id	sid	tid	token	lemma	pos
cana-5932	1	1	communications	communication	NOUN
cana-5932	1	2	on	on	ADP
cana-5932	1	3	applied	apply	VERB
cana-5932	1	4	nonlinear	nonlinear	ADJ
cana-5932	1	5	analysis	analysis	NOUN
cana-5932	1	6	issn	issn	NOUN
cana-5932	1	7	:	:	PUNCT
cana-5932	1	8	1074	1074	NUM
cana-5932	1	9	-	-	PUNCT
cana-5932	1	10	133x	133x	NUM
cana-5932	1	11	vol	vol	VERB
cana-5932	1	12	32	32	NUM
cana-5932	1	13	no	no	NOUN
cana-5932	1	14	.	.	PUNCT
cana-5932	2	1	10s	10	NOUN
cana-5932	2	2	(	(	PUNCT
cana-5932	2	3	2025	2025	NUM
cana-5932	2	4	)	)	PUNCT
cana-5932	2	5	3127	3127	NUM
cana-5932	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	2	7	fixed	fix	VERB
cana-5932	2	8	point	point	NOUN
cana-5932	2	9	theorem	theorem	NOUN
cana-5932	2	10	for	for	ADP
cana-5932	2	11	h	h	NOUN
cana-5932	2	12	–	–	PUNCT
cana-5932	2	13	f	f	X
cana-5932	2	14	–	–	PUNCT
cana-5932	2	15	contractive	contractive	ADJ
cana-5932	2	16	mapping	mapping	NOUN
cana-5932	2	17	in	in	ADP
cana-5932	2	18	complete	complete	ADJ
cana-5932	2	19	fuzzy	fuzzy	ADJ
cana-5932	2	20	metric	metric	ADJ
cana-5932	2	21	spaces	space	NOUN
cana-5932	2	22	j.	j.	PROPN
cana-5932	2	23	ravinder1	ravinder1	PROPN
cana-5932	2	24	,	,	PUNCT
cana-5932	2	25	a.	a.	NOUN
cana-5932	2	26	bernick	bernick	PROPN
cana-5932	2	27	raj2	raj2	PROPN
cana-5932	2	28	*	*	PROPN
cana-5932	2	29	1	1	NUM
cana-5932	2	30	&	&	CCONJ
cana-5932	2	31	2	2	NUM
cana-5932	2	32	department	department	NOUN
cana-5932	2	33	of	of	ADP
cana-5932	2	34	mathematics	mathematic	NOUN
cana-5932	2	35	and	and	CCONJ
cana-5932	2	36	actuarial	actuarial	ADJ
cana-5932	2	37	science	science	NOUN
cana-5932	2	38	,	,	PUNCT
cana-5932	2	39	b.s	b.s	PROPN
cana-5932	2	40	.	.	PROPN
cana-5932	2	41	abdur	abdur	PROPN
cana-5932	2	42	rahman	rahman	PROPN
cana-5932	2	43	crescent	crescent	PROPN
cana-5932	2	44	institute	institute	PROPN
cana-5932	2	45	of	of	ADP
cana-5932	2	46	science	science	NOUN
cana-5932	2	47	and	and	CCONJ
cana-5932	2	48	technology	technology	NOUN
cana-5932	2	49	,	,	PUNCT
cana-5932	2	50	chennai	chennai	PROPN
cana-5932	2	51	,	,	PUNCT
cana-5932	2	52	tamil	tamil	PROPN
cana-5932	2	53	nadu	nadu	PROPN
cana-5932	2	54	,	,	PUNCT
cana-5932	2	55	india	india	PROPN
cana-5932	2	56	,	,	PUNCT
cana-5932	2	57	600048	600048	NUM
cana-5932	2	58	.	.	PUNCT
cana-5932	3	1	email	email	NOUN
cana-5932	3	2	:	:	PUNCT
cana-5932	3	3	1	1	NUM
cana-5932	3	4	ravinder.maths@gmail.com	ravinder.maths@gmail.com	NOUN
cana-5932	3	5	,	,	PUNCT
cana-5932	3	6	2	2	NUM
cana-5932	3	7	bernickraj@crescent.education	bernickraj@crescent.education	NOUN
cana-5932	3	8	article	article	NOUN
cana-5932	3	9	history	history	NOUN
cana-5932	3	10	:	:	PUNCT
cana-5932	3	11	received	receive	VERB
cana-5932	3	12	:	:	PUNCT
cana-5932	3	13	02	02	NUM
cana-5932	3	14	-	-	PUNCT
cana-5932	3	15	01	01	NUM
cana-5932	3	16	-	-	PUNCT
cana-5932	3	17	2025	2025	NUM
cana-5932	3	18	revised	revise	VERB
cana-5932	3	19	:	:	PUNCT
cana-5932	3	20	25	25	NUM
cana-5932	3	21	-	-	PUNCT
cana-5932	3	22	02	02	NUM
cana-5932	3	23	-	-	PUNCT
cana-5932	3	24	2025	2025	NUM
cana-5932	3	25	accepted	accept	VERB
cana-5932	3	26	:	:	PUNCT
cana-5932	3	27	20	20	NUM
cana-5932	3	28	-	-	SYM
cana-5932	3	29	03	03	NUM
cana-5932	3	30	-	-	PUNCT
cana-5932	3	31	2025	2025	NUM
cana-5932	3	32	abstract	abstract	NOUN
cana-5932	3	33	:	:	PUNCT
cana-5932	3	34	this	this	DET
cana-5932	3	35	study	study	NOUN
cana-5932	3	36	introduces	introduce	VERB
cana-5932	3	37	a	a	DET
cana-5932	3	38	new	new	ADJ
cana-5932	3	39	class	class	NOUN
cana-5932	3	40	of	of	ADP
cana-5932	3	41	contractive	contractive	ADJ
cana-5932	3	42	mappings	mapping	NOUN
cana-5932	3	43	,	,	PUNCT
cana-5932	3	44	namely	namely	ADV
cana-5932	3	45	h	h	NOUN
cana-5932	3	46	–	–	PUNCT
cana-5932	3	47	f	f	X
cana-5932	3	48	–	–	PUNCT
cana-5932	3	49	contractive	contractive	ADJ
cana-5932	3	50	mappings	mapping	NOUN
cana-5932	3	51	,	,	PUNCT
cana-5932	3	52	in	in	ADP
cana-5932	3	53	the	the	DET
cana-5932	3	54	setting	setting	NOUN
cana-5932	3	55	of	of	ADP
cana-5932	3	56	complete	complete	ADJ
cana-5932	3	57	fuzzy	fuzzy	ADJ
cana-5932	3	58	metric	metric	ADJ
cana-5932	3	59	spaces	space	NOUN
cana-5932	3	60	.	.	PUNCT
cana-5932	4	1	we	we	PRON
cana-5932	4	2	provide	provide	VERB
cana-5932	4	3	the	the	DET
cana-5932	4	4	sufficient	sufficient	ADJ
cana-5932	4	5	conditions	condition	NOUN
cana-5932	4	6	under	under	ADP
cana-5932	4	7	which	which	PRON
cana-5932	4	8	such	such	ADJ
cana-5932	4	9	mappings	mapping	NOUN
cana-5932	4	10	admit	admit	VERB
cana-5932	4	11	unique	unique	ADJ
cana-5932	4	12	fixed	fix	VERB
cana-5932	4	13	points	point	NOUN
cana-5932	4	14	.	.	PUNCT
cana-5932	5	1	our	our	PRON
cana-5932	5	2	approach	approach	NOUN
cana-5932	5	3	builds	build	VERB
cana-5932	5	4	on	on	ADP
cana-5932	5	5	classical	classical	ADJ
cana-5932	5	6	results	result	NOUN
cana-5932	5	7	by	by	ADP
cana-5932	5	8	employing	employ	VERB
cana-5932	5	9	a	a	DET
cana-5932	5	10	novel	novel	ADJ
cana-5932	5	11	functional	functional	ADJ
cana-5932	5	12	inequality	inequality	NOUN
cana-5932	5	13	that	that	PRON
cana-5932	5	14	involves	involve	VERB
cana-5932	5	15	a	a	DET
cana-5932	5	16	strictly	strictly	ADV
cana-5932	5	17	increasing	increase	VERB
cana-5932	5	18	function	function	NOUN
cana-5932	5	19	f	f	PROPN
cana-5932	5	20	∈	∈	PROPN
cana-5932	5	21	f	f	PROPN
cana-5932	5	22	and	and	CCONJ
cana-5932	5	23	auxiliary	auxiliary	ADJ
cana-5932	5	24	function	function	NOUN
cana-5932	5	25	h	h	PROPN
cana-5932	5	26	∈	∈	PROPN
cana-5932	5	27	h.	h.	PROPN
cana-5932	6	1	an	an	DET
cana-5932	6	2	illustrative	illustrative	ADJ
cana-5932	6	3	example	example	NOUN
cana-5932	6	4	demonstrates	demonstrate	VERB
cana-5932	6	5	the	the	DET
cana-5932	6	6	applicability	applicability	NOUN
cana-5932	6	7	of	of	ADP
cana-5932	6	8	the	the	DET
cana-5932	6	9	main	main	ADJ
cana-5932	6	10	results	result	NOUN
cana-5932	6	11	.	.	PUNCT
cana-5932	7	1	this	this	DET
cana-5932	7	2	work	work	NOUN
cana-5932	7	3	enriches	enrich	VERB
cana-5932	7	4	the	the	DET
cana-5932	7	5	framework	framework	NOUN
cana-5932	7	6	of	of	ADP
cana-5932	7	7	the	the	DET
cana-5932	7	8	fixed	fix	VERB
cana-5932	7	9	point	point	NOUN
cana-5932	7	10	theory	theory	NOUN
cana-5932	7	11	in	in	ADP
cana-5932	7	12	fuzzy	fuzzy	ADJ
cana-5932	7	13	metric	metric	ADJ
cana-5932	7	14	spaces	space	NOUN
cana-5932	7	15	and	and	CCONJ
cana-5932	7	16	opens	open	VERB
cana-5932	7	17	pathways	pathway	NOUN
cana-5932	7	18	for	for	ADP
cana-5932	7	19	further	further	ADJ
cana-5932	7	20	exploration	exploration	NOUN
cana-5932	7	21	.	.	PUNCT
cana-5932	8	1	keywords	keyword	NOUN
cana-5932	8	2	:	:	PUNCT
cana-5932	8	3	fixed	fixed	ADJ
cana-5932	8	4	point	point	NOUN
cana-5932	8	5	,	,	PUNCT
cana-5932	8	6	fuzzy	fuzzy	ADJ
cana-5932	8	7	metric	metric	ADJ
cana-5932	8	8	space	space	NOUN
cana-5932	8	9	,	,	PUNCT
cana-5932	8	10	h	h	NOUN
cana-5932	8	11	–	–	PUNCT
cana-5932	8	12	f	f	X
cana-5932	8	13	–	–	PUNCT
cana-5932	8	14	contractive	contractive	ADJ
cana-5932	8	15	mapping	mapping	NOUN
cana-5932	8	16	,	,	PUNCT
cana-5932	8	17	fuzzy	fuzzy	ADJ
cana-5932	8	18	analysis	analysis	NOUN
cana-5932	8	19	,	,	PUNCT
cana-5932	8	20	contraction	contraction	NOUN
cana-5932	8	21	principle	principle	NOUN
cana-5932	8	22	.	.	PUNCT
cana-5932	9	1	1	1	X
cana-5932	9	2	.	.	X
cana-5932	9	3	introduction	introduction	NOUN
cana-5932	9	4	the	the	DET
cana-5932	9	5	theory	theory	NOUN
cana-5932	9	6	of	of	ADP
cana-5932	9	7	fuzzy	fuzzy	ADJ
cana-5932	9	8	metric	metric	ADJ
cana-5932	9	9	spaces	space	NOUN
cana-5932	9	10	,	,	PUNCT
cana-5932	9	11	initiated	initiate	VERB
cana-5932	9	12	by	by	ADP
cana-5932	9	13	kramosil	kramosil	NOUN
cana-5932	9	14	and	and	CCONJ
cana-5932	9	15	michalek	michalek	NOUN
cana-5932	9	16	[	[	X
cana-5932	9	17	1	1	NUM
cana-5932	9	18	]	]	PUNCT
cana-5932	9	19	and	and	CCONJ
cana-5932	9	20	further	far	ADV
cana-5932	9	21	developed	develop	VERB
cana-5932	9	22	by	by	ADP
cana-5932	9	23	george	george	PROPN
cana-5932	9	24	and	and	CCONJ
cana-5932	9	25	veeramani	veeramani	NOUN
cana-5932	10	1	[	[	X
cana-5932	10	2	2	2	NUM
cana-5932	10	3	]	]	PUNCT
cana-5932	10	4	,	,	PUNCT
cana-5932	10	5	has	have	AUX
cana-5932	10	6	received	receive	VERB
cana-5932	10	7	considerable	considerable	ADJ
cana-5932	10	8	attention	attention	NOUN
cana-5932	10	9	for	for	ADP
cana-5932	10	10	addressing	address	VERB
cana-5932	10	11	the	the	DET
cana-5932	10	12	uncertainties	uncertainty	NOUN
cana-5932	10	13	in	in	ADP
cana-5932	10	14	mathematical	mathematical	ADJ
cana-5932	10	15	models	model	NOUN
cana-5932	10	16	.	.	PUNCT
cana-5932	11	1	several	several	ADJ
cana-5932	11	2	classical	classical	ADJ
cana-5932	11	3	fixed	fix	VERB
cana-5932	11	4	point	point	NOUN
cana-5932	11	5	results	result	NOUN
cana-5932	11	6	,	,	PUNCT
cana-5932	11	7	such	such	ADJ
cana-5932	11	8	as	as	ADP
cana-5932	11	9	banach	banach	NOUN
