	43	𝑛	𝑛	PRON
cana-2868	52	44	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-2868	52	45	𝑝	𝑝	PROPN
cana-2868	52	46	,	,	PUNCT
cana-2868	52	47	𝑞	𝑞	PROPN
cana-2868	52	48	∈	∈	PROPN
cana-2868	52	49	n0	n0	PROPN
cana-2868	52	50	}	}	PUNCT
cana-2868	52	51	,	,	PUNCT
cana-2868	52	52	for	for	ADP
cana-2868	52	53	n	n	DET
cana-2868	52	54	∈	∈	PROPN
cana-2868	52	55	n0	n0	PROPN
cana-2868	52	56	.	.	PUNCT
cana-2868	53	1	if	if	SCONJ
cana-2868	53	2	t	t	PROPN
cana-2868	53	3	satisfies	satisfy	VERB
cana-2868	53	4	the	the	DET
cana-2868	53	5	contractivity	contractivity	NOUN
cana-2868	53	6	condition	condition	NOUN
cana-2868	53	7	in	in	ADP
cana-2868	53	8	(	(	PUNCT
cana-2868	53	9	1	1	NUM
cana-2868	53	10	)	)	PUNCT
cana-2868	53	11	,	,	PUNCT
cana-2868	53	12	then	then	ADV
cana-2868	53	13	{	{	PUNCT
cana-2868	53	14	an	an	PRON
cana-2868	53	15	}	}	PUNCT
cana-2868	53	16	is	be	AUX
cana-2868	53	17	bounded	bound	VERB
cana-2868	53	18	.	.	PUNCT
cana-2868	54	1	in	in	ADP
cana-2868	54	2	2024	2024	NUM
cana-2868	54	3	,	,	PUNCT
cana-2868	54	4	kalo.et.al	kalo.et.al	PROPN
cana-2868	54	5	[	[	X
cana-2868	54	6	1	1	NUM
cana-2868	54	7	]	]	PUNCT
cana-2868	54	8	proved	prove	VERB
cana-2868	54	9	fixed	fix	VERB
cana-2868	54	10	point	point	NOUN
cana-2868	54	11	theorems	theorem	NOUN
cana-2868	54	12	in	in	ADP
cana-2868	54	13	geraghty	geraghty	NOUN
cana-2868	54	14	-	-	PUNCT
cana-2868	54	15	ciric	ciric	ADJ
cana-2868	54	16	-	-	PUNCT
cana-2868	54	17	type	type	NOUN
cana-2868	54	18	contraction	contraction	NOUN
cana-2868	54	19	mapping	mapping	NOUN
cana-2868	54	20	in	in	ADP
cana-2868	54	21	b	b	PROPN
cana-2868	54	22	metric	metric	ADJ
cana-2868	54	23	spaces	space	NOUN
cana-2868	54	24	theorem	theorem	VERB
cana-2868	54	25	1.10[1	1.10[1	X
cana-2868	54	26	]	]	PUNCT
cana-2868	54	27	let	let	VERB
cana-2868	54	28	(	(	PUNCT
cana-2868	54	29	m	m	NOUN
cana-2868	54	30	,	,	PUNCT
cana-2868	54	31	d	d	NOUN
cana-2868	54	32	,	,	PUNCT
cana-2868	54	33	v	v	NOUN
cana-2868	54	34	)	)	PUNCT
cana-2868	54	35	be	be	AUX
cana-2868	54	36	a	a	DET
cana-2868	54	37	complete	complete	ADJ
cana-2868	54	38	b	b	X
cana-2868	54	39	-	-	PUNCT
cana-2868	54	40	metric	metric	ADJ
cana-2868	54	41	space	space	NOUN
cana-2868	54	42	with	with	ADP
cana-2868	54	43	t	t	PROPN
cana-2868	54	44	≥	≥	NUM
cana-2868	54	45	1	1	NUM
cana-2868	54	46	and	and	CCONJ
cana-2868	54	47	let	let	VERB
cana-2868	54	48	t	t	NOUN
cana-2868	54	49	:	:	PUNCT
cana-2868	54	50	m	m	PROPN
cana-2868	54	51	→	→	PUNCT
cana-2868	54	52	m	m	AUX
cana-2868	54	53	be	be	AUX
cana-2868	54	54	a	a	DET
cana-2868	54	55	selfmapping	selfmapping	ADJ
cana-2868	54	56	ciric	ciric	ADJ
cana-2868	54	57	-type	-type	NOUN
cana-2868	54	58	geraghty	geraghty	NOUN
cana-2868	54	59	contraction	contraction	NOUN
cana-2868	54	60	(	(	PUNCT
cana-2868	54	61	1	1	NUM
cana-2868	54	62	)	)	PUNCT
cana-2868	54	63	,	,	PUNCT
cana-2868	54	64	then	then	ADV
cana-2868	54	65	t	t	PROPN
cana-2868	54	66	has	have	VERB
cana-2868	54	67	a	a	DET
cana-2868	54	68	unique	unique	ADJ
cana-2868	54	69	fixed	fix	VERB
cana-2868	54	70	point	point	NOUN
cana-2868	54	71	x	x	NOUN
cana-2868	54	72	*	*	PUNCT
cana-2868	54	73	in	in	ADP
cana-2868	54	74	k.	k.	PROPN
cana-2868	55	1	this	this	DET
cana-2868	55	2	lemma	lemma	PROPN
cana-2868	55	3	can	can	AUX
cana-2868	55	4	use	use	VERB
cana-2868	55	5	to	to	PART
cana-2868	55	6	prove	prove	VERB
cana-2868	55	7	results	result	NOUN
cana-2868	55	8	.	.	PUNCT
cana-2868	56	1	lemma	lemma	PROPN
cana-2868	56	2	[	[	PUNCT
cana-2868	56	3	1.11	1.11	NUM
cana-2868	56	4	]	]	PUNCT
cana-2868	56	5	.	.	PUNCT
cana-2868	57	1	let	let	VERB
cana-2868	57	2	(	(	PUNCT
cana-2868	57	3	m	m	NOUN
cana-2868	57	4	,	,	PUNCT
cana-2868	57	5	d	d	NOUN
cana-2868	57	6	,	,	PUNCT
cana-2868	57	7	v	v	NOUN
cana-2868	57	8	)	)	PUNCT
cana-2868	57	9	be	be	AUX
cana-2868	57	10	a	a	DET
cana-2868	57	11	b	b	NOUN
cana-2868	57	12	-	-	PUNCT
cana-2868	57	13	metric	metric	ADJ
cana-2868	57	14	space	space	NOUN
cana-2868	57	15	with	with	ADP
cana-2868	57	16	t	t	PROPN
cana-2868	57	17	≥	≥	NUM
cana-2868	57	18	1	1	NUM
cana-2868	57	19	and	and	CCONJ
cana-2868	57	20	let	let	VERB
cana-2868	57	21	{	{	PUNCT
cana-2868	57	22	an	an	NOUN
cana-2868	57	23	}	}	PUNCT
cana-2868	57	24	and	and	CCONJ
cana-2868	57	25	{	{	PUNCT
cana-2868	57	26	bn	bn	X
cana-2868	57	27	}	}	PUNCT
cana-2868	57	28	be	be	AUX
cana-2868	57	29	bconvergent	bconvergent	ADJ
cana-2868	57	30	to	to	ADP
cana-2868	57	31	x	x	PRON
cana-2868	57	32	,	,	PUNCT
cana-2868	57	33	y	y	PROPN
cana-2868	57	34	in	in	ADP
cana-2868	57	35	m	m	PROPN
cana-2868	57	36	,	,	PUNCT
cana-2868	57	37	then	then	ADV
cana-2868	57	38	we	we	PRON
cana-2868	57	39	have	have	VERB
cana-2868	57	40	1	1	NUM
cana-2868	57	41	𝑣2	𝑣2	NUM
cana-2868	57	42	𝑑(𝑎	𝑑(𝑎	PROPN
cana-2868	57	43	,	,	PUNCT
cana-2868	57	44	𝑏	𝑏	NOUN
cana-2868	57	45	)	)	PUNCT
cana-2868	57	46	≤	≤	NOUN
cana-2868	57	47	liminf	liminf	ADJ
cana-2868	57	48	𝑛→∞	𝑛→∞	NUM
cana-2868	57	49	𝑑(𝑎𝑛	𝑑(𝑎𝑛	NOUN
cana-2868	57	50	,	,	PUNCT
cana-2868	57	51	𝑏𝑛	𝑏𝑛	NOUN
cana-2868	57	52	)	)	PUNCT
cana-2868	57	53	≤	≤	NOUN
cana-2868	57	54	limsup	limsup	NOUN
cana-2868	57	55	𝑛→∞	𝑛→∞	NUM
cana-2868	57	56	𝑑(𝑎𝑛	𝑑(𝑎𝑛	NUM
cana-2868	57	57	,	,	PUNCT
cana-2868	57	58	𝑏𝑦𝑛	𝑏𝑦𝑛	NOUN
cana-2868	57	59	)	)	PUNCT
cana-2868	57	60	≤	≤	NOUN
cana-2868	57	61	𝑣2	𝑣2	NUM
cana-2868	57	62	𝑑(𝑎	𝑑(𝑎	PROPN
cana-2868	57	63	,	,	PUNCT
cana-2868	57	64	𝑏	𝑏	NOUN
cana-2868	57	65	)	)	PUNCT
cana-2868	57	66	in	in	ADP
cana-2868	57	67	particular	particular	ADJ
cana-2868	57	68	,	,	PUNCT
cana-2868	57	69	if	if	SCONJ
cana-2868	57	70	a	a	DET
cana-2868	57	71	=	=	NOUN
cana-2868	57	72	b	b	NOUN
cana-2868	57	73	,	,	PUNCT
cana-2868	57	74	we	we	PRON
cana-2868	57	75	have	have	VERB
cana-2868	57	76	lim	lim	NOUN
cana-2868	57	77	𝑛→∞	𝑛→∞	NUM
cana-2868	57	78	𝑑(𝑎𝑛	𝑑(𝑎𝑛	PROPN
cana-2868	57	79	,	,	PUNCT
cana-2868	57	80	𝑏𝑛	𝑏𝑛	NOUN
cana-2868	57	81	)	)	PUNCT
cana-2868	57	82	=	=	SYM
cana-2868	57	83	0	0	X
cana-2868	57	84	.	.	PUNCT
cana-2868	58	1	and	and	CCONJ
cana-2868	58	2	for	for	ADP
cana-2868	58	3	any	any	DET
cana-2868	58	4	c	c	NOUN
cana-2868	58	5	in	in	ADP
cana-2868	58	6	m	m	PROPN
cana-2868	58	7	,	,	PUNCT
cana-2868	58	8	1	1	NUM
cana-2868	58	9	𝑣	𝑣	ADP
cana-2868	58	10	𝑑(𝑎	𝑑(𝑎	PROPN
cana-2868	58	11	,	,	PUNCT
cana-2868	58	12	𝑐	𝑐	NOUN
cana-2868	58	13	)	)	PUNCT
cana-2868	58	14	≤	≤	NOUN
cana-2868	58	15	liminf	liminf	ADJ
cana-2868	58	16	𝑛→∞	𝑛→∞	NUM
cana-2868	58	17	𝑑(𝑎𝑛	𝑑(𝑎𝑛	NUM
cana-2868	58	18	,	,	PUNCT
cana-2868	58	19	𝑐	𝑐	NOUN
cana-2868	58	20	)	)	PUNCT
cana-2868	58	21	≤	≤	NOUN
cana-2868	58	22	limsup	limsup	NOUN
cana-2868	58	23	𝑛→∞	𝑛→∞	NUM
cana-2868	58	24	𝑑(𝑎𝑛	𝑑(𝑎𝑛	NUM
cana-2868	58	25	,	,	PUNCT
cana-2868	58	26	𝑐𝑧	𝑐𝑧	NOUN
cana-2868	58	27	)	)	PUNCT
cana-2868	58	28	≤	≤	NOUN
cana-2868	58	29	𝑣	𝑣	ADP
cana-2868	58	30	𝑑(𝑎	𝑑(𝑎	PROPN
cana-2868	58	31	,	,	PUNCT
cana-2868	58	32	𝑐	𝑐	NOUN
cana-2868	58	33	)	)	PUNCT
cana-2868	58	34	.	.	PUNCT
cana-2868	59	1	main	main	ADJ
cana-2868	59	2	results	result	NOUN
cana-2868	59	3	:	:	PUNCT
cana-2868	59	4	now	now	ADV
cana-2868	59	5	,	,	PUNCT
cana-2868	59	6	we	we	PRON
cana-2868	59	7	define	define	VERB
cana-2868	59	8	ciric	ciric	ADJ
cana-2868	59	9	type	type	NOUN
cana-2868	59	10	geraghty	geraghty	PROPN
cana-2868	59	11	contraction	contraction	NOUN
cana-2868	59	12	with	with	ADP
cana-2868	59	13	rational	rational	ADJ
cana-2868	59	14	type	type	NOUN
cana-2868	59	15	of	of	ADP
cana-2868	59	16	expressions	expression	NOUN
cana-2868	59	17	in	in	ADP
cana-2868	59	18	b	b	NOUN
cana-2868	59	19	metric	metric	ADJ
cana-2868	59	20	spaces	space	NOUN
cana-2868	59	21	.	.	PUNCT
cana-2868	60	1	definition	definition	NOUN
cana-2868	60	2	2.1	2.1	NUM
cana-2868	60	3	:	:	PUNCT
cana-2868	60	4	a	a	DET
cana-2868	60	5	mapping	mapping	NOUN
cana-2868	60	6	h	h	NOUN
cana-2868	60	7	:	:	PUNCT
cana-2868	60	8	k	k	X
cana-2868	60	9	→	→	PUNCT
cana-2868	60	10	𝐾	𝐾	PROPN
cana-2868	60	11	on	on	ADP
cana-2868	60	12	a	a	DET
cana-2868	60	13	bmetric	bmetric	ADJ
cana-2868	60	14	space	space	NOUN
cana-2868	60	15	(	(	PUNCT
cana-2868	60	16	k	k	X
cana-2868	60	17	,	,	PUNCT
cana-2868	60	18	d	d	PROPN
cana-2868	60	19	,	,	PUNCT
cana-2868	60	20	t	t	PROPN
cana-2868	60	21	)	)	PUNCT
cana-2868	60	22	with	with	ADP
cana-2868	60	23	t	t	PROPN
cana-2868	60	24	≥	≥	NOUN
cana-2868	60	25	1	1	NUM
cana-2868	60	26	is	be	AUX
cana-2868	60	27	called	call	VERB
cana-2868	60	28	a	a	DET
cana-2868	60	29	–	–	PUNCT
cana-2868	60	30	ciric	ciric	ADJ
cana-2868	60	31	type	type	NOUN
cana-2868	60	32	geraghty	geraghty	PROPN
cana-2868	60	33	contraction	contraction	NOUN
cana-2868	60	34	with	with	ADP
cana-2868	60	35	rational	rational	ADJ
cana-2868	60	36	mapping	mapping	NOUN
cana-2868	60	37	,	,	PUNCT
cana-2868	60	38	if	if	SCONJ
cana-2868	60	39	there	there	PRON
cana-2868	60	40	exists	exist	VERB
cana-2868	60	41	𝛽	𝛽	PROPN
cana-2868	60	42	∈	∈	NOUN
cana-2868	60	43	s	s	VERB
cana-2868	60	44	such	such	ADJ
cana-2868	60	45	that	that	SCONJ
cana-2868	60	46	𝑑(𝐻𝑢	𝑑(𝐻𝑢	NOUN
cana-2868	60	47	,	,	PUNCT
cana-2868	60	48	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	60	49	)	)	PUNCT
cana-2868	60	50	≤	≤	NOUN
cana-2868	60	51	𝑀(𝑢	𝑀(𝑢	PROPN
cana-2868	60	52	,	,	PUNCT
cana-2868	60	53	𝑣	𝑣	NOUN
cana-2868	60	54	)	)	PUNCT
cana-2868	60	55	,	,	PUNCT
cana-2868	60	56	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2868	60	57	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-2868	60	58	𝑢	𝑢	PROPN
cana-2868	60	59	,	,	PUNCT
cana-2868	60	60	𝑣	𝑣	PRON
cana-2868	60	61	∈	∈	PROPN
cana-2868	60	62	𝐾,	𝐾,	PROPN
cana-2868	60	63	…	…	PUNCT
cana-2868	60	64	…	…	PUNCT
cana-2868	60	65	2.1.1	2.1.1	NUM
cana-2868	60	66	where	where	SCONJ
cana-2868	60	67	𝑀(𝑢	𝑀(𝑢	NOUN
cana-2868	60	68	,	,	PUNCT
cana-2868	60	69	𝑣	𝑣	NOUN
cana-2868	60	70	)	)	PUNCT
cana-2868	60	71	=	=	PUNCT
cana-2868	60	72	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2868	60	73	{	{	PUNCT
cana-2868	60	74	𝛽	𝛽	NOUN
cana-2868	60	75	(	(	PUNCT
cana-2868	60	76	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	PROPN
cana-2868	60	77	)	)	PUNCT
cana-2868	60	78	]	]	PUNCT
cana-2868	61	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	61	2	)	)	PUNCT
cana-2868	61	3	)	)	PUNCT
cana-2868	62	1	(	(	PUNCT
cana-2868	62	2	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	NOUN
cana-2868	62	3	)	)	PUNCT
cana-2868	62	4	]	]	PUNCT
cana-2868	63	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	63	2	)	)	PUNCT
cana-2868	63	3	)	)	PUNCT
cana-2868	63	4	,	,	PUNCT
cana-2868	63	5	𝛽	𝛽	NOUN
cana-2868	63	6	(	(	PUNCT
cana-2868	63	7	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	PROPN
cana-2868	63	8	)	)	PUNCT
cana-2868	63	9	]	]	PUNCT
cana-2868	64	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	64	2	)	)	PUNCT
cana-2868	64	3	)	)	PUNCT
cana-2868	65	1	(	(	PUNCT
cana-2868	65	2	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	NOUN
cana-2868	65	3	)	)	PUNCT
cana-2868	65	4	]	]	PUNCT
cana-2868	66	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	66	2	)	)	PUNCT
cana-2868	66	3	)	)	PUNCT
cana-2868	66	4	,	,	PUNCT
cana-2868	66	5	𝛽	𝛽	NOUN
cana-2868	66	6	(	(	PUNCT
cana-2868	66	7	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	PROPN
cana-2868	66	8	)	)	PUNCT
cana-2868	66	9	]	]	PUNCT
cana-2868	67	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	67	2	)	)	PUNCT
cana-2868	67	3	)	)	PUNCT
cana-2868	68	1	(	(	PUNCT
cana-2868	68	2	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	NOUN
cana-2868	68	3	)	)	PUNCT
cana-2868	68	4	]	]	PUNCT
cana-2868	69	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	69	2	)	)	PUNCT
cana-2868	69	3	)	)	PUNCT
cana-2868	69	4	}	}	PUNCT
cana-2868	69	5	first	first	ADV
cana-2868	69	6	,	,	PUNCT
cana-2868	69	7	we	we	PRON
cana-2868	69	8	prove	prove	VERB
cana-2868	69	9	that	that	SCONJ
cana-2868	69	10	sequence	sequence	NOUN
cana-2868	69	11	is	be	AUX
cana-2868	69	12	{	{	PUNCT
cana-2868	69	13	bn	bn	AUX
cana-2868	69	14	}	}	PUNCT
cana-2868	69	15	is	be	AUX
cana-2868	69	16	bounded	bound	VERB
cana-2868	69	17	.	.	PUNCT
cana-2868	70	1	theorem	theorem	NOUN
cana-2868	70	2	2	2	NUM
cana-2868	70	3	.	.	NOUN
cana-2868	70	4	2	2	NUM
cana-2868	70	5	:	:	PUNCT
cana-2868	70	6	let	let	VERB
cana-2868	70	7	(	(	PUNCT
cana-2868	70	8	k	k	X
cana-2868	70	9	,	,	PUNCT
cana-2868	70	10	d	d	PROPN
cana-2868	70	11	,	,	PUNCT
cana-2868	70	12	t	t	PROPN
cana-2868	70	13	)	)	PUNCT
cana-2868	70	14	be	be	AUX
cana-2868	70	15	a	a	DET
cana-2868	70	16	b	b	NOUN
cana-2868	70	17	-	-	PUNCT
cana-2868	70	18	metric	metric	ADJ
cana-2868	70	19	space	space	NOUN
cana-2868	70	20	with	with	ADP
cana-2868	70	21	t	t	PROPN
cana-2868	70	22	≥	≥	NUM
cana-2868	70	23	1	1	NUM
cana-2868	70	24	and	and	CCONJ
cana-2868	70	25	let	let	VERB
cana-2868	70	26	h	h	NOUN
cana-2868	70	27	:	:	PUNCT
cana-2868	70	28	k	k	X
cana-2868	70	29	→	→	PUNCT
cana-2868	70	30	k	k	X
cana-2868	70	31	be	be	AUX
cana-2868	70	32	a	a	DET
cana-2868	70	33	selfmapping	selfmapping	NOUN
cana-2868	70	34	.	.	PUNCT
cana-2868	71	1	let	let	VERB
cana-2868	71	2	s0	s0	PROPN
cana-2868	71	3	∈	∈	PROPN
cana-2868	71	4	k	k	PROPN
cana-2868	71	5	be	be	AUX
cana-2868	71	6	given	give	VERB
cana-2868	71	7	and	and	CCONJ
cana-2868	71	8	{	{	PUNCT
cana-2868	71	9	sn	sn	AUX
cana-2868	71	10	}	}	PUNCT
cana-2868	71	11	be	be	AUX
cana-2868	71	12	a	a	DET
cana-2868	71	13	sequence	sequence	NOUN
cana-2868	71	14	in	in	ADP
cana-2868	71	15	k	k	PROPN
cana-2868	71	16	,	,	PUNCT
cana-2868	71	17	sn	sn	PROPN
cana-2868	71	18	=	=	PUNCT
cana-2868	72	1	h	h	PROPN
cana-2868	72	2	sn–1	sn–1	VERB
cana-2868	72	3	for	for	ADP
cana-2868	72	4	all	all	PRON
cana-2868	72	5	n	n	DET
cana-2868	72	6	∈	∈	PROPN
cana-2868	72	7	n.	n.	NOUN
cana-2868	72	8	the	the	DET
cana-2868	72	9	sequence	sequence	NOUN
cana-2868	72	10	bn	bn	PROPN
cana-2868	72	11	=	=	ADJ
cana-2868	73	1	max	max	X
cana-2868	73	2	{	{	PUNCT
cana-2868	73	3	d(sp	d(sp	PROPN
cana-2868	73	4	,	,	PUNCT
cana-2868	73	5	sq)|0	sq)|0	PROPN
cana-2868	73	6	≤	≤	PUNCT
cana-2868	73	7	p	p	X
cana-2868	73	8	,	,	PUNCT
cana-2868	73	9	q	q	PROPN
cana-2868	73	10	≤	≤	NOUN
cana-2868	73	11	n	n	CCONJ
cana-2868	73	12	and	and	CCONJ
cana-2868	73	13	p	p	X
cana-2868	73	14	,	,	PUNCT
cana-2868	73	15	q	q	PROPN
cana-2868	73	16	∈	∈	PROPN
cana-2868	73	17	n0	n0	PROPN
cana-2868	73	18	}	}	PUNCT
cana-2868	73	19	(	(	PUNCT
cana-2868	73	20	2.2.1	2.2.1	NUM
cana-2868	73	21	)	)	PUNCT
cana-2868	73	22	communications	communication	NOUN
cana-2868	73	23	on	on	ADP
cana-2868	73	24	applied	apply	VERB
cana-2868	73	25	nonlinear	nonlinear	ADJ
cana-2868	73	26	analysis	analysis	NOUN
cana-2868	73	27	issn	issn	NOUN
cana-2868	73	28	:	:	PUNCT
cana-2868	73	29	1074	1074	NUM
cana-2868	73	30	-	-	PUNCT
cana-2868	73	31	133x	133x	NUM
cana-2868	73	32	vol	vol	NOUN
cana-2868	73	33	32	32	NUM
cana-2868	73	34	no	no	NOUN
cana-2868	73	35	.	.	PUNCT
cana-2868	74	1	4s	4s	NUM
cana-2868	74	2	(	(	PUNCT
cana-2868	74	3	2025	2025	NUM
cana-2868	74	4	)	)	PUNCT
cana-2868	74	5	516	516	NUM
cana-2868	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	74	7	for	for	ADP
cana-2868	74	8	n	n	DET
cana-2868	74	9	∈	∈	PROPN
cana-2868	74	10	n0	n0	PROPN
cana-2868	74	11	.	.	PUNCT
cana-2868	75	1	if	if	SCONJ
cana-2868	75	2	h	h	NOUN
cana-2868	75	3	satisfies	satisfy	VERB
cana-2868	75	4	the	the	DET
cana-2868	75	5	contractivity	contractivity	NOUN
cana-2868	75	6	condition	condition	NOUN
cana-2868	75	7	in	in	ADP
cana-2868	75	8	(	(	PUNCT
cana-2868	75	9	2.1.1	2.1.1	NUM
cana-2868	75	10	)	)	PUNCT
cana-2868	75	11	,	,	PUNCT
cana-2868	75	12	then	then	ADV
cana-2868	75	13	{	{	PUNCT
cana-2868	75	14	bn	bn	X
cana-2868	75	15	}	}	PUNCT
cana-2868	75	16	is	be	AUX
cana-2868	75	17	bounded	bound	VERB
cana-2868	75	18	{	{	PUNCT
cana-2868	75	19	d(sp	d(sp	INTJ
cana-2868	75	20	,	,	PUNCT
cana-2868	75	21	sq)/	sq)/	NOUN
cana-2868	75	22	0	0	NUM
cana-2868	75	23	≤	≤	PROPN
cana-2868	75	24	p	p	X
cana-2868	75	25	,	,	PUNCT
cana-2868	75	26	q	q	PROPN
cana-2868	75	27	≤	≤	NOUN
cana-2868	75	28	n	n	CCONJ
cana-2868	75	29	and	and	CCONJ
cana-2868	75	30	p	p	X
cana-2868	75	31	,	,	PUNCT
cana-2868	75	32	q	q	PROPN
cana-2868	75	33	∈	∈	PROPN
cana-2868	75	34	n0	n0	PROPN
cana-2868	75	35	}	}	PUNCT