cana-5932	11	10	,	,	PUNCT
cana-5932	11	11	kannan	kannan	PROPN
cana-5932	11	12	,	,	PUNCT
cana-5932	11	13	and	and	CCONJ
cana-5932	11	14	chatterjea	chatterjea	ADJ
cana-5932	11	15	contractions	contraction	NOUN
cana-5932	11	16	,	,	PUNCT
cana-5932	11	17	have	have	AUX
cana-5932	11	18	been	be	AUX
cana-5932	11	19	generalized	generalize	VERB
cana-5932	11	20	to	to	ADP
cana-5932	11	21	fuzzy	fuzzy	ADJ
cana-5932	11	22	metric	metric	ADJ
cana-5932	11	23	settings	setting	NOUN
cana-5932	11	24	[	[	X
cana-5932	11	25	3	3	NUM
cana-5932	11	26	,	,	PUNCT
cana-5932	11	27	4	4	NUM
cana-5932	11	28	,	,	PUNCT
cana-5932	11	29	5	5	NUM
cana-5932	11	30	,	,	PUNCT
cana-5932	11	31	6	6	NUM
cana-5932	11	32	,	,	PUNCT
cana-5932	11	33	7	7	NUM
cana-5932	11	34	]	]	PUNCT
cana-5932	11	35	.	.	PUNCT
cana-5932	12	1	in	in	ADP
cana-5932	12	2	this	this	DET
cana-5932	12	3	context	context	NOUN
cana-5932	12	4	,	,	PUNCT
cana-5932	12	5	new	new	ADJ
cana-5932	12	6	con	con	NOUN
cana-5932	12	7	traction	traction	PROPN
cana-5932	12	8	types	type	NOUN
cana-5932	12	9	,	,	PUNCT
cana-5932	12	10	such	such	ADJ
cana-5932	12	11	as	as	ADP
cana-5932	12	12	f	f	PROPN
cana-5932	12	13	-contractions	-contractions	PROPN
cana-5932	12	14	and	and	CCONJ
cana-5932	12	15	simulation	simulation	NOUN
cana-5932	12	16	functions	function	NOUN
cana-5932	12	17	have	have	AUX
cana-5932	12	18	broadened	broaden	VERB
cana-5932	12	19	the	the	DET
cana-5932	12	20	scope	scope	NOUN
cana-5932	12	21	of	of	ADP
cana-5932	12	22	fixed	fix	VERB
cana-5932	12	23	point	point	NOUN
cana-5932	12	24	analysis	analysis	NOUN
cana-5932	12	25	.	.	PUNCT
cana-5932	13	1	motivated	motivate	VERB
cana-5932	13	2	by	by	ADP
cana-5932	13	3	these	these	DET
cana-5932	13	4	developments	development	NOUN
cana-5932	13	5	,	,	PUNCT
cana-5932	13	6	we	we	PRON
cana-5932	13	7	propose	propose	VERB
cana-5932	13	8	the	the	DET
cana-5932	13	9	concept	concept	NOUN
cana-5932	13	10	of	of	ADP
cana-5932	13	11	h	h	NOUN
cana-5932	13	12	–	–	PUNCT
cana-5932	13	13	f	f	X
cana-5932	13	14	–	–	PUNCT
cana-5932	13	15	contractive	contractive	ADJ
cana-5932	13	16	mappings	mapping	NOUN
cana-5932	13	17	.	.	PUNCT
cana-5932	14	1	these	these	DET
cana-5932	14	2	mappings	mapping	NOUN
cana-5932	14	3	incorporate	incorporate	VERB
cana-5932	14	4	an	an	DET
cana-5932	14	5	auxiliary	auxiliary	ADJ
cana-5932	14	6	function	function	NOUN
cana-5932	14	7	h	h	NOUN
cana-5932	14	8	and	and	CCONJ
cana-5932	14	9	transformation	transformation	NOUN
cana-5932	14	10	function	function	NOUN
cana-5932	14	11	f	f	NOUN
cana-5932	14	12	,	,	PUNCT
cana-5932	14	13	which	which	PRON
cana-5932	14	14	allows	allow	VERB
cana-5932	14	15	us	we	PRON
cana-5932	14	16	to	to	PART
cana-5932	14	17	derive	derive	VERB
cana-5932	14	18	a	a	DET
cana-5932	14	19	fixed	fix	VERB
cana-5932	14	20	point	point	NOUN
cana-5932	14	21	theorem	theorem	NOUN
cana-5932	14	22	that	that	PRON
cana-5932	14	23	generalizes	generalize	VERB
cana-5932	14	24	several	several	ADJ
cana-5932	14	25	known	know	VERB
cana-5932	14	26	results	result	NOUN
cana-5932	14	27	.	.	PUNCT
cana-5932	15	1	our	our	PRON
cana-5932	15	2	main	main	ADJ
cana-5932	15	3	contribution	contribution	NOUN
cana-5932	15	4	lies	lie	VERB
cana-5932	15	5	in	in	ADP
cana-5932	15	6	establishing	establish	VERB
cana-5932	15	7	a	a	DET
cana-5932	15	8	unique	unique	ADJ
cana-5932	15	9	fixed	fix	VERB
cana-5932	15	10	point	point	NOUN
cana-5932	15	11	under	under	ADP
cana-5932	15	12	the	the	DET
cana-5932	15	13	new	new	ADJ
cana-5932	15	14	contractive	contractive	ADJ
cana-5932	15	15	condition	condition	NOUN
cana-5932	15	16	.	.	PUNCT
cana-5932	16	1	2	2	X
cana-5932	16	2	.	.	X
cana-5932	16	3	preliminaries	preliminary	NOUN
cana-5932	16	4	here	here	ADV
cana-5932	16	5	,	,	PUNCT
cana-5932	16	6	we	we	PRON
cana-5932	16	7	recall	recall	VERB
cana-5932	16	8	the	the	DET
cana-5932	16	9	key	key	ADJ
cana-5932	16	10	definitions	definition	NOUN
cana-5932	16	11	required	require	VERB
cana-5932	16	12	for	for	ADP
cana-5932	16	13	our	our	PRON
cana-5932	16	14	analysis	analysis	NOUN
cana-5932	16	15	.	.	PUNCT
cana-5932	17	1	throughout	throughout	ADP
cana-5932	17	2	this	this	DET
cana-5932	17	3	paper	paper	NOUN
cana-5932	17	4	,	,	PUNCT
cana-5932	17	5	n	n	PRON
cana-5932	17	6	denotes	denote	VERB
cana-5932	17	7	the	the	DET
cana-5932	17	8	set	set	NOUN
cana-5932	17	9	of	of	ADP
cana-5932	17	10	all	all	DET
cana-5932	17	11	positive	positive	ADJ
cana-5932	17	12	integers	integer	NOUN
cana-5932	17	13	,	,	PUNCT
cana-5932	17	14	n0	n0	NUM
cana-5932	17	15	denotes	denote	VERB
cana-5932	17	16	the	the	DET
cana-5932	17	17	set	set	NOUN
cana-5932	17	18	of	of	ADP
cana-5932	17	19	all	all	DET
cana-5932	17	20	non	non	ADJ
cana-5932	17	21	-	-	ADJ
cana-5932	17	22	negative	negative	ADJ
cana-5932	17	23	integers	integer	NOUN
cana-5932	17	24	,	,	PUNCT
cana-5932	17	25	r+	r+	PUNCT
cana-5932	17	26	=	=	PUNCT
cana-5932	17	27	(	(	PUNCT
cana-5932	17	28	0	0	NUM
cana-5932	17	29	,	,	PUNCT
cana-5932	17	30	∞	∞	NUM
cana-5932	17	31	)	)	PUNCT
cana-5932	17	32	and	and	CCONJ
cana-5932	17	33	i	i	PRON
cana-5932	17	34	=	=	PUNCT
cana-5932	18	1	[	[	X
cana-5932	18	2	0	0	NUM
cana-5932	18	3	,	,	PUNCT
cana-5932	18	4	1	1	NUM
cana-5932	18	5	]	]	PUNCT
cana-5932	18	6	.	.	PUNCT
cana-5932	19	1	definition	definition	NOUN
cana-5932	19	2	1	1	NUM
cana-5932	19	3	.	.	PUNCT
cana-5932	20	1	[	[	X
cana-5932	20	2	8	8	NUM
cana-5932	20	3	]	]	PUNCT
cana-5932	20	4	a	a	DET
cana-5932	20	5	binary	binary	ADJ
cana-5932	20	6	operation	operation	NOUN
cana-5932	20	7	∗	∗	NOUN
cana-5932	20	8	:	:	PUNCT
cana-5932	21	1	i	i	PRON
cana-5932	21	2	×	×	VERB
cana-5932	21	3	i	i	PRON
cana-5932	21	4	→	→	PUNCT
cana-5932	21	5	i	i	PRON
cana-5932	21	6	is	be	AUX
cana-5932	21	7	called	call	VERB
cana-5932	21	8	a	a	DET
cana-5932	21	9	continuous	continuous	ADJ
cana-5932	21	10	t	t	NOUN
cana-5932	21	11	-	-	PUNCT
cana-5932	21	12	norm	norm	NOUN
cana-5932	21	13	,	,	PUNCT
cana-5932	21	14	if	if	SCONJ
cana-5932	21	15	the	the	DET
cana-5932	21	16	following	follow	VERB
cana-5932	21	17	conditions	condition	NOUN
cana-5932	21	18	holds	hold	VERB
cana-5932	21	19	:	:	PUNCT
cana-5932	21	20	(	(	PUNCT
cana-5932	21	21	i	i	NOUN
cana-5932	21	22	)	)	PUNCT
cana-5932	21	23	∗	∗	NOUN
cana-5932	21	24	is	be	AUX
cana-5932	21	25	associative	associative	ADJ
cana-5932	21	26	and	and	CCONJ
cana-5932	21	27	commutative	commutative	ADJ
cana-5932	21	28	.	.	PUNCT
cana-5932	22	1	(	(	PUNCT
cana-5932	22	2	ii	ii	NOUN
cana-5932	22	3	)	)	PUNCT
cana-5932	22	4	for	for	ADP
cana-5932	22	5	any	any	DET
cana-5932	22	6	a	a	DET
cana-5932	22	7	∈	∈	NOUN
cana-5932	23	1	i	i	PRON
cana-5932	23	2	,	,	PUNCT
cana-5932	23	3	a	a	DET
cana-5932	23	4	∗	∗	NOUN
cana-5932	23	5	1	1	NUM
cana-5932	23	6	=	=	SYM
cana-5932	23	7	a.	a.	NOUN
cana-5932	23	8	(	(	PUNCT
cana-5932	23	9	iii	iii	NOUN
cana-5932	23	10	)	)	PUNCT
cana-5932	23	11	for	for	ADP
cana-5932	23	12	a	a	DET
cana-5932	23	13	,	,	PUNCT
cana-5932	23	14	b	b	NOUN
cana-5932	23	15	,	,	PUNCT
cana-5932	23	16	c	c	NOUN
cana-5932	23	17	,	,	PUNCT
cana-5932	23	18	d	d	PROPN
cana-5932	23	19	∈	∈	PROPN
cana-5932	23	20	i	i	PRON
cana-5932	23	21	with	with	ADP
cana-5932	23	22	a	a	DET
cana-5932	23	23	≤	≤	NUM
cana-5932	23	24	b	b	NOUN
cana-5932	23	25	and	and	CCONJ
cana-5932	23	26	c	c	NOUN
cana-5932	23	27	≤	≤	NUM
cana-5932	23	28	d	d	PROPN
cana-5932	23	29	,	,	PUNCT
cana-5932	23	30	a	a	DET
cana-5932	23	31	∗	∗	NOUN
cana-5932	23	32	c	c	NOUN
cana-5932	23	33	≤	≤	NUM
cana-5932	23	34	b	b	PROPN
cana-5932	23	35	∗	∗	X
cana-5932	23	36	d.	d.	PROPN
cana-5932	23	37	communications	communication	NOUN
cana-5932	23	38	on	on	ADP
cana-5932	23	39	applied	apply	VERB
cana-5932	23	40	nonlinear	nonlinear	ADJ
cana-5932	23	41	analysis	analysis	NOUN
cana-5932	23	42	issn	issn	NOUN
cana-5932	23	43	:	:	PUNCT
cana-5932	23	44	1074	1074	NUM
cana-5932	23	45	-	-	PUNCT
cana-5932	23	46	133x	133x	NUM
cana-5932	23	47	vol	vol	VERB
cana-5932	23	48	32	32	NUM
cana-5932	23	49	no	no	NOUN
cana-5932	23	50	.	.	PUNCT
cana-5932	24	1	10s	10	NOUN
cana-5932	24	2	(	(	PUNCT
cana-5932	24	3	2025	2025	NUM
cana-5932	24	4	)	)	PUNCT
cana-5932	24	5	3128	3128	NUM
cana-5932	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	24	7	(	(	PUNCT
cana-5932	24	8	iv	iv	X
cana-5932	24	9	)	)	PUNCT
cana-5932	24	10	∗	∗	NOUN
cana-5932	24	11	is	be	AUX
cana-5932	24	12	continuous	continuous	ADJ
cana-5932	24	13	.	.	PUNCT
cana-5932	25	1	for	for	ADP
cana-5932	25	2	example	example	NOUN
cana-5932	25	3	,	,	PUNCT
cana-5932	25	4	a	a	DET
cana-5932	25	5	∗1	∗1	PROPN
cana-5932	25	6	b	b	PROPN
cana-5932	25	7	=	=	SYM
cana-5932	25	8	min{a	min{a	PROPN
cana-5932	25	9	,	,	PUNCT
cana-5932	25	10	b	b	NOUN
cana-5932	25	11	}	}	PUNCT
cana-5932	25	12	,	,	PUNCT
cana-5932	25	13	a	a	DET
cana-5932	25	14	∗2	∗2	PROPN
cana-5932	25	15	b	b	X
cana-5932	25	16	=	=	PUNCT
cana-5932	25	17	ab	ab	PROPN
cana-5932	25	18	and	and	CCONJ
cana-5932	25	19	a	a	DET
cana-5932	25	20	∗3	∗3	PROPN
cana-5932	25	21	b	b	PROPN
cana-5932	25	22	=	=	SYM
cana-5932	25	23	max{a	max{a	PROPN
cana-5932	25	24	+	+	CCONJ
cana-5932	25	25	b	b	NOUN
cana-5932	26	1	−	−	PROPN
cana-5932	26	2	1	1	NUM
cana-5932	26	3	,	,	PUNCT
cana-5932	26	4	0	0	NUM
cana-5932	26	5	}	}	PUNCT
cana-5932	26	6	are	be	AUX
cana-5932	26	7	commonly	commonly	ADV
cana-5932	26	8	used	use	VERB
cana-5932	26	9	continuous	continuous	ADJ
cana-5932	26	10	t	t	NOUN
cana-5932	26	11	-	-	PUNCT
cana-5932	26	12	norms	norm	NOUN
cana-5932	26	13	.	.	PUNCT
cana-5932	27	1	definition	definition	NOUN
cana-5932	27	2	2	2	NUM
cana-5932	27	3	.	.	PUNCT
cana-5932	28	1	[	[	X
cana-5932	28	2	2	2	X
cana-5932	28	3	]	]	PUNCT
cana-5932	28	4	the	the	DET
cana-5932	28	5	3	3	NUM
cana-5932	28	6	-	-	PUNCT
cana-5932	28	7	tuple	tuple	NOUN
cana-5932	28	8	(	(	PUNCT
cana-5932	28	9	x	x	X
cana-5932	28	10	,	,	PUNCT
cana-5932	28	11	m	m	PROPN
cana-5932	28	12	,	,	PUNCT
cana-5932	28	13	∗	∗	NOUN
cana-5932	28	14	)	)	PUNCT
cana-5932	28	15	is	be	AUX
cana-5932	28	16	said	say	VERB
cana-5932	28	17	to	to	PART
cana-5932	28	18	be	be	AUX
cana-5932	28	19	a	a	DET
cana-5932	28	20	fuzzy	fuzzy	ADJ
cana-5932	28	21	metric	metric	ADJ
cana-5932	28	22	space	space	NOUN
cana-5932	28	23	(	(	PUNCT
cana-5932	28	24	referred	refer	VERB
cana-5932	28	25	to	to	ADP
cana-5932	28	26	as	as	ADP
cana-5932	28	27	gv	gv	ADP
cana-5932	28	28	fuzzy	fuzzy	ADJ
cana-5932	28	29	metric	metric	ADJ
cana-5932	28	30	space	space	NOUN
cana-5932	28	31	)	)	PUNCT
cana-5932	28	32	,	,	PUNCT
cana-5932	28	33	if	if	SCONJ
cana-5932	28	34	x	x	PRON
cana-5932	28	35	is	be	AUX
cana-5932	28	36	an	an	DET
cana-5932	28	37	arbitrary	arbitrary	ADJ
cana-5932	28	38	non	non	ADJ
cana-5932	28	39	-	-	ADJ
cana-5932	28	40	empty	empty	ADJ
cana-5932	28	41	set	set	NOUN
cana-5932	28	42	,	,	PUNCT
cana-5932	28	43	∗	∗	NOUN
cana-5932	28	44	is	be	AUX
cana-5932	28	45	a	a	DET
cana-5932	28	46	continuous	continuous	ADJ
cana-5932	28	47	t	t	NOUN
cana-5932	28	48	-	-	PUNCT
cana-5932	28	49	norm	norm	NOUN
cana-5932	28	50	and	and	CCONJ
cana-5932	28	51	m	m	NOUN
cana-5932	28	52	is	be	AUX
cana-5932	28	53	a	a	DET
cana-5932	28	54	fuzzy	fuzzy	ADJ
cana-5932	28	55	set	set	NOUN
cana-5932	28	56	defined	define	VERB
cana-5932	28	57	on	on	ADP
cana-5932	28	58	x2	x2	PROPN
cana-5932	28	59	×	×	PROPN
cana-5932	28	60	r+	r+	PUNCT
cana-5932	28	61	satisfying	satisfy	VERB
cana-5932	28	62	the	the	DET
cana-5932	28	63	following	follow	VERB
cana-5932	28	64	conditions	condition	NOUN
cana-5932	28	65	;	;	PUNCT
cana-5932	28	66	(	(	PUNCT
cana-5932	28	67	gv-1	gv-1	X
cana-5932	28	68	)	)	PUNCT
cana-5932	28	69	m	m	VERB
cana-5932	28	70	(	(	PUNCT
cana-5932	28	71	x	x	X
cana-5932	28	72	,	,	PUNCT
cana-5932	28	73	y	y	PROPN
cana-5932	28	74	,	,	PUNCT
cana-5932	28	75	0	0	NUM
cana-5932	28	76	)	)	PUNCT
cana-5932	28	77	>	>	X
cana-5932	28	78	0	0	PUNCT
cana-5932	28	79	for	for	ADP
cana-5932	28	80	all	all	DET
cana-5932	28	81	x	x	NOUN
cana-5932	28	82	,	,	PUNCT
cana-5932	28	83	y	y	PROPN
cana-5932	28	84	∈	∈	PROPN
cana-5932	28	85	x.	x.	NOUN
cana-5932	28	86	(	(	PUNCT
cana-5932	28	87	gv-2	gv-2	NOUN
cana-5932	28	88	)	)	PUNCT
cana-5932	28	89	m	m	PROPN
cana-5932	28	90	(	(	PUNCT
cana-5932	28	91	x	x	X
cana-5932	28	92	,	,	PUNCT
cana-5932	28	93	y	y	PROPN
cana-5932	28	94	,	,	PUNCT
cana-5932	28	95	t	t	PROPN
cana-5932	28	96	)	)	PUNCT
cana-5932	28	97	=	=	SYM
cana-5932	28	98	1	1	NUM
cana-5932	28	99	for	for	ADP
cana-5932	28	100	all	all	DET
cana-5932	28	101	t	t	NOUN
cana-5932	28	102	∈	∈	PROPN
cana-5932	28	103	r+	r+	NOUN
cana-5932	28	104	iff	iff	PROPN
cana-5932	28	105	x	x	PROPN
cana-5932	28	106	=	=	SYM
cana-5932	28	107	y.	y.	PROPN
cana-5932	28	108	(	(	PUNCT
cana-5932	28	109	gv-3	gv-3	ADJ
cana-5932	28	110	)	)	PUNCT
cana-5932	28	111	m	m	VERB
cana-5932	28	112	(	(	PUNCT
cana-5932	28	113	x	x	X
cana-5932	28	114	,	,	PUNCT
cana-5932	28	115	y	y	PROPN
cana-5932	28	116	,	,	PUNCT
cana-5932	28	117	t	t	PROPN
cana-5932	28	118	)	)	PUNCT
cana-5932	28	119	=	=	SYM
cana-5932	28	120	m	m	PROPN
cana-5932	28	121	(	(	PUNCT
cana-5932	28	122	y	y	PROPN
cana-5932	28	123	,	,	PUNCT
cana-5932	28	124	x	x	PROPN
cana-5932	28	125	,	,	PUNCT
cana-5932	28	126	t	t	PROPN
cana-5932	28	127	)	)	PUNCT
cana-5932	28	128	for	for	ADP
cana-5932	28	129	any	any	DET
cana-5932	28	130	t	t	NOUN
cana-5932	28	131	∈	∈	PROPN
cana-5932	28	132	r+	r+	X
cana-5932	28	133	.	.	PUNCT
cana-5932	29	1	(	(	PUNCT
cana-5932	29	2	gv-4	gv-4	X
cana-5932	29	3	)	)	PUNCT
cana-5932	29	4	m	m	VERB
cana-5932	29	5	(	(	PUNCT
cana-5932	29	6	x	x	X
cana-5932	29	7	,	,	PUNCT
cana-5932	29	8	z	z	PROPN
cana-5932	29	9	,	,	PUNCT
cana-5932	29	10	t+s	t+s	NUM
cana-5932	29	11	)	)	PUNCT
cana-5932	29	12	≥	≥	NOUN
cana-5932	29	13	m	m	VERB
cana-5932	29	14	(	(	PUNCT
cana-5932	29	15	x	x	X
cana-5932	29	16	,	,	PUNCT
cana-5932	29	17	y	y	PROPN
cana-5932	29	18	,	,	PUNCT
cana-5932	29	19	t)∗m	t)∗m	NOUN
cana-5932	29	20	(	(	PUNCT
cana-5932	29	21	y	y	PROPN
cana-5932	29	22	,	,	PUNCT
cana-5932	29	23	z	z	PROPN
cana-5932	29	24	,	,	PUNCT
cana-5932	29	25	s	s	NOUN
cana-5932	29	26	)	)	PUNCT
cana-5932	29	27	for	for	ADP
cana-5932	29	28	all	all	DET
cana-5932	29	29	x	x	NOUN
cana-5932	29	30	,	,	PUNCT
cana-5932	29	31	y	y	PROPN
cana-5932	29	32	,	,	PUNCT
cana-5932	29	33	z	z	NOUN
cana-5932	29	34	∈	∈	PROPN
cana-5932	29	35	x	x	X
cana-5932	29	36	and	and	CCONJ
cana-5932	29	37	t	t	PROPN
cana-5932	29	38	,	,	PUNCT
cana-5932	29	39	s	s	PART
cana-5932	29	40	∈	∈	PROPN
cana-5932	29	41	r+	r+	NOUN
cana-5932	29	42	.	.	PUNCT
cana-5932	30	1	(	(	PUNCT
cana-5932	30	2	gv-5	gv-5	INTJ
cana-5932	30	3	)	)	PUNCT
cana-5932	30	4	m	m	VERB
cana-5932	30	5	(	(	PUNCT
cana-5932	30	6	x	x	X
cana-5932	30	7	,	,	PUNCT
cana-5932	30	8	y	y	PROPN
cana-5932	30	9	,	,	PUNCT
cana-5932	30	10	.	.	PUNCT
cana-5932	30	11	)	)	PUNCT
cana-5932	31	1	:	:	PUNCT
cana-5932	31	2	r+	r+	X
cana-5932	31	3	→	→	PUNCT
cana-5932	31	4	i	i	PRON
cana-5932	31	5	is	be	AUX
cana-5932	31	6	continuous	continuous	ADJ
cana-5932	31	7	.	.	PUNCT
cana-5932	32	1	remarks	remark	VERB
cana-5932	32	2	:	:	PUNCT
cana-5932	33	1	1	1	X
cana-5932	33	2	.	.	PUNCT
cana-5932	34	1	[	[	X
cana-5932	34	2	2	2	NUM
cana-5932	34	3	]	]	PUNCT
cana-5932	34	4	for	for	ADP
cana-5932	34	5	any	any	DET
cana-5932	34	6	r	r	NOUN
cana-5932	34	7	∈	∈	PROPN
cana-5932	34	8	(	(	PUNCT
cana-5932	34	9	0	0	NUM
cana-5932	34	10	,	,	PUNCT
cana-5932	34	11	1	1	NUM
cana-5932	34	12	)	)	PUNCT
cana-5932	34	13	,	,	PUNCT
cana-5932	34	14	whenever	whenever	SCONJ
cana-5932	34	15	m	m	VERB
cana-5932	34	16	(	(	PUNCT
cana-5932	34	17	x	x	NOUN
cana-5932	34	18	,	,	PUNCT
cana-5932	34	19	y	y	PROPN
cana-5932	34	20	,	,	PUNCT
cana-5932	34	21	t	t	PROPN
cana-5932	34	22	)	)	PUNCT
cana-5932	34	23	>	>	X
cana-5932	34	24	1	1	NUM
cana-5932	34	25	−	−	NOUN
cana-5932	34	26	r	r	NOUN
cana-5932	34	27	for	for	ADP
cana-5932	34	28	x	x	PROPN
cana-5932	34	29	,	,	PUNCT
cana-5932	34	30	y	y	PROPN
cana-5932	34	31	∈	∈	PROPN
cana-5932	34	32	x	x	X
cana-5932	34	33	and	and	CCONJ
cana-5932	34	34	t	t	PROPN
cana-5932	34	35	∈	∈	PROPN
cana-5932	34	36	r+	r+	ADV
cana-5932	34	37	,	,	PUNCT
cana-5932	34	38	there	there	PRON
cana-5932	34	39	exists	exist	VERB
cana-5932	34	40	t0	t0	PROPN
cana-5932	34	41	∈	∈	PROPN
cana-5932	34	42	(	(	PUNCT
cana-5932	34	43	0	0	NUM
cana-5932	34	44	,	,	PUNCT
cana-5932	34	45	t	t	PROPN
cana-5932	34	46	)	)	PUNCT
cana-5932	35	1	such	such	ADJ
cana-5932	35	2	that	that	SCONJ
cana-5932	35	3	m	m	VERB
cana-5932	35	4	(	(	PUNCT
cana-5932	35	5	x	x	X
cana-5932	35	6	,	,	PUNCT
cana-5932	35	7	y	y	PROPN
cana-5932	35	8	,	,	PUNCT
cana-5932	35	9	t0	t0	PROPN
cana-5932	35	10	)	)	PUNCT
cana-5932	35	11	>	>	X
cana-5932	35	12	1	1	NUM
cana-5932	35	13	−	−	PROPN
cana-5932	35	14	r.	r.	PROPN
cana-5932	35	15	2	2	NUM
cana-5932	35	16	.	.	PUNCT
cana-5932	35	17	for	for	ADP
cana-5932	35	18	any	any	DET
cana-5932	35	19	r1	r1	PROPN
cana-5932	35	20	>	>	PUNCT
cana-5932	35	21	r2	r2	PROPN
cana-5932	35	22	,	,	PUNCT
cana-5932	35	23	there	there	PRON
cana-5932	35	24	exists	exist	VERB
cana-5932	35	25	r3	r3	PROPN
cana-5932	35	26	such	such	ADJ
cana-5932	35	27	that	that	SCONJ
cana-5932	35	28	r1	r1	PROPN
cana-5932	35	29	∗	∗	NOUN
cana-5932	35	30	r3	r3	PROPN
cana-5932	35	31	>	>	X
cana-5932	35	32	r2	r2	PROPN
cana-5932	35	33	,	,	PUNCT
cana-5932	35	34	and	and	CCONJ
cana-5932	35	35	for	for	ADP
cana-5932	35	36	any	any	DET
cana-5932	35	37	r4	r4	NOUN
cana-5932	35	38	,	,	PUNCT
cana-5932	35	39	we	we	PRON
cana-5932	35	40	can	can	AUX
cana-5932	35	41	find	find	VERB
cana-5932	35	42	an	an	DET
cana-5932	35	43	r5	r5	PROPN
cana-5932	35	44	such	such	ADJ
cana-5932	35	45	that	that	SCONJ
cana-5932	35	46	r5	r5	PROPN
cana-5932	35	47	∗	∗	VERB
cana-5932	35	48	r5	r5	PROPN
cana-5932	35	49	>	>	X
cana-5932	35	50	r4	r4	PROPN
cana-5932	35	51	,	,	PUNCT
cana-5932	35	52	where	where	SCONJ