cana-2868	75	36	for	for	ADP
cana-2868	75	37	n	n	DET
cana-2868	75	38	∈	∈	PROPN
cana-2868	75	39	n0	n0	PROPN
cana-2868	75	40	.	.	PUNCT
cana-2868	76	1	if	if	SCONJ
cana-2868	76	2	k	k	PROPN
cana-2868	76	3	satisfies	satisfy	VERB
cana-2868	76	4	condition	condition	NOUN
cana-2868	76	5	of	of	ADP
cana-2868	76	6	(	(	PUNCT
cana-2868	76	7	2.1.1	2.1.1	NUM
cana-2868	76	8	)	)	PUNCT
cana-2868	76	9	,	,	PUNCT
cana-2868	76	10	then	then	ADV
cana-2868	76	11	{	{	PUNCT
cana-2868	76	12	bn	bn	NOUN
cana-2868	76	13	}	}	PUNCT
cana-2868	76	14	is	be	AUX
cana-2868	76	15	bounded	bound	VERB
cana-2868	76	16	.	.	PUNCT
cana-2868	77	1	proof	proof	NOUN
cana-2868	77	2	:	:	PUNCT
cana-2868	77	3	let	let	VERB
cana-2868	77	4	n	n	PRON
cana-2868	77	5	∈	∈	PROPN
cana-2868	77	6	n.	n.	NOUN
cana-2868	77	7	then	then	ADV
cana-2868	77	8	for	for	ADP
cana-2868	77	9	any	any	DET
cana-2868	77	10	p	p	NOUN
cana-2868	77	11	,	,	PUNCT
cana-2868	77	12	q	q	PUNCT
cana-2868	77	13	∈	∈	PROPN
cana-2868	77	14	n	n	NOUN
cana-2868	77	15	with	with	ADP
cana-2868	77	16	1	1	NUM
cana-2868	77	17	≤	≤	NUM
cana-2868	77	18	𝑝	𝑝	NOUN
cana-2868	77	19	,	,	PUNCT
cana-2868	77	20	𝑞	𝑞	PROPN
cana-2868	77	21	≤	≤	PROPN
cana-2868	77	22	𝑛	𝑛	PROPN
cana-2868	77	23	,	,	PUNCT
cana-2868	77	24	from	from	ADP
cana-2868	77	25	condition	condition	NOUN
cana-2868	77	26	(	(	PUNCT
cana-2868	77	27	2.1.1	2.1.1	NUM
cana-2868	77	28	)	)	PUNCT
cana-2868	77	29	,	,	PUNCT
cana-2868	77	30	we	we	PRON
cana-2868	77	31	have	have	VERB
cana-2868	77	32	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	77	33	,	,	PUNCT
cana-2868	77	34	s𝑞	s𝑞	ADP
cana-2868	77	35	)	)	PUNCT
cana-2868	78	1	=	=	SYM
cana-2868	78	2	𝑑(𝐻𝑠𝑝−1	𝑑(𝐻𝑠𝑝−1	PROPN
cana-2868	78	3	,	,	PUNCT
cana-2868	78	4	𝐻𝑠𝑞−1	𝐻𝑠𝑞−1	NOUN
cana-2868	78	5	)	)	PUNCT
cana-2868	78	6	≤	≤	NUM
cana-2868	78	7	𝑀(𝑠𝑝−1	𝑀(𝑠𝑝−1	PROPN
cana-2868	78	8	,	,	PUNCT
cana-2868	78	9	𝑠𝑞−1	𝑠𝑞−1	NOUN
cana-2868	78	10	)	)	PUNCT
cana-2868	78	11	=	=	SYM
cana-2868	78	12	max	max	PROPN
cana-2868	78	13	{	{	PUNCT
cana-2868	78	14	𝛽	𝛽	PROPN
cana-2868	78	15	(	(	PUNCT
cana-2868	78	16	𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1	𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1	PROPN
cana-2868	78	17	)	)	PUNCT
cana-2868	78	18	]	]	PUNCT
cana-2868	78	19	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NUM
cana-2868	78	20	)	)	PUNCT
cana-2868	78	21	)	)	PUNCT
cana-2868	78	22	(	(	PUNCT
cana-2868	78	23	𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1	𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1	PROPN
cana-2868	78	24	)	)	PUNCT
cana-2868	78	25	]	]	PUNCT
cana-2868	78	26	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NUM
cana-2868	78	27	)	)	PUNCT
cana-2868	78	28	)	)	PUNCT
cana-2868	78	29	,	,	PUNCT
cana-2868	78	30	𝛽	𝛽	NOUN
cana-2868	78	31	(	(	PUNCT
cana-2868	78	32	𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NOUN
cana-2868	78	33	)	)	PUNCT
cana-2868	78	34	]	]	PUNCT
cana-2868	78	35	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	NUM
cana-2868	78	36	)	)	PUNCT
cana-2868	78	37	)	)	PUNCT
cana-2868	78	38	(	(	PUNCT
cana-2868	78	39	𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NOUN
cana-2868	78	40	)	)	PUNCT
cana-2868	78	41	]	]	PUNCT
cana-2868	78	42	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	NUM
cana-2868	78	43	)	)	PUNCT
cana-2868	78	44	)	)	PUNCT
cana-2868	78	45	,	,	PUNCT
cana-2868	78	46	𝛽	𝛽	NOUN
cana-2868	78	47	(	(	PUNCT
cana-2868	78	48	𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1	𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1	PROPN
cana-2868	78	49	)	)	PUNCT
cana-2868	78	50	]	]	PUNCT
cana-2868	78	51	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	NUM
cana-2868	78	52	)	)	PUNCT
cana-2868	78	53	)	)	PUNCT
cana-2868	79	1	(	(	PUNCT
cana-2868	79	2	𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1	𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1	PROPN
cana-2868	79	3	)	)	PUNCT
cana-2868	79	4	]	]	PUNCT
cana-2868	79	5	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1	NUM
cana-2868	79	6	)	)	PUNCT
cana-2868	79	7	)	)	PUNCT
cana-2868	79	8	}	}	PUNCT
cana-2868	80	1	<	<	X
cana-2868	80	2	1	1	NUM
cana-2868	80	3	𝑡	𝑡	PROPN
cana-2868	80	4	max	max	PROPN
cana-2868	80	5	{	{	PUNCT
cana-2868	80	6	(	(	PUNCT
cana-2868	80	7	𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1	𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1	PROPN
cana-2868	80	8	)	)	PUNCT
cana-2868	80	9	]	]	PUNCT
cana-2868	80	10	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NUM
cana-2868	80	11	)	)	PUNCT
cana-2868	80	12	)	)	PUNCT
cana-2868	80	13	,	,	PUNCT
cana-2868	80	14	(	(	PUNCT
cana-2868	80	15	𝑑(𝑠𝑞−1	𝑑(𝑠𝑞−1	PROPN
cana-2868	80	16	,	,	PUNCT
cana-2868	80	17	𝐻𝑠𝑝−1)[1	𝐻𝑠𝑝−1)[1	PROPN
cana-2868	80	18	+	+	SYM
cana-2868	80	19	𝑑(𝑠𝑞−1	𝑑(𝑠𝑞−1	PROPN
cana-2868	80	20	,	,	PUNCT
cana-2868	80	21	𝐻𝑠𝑞−1	𝐻𝑠𝑞−1	NOUN
cana-2868	80	22	)	)	PUNCT
cana-2868	80	23	]	]	PUNCT
cana-2868	80	24	1	1	NUM
cana-2868	80	25	+	+	CCONJ
cana-2868	80	26	𝑑(𝑠𝑝−1	𝑑(𝑠𝑝−1	PROPN
cana-2868	80	27	,	,	PUNCT
cana-2868	80	28	𝐻𝑠𝑞−1	𝐻𝑠𝑞−1	NOUN
cana-2868	80	29	)	)	PUNCT
cana-2868	80	30	)	)	PUNCT
cana-2868	80	31	,	,	PUNCT
cana-2868	80	32	𝑑(𝑠𝑞−1	𝑑(𝑠𝑞−1	PROPN
cana-2868	80	33	,	,	PUNCT
cana-2868	80	34	𝐻𝑠𝑞−1)[1	𝐻𝑠𝑞−1)[1	PROPN
cana-2868	80	35	+	+	CCONJ
cana-2868	80	36	𝑑(𝑠𝑞−1	𝑑(𝑠𝑞−1	PROPN
cana-2868	80	37	,	,	PUNCT
cana-2868	80	38	𝐻𝑠𝑝−1	𝐻𝑠𝑝−1	PROPN
cana-2868	80	39	)	)	PUNCT
cana-2868	80	40	]	]	PUNCT
cana-2868	80	41	1	1	NUM
cana-2868	80	42	+	+	CCONJ
cana-2868	80	43	𝑑(𝑠𝑝−1	𝑑(𝑠𝑝−1	PROPN
cana-2868	80	44	,	,	PUNCT
cana-2868	80	45	𝐻𝑠𝑞−1	𝐻𝑠𝑞−1	X
cana-2868	80	46	)	)	PUNCT
cana-2868	80	47	}	}	PUNCT
cana-2868	80	48	≤	≤	NUM
cana-2868	80	49	b𝑛	b𝑛	NOUN
cana-2868	80	50	,	,	PUNCT
cana-2868	80	51	so	so	SCONJ
cana-2868	80	52	that	that	SCONJ
cana-2868	80	53	max	max	PROPN
cana-2868	80	54	{	{	PUNCT
cana-2868	80	55	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	80	56	,	,	PUNCT
cana-2868	80	57	s𝑞)|	s𝑞)|	NOUN
cana-2868	80	58	0	0	NUM
cana-2868	80	59	≤	≤	NUM
cana-2868	80	60	𝑝	𝑝	NOUN
cana-2868	80	61	,	,	PUNCT
cana-2868	80	62	𝑞	𝑞	PROPN
cana-2868	80	63	≤	≤	NOUN
cana-2868	80	64	𝑛	𝑛	PRON
cana-2868	80	65	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-2868	80	66	𝑝	𝑝	PROPN
cana-2868	80	67	,	,	PUNCT
cana-2868	80	68	𝑞	𝑞	PROPN
cana-2868	80	69	∈	∈	PROPN
cana-2868	80	70	n0	n0	PROPN
cana-2868	80	71	}	}	PUNCT
cana-2868	80	72	<	<	X
cana-2868	80	73	bn	bn	X
cana-2868	80	74	.	.	PUNCT
cana-2868	81	1	consequently	consequently	ADV
cana-2868	81	2	,	,	PUNCT
cana-2868	81	3	there	there	PRON
cana-2868	81	4	is	be	VERB
cana-2868	81	5	𝑤𝑛	𝑤𝑛	ADP
cana-2868	81	6	∈	∈	PROPN
cana-2868	81	7	𝑁	𝑁	PROPN
cana-2868	81	8	,	,	PUNCT
cana-2868	81	9	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-2868	81	10	1	1	NUM
cana-2868	81	11	≤	≤	NUM
cana-2868	81	12	𝑤𝑛	𝑤𝑛	ADP
cana-2868	81	13	≤	≤	NUM
cana-2868	81	14	𝑛	𝑛	ADP
cana-2868	81	15	such	such	ADJ
cana-2868	81	16	that	that	PRON
cana-2868	81	17	bn	bn	NOUN
cana-2868	81	18	=	=	NOUN
cana-2868	81	19	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	81	20	,	,	PUNCT
cana-2868	81	21	s𝑤𝑛	s𝑤𝑛	PROPN
cana-2868	81	22	}	}	PUNCT
cana-2868	81	23	here	here	ADV
cana-2868	81	24	,	,	PUNCT
cana-2868	81	25	we	we	PRON
cana-2868	81	26	can	can	AUX
cana-2868	81	27	see	see	VERB
cana-2868	81	28	that	that	SCONJ
cana-2868	81	29	0	0	NUM
cana-2868	81	30	≤	≤	NUM
cana-2868	81	31	b𝑛	b𝑛	NOUN
cana-2868	81	32	≤	≤	NUM
cana-2868	81	33	𝐵𝑛+1	𝐵𝑛+1	X
cana-2868	81	34	for	for	ADP
cana-2868	81	35	all	all	DET
cana-2868	81	36	𝑛	𝑛	PRON
cana-2868	81	37	∈	∈	NOUN
cana-2868	81	38	𝑁.	𝑁.	PROPN
cana-2868	81	39	now	now	ADV
cana-2868	81	40	we	we	PRON
cana-2868	81	41	have	have	VERB
cana-2868	81	42	to	to	PART
cana-2868	81	43	prove	prove	VERB
cana-2868	81	44	sequence	sequence	NOUN
cana-2868	81	45	{	{	PUNCT
cana-2868	81	46	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	81	47	}	}	PUNCT
cana-2868	81	48	is	be	AUX
cana-2868	81	49	bounded	bound	VERB
cana-2868	81	50	.	.	PUNCT
cana-2868	82	1	on	on	ADP
cana-2868	82	2	the	the	DET
cana-2868	82	3	contrary	contrary	NOUN
cana-2868	82	4	,	,	PUNCT
cana-2868	82	5	we	we	PRON
cana-2868	82	6	assume	assume	VERB
cana-2868	82	7	that	that	SCONJ
cana-2868	82	8	{	{	PUNCT
cana-2868	82	9	𝐵𝑛	𝐵𝑛	NOUN
cana-2868	82	10	}	}	PUNCT
cana-2868	82	11	is	be	AUX
cana-2868	82	12	not	not	PART
cana-2868	82	13	bounded	bound	VERB
cana-2868	82	14	.	.	PUNCT
cana-2868	83	1	since	since	SCONJ
cana-2868	83	2	{	{	PUNCT
cana-2868	83	3	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	83	4	}	}	PUNCT
cana-2868	83	5	is	be	AUX
cana-2868	83	6	non	non	ADJ
cana-2868	83	7	decreasing	decrease	VERB
cana-2868	83	8	sequence	sequence	NOUN
cana-2868	83	9	of	of	ADP
cana-2868	83	10	non	non	ADJ
cana-2868	83	11	negative	negative	ADJ
cana-2868	83	12	reals	real	NOUN
cana-2868	83	13	,	,	PUNCT
cana-2868	83	14	we	we	PRON
cana-2868	83	15	have	have	VERB
cana-2868	83	16	lim	lim	PROPN
cana-2868	83	17	𝑛→∞	𝑛→∞	NUM
cana-2868	83	18	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	83	19	=	=	PUNCT
cana-2868	83	20	∞.	∞.	PROPN
cana-2868	83	21	now	now	ADV
cana-2868	83	22	,	,	PUNCT
cana-2868	83	23	by	by	ADP
cana-2868	83	24	using	use	VERB
cana-2868	83	25	btriangular	btriangular	ADJ
cana-2868	83	26	inequality	inequality	NOUN
cana-2868	83	27	on	on	ADP
cana-2868	83	28	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	83	29	,	,	PUNCT
cana-2868	83	30	s𝑤𝑛	s𝑤𝑛	PROPN
cana-2868	83	31	}	}	PUNCT
cana-2868	83	32	and	and	CCONJ
cana-2868	83	33	using	use	VERB
cana-2868	83	34	the	the	DET
cana-2868	83	35	inequality	inequality	NOUN
cana-2868	83	36	(	(	PUNCT
cana-2868	83	37	1	1	NUM
cana-2868	83	38	)	)	PUNCT
cana-2868	83	39	,	,	PUNCT
cana-2868	84	1	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	84	2	=	=	PUNCT
cana-2868	84	3	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	84	4	,	,	PUNCT
cana-2868	84	5	s𝑤𝑛	s𝑤𝑛	PROPN
cana-2868	84	6	}	}	PUNCT
cana-2868	84	7	≤	≤	NUM
cana-2868	84	8	𝑡[𝑑(𝑠0	𝑡[𝑑(𝑠0	PROPN
cana-2868	84	9	,	,	PUNCT
cana-2868	84	10	𝑠𝑤𝑛	𝑠𝑤𝑛	NOUN
cana-2868	84	11	)	)	PUNCT
cana-2868	84	12	≤	≤	PUNCT
cana-2868	85	1	𝑣[𝑑(𝑠0	𝑣[𝑑(𝑠0	PROPN
cana-2868	85	2	,	,	PUNCT
cana-2868	85	3	𝑠1	𝑠1	PROPN
cana-2868	85	4	)	)	PUNCT
cana-2868	86	1	+	+	CCONJ
cana-2868	86	2	𝑑(𝑠1	𝑑(𝑠1	ADJ
cana-2868	86	3	,	,	PUNCT
cana-2868	86	4	𝑠𝑤𝑛	𝑠𝑤𝑛	NOUN
cana-2868	86	5	)	)	PUNCT
cana-2868	86	6	]	]	PUNCT
cana-2868	86	7	---(i	---(i	PROPN
cana-2868	86	8	)	)	PUNCT
cana-2868	86	9	=	=	SYM
cana-2868	86	10	𝑡[𝑑(𝑠0	𝑡[𝑑(𝑠0	PROPN
cana-2868	86	11	,	,	PUNCT
cana-2868	86	12	𝑠1	𝑠1	PROPN
cana-2868	86	13	)	)	PUNCT
cana-2868	87	1	+	+	CCONJ
cana-2868	87	2	𝑣𝑀(𝑠0	𝑣𝑀(𝑠0	PROPN
cana-2868	87	3	,	,	PUNCT
cana-2868	87	4	𝑠𝑤𝑛−1	𝑠𝑤𝑛−1	PROPN
cana-2868	87	5	)	)	PUNCT
cana-2868	87	6	]	]	PUNCT
cana-2868	87	7	where	where	SCONJ
cana-2868	87	8	,	,	PUNCT
cana-2868	87	9	𝑀(𝑠0	𝑀(𝑠0	PROPN
cana-2868	87	10	,	,	PUNCT
cana-2868	87	11	𝑠𝑤𝑛−1)=	𝑠𝑤𝑛−1)=	PROPN
cana-2868	87	12	max	max	PROPN
cana-2868	87	13	{	{	PUNCT
cana-2868	87	14	𝛽	𝛽	PROPN
cana-2868	87	15	(	(	PUNCT
cana-2868	87	16	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0	PROPN
cana-2868	87	17	)	)	PUNCT
cana-2868	87	18	]	]	PUNCT
cana-2868	87	19	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NUM
cana-2868	87	20	)	)	PUNCT
cana-2868	87	21	)	)	PUNCT
cana-2868	87	22	(	(	PUNCT
cana-2868	87	23	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0	NOUN
cana-2868	87	24	)	)	PUNCT
cana-2868	87	25	]	]	PUNCT
cana-2868	88	1	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1	NUM
cana-2868	88	2	)	)	PUNCT
cana-2868	88	3	)	)	PUNCT
cana-2868	88	4	,	,	PUNCT
cana-2868	88	5	𝛽	𝛽	PROPN
cana-2868	88	6	(	(	PUNCT
cana-2868	88	7	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	NOUN
cana-2868	88	8	)	)	PUNCT
cana-2868	88	9	]	]	PUNCT
cana-2868	88	10	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	88	11	)	)	PUNCT
cana-2868	88	12	)	)	PUNCT
cana-2868	89	1	(	(	PUNCT
cana-2868	89	2	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	NOUN
cana-2868	89	3	)	)	PUNCT
cana-2868	89	4	]	]	PUNCT
cana-2868	90	1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	90	2	)	)	PUNCT
cana-2868	90	3	)	)	PUNCT
cana-2868	90	4	,	,	PUNCT
cana-2868	90	5	𝛽	𝛽	PROPN
cana-2868	90	6	(	(	PUNCT
cana-2868	90	7	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	PROPN
cana-2868	90	8	)	)	PUNCT
cana-2868	90	9	]	]	PUNCT
cana-2868	90	10	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	90	11	)	)	PUNCT
cana-2868	90	12	)	)	PUNCT
cana-2868	90	13	(	(	PUNCT
cana-2868	90	14	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	NOUN
cana-2868	90	15	)	)	PUNCT
cana-2868	90	16	]	]	PUNCT
cana-2868	90	17	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	90	18	)	)	PUNCT
cana-2868	90	19	)	)	PUNCT
cana-2868	90	20	}	}	PUNCT
cana-2868	90	21	here	here	ADV
cana-2868	90	22	observe	observe	VERB
cana-2868	90	23	that	that	SCONJ
cana-2868	90	24	the	the	DET
cana-2868	90	25	sequences	sequence	NOUN
cana-2868	90	26	{	{	PUNCT
cana-2868	90	27	𝛽	𝛽	NOUN
cana-2868	90	28	(	(	PUNCT
cana-2868	90	29	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0	PROPN
cana-2868	90	30	)	)	PUNCT
cana-2868	90	31	]	]	PUNCT
cana-2868	90	32	1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	90	33	)	)	PUNCT
cana-2868	90	34	)	)	PUNCT
cana-2868	90	35	}	}	PUNCT
cana-2868	90	36	,	,	PUNCT
cana-2868	90	37	𝛽	𝛽	PROPN
cana-2868	90	38	(	(	PUNCT
cana-2868	90	39	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	NOUN
cana-2868	90	40	)	)	PUNCT
cana-2868	90	41	]	]	PUNCT
cana-2868	90	42	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	90	43	)	)	PUNCT
cana-2868	90	44	)	)	PUNCT
cana-2868	90	45	}	}	PUNCT
cana-2868	90	46	,	,	PUNCT
cana-2868	90	47	{	{	PUNCT
cana-2868	90	48	𝛽	𝛽	NOUN
cana-2868	90	49	(	(	PUNCT
cana-2868	90	50	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	PROPN
cana-2868	90	51	)	)	PUNCT
cana-2868	90	52	]	]	PUNCT
cana-2868	90	53	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	90	54	)	)	PUNCT
cana-2868	90	55	)	)	PUNCT
cana-2868	90	56	}	}	PUNCT
cana-2868	90	57	are	be	AUX
cana-2868	90	58	the	the	DET
cana-2868	90	59	sequences	sequence	NOUN
cana-2868	90	60	of	of	ADP
cana-2868	90	61	real	real	ADJ
cana-2868	90	62	communications	communication	NOUN
cana-2868	90	63	on	on	ADP
cana-2868	90	64	applied	apply	VERB
cana-2868	90	65	nonlinear	nonlinear	ADJ
cana-2868	90	66	analysis	analysis	NOUN
cana-2868	90	67	issn	issn	NOUN
cana-2868	90	68	:	:	PUNCT
cana-2868	90	69	1074	1074	NUM
cana-2868	90	70	-	-	PUNCT
cana-2868	90	71	133x	133x	NUM
cana-2868	90	72	vol	vol	NOUN
cana-2868	90	73	32	32	NUM
cana-2868	90	74	no	no	NOUN
cana-2868	90	75	.	.	PUNCT
cana-2868	91	1	4s	4s	NUM
cana-2868	91	2	(	(	PUNCT
cana-2868	91	3	2025	2025	NUM
cana-2868	91	4	)	)	PUNCT
cana-2868	91	5	517	517	NUM
cana-2868	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	91	7	numbers	number	NOUN
cana-2868	91	8	and	and	CCONJ
cana-2868	91	9	the	the	DET
cana-2868	91	10	sub	sub	NOUN
cana-2868	91	11	sequences	sequence	NOUN
cana-2868	91	12	{	{	PUNCT
cana-2868	91	13	𝛽	𝛽	NOUN
cana-2868	91	14	(	(	PUNCT
cana-2868	91	15	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	NOUN
cana-2868	91	16	)	)	PUNCT
cana-2868	91	17	]	]	PUNCT
cana-2868	91	18	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	91	19	)	)	PUNCT
cana-2868	91	20	)	)	PUNCT
cana-2868	92	1	}	}	PUNCT
cana-2868	92	2	,	,	PUNCT
cana-2868	92	3	𝛽	𝛽	PROPN
cana-2868	92	4	(	(	PUNCT
cana-2868	92	5	𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠𝑤𝑛𝑘−1	𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠𝑤𝑛𝑘−1	PROPN
cana-2868	92	6	)	)	PUNCT
cana-2868	92	7	]	]	PUNCT
cana-2868	92	8	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛𝑘−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛𝑘−1	NUM
cana-2868	92	9	)	)	PUNCT
cana-2868	92	10	)	)	PUNCT
cana-2868	92	11	}	}	PUNCT
cana-2868	92	12	,	,	PUNCT
cana-2868	92	13	{	{	PUNCT
cana-2868	92	14	𝛽	𝛽	NOUN
cana-2868	92	15	(	(	PUNCT
cana-2868	92	16	𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠𝑤𝑛𝑘−1)[1+𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠0	𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠𝑤𝑛𝑘−1)[1+𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠0	PROPN
cana-2868	92	17	)	)	PUNCT
cana-2868	92	18	]	]	PUNCT
cana-2868	92	19	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛𝑘−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛𝑘−1	NUM
cana-2868	92	20	)	)	PUNCT
cana-2868	92	21	)	)	PUNCT
cana-2868	92	22	}	}	PUNCT
cana-2868	92	23	case	case	NOUN
cana-2868	92	24	(	(	PUNCT
cana-2868	92	25	i	i	NOUN
cana-2868	92	26	):	):	PUNCT
cana-2868	92	27	suppose	suppose	VERB
cana-2868	92	28	that	that	SCONJ
cana-2868	92	29	𝑀(𝑠0	𝑀(𝑠0	PROPN
cana-2868	92	30	,	,	PUNCT
cana-2868	92	31	𝑠𝑤𝑛−1	𝑠𝑤𝑛−1	PROPN
cana-2868	92	32	)	)	PUNCT
cana-2868	92	33	=	=	SYM
cana-2868	92	34	{	{	PUNCT
cana-2868	92	35	𝛽	𝛽	NOUN
cana-2868	92	36	(	(	PUNCT
cana-2868	92	37	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	NOUN
cana-2868	92	38	)	)	PUNCT
cana-2868	92	39	]	]	PUNCT
cana-2868	92	40	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	92	41	)	)	PUNCT
cana-2868	92	42	)	)	PUNCT
cana-2868	92	43	}	}	PUNCT
cana-2868	92	44	------(ii	------(ii	PROPN
cana-2868	92	45	)	)	PUNCT
cana-2868	92	46	then	then	ADV
cana-2868	92	47	from	from	ADP
cana-2868	92	48	(	(	PUNCT
cana-2868	92	49	i	i	NOUN
cana-2868	92	50	)	)	PUNCT
cana-2868	92	51	and	and	CCONJ
cana-2868	92	52	(	(	PUNCT
cana-2868	92	53	ii	ii	NOUN
cana-2868	92	54	)	)	PUNCT
cana-2868	92	55	,	,	PUNCT
cana-2868	92	56	we	we	PRON
cana-2868	92	57	get	get	VERB
cana-2868	92	58	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	92	59	≤	≤	NOUN
cana-2868	92	60	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	92	61	,	,	PUNCT
cana-2868	92	62	s𝑤𝑛	s𝑤𝑛	PROPN
cana-2868	92	63	}	}	PUNCT
cana-2868	92	64	≤	≤	NUM
cana-2868	92	65	𝑡[𝑑(𝑠0	𝑡[𝑑(𝑠0	PROPN
cana-2868	92	66	,	,	PUNCT
cana-2868	92	67	𝑠𝑤𝑛	𝑠𝑤𝑛	NOUN
cana-2868	92	68	)	)	PUNCT
cana-2868	92	69	≤	≤	PUNCT
cana-2868	93	1	𝑣[𝑑(𝑠0	𝑣[𝑑(𝑠0	PROPN
cana-2868	93	2	,	,	PUNCT
cana-2868	93	3	𝑠1	𝑠1	PROPN
cana-2868	93	4	)	)	PUNCT
cana-2868	94	1	+	+	CCONJ
cana-2868	94	2	𝑑(𝑠1	𝑑(𝑠1	ADJ
cana-2868	94	3	,	,	PUNCT
cana-2868	94	4	𝑠𝑤𝑛	𝑠𝑤𝑛	NOUN
cana-2868	94	5	)	)	PUNCT
cana-2868	94	6	]	]	PUNCT
cana-2868	94	7	---(i	---(i	NOUN
cana-2868	94	8	)	)	PUNCT
cana-2868	94	9	≤	≤	NUM
cana-2868	94	10	𝑡[𝑑(𝑠0	𝑡[𝑑(𝑠0	PROPN
cana-2868	94	11	,	,	PUNCT
cana-2868	94	12	𝑠1	𝑠1	PROPN
cana-2868	94	13	)	)	PUNCT
cana-2868	95	1	+	+	CCONJ
cana-2868	95	2	𝑣𝑀(𝑠0	𝑣𝑀(𝑠0	PROPN
cana-2868	95	3	,	,	PUNCT
cana-2868	95	4	𝑠𝑤𝑛−1	𝑠𝑤𝑛−1	PROPN
cana-2868	95	5	)	)	PUNCT
cana-2868	95	6	]	]	PUNCT
cana-2868	96	1	≤	≤	NUM
cana-2868	96	2	𝑡	𝑡	PROPN
cana-2868	96	3	[	[	X
cana-2868	96	4	𝑑(𝑠0	𝑑(𝑠0	X
cana-2868	96	5	,	,	PUNCT
cana-2868	96	6	𝑠1	𝑠1	PROPN
cana-2868	96	7	)	)	PUNCT
cana-2868	96	8	+	+	CCONJ
cana-2868	96	9	𝑣	𝑣	PRON
cana-2868	96	10	{	{	PUNCT
cana-2868	96	11	𝛽	𝛽	NOUN
cana-2868	96	12	(	(	PUNCT
cana-2868	96	13	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	NOUN
cana-2868	96	14	)	)	PUNCT
cana-2868	96	15	]	]	PUNCT
cana-2868	97	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	97	2	)	)	PUNCT
cana-2868	97	3	)	)	PUNCT
cana-2868	97	4	}	}	PUNCT
cana-2868	97	5	𝐵𝑛𝑘]----(iii	𝐵𝑛𝑘]----(iii	NOUN
cana-2868	97	6	)	)	PUNCT
cana-2868	97	7	so	so	SCONJ
cana-2868	97	8	that	that	SCONJ
cana-2868	97	9	,	,	PUNCT
cana-2868	97	10	1	1	NUM
cana-2868	97	11	𝑡	𝑡	NOUN
cana-2868	97	12	−	−	NOUN
cana-2868	97	13	𝑑(𝑠0,𝑠1	𝑑(𝑠0,𝑠1	NOUN
cana-2868	97	14	)	)	PUNCT
cana-2868	97	15	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	97	16	≤	≤	NOUN
cana-2868	97	17	𝛽	𝛽	PROPN
cana-2868	97	18	(	(	PUNCT
cana-2868	97	19	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	NOUN
cana-2868	97	20	)	)	PUNCT
cana-2868	97	21	]	]	PUNCT
cana-2868	98	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	98	2	)	)	PUNCT
cana-2868	98	3	)	)	PUNCT
cana-2868	99	1	<	<	X
cana-2868	99	2	1	1	NUM
cana-2868	99	3	𝑡	𝑡	NOUN
cana-2868	99	4	since	since	SCONJ
cana-2868	99	5	as	as	ADP
cana-2868	99	6	𝑘	𝑘	PRON
cana-2868	99	7	→	→	SYM
cana-2868	99	8	∞	∞	PROPN
cana-2868	99	9	,	,	PUNCT
cana-2868	99	10	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	99	11	→	→	SYM
cana-2868	99	12	∞	∞	PROPN
cana-2868	99	13	,	,	PUNCT
cana-2868	99	14	so	so	SCONJ
cana-2868	99	15	that	that	SCONJ
cana-2868	99	16	lim	lim	PROPN
cana-2868	99	17	𝑛→∞	𝑛→∞	NUM
cana-2868	99	18	(	(	PUNCT
cana-2868	99	19	1	1	NUM
cana-2868	99	20	𝑡	𝑡	NOUN
cana-2868	99	21	−	−	NOUN
cana-2868	99	22	𝑑(𝑠0,𝑠1	𝑑(𝑠0,𝑠1	NOUN
cana-2868	99	23	)	)	PUNCT
cana-2868	99	24	𝐵𝑛𝑘	𝐵𝑛𝑘	NOUN
cana-2868	99	25	)	)	PUNCT
cana-2868	99	26	=	=	SYM
cana-2868	99	27	1	1	NUM
cana-2868	99	28	𝑡	𝑡	NOUN
cana-2868	99	29	.	.	PUNCT
cana-2868	100	1	therefore	therefore	ADV
cana-2868	100	2	lim	lim	PROPN
cana-2868	100	3	sup	sup	PROPN
cana-2868	100	4	𝑘→∞	𝑘→∞	NUM
cana-2868	100	5	𝛽	𝛽	PROPN
cana-2868	100	6	(	(	PUNCT
cana-2868	100	7	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	NOUN
cana-2868	100	8	)	)	PUNCT
cana-2868	100	9	]	]	PUNCT
cana-2868	101	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	101	2	)	)	PUNCT
cana-2868	101	3	)	)	PUNCT
cana-2868	102	1	=	=	SYM
cana-2868	102	2	1	1	NUM
cana-2868	102	3	𝑡	𝑡	NOUN
cana-2868	102	4	.	.	PUNCT
cana-2868	103	1	hence	hence	ADV
cana-2868	103	2	lim	lim	PROPN
cana-2868	103	3	𝑛→∞	𝑛→∞	NUM
cana-2868	103	4	𝑑(𝑠0	𝑑(𝑠0	PROPN
cana-2868	103	5	,	,	PUNCT
cana-2868	103	6	𝑠𝑤𝑛𝑘−1	𝑠𝑤𝑛𝑘−1	PROPN
cana-2868	103	7	)	)	PUNCT
cana-2868	103	8	=	=	SYM
cana-2868	103	9	0	0	X
cana-2868	103	10	.	.	PUNCT
cana-2868	104	1	since	since	SCONJ
cana-2868	104	2	𝛽	𝛽	NOUN
cana-2868	104	3	is	be	AUX
cana-2868	104	4	the	the	DET
cana-2868	104	5	class	class	NOUN
cana-2868	104	6	of	of	ADP
cana-2868	104	7	functions	function	NOUN
cana-2868	104	8	𝑆	𝑆	PROPN
cana-2868	104	9	,	,	PUNCT
cana-2868	104	10	taking	take	VERB
cana-2868	104	11	limit	limit	NOUN
cana-2868	104	12	on	on	ADP
cana-2868	104	13	(	(	PUNCT
cana-2868	104	14	iii	iii	NOUN
cana-2868	104	15	)	)	PUNCT
cana-2868	104	16	,	,	PUNCT
cana-2868	104	17	we	we	PRON
cana-2868	104	18	get	get	VERB
cana-2868	104	19	lim	lim	PROPN
cana-2868	104	20	bnk	bnk	PROPN
cana-2868	104	21	≤	≤	NOUN
cana-2868	105	1	𝑘→∞	𝑘→∞	NUM
cana-2868	105	2	𝑡	𝑡	PROPN
cana-2868	105	3	[	[	X
cana-2868	105	4	𝑑(𝑠0	𝑑(𝑠0	X
cana-2868	105	5	,	,	PUNCT
cana-2868	105	6	𝑠1	𝑠1	PROPN
cana-2868	105	7	)	)	PUNCT
cana-2868	105	8	+	+	CCONJ
cana-2868	105	9	𝑡	𝑡	PROPN
cana-2868	105	10	{	{	PUNCT
cana-2868	105	11	𝛽	𝛽	PROPN
cana-2868	105	12	(	(	PUNCT
cana-2868	105	13	𝑑(𝑠0	𝑑(𝑠0	PROPN
cana-2868	105	14	,	,	PUNCT
cana-2868	105	15	𝐻𝑠0)[1	𝐻𝑠0)[1	PROPN
cana-2868	105	16	+	+	SYM
cana-2868	105	17	𝑑(𝐻𝑠𝑞𝑘−1	𝑑(𝐻𝑠𝑞𝑘−1	PROPN
cana-2868	105	18	,	,	PUNCT
cana-2868	105	19	𝐻𝑠0	𝐻𝑠0	NOUN
cana-2868	105	20	)	)	PUNCT
cana-2868	105	21	]	]	PUNCT
cana-2868	105	22	1	1	NUM
cana-2868	105	23	+	+	NUM
cana-2868	105	24	𝑑(𝑣	𝑑(𝑣	NOUN
cana-2868	105	25	,	,	PUNCT
cana-2868	105	26	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	105	27	)	)	PUNCT
cana-2868	105	28	)	)	PUNCT
cana-2868	105	29	}	}	PUNCT
cana-2868	105	30	]	]	PUNCT
cana-2868	105	31	𝑡.	𝑡.	PROPN
cana-2868	105	32	𝑑(𝑠0	𝑑(𝑠0	PROPN
cana-2868	105	33	,	,	PUNCT
cana-2868	105	34	𝑠1	𝑠1	PROPN
cana-2868	105	35	)	)	PUNCT
cana-2868	105	36	,	,	PUNCT
cana-2868	105	37	contradiction	contradiction	NOUN
cana-2868	105	38	,	,	PUNCT
cana-2868	105	39	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	105	40	→	→	SYM
cana-2868	105	41	∞	∞	PROPN
cana-2868	105	42	,	,	PUNCT
cana-2868	105	43	as	as	ADP
cana-2868	105	44	𝑘	𝑘	PROPN
cana-2868	105	45	→	→	SYM
cana-2868	105	46	∞.	∞.	PROPN
cana-2868	105	47	case	case	NOUN
cana-2868	105	48	(	(	PUNCT
cana-2868	105	49	ii	ii	NOUN
cana-2868	105	50	)	)	PUNCT
cana-2868	105	51	suppose	suppose	VERB
cana-2868	105	52	that	that	SCONJ
cana-2868	105	53	𝑀(𝑠0	𝑀(𝑠0	PROPN
cana-2868	105	54	,	,	PUNCT
cana-2868	105	55	𝑠𝑤𝑛−1	𝑠𝑤𝑛−1	PROPN
cana-2868	105	56	)	)	PUNCT
cana-2868	105	57	=	=	PRON
cana-2868	105	58	{	{	PUNCT
cana-2868	105	59	𝛽	𝛽	PROPN
cana-2868	105	60	(	(	PUNCT
cana-2868	105	61	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1	NOUN
cana-2868	105	62	)	)	PUNCT
cana-2868	105	63	]	]	PUNCT
cana-2868	105	64	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	105	65	)	)	PUNCT
cana-2868	105	66	)	)	PUNCT
cana-2868	105	67	}	}	PUNCT
cana-2868	105	68	------(v	------(v	NUM
cana-2868	105	69	)	)	PUNCT
cana-2868	105	70	then	then	ADV
cana-2868	105	71	from	from	ADP
cana-2868	105	72	(	(	PUNCT
cana-2868	105	73	i	i	NOUN
cana-2868	105	74	)	)	PUNCT
cana-2868	105	75	and	and	CCONJ
cana-2868	105	76	(	(	PUNCT
cana-2868	105	77	v	v	NOUN
cana-2868	105	78	)	)	PUNCT
cana-2868	105	79	,	,	PUNCT
cana-2868	105	80	we	we	PRON
cana-2868	105	81	get	get	VERB
cana-2868	105	82	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	105	83	≤	≤	NUM
cana-2868	105	84	𝑡	𝑡	PROPN
cana-2868	106	1	[	[	X
cana-2868	106	2	𝑑(𝑠0	𝑑(𝑠0	X
cana-2868	106	3	,	,	PUNCT
cana-2868	106	4	𝑠1	𝑠1	PROPN
cana-2868	106	5	)	)	PUNCT
cana-2868	106	6	+	+	CCONJ
cana-2868	106	7	𝑡	𝑡	PROPN
cana-2868	106	8	{	{	PUNCT
cana-2868	106	9	𝛽	𝛽	NOUN
cana-2868	106	10	(	(	PUNCT
cana-2868	106	11	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0	NOUN
cana-2868	106	12	)	)	PUNCT
cana-2868	106	13	]	]	PUNCT
cana-2868	106	14	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	106	15	)	)	PUNCT
cana-2868	106	16	)	)	PUNCT
cana-2868	106	17	}	}	PUNCT
cana-2868	106	18	𝐵𝑛𝑘]----(iii	𝐵𝑛𝑘]----(iii	PROPN
cana-2868	106	19	)	)	PUNCT
cana-2868	106	20	<	<	X
cana-2868	106	21	(	(	PUNCT
cana-2868	106	22	t+1	t+1	PROPN
cana-2868	106	23	)	)	PUNCT
cana-2868	106	24	𝑑(𝑠0	𝑑(𝑠0	PROPN
cana-2868	106	25	,	,	PUNCT
cana-2868	106	26	𝑠1	𝑠1	PROPN
cana-2868	106	27	)	)	PUNCT
cana-2868	106	28	,	,	PUNCT
cana-2868	106	29	which	which	PRON
cana-2868	106	30	is	be	AUX
cana-2868	106	31	contradiction	contradiction	NOUN
cana-2868	106	32	to	to	ADP
cana-2868	106	33	,	,	PUNCT
cana-2868	106	34	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	106	35	→	→	SYM
cana-2868	106	36	∞	∞	PROPN
cana-2868	106	37	,	,	PUNCT
cana-2868	106	38	as	as	ADP
cana-2868	106	39	𝑘	𝑘	PROPN
cana-2868	106	40	→	→	SYM
cana-2868	106	41	∞.	∞.	PROPN
cana-2868	106	42	case	case	NOUN
cana-2868	106	43	(	(	PUNCT
cana-2868	106	44	iii	iii	NOUN
cana-2868	106	45	)	)	PUNCT
cana-2868	106	46	suppose	suppose	VERB
cana-2868	106	47	that	that	SCONJ
cana-2868	106	48	𝑀(𝑠0	𝑀(𝑠0	PROPN
cana-2868	106	49	,	,	PUNCT
cana-2868	106	50	𝑠𝑤𝑛−1	𝑠𝑤𝑛−1	PROPN
cana-2868	106	51	)	)	PUNCT
cana-2868	106	52	=	=	SYM
cana-2868	106	53	𝛽	𝛽	PROPN
cana-2868	106	54	(	(	PUNCT
cana-2868	106	55	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	PROPN
cana-2868	106	56	)	)	PUNCT
cana-2868	106	57	]	]	PUNCT
cana-2868	106	58	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	106	59	)	)	PUNCT
cana-2868	106	60	)	)	PUNCT
cana-2868	106	61	------(vi	------(vi	PROPN
cana-2868	106	62	)	)	PUNCT
cana-2868	106	63	then	then	ADV
cana-2868	106	64	from	from	ADP
cana-2868	106	65	(	(	PUNCT
cana-2868	106	66	i	i	NOUN
cana-2868	106	67	)	)	PUNCT
cana-2868	106	68	and	and	CCONJ
cana-2868	106	69	(	(	PUNCT
cana-2868	106	70	vi	vi	X
cana-2868	106	71	)	)	PUNCT
cana-2868	106	72	,	,	PUNCT
cana-2868	106	73	we	we	PRON
cana-2868	106	74	get	get	VERB
cana-2868	106	75	communications	communication	NOUN
cana-2868	106	76	on	on	ADP
cana-2868	106	77	applied	apply	VERB
cana-2868	106	78	nonlinear	nonlinear	ADJ
cana-2868	106	79	analysis	analysis	NOUN
cana-2868	106	80	issn	issn	NOUN
cana-2868	106	81	:	:	PUNCT
cana-2868	106	82	1074	1074	NUM
cana-2868	106	83	-	-	PUNCT
cana-2868	106	84	133x	133x	NUM
cana-2868	106	85	vol	vol	NOUN
cana-2868	106	86	32	32	NUM
cana-2868	106	87	no	no	NOUN
cana-2868	106	88	.	.	PUNCT
cana-2868	107	1	4s	4s	NUM
cana-2868	107	2	(	(	PUNCT
cana-2868	107	3	2025	2025	NUM
cana-2868	107	4	)	)	PUNCT
cana-2868	107	5	518	518	NUM
cana-2868	107	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	107	7	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	107	8	≤	≤	NUM
cana-2868	107	9	𝑡	𝑡	PROPN
cana-2868	108	1	[	[	X
cana-2868	108	2	𝑑(𝑠0	𝑑(𝑠0	X
cana-2868	108	3	,	,	PUNCT
cana-2868	108	4	𝑠1	𝑠1	PROPN
cana-2868	108	5	)	)	PUNCT
cana-2868	108	6	+	+	CCONJ
cana-2868	108	7	𝑡	𝑡	PROPN
cana-2868	108	8	{	{	PUNCT
cana-2868	108	9	𝛽	𝛽	PROPN
cana-2868	108	10	(	(	PUNCT
cana-2868	108	11	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0	PROPN
cana-2868	108	12	)	)	PUNCT
cana-2868	108	13	]	]	PUNCT
cana-2868	108	14	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1	NUM
cana-2868	108	15	)	)	PUNCT
cana-2868	108	16	)	)	PUNCT
cana-2868	108	17	}	}	PUNCT
cana-2868	108	18	𝐵𝑛𝑘	𝐵𝑛𝑘	NOUN
cana-2868	108	19	]	]	PUNCT
cana-2868	108	20	<	<	X
cana-2868	108	21	(	(	PUNCT
cana-2868	108	22	t+1	t+1	PROPN
cana-2868	108	23	)	)	PUNCT
cana-2868	108	24	𝑑(𝑠0	𝑑(𝑠0	PROPN
cana-2868	108	25	,	,	PUNCT
cana-2868	108	26	𝑠1	𝑠1	PROPN
cana-2868	108	27	)	)	PUNCT
cana-2868	108	28	,	,	PUNCT
cana-2868	108	29	which	which	PRON
cana-2868	108	30	is	be	AUX
cana-2868	108	31	contradiction	contradiction	NOUN
cana-2868	108	32	to	to	ADP
cana-2868	108	33	,	,	PUNCT
cana-2868	108	34	𝐵𝑛𝑘	𝐵𝑛𝑘	PROPN
cana-2868	108	35	→	→	SYM
cana-2868	108	36	∞	∞	PROPN
cana-2868	108	37	,	,	PUNCT
cana-2868	108	38	as	as	ADP
cana-2868	108	39	𝑘	𝑘	PROPN
cana-2868	108	40	→	→	SYM
cana-2868	108	41	∞.	∞.	PROPN
cana-2868	108	42	so	so	SCONJ
cana-2868	108	43	that	that	SCONJ
cana-2868	108	44	we	we	PRON
cana-2868	108	45	are	be	AUX
cana-2868	108	46	getting	get	VERB
cana-2868	108	47	contradictions	contradiction	NOUN
cana-2868	108	48	in	in	ADP
cana-2868	108	49	all	all	DET
cana-2868	108	50	the	the	DET
cana-2868	108	51	cases	case	NOUN
cana-2868	108	52	and	and	CCONJ
cana-2868	108	53	hence	hence	ADV
cana-2868	108	54	the	the	DET
cana-2868	108	55	sequence	sequence	NOUN
cana-2868	108	56	{	{	PUNCT
cana-2868	108	57	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	108	58	}	}	PUNCT
cana-2868	108	59	not	not	PART
cana-2868	108	60	bounded	bound	VERB
cana-2868	108	61	.	.	PUNCT
cana-2868	109	1	now	now	ADV
cana-2868	109	2	we	we	PRON
cana-2868	109	3	prove	prove	VERB
cana-2868	109	4	the	the	DET
cana-2868	109	5	existence	existence	NOUN
cana-2868	109	6	of	of	ADP
cana-2868	109	7	the	the	DET
cana-2868	109	8	fixed	fix	VERB
cana-2868	109	9	point	point	NOUN
cana-2868	109	10	with	with	ADP
cana-2868	109	11	unique	unique	ADJ
cana-2868	109	12	in	in	ADP
cana-2868	109	13	–	–	PUNCT
cana-2868	109	14	ciric	ciric	ADJ
cana-2868	109	15	-type	-type	NOUN
cana-2868	109	16	geraghty	geraghty	NOUN
cana-2868	109	17	contraction	contraction	NOUN
cana-2868	109	18	with	with	ADP
cana-2868	109	19	rational	rational	ADJ
cana-2868	109	20	terms	term	NOUN
cana-2868	109	21	in	in	ADP
cana-2868	109	22	b	b	NOUN
cana-2868	109	23	–	–	PUNCT
cana-2868	109	24	metric	metric	ADJ
cana-2868	109	25	spaces	space	NOUN
cana-2868	109	26	.	.	PUNCT
cana-2868	110	1	theorem	theorem	VERB
cana-2868	110	2	2.3	2.3	NUM
cana-2868	110	3	:	:	PUNCT
cana-2868	110	4	let	let	VERB
cana-2868	110	5	(	(	PUNCT
cana-2868	110	6	k	k	X
cana-2868	110	7	,	,	PUNCT
cana-2868	110	8	d	d	PROPN
cana-2868	110	9	,	,	PUNCT
cana-2868	110	10	t	t	PROPN
cana-2868	110	11	)	)	PUNCT
cana-2868	110	12	be	be	AUX
cana-2868	110	13	a	a	DET
cana-2868	110	14	cbms	cbms	NOUN
cana-2868	110	15	(	(	PUNCT
cana-2868	110	16	complete	complete	ADJ
cana-2868	110	17	bmetric	bmetric	ADJ
cana-2868	110	18	space	space	NOUN
cana-2868	110	19	)	)	PUNCT
cana-2868	110	20	space	space	NOUN
cana-2868	110	21	with	with	ADP
cana-2868	110	22	t	t	PROPN
cana-2868	110	23	≥	≥	NUM
cana-2868	110	24	1	1	NUM
cana-2868	110	25	and	and	CCONJ
cana-2868	110	26	let	let	VERB
cana-2868	110	27	h	h	NOUN
cana-2868	110	28	:	:	PUNCT
cana-2868	110	29	k	k	X
cana-2868	110	30	→	→	PUNCT
cana-2868	110	31	k	k	X
cana-2868	110	32	be	be	AUX
cana-2868	110	33	a	a	DET
cana-2868	110	34	map	map	NOUN
cana-2868	110	35	–	–	PUNCT
cana-2868	110	36	ciric	ciric	ADJ
cana-2868	110	37	-type	-type	NOUN
cana-2868	110	38	geraghty	geraghty	NOUN
cana-2868	110	39	contraction	contraction	NOUN
cana-2868	110	40	with	with	ADP
cana-2868	110	41	rational	rational	ADJ
cana-2868	110	42	terms	term	NOUN
cana-2868	110	43	(	(	PUNCT