cana-5932	35	53	r1	r1	PROPN
cana-5932	35	54	,	,	PUNCT
cana-5932	35	55	r2	r2	PROPN
cana-5932	35	56	,	,	PUNCT
cana-5932	35	57	r3	r3	PROPN
cana-5932	35	58	,	,	PUNCT
cana-5932	35	59	r4	r4	NOUN
cana-5932	35	60	,	,	PUNCT
cana-5932	35	61	r5	r5	PROPN
cana-5932	35	62	∈	∈	PROPN
cana-5932	35	63	(	(	PUNCT
cana-5932	35	64	0	0	NUM
cana-5932	35	65	,	,	PUNCT
cana-5932	35	66	1	1	NUM
cana-5932	35	67	)	)	PUNCT
cana-5932	35	68	.	.	PUNCT
cana-5932	36	1	3	3	X
cana-5932	36	2	.	.	PUNCT
cana-5932	37	1	[	[	X
cana-5932	37	2	3	3	NUM
cana-5932	37	3	,	,	PUNCT
cana-5932	37	4	2	2	NUM
cana-5932	37	5	]	]	PUNCT
cana-5932	37	6	it	it	PRON
cana-5932	37	7	is	be	AUX
cana-5932	37	8	well	well	ADV
cana-5932	37	9	known	known	ADJ
cana-5932	37	10	and	and	CCONJ
cana-5932	37	11	easy	easy	ADJ
cana-5932	37	12	to	to	PART
cana-5932	37	13	verify	verify	VERB
cana-5932	37	14	that	that	PRON
cana-5932	37	15	for	for	ADP
cana-5932	37	16	every	every	DET
cana-5932	37	17	x	x	NOUN
cana-5932	37	18	,	,	PUNCT
cana-5932	37	19	y	y	PROPN
cana-5932	37	20	∈	∈	PROPN
cana-5932	37	21	x	x	X
cana-5932	37	22	,	,	PUNCT
cana-5932	37	23	m	m	VERB
cana-5932	37	24	(	(	PUNCT
cana-5932	37	25	x	x	X
cana-5932	37	26	,	,	PUNCT
cana-5932	37	27	y	y	PROPN
cana-5932	37	28	,	,	PUNCT
cana-5932	37	29	.	.	PUNCT
cana-5932	37	30	)	)	PUNCT
cana-5932	37	31	is	be	AUX
cana-5932	37	32	a	a	DET
cana-5932	37	33	nondecreasing	nondecrease	VERB
cana-5932	37	34	continuous	continuous	ADJ
cana-5932	37	35	function	function	NOUN
cana-5932	37	36	of	of	ADP
cana-5932	37	37	r+	r+	X
cana-5932	37	38	.	.	PUNCT
cana-5932	38	1	definition	definition	NOUN
cana-5932	38	2	3	3	X
cana-5932	38	3	.	.	PUNCT
cana-5932	39	1	let	let	AUX
cana-5932	39	2	(	(	PUNCT
cana-5932	39	3	x	x	X
cana-5932	39	4	,	,	PUNCT
cana-5932	39	5	m	m	PROPN
cana-5932	39	6	,	,	PUNCT
cana-5932	39	7	∗	∗	NOUN
cana-5932	39	8	)	)	PUNCT
cana-5932	39	9	be	be	VERB
cana-5932	39	10	a	a	DET
cana-5932	39	11	fuzzy	fuzzy	ADJ
cana-5932	39	12	metric	metric	ADJ
cana-5932	39	13	space	space	NOUN
cana-5932	39	14	.	.	PUNCT
cana-5932	40	1	(	(	PUNCT
cana-5932	40	2	i	i	NOUN
cana-5932	40	3	)	)	PUNCT
cana-5932	40	4	a	a	DET
cana-5932	40	5	sequence	sequence	NOUN
cana-5932	40	6	{	{	PUNCT
cana-5932	40	7	xn	xn	NOUN
cana-5932	40	8	}	}	PUNCT
cana-5932	40	9	converges	converge	NOUN
cana-5932	40	10	to	to	ADP
cana-5932	40	11	x	x	SYM
cana-5932	40	12	∈	∈	PROPN
cana-5932	40	13	x	x	SYM
cana-5932	40	14	,	,	PUNCT
cana-5932	40	15	if	if	SCONJ
cana-5932	40	16	every	every	DET
cana-5932	40	17	δ	δ	PROPN
cana-5932	40	18	∈	∈	PROPN
cana-5932	40	19	(	(	PUNCT
cana-5932	40	20	0	0	NUM
cana-5932	40	21	,	,	PUNCT
cana-5932	40	22	1	1	NUM
cana-5932	40	23	)	)	PUNCT
cana-5932	40	24	and	and	CCONJ
cana-5932	40	25	t	t	X
cana-5932	40	26	>	>	X
cana-5932	40	27	0	0	PROPN
cana-5932	40	28	,	,	PUNCT
cana-5932	40	29	there	there	PRON
cana-5932	40	30	exists	exist	VERB
cana-5932	40	31	n0	n0	PROPN
cana-5932	40	32	∈	∈	PROPN
cana-5932	40	33	n	n	PRON
cana-5932	40	34	such	such	ADJ
cana-5932	40	35	that	that	SCONJ
cana-5932	40	36	m	m	PROPN
cana-5932	40	37	(	(	PUNCT
cana-5932	40	38	xn	xn	PROPN
cana-5932	40	39	,	,	PUNCT
cana-5932	40	40	x	x	X
cana-5932	40	41	,	,	PUNCT
cana-5932	40	42	t	t	PROPN
cana-5932	40	43	)	)	PUNCT
cana-5932	40	44	>	>	X
cana-5932	41	1	1	1	NUM
cana-5932	41	2	−	−	PROPN
cana-5932	41	3	δ	δ	PROPN
cana-5932	41	4	for	for	ADP
cana-5932	41	5	all	all	DET
cana-5932	41	6	n	n	PRON
cana-5932	41	7	≥	≥	NOUN
cana-5932	41	8	n0	n0	NUM
cana-5932	41	9	.	.	PUNCT
cana-5932	42	1	this	this	PRON
cana-5932	42	2	implies	imply	VERB
cana-5932	42	3	that	that	SCONJ
cana-5932	42	4	lim	lim	PROPN
cana-5932	42	5	(	(	PUNCT
cana-5932	42	6	,	,	PUNCT
cana-5932	42	7	,	,	PUNCT
cana-5932	42	8	)	)	PUNCT
cana-5932	42	9	1.n	1.n	NUM
cana-5932	42	10	n	n	CCONJ
cana-5932	42	11	m	m	VERB
cana-5932	42	12	x	x	X
cana-5932	42	13	x	x	X
cana-5932	42	14	t	t	NOUN
cana-5932	42	15	→	→	PUNCT
cana-5932	42	16	=	=	SYM
cana-5932	42	17	(	(	PUNCT
cana-5932	42	18	ii	ii	NOUN
cana-5932	42	19	)	)	PUNCT
cana-5932	42	20	a	a	DET
cana-5932	42	21	sequence	sequence	NOUN
cana-5932	42	22	{	{	PUNCT
cana-5932	42	23	xn	xn	NOUN
cana-5932	42	24	}	}	PUNCT
cana-5932	42	25	is	be	AUX
cana-5932	42	26	said	say	VERB
cana-5932	42	27	to	to	PART
cana-5932	42	28	be	be	AUX
cana-5932	42	29	a	a	DET
cana-5932	42	30	cauchy	cauchy	NOUN
cana-5932	42	31	,	,	PUNCT
cana-5932	42	32	if	if	SCONJ
cana-5932	42	33	for	for	ADP
cana-5932	42	34	every	every	PRON
cana-5932	42	35	ϵ	ϵ	X
cana-5932	42	36	>	>	X
cana-5932	42	37	0	0	NUM
cana-5932	42	38	and	and	CCONJ
cana-5932	42	39	t	t	PROPN
cana-5932	42	40	∈	∈	PROPN
cana-5932	42	41	r+	r+	ADV
cana-5932	42	42	,	,	PUNCT
cana-5932	42	43	there	there	PRON
cana-5932	42	44	exists	exist	VERB
cana-5932	42	45	n0	n0	PROPN
cana-5932	42	46	∈	∈	PROPN
cana-5932	42	47	n	n	PRON
cana-5932	42	48	such	such	ADJ
cana-5932	42	49	that	that	SCONJ
cana-5932	42	50	m	m	PROPN
cana-5932	42	51	(	(	PUNCT
cana-5932	42	52	xm	xm	PROPN
cana-5932	42	53	,	,	PUNCT
cana-5932	42	54	xn	xn	PROPN
cana-5932	42	55	,	,	PUNCT
cana-5932	42	56	t	t	PROPN
cana-5932	42	57	)	)	PUNCT
cana-5932	42	58	>	>	X
cana-5932	43	1	1	1	NUM
cana-5932	43	2	−	−	NOUN
cana-5932	43	3	ϵ	ϵ	NOUN
cana-5932	43	4	for	for	ADP
cana-5932	43	5	all	all	DET
cana-5932	43	6	m	m	VERB
cana-5932	43	7	>	>	X
cana-5932	43	8	n	n	PRON
cana-5932	43	9	≥	≥	PROPN
cana-5932	43	10	n0	n0	NUM
cana-5932	43	11	.	.	PUNCT
cana-5932	44	1	moreover	moreover	ADV
cana-5932	44	2	,	,	PUNCT
cana-5932	44	3	lim	lim	PROPN
cana-5932	44	4	(	(	PUNCT
cana-5932	44	5	,	,	PUNCT
cana-5932	44	6	,	,	PUNCT
cana-5932	44	7	)	)	PUNCT
cana-5932	44	8	1,n	1,n	PROPN
cana-5932	45	1	p	p	NOUN
cana-5932	45	2	n	n	CCONJ
cana-5932	45	3	n	n	ADV
cana-5932	45	4	m	m	VERB
cana-5932	45	5	x	x	NOUN
cana-5932	45	6	x	x	X
cana-5932	45	7	t+	t+	NOUN
cana-5932	45	8	→	→	PUNCT
cana-5932	45	9	=	=	PUNCT
cana-5932	45	10	for	for	ADP
cana-5932	45	11	every	every	DET
cana-5932	45	12	p	p	PROPN
cana-5932	45	13	∈	∈	PROPN
cana-5932	45	14	n	n	NOUN
cana-5932	45	15	and	and	CCONJ
cana-5932	45	16	t	t	PROPN
cana-5932	45	17	∈	∈	PROPN
cana-5932	45	18	r+	r+	X
cana-5932	45	19	.	.	PUNCT
cana-5932	46	1	(	(	PUNCT
cana-5932	46	2	iii	iii	X
cana-5932	46	3	)	)	PUNCT
cana-5932	46	4	a	a	DET
cana-5932	46	5	fuzzy	fuzzy	ADJ
cana-5932	46	6	metric	metric	ADJ
cana-5932	46	7	space	space	NOUN
cana-5932	46	8	in	in	ADP
cana-5932	46	9	which	which	PRON
cana-5932	46	10	every	every	DET
cana-5932	46	11	cauchy	cauchy	ADJ
cana-5932	46	12	sequence	sequence	NOUN
cana-5932	46	13	is	be	AUX
cana-5932	46	14	convergent	convergent	NOUN
cana-5932	46	15	is	be	AUX
cana-5932	46	16	called	call	VERB
cana-5932	46	17	the	the	DET
cana-5932	46	18	complete	complete	ADJ
cana-5932	46	19	fuzzy	fuzzy	ADJ
cana-5932	46	20	metric	metric	ADJ
cana-5932	46	21	space	space	NOUN
cana-5932	46	22	.	.	PUNCT
cana-5932	47	1	3	3	X
cana-5932	47	2	.	.	X
cana-5932	47	3	main	main	ADJ
cana-5932	47	4	result	result	NOUN
cana-5932	47	5	in	in	ADP
cana-5932	47	6	the	the	DET
cana-5932	47	7	following	following	NOUN
cana-5932	47	8	,	,	PUNCT
cana-5932	47	9	f	f	PROPN
cana-5932	47	10	denotes	denote	VERB
cana-5932	47	11	the	the	DET
cana-5932	47	12	set	set	NOUN
cana-5932	47	13	of	of	ADP
cana-5932	47	14	all	all	DET
cana-5932	47	15	strictly	strictly	ADV
cana-5932	47	16	increasing	increase	VERB
cana-5932	47	17	functions	function	NOUN
cana-5932	47	18	f:(0,1)→r	f:(0,1)→r	NOUN
cana-5932	47	19	satisfying	satisfy	VERB
cana-5932	47	20	conditions	condition	NOUN
cana-5932	47	21	0	0	NUM
cana-5932	48	1	lim	lim	PROPN
cana-5932	48	2	(	(	PUNCT
cana-5932	48	3	)	)	PUNCT
cana-5932	49	1	p	p	NOUN
cana-5932	49	2	f	f	X
cana-5932	50	1	p	p	X
cana-5932	50	2	+	+	PROPN
cana-5932	50	3	→	→	SYM
cana-5932	50	4	=	=	SYM
cana-5932	50	5	−	−	NOUN
cana-5932	50	6	and	and	CCONJ
cana-5932	50	7	1	1	NUM
cana-5932	50	8	lim	lim	NOUN
cana-5932	50	9	(	(	PUNCT
cana-5932	50	10	)	)	PUNCT
cana-5932	50	11	.	.	PUNCT
cana-5932	51	1	p	p	X
cana-5932	51	2	f	f	X
cana-5932	51	3	p	p	NOUN
cana-5932	51	4	−→	−→	NOUN
cana-5932	51	5	=	=	SYM
cana-5932	51	6			NOUN
cana-5932	51	7	in	in	ADP
cana-5932	51	8	addition	addition	NOUN
cana-5932	51	9	,	,	PUNCT
cana-5932	51	10	h	h	PROPN
cana-5932	51	11	denotes	denote	VERB
cana-5932	51	12	the	the	DET
cana-5932	51	13	set	set	NOUN
cana-5932	51	14	of	of	ADP
cana-5932	51	15	mappings	mapping	NOUN
cana-5932	51	16	h	h	NOUN
cana-5932	51	17	:	:	PUNCT
cana-5932	51	18	x×x→i	x×x→i	PROPN
cana-5932	51	19	satisfies	satisfy	VERB
cana-5932	51	20	the	the	DET
cana-5932	51	21	conditions	condition	NOUN
cana-5932	51	22	,	,	PUNCT
cana-5932	51	23	for	for	ADP
cana-5932	51	24	any	any	DET
cana-5932	51	25	sequence	sequence	NOUN
cana-5932	51	26	{	{	PUNCT
cana-5932	51	27	xn	xn	NOUN
cana-5932	51	28	}	}	PUNCT
cana-5932	51	29	⊂	⊂	PROPN
cana-5932	51	30	x	x	X
cana-5932	51	31	,	,	PUNCT
cana-5932	51	32	1lim	1lim	NUM
cana-5932	51	33	(	(	PUNCT