cana-2868	110	44	2.11	2.11	NUM
cana-2868	110	45	)	)	PUNCT
cana-2868	110	46	,	,	PUNCT
cana-2868	110	47	then	then	ADV
cana-2868	110	48	h	h	PROPN
cana-2868	110	49	has	have	VERB
cana-2868	110	50	a	a	DET
cana-2868	110	51	unique	unique	ADJ
cana-2868	110	52	fixed	fix	VERB
cana-2868	110	53	point	point	NOUN
cana-2868	110	54	s	s	NOUN
cana-2868	110	55	in	in	ADP
cana-2868	110	56	k.	k.	PROPN
cana-2868	110	57	proof	proof	NOUN
cana-2868	110	58	:	:	PUNCT
cana-2868	110	59	let𝑠0	let𝑠0	PROPN
cana-2868	110	60	∈	∈	PROPN
cana-2868	110	61	𝐾.	𝐾.	PROPN
cana-2868	110	62	now	now	ADV
cana-2868	110	63	consider	consider	VERB
cana-2868	110	64	a	a	DET
cana-2868	110	65	sequence	sequence	NOUN
cana-2868	110	66	{	{	PUNCT
cana-2868	110	67	𝑠𝑛	𝑠𝑛	NOUN
cana-2868	110	68	}	}	PUNCT
cana-2868	110	69	in	in	ADP
cana-2868	110	70	k	k	PROPN
cana-2868	110	71	by	by	ADP
cana-2868	110	72	defining	define	VERB
cana-2868	110	73	𝑠𝑛	𝑠𝑛	NOUN
cana-2868	110	74	=	=	SYM
cana-2868	110	75	𝐾𝑠𝑛−1	𝐾𝑠𝑛−1	NOUN
cana-2868	110	76	=	=	PUNCT
cana-2868	110	77	𝐾𝑛𝑠0	𝐾𝑛𝑠0	NOUN
cana-2868	110	78	∀	∀	X
cana-2868	110	79	𝑛	𝑛	PRON
cana-2868	110	80	∈	∈	PROPN
cana-2868	110	81	𝑁.	𝑁.	PROPN
cana-2868	110	82	now	now	ADV
cana-2868	110	83	we	we	PRON
cana-2868	110	84	show	show	VERB
cana-2868	110	85	that	that	SCONJ
cana-2868	110	86	{	{	PUNCT
cana-2868	110	87	𝑠𝑛}𝑛	𝑠𝑛}𝑛	PROPN
cana-2868	110	88	∈	∈	PROPN
cana-2868	110	89	𝑁	𝑁	PROPN
cana-2868	110	90	is	be	AUX
cana-2868	110	91	a	a	DET
cana-2868	110	92	bcauchy	bcauchy	ADJ
cana-2868	110	93	sequence	sequence	NOUN
cana-2868	110	94	in	in	ADP
cana-2868	110	95	k.	k.	NOUN
cana-2868	110	96	we	we	PRON
cana-2868	110	97	take	take	VERB
cana-2868	110	98	a	a	DET
cana-2868	110	99	sequence	sequence	NOUN
cana-2868	110	100	as	as	ADP
cana-2868	110	101	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	110	102	=	=	SYM
cana-2868	110	103	𝑚𝑎𝑥{𝑑(𝑠𝑝	𝑚𝑎𝑥{𝑑(𝑠𝑝	PROPN
cana-2868	110	104	,	,	PUNCT
cana-2868	110	105	𝑠𝑞)|0	𝑠𝑞)|0	PROPN
cana-2868	110	106	≤	≤	PROPN
cana-2868	110	107	𝑝	𝑝	PROPN
cana-2868	110	108	,	,	PUNCT
cana-2868	110	109	𝑞	𝑞	PROPN
cana-2868	110	110	≤	≤	PROPN
cana-2868	110	111	𝑛	𝑛	PROPN
cana-2868	110	112	,	,	PUNCT
cana-2868	110	113	𝑝	𝑝	PROPN
cana-2868	110	114	,	,	PUNCT
cana-2868	110	115	𝑞	𝑞	PROPN
cana-2868	110	116	∈	∈	PROPN
cana-2868	110	117	𝑁	𝑁	PROPN
cana-2868	110	118	}	}	PUNCT
cana-2868	110	119	by	by	ADP
cana-2868	110	120	using	use	VERB
cana-2868	110	121	the	the	DET
cana-2868	110	122	above	above	ADJ
cana-2868	110	123	theorem	theorem	NOUN
cana-2868	110	124	2	2	NUM
cana-2868	110	125	.	.	NUM
cana-2868	110	126	2	2	NUM
cana-2868	110	127	,	,	PUNCT
cana-2868	110	128	∃m	∃m	PROPN
cana-2868	110	129	>	>	X
cana-2868	110	130	0	0	NUM
cana-2868	110	131	such	such	ADJ
cana-2868	110	132	that	that	SCONJ
cana-2868	110	133	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	110	134	≤	≤	PROPN
cana-2868	110	135	𝑀	𝑀	PROPN
cana-2868	110	136	,	,	PUNCT
cana-2868	110	137	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2868	110	138	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-2868	110	139	𝑛	𝑛	DET
cana-2868	110	140	∈	∈	PROPN
cana-2868	110	141	𝑁	𝑁	PROPN
cana-2868	110	142	.	.	PUNCT
cana-2868	111	1	as	as	SCONJ
cana-2868	111	2	{	{	PUNCT
cana-2868	111	3	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	111	4	}	}	PUNCT
cana-2868	111	5	is	be	AUX
cana-2868	111	6	an	an	DET
cana-2868	111	7	increasing	increase	VERB
cana-2868	111	8	sequence	sequence	NOUN
cana-2868	111	9	,	,	PUNCT
cana-2868	111	10	we	we	PRON
cana-2868	111	11	have	have	VERB
cana-2868	111	12	lim	lim	PROPN
cana-2868	111	13	𝑛→∞	𝑛→∞	NUM
cana-2868	111	14	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	111	15	≤	≤	ADV
cana-2868	111	16	𝑀.	𝑀.	PROPN
cana-2868	111	17	we	we	PRON
cana-2868	111	18	take	take	VERB
cana-2868	111	19	a	a	DET
cana-2868	111	20	sequence	sequence	NOUN
cana-2868	111	21	{	{	PUNCT
cana-2868	111	22	𝜆𝑛	𝜆𝑛	NOUN
cana-2868	111	23	}	}	PUNCT
cana-2868	111	24	on	on	ADP
cana-2868	111	25	a	a	DET
cana-2868	111	26	bmetric	bmetric	ADJ
cana-2868	111	27	space	space	NOUN
cana-2868	111	28	(	(	PUNCT
cana-2868	111	29	k	k	NOUN
cana-2868	111	30	,	,	PUNCT
cana-2868	111	31	d	d	PROPN
cana-2868	111	32	,	,	PUNCT
cana-2868	111	33	t	t	PROPN
cana-2868	111	34	)	)	PUNCT
cana-2868	111	35	by	by	ADP
cana-2868	111	36	{	{	PUNCT
cana-2868	111	37	𝜆𝑛	𝜆𝑛	NOUN
cana-2868	111	38	}	}	PUNCT
cana-2868	111	39	=	=	SYM
cana-2868	111	40	sup	sup	NOUN
cana-2868	111	41	{	{	PUNCT
cana-2868	111	42	𝑑(𝑠𝑝	𝑑(𝑠𝑝	ADP
cana-2868	111	43	,	,	PUNCT
cana-2868	111	44	𝑠𝑞)|𝑝	𝑠𝑞)|𝑝	ADJ
cana-2868	111	45	,	,	PUNCT
cana-2868	111	46	𝑞	𝑞	PROPN
cana-2868	111	47	≥	≥	PROPN
cana-2868	111	48	𝑛	𝑛	PROPN
cana-2868	111	49	,	,	PUNCT
cana-2868	111	50	𝑝	𝑝	PROPN
cana-2868	111	51	,	,	PUNCT
cana-2868	111	52	𝑞	𝑞	X
cana-2868	111	53	∈	∈	PROPN
cana-2868	111	54	𝑁	𝑁	PROPN
cana-2868	111	55	}	}	PUNCT
cana-2868	111	56	,	,	PUNCT
cana-2868	111	57	then	then	ADV
cana-2868	111	58	we	we	PRON
cana-2868	111	59	have	have	VERB
cana-2868	111	60	0	0	NUM
cana-2868	111	61	≤	≤	NUM
cana-2868	111	62	𝜆𝑛	𝜆𝑛	ADP
cana-2868	111	63	≤	≤	NUM
cana-2868	111	64	𝜆𝑛−1	𝜆𝑛−1	NOUN
cana-2868	111	65	≤	≤	NOUN
cana-2868	111	66	𝜆𝑛−2	𝜆𝑛−2	VERB
cana-2868	111	67	≤	≤	NOUN
cana-2868	111	68	⋯	⋯	ADP
cana-2868	111	69	≤	≤	NOUN
cana-2868	111	70	𝜆0	𝜆0	NOUN
cana-2868	112	1	=	=	SYM
cana-2868	112	2	lim	lim	PROPN
cana-2868	112	3	𝑛→∞	𝑛→∞	NUM
cana-2868	112	4	𝐵𝑛	𝐵𝑛	PROPN
cana-2868	112	5	≤	≤	PROPN
cana-2868	112	6	𝑀	𝑀	PROPN
cana-2868	112	7	∀	∀	X
cana-2868	112	8	𝑛	𝑛	PRON
cana-2868	112	9	∈	∈	NOUN
cana-2868	112	10	𝑁.	𝑁.	PROPN
cana-2868	112	11	the	the	DET
cana-2868	112	12	sequence	sequence	NOUN
cana-2868	112	13	{	{	PUNCT
cana-2868	112	14	𝜆𝑛	𝜆𝑛	NOUN
cana-2868	112	15	}	}	PUNCT
cana-2868	112	16	is	be	AUX
cana-2868	112	17	a	a	DET
cana-2868	112	18	decreasing	decrease	VERB
cana-2868	112	19	and	and	CCONJ
cana-2868	112	20	bounded	bound	VERB
cana-2868	112	21	sequence	sequence	NOUN
cana-2868	112	22	of	of	ADP
cana-2868	112	23	non	non	ADJ
cana-2868	112	24	negative	negative	ADJ
cana-2868	112	25	real	real	ADJ
cana-2868	112	26	numbers	number	NOUN
cana-2868	112	27	,	,	PUNCT
cana-2868	112	28	and	and	CCONJ
cana-2868	112	29	hence	hence	ADV
cana-2868	112	30	it	it	PRON
cana-2868	112	31	is	be	AUX
cana-2868	112	32	converges	converge	NOUN
cana-2868	112	33	to	to	ADP
cana-2868	112	34	some	some	DET
cana-2868	112	35	𝑙	𝑙	PRON
cana-2868	112	36	≥0	≥0	NOUN
cana-2868	112	37	,	,	PUNCT
cana-2868	112	38	that	that	PRON
cana-2868	112	39	is	is	ADV
cana-2868	112	40	lim	lim	NOUN
cana-2868	112	41	𝑛→∞	𝑛→∞	NUM
cana-2868	112	42	𝜆𝑛	𝜆𝑛	PROPN
cana-2868	113	1	=	=	NOUN
cana-2868	113	2	𝑙.	𝑙.	NOUN
cana-2868	114	1	then	then	ADV
cana-2868	114	2	there	there	PRON
cana-2868	114	3	exist	exist	VERB
cana-2868	114	4	two	two	NUM
cana-2868	114	5	sub	sub	NOUN
cana-2868	114	6	sequences	sequence	NOUN
cana-2868	114	7	{	{	PUNCT
cana-2868	114	8	𝑠𝑝𝑘	𝑠𝑝𝑘	NOUN
cana-2868	114	9	}	}	PUNCT
cana-2868	114	10	and	and	CCONJ
cana-2868	114	11	{	{	PUNCT
cana-2868	114	12	𝑠𝑞𝑘	𝑠𝑞𝑘	NOUN
cana-2868	114	13	}	}	PUNCT
cana-2868	114	14	of	of	ADP
cana-2868	114	15	{	{	PUNCT
cana-2868	114	16	𝑠𝑛	𝑠𝑛	NOUN
cana-2868	114	17	}	}	PUNCT
cana-2868	114	18	with	with	ADP
cana-2868	114	19	𝑞𝑘	𝑞𝑘	ADP
cana-2868	114	20	>	>	X
cana-2868	114	21	𝑝𝑘	𝑝𝑘	X
cana-2868	114	22	≥	≥	NOUN
cana-2868	114	23	𝑘	𝑘	PROPN
cana-2868	114	24	for	for	ADP
cana-2868	114	25	𝑘	𝑘	DET
cana-2868	114	26	∈	∈	NOUN
cana-2868	114	27	𝑁	𝑁	PROPN
cana-2868	114	28	such	such	ADJ
cana-2868	114	29	that	that	DET
cana-2868	114	30	𝑑(𝑠𝑝𝑘	𝑑(𝑠𝑝𝑘	PROPN
cana-2868	114	31	,	,	PUNCT
cana-2868	114	32	𝑠𝑞𝑘	𝑠𝑞𝑘	NOUN
cana-2868	114	33	)	)	PUNCT
cana-2868	114	34	→	→	SYM
cana-2868	114	35	𝑙	𝑙	X
cana-2868	114	36	𝑎𝑠	𝑎𝑠	NOUN
cana-2868	114	37	𝑘	𝑘	X
cana-2868	114	38	→	→	SYM
cana-2868	114	39	∞.	∞.	PROPN
cana-2868	114	40	2.3.1	2.3.1	NUM
cana-2868	114	41	now	now	ADV
cana-2868	114	42	we	we	PRON
cana-2868	114	43	have	have	VERB
cana-2868	114	44	to	to	PART
cana-2868	114	45	prove	prove	VERB
cana-2868	114	46	that𝑙	that𝑙	NOUN
cana-2868	114	47	=	=	SYM
cana-2868	114	48	0	0	X
cana-2868	114	49	.	.	PUNCT
cana-2868	115	1	on	on	ADP
cana-2868	115	2	the	the	DET
cana-2868	115	3	contrary	contrary	NOUN
cana-2868	115	4	,	,	PUNCT
cana-2868	115	5	assume	assume	VERB
cana-2868	115	6	that	that	SCONJ
cana-2868	115	7	𝑙	𝑙	X
cana-2868	115	8	>	>	X
cana-2868	115	9	0	0	X
cana-2868	115	10	.	.	PUNCT
cana-2868	116	1	put	put	VERB
cana-2868	116	2	𝑢	𝑢	NOUN
cana-2868	116	3	=	=	SYM
cana-2868	116	4	𝑠𝑝𝑘−1	𝑠𝑝𝑘−1	PROPN
cana-2868	116	5	,	,	PUNCT
cana-2868	116	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-2868	116	7	𝑣	𝑣	X
cana-2868	116	8	=	=	SYM
cana-2868	116	9	𝑠𝑞𝑘−1	𝑠𝑞𝑘−1	PROPN
cana-2868	116	10	in	in	ADP
cana-2868	116	11	the	the	DET
cana-2868	116	12	inequality	inequality	NOUN
cana-2868	116	13	.	.	PUNCT
cana-2868	117	1	we	we	PRON
cana-2868	117	2	have	have	VERB
cana-2868	117	3	𝑑(𝑠𝑝𝑘	𝑑(𝑠𝑝𝑘	NOUN
cana-2868	117	4	,	,	PUNCT
cana-2868	117	5	𝑠𝑞𝑘	𝑠𝑞𝑘	NOUN
cana-2868	117	6	)	)	PUNCT
cana-2868	117	7	=	=	SYM
cana-2868	117	8	𝑑(𝐻𝑠𝑝𝑘−1	𝑑(𝐻𝑠𝑝𝑘−1	PROPN
cana-2868	117	9	,	,	PUNCT
cana-2868	117	10	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	PROPN
cana-2868	117	11	)	)	PUNCT
cana-2868	117	12	≤	≤	NUM
cana-2868	117	13	𝑀(𝑠𝑝𝑘−1	𝑀(𝑠𝑝𝑘−1	PROPN
cana-2868	117	14	,	,	PUNCT
cana-2868	117	15	𝑠𝑞𝑘−1	𝑠𝑞𝑘−1	PROPN
cana-2868	117	16	)	)	PUNCT
cana-2868	117	17	2.3.2	2.3.2	NUM
cana-2868	118	1	where	where	SCONJ
cana-2868	118	2	(	(	PUNCT
cana-2868	118	3	𝑠𝑝𝑘−1	𝑠𝑝𝑘−1	PROPN
cana-2868	118	4	,	,	PUNCT
cana-2868	118	5	𝑠𝑞𝑘−1	𝑠𝑞𝑘−1	PROPN
cana-2868	118	6	)	)	PUNCT
cana-2868	118	7	=	=	SYM
cana-2868	118	8	max	max	PROPN
cana-2868	118	9	{	{	PUNCT
cana-2868	118	10	𝛽	𝛽	PROPN
cana-2868	118	11	(	(	PUNCT
cana-2868	118	12	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	118	13	,	,	PUNCT
cana-2868	118	14	,	,	PUNCT
cana-2868	118	15	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	AUX
cana-2868	118	16	,	,	PUNCT
cana-2868	118	17	)	)	PUNCT
cana-2868	119	1	[	[	X
cana-2868	119	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	119	3	,	,	PUNCT
cana-2868	119	4	)	)	PUNCT
cana-2868	119	5	]	]	PUNCT
cana-2868	120	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	120	2	)	)	PUNCT
cana-2868	120	3	)	)	PUNCT
cana-2868	121	1	(	(	PUNCT
cana-2868	121	2	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	121	3	,	,	PUNCT
cana-2868	121	4	,	,	PUNCT
cana-2868	121	5	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	121	6	,	,	PUNCT
cana-2868	121	7	)	)	PUNCT
cana-2868	122	1	[	[	X
cana-2868	122	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	122	3	,	,	PUNCT
cana-2868	122	4	)	)	PUNCT
cana-2868	122	5	]	]	PUNCT
cana-2868	123	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	123	2	)	)	PUNCT
cana-2868	123	3	)	)	PUNCT
cana-2868	123	4	communications	communication	NOUN
cana-2868	123	5	on	on	ADP
cana-2868	123	6	applied	apply	VERB
cana-2868	123	7	nonlinear	nonlinear	ADJ
cana-2868	123	8	analysis	analysis	NOUN
cana-2868	123	9	issn	issn	NOUN
cana-2868	123	10	:	:	PUNCT
cana-2868	123	11	1074	1074	NUM
cana-2868	123	12	-	-	PUNCT
cana-2868	123	13	133x	133x	NUM
cana-2868	123	14	vol	vol	NOUN
cana-2868	123	15	32	32	NUM
cana-2868	123	16	no	no	NOUN
cana-2868	123	17	.	.	PUNCT
cana-2868	124	1	4s	4s	NUM
cana-2868	124	2	(	(	PUNCT
cana-2868	124	3	2025	2025	NUM
cana-2868	124	4	)	)	PUNCT
cana-2868	124	5	519	519	NUM
cana-2868	124	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	124	7	𝛽	𝛽	PROPN
cana-2868	124	8	(	(	PUNCT
cana-2868	124	9	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	PROPN
cana-2868	124	10	,	,	PUNCT
cana-2868	124	11	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	124	12	,	,	PUNCT
cana-2868	124	13	)	)	PUNCT
cana-2868	125	1	[	[	X
cana-2868	125	2	1	1	NUM
cana-2868	125	3	+	+	NUM
cana-2868	125	4	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	NOUN
cana-2868	125	5	,	,	PUNCT
cana-2868	125	6	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	NOUN
cana-2868	125	7	)	)	PUNCT
cana-2868	125	8	]	]	PUNCT
cana-2868	125	9	1	1	NUM
cana-2868	125	10	+	+	CCONJ
cana-2868	125	11	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	125	12	,	,	PUNCT
cana-2868	125	13	,	,	PUNCT
cana-2868	125	14	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	PROPN
cana-2868	125	15	)	)	PUNCT
cana-2868	125	16	)	)	PUNCT
cana-2868	125	17	(	(	PUNCT
cana-2868	125	18	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	PROPN
cana-2868	125	19	,	,	PUNCT
cana-2868	125	20	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	125	21	,	,	PUNCT
cana-2868	125	22	)	)	PUNCT
cana-2868	126	1	[	[	X
cana-2868	126	2	1	1	NUM
cana-2868	126	3	+	+	NUM
cana-2868	126	4	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	NOUN
cana-2868	126	5	,	,	PUNCT
cana-2868	126	6	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	NOUN
cana-2868	126	7	)	)	PUNCT
cana-2868	126	8	]	]	PUNCT
cana-2868	126	9	1	1	NUM
cana-2868	126	10	+	+	CCONJ
cana-2868	126	11	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	126	12	,	,	PUNCT
cana-2868	126	13	,	,	PUNCT
cana-2868	126	14	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	NUM
cana-2868	126	15	)	)	PUNCT
cana-2868	126	16	)	)	PUNCT
cana-2868	127	1	𝛽	𝛽	NOUN
cana-2868	127	2	(	(	PUNCT
cana-2868	127	3	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	PROPN
cana-2868	127	4	,	,	PUNCT
cana-2868	127	5	𝐻𝑠𝑞𝑘−1)[1	𝐻𝑠𝑞𝑘−1)[1	NOUN
cana-2868	127	6	+	+	SYM
cana-2868	127	7	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	NOUN
cana-2868	127	8	,	,	PUNCT
cana-2868	127	9	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	127	10	,	,	PUNCT
cana-2868	127	11	)	)	PUNCT
cana-2868	127	12	]	]	PUNCT
cana-2868	128	1	1	1	NUM
cana-2868	128	2	+	+	CCONJ
cana-2868	128	3	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	128	4	,	,	PUNCT
cana-2868	128	5	,	,	PUNCT
cana-2868	128	6	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	PROPN
cana-2868	128	7	)	)	PUNCT
cana-2868	128	8	)	)	PUNCT
cana-2868	128	9	(	(	PUNCT
cana-2868	128	10	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	PROPN
cana-2868	128	11	,	,	PUNCT
cana-2868	128	12	𝐻𝑠𝑞𝑘−1)[1	𝐻𝑠𝑞𝑘−1)[1	NOUN
cana-2868	128	13	+	+	SYM
cana-2868	128	14	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	NOUN
cana-2868	128	15	,	,	PUNCT
cana-2868	128	16	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	128	17	,	,	PUNCT
cana-2868	128	18	)	)	PUNCT
cana-2868	128	19	]	]	PUNCT
cana-2868	129	1	1	1	NUM
cana-2868	129	2	+	+	CCONJ
cana-2868	129	3	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	129	4	,	,	PUNCT
cana-2868	129	5	,	,	PUNCT
cana-2868	129	6	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	PROPN
cana-2868	129	7	)	)	PUNCT
cana-2868	129	8	)	)	PUNCT
cana-2868	129	9	}	}	PUNCT
cana-2868	129	10	now	now	ADV
cana-2868	129	11	the	the	DET
cana-2868	129	12	maximum	maximum	NOUN
cana-2868	129	13	is	be	AUX
cana-2868	129	14	one	one	NUM
cana-2868	129	15	of	of	ADP
cana-2868	129	16	the	the	DET
cana-2868	129	17	terms	term	NOUN
cana-2868	129	18	of	of	ADP
cana-2868	129	19	r.h.s	r.h.s	NOUN
cana-2868	129	20	of	of	ADP
cana-2868	129	21	𝑀(𝑠𝑝𝑘−1	𝑀(𝑠𝑝𝑘−1	PROPN
cana-2868	129	22	,	,	PUNCT
cana-2868	129	23	𝑠𝑞𝑘−1	𝑠𝑞𝑘−1	PROPN
cana-2868	129	24	)	)	PUNCT
cana-2868	129	25	now	now	ADV
cana-2868	129	26	we	we	PRON
cana-2868	129	27	consider	consider	VERB
cana-2868	129	28	three	three	NUM
cana-2868	129	29	as	as	ADP
cana-2868	129	30	cases	case	NOUN
cana-2868	129	31	as	as	ADP
cana-2868	129	32	the	the	DET
cana-2868	129	33	possibility	possibility	NOUN
cana-2868	129	34	of	of	ADP
cana-2868	129	35	each	each	DET
cana-2868	129	36	one	one	NUM
cana-2868	129	37	term	term	NOUN
cana-2868	129	38	of	of	ADP
cana-2868	129	39	r	r	NOUN
cana-2868	129	40	h	h	NOUN
cana-2868	129	41	s.	s.	PROPN
cana-2868	129	42	suppose	suppose	VERB
cana-2868	129	43	that	that	SCONJ
cana-2868	129	44	𝑀(𝑠𝑝𝑘−1	𝑀(𝑠𝑝𝑘−1	PROPN
cana-2868	129	45	,	,	PUNCT
cana-2868	129	46	𝑠𝑞𝑘−1)=	𝑠𝑞𝑘−1)=	PROPN
cana-2868	129	47	𝛽	𝛽	PROPN
cana-2868	129	48	(	(	PUNCT
cana-2868	129	49	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	129	50	,	,	PUNCT
cana-2868	129	51	,	,	PUNCT
cana-2868	129	52	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	129	53	,	,	PUNCT
cana-2868	129	54	)	)	PUNCT
cana-2868	130	1	[	[	X
cana-2868	130	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	130	3	,	,	PUNCT
cana-2868	130	4	)	)	PUNCT
cana-2868	130	5	]	]	PUNCT
cana-2868	131	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	131	2	)	)	PUNCT
cana-2868	131	3	)	)	PUNCT
cana-2868	132	1	(	(	PUNCT
cana-2868	132	2	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	132	3	,	,	PUNCT
cana-2868	132	4	,	,	PUNCT
cana-2868	132	5	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	132	6	,	,	PUNCT
cana-2868	132	7	)	)	PUNCT
cana-2868	133	1	[	[	X
cana-2868	133	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	133	3	,	,	PUNCT
cana-2868	133	4	)	)	PUNCT
cana-2868	133	5	]	]	PUNCT
cana-2868	134	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	134	2	)	)	PUNCT
cana-2868	134	3	)	)	PUNCT
cana-2868	134	4	for	for	ADP
cana-2868	134	5	all	all	DET
cana-2868	134	6	k	k	PROPN
cana-2868	134	7	in	in	ADP
cana-2868	134	8	n.	n.	PROPN
cana-2868	134	9	therefore	therefore	ADV
cana-2868	134	10	,	,	PUNCT
cana-2868	134	11	from	from	ADP
cana-2868	134	12	2.32𝑑(𝑠𝑝𝑘	2.32𝑑(𝑠𝑝𝑘	NUM
cana-2868	134	13	,	,	PUNCT
cana-2868	134	14	𝑠𝑞𝑘	𝑠𝑞𝑘	NOUN
cana-2868	134	15	)	)	PUNCT
cana-2868	135	1	≤	≤	NOUN
cana-2868	135	2	𝛽	𝛽	PROPN
cana-2868	135	3	(	(	PUNCT
cana-2868	135	4	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	135	5	,	,	PUNCT
cana-2868	135	6	,	,	PUNCT
cana-2868	135	7	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	135	8	,	,	PUNCT
cana-2868	135	9	)	)	PUNCT
cana-2868	136	1	[	[	X
cana-2868	136	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	136	3	,	,	PUNCT
cana-2868	136	4	)	)	PUNCT
cana-2868	136	5	]	]	PUNCT
cana-2868	137	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	137	2	)	)	PUNCT
cana-2868	137	3	)	)	PUNCT
cana-2868	138	1	(	(	PUNCT
cana-2868	138	2	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	138	3	,	,	PUNCT
cana-2868	138	4	,	,	PUNCT
cana-2868	138	5	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	138	6	,	,	PUNCT
cana-2868	138	7	)	)	PUNCT
cana-2868	139	1	[	[	X
cana-2868	139	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	139	3	,	,	PUNCT
cana-2868	139	4	)	)	PUNCT
cana-2868	139	5	]	]	PUNCT
cana-2868	140	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	140	2	)	)	PUNCT
cana-2868	140	3	)	)	PUNCT
cana-2868	141	1	≤	≤	NUM
cana-2868	141	2	𝛽	𝛽	NOUN
cana-2868	141	3	(	(	PUNCT
cana-2868	141	4	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	141	5	,	,	PUNCT
cana-2868	141	6	,	,	PUNCT
cana-2868	141	7	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	141	8	,	,	PUNCT
cana-2868	141	9	)	)	PUNCT
cana-2868	142	1	[	[	X
cana-2868	142	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	142	3	,	,	PUNCT
cana-2868	142	4	)	)	PUNCT
cana-2868	142	5	]	]	PUNCT
cana-2868	142	6	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	142	7	)	)	PUNCT
cana-2868	142	8	)	)	PUNCT
cana-2868	143	1	𝜆𝑘−1	𝜆𝑘−1	PUNCT
cana-2868	143	2	…	…	PUNCT
cana-2868	143	3	..	..	PUNCT
cana-2868	143	4	2.3.3	2.3.3	NUM
cana-2868	143	5	taking	take	VERB
cana-2868	143	6	limit	limit	NOUN
cana-2868	143	7	suprimum	suprimum	ADV
cana-2868	143	8	𝑎𝑠	𝑎𝑠	PROPN
cana-2868	143	9	𝑘	𝑘	PROPN
cana-2868	143	10	→	→	SYM
cana-2868	143	11	∞	∞	NUM
cana-2868	143	12	on	on	ADP
cana-2868	143	13	both	both	DET
cana-2868	143	14	sides	side	NOUN
cana-2868	143	15	of	of	ADP
cana-2868	143	16	(	(	PUNCT
cana-2868	143	17	2.3.2	2.3.2	NUM
cana-2868	143	18	)	)	PUNCT
cana-2868	143	19	,	,	PUNCT
cana-2868	143	20	we	we	PRON
cana-2868	143	21	get	get	VERB
cana-2868	143	22	lim	lim	PROPN
cana-2868	143	23	𝑘→∞	𝑘→∞	NUM
cana-2868	143	24	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-2868	143	25	𝑑(𝑠𝑝𝑘	𝑑(𝑠𝑝𝑘	X
cana-2868	143	26	,	,	PUNCT
cana-2868	143	27	𝑠𝑞𝑘	𝑠𝑞𝑘	NOUN
cana-2868	143	28	)	)	PUNCT
cana-2868	143	29	≤	≤	NOUN
cana-2868	143	30	lim	lim	PROPN
cana-2868	143	31	𝑘→∞	𝑘→∞	NUM
cana-2868	143	32	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-2868	143	33	𝛽	𝛽	PROPN
cana-2868	143	34	(	(	PUNCT
cana-2868	143	35	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	143	36	,	,	PUNCT
cana-2868	143	37	,	,	PUNCT
cana-2868	143	38	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	143	39	,	,	PUNCT
cana-2868	143	40	)	)	PUNCT
cana-2868	144	1	[	[	X
cana-2868	144	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	144	3	,	,	PUNCT
cana-2868	144	4	)	)	PUNCT
cana-2868	144	5	]	]	PUNCT
cana-2868	145	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	145	2	)	)	PUNCT
cana-2868	145	3	)	)	PUNCT
cana-2868	146	1	𝜆𝑘−1	𝜆𝑘−1	VERB
cana-2868	146	2	from	from	ADP
cana-2868	146	3	2.3.1	2.3.1	NUM
cana-2868	146	4	,	,	PUNCT
cana-2868	146	5	we	we	PRON
cana-2868	146	6	get	get	VERB
cana-2868	146	7	𝑙	𝑙	DET
cana-2868	146	8	≤	≤	NUM
cana-2868	146	9	𝛽	𝛽	NOUN
cana-2868	146	10	(	(	PUNCT
cana-2868	146	11	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	146	12	,	,	PUNCT
cana-2868	146	13	,	,	PUNCT
cana-2868	146	14	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	146	15	,	,	PUNCT
cana-2868	146	16	)	)	PUNCT
cana-2868	147	1	[	[	X
cana-2868	147	2	1	1	NUM
cana-2868	147	3	+	+	NUM
cana-2868	147	4	𝑑(𝐻𝑠𝑞𝑘−1	𝑑(𝐻𝑠𝑞𝑘−1	PROPN
cana-2868	147	5	,	,	PUNCT
cana-2868	147	6	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	147	7	,	,	PUNCT
cana-2868	147	8	)	)	PUNCT
cana-2868	147	9	]	]	PUNCT
cana-2868	147	10	1	1	NUM
cana-2868	147	11	+	+	NUM
cana-2868	147	12	𝑑(𝑠𝑞𝑘−1	𝑑(𝑠𝑞𝑘−1	NOUN
cana-2868	147	13	,	,	PUNCT
cana-2868	147	14	𝐻𝑠𝑞𝑘−1	𝐻𝑠𝑞𝑘−1	NUM
cana-2868	147	15	)	)	PUNCT
cana-2868	147	16	)	)	PUNCT
cana-2868	147	17	𝑙	𝑙	X
cana-2868	147	18	,	,	PUNCT
cana-2868	147	19	1	1	NUM
cana-2868	147	20	𝑡	𝑡	NOUN
cana-2868	147	21	≤	≤	NUM
cana-2868	147	22	1	1	NUM
cana-2868	147	23	≤	≤	NOUN
cana-2868	147	24	lim	lim	PROPN
cana-2868	147	25	𝑘→∞	𝑘→∞	NUM
cana-2868	147	26	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-2868	147	27	𝛽	𝛽	PROPN
cana-2868	147	28	(	(	PUNCT
cana-2868	147	29	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	147	30	,	,	PUNCT
cana-2868	147	31	,	,	PUNCT
cana-2868	147	32	𝐻𝑠𝑝𝑘−1	𝐻𝑠𝑝𝑘−1	INTJ
cana-2868	147	33	,	,	PUNCT
cana-2868	147	34	)	)	PUNCT
cana-2868	148	1	[	[	X
cana-2868	148	2	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1	NUM
cana-2868	148	3	,	,	PUNCT
cana-2868	148	4	)	)	PUNCT
cana-2868	148	5	]	]	PUNCT
cana-2868	149	1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1	NUM
cana-2868	149	2	)	)	PUNCT
cana-2868	149	3	)	)	PUNCT
cana-2868	150	1	<	<	X
cana-2868	150	2	1	1	NUM
cana-2868	150	3	𝑡	𝑡	NOUN
cana-2868	150	4	since	since	SCONJ
cana-2868	150	5	𝛽	𝛽	NOUN
cana-2868	150	6	is	be	AUX
cana-2868	150	7	the	the	DET
cana-2868	150	8	class	class	NOUN
cana-2868	150	9	of	of	ADP
cana-2868	150	10	functions	function	NOUN
cana-2868	150	11	s	s	PART
cana-2868	150	12	have	have	VERB
cana-2868	150	13	lim	lim	PROPN
cana-2868	150	14	𝑘→∞	𝑘→∞	NUM
cana-2868	150	15	𝑑(𝑠𝑝𝑘−1	𝑑(𝑠𝑝𝑘−1	PROPN
cana-2868	150	16	,	,	PUNCT
cana-2868	150	17	𝑠𝑞𝑘−1	𝑠𝑞𝑘−1	PROPN
cana-2868	150	18	)	)	PUNCT
cana-2868	150	19	=	=	PUNCT
cana-2868	150	20	0	0	PUNCT
cana-2868	150	21	using	use	VERB
cana-2868	150	22	2.3.1	2.3.1	NUM
cana-2868	150	23	and	and	CCONJ
cana-2868	150	24	2.3.3	2.3.3	NUM
cana-2868	150	25	we	we	PRON
cana-2868	150	26	get	get	VERB
cana-2868	150	27	𝑙	𝑙	PRON
cana-2868	150	28	=	=	SYM
cana-2868	150	29	lim	lim	PROPN
cana-2868	150	30	𝑘→∞	𝑘→∞	NUM
cana-2868	150	31	𝑑(𝑠𝑝𝑘	𝑑(𝑠𝑝𝑘	PROPN
cana-2868	150	32	,	,	PUNCT
cana-2868	150	33	𝑠𝑞𝑘	𝑠𝑞𝑘	NOUN
cana-2868	150	34	)	)	PUNCT
cana-2868	150	35	=	=	SYM
cana-2868	150	36	0	0	NUM
cana-2868	150	37	,	,	PUNCT
cana-2868	150	38	which	which	PRON
cana-2868	150	39	is	be	AUX
cana-2868	150	40	contradiction	contradiction	NOUN
cana-2868	150	41	to	to	ADP
cana-2868	150	42	our	our	PRON
cana-2868	150	43	assumption	assumption	NOUN
cana-2868	150	44	𝑙	𝑙	X
cana-2868	150	45	>	>	X
cana-2868	150	46	0	0	NUM
cana-2868	150	47	.	.	PUNCT
cana-2868	151	1	hence	hence	ADV
cana-2868	151	2	,	,	PUNCT
cana-2868	151	3	lim	lim	PROPN
cana-2868	151	4	𝑛→∞	𝑛→∞	NUM
cana-2868	151	5	𝜆𝑛	𝜆𝑛	PROPN
cana-2868	152	1	=	=	SYM
cana-2868	152	2	𝑙	𝑙	NOUN
cana-2868	152	3	=	=	SYM
cana-2868	152	4	0	0	X
cana-2868	152	5	.	.	PUNCT
cana-2868	153	1	similarly	similarly	ADV
cana-2868	153	2	,	,	PUNCT
cana-2868	153	3	we	we	PRON
cana-2868	153	4	can	can	AUX
cana-2868	153	5	show	show	VERB
cana-2868	153	6	that	that	SCONJ
cana-2868	153	7	in	in	ADP
cana-2868	153	8	the	the	DET
cana-2868	153	9	remaining	remain	VERB
cana-2868	153	10	two	two	NUM
cana-2868	153	11	cases	case	NOUN
cana-2868	153	12	𝑙	𝑙	DET
cana-2868	153	13	=	=	SYM
cana-2868	153	14	lim	lim	PROPN
cana-2868	153	15	𝑛→∞	𝑛→∞	NUM
cana-2868	153	16	𝜆𝑛	𝜆𝑛	PROPN
cana-2868	153	17	=	=	NOUN
cana-2868	153	18	0	0	X
cana-2868	153	19	.	.	PUNCT
cana-2868	154	1	now	now	ADV
cana-2868	154	2	,	,	PUNCT
cana-2868	154	3	let	let	VERB
cana-2868	154	4	𝑚	𝑚	PRON
cana-2868	154	5	,	,	PUNCT
cana-2868	154	6	𝑛	𝑛	DET
cana-2868	154	7	∈	∈	NOUN
cana-2868	154	8	𝑁	𝑁	PROPN
cana-2868	154	9	,	,	PUNCT
cana-2868	154	10	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-2868	154	11	𝑚	𝑚	X
cana-2868	154	12	>	>	X
cana-2868	154	13	𝑛	𝑛	PROPN
cana-2868	154	14	,	,	PUNCT
cana-2868	154	15	we	we	PRON
cana-2868	154	16	get	get	VERB
cana-2868	154	17	lim	lim	NOUN
cana-2868	154	18	𝑛→∞	𝑛→∞	NUM
cana-2868	154	19	𝑑(𝑠𝑛	𝑑(𝑠𝑛	PROPN
cana-2868	154	20	,	,	PUNCT
cana-2868	154	21	𝑠𝑚	𝑠𝑚	NOUN
cana-2868	154	22	)	)	PUNCT
cana-2868	154	23	≤	≤	NOUN
cana-2868	154	24	lim	lim	NOUN
cana-2868	154	25	𝑛→∞	𝑛→∞	NUM
cana-2868	154	26	𝜆𝑛	𝜆𝑛	PROPN
cana-2868	155	1	=	=	NOUN
cana-2868	155	2	0	0	NUM
cana-2868	155	3	.	.	PUNCT
cana-2868	156	1	hence	hence	ADV
cana-2868	156	2	,	,	PUNCT
cana-2868	156	3	the	the	DET
cana-2868	156	4	sequence	sequence	NOUN
cana-2868	156	5	{	{	PUNCT
cana-2868	156	6	𝑠𝑛	𝑠𝑛	NOUN
cana-2868	156	7	}	}	PUNCT
cana-2868	156	8	is	be	AUX
cana-2868	156	9	a	a	DET
cana-2868	156	10	bcauchy	bcauchy	ADJ
cana-2868	156	11	sequence	sequence	NOUN
cana-2868	156	12	in	in	ADP
cana-2868	156	13	k.	k.	PROPN
cana-2868	156	14	since	since	SCONJ
cana-2868	156	15	k	k	PROPN
cana-2868	156	16	is	be	AUX
cana-2868	156	17	complete	complete	ADJ
cana-2868	156	18	,	,	PUNCT
cana-2868	156	19	the	the	DET
cana-2868	156	20	sequence	sequence	NOUN
cana-2868	156	21	{	{	PUNCT
cana-2868	156	22	𝑠𝑛	𝑠𝑛	NOUN
cana-2868	156	23	}	}	PUNCT
cana-2868	156	24	is	be	AUX
cana-2868	156	25	convergent	convergent	ADJ
cana-2868	156	26	to	to	ADP
cana-2868	156	27	some	some	DET
cana-2868	156	28	s	s	NOUN
cana-2868	156	29	*	*	PUNCT
cana-2868	156	30	in	in	ADP
cana-2868	156	31	m.	m.	NOUN
cana-2868	156	32	communications	communication	NOUN
cana-2868	156	33	on	on	ADP
cana-2868	156	34	applied	apply	VERB
cana-2868	156	35	nonlinear	nonlinear	ADJ
cana-2868	156	36	analysis	analysis	NOUN
cana-2868	156	37	issn	issn	NOUN
cana-2868	156	38	:	:	PUNCT
cana-2868	156	39	1074	1074	NUM
cana-2868	156	40	-	-	PUNCT
cana-2868	156	41	133x	133x	NUM
cana-2868	156	42	vol	vol	NOUN
cana-2868	156	43	32	32	NUM
cana-2868	156	44	no	no	NOUN
cana-2868	156	45	.	.	PUNCT
cana-2868	157	1	4s	4s	NUM
cana-2868	157	2	(	(	PUNCT
cana-2868	157	3	2025	2025	NUM
cana-2868	157	4	)	)	PUNCT
cana-2868	157	5	520	520	NUM
cana-2868	157	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	157	7	now	now	ADV
cana-2868	157	8	we	we	PRON
cana-2868	157	9	prove	prove	VERB
cana-2868	157	10	that	that	SCONJ
cana-2868	157	11	s	s	VERB
cana-2868	157	12	*	*	VERB
cana-2868	157	13	is	be	AUX
cana-2868	157	14	a	a	DET
cana-2868	157	15	fixed	fix	VERB
cana-2868	157	16	point	point	NOUN
cana-2868	157	17	of	of	ADP
cana-2868	157	18	h.	h.	PROPN
cana-2868	157	19	assume	assume	VERB
cana-2868	157	20	that	that	SCONJ
cana-2868	157	21	hs	hs	PROPN
cana-2868	157	22	*	*	NOUN
cana-2868	157	23	≠s	≠s	NOUN
cana-2868	157	24	*	*	SYM
cana-2868	157	25	,	,	PUNCT
cana-2868	157	26	(	(	PUNCT
cana-2868	157	27	hs∗	hs∗	NOUN
cana-2868	157	28	,	,	PUNCT
cana-2868	157	29	s∗	s∗	PROPN
cana-2868	157	30	)	)	PUNCT
cana-2868	157	31	>	>	X
cana-2868	157	32	0	0	PUNCT
cana-2868	157	33	,	,	PUNCT
cana-2868	157	34	taking	take	VERB
cana-2868	157	35	u=	u=	ADV
cana-2868	157	36	sn	sn	PROPN
cana-2868	157	37	,	,	PUNCT
cana-2868	157	38	v	v	NOUN
cana-2868	157	39	=	=	NUM
cana-2868	157	40	s∗	s∗	PROPN
cana-2868	157	41	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	PROPN
cana-2868	157	42	,	,	PUNCT
cana-2868	157	43	hs∗	hs∗	NOUN
cana-2868	157	44	)	)	PUNCT
cana-2868	157	45	=	=	SYM
cana-2868	158	1	𝑑(𝐻𝑠𝑛	𝑑(𝐻𝑠𝑛	NOUN
cana-2868	158	2	,	,	PUNCT
cana-2868	158	3	hs∗	hs∗	NOUN
cana-2868	158	4	)	)	PUNCT
cana-2868	158	5	≤	≤	NOUN
cana-2868	158	6	m(sn	m(sn	NOUN
cana-2868	158	7	,	,	PUNCT
cana-2868	158	8	s∗)	s∗)	NOUN
cana-2868	158	9	…	…	SYM
cana-2868	158	10	…	…	PUNCT
cana-2868	158	11	2.3.4	2.3.4	NUM
cana-2868	158	12	where	where	SCONJ
cana-2868	158	13	m(sn	m(sn	NOUN
cana-2868	158	14	,	,	PUNCT
cana-2868	158	15	s∗	s∗	PROPN
cana-2868	158	16	)	)	PUNCT
cana-2868	158	17	=	=	PUNCT
cana-2868	158	18	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2868	158	19	{	{	PUNCT
cana-2868	158	20	𝛽	𝛽	NOUN
cana-2868	158	21	(	(	PUNCT
cana-2868	158	22	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	158	23	)	)	PUNCT
cana-2868	158	24	]	]	PUNCT
cana-2868	158	25	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	158	26	)	)	PUNCT
cana-2868	158	27	)	)	PUNCT
cana-2868	158	28	(	(	PUNCT
cana-2868	158	29	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	158	30	)	)	PUNCT
cana-2868	158	31	]	]	PUNCT
cana-2868	159	1	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	159	2	)	)	PUNCT
cana-2868	159	3	)	)	PUNCT
cana-2868	160	1	,	,	PUNCT
cana-2868	160	2	𝛽	𝛽	NOUN
cana-2868	160	3	(	(	PUNCT
cana-2868	160	4	𝑑(s∗,𝐻sn)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,𝐻sn)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	160	5	)	)	PUNCT
cana-2868	160	6	]	]	PUNCT
cana-2868	160	7	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	8	)	)	PUNCT
cana-2868	160	9	)	)	PUNCT
cana-2868	160	10	(	(	PUNCT
cana-2868	160	11	𝑑(s∗,𝐻sn)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,𝐻sn)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	160	12	)	)	PUNCT
cana-2868	160	13	]	]	PUNCT
cana-2868	160	14	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	15	)	)	PUNCT
cana-2868	160	16	)	)	PUNCT
cana-2868	160	17	,	,	PUNCT
cana-2868	160	18	𝛽	𝛽	PROPN
cana-2868	160	19	(	(	PUNCT
cana-2868	160	20	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	160	21	)	)	PUNCT
cana-2868	160	22	]	]	PUNCT
cana-2868	160	23	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	24	)	)	PUNCT
cana-2868	160	25	)	)	PUNCT
cana-2868	160	26	(	(	PUNCT
cana-2868	160	27	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	160	28	)	)	PUNCT
cana-2868	160	29	]	]	PUNCT
cana-2868	160	30	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	31	)	)	PUNCT
cana-2868	160	32	)	)	PUNCT
cana-2868	160	33	}	}	PUNCT
cana-2868	160	34	=	=	PUNCT
cana-2868	160	35	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2868	160	36	{	{	PUNCT
cana-2868	160	37	𝛽	𝛽	NOUN
cana-2868	160	38	(	(	PUNCT
cana-2868	160	39	𝑑(sn	𝑑(sn	PROPN
cana-2868	160	40	,	,	PUNCT
cana-2868	160	41	sn+1)[1+𝑑(𝐻s∗,sn+1	sn+1)[1+𝑑(𝐻s∗,sn+1	PROPN
cana-2868	160	42	)	)	PUNCT
cana-2868	160	43	]	]	PUNCT
cana-2868	160	44	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	160	45	)	)	PUNCT
cana-2868	160	46	)	)	PUNCT
cana-2868	160	47	(	(	PUNCT
cana-2868	160	48	𝑑(sn	𝑑(sn	PROPN
cana-2868	160	49	,	,	PUNCT
cana-2868	160	50	sn+1)[1+𝑑(𝐻s∗,sn+1	sn+1)[1+𝑑(𝐻s∗,sn+1	PROPN
cana-2868	160	51	)	)	PUNCT
cana-2868	160	52	]	]	PUNCT
cana-2868	160	53	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	160	54	)	)	PUNCT
cana-2868	160	55	)	)	PUNCT
cana-2868	160	56	,	,	PUNCT
cana-2868	160	57	𝛽	𝛽	PROPN
cana-2868	160	58	(	(	PUNCT
cana-2868	160	59	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	160	60	)	)	PUNCT
cana-2868	160	61	]	]	PUNCT
cana-2868	160	62	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	63	)	)	PUNCT
cana-2868	160	64	)	)	PUNCT
cana-2868	160	65	(	(	PUNCT