cana-5932	51	34	,	,	PUNCT
cana-5932	51	35	)	)	PUNCT
cana-5932	51	36	0n	0n	NOUN
cana-5932	51	37	n	n	CCONJ
cana-5932	51	38	n	n	NOUN
cana-5932	51	39	h	h	NOUN
cana-5932	51	40	x	x	PUNCT
cana-5932	51	41	x	x	PUNCT
cana-5932	51	42	+	+	PUNCT
cana-5932	51	43	→	→	PUNCT
cana-5932	51	44	=	=	SYM
cana-5932	51	45	and	and	CCONJ
cana-5932	51	46	h(x	h(x	PROPN
cana-5932	51	47	y	y	PROPN
cana-5932	51	48	)	)	PUNCT
cana-5932	51	49	=	=	SYM
cana-5932	52	1	0	0	NUM
cana-5932	52	2	,	,	PUNCT
cana-5932	52	3	if	if	SCONJ
cana-5932	52	4	x	x	PRON
cana-5932	52	5	=	=	SYM
cana-5932	52	6	y.	y.	NOUN
cana-5932	52	7	definition	definition	NOUN
cana-5932	52	8	4	4	NUM
cana-5932	52	9	.	.	X
cana-5932	53	1	for	for	ADP
cana-5932	53	2	any	any	DET
cana-5932	53	3	f	f	PROPN
cana-5932	53	4	∈	∈	PROPN
cana-5932	53	5	f	f	PROPN
cana-5932	53	6	,	,	PUNCT
cana-5932	53	7	a	a	DET
cana-5932	53	8	mapping	mapping	NOUN
cana-5932	53	9	t	t	NOUN
cana-5932	53	10	:	:	PUNCT
cana-5932	53	11	x	x	X
cana-5932	53	12	→	→	PUNCT
cana-5932	53	13	x	x	X
cana-5932	53	14	is	be	AUX
cana-5932	53	15	said	say	VERB
cana-5932	53	16	to	to	PART
cana-5932	53	17	be	be	AUX
cana-5932	53	18	h	h	NOUN
cana-5932	53	19	–	–	PUNCT
cana-5932	53	20	f	f	X
cana-5932	53	21	–	–	PUNCT
cana-5932	53	22	contractive	contractive	ADJ
cana-5932	53	23	if	if	SCONJ
cana-5932	53	24	there	there	PRON
cana-5932	53	25	exists	exist	VERB
cana-5932	53	26	a	a	DET
cana-5932	53	27	function	function	NOUN
cana-5932	53	28	h	h	NOUN
cana-5932	53	29	∈	∈	PROPN
cana-5932	53	30	h	h	NOUN
cana-5932	53	31	such	such	ADJ
cana-5932	54	1	that	that	SCONJ
cana-5932	54	2	f	f	PROPN
cana-5932	54	3	(	(	PUNCT
cana-5932	54	4	m	m	PROPN
cana-5932	54	5	(	(	PUNCT
cana-5932	54	6	tx	tx	PROPN
cana-5932	54	7	,	,	PUNCT
cana-5932	54	8	ty	ty	PROPN
cana-5932	54	9	,	,	PUNCT
cana-5932	54	10	t	t	PROPN
cana-5932	54	11	)	)	PUNCT
cana-5932	54	12	)	)	PUNCT
cana-5932	54	13	≥	≥	PROPN
cana-5932	54	14	h(x	h(x	PROPN
cana-5932	54	15	,	,	PUNCT
cana-5932	54	16	y	y	PROPN
cana-5932	54	17	)	)	PUNCT
cana-5932	55	1	+	+	CCONJ
cana-5932	55	2	f	f	X
cana-5932	55	3	(	(	PUNCT
cana-5932	55	4	m	m	PROPN
cana-5932	55	5	(	(	PUNCT
cana-5932	55	6	x	x	NOUN
cana-5932	55	7	,	,	PUNCT
cana-5932	55	8	y	y	PROPN
cana-5932	55	9	,	,	PUNCT
cana-5932	55	10	t	t	PROPN
cana-5932	55	11	)	)	PUNCT
cana-5932	55	12	)	)	PUNCT
cana-5932	55	13	(	(	PUNCT
cana-5932	55	14	1	1	X
cana-5932	55	15	)	)	PUNCT
cana-5932	55	16	communications	communication	NOUN
cana-5932	55	17	on	on	ADP
cana-5932	55	18	applied	apply	VERB
cana-5932	55	19	nonlinear	nonlinear	ADJ
cana-5932	55	20	analysis	analysis	NOUN
cana-5932	55	21	issn	issn	NOUN
cana-5932	55	22	:	:	PUNCT
cana-5932	55	23	1074	1074	NUM
cana-5932	55	24	-	-	PUNCT
cana-5932	55	25	133x	133x	NUM
cana-5932	55	26	vol	vol	VERB
cana-5932	55	27	32	32	NUM
cana-5932	55	28	no	no	NOUN
cana-5932	55	29	.	.	PUNCT
cana-5932	56	1	10s	10	NOUN
cana-5932	56	2	(	(	PUNCT
cana-5932	56	3	2025	2025	NUM
cana-5932	56	4	)	)	PUNCT
cana-5932	56	5	3129	3129	NUM
cana-5932	56	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	56	7	for	for	ADP
cana-5932	56	8	all	all	DET
cana-5932	56	9	x	x	NOUN
cana-5932	56	10	,	,	PUNCT
cana-5932	56	11	y	y	PROPN
cana-5932	56	12	∈	∈	PROPN
cana-5932	56	13	x	x	PUNCT
cana-5932	56	14	with	with	ADP
cana-5932	56	15	.x	.x	PROPN
cana-5932	56	16	y	y	PROPN
cana-5932	56	17	theorem	theorem	NOUN
cana-5932	56	18	1	1	NUM
cana-5932	56	19	.	.	PUNCT
cana-5932	57	1	let	let	AUX
cana-5932	57	2	(	(	PUNCT
cana-5932	57	3	x	x	X
cana-5932	57	4	,	,	PUNCT
cana-5932	57	5	m	m	PROPN
cana-5932	57	6	,	,	PUNCT
cana-5932	57	7	∗	∗	NOUN
cana-5932	57	8	)	)	PUNCT
cana-5932	57	9	be	be	VERB
cana-5932	57	10	a	a	DET
cana-5932	57	11	complete	complete	ADJ
cana-5932	57	12	fuzzy	fuzzy	ADJ
cana-5932	57	13	metric	metric	ADJ
cana-5932	57	14	space	space	NOUN
cana-5932	57	15	.	.	PUNCT
cana-5932	58	1	if	if	SCONJ
cana-5932	58	2	t	t	NOUN
cana-5932	58	3	:	:	PUNCT
cana-5932	58	4	x	x	X
cana-5932	58	5	→	→	PUNCT
cana-5932	58	6	x	x	X
cana-5932	58	7	is	be	AUX
cana-5932	58	8	an	an	DET
cana-5932	58	9	h	h	NOUN
cana-5932	58	10	–	–	PUNCT
cana-5932	58	11	f	f	X
cana-5932	58	12	–	–	PUNCT
cana-5932	58	13	contractive	contractive	ADJ
cana-5932	58	14	mapping	mapping	NOUN
cana-5932	58	15	,	,	PUNCT
cana-5932	58	16	then	then	ADV
cana-5932	58	17	t	t	PROPN
cana-5932	58	18	has	have	VERB
cana-5932	58	19	a	a	DET
cana-5932	58	20	unique	unique	ADJ
cana-5932	58	21	fixed	fix	VERB
cana-5932	58	22	point	point	NOUN
cana-5932	58	23	in	in	ADP
cana-5932	58	24	x.	x.	NOUN
cana-5932	58	25	proof	proof	NOUN
cana-5932	58	26	:	:	PUNCT
cana-5932	58	27	let	let	VERB
cana-5932	58	28	x0	x0	PROPN
cana-5932	58	29	∈	∈	PROPN
cana-5932	58	30	x	x	X
cana-5932	58	31	and	and	CCONJ
cana-5932	58	32	xn+1	xn+1	NUM
cana-5932	58	33	=	=	SYM
cana-5932	58	34	txn	txn	VERB
cana-5932	58	35	for	for	ADP
cana-5932	58	36	all	all	DET
cana-5932	58	37	n	n	PRON
cana-5932	58	38	∈	∈	PROPN
cana-5932	58	39	n0	n0	PROPN
cana-5932	58	40	.	.	PROPN
cana-5932	58	41	assume	assume	VERB
cana-5932	58	42	that	that	SCONJ
cana-5932	58	43	t	t	NOUN
cana-5932	58	44	:	:	PUNCT
cana-5932	58	45	x	x	X
cana-5932	58	46	→	→	PUNCT
cana-5932	58	47	x	x	X
cana-5932	58	48	is	be	AUX
cana-5932	58	49	an	an	DET
cana-5932	58	50	h	h	NOUN
cana-5932	58	51	–	–	PUNCT
cana-5932	58	52	f	f	X
cana-5932	58	53	–	–	PUNCT
cana-5932	58	54	contractive	contractive	ADJ
cana-5932	58	55	mapping	mapping	NOUN
cana-5932	58	56	.	.	PUNCT
cana-5932	59	1	for	for	ADP
cana-5932	59	2	some	some	DET
cana-5932	59	3	n	n	PRON
cana-5932	59	4	∈	∈	PROPN
cana-5932	59	5	n0	n0	NOUN
cana-5932	59	6	,	,	PUNCT
cana-5932	59	7	if	if	SCONJ
cana-5932	59	8	xn+1	xn+1	ADV
cana-5932	59	9	=	=	SYM
cana-5932	59	10	txn	txn	NOUN
cana-5932	59	11	=	=	SYM
cana-5932	59	12	xn	xn	PROPN
cana-5932	59	13	,	,	PUNCT
cana-5932	59	14	then	then	ADV
cana-5932	59	15	xn	xn	PROPN
cana-5932	59	16	is	be	AUX
cana-5932	59	17	a	a	DET
cana-5932	59	18	fixed	fix	VERB
cana-5932	59	19	point	point	NOUN
cana-5932	59	20	.	.	PUNCT
cana-5932	60	1	assume	assume	VERB
cana-5932	60	2	that	that	SCONJ
cana-5932	60	3	for	for	ADP
cana-5932	60	4	any	any	DET
cana-5932	60	5	n	n	PRON
cana-5932	60	6	∈	∈	PROPN
cana-5932	60	7	n0	n0	NOUN
cana-5932	60	8	,	,	PUNCT
cana-5932	60	9	txn	txn	PROPN
cana-5932	60	10	=	=	SYM
cana-5932	60	11	xn+1	xn+1	PROPN
cana-5932	60	12			PROPN
cana-5932	60	13	xn	xn	PROPN
cana-5932	60	14	.	.	PROPN
cana-5932	61	1	from	from	ADP
cana-5932	61	2	equation	equation	NOUN
cana-5932	61	3	(	(	PUNCT
cana-5932	61	4	1	1	NUM
cana-5932	61	5	)	)	PUNCT
cana-5932	61	6	,	,	PUNCT
cana-5932	61	7	for	for	ADP
cana-5932	61	8	every	every	DET
cana-5932	61	9	n	n	CCONJ
cana-5932	61	10	∈	∈	PROPN
cana-5932	61	11	n0	n0	NOUN
cana-5932	61	12	and	and	CCONJ
cana-5932	61	13	t	t	PROPN
cana-5932	61	14	>	>	X
cana-5932	61	15	0	0	PROPN
cana-5932	61	16	,	,	PUNCT
cana-5932	61	17	we	we	PRON
cana-5932	61	18	have	have	VERB
cana-5932	61	19	f	f	PROPN
cana-5932	61	20	(	(	PUNCT
cana-5932	61	21	m	m	PROPN
cana-5932	61	22	(	(	PUNCT
cana-5932	61	23	txn	txn	NOUN
cana-5932	61	24	,	,	PUNCT
cana-5932	61	25	tnn+1	tnn+1	PROPN
cana-5932	61	26	,	,	PUNCT
cana-5932	61	27	t	t	PROPN
cana-5932	61	28	)	)	PUNCT
cana-5932	61	29	)	)	PUNCT
cana-5932	61	30	≥	≥	NOUN
cana-5932	61	31	h(xn	h(xn	NOUN
cana-5932	61	32	,	,	PUNCT
cana-5932	61	33	xn+1	xn+1	NUM
cana-5932	61	34	)	)	PUNCT
cana-5932	62	1	+	+	CCONJ
cana-5932	62	2	f	f	X
cana-5932	62	3	(	(	PUNCT
cana-5932	62	4	m	m	PROPN
cana-5932	62	5	(	(	PUNCT
cana-5932	62	6	xn	xn	PROPN
cana-5932	62	7	,	,	PUNCT
cana-5932	62	8	xn+1	xn+1	PROPN
cana-5932	62	9	,	,	PUNCT
cana-5932	62	10	t	t	PROPN
cana-5932	62	11	)	)	PUNCT
cana-5932	62	12	)	)	PUNCT
cana-5932	62	13	.	.	PUNCT
cana-5932	63	1	(	(	PUNCT
cana-5932	63	2	2	2	X
cana-5932	63	3	)	)	PUNCT
cana-5932	63	4	because	because	SCONJ
cana-5932	63	5	h(xn	h(xn	X
cana-5932	63	6	,	,	PUNCT
cana-5932	63	7	xn+1	xn+1	NUM
cana-5932	63	8	)	)	PUNCT
cana-5932	63	9	>	>	X
cana-5932	63	10	0	0	PUNCT
cana-5932	64	1	for	for	ADP
cana-5932	64	2	all	all	PRON
cana-5932	64	3	n	n	PRON
cana-5932	64	4	∈	∈	PROPN
cana-5932	64	5	n	n	CCONJ
cana-5932	64	6	,	,	PUNCT
cana-5932	64	7	f	f	PROPN
cana-5932	64	8	(	(	PUNCT
cana-5932	64	9	m	m	PROPN
cana-5932	64	10	(	(	PUNCT
cana-5932	64	11	txn	txn	NOUN
cana-5932	64	12	,	,	PUNCT
cana-5932	64	13	tnn+1	tnn+1	PROPN
cana-5932	64	14	,	,	PUNCT
cana-5932	64	15	t	t	PROPN
cana-5932	64	16	)	)	PUNCT
cana-5932	64	17	)	)	PUNCT
cana-5932	64	18	≥	≥	NOUN
cana-5932	64	19	h(xn	h(xn	NOUN
cana-5932	64	20	,	,	PUNCT
cana-5932	64	21	xn+1)+f	xn+1)+f	PUNCT
cana-5932	64	22	(	(	PUNCT
cana-5932	64	23	m	m	PROPN
cana-5932	64	24	(	(	PUNCT
cana-5932	64	25	xn	xn	PROPN
cana-5932	64	26	,	,	PUNCT
cana-5932	64	27	xn+1	xn+1	PROPN
cana-5932	64	28	,	,	PUNCT
cana-5932	64	29	t	t	PROPN
cana-5932	64	30	)	)	PUNCT
cana-5932	64	31	)	)	PUNCT
cana-5932	65	1	>	>	X