cana-2868	160	66	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	160	67	)	)	PUNCT
cana-2868	160	68	]	]	PUNCT
cana-2868	160	69	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	70	)	)	PUNCT
cana-2868	160	71	)	)	PUNCT
cana-2868	160	72	,	,	PUNCT
cana-2868	160	73	𝛽	𝛽	PROPN
cana-2868	160	74	(	(	PUNCT
cana-2868	160	75	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,sn+1	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,sn+1	ADJ
cana-2868	160	76	)	)	PUNCT
cana-2868	160	77	]	]	PUNCT
cana-2868	160	78	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	79	)	)	PUNCT
cana-2868	160	80	)	)	PUNCT
cana-2868	160	81	(	(	PUNCT
cana-2868	160	82	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,sn+1	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,sn+1	ADJ
cana-2868	160	83	)	)	PUNCT
cana-2868	160	84	]	]	PUNCT
cana-2868	160	85	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	160	86	)	)	PUNCT
cana-2868	160	87	)	)	PUNCT
cana-2868	160	88	}	}	PUNCT
cana-2868	160	89	again	again	ADV
cana-2868	160	90	,	,	PUNCT
cana-2868	160	91	three	three	NUM
cana-2868	160	92	cases	case	NOUN
cana-2868	160	93	will	will	AUX
cana-2868	160	94	arise	arise	VERB
cana-2868	160	95	:	:	PUNCT
cana-2868	160	96	case	case	NOUN
cana-2868	160	97	(	(	PUNCT
cana-2868	160	98	i	i	NOUN
cana-2868	160	99	)	)	PUNCT
cana-2868	160	100	:	:	PUNCT
cana-2868	160	101	suppose	suppose	VERB
cana-2868	160	102	that	that	SCONJ
cana-2868	160	103	m(sn	m(sn	PROPN
cana-2868	160	104	,	,	PUNCT
cana-2868	160	105	s∗	s∗	PROPN
cana-2868	160	106	)	)	PUNCT
cana-2868	160	107	=	=	SYM
cana-2868	160	108	𝛽	𝛽	PROPN
cana-2868	160	109	(	(	PUNCT
cana-2868	160	110	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	160	111	)	)	PUNCT
cana-2868	160	112	]	]	PUNCT
cana-2868	160	113	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	160	114	)	)	PUNCT
cana-2868	160	115	)	)	PUNCT
cana-2868	160	116	(	(	PUNCT
cana-2868	160	117	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	160	118	)	)	PUNCT
cana-2868	160	119	]	]	PUNCT
cana-2868	160	120	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	160	121	)	)	PUNCT
cana-2868	160	122	)	)	PUNCT
cana-2868	160	123	then	then	ADV
cana-2868	160	124	from	from	ADP
cana-2868	160	125	2.3.4	2.3.4	NUM
cana-2868	160	126	,	,	PUNCT
cana-2868	160	127	we	we	PRON
cana-2868	160	128	have	have	VERB
cana-2868	160	129	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	PROPN
cana-2868	160	130	,	,	PUNCT
cana-2868	160	131	hs∗	hs∗	NOUN
cana-2868	160	132	)	)	PUNCT
cana-2868	161	1	=	=	SYM
cana-2868	161	2	𝛽	𝛽	NOUN
cana-2868	161	3	(	(	PUNCT
cana-2868	161	4	𝑑(sn	𝑑(sn	PROPN
cana-2868	161	5	,	,	PUNCT
cana-2868	161	6	𝐻sn)[1	𝐻sn)[1	PRON
cana-2868	161	7	+	+	CCONJ
cana-2868	161	8	𝑑(𝐻s∗	𝑑(𝐻s∗	ADJ
cana-2868	161	9	,	,	PUNCT
cana-2868	161	10	𝐻sn	𝐻sn	PROPN
cana-2868	161	11	)	)	PUNCT
cana-2868	161	12	]	]	PUNCT
cana-2868	161	13	1	1	NUM
cana-2868	161	14	+	+	CCONJ
cana-2868	161	15	𝑑(s∗	𝑑(s∗	ADJ
cana-2868	161	16	,	,	PUNCT
cana-2868	161	17	𝐻s∗	𝐻s∗	X
cana-2868	161	18	)	)	PUNCT
cana-2868	161	19	)	)	PUNCT
cana-2868	161	20	(	(	PUNCT
cana-2868	161	21	𝑑(sn	𝑑(sn	PROPN
cana-2868	161	22	,	,	PUNCT
cana-2868	161	23	𝐻sn)[1	𝐻sn)[1	PRON
cana-2868	161	24	+	+	CCONJ
cana-2868	161	25	𝑑(𝐻s∗	𝑑(𝐻s∗	ADJ
cana-2868	161	26	,	,	PUNCT
cana-2868	161	27	𝐻sn	𝐻sn	PROPN
cana-2868	161	28	)	)	PUNCT
cana-2868	161	29	]	]	PUNCT
cana-2868	161	30	1	1	NUM
cana-2868	161	31	+	+	CCONJ
cana-2868	161	32	𝑑(s∗	𝑑(s∗	ADJ
cana-2868	161	33	,	,	PUNCT
cana-2868	161	34	𝐻s∗	𝐻s∗	X
cana-2868	161	35	)	)	PUNCT
cana-2868	161	36	)	)	PUNCT
cana-2868	161	37	as	as	ADP
cana-2868	161	38	limit	limit	VERB
cana-2868	161	39	𝑛	𝑛	ADP
cana-2868	161	40	→	→	SYM
cana-2868	161	41	∞	∞	NUM
cana-2868	161	42	on	on	ADP
cana-2868	161	43	both	both	DET
cana-2868	161	44	sides	side	NOUN
cana-2868	161	45	of	of	ADP
cana-2868	161	46	the	the	DET
cana-2868	161	47	above	above	ADJ
cana-2868	161	48	inequality	inequality	NOUN
cana-2868	161	49	and	and	CCONJ
cana-2868	161	50	from	from	ADP
cana-2868	161	51	the	the	DET
cana-2868	161	52	theorem	theorem	NOUN
cana-2868	161	53	(	(	PUNCT
cana-2868	161	54	2.2.1	2.2.1	NUM
cana-2868	161	55	)	)	PUNCT
cana-2868	161	56	,	,	PUNCT
cana-2868	161	57	we	we	PRON
cana-2868	161	58	get	get	VERB
cana-2868	161	59	lim	lim	PROPN
cana-2868	161	60	𝑛→∞	𝑛→∞	NUM
cana-2868	161	61	(	(	PUNCT
cana-2868	161	62	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	161	63	)	)	PUNCT
cana-2868	161	64	]	]	PUNCT
cana-2868	161	65	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	161	66	)	)	PUNCT
cana-2868	161	67	)	)	PUNCT
cana-2868	162	1	=	=	PUNCT
cana-2868	162	2	0	0	NUM
cana-2868	162	3	,	,	PUNCT
cana-2868	162	4	so	so	SCONJ
cana-2868	162	5	that	that	SCONJ
cana-2868	162	6	1	1	NUM
cana-2868	162	7	𝑡	𝑡	PROPN
cana-2868	162	8	𝑑(𝑠∗	𝑑(𝑠∗	NOUN
cana-2868	162	9	,	,	PUNCT
cana-2868	162	10	𝐻𝑠∗	𝐻𝑠∗	NOUN
cana-2868	162	11	)	)	PUNCT
cana-2868	162	12	≤	≤	NOUN
cana-2868	162	13	lim	lim	NOUN
cana-2868	162	14	𝑛→∞	𝑛→∞	NUM
cana-2868	162	15	sup	sup	NOUN
cana-2868	162	16	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	NOUN
cana-2868	162	17	,	,	PUNCT
cana-2868	162	18	hs∗	hs∗	NOUN
cana-2868	162	19	)	)	PUNCT
cana-2868	162	20	≤	≤	NOUN
cana-2868	162	21	lim	lim	NOUN
cana-2868	162	22	𝑛→∞	𝑛→∞	NUM
cana-2868	162	23	𝑠𝑢𝑝	𝑠𝑢𝑝	PROPN
cana-2868	162	24	𝛽	𝛽	PROPN
cana-2868	162	25	(	(	PUNCT
cana-2868	162	26	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	162	27	)	)	PUNCT
cana-2868	162	28	]	]	PUNCT
cana-2868	162	29	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	162	30	)	)	PUNCT
cana-2868	162	31	)	)	PUNCT
cana-2868	163	1	lim	lim	NOUN
cana-2868	163	2	𝑛→∞	𝑛→∞	NUM
cana-2868	163	3	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-2868	163	4	(	(	PUNCT
cana-2868	163	5	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn	NUM
cana-2868	163	6	)	)	PUNCT
cana-2868	163	7	]	]	PUNCT
cana-2868	163	8	1+𝑑(s∗,𝐻s∗	1+𝑑(s∗,𝐻s∗	NUM
cana-2868	163	9	)	)	PUNCT
cana-2868	163	10	)	)	PUNCT
cana-2868	164	1	=	=	SYM
cana-2868	164	2	0	0	PUNCT
cana-2868	164	3	and	and	CCONJ
cana-2868	164	4	hence	hence	ADV
cana-2868	164	5	consequently	consequently	ADV
cana-2868	164	6	,	,	PUNCT
cana-2868	164	7	we	we	PRON
cana-2868	164	8	get	get	VERB
cana-2868	164	9	𝑑(𝑠∗	𝑑(𝑠∗	NOUN
cana-2868	164	10	,	,	PUNCT
cana-2868	164	11	𝐻𝑠∗	𝐻𝑠∗	NOUN
cana-2868	164	12	)	)	PUNCT
cana-2868	165	1	=	=	SYM
cana-2868	165	2	0	0	NUM
cana-2868	165	3	,	,	PUNCT
cana-2868	165	4	which	which	PRON
cana-2868	165	5	contradicts	contradict	VERB
cana-2868	165	6	our	our	PRON
cana-2868	165	7	assumption	assumption	NOUN
cana-2868	165	8	𝑑(hs∗	𝑑(hs∗	PROPN
cana-2868	165	9	,	,	PUNCT
cana-2868	165	10	s∗	s∗	PROPN
cana-2868	165	11	)	)	PUNCT
cana-2868	165	12	>	>	X
cana-2868	165	13	0	0	PUNCT
cana-2868	165	14	.	.	PUNCT
cana-2868	166	1	case	case	NOUN
cana-2868	166	2	(	(	PUNCT
cana-2868	166	3	ii	ii	NOUN
cana-2868	166	4	)	)	PUNCT
cana-2868	166	5	:	:	PUNCT
cana-2868	166	6	suppose	suppose	VERB
cana-2868	166	7	that	that	SCONJ
cana-2868	166	8	m(sn	m(sn	PROPN
cana-2868	166	9	,	,	PUNCT
cana-2868	166	10	s∗	s∗	PROPN
cana-2868	166	11	)	)	PUNCT
cana-2868	166	12	=	=	SYM
cana-2868	166	13	𝛽	𝛽	PROPN
cana-2868	166	14	(	(	PUNCT
cana-2868	166	15	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	166	16	)	)	PUNCT
cana-2868	166	17	]	]	PUNCT
cana-2868	166	18	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	166	19	)	)	PUNCT
cana-2868	166	20	)	)	PUNCT
cana-2868	166	21	(	(	PUNCT
cana-2868	166	22	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	166	23	)	)	PUNCT
cana-2868	166	24	]	]	PUNCT
cana-2868	166	25	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	166	26	)	)	PUNCT
cana-2868	166	27	)	)	PUNCT
cana-2868	166	28	then	then	ADV
cana-2868	166	29	from	from	ADP
cana-2868	166	30	2.3.4	2.3.4	NUM
cana-2868	166	31	,	,	PUNCT
cana-2868	166	32	we	we	PRON
cana-2868	166	33	have	have	VERB
cana-2868	166	34	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	PROPN
cana-2868	166	35	,	,	PUNCT
cana-2868	166	36	hs∗	hs∗	NOUN
cana-2868	166	37	)	)	PUNCT
cana-2868	167	1	=	=	SYM
cana-2868	167	2	𝛽	𝛽	PROPN
cana-2868	167	3	(	(	PUNCT
cana-2868	167	4	𝑑(s∗	𝑑(s∗	X
cana-2868	167	5	,	,	PUNCT
cana-2868	167	6	sn+1)[1	sn+1)[1	PROPN
cana-2868	167	7	+	+	SYM
cana-2868	167	8	𝑑(s∗	𝑑(s∗	ADJ
cana-2868	167	9	,	,	PUNCT
cana-2868	167	10	𝐻s∗	𝐻s∗	X
cana-2868	167	11	)	)	PUNCT
cana-2868	167	12	]	]	PUNCT
cana-2868	167	13	1	1	NUM
cana-2868	167	14	+	+	CCONJ
cana-2868	167	15	𝑑(sn	𝑑(sn	NUM
cana-2868	167	16	,	,	PUNCT
cana-2868	167	17	𝐻s∗	𝐻s∗	X
cana-2868	167	18	)	)	PUNCT
cana-2868	167	19	)	)	PUNCT
cana-2868	167	20	(	(	PUNCT
cana-2868	167	21	𝑑(s∗	𝑑(s∗	X
cana-2868	167	22	,	,	PUNCT
cana-2868	167	23	sn+1)[1	sn+1)[1	PROPN
cana-2868	167	24	+	+	SYM
cana-2868	167	25	𝑑(s∗	𝑑(s∗	ADJ
cana-2868	167	26	,	,	PUNCT
cana-2868	167	27	𝐻s∗	𝐻s∗	X
cana-2868	167	28	)	)	PUNCT
cana-2868	167	29	]	]	PUNCT
cana-2868	167	30	1	1	NUM
cana-2868	167	31	+	+	CCONJ
cana-2868	167	32	𝑑(sn	𝑑(sn	NUM
cana-2868	167	33	,	,	PUNCT
cana-2868	167	34	𝐻s∗	𝐻s∗	X
cana-2868	167	35	)	)	PUNCT
cana-2868	167	36	)	)	PUNCT
cana-2868	167	37	as	as	SCONJ
cana-2868	167	38	limit	limit	VERB
cana-2868	167	39	𝑛	𝑛	ADP
cana-2868	167	40	→	→	SYM
cana-2868	167	41	∞	∞	NUM
cana-2868	167	42	on	on	ADP
cana-2868	167	43	both	both	DET
cana-2868	167	44	sides	side	NOUN
cana-2868	167	45	of	of	ADP
cana-2868	167	46	the	the	DET
cana-2868	167	47	above	above	ADJ
cana-2868	167	48	inequality	inequality	NOUN
cana-2868	167	49	and	and	CCONJ
cana-2868	167	50	from	from	ADP
cana-2868	167	51	the	the	DET
cana-2868	167	52	theorem	theorem	NOUN
cana-2868	167	53	(	(	PUNCT
cana-2868	167	54	i	i	NOUN
cana-2868	167	55	)	)	PUNCT
cana-2868	167	56	,	,	PUNCT
cana-2868	167	57	we	we	PRON
cana-2868	167	58	get	get	VERB
cana-2868	167	59	lim	lim	PROPN
cana-2868	167	60	𝑛→∞	𝑛→∞	NUM
cana-2868	167	61	(	(	PUNCT
cana-2868	167	62	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	167	63	)	)	PUNCT
cana-2868	167	64	]	]	PUNCT
cana-2868	167	65	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	167	66	)	)	PUNCT
cana-2868	167	67	)	)	PUNCT
cana-2868	168	1	=	=	PUNCT
cana-2868	168	2	0	0	NUM
cana-2868	168	3	,	,	PUNCT
cana-2868	168	4	so	so	SCONJ
cana-2868	168	5	that	that	SCONJ
cana-2868	168	6	1	1	NUM
cana-2868	168	7	𝑡	𝑡	PROPN
cana-2868	168	8	𝑑(𝑠∗	𝑑(𝑠∗	NOUN
cana-2868	168	9	,	,	PUNCT
cana-2868	168	10	𝐻𝑠∗	𝐻𝑠∗	NOUN
cana-2868	168	11	)	)	PUNCT
cana-2868	168	12	≤	≤	NOUN
cana-2868	168	13	lim	lim	NOUN
cana-2868	168	14	𝑛→∞	𝑛→∞	NUM
cana-2868	168	15	sup	sup	NOUN
cana-2868	168	16	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	NOUN
cana-2868	168	17	,	,	PUNCT
cana-2868	168	18	hs∗	hs∗	NOUN
cana-2868	168	19	)	)	PUNCT
cana-2868	168	20	≤	≤	NOUN
cana-2868	168	21	lim	lim	NOUN
cana-2868	168	22	𝑛→∞	𝑛→∞	NUM
cana-2868	168	23	𝑠𝑢𝑝	𝑠𝑢𝑝	PROPN
cana-2868	168	24	𝛽	𝛽	PROPN
cana-2868	168	25	(	(	PUNCT
cana-2868	168	26	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	168	27	)	)	PUNCT
cana-2868	168	28	]	]	PUNCT
cana-2868	168	29	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	168	30	)	)	PUNCT
cana-2868	168	31	)	)	PUNCT
cana-2868	168	32	(	(	PUNCT
cana-2868	168	33	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗	PROPN
cana-2868	168	34	)	)	PUNCT
cana-2868	168	35	]	]	PUNCT
cana-2868	168	36	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	168	37	)	)	PUNCT
cana-2868	168	38	)	)	PUNCT
cana-2868	169	1	=	=	SYM
cana-2868	169	2	0	0	NUM
cana-2868	169	3	communications	communication	NOUN
cana-2868	169	4	on	on	ADP
cana-2868	169	5	applied	apply	VERB
cana-2868	169	6	nonlinear	nonlinear	ADJ
cana-2868	169	7	analysis	analysis	NOUN
cana-2868	169	8	issn	issn	NOUN
cana-2868	169	9	:	:	PUNCT
cana-2868	169	10	1074	1074	NUM
cana-2868	169	11	-	-	PUNCT
cana-2868	169	12	133x	133x	NUM
cana-2868	169	13	vol	vol	NOUN
cana-2868	169	14	32	32	NUM
cana-2868	169	15	no	no	NOUN
cana-2868	169	16	.	.	PUNCT
cana-2868	170	1	4s	4s	NUM
cana-2868	170	2	(	(	PUNCT
cana-2868	170	3	2025	2025	NUM
cana-2868	170	4	)	)	PUNCT
cana-2868	170	5	521	521	NUM
cana-2868	170	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	170	7	and	and	CCONJ
cana-2868	170	8	hence	hence	ADV
cana-2868	170	9	consequently	consequently	ADV
cana-2868	170	10	,	,	PUNCT
cana-2868	170	11	we	we	PRON
cana-2868	170	12	get	get	VERB
cana-2868	170	13	𝑑(𝑠∗	𝑑(𝑠∗	NOUN
cana-2868	170	14	,	,	PUNCT
cana-2868	170	15	𝐻𝑠∗	𝐻𝑠∗	NOUN
cana-2868	170	16	)	)	PUNCT
cana-2868	170	17	=	=	SYM
cana-2868	171	1	0	0	NUM
cana-2868	171	2	,	,	PUNCT
cana-2868	171	3	which	which	PRON
cana-2868	171	4	contradicts	contradict	VERB
cana-2868	171	5	our	our	PRON
cana-2868	171	6	assumption	assumption	NOUN
cana-2868	171	7	𝑑(hs∗	𝑑(hs∗	PROPN
cana-2868	171	8	,	,	PUNCT
cana-2868	171	9	s∗	s∗	PROPN
cana-2868	171	10	)	)	PUNCT
cana-2868	171	11	>	>	X
cana-2868	171	12	0	0	PUNCT
cana-2868	171	13	.	.	PUNCT
cana-2868	172	1	case	case	NOUN
cana-2868	172	2	(	(	PUNCT
cana-2868	172	3	iii	iii	NOUN
cana-2868	172	4	)	)	PUNCT
cana-2868	172	5	:	:	PUNCT
cana-2868	172	6	suppose	suppose	VERB
cana-2868	172	7	that	that	SCONJ
cana-2868	172	8	m(sn	m(sn	PROPN
cana-2868	172	9	,	,	PUNCT
cana-2868	172	10	s∗	s∗	PROPN
cana-2868	172	11	)	)	PUNCT
cana-2868	172	12	=	=	SYM
cana-2868	172	13	𝛽	𝛽	PROPN
cana-2868	172	14	(	(	PUNCT
cana-2868	172	15	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	172	16	)	)	PUNCT
cana-2868	172	17	]	]	PUNCT
cana-2868	172	18	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	172	19	)	)	PUNCT
cana-2868	172	20	)	)	PUNCT
cana-2868	172	21	(	(	PUNCT
cana-2868	172	22	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	172	23	)	)	PUNCT
cana-2868	172	24	]	]	PUNCT
cana-2868	172	25	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	172	26	)	)	PUNCT
cana-2868	172	27	)	)	PUNCT
cana-2868	172	28	then	then	ADV
cana-2868	172	29	from	from	ADP
cana-2868	172	30	2.3.4	2.3.4	NUM
cana-2868	172	31	,	,	PUNCT
cana-2868	172	32	we	we	PRON
cana-2868	172	33	have	have	VERB
cana-2868	172	34	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	PROPN
cana-2868	172	35	,	,	PUNCT
cana-2868	172	36	hs∗	hs∗	NOUN
cana-2868	172	37	)	)	PUNCT
cana-2868	173	1	=	=	SYM
cana-2868	173	2	𝛽	𝛽	PROPN
cana-2868	173	3	(	(	PUNCT
cana-2868	173	4	𝑑(s∗	𝑑(s∗	PROPN
cana-2868	173	5	,	,	PUNCT
cana-2868	173	6	𝐻s∗)[1	𝐻s∗)[1	PROPN
cana-2868	173	7	+	+	CCONJ
cana-2868	173	8	𝑑(s∗	𝑑(s∗	ADJ
cana-2868	173	9	,	,	PUNCT
cana-2868	173	10	𝐻sn	𝐻sn	PROPN
cana-2868	173	11	)	)	PUNCT
cana-2868	173	12	]	]	PUNCT
cana-2868	173	13	1	1	NUM
cana-2868	173	14	+	+	CCONJ
cana-2868	173	15	𝑑(sn	𝑑(sn	NUM
cana-2868	173	16	,	,	PUNCT
cana-2868	173	17	𝐻s∗	𝐻s∗	X
cana-2868	173	18	)	)	PUNCT
cana-2868	173	19	)	)	PUNCT
cana-2868	173	20	(	(	PUNCT
cana-2868	173	21	𝑑(s∗	𝑑(s∗	X
cana-2868	173	22	,	,	PUNCT
cana-2868	173	23	𝐻s∗)[1	𝐻s∗)[1	PROPN
cana-2868	173	24	+	+	CCONJ
cana-2868	173	25	𝑑(s∗	𝑑(s∗	ADJ
cana-2868	173	26	,	,	PUNCT
cana-2868	173	27	𝐻sn	𝐻sn	PROPN
cana-2868	173	28	)	)	PUNCT
cana-2868	173	29	]	]	PUNCT
cana-2868	173	30	1	1	NUM
cana-2868	173	31	+	+	CCONJ
cana-2868	173	32	𝑑(sn	𝑑(sn	NUM
cana-2868	173	33	,	,	PUNCT
cana-2868	173	34	𝐻s∗	𝐻s∗	X
cana-2868	173	35	)	)	PUNCT
cana-2868	173	36	)	)	PUNCT
cana-2868	173	37	as	as	SCONJ
cana-2868	173	38	limit	limit	VERB
cana-2868	173	39	𝑛	𝑛	ADP
cana-2868	173	40	→	→	SYM
cana-2868	173	41	∞	∞	PROPN
cana-2868	173	42	in	in	ADP
cana-2868	173	43	the	the	DET
cana-2868	173	44	above	above	NOUN
cana-2868	173	45	and	and	CCONJ
cana-2868	173	46	from	from	ADP
cana-2868	173	47	the	the	DET
cana-2868	173	48	theorem	theorem	NOUN
cana-2868	173	49	(	(	PUNCT
cana-2868	173	50	i	i	NOUN
cana-2868	173	51	)	)	PUNCT
cana-2868	173	52	,	,	PUNCT
cana-2868	173	53	we	we	PRON
cana-2868	173	54	get	get	VERB
cana-2868	173	55	lim	lim	PROPN
cana-2868	173	56	𝑛→∞	𝑛→∞	NUM
cana-2868	173	57	(	(	PUNCT
cana-2868	173	58	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	173	59	)	)	PUNCT
cana-2868	173	60	]	]	PUNCT
cana-2868	173	61	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	173	62	)	)	PUNCT
cana-2868	173	63	)	)	PUNCT
cana-2868	174	1	=	=	PUNCT
cana-2868	174	2	0	0	NUM
cana-2868	174	3	,	,	PUNCT
cana-2868	174	4	so	so	SCONJ
cana-2868	174	5	that	that	SCONJ
cana-2868	174	6	1	1	NUM
cana-2868	174	7	𝑡	𝑡	PROPN
cana-2868	174	8	𝑑(𝑠∗	𝑑(𝑠∗	NOUN
cana-2868	174	9	,	,	PUNCT
cana-2868	174	10	𝐻𝑠∗	𝐻𝑠∗	NOUN
cana-2868	174	11	)	)	PUNCT
cana-2868	174	12	≤	≤	NOUN
cana-2868	174	13	lim	lim	NOUN
cana-2868	174	14	𝑛→∞	𝑛→∞	NUM
cana-2868	174	15	sup	sup	NOUN
cana-2868	174	16	𝑑(𝑠𝑛+1	𝑑(𝑠𝑛+1	NOUN
cana-2868	174	17	,	,	PUNCT
cana-2868	174	18	hs∗	hs∗	NOUN
cana-2868	174	19	)	)	PUNCT
cana-2868	174	20	≤	≤	NOUN
cana-2868	174	21	lim	lim	NOUN
cana-2868	174	22	𝑛→∞	𝑛→∞	NUM
cana-2868	174	23	𝑠𝑢𝑝	𝑠𝑢𝑝	PROPN
cana-2868	174	24	𝛽	𝛽	PROPN
cana-2868	174	25	(	(	PUNCT
cana-2868	174	26	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	174	27	)	)	PUNCT
cana-2868	174	28	]	]	PUNCT
cana-2868	174	29	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	174	30	)	)	PUNCT
cana-2868	174	31	)	)	PUNCT
cana-2868	174	32	(	(	PUNCT
cana-2868	174	33	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn	PROPN
cana-2868	174	34	)	)	PUNCT
cana-2868	174	35	]	]	PUNCT
cana-2868	174	36	1+𝑑(sn,𝐻s∗	1+𝑑(sn,𝐻s∗	NUM
cana-2868	174	37	)	)	PUNCT
cana-2868	174	38	)	)	PUNCT
cana-2868	175	1	=	=	SYM
cana-2868	175	2	0	0	PUNCT
cana-2868	175	3	and	and	CCONJ
cana-2868	175	4	hence	hence	ADV
cana-2868	175	5	consequently	consequently	ADV
cana-2868	175	6	,	,	PUNCT
cana-2868	175	7	we	we	PRON
cana-2868	175	8	get	get	VERB
cana-2868	175	9	𝑑(𝑠∗	𝑑(𝑠∗	NOUN
cana-2868	175	10	,	,	PUNCT
cana-2868	175	11	𝐻𝑠∗	𝐻𝑠∗	NOUN
cana-2868	175	12	)	)	PUNCT
cana-2868	176	1	=	=	SYM
cana-2868	176	2	0	0	NUM
cana-2868	176	3	,	,	PUNCT
cana-2868	176	4	this	this	PRON
cana-2868	176	5	is	be	AUX
cana-2868	176	6	contradiction	contradiction	NOUN
cana-2868	176	7	to	to	ADP
cana-2868	176	8	our	our	PRON
cana-2868	176	9	assumption	assumption	NOUN
cana-2868	176	10	,	,	PUNCT
cana-2868	176	11	𝑑(hs∗	𝑑(hs∗	PROPN
cana-2868	176	12	,	,	PUNCT
cana-2868	176	13	s∗	s∗	PROPN
cana-2868	176	14	)	)	PUNCT
cana-2868	176	15	>	>	X
cana-2868	176	16	0	0	PUNCT
cana-2868	176	17	.	.	PUNCT
cana-2868	177	1	therefore	therefore	ADV
cana-2868	177	2	from	from	ADP
cana-2868	177	3	all	all	DET
cana-2868	177	4	these	these	DET
cana-2868	177	5	cases	case	NOUN
cana-2868	177	6	,	,	PUNCT
cana-2868	177	7	we	we	PRON
cana-2868	177	8	can	can	AUX
cana-2868	177	9	get	get	VERB
cana-2868	177	10	𝑠∗	𝑠∗	NOUN
cana-2868	177	11	=	=	PRON
cana-2868	178	1	𝐻𝑠∗.	𝐻𝑠∗.	NOUN
cana-2868	178	2	hence	hence	ADV
cana-2868	178	3	𝑠∗	𝑠∗	NOUN
cana-2868	178	4	is	be	AUX
cana-2868	178	5	a	a	DET
cana-2868	178	6	fixed	fix	VERB
cana-2868	178	7	point	point	NOUN
cana-2868	178	8	of	of	ADP
cana-2868	178	9	h.	h.	PROPN
cana-2868	178	10	now	now	ADV
cana-2868	178	11	,	,	PUNCT
cana-2868	178	12	our	our	PRON
cana-2868	178	13	aim	aim	NOUN
cana-2868	178	14	is	be	AUX
cana-2868	178	15	to	to	PART
cana-2868	178	16	prove	prove	VERB
cana-2868	178	17	that	that	SCONJ
cana-2868	178	18	the	the	DET
cana-2868	178	19	uniqueness	uniqueness	NOUN
cana-2868	178	20	of	of	ADP
cana-2868	178	21	the	the	DET
cana-2868	178	22	fixed	fix	VERB
cana-2868	178	23	point	point	NOUN
cana-2868	178	24	.	.	PUNCT
cana-2868	179	1	assume	assume	VERB
cana-2868	179	2	that	that	SCONJ
cana-2868	179	3	𝑟	𝑟	X
cana-2868	179	4	∈	∈	PROPN
cana-2868	179	5	𝐻	𝐻	PROPN
cana-2868	179	6	is	be	AUX
cana-2868	179	7	other	other	ADJ
cana-2868	179	8	fixed	fix	VERB
cana-2868	179	9	point	point	NOUN
cana-2868	179	10	of	of	ADP
cana-2868	179	11	h	h	PROPN
cana-2868	179	12	such	such	ADJ
cana-2868	179	13	that𝑠∗	that𝑠∗	PROPN
cana-2868	179	14	≠	≠	PROPN
cana-2868	179	15	𝑟	𝑟	NOUN
cana-2868	179	16	,	,	PUNCT
cana-2868	179	17	𝑑(𝑠∗	𝑑(𝑠∗	NUM
cana-2868	179	18	,	,	PUNCT
cana-2868	179	19	𝑟	𝑟	NOUN
cana-2868	179	20	)	)	PUNCT
cana-2868	179	21	>	>	X
cana-2868	180	1	0	0	X
cana-2868	180	2	.	.	PUNCT
cana-2868	181	1	from	from	ADP
cana-2868	181	2	(	(	PUNCT
cana-2868	181	3	2.3.1	2.3.1	NUM
cana-2868	181	4	)	)	PUNCT
cana-2868	181	5	𝑑(𝑠∗	𝑑(𝑠∗	PROPN
cana-2868	181	6	,	,	PUNCT
cana-2868	181	7	𝑟	𝑟	X
cana-2868	181	8	)	)	PUNCT
cana-2868	181	9	=	=	SYM
cana-2868	181	10	𝑑(𝐻𝑠∗	𝑑(𝐻𝑠∗	X
cana-2868	181	11	,	,	PUNCT
cana-2868	181	12	𝑟	𝑟	NOUN
cana-2868	181	13	)	)	PUNCT
cana-2868	181	14	≤	≤	NOUN
cana-2868	181	15	𝑀(𝑠∗	𝑀(𝑠∗	PROPN
cana-2868	181	16	,	,	PUNCT
cana-2868	181	17	𝑟	𝑟	NOUN
cana-2868	181	18	)	)	PUNCT
cana-2868	181	19	=	=	SYM
cana-2868	181	20	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2868	181	21	{	{	PUNCT
cana-2868	181	22	𝛽	𝛽	NOUN
cana-2868	181	23	(	(	PUNCT
cana-2868	181	24	𝑑(𝑠∗,𝐻𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	𝑑(𝑠∗,𝐻𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	PROPN
cana-2868	181	25	)	)	PUNCT
cana-2868	181	26	]	]	X
cana-2868	181	27	1+𝑑(𝑟,𝐻𝑟	1+𝑑(𝑟,𝐻𝑟	X
cana-2868	181	28	)	)	PUNCT
cana-2868	181	29	)	)	PUNCT
cana-2868	181	30	(	(	PUNCT
cana-2868	181	31	𝑑(𝑠∗,𝐻𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	𝑑(𝑠∗,𝐻𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	PROPN
cana-2868	181	32	)	)	PUNCT
cana-2868	181	33	]	]	X
cana-2868	181	34	1+𝑑(𝑟,𝐻𝑟	1+𝑑(𝑟,𝐻𝑟	X
cana-2868	181	35	)	)	PUNCT
cana-2868	181	36	)	)	PUNCT
cana-2868	181	37	,	,	PUNCT
cana-2868	181	38	𝛽	𝛽	PROPN
cana-2868	181	39	(	(	PUNCT
cana-2868	181	40	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝐻𝑟	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝐻𝑟	NUM
cana-2868	181	41	)	)	PUNCT
cana-2868	181	42	]	]	PUNCT
cana-2868	181	43	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	44	)	)	PUNCT
cana-2868	181	45	)	)	PUNCT
cana-2868	181	46	(	(	PUNCT
cana-2868	181	47	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝐻𝑟	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝐻𝑟	NOUN
cana-2868	181	48	)	)	PUNCT
cana-2868	181	49	]	]	PUNCT
cana-2868	181	50	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	51	)	)	PUNCT
cana-2868	181	52	)	)	PUNCT
cana-2868	181	53	,	,	PUNCT
cana-2868	181	54	𝛽	𝛽	PROPN
cana-2868	181	55	(	(	PUNCT
cana-2868	181	56	𝑑(𝑟,𝐻𝑟)[1+𝑑(𝑟,𝐻𝑠∗	𝑑(𝑟,𝐻𝑟)[1+𝑑(𝑟,𝐻𝑠∗	NOUN
cana-2868	181	57	)	)	PUNCT
cana-2868	181	58	]	]	PUNCT
cana-2868	181	59	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	60	)	)	PUNCT
cana-2868	181	61	)	)	PUNCT
cana-2868	181	62	(	(	PUNCT
cana-2868	181	63	𝑑(𝑟,𝐻𝑟)[1+𝑑(𝑟,𝐻𝑠∗	𝑑(𝑟,𝐻𝑟)[1+𝑑(𝑟,𝐻𝑠∗	NOUN
cana-2868	181	64	)	)	PUNCT
cana-2868	181	65	]	]	PUNCT
cana-2868	181	66	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	67	)	)	PUNCT
cana-2868	181	68	)	)	PUNCT
cana-2868	181	69	}	}	PUNCT
cana-2868	181	70	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2868	181	71	{	{	PUNCT
cana-2868	181	72	𝛽	𝛽	NOUN
cana-2868	181	73	(	(	PUNCT
cana-2868	181	74	𝑑(𝑠∗,𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	𝑑(𝑠∗,𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	NOUN
cana-2868	181	75	)	)	PUNCT
cana-2868	181	76	]	]	X
cana-2868	181	77	1+𝑑(𝑟,𝑟	1+𝑑(𝑟,𝑟	NUM
cana-2868	181	78	)	)	PUNCT
cana-2868	181	79	)	)	PUNCT
cana-2868	181	80	(	(	PUNCT
cana-2868	181	81	𝑑(𝑠∗,𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	𝑑(𝑠∗,𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗	NOUN
cana-2868	181	82	)	)	PUNCT
cana-2868	181	83	]	]	X
cana-2868	181	84	1+𝑑(𝑟,𝑟	1+𝑑(𝑟,𝑟	NUM
cana-2868	181	85	)	)	PUNCT
cana-2868	181	86	)	)	PUNCT
cana-2868	181	87	,	,	PUNCT
cana-2868	181	88	𝛽	𝛽	NOUN
cana-2868	181	89	(	(	PUNCT
cana-2868	181	90	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟	PROPN
cana-2868	181	91	)	)	PUNCT
cana-2868	181	92	]	]	PUNCT
cana-2868	181	93	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	94	)	)	PUNCT
cana-2868	181	95	)	)	PUNCT
cana-2868	181	96	(	(	PUNCT
cana-2868	181	97	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟	PROPN
cana-2868	181	98	)	)	PUNCT
cana-2868	181	99	]	]	PUNCT
cana-2868	181	100	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	101	)	)	PUNCT
cana-2868	181	102	)	)	PUNCT
cana-2868	181	103	,	,	PUNCT
cana-2868	181	104	𝛽	𝛽	PROPN
cana-2868	181	105	(	(	PUNCT
cana-2868	181	106	𝑑(𝑟,𝑟)[1+𝑑(𝑟,𝐻𝑠∗	𝑑(𝑟,𝑟)[1+𝑑(𝑟,𝐻𝑠∗	X
cana-2868	181	107	)	)	PUNCT
cana-2868	181	108	]	]	PUNCT
cana-2868	181	109	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	110	)	)	PUNCT
cana-2868	181	111	)	)	PUNCT
cana-2868	181	112	(	(	PUNCT
cana-2868	181	113	𝑑(𝑟,𝑟)[1+𝑑(𝑟,𝐻𝑠∗	𝑑(𝑟,𝑟)[1+𝑑(𝑟,𝐻𝑠∗	X
cana-2868	181	114	)	)	PUNCT
cana-2868	181	115	]	]	PUNCT
cana-2868	181	116	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	181	117	)	)	PUNCT
cana-2868	181	118	)	)	PUNCT
cana-2868	181	119	}	}	PUNCT
cana-2868	181	120	=	=	SYM
cana-2868	181	121	𝛽	𝛽	NOUN
cana-2868	181	122	(	(	PUNCT
cana-2868	181	123	𝑑(𝑟	𝑑(𝑟	PROPN
cana-2868	181	124	,	,	PUNCT
cana-2868	181	125	𝐻𝑠∗)[1	𝐻𝑠∗)[1	NOUN
cana-2868	181	126	+	+	CCONJ
cana-2868	181	127	𝑑(𝑟	𝑑(𝑟	ADJ
cana-2868	181	128	,	,	PUNCT
cana-2868	181	129	𝑟	𝑟	NOUN
cana-2868	181	130	)	)	PUNCT
cana-2868	181	131	]	]	PUNCT
cana-2868	181	132	1	1	NUM
cana-2868	181	133	+	+	CCONJ
cana-2868	181	134	𝑑(𝑠∗	𝑑(𝑠∗	ADJ
cana-2868	181	135	,	,	PUNCT
cana-2868	181	136	𝐻𝑟	𝐻𝑟	PROPN
cana-2868	181	137	)	)	PUNCT
cana-2868	181	138	)	)	PUNCT
cana-2868	182	1	(	(	PUNCT
cana-2868	182	2	𝑑(𝑟	𝑑(𝑟	ADJ
cana-2868	182	3	,	,	PUNCT
cana-2868	182	4	𝐻𝑠∗)[1	𝐻𝑠∗)[1	NOUN
cana-2868	182	5	+	+	CCONJ
cana-2868	182	6	𝑑(𝑟	𝑑(𝑟	ADJ
cana-2868	182	7	,	,	PUNCT
cana-2868	182	8	𝑟	𝑟	NOUN
cana-2868	182	9	)	)	PUNCT
cana-2868	182	10	]	]	PUNCT
cana-2868	182	11	1	1	NUM
cana-2868	182	12	+	+	CCONJ
cana-2868	182	13	𝑑(𝑠∗	𝑑(𝑠∗	ADJ
cana-2868	182	14	,	,	PUNCT
cana-2868	182	15	𝐻𝑟	𝐻𝑟	PROPN
cana-2868	182	16	)	)	PUNCT
cana-2868	182	17	)	)	PUNCT
cana-2868	182	18	≤	≤	NOUN
cana-2868	182	19	(	(	PUNCT
cana-2868	182	20	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟	𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟	PROPN
cana-2868	182	21	)	)	PUNCT
cana-2868	182	22	]	]	PUNCT
cana-2868	182	23	1+𝑑(𝑠∗,𝐻𝑟	1+𝑑(𝑠∗,𝐻𝑟	NUM
cana-2868	182	24	)	)	PUNCT
cana-2868	182	25	)	)	PUNCT
cana-2868	183	1	<	<	X
cana-2868	183	2	1	1	NUM
cana-2868	183	3	𝑡	𝑡	PROPN
cana-2868	183	4	𝑑(𝑠∗	𝑑(𝑠∗	PROPN
cana-2868	183	5	,	,	PUNCT
cana-2868	183	6	𝑟	𝑟	NOUN
cana-2868	183	7	)	)	PUNCT
cana-2868	183	8	,	,	PUNCT
cana-2868	183	9	which	which	PRON
cana-2868	183	10	is	be	AUX
cana-2868	183	11	contradiction	contradiction	NOUN
cana-2868	183	12	.	.	PUNCT
cana-2868	184	1	so	so	ADV
cana-2868	184	2	that	that	DET
cana-2868	184	3	𝑠∗	𝑠∗	NOUN
cana-2868	185	1	=	=	NOUN
cana-2868	185	2	𝑟	𝑟	NOUN
cana-2868	185	3	hence	hence	ADV
cana-2868	185	4	𝑠∗	𝑠∗	NOUN
cana-2868	185	5	is	be	AUX
cana-2868	185	6	the	the	DET
cana-2868	185	7	only	only	ADJ
cana-2868	185	8	fixed	fix	VERB
cana-2868	185	9	point	point	NOUN
cana-2868	185	10	of	of	ADP
cana-2868	185	11	h	h	NOUN
cana-2868	185	12	in	in	ADP
cana-2868	185	13	m.	m.	NOUN
cana-2868	185	14	we	we	PRON
cana-2868	185	15	derive	derive	VERB
cana-2868	185	16	corollaries	corollary	NOUN
cana-2868	185	17	from	from	ADP
cana-2868	185	18	theorem2.3	theorem2.3	NOUN
cana-2868	185	19	corollaries	corollary	NOUN
cana-2868	185	20	:	:	PUNCT
cana-2868	185	21	corollary	corollary	ADJ
cana-2868	185	22	2.4	2.4	NUM
cana-2868	185	23	.	.	PUNCT
cana-2868	186	1	let	let	AUX
cana-2868	186	2	(	(	PUNCT
cana-2868	186	3	𝐾	𝐾	PROPN
cana-2868	186	4	,	,	PUNCT
cana-2868	186	5	𝑑	𝑑	PROPN
cana-2868	186	6	,	,	PUNCT
cana-2868	186	7	𝑡	𝑡	NOUN
cana-2868	186	8	)	)	PUNCT
cana-2868	186	9	be	be	VERB
cana-2868	186	10	a	a	DET
cana-2868	186	11	cbms	cbms	NOUN
cana-2868	186	12	with	with	ADP
cana-2868	186	13	𝑡	𝑡	PROPN
cana-2868	186	14	≥	≥	PROPN
cana-2868	186	15	1	1	NUM
cana-2868	186	16	.	.	PUNCT
cana-2868	187	1	suppose	suppose	VERB
cana-2868	187	2	that	that	SCONJ
cana-2868	187	3	𝐻	𝐻	PROPN
cana-2868	187	4	:	:	PUNCT
cana-2868	187	5	𝐾	𝐾	PROPN
cana-2868	187	6	→	→	SYM
cana-2868	187	7	𝐾	𝐾	PROPN
cana-2868	187	8	be	be	AUX
cana-2868	187	9	a	a	DET
cana-2868	187	10	self	self	NOUN
cana-2868	187	11	map	map	NOUN
cana-2868	187	12	and	and	CCONJ
cana-2868	187	13	𝛽	𝛽	NOUN
cana-2868	187	14	is	be	AUX
cana-2868	187	15	the	the	DET
cana-2868	187	16	class	class	NOUN
cana-2868	187	17	of	of	ADP
cana-2868	187	18	geraghty	geraghty	PROPN
cana-2868	187	19	functions	function	NOUN
cana-2868	187	20	𝑆	𝑆	PROPN
cana-2868	187	21	then	then	ADV
cana-2868	187	22	for	for	ADP
cana-2868	187	23	any	any	DET
cana-2868	187	24	𝑢	𝑢	NOUN
cana-2868	187	25	,	,	PUNCT
cana-2868	187	26	𝑣	𝑣	PRON
cana-2868	187	27	∈	∈	PROPN
cana-2868	187	28	𝐾	𝐾	PROPN
cana-2868	187	29	communications	communication	NOUN
cana-2868	187	30	on	on	ADP
cana-2868	187	31	applied	apply	VERB
cana-2868	187	32	nonlinear	nonlinear	ADJ
cana-2868	187	33	analysis	analysis	NOUN
cana-2868	187	34	issn	issn	NOUN
cana-2868	187	35	:	:	PUNCT
cana-2868	187	36	1074	1074	NUM
cana-2868	187	37	-	-	PUNCT
cana-2868	187	38	133x	133x	NUM
cana-2868	187	39	vol	vol	NOUN
cana-2868	187	40	32	32	NUM
cana-2868	187	41	no	no	NOUN
cana-2868	187	42	.	.	PUNCT
cana-2868	188	1	4s	4s	NUM
cana-2868	188	2	(	(	PUNCT
cana-2868	188	3	2025	2025	NUM
cana-2868	188	4	)	)	PUNCT
cana-2868	188	5	522	522	NUM
cana-2868	188	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2868	188	7	𝑑(𝐻𝑢	𝑑(𝐻𝑢	NOUN
cana-2868	188	8	,	,	PUNCT
cana-2868	188	9	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	188	10	)	)	PUNCT
cana-2868	188	11	≤	≤	NOUN
cana-2868	189	1	𝛽	𝛽	ADP
cana-2868	189	2	(	(	PUNCT
cana-2868	189	3	𝑁(𝑢	𝑁(𝑢	PROPN
cana-2868	189	4	,	,	PUNCT
cana-2868	189	5	𝑣))𝑁(𝑢	𝑣))𝑁(𝑢	PROPN
cana-2868	189	6	,	,	PUNCT
cana-2868	189	7	𝑣	𝑣	NOUN
cana-2868	189	8	)	)	PUNCT
cana-2868	189	9	where	where	SCONJ
cana-2868	189	10	𝑁(𝑢	𝑁(𝑢	NOUN
cana-2868	189	11	,	,	PUNCT
cana-2868	189	12	𝑣)=	𝑣)=	NOUN
cana-2868	189	13	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-2868	189	14	{	{	PUNCT
cana-2868	189	15	(	(	PUNCT
cana-2868	189	16	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	NOUN
cana-2868	189	17	)	)	PUNCT
cana-2868	189	18	]	]	PUNCT
cana-2868	190	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	190	2	)	)	PUNCT
cana-2868	190	3	)	)	PUNCT
cana-2868	190	4	,	,	PUNCT
cana-2868	190	5	(	(	PUNCT
cana-2868	190	6	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	NOUN
cana-2868	190	7	)	)	PUNCT
cana-2868	190	8	]	]	PUNCT
cana-2868	191	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	191	2	)	)	PUNCT
cana-2868	191	3	)	)	PUNCT
cana-2868	191	4	,	,	PUNCT
cana-2868	191	5	(	(	PUNCT
cana-2868	191	6	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	NOUN
cana-2868	191	7	)	)	PUNCT
cana-2868	191	8	]	]	PUNCT
cana-2868	192	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	192	2	)	)	PUNCT
cana-2868	192	3	)	)	PUNCT
cana-2868	192	4	}	}	PUNCT
cana-2868	192	5	.	.	PUNCT
cana-2868	193	1	then	then	ADV
cana-2868	193	2	h	h	PROPN
cana-2868	193	3	has	have	VERB
cana-2868	193	4	a	a	DET
cana-2868	193	5	unique	unique	ADJ
cana-2868	193	6	fixed	fix	VERB
cana-2868	193	7	point	point	NOUN
cana-2868	193	8	𝑢∗	𝑢∗	NOUN
cana-2868	193	9	∈	∈	PROPN
cana-2868	193	10	𝐾.	𝐾.	PROPN
cana-2868	193	11	𝑃𝑟𝑜𝑜𝑓	𝑃𝑟𝑜𝑜𝑓	PROPN
cana-2868	193	12	:	:	PUNCT
cana-2868	193	13	for	for	ADP
cana-2868	193	14	any	any	DET
cana-2868	193	15	u	u	NOUN
cana-2868	193	16	,	,	PUNCT
cana-2868	193	17	v	v	NOUN
cana-2868	193	18	in	in	ADP
cana-2868	193	19	k	k	PROPN
cana-2868	193	20	,	,	PUNCT
cana-2868	193	21	n(u	n(u	PROPN
cana-2868	193	22	,	,	PUNCT
cana-2868	193	23	v	v	NOUN
cana-2868	193	24	)	)	PUNCT
cana-2868	193	25	one	one	NUM
cana-2868	193	26	of	of	ADP
cana-2868	193	27	the	the	DET
cana-2868	193	28	term	term	NOUN
cana-2868	193	29	of	of	ADP
cana-2868	193	30	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	NOUN
cana-2868	193	31	)	)	PUNCT
cana-2868	193	32	]	]	PUNCT
cana-2868	194	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	194	2	)	)	PUNCT
cana-2868	194	3	)	)	PUNCT
cana-2868	194	4	,	,	PUNCT
cana-2868	194	5	(	(	PUNCT
cana-2868	194	6	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	NOUN
cana-2868	194	7	)	)	PUNCT
cana-2868	194	8	]	]	PUNCT
cana-2868	195	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	195	2	)	)	PUNCT
cana-2868	195	3	)	)	PUNCT
cana-2868	195	4	,	,	PUNCT
cana-2868	195	5	(	(	PUNCT
cana-2868	195	6	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	NOUN
cana-2868	195	7	)	)	PUNCT