cana-5932	66	1	f	f	X
cana-5932	66	2	(	(	PUNCT
cana-5932	66	3	m	m	PROPN
cana-5932	66	4	(	(	PUNCT
cana-5932	66	5	xn	xn	PROPN
cana-5932	66	6	,	,	PUNCT
cana-5932	66	7	xn+1	xn+1	PROPN
cana-5932	66	8	,	,	PUNCT
cana-5932	66	9	t	t	PROPN
cana-5932	66	10	)	)	PUNCT
cana-5932	66	11	)	)	PUNCT
cana-5932	66	12	.	.	PUNCT
cana-5932	67	1	(	(	PUNCT
cana-5932	67	2	3	3	X
cana-5932	67	3	)	)	PUNCT
cana-5932	67	4	we	we	PRON
cana-5932	67	5	know	know	VERB
cana-5932	67	6	that	that	SCONJ
cana-5932	67	7	f	f	PROPN
cana-5932	67	8	is	be	AUX
cana-5932	67	9	a	a	DET
cana-5932	67	10	strictly	strictly	ADV
cana-5932	67	11	increasing	increase	VERB
cana-5932	67	12	function	function	NOUN
cana-5932	67	13	,	,	PUNCT
cana-5932	67	14	therefore	therefore	ADV
cana-5932	67	15	,	,	PUNCT
cana-5932	67	16	m	m	VERB
cana-5932	67	17	(	(	PUNCT
cana-5932	67	18	txn	txn	NOUN
cana-5932	67	19	,	,	PUNCT
cana-5932	67	20	tnn+1	tnn+1	PROPN
cana-5932	67	21	,	,	PUNCT
cana-5932	67	22	t	t	PROPN
cana-5932	67	23	)	)	PUNCT
cana-5932	67	24	=	=	SYM
cana-5932	68	1	m	m	PROPN
cana-5932	68	2	(	(	PUNCT
cana-5932	68	3	xn+1	xn+1	PROPN
cana-5932	68	4	,	,	PUNCT
cana-5932	68	5	xn+2	xn+2	NUM
cana-5932	68	6	,	,	PUNCT
cana-5932	68	7	t	t	PROPN
cana-5932	68	8	)	)	PUNCT
cana-5932	68	9	>	>	X
cana-5932	68	10	m	m	PROPN
cana-5932	68	11	(	(	PUNCT
cana-5932	68	12	xn	xn	PROPN
cana-5932	68	13	,	,	PUNCT
cana-5932	68	14	xn+1	xn+1	PROPN
cana-5932	68	15	,	,	PUNCT
cana-5932	68	16	t	t	PROPN
cana-5932	68	17	)	)	PUNCT
cana-5932	68	18	.	.	PUNCT
cana-5932	69	1	thus	thus	ADV
cana-5932	69	2	,	,	PUNCT
cana-5932	69	3	for	for	ADP
cana-5932	69	4	every	every	DET
cana-5932	69	5	t	t	NOUN
cana-5932	69	6	>	>	X
cana-5932	69	7	0	0	NUM
cana-5932	69	8	,	,	PUNCT
cana-5932	69	9	{	{	PUNCT
cana-5932	69	10	m	m	PROPN
cana-5932	69	11	(	(	PUNCT
cana-5932	69	12	xn	xn	PROPN
cana-5932	69	13	,	,	PUNCT
cana-5932	69	14	xn+1	xn+1	PROPN
cana-5932	69	15	,	,	PUNCT
cana-5932	69	16	t	t	PROPN
cana-5932	69	17	)	)	PUNCT
cana-5932	69	18	}	}	PUNCT
cana-5932	69	19	is	be	AUX
cana-5932	69	20	an	an	DET
cana-5932	69	21	increasing	increase	VERB
cana-5932	69	22	sequence	sequence	NOUN
cana-5932	69	23	bounded	bound	VERB
cana-5932	69	24	from	from	ADP
cana-5932	69	25	above	above	ADV
cana-5932	69	26	in	in	ADP
cana-5932	69	27	i.	i.	NOUN
cana-5932	69	28	hence	hence	ADV
cana-5932	69	29	,	,	PUNCT
cana-5932	69	30	for	for	ADP
cana-5932	69	31	every	every	DET
cana-5932	69	32	t	t	NOUN
cana-5932	69	33	>	>	X
cana-5932	69	34	0	0	NUM
cana-5932	69	35	,	,	PUNCT
cana-5932	69	36	{	{	PUNCT
cana-5932	69	37	m	m	PROPN
cana-5932	69	38	(	(	PUNCT
cana-5932	69	39	xn	xn	PROPN
cana-5932	69	40	,	,	PUNCT
cana-5932	69	41	xn+1	xn+1	PROPN
cana-5932	69	42	,	,	PUNCT
cana-5932	69	43	t	t	PROPN
cana-5932	69	44	)	)	PUNCT
cana-5932	69	45	}	}	PUNCT
cana-5932	69	46	converges	converge	VERB
cana-5932	69	47	in	in	ADP
cana-5932	69	48	i.	i.	NOUN
cana-5932	69	49	that	that	PRON
cana-5932	69	50	is	be	AUX
cana-5932	69	51	,	,	PUNCT
cana-5932	69	52	for	for	ADP
cana-5932	69	53	every	every	DET
cana-5932	69	54	t	t	NOUN
cana-5932	69	55	>	>	X
cana-5932	69	56	0	0	NUM
cana-5932	69	57	,	,	PUNCT
cana-5932	69	58	there	there	PRON
cana-5932	69	59	exists	exist	VERB
cana-5932	69	60	δ(t	δ(t	NOUN
cana-5932	69	61	)	)	PUNCT
cana-5932	69	62	∈	∈	PROPN
cana-5932	69	63	(	(	PUNCT
cana-5932	69	64	0	0	NUM
cana-5932	69	65	,	,	PUNCT
cana-5932	69	66	1	1	NUM
cana-5932	69	67	)	)	PUNCT
cana-5932	69	68	and	and	CCONJ
cana-5932	69	69	n	n	PRON
cana-5932	69	70	∈	∈	PROPN
cana-5932	69	71	n0	n0	PROPN
cana-5932	69	72	,	,	PUNCT
cana-5932	70	1	such	such	ADJ
cana-5932	70	2	that	that	SCONJ
cana-5932	70	3	m	m	PROPN
cana-5932	70	4	(	(	PUNCT
cana-5932	70	5	xn	xn	PROPN
cana-5932	70	6	,	,	PUNCT
cana-5932	70	7	xn+1	xn+1	PROPN
cana-5932	70	8	,	,	PUNCT
cana-5932	70	9	t	t	PROPN
cana-5932	70	10	)	)	PUNCT
cana-5932	70	11	>	>	X
cana-5932	70	12	1	1	NUM
cana-5932	70	13	−	−	PROPN
cana-5932	70	14	δ(t	δ(t	PROPN
cana-5932	70	15	)	)	PUNCT
cana-5932	70	16	for	for	ADP
cana-5932	70	17	all	all	DET
cana-5932	70	18	n	n	PRON
cana-5932	70	19	≥	≥	NOUN
cana-5932	70	20	n.	n.	NOUN
cana-5932	70	21	moreover	moreover	ADV
cana-5932	70	22	,	,	PUNCT
cana-5932	70	23	for	for	ADP
cana-5932	70	24	every	every	DET
cana-5932	70	25	t	t	NOUN
cana-5932	70	26	>	>	X
cana-5932	70	27	0	0	PROPN
cana-5932	70	28	,	,	PUNCT
cana-5932	70	29	suppose	suppose	VERB
cana-5932	70	30	that	that	SCONJ
cana-5932	70	31	there	there	PRON
cana-5932	70	32	exists	exist	VERB
cana-5932	70	33	α(t	α(t	NOUN
cana-5932	70	34	)	)	PUNCT
cana-5932	70	35	∈	∈	PROPN
cana-5932	71	1	i	i	PRON
cana-5932	71	2	such	such	ADJ
cana-5932	71	3	that	that	PRON
cana-5932	71	4	as	as	ADP
cana-5932	71	5	n	n	PROPN
cana-5932	71	6	→	→	SYM
cana-5932	71	7	∞	∞	PROPN
cana-5932	71	8	the	the	DET
cana-5932	71	9	sequence	sequence	NOUN
cana-5932	71	10	{	{	PUNCT
cana-5932	71	11	m	m	PROPN
cana-5932	71	12	(	(	PUNCT
cana-5932	71	13	xn	xn	PROPN
cana-5932	71	14	,	,	PUNCT
cana-5932	71	15	xn+1	xn+1	PROPN
cana-5932	71	16	,	,	PUNCT
cana-5932	71	17	t	t	PROPN
cana-5932	71	18	)	)	PUNCT
cana-5932	71	19	}	}	PUNCT
cana-5932	71	20	approaches	approach	VERB
cana-5932	71	21	to	to	ADP
cana-5932	71	22	its	its	PRON
cana-5932	71	23	limit	limit	NOUN
cana-5932	71	24	α(t	α(t	NOUN
cana-5932	71	25	)	)	PUNCT
cana-5932	71	26	from	from	ADP
cana-5932	71	27	its	its	PRON
cana-5932	71	28	left	left	ADJ
cana-5932	71	29	side	side	NOUN
cana-5932	71	30	.	.	PUNCT
cana-5932	72	1	1lim	1lim	NUM
cana-5932	72	2	(	(	PUNCT
cana-5932	72	3	,	,	PUNCT
cana-5932	72	4	,	,	PUNCT
cana-5932	72	5	)	)	PUNCT
cana-5932	72	6	(	(	PUNCT
cana-5932	72	7	)	)	PUNCT
cana-5932	72	8	.n	.n	PROPN
cana-5932	72	9	n	n	CCONJ
cana-5932	72	10	n	n	ADV
cana-5932	72	11	m	m	VERB
cana-5932	72	12	x	x	X
cana-5932	72	13	x	x	SYM
cana-5932	72	14	t	t	NOUN
cana-5932	72	15	t	t	NOUN
cana-5932	72	16	−	−	PROPN
cana-5932	72	17	+	+	PROPN
cana-5932	72	18	→	→	PUNCT
cana-5932	72	19			NOUN
cana-5932	72	20	=	=	NOUN
cana-5932	72	21	and	and	CCONJ
cana-5932	72	22	,	,	PUNCT
cana-5932	72	23	this	this	PRON
cana-5932	72	24	imply	imply	VERB
cana-5932	72	25	(	(	PUNCT
cana-5932	72	26	)	)	PUNCT
cana-5932	72	27	(	(	PUNCT
cana-5932	72	28	)	)	PUNCT
cana-5932	72	29	1lim	1lim	NUM
cana-5932	72	30	(	(	PUNCT
cana-5932	72	31	,	,	PUNCT
cana-5932	72	32	,	,	PUNCT
cana-5932	72	33	)	)	PUNCT
cana-5932	72	34	(	(	PUNCT
cana-5932	72	35	)	)	PUNCT
cana-5932	72	36	.n	.n	PROPN
cana-5932	73	1	n	n	CCONJ
cana-5932	73	2	n	n	ADV
cana-5932	73	3	f	f	NOUN
cana-5932	73	4	m	m	VERB
cana-5932	73	5	x	x	X
cana-5932	73	6	x	x	SYM
cana-5932	74	1	t	t	NOUN
cana-5932	74	2	f	f	PROPN
cana-5932	74	3	t	t	NOUN
cana-5932	74	4	−	−	PROPN
cana-5932	74	5	+	+	CCONJ
cana-5932	74	6	→	→	PUNCT
cana-5932	74	7	=	=	PUNCT
cana-5932	74	8	by	by	ADP
cana-5932	74	9	taking	take	VERB
cana-5932	74	10	limit	limit	NOUN
cana-5932	74	11	both	both	DET
cana-5932	74	12	sides	side	NOUN
cana-5932	74	13	of	of	ADP
cana-5932	74	14	equation	equation	NOUN
cana-5932	74	15	(	(	PUNCT
cana-5932	74	16	3	3	NUM
cana-5932	74	17	)	)	PUNCT
cana-5932	74	18	,	,	PUNCT
cana-5932	74	19	we	we	PRON
cana-5932	74	20	get	get	VERB
cana-5932	74	21	(	(	PUNCT
cana-5932	74	22	)	)	PUNCT
cana-5932	74	23	(	(	PUNCT
cana-5932	74	24	)	)	PUNCT
cana-5932	74	25	(	(	PUNCT
cana-5932	74	26	)	)	PUNCT
cana-5932	74	27	1	1	NUM
cana-5932	74	28	(	(	PUNCT
cana-5932	74	29	)	)	PUNCT
cana-5932	74	30	lim	lim	PROPN
cana-5932	74	31	(	(	PUNCT
cana-5932	74	32	,	,	PUNCT
cana-5932	74	33	)	)	PUNCT
cana-5932	74	34	(	(	PUNCT
cana-5932	74	35	)	)	PUNCT
cana-5932	74	36	(	(	PUNCT
cana-5932	74	37	)	)	PUNCT
cana-5932	74	38	.n	.n	PROPN
cana-5932	75	1	n	n	CCONJ
cana-5932	75	2	n	n	PROPN
cana-5932	76	1	f	f	NOUN
cana-5932	76	2	t	t	NOUN
cana-5932	76	3	h	h	NOUN
cana-5932	76	4	x	x	PUNCT
cana-5932	76	5	x	x	PUNCT
cana-5932	76	6	f	f	NOUN
cana-5932	76	7	t	t	PROPN
cana-5932	76	8	f	f	PROPN
cana-5932	76	9	t	t	PROPN
cana-5932	76	10			X
cana-5932	76	11	−	−	X
cana-5932	76	12	−	−	PROPN
cana-5932	76	13	−	−	PROPN
cana-5932	76	14	+	+	CCONJ
cana-5932	76	15	→	→	PUNCT
cana-5932	76	16			NUM
cana-5932	76	17	+	+	CCONJ
cana-5932	76	18			VERB
cana-5932	76	19	this	this	PRON
cana-5932	76	20	is	be	AUX
cana-5932	76	21	possible	possible	ADJ
cana-5932	76	22	when	when	SCONJ
cana-5932	76	23	1lim	1lim	NUM
cana-5932	76	24	(	(	PUNCT
cana-5932	76	25	,	,	PUNCT
cana-5932	76	26	)	)	PUNCT
cana-5932	76	27	0n	0n	NOUN
cana-5932	76	28	n	n	CCONJ
cana-5932	76	29	n	n	NOUN
cana-5932	76	30	h	h	NOUN
cana-5932	76	31	x	x	PUNCT
cana-5932	77	1	x	x	PUNCT
cana-5932	78	1	+	+	PUNCT
cana-5932	78	2	→	→	PUNCT
cana-5932	78	3	=	=	NOUN
cana-5932	78	4	(	(	PUNCT
cana-5932	78	5	4	4	NUM
cana-5932	78	6	)	)	PUNCT
cana-5932	78	7	and	and	CCONJ
cana-5932	78	8	(	(	PUNCT
cana-5932	78	9	)	)	PUNCT
cana-5932	78	10	(	(	PUNCT
cana-5932	78	11	)	)	PUNCT
cana-5932	78	12	0,f	0,f	NOUN
cana-5932	79	1	t	t	NOUN