cana-2868	195	8	]	]	PUNCT
cana-2868	196	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	196	2	)	)	PUNCT
cana-2868	196	3	)	)	PUNCT
cana-2868	197	1	then	then	ADV
cana-2868	197	2	it	it	PRON
cana-2868	197	3	follows	follow	VERB
cana-2868	197	4	that	that	SCONJ
cana-2868	197	5	𝑑(𝐻𝑢	𝑑(𝐻𝑢	ADV
cana-2868	197	6	,	,	PUNCT
cana-2868	197	7	𝐻𝑣	𝐻𝑣	PROPN
cana-2868	197	8	)	)	PUNCT
cana-2868	197	9	≤	≤	NOUN
cana-2868	197	10	𝛽	𝛽	ADP
cana-2868	197	11	(	(	PUNCT
cana-2868	197	12	𝑁(𝑢	𝑁(𝑢	PROPN
cana-2868	197	13	,	,	PUNCT
cana-2868	197	14	𝑣))𝑁(𝑢	𝑣))𝑁(𝑢	PROPN
cana-2868	197	15	,	,	PUNCT
cana-2868	197	16	𝑣	𝑣	NOUN
cana-2868	197	17	)	)	PUNCT
cana-2868	197	18	≤	≤	NOUN
cana-2868	197	19	𝛽	𝛽	ADP
cana-2868	197	20	(	(	PUNCT
cana-2868	197	21	𝑁(𝑢	𝑁(𝑢	PROPN
cana-2868	197	22	,	,	PUNCT
cana-2868	197	23	𝑣))𝑁(𝑢	𝑣))𝑁(𝑢	PROPN
cana-2868	197	24	,	,	PUNCT
cana-2868	197	25	𝑣	𝑣	NOUN
cana-2868	197	26	)	)	PUNCT
cana-2868	197	27	≤	≤	NUM
cana-2868	197	28	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2868	197	29	{	{	PUNCT
cana-2868	197	30	𝛽	𝛽	NOUN
cana-2868	197	31	(	(	PUNCT
cana-2868	197	32	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	PROPN
cana-2868	197	33	)	)	PUNCT
cana-2868	197	34	]	]	PUNCT
cana-2868	197	35	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	197	36	)	)	PUNCT
cana-2868	197	37	)	)	PUNCT
cana-2868	197	38	(	(	PUNCT
cana-2868	197	39	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢	NOUN
cana-2868	197	40	)	)	PUNCT
cana-2868	197	41	]	]	PUNCT
cana-2868	198	1	1+𝑑(𝑣,𝐻𝑣	1+𝑑(𝑣,𝐻𝑣	NUM
cana-2868	198	2	)	)	PUNCT
cana-2868	198	3	)	)	PUNCT
cana-2868	198	4	,	,	PUNCT
cana-2868	198	5	𝛽	𝛽	NOUN
cana-2868	198	6	(	(	PUNCT
cana-2868	198	7	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	PROPN
cana-2868	198	8	)	)	PUNCT
cana-2868	198	9	]	]	PUNCT
cana-2868	199	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	199	2	)	)	PUNCT
cana-2868	199	3	)	)	PUNCT
cana-2868	200	1	(	(	PUNCT
cana-2868	200	2	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣	NOUN
cana-2868	200	3	)	)	PUNCT
cana-2868	200	4	]	]	PUNCT
cana-2868	201	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	201	2	)	)	PUNCT
cana-2868	201	3	)	)	PUNCT
cana-2868	201	4	,	,	PUNCT
cana-2868	201	5	𝛽	𝛽	NOUN
cana-2868	201	6	(	(	PUNCT
cana-2868	201	7	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	PROPN
cana-2868	201	8	)	)	PUNCT
cana-2868	201	9	]	]	PUNCT
cana-2868	202	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	202	2	)	)	PUNCT
cana-2868	202	3	)	)	PUNCT
cana-2868	203	1	(	(	PUNCT
cana-2868	203	2	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢	NOUN
cana-2868	203	3	)	)	PUNCT
cana-2868	203	4	]	]	PUNCT
cana-2868	204	1	1+𝑑(𝑢,𝐻𝑣	1+𝑑(𝑢,𝐻𝑣	NOUN
cana-2868	204	2	)	)	PUNCT
cana-2868	204	3	)	)	PUNCT
cana-2868	204	4	}	}	PUNCT
cana-2868	205	1	=	=	SYM
cana-2868	205	2	m(u	m(u	PROPN
cana-2868	205	3	,	,	PUNCT
cana-2868	205	4	v	v	NOUN
cana-2868	205	5	)	)	PUNCT
cana-2868	205	6	,	,	PUNCT
cana-2868	205	7	and	and	CCONJ
cana-2868	205	8	observe	observe	VERB
cana-2868	205	9	that	that	SCONJ
cana-2868	205	10	all	all	DET
cana-2868	205	11	the	the	DET
cana-2868	205	12	hypothesis	hypothesis	NOUN
cana-2868	205	13	of	of	ADP
cana-2868	205	14	theorem	theorem	ADJ
cana-2868	205	15	2	2	NUM
cana-2868	205	16	are	be	AUX
cana-2868	205	17	satisfies	satisfie	NOUN
cana-2868	205	18	,	,	PUNCT
cana-2868	205	19	and	and	CCONJ
cana-2868	205	20	hence	hence	ADV
cana-2868	205	21	we	we	PRON
cana-2868	205	22	can	can	AUX
cana-2868	205	23	have	have	VERB
cana-2868	205	24	that	that	PRON
cana-2868	205	25	h	h	NOUN
cana-2868	205	26	has	have	AUX
cana-2868	205	27	unique	unique	ADJ
cana-2868	205	28	fixed	fix	VERB
cana-2868	205	29	point	point	NOUN
cana-2868	205	30	.	.	PUNCT
cana-2868	206	1	the	the	DET
cana-2868	206	2	following	follow	VERB
cana-2868	206	3	is	be	AUX
cana-2868	206	4	an	an	DET
cana-2868	206	5	example	example	NOUN
cana-2868	206	6	in	in	ADP
cana-2868	206	7	support	support	NOUN
cana-2868	206	8	our	our	PRON
cana-2868	206	9	main	main	ADJ
cana-2868	206	10	result	result	NOUN
cana-2868	206	11	.	.	PUNCT
cana-2868	207	1	examples	example	NOUN
cana-2868	207	2	.	.	PUNCT
cana-2868	208	1	example	example	NOUN
cana-2868	208	2	2.5	2.5	NUM
cana-2868	208	3	:	:	PUNCT
cana-2868	208	4	let	let	VERB
cana-2868	208	5	𝐾	𝐾	PROPN
cana-2868	208	6	=	=	SYM
cana-2868	208	7	{	{	PUNCT
cana-2868	208	8	1	1	NUM
cana-2868	208	9	,	,	PUNCT
cana-2868	208	10	1	1	NUM
cana-2868	208	11	2	2	NUM
cana-2868	208	12	,	,	PUNCT
cana-2868	208	13	1	1	NUM
cana-2868	208	14	4	4	NUM
cana-2868	208	15	}	}	PUNCT
cana-2868	208	16	𝑈{0	𝑈{0	VERB
cana-2868	208	17	}	}	PUNCT
cana-2868	208	18	and	and	CCONJ
cana-2868	208	19	define	define	VERB
cana-2868	208	20	the	the	DET
cana-2868	208	21	function	function	NOUN
cana-2868	208	22	from	from	ADP
cana-2868	208	23	𝑑	𝑑	NOUN
cana-2868	208	24	:	:	PUNCT
cana-2868	208	25	𝐾𝑋𝐾	𝐾𝑋𝐾	PROPN
cana-2868	208	26	→	→	SYM
cana-2868	208	27	𝑅+	𝑅+	NUM
cana-2868	208	28	by	by	ADP
cana-2868	208	29	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2868	208	30	,	,	PUNCT
cana-2868	208	31	𝑦	𝑦	X
cana-2868	208	32	)	)	PUNCT
cana-2868	208	33	=	=	SYM
cana-2868	209	1	(	(	PUNCT
cana-2868	209	2	𝑥	𝑥	NOUN
cana-2868	209	3	−	−	NOUN
cana-2868	209	4	𝑦)2	𝑦)2	NOUN
cana-2868	209	5	,	,	PUNCT
cana-2868	209	6	𝐻(𝑥	𝐻(𝑥	NOUN
cana-2868	209	7	)	)	PUNCT
cana-2868	209	8	=	=	SYM
cana-2868	209	9	1	1	NUM
cana-2868	209	10	𝑡2	𝑡2	PROPN
cana-2868	209	11	+	+	PROPN
cana-2868	209	12	1	1	NUM
cana-2868	209	13	,	,	PUNCT
cana-2868	209	14	𝑖𝑓	𝑖𝑓	ADP
cana-2868	209	15	𝑥	𝑥	NOUN
cana-2868	210	1	=	=	SYM
cana-2868	210	2	1	1	NUM
cana-2868	210	3	𝑡	𝑡	NOUN
cana-2868	210	4	,	,	PUNCT
cana-2868	210	5	0	0	PUNCT
cana-2868	210	6	if	if	SCONJ
cana-2868	210	7	x=0	x=0	PROPN
cana-2868	210	8	.	.	PUNCT
cana-2868	211	1	we	we	PRON
cana-2868	211	2	define𝛽(𝛼	define𝛽(𝛼	VERB
cana-2868	211	3	)	)	PUNCT
cana-2868	211	4	=	=	SYM
cana-2868	211	5	𝑒−𝛼	𝑒−𝛼	PROPN
cana-2868	211	6	,	,	PUNCT
cana-2868	211	7	𝛼	𝛼	X
cana-2868	211	8	>	>	X
cana-2868	211	9	0	0	NUM
cana-2868	211	10	,	,	PUNCT
cana-2868	211	11	𝛽(0	𝛽(0	PROPN
cana-2868	211	12	)	)	PUNCT
cana-2868	211	13	=	=	SYM
cana-2868	211	14	0	0	NUM
cana-2868	211	15	,	,	PUNCT
cana-2868	211	16	all	all	DET
cana-2868	211	17	the	the	DET
cana-2868	211	18	conditions	condition	NOUN
cana-2868	211	19	holds	hold	VERB
cana-2868	211	20	and	and	CCONJ
cana-2868	211	21	‘	'	PUNCT
cana-2868	211	22	0	0	NUM
cana-2868	211	23	’	'	PUNCT
cana-2868	211	24	fixed	fix	VERB
cana-2868	211	25	point	point	NOUN
cana-2868	211	26	and	and	CCONJ
cana-2868	211	27	is	be	AUX
cana-2868	211	28	unique	unique	ADJ
cana-2868	211	29	.	.	PUNCT
cana-2868	212	1	conclusions	conclusion	NOUN
cana-2868	212	2	:	:	PUNCT
cana-2868	212	3	in	in	ADP
cana-2868	212	4	this	this	PRON
cana-2868	212	5	we	we	PRON
cana-2868	212	6	proved	prove	VERB
cana-2868	212	7	the	the	DET
cana-2868	212	8	existence	existence	NOUN
cana-2868	212	9	and	and	CCONJ
cana-2868	212	10	uniqueness	uniqueness	NOUN
cana-2868	212	11	of	of	ADP
cana-2868	212	12	the	the	DET
cana-2868	212	13	fixed	fix	VERB
cana-2868	212	14	points	point	NOUN
cana-2868	212	15	generalized	generalize	VERB
cana-2868	212	16	ciric	ciric	ADJ
cana-2868	212	17	type	type	NOUN
cana-2868	212	18	geraghty	geraghty	VERB
cana-2868	212	19	rational	rational	ADJ
cana-2868	212	20	contractions	contraction	NOUN
cana-2868	212	21	in	in	ADP
cana-2868	212	22	b	b	NOUN
cana-2868	212	23	-	-	ADJ
cana-2868	212	24	metric	metric	ADJ
cana-2868	212	25	spaces	space	NOUN
cana-2868	212	26	,	,	PUNCT
cana-2868	212	27	our	our	PRON
cana-2868	212	28	results	result	NOUN
cana-2868	212	29	extend	extend	VERB
cana-2868	212	30	some	some	PRON
cana-2868	212	31	of	of	ADP
cana-2868	212	32	the	the	DET
cana-2868	212	33	known	know	VERB
cana-2868	212	34	theorems	theorem	NOUN
cana-2868	212	35	,	,	PUNCT
cana-2868	212	36	kalo.et.al	kalo.et.al	PROPN
cana-2868	213	1	[	[	X
cana-2868	213	2	1	1	NUM
cana-2868	213	3	]	]	PUNCT
cana-2868	213	4	proved	prove	VERB
cana-2868	213	5	fixed	fix	VERB
cana-2868	213	6	point	point	NOUN
cana-2868	213	7	theorems	theorem	NOUN
cana-2868	213	8	in	in	ADP
cana-2868	213	9	geraghty	geraghty	NOUN
cana-2868	213	10	-	-	PUNCT
cana-2868	213	11	ciric	ciric	ADJ
cana-2868	213	12	-	-	PUNCT
cana-2868	213	13	type	type	NOUN
cana-2868	213	14	contraction	contraction	NOUN
cana-2868	213	15	mapping	mapping	NOUN
cana-2868	213	16	in	in	ADP
cana-2868	213	17	bmetric	bmetric	ADJ
cana-2868	213	18	spaces	space	NOUN
cana-2868	213	19	.	.	PUNCT
cana-2868	214	1	we	we	PRON
cana-2868	214	2	derived	derive	VERB
cana-2868	214	3	some	some	DET
cana-2868	214	4	corollaries	corollary	NOUN
cana-2868	214	5	and	and	CCONJ
cana-2868	214	6	given	give	VERB
cana-2868	214	7	examples	example	NOUN
cana-2868	214	8	in	in	ADP
cana-2868	214	9	support	support	NOUN
cana-2868	214	10	our	our	PRON
cana-2868	214	11	main	main	ADJ
cana-2868	214	12	result	result	NOUN
cana-2868	214	13	.	.	PUNCT
cana-2868	215	1	references	reference	NOUN
cana-2868	215	2	[	[	X
cana-2868	215	3	1	1	NUM
cana-2868	215	4	]	]	PUNCT
cana-2868	215	5	albray	albray	ADJ
cana-2868	215	6	gebremariam	gebremariam	PROPN
cana-2868	215	7	kalo	kalo	PROPN
cana-2868	215	8	.	.	PROPN
cana-2868	215	9	,	,	PUNCT
cana-2868	215	10	kidane	kidane	PROPN
cana-2868	215	11	tola	tola	PROPN
cana-2868	215	12	and	and	CCONJ
cana-2868	215	13	haider	haider	PROPN
cana-2868	215	14	ebrahim	ebrahim	PROPN
cana-2868	215	15	yesuf	yesuf	PROPN
cana-2868	215	16	:	:	PUNCT
cana-2868	215	17	fixed	fix	VERB
cana-2868	215	18	point	point	NOUN
cana-2868	215	19	results	result	NOUN
cana-2868	215	20	of	of	ADP
cana-2868	215	21	geraghtyciric	geraghtyciric	ADJ
cana-2868	215	22	–	–	PUNCT
cana-2868	215	23	type	type	NOUN
cana-2868	215	24	contraction	contraction	NOUN
cana-2868	215	25	mappings	mapping	NOUN
cana-2868	215	26	in	in	ADP
cana-2868	215	27	b	b	NOUN
cana-2868	215	28	-	-	PUNCT
cana-2868	215	29	metric	metric	ADJ
cana-2868	215	30	space	space	NOUN
cana-2868	215	31	with	with	ADP
cana-2868	215	32	applications	application	NOUN
cana-2868	215	33	,	,	PUNCT
cana-2868	215	34	fixed	fix	VERB
cana-2868	215	35	point	point	NOUN
cana-2868	215	36	theory	theory	NOUN
cana-2868	215	37	algorithms	algorithm	VERB
cana-2868	215	38	sci	sci	PROPN
cana-2868	215	39	eng	eng	PROPN
cana-2868	215	40	,	,	PUNCT
cana-2868	215	41	2024:8	2024:8	NUM
cana-2868	215	42	,	,	PUNCT
cana-2868	215	43	doi.org/10.1186/s	doi.org/10.1186/s	PROPN
cana-2868	215	44	1336	1336	NUM
cana-2868	215	45	-	-	PUNCT
cana-2868	215	46	024	024	NUM
cana-2868	215	47	-	-	PUNCT
cana-2868	215	48	00764	00764	NUM
cana-2868	215	49	-	-	PUNCT
cana-2868	215	50	3	3	NUM
cana-2868	215	51	.	.	PUNCT
cana-2868	216	1	[	[	X
cana-2868	216	2	2	2	NUM
cana-2868	216	3	]	]	X
cana-2868	216	4	banach	banach	NOUN
cana-2868	216	5	,	,	PUNCT
cana-2868	216	6	s	s	PART
cana-2868	216	7	:	:	PUNCT
cana-2868	216	8	sur	sur	PROPN
cana-2868	216	9	les	les	PROPN
cana-2868	216	10	operations	operation	NOUN
cana-2868	216	11	dans	dan	NOUN
cana-2868	216	12	les	le	NOUN
cana-2868	216	13	ensembles	ensemble	NOUN
cana-2868	216	14	abstraits	abstrait	NOUN
cana-2868	216	15	et	et	PROPN
cana-2868	216	16	leur	leur	PROPN
cana-2868	216	17	applications	applications	PROPN
cana-2868	216	18	aux	aux	PROPN
cana-2868	216	19	equations	equation	NOUN
cana-2868	216	20	integrals	integral	NOUN
cana-2868	216	21	.	.	PUNCT
cana-2868	217	1	fundam.math.3.133	fundam.math.3.133	NOUN
cana-2868	217	2	-	-	NOUN
cana-2868	217	3	181(1922	181(1922	NUM
cana-2868	217	4	)	)	PUNCT
cana-2868	217	5	.	.	PUNCT
cana-2868	218	1	[	[	X
cana-2868	218	2	3	3	X
cana-2868	218	3	]	]	X
cana-2868	218	4	b.k	b.k	PROPN
cana-2868	218	5	.	.	PROPN
cana-2868	218	6	dass	dass	PROPN
cana-2868	218	7	,	,	PUNCT
cana-2868	218	8	s.	s.	PROPN
cana-2868	218	9	gupta	gupta	PROPN
cana-2868	218	10	,	,	PUNCT
cana-2868	218	11	an	an	DET
cana-2868	218	12	extension	extension	NOUN
cana-2868	218	13	of	of	ADP
cana-2868	218	14	banach	banach	NOUN
cana-2868	218	15	contraction	contraction	NOUN
cana-2868	218	16	principle	principle	NOUN
cana-2868	218	17	through	through	ADP
cana-2868	218	18	rational	rational	ADJ
cana-2868	218	19	expressions	expression	NOUN
cana-2868	218	20	,	,	PUNCT
cana-2868	218	21	indian	indian	ADJ
cana-2868	218	22	j.	j.	PROPN
cana-2868	218	23	pure	pure	PROPN
cana-2868	218	24	and	and	CCONJ
cana-2868	218	25	appl	appl	PROPN
cana-2868	218	26	.	.	PROPN
cana-2868	218	27	math	math	PROPN
cana-2868	218	28	.	.	PUNCT
cana-2868	218	29	,	,	PUNCT
cana-2868	218	30	1975	1975	NUM
cana-2868	218	31	,	,	PUNCT
cana-2868	218	32	6	6	NUM
cana-2868	218	33	:	:	SYM
cana-2868	218	34	1455	1455	NUM
cana-2868	218	35	-	-	SYM
cana-2868	218	36	1458	1458	NUM
cana-2868	218	37	.	.	PUNCT
cana-2868	219	1	[	[	X
cana-2868	219	2	4	4	NUM
cana-2868	219	3	]	]	X
cana-2868	219	4	ciric	ciric	ADJ
cana-2868	219	5	,	,	PUNCT
cana-2868	219	6	l.b	l.b	PROPN
cana-2868	219	7	,	,	PUNCT
cana-2868	219	8	:	:	PUNCT
cana-2868	219	9	generalized	generalized	ADJ
cana-2868	219	10	contractions	contraction	NOUN
cana-2868	219	11	and	and	CCONJ
cana-2868	219	12	fixed	fix	VERB
cana-2868	219	13	point	point	NOUN
cana-2868	219	14	theorems	theorem	NOUN
cana-2868	219	15	.	.	PUNCT
cana-2868	220	1	publ.inst.math,12(26),19	publ.inst.math,12(26),19	ADV
cana-2868	220	2	-	-	PUNCT
cana-2868	220	3	26(1971	26(1971	NUM
cana-2868	220	4	)	)	PUNCT
cana-2868	220	5	.	.	PUNCT
cana-2868	221	1	[	[	X
cana-2868	221	2	5	5	NUM
cana-2868	221	3	]	]	X
cana-2868	221	4	ciric	ciric	ADJ
cana-2868	221	5	,	,	PUNCT
cana-2868	221	6	l.b	l.b	PROPN
cana-2868	221	7	,	,	PUNCT
cana-2868	221	8	:	:	PUNCT
cana-2868	221	9	generalization	generalization	NOUN
cana-2868	221	10	of	of	ADP
cana-2868	221	11	banach	banach	NOUN
cana-2868	221	12	contraction	contraction	NOUN
cana-2868	221	13	principle	principle	NOUN
cana-2868	221	14	.	.	PUNCT
cana-2868	222	1	proc.am.maths.soc.45(2),267	proc.am.maths.soc.45(2),267	NOUN
cana-2868	222	2	-	-	PUNCT
cana-2868	222	3	273(1974	273(1974	NUM
cana-2868	222	4	)	)	PUNCT
cana-2868	222	5	.	.	PUNCT
cana-2868	223	1	[	[	X
cana-2868	223	2	6	6	NUM
cana-2868	223	3	]	]	SYM
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cana-2868	229	36	,	,	PUNCT
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cana-2868	243	10	some	some	DET
cana-2868	243	11	fixed	fix	VERB
cana-2868	243	12	point	point	NOUN
cana-2868	243	13	theorems	theorem	NOUN
cana-2868	243	14	for	for	ADP
cana-2868	243	15	rational	rational	ADJ
cana-2868	243	16	type	type	NOUN
cana-2868	243	17	geraghty	geraghty	PROPN
cana-2868	243	18	contractive	contractive	ADJ
cana-2868	243	19	mappings	mapping	NOUN
cana-2868	243	20	in	in	ADP
cana-2868	243	21	ordered	order	VERB
cana-2868	243	22	b	b	X
cana-2868	243	23	-	-	ADJ
cana-2868	243	24	metric	metric	ADJ
cana-2868	243	25	spaces	space	NOUN
cana-2868	243	26	.	.	PUNCT
cana-2868	244	1	inequal	inequal	ADJ
cana-2868	244	2	.	.	PUNCT
cana-2868	245	1	appl	appl	PROPN
cana-2868	245	2	.	.	PROPN
cana-2868	245	3	2014(2014	2014(2014	NUM
cana-2868	245	4	)	)	PUNCT
cana-2868	245	5	,	,	PUNCT
cana-2868	245	6	paperno.373	paperno.373	PROPN
cana-2868	245	7	.	.	PUNCT
cana-2868	246	1	[	[	X
cana-2868	246	2	17	17	NUM
cana-2868	246	3	]	]	X
cana-2868	246	4	s.	s.	PROPN
cana-2868	246	5	czerwik	czerwik	PROPN
cana-2868	246	6	,	,	PUNCT
cana-2868	246	7	nonlinear	nonlinear	PROPN
cana-2868	246	8	set	set	VERB
cana-2868	246	9	valued	value	VERB
cana-2868	246	10	contraction	contraction	NOUN
cana-2868	246	11	mappings	mapping	NOUN
cana-2868	246	12	in	in	ADP
cana-2868	246	13	b	b	NOUN
cana-2868	246	14	metric	metric	ADJ
cana-2868	246	15	spaces	space	NOUN
cana-2868	246	16	,	,	PUNCT
cana-2868	246	17	atti	atti	PROPN
cana-2868	246	18	semin	semin	PROPN
cana-2868	246	19	,	,	PUNCT
cana-2868	246	20	mat.fis.univ.modena	mat.fis.univ.modena	PROPN
cana-2868	246	21	46(1998	46(1998	NUM
cana-2868	246	22	)	)	PUNCT
cana-2868	246	23	,	,	PUNCT
cana-2868	246	24	no	no	DET
cana-2868	246	25	2	2	NUM
cana-2868	246	26	,	,	PUNCT
cana-2868	246	27	263	263	NUM
cana-2868	246	28	-	-	SYM
cana-2868	246	29	276	276	NUM
cana-2868	246	30	.	.	PUNCT