cana-5932	79	2	−	−	X
cana-5932	79	3	=	=	PUNCT
cana-5932	79	4	which	which	PRON
cana-5932	79	5	contradict	contradict	VERB
cana-5932	79	6	with	with	ADP
cana-5932	79	7	(	(	PUNCT
cana-5932	79	8	)	)	PUNCT
cana-5932	79	9	(	(	PUNCT
cana-5932	79	10	)	)	PUNCT
cana-5932	79	11	0.f	0.f	NUM
cana-5932	80	1	t	t	NOUN
cana-5932	80	2	−	−	PROPN
cana-5932	80	3			VERB
cana-5932	80	4	therefore	therefore	ADV
cana-5932	80	5	,	,	PUNCT
cana-5932	80	6	we	we	PRON
cana-5932	80	7	have	have	VERB
cana-5932	80	8	1lim	1lim	NUM
cana-5932	80	9	(	(	PUNCT
cana-5932	80	10	,	,	PUNCT
cana-5932	80	11	,	,	PUNCT
cana-5932	80	12	)	)	PUNCT
cana-5932	80	13	1	1	NUM
cana-5932	80	14	.n	.n	NOUN
cana-5932	80	15	n	n	CCONJ
cana-5932	80	16	n	n	NOUN
cana-5932	80	17	m	m	VERB
cana-5932	80	18	x	x	X
cana-5932	80	19	x	x	X
cana-5932	80	20	t	t	NOUN
cana-5932	80	21	−	−	PROPN
cana-5932	80	22	+	+	CCONJ
cana-5932	80	23	→	→	PUNCT
cana-5932	80	24	=	=	SYM
cana-5932	80	25	(	(	PUNCT
cana-5932	80	26	5	5	NUM
cana-5932	80	27	)	)	PUNCT
cana-5932	80	28	furthermore	furthermore	ADV
cana-5932	80	29	,	,	PUNCT
cana-5932	80	30	we	we	PRON
cana-5932	80	31	need	need	VERB
cana-5932	80	32	to	to	PART
cana-5932	80	33	prove	prove	VERB
cana-5932	80	34	that	that	SCONJ
cana-5932	80	35	{	{	PUNCT
cana-5932	80	36	xn	xn	X
cana-5932	80	37	}	}	PUNCT
cana-5932	80	38	is	be	AUX
cana-5932	80	39	a	a	DET
cana-5932	80	40	cauchy	cauchy	ADJ
cana-5932	80	41	sequence	sequence	NOUN
cana-5932	80	42	.	.	PUNCT
cana-5932	81	1	for	for	ADP
cana-5932	81	2	m	m	PROPN
cana-5932	81	3	,	,	PUNCT
cana-5932	81	4	n	n	PROPN
cana-5932	81	5	∈	∈	PROPN
cana-5932	81	6	n0	n0	NOUN
cana-5932	81	7	and	and	CCONJ
cana-5932	81	8	equation	equation	NOUN
cana-5932	81	9	(	(	PUNCT
cana-5932	81	10	2	2	NUM
cana-5932	81	11	)	)	PUNCT
cana-5932	81	12	implies	imply	VERB
cana-5932	81	13	that	that	SCONJ
cana-5932	81	14	(	(	PUNCT
cana-5932	81	15	)	)	PUNCT
cana-5932	81	16	(	(	PUNCT
cana-5932	81	17	)	)	PUNCT
cana-5932	81	18	(	(	PUNCT
cana-5932	81	19	)	)	PUNCT
cana-5932	81	20	(	(	PUNCT
cana-5932	81	21	)	)	PUNCT
cana-5932	81	22	(	(	PUNCT
cana-5932	81	23	)	)	PUNCT
cana-5932	82	1	1	1	NUM
cana-5932	82	2	1	1	NUM
cana-5932	82	3	1	1	NUM
cana-5932	82	4	1	1	NUM
cana-5932	82	5	1	1	NUM
cana-5932	82	6	2	2	NUM
cana-5932	82	7	1	1	NUM
cana-5932	82	8	2	2	NUM
cana-5932	82	9	2	2	NUM
cana-5932	82	10	1	1	NUM
cana-5932	82	11	1	1	NUM
cana-5932	82	12	0	0	NUM
cana-5932	82	13	(	(	PUNCT
cana-5932	82	14	,	,	PUNCT
cana-5932	82	15	,	,	PUNCT
cana-5932	82	16	)	)	PUNCT
cana-5932	82	17	(	(	PUNCT
cana-5932	82	18	,	,	PUNCT
cana-5932	82	19	,	,	PUNCT
cana-5932	82	20	)	)	PUNCT
cana-5932	82	21	(	(	PUNCT
cana-5932	82	22	,	,	PUNCT
cana-5932	82	23	)	)	PUNCT
cana-5932	82	24	(	(	PUNCT
cana-5932	82	25	,	,	PUNCT
cana-5932	82	26	,	,	PUNCT
cana-5932	82	27	)	)	PUNCT
cana-5932	82	28	(	(	PUNCT
cana-5932	82	29	,	,	PUNCT
cana-5932	82	30	)	)	PUNCT
cana-5932	82	31	(	(	PUNCT
cana-5932	82	32	,	,	PUNCT
cana-5932	82	33	)	)	PUNCT
cana-5932	82	34	(	(	PUNCT
cana-5932	82	35	,	,	PUNCT
cana-5932	82	36	,	,	PUNCT
cana-5932	82	37	)	)	PUNCT
cana-5932	82	38	.	.	PUNCT
cana-5932	82	39	.	.	PUNCT
cana-5932	82	40	.	.	PUNCT
cana-5932	83	1	(	(	PUNCT
cana-5932	83	2	,	,	PUNCT
cana-5932	83	3	)	)	PUNCT
cana-5932	83	4	(	(	PUNCT
cana-5932	83	5	,	,	PUNCT
cana-5932	83	6	,	,	PUNCT
cana-5932	83	7	)	)	PUNCT
cana-5932	83	8	n	n	CCONJ
cana-5932	83	9	m	m	VERB
cana-5932	83	10	n	n	VERB
cana-5932	83	11	m	m	VERB
cana-5932	83	12	n	n	ADV
cana-5932	83	13	m	m	NOUN
cana-5932	83	14	n	n	ADV
cana-5932	83	15	m	m	NOUN
cana-5932	83	16	n	n	ADV
cana-5932	83	17	m	m	NOUN
cana-5932	83	18	n	n	ADV
cana-5932	83	19	m	m	NOUN
cana-5932	83	20	n	n	ADV
cana-5932	83	21	m	m	NOUN
cana-5932	83	22	n	n	ADV
cana-5932	83	23	m	m	NOUN
cana-5932	83	24	n	n	ADV
cana-5932	83	25	m	m	NOUN
cana-5932	83	26	n	n	ADV
cana-5932	83	27	m	m	NOUN
cana-5932	83	28	n	n	ADV
cana-5932	83	29	m	m	NOUN
cana-5932	83	30	n	n	ADV
cana-5932	83	31	m	m	NOUN
cana-5932	83	32	n	n	ADV
cana-5932	83	33	m	m	NOUN
cana-5932	83	34	n	n	PRON
cana-5932	83	35	m	m	NOUN
cana-5932	83	36	m	m	VERB
cana-5932	83	37	n	n	ADV
cana-5932	83	38	k	k	PROPN
cana-5932	84	1	n	n	CCONJ
cana-5932	85	1	k	k	PROPN
cana-5932	86	1	n	n	CCONJ
cana-5932	87	1	n	n	PROPN
cana-5932	88	1	k	k	NOUN
cana-5932	89	1	f	f	X
cana-5932	89	2	m	m	VERB
cana-5932	89	3	x	x	X
cana-5932	89	4	x	x	SYM
cana-5932	89	5	t	t	X
cana-5932	89	6	f	f	PROPN
cana-5932	89	7	m	m	VERB
cana-5932	89	8	tx	tx	PROPN
cana-5932	89	9	tx	tx	PROPN
cana-5932	89	10	t	t	PROPN
cana-5932	89	11	h	h	NOUN
cana-5932	90	1	x	x	PUNCT
cana-5932	90	2	x	x	PUNCT
cana-5932	90	3	f	f	X
cana-5932	90	4	m	m	VERB
cana-5932	90	5	x	x	X
cana-5932	90	6	x	x	X
cana-5932	90	7	t	t	NOUN
cana-5932	90	8	h	h	NOUN
cana-5932	91	1	x	x	PUNCT
cana-5932	91	2	x	x	PUNCT
cana-5932	91	3	h	h	NOUN
cana-5932	91	4	x	x	X
cana-5932	91	5	x	x	X
cana-5932	92	1	f	f	X
cana-5932	92	2	m	m	VERB
cana-5932	92	3	x	x	X
cana-5932	92	4	x	x	X
cana-5932	92	5	t	t	NOUN
cana-5932	92	6	h	h	NOUN
cana-5932	93	1	x	x	PUNCT
cana-5932	93	2	x	x	X
cana-5932	94	1	f	f	X
cana-5932	94	2	m	m	VERB
cana-5932	94	3	x	x	X
cana-5932	94	4	x	x	X
cana-5932	94	5	t	t	NOUN
cana-5932	94	6	+	+	CCONJ
cana-5932	95	1	+	+	PUNCT
cana-5932	95	2	+	+	PUNCT
cana-5932	95	3	+	+	CCONJ
cana-5932	95	4	−	−	X
cana-5932	96	1	+	+	CCONJ
cana-5932	96	2	+	+	CCONJ
cana-5932	96	3	−	−	X
cana-5932	97	1	+	+	CCONJ
cana-5932	97	2	+	+	CCONJ
cana-5932	97	3	−	−	X
cana-5932	98	1	+	+	CCONJ
cana-5932	98	2	+	+	CCONJ
cana-5932	98	3	−	−	X
cana-5932	99	1	+	+	CCONJ
cana-5932	99	2	+	+	CCONJ
cana-5932	99	3	−	−	X
cana-5932	99	4	+	+	CCONJ
cana-5932	99	5	−	−	PROPN
cana-5932	100	1	+	+	CCONJ
cana-5932	100	2	−	−	PROPN
cana-5932	101	1	+	+	CCONJ
cana-5932	101	2	−	−	PROPN
cana-5932	102	1	+	+	CCONJ
cana-5932	102	2	−	−	PROPN
cana-5932	103	1	+	+	CCONJ
cana-5932	103	2	+	+	PUNCT
cana-5932	103	3	=	=	SYM
cana-5932	103	4	=	=	SYM
cana-5932	103	5			NUM
cana-5932	103	6	+	+	SYM
cana-5932	103	7			NUM
cana-5932	103	8	+	+	CCONJ
cana-5932	104	1	+	+	CCONJ
cana-5932	104	2			NUM
cana-5932	104	3	+	+	NOUN
cana-5932	104	4			NOUN
cana-5932	104	5	communications	communication	NOUN
cana-5932	104	6	on	on	ADP
cana-5932	104	7	applied	apply	VERB
cana-5932	104	8	nonlinear	nonlinear	ADJ
cana-5932	104	9	analysis	analysis	NOUN
cana-5932	104	10	issn	issn	NOUN
cana-5932	104	11	:	:	PUNCT
cana-5932	104	12	1074	1074	NUM
cana-5932	104	13	-	-	PUNCT
cana-5932	104	14	133x	133x	NUM
cana-5932	104	15	vol	vol	VERB
cana-5932	104	16	32	32	NUM
cana-5932	104	17	no	no	NOUN
cana-5932	104	18	.	.	PUNCT
cana-5932	105	1	10s	10	NOUN
cana-5932	105	2	(	(	PUNCT
cana-5932	105	3	2025	2025	NUM
cana-5932	105	4	)	)	PUNCT
cana-5932	105	5	3130	3130	NUM
cana-5932	105	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	105	7	by	by	ADP
cana-5932	105	8	taking	take	VERB
cana-5932	105	9	the	the	DET
cana-5932	105	10	limits	limit	NOUN
cana-5932	105	11	of	of	ADP
cana-5932	105	12	both	both	DET
cana-5932	105	13	sides	side	NOUN
cana-5932	105	14	and	and	CCONJ
cana-5932	105	15	together	together	ADV
cana-5932	105	16	from	from	ADP
cana-5932	105	17	equations	equation	NOUN
cana-5932	105	18	(	(	PUNCT
cana-5932	105	19	3	3	NUM
cana-5932	105	20	)	)	PUNCT
cana-5932	105	21	and	and	CCONJ
cana-5932	105	22	(	(	PUNCT
cana-5932	105	23	5	5	NUM
cana-5932	105	24	)	)	PUNCT
cana-5932	105	25	,	,	PUNCT
cana-5932	105	26	we	we	PRON
cana-5932	105	27	obtain	obtain	VERB
cana-5932	105	28	(	(	PUNCT
cana-5932	105	29	)	)	PUNCT
cana-5932	105	30	(	(	PUNCT
cana-5932	105	31	)	)	PUNCT
cana-5932	105	32	(	(	PUNCT
cana-5932	105	33	)	)	SYM
cana-5932	105	34	1	1	NUM
cana-5932	105	35	1	1	NUM
cana-5932	105	36	0	0	NUM
cana-5932	105	37	lim	lim	NOUN
cana-5932	105	38	(	(	PUNCT
cana-5932	105	39	,	,	PUNCT
cana-5932	105	40	,	,	PUNCT
cana-5932	105	41	)	)	PUNCT
cana-5932	105	42	lim	lim	PROPN
cana-5932	105	43	(	(	PUNCT
cana-5932	105	44	,	,	PUNCT
cana-5932	105	45	)	)	PUNCT
cana-5932	105	46	1	1	NUM
cana-5932	105	47	1	1	NUM
cana-5932	105	48	.	.	PUNCT
cana-5932	106	1	m	m	VERB
cana-5932	106	2	n	n	VERB
cana-5932	106	3	m	m	VERB
cana-5932	106	4	n	n	ADV
cana-5932	106	5	m	m	PROPN
cana-5932	106	6	n	n	ADV
cana-5932	106	7	k	k	PROPN
cana-5932	106	8	n	n	CCONJ
cana-5932	106	9	k	k	PROPN
cana-5932	106	10	n	n	CCONJ
cana-5932	106	11	n	n	PROPN
cana-5932	106	12	k	k	NOUN
cana-5932	107	1	f	f	X
cana-5932	107	2	m	m	VERB
cana-5932	107	3	x	x	X
cana-5932	107	4	x	x	X
cana-5932	107	5	t	t	NOUN
cana-5932	107	6	h	h	NOUN
cana-5932	108	1	x	x	PUNCT
cana-5932	108	2	x	x	PUNCT
cana-5932	108	3	f	f	X
cana-5932	108	4	f−	f−	NOUN
cana-5932	108	5	−	−	PROPN
cana-5932	109	1	+	+	PROPN
cana-5932	110	1	+	+	PUNCT
cana-5932	110	2	+	+	PUNCT
cana-5932	110	3	+	+	CCONJ
cana-5932	110	4	−	−	PROPN
cana-5932	110	5	+	+	CCONJ
cana-5932	110	6	→	→	PUNCT
cana-5932	110	7	→	→	PUNCT
cana-5932	110	8	=	=	NOUN
cana-5932	110	9			X
cana-5932	110	10	+	+	CCONJ
cana-5932	110	11			NOUN
cana-5932	110	12	from	from	ADP
cana-5932	110	13	equation	equation	NOUN
cana-5932	110	14	(	(	PUNCT
cana-5932	110	15	4	4	NUM
cana-5932	110	16	)	)	PUNCT
cana-5932	110	17	,	,	PUNCT
cana-5932	110	18	it	it	PRON
cana-5932	110	19	is	be	AUX
cana-5932	110	20	clear	clear	ADJ
cana-5932	110	21	that	that	SCONJ
cana-5932	110	22	for	for	ADP
cana-5932	110	23	every	every	DET
cana-5932	110	24	k	k	PROPN
cana-5932	110	25	∈	∈	PROPN
cana-5932	110	26	n0	n0	PROPN
cana-5932	110	27	,	,	PUNCT
cana-5932	110	28	1lim	1lim	NUM
cana-5932	110	29	(	(	PUNCT
cana-5932	110	30	,	,	PUNCT
cana-5932	110	31	)	)	PUNCT
cana-5932	110	32	0.n	0.n	X
cana-5932	111	1	k	k	PROPN
cana-5932	111	2	n	n	CCONJ
cana-5932	111	3	k	k	PROPN
cana-5932	111	4	n	n	ADV
cana-5932	111	5	h	h	NOUN
cana-5932	112	1	x	x	NOUN
cana-5932	112	2	x+	x+	PUNCT
cana-5932	112	3	−	−	PROPN
cana-5932	112	4	+	+	NOUN
cana-5932	112	5	→	→	PUNCT
cana-5932	112	6	=	=	VERB
cana-5932	112	7	hence	hence	ADV
cana-5932	112	8	(	(	PUNCT
cana-5932	112	9	)	)	PUNCT
cana-5932	112	10	(	(	PUNCT
cana-5932	112	11	)	)	PUNCT
cana-5932	112	12	1lim	1lim	NUM
cana-5932	112	13	(	(	PUNCT
cana-5932	112	14	,	,	PUNCT
cana-5932	112	15	,	,	PUNCT
cana-5932	112	16	)	)	PUNCT
cana-5932	112	17	1	1	NUM
cana-5932	112	18	,	,	PUNCT
cana-5932	112	19	n	n	CCONJ
cana-5932	112	20	m	m	VERB
cana-5932	112	21	n	n	NOUN
cana-5932	112	22	m	m	NOUN
cana-5932	112	23	n	n	ADV
cana-5932	112	24	f	f	NOUN
cana-5932	112	25	m	m	VERB
cana-5932	112	26	x	x	X
cana-5932	112	27	x	x	X
cana-5932	112	28	t	t	NOUN
cana-5932	112	29	f	f	X
cana-5932	112	30	−	−	PROPN
cana-5932	113	1	+	+	CCONJ
cana-5932	113	2	+	+	CCONJ
cana-5932	113	3	+	+	NUM
cana-5932	113	4	→	→	PUNCT
cana-5932	113	5	=	=	PUNCT
cana-5932	113	6	and	and	CCONJ
cana-5932	113	7	hence	hence	ADV
cana-5932	113	8	1lim	1lim	NUM
cana-5932	113	9	(	(	PUNCT
cana-5932	113	10	,	,	PUNCT
cana-5932	113	11	,	,	PUNCT
cana-5932	113	12	)	)	PUNCT
cana-5932	113	13	1	1	NUM
cana-5932	113	14	.n	.n	NOUN
cana-5932	113	15	m	m	VERB
cana-5932	113	16	n	n	NOUN
cana-5932	113	17	m	m	VERB
cana-5932	113	18	n	n	ADV
cana-5932	113	19	m	m	NOUN
cana-5932	113	20	x	x	X
cana-5932	113	21	x	x	X
cana-5932	113	22	t	t	NOUN
cana-5932	113	23	−	−	NOUN
cana-5932	114	1	+	+	CCONJ
cana-5932	114	2	+	+	CCONJ
cana-5932	114	3	+	+	NUM
cana-5932	114	4	→	→	PUNCT
cana-5932	114	5	=	=	PRON
cana-5932	114	6	therefore	therefore	ADV
cana-5932	114	7	{	{	PUNCT
cana-5932	114	8	xn	xn	X
cana-5932	114	9	}	}	PUNCT
cana-5932	114	10	is	be	AUX
cana-5932	114	11	a	a	DET
cana-5932	114	12	cauchy	cauchy	ADJ
cana-5932	114	13	sequence	sequence	NOUN
cana-5932	114	14	.	.	PUNCT
cana-5932	115	1	since	since	SCONJ
cana-5932	115	2	(	(	PUNCT
cana-5932	115	3	x	x	X
cana-5932	115	4	,	,	PUNCT
cana-5932	115	5	m	m	PROPN
cana-5932	115	6	,	,	PUNCT
cana-5932	115	7	∗	∗	NOUN
cana-5932	115	8	)	)	PUNCT
cana-5932	115	9	is	be	AUX
cana-5932	115	10	complete	complete	ADJ
cana-5932	115	11	,	,	PUNCT
cana-5932	115	12	therefore	therefore	ADV
cana-5932	115	13	there	there	PRON
cana-5932	115	14	exists	exist	VERB
cana-5932	115	15	z	z	NOUN
cana-5932	115	16	∈	∈	PROPN
cana-5932	115	17	x	x	PUNCT
cana-5932	115	18	such	such	ADJ
cana-5932	115	19	that	that	SCONJ
cana-5932	115	20	lim	lim	PROPN
cana-5932	115	21	.n	.n	PROPN
cana-5932	116	1	n	n	ADP
cana-5932	116	2	x	x	SYM
cana-5932	116	3	z	z	NOUN
cana-5932	116	4	→	→	PUNCT
cana-5932	116	5	=	=	NOUN
cana-5932	116	6	(	(	PUNCT
cana-5932	116	7	6	6	NUM
cana-5932	116	8	)	)	PUNCT
cana-5932	116	9	it	it	PRON
cana-5932	116	10	is	be	AUX
cana-5932	116	11	easy	easy	ADJ
cana-5932	116	12	to	to	PART
cana-5932	116	13	verify	verify	VERB
cana-5932	116	14	that	that	SCONJ
cana-5932	116	15	z	z	NOUN
cana-5932	116	16	is	be	AUX
cana-5932	116	17	a	a	DET
cana-5932	116	18	fixed	fix	VERB
cana-5932	116	19	point	point	NOUN
cana-5932	116	20	of	of	ADP
cana-5932	116	21	t.	t.	PROPN
cana-5932	116	22	since	since	SCONJ
cana-5932	116	23	t	t	PROPN
cana-5932	116	24	is	be	AUX
cana-5932	116	25	continuous	continuous	ADJ
cana-5932	116	26	and	and	CCONJ
cana-5932	116	27	equation	equation	NOUN
cana-5932	116	28	(	(	PUNCT
cana-5932	116	29	7	7	NUM
cana-5932	116	30	)	)	PUNCT
cana-5932	117	1	,	,	PUNCT
cana-5932	117	2	implies	imply	VERB
cana-5932	117	3	that	that	SCONJ
cana-5932	117	4	(	(	PUNCT
cana-5932	117	5	)	)	PUNCT
cana-5932	117	6	1	1	NUM
cana-5932	117	7	(	(	PUNCT
cana-5932	117	8	)	)	PUNCT
cana-5932	117	9	lim	lim	PROPN
cana-5932	117	10	lim	lim	PROPN
cana-5932	117	11	lim	lim	PROPN
cana-5932	117	12	.n	.n	PROPN
cana-5932	117	13	n	n	CCONJ
cana-5932	117	14	n	n	CCONJ
cana-5932	117	15	n	n	CCONJ
cana-5932	117	16	n	n	CCONJ
cana-5932	117	17	n	n	NOUN
cana-5932	117	18	t	t	PROPN
cana-5932	117	19	z	z	PROPN
cana-5932	117	20	t	t	PROPN
cana-5932	117	21	x	x	PUNCT
cana-5932	117	22	tx	tx	PROPN
cana-5932	117	23	x	x	SYM
cana-5932	117	24	z+	z+	NUM
cana-5932	117	25	→	→	PUNCT
cana-5932	117	26	→	→	PUNCT
cana-5932	117	27	→	→	PUNCT
cana-5932	118	1	=	=	NOUN
cana-5932	118	2	=	=	PUNCT
cana-5932	119	1	=	=	SYM
cana-5932	119	2	=	=	SYM
cana-5932	119	3	(	(	PUNCT
cana-5932	119	4	7	7	NUM
cana-5932	119	5	)	)	PUNCT
cana-5932	119	6	finally	finally	ADV
cana-5932	119	7	,	,	PUNCT
cana-5932	119	8	to	to	PART
cana-5932	119	9	prove	prove	VERB
cana-5932	119	10	the	the	DET
cana-5932	119	11	uniqueness	uniqueness	NOUN
cana-5932	119	12	of	of	ADP
cana-5932	119	13	the	the	DET
cana-5932	119	14	fixed	fixed	ADJ
cana-5932	119	15	point	point	NOUN
cana-5932	119	16	z	z	PROPN
cana-5932	119	17	of	of	ADP
cana-5932	119	18	t	t	PROPN
cana-5932	119	19	,	,	PUNCT
cana-5932	119	20	assume	assume	VERB
cana-5932	119	21	that	that	SCONJ
cana-5932	119	22	there	there	PRON
cana-5932	119	23	exists	exist	VERB
cana-5932	119	24	another	another	DET
cana-5932	119	25	fixed	fix	VERB
cana-5932	119	26	point	point	NOUN
cana-5932	119	27	u	u	PROPN
cana-5932	119	28	of	of	ADP
cana-5932	119	29	t	t	PROPN
cana-5932	119	30	in	in	ADP
cana-5932	119	31	x.	x.	NOUN
cana-5932	119	32	that	that	PRON
cana-5932	119	33	is	be	AUX
cana-5932	119	34	t	t	PROPN
cana-5932	119	35	(	(	PUNCT
cana-5932	119	36	u	u	NOUN
cana-5932	119	37	)	)	PUNCT
cana-5932	119	38	=	=	SYM
cana-5932	119	39	u	u	NOUN
cana-5932	119	40	,	,	PUNCT
cana-5932	119	41	where	where	SCONJ
cana-5932	119	42	z	z	PROPN
cana-5932	119	43			VERB
cana-5932	119	44	u.	u.	ADV
cana-5932	119	45	now	now	ADV
cana-5932	119	46	,	,	PUNCT
cana-5932	119	47	by	by	ADP
cana-5932	119	48	using	use	VERB
cana-5932	119	49	equation	equation	NOUN
cana-5932	119	50	(	(	PUNCT
cana-5932	119	51	1	1	NUM
cana-5932	119	52	)	)	PUNCT
cana-5932	119	53	and	and	CCONJ
cana-5932	119	54	(	(	PUNCT
cana-5932	119	55	3	3	NUM
cana-5932	119	56	)	)	PUNCT
cana-5932	119	57	,	,	PUNCT
cana-5932	119	58	we	we	PRON
cana-5932	119	59	get	get	VERB
cana-5932	119	60	f	f	PROPN
cana-5932	119	61	(	(	PUNCT
cana-5932	119	62	m	m	PROPN
cana-5932	119	63	(	(	PUNCT
cana-5932	119	64	tz	tz	PROPN
cana-5932	119	65	,	,	PUNCT
cana-5932	119	66	tu	tu	PROPN
cana-5932	119	67	,	,	PUNCT
cana-5932	119	68	t	t	PROPN
cana-5932	119	69	)	)	PUNCT
cana-5932	119	70	)	)	PUNCT
cana-5932	119	71	≥	≥	NOUN
cana-5932	120	1	h(z	h(z	NOUN
cana-5932	120	2	,	,	PUNCT
cana-5932	120	3	u	u	NOUN
cana-5932	120	4	)	)	PUNCT
cana-5932	121	1	+	+	NUM
cana-5932	121	2	f	f	X
cana-5932	121	3	(	(	PUNCT
cana-5932	121	4	m	m	PROPN
cana-5932	121	5	(	(	PUNCT
cana-5932	121	6	z	z	PROPN
cana-5932	121	7	,	,	PUNCT
cana-5932	121	8	u	u	PROPN
cana-5932	121	9	,	,	PUNCT
cana-5932	121	10	t	t	PROPN
cana-5932	121	11	)	)	PUNCT
cana-5932	121	12	)	)	PUNCT
cana-5932	122	1	>	>	X
cana-5932	123	1	f	f	X
cana-5932	123	2	(	(	PUNCT
cana-5932	123	3	m	m	PROPN
cana-5932	123	4	(	(	PUNCT
cana-5932	123	5	z	z	PROPN
cana-5932	123	6	,	,	PUNCT
cana-5932	123	7	u	u	PROPN
cana-5932	123	8	,	,	PUNCT
cana-5932	123	9	t	t	PROPN
cana-5932	123	10	)	)	PUNCT
cana-5932	123	11	)	)	PUNCT
cana-5932	124	1	⇒	⇒	PROPN
cana-5932	124	2	f	f	PROPN
cana-5932	124	3	(	(	PUNCT
cana-5932	124	4	m	m	PROPN
cana-5932	124	5	(	(	PUNCT
cana-5932	124	6	z	z	PROPN
cana-5932	124	7	,	,	PUNCT
cana-5932	124	8	u	u	PROPN
cana-5932	124	9	,	,	PUNCT
cana-5932	124	10	t	t	PROPN
cana-5932	124	11	)	)	PUNCT
cana-5932	124	12	)	)	PUNCT
cana-5932	124	13	≥	≥	NOUN
cana-5932	125	1	h(z	h(z	NOUN
cana-5932	125	2	,	,	PUNCT
cana-5932	125	3	u	u	NOUN
cana-5932	125	4	)	)	PUNCT
cana-5932	126	1	+	+	NUM
cana-5932	126	2	f	f	X
cana-5932	126	3	(	(	PUNCT
cana-5932	126	4	m	m	PROPN
cana-5932	126	5	(	(	PUNCT
cana-5932	126	6	z	z	PROPN
cana-5932	126	7	,	,	PUNCT
cana-5932	126	8	u	u	PROPN
cana-5932	126	9	,	,	PUNCT
cana-5932	126	10	t	t	PROPN
cana-5932	126	11	)	)	PUNCT
cana-5932	126	12	)	)	PUNCT
cana-5932	127	1	>	>	X
cana-5932	128	1	f	f	X
cana-5932	128	2	(	(	PUNCT
cana-5932	128	3	m	m	PROPN
cana-5932	128	4	(	(	PUNCT
cana-5932	128	5	z	z	PROPN
cana-5932	128	6	,	,	PUNCT
cana-5932	128	7	u	u	PROPN
cana-5932	128	8	,	,	PUNCT
cana-5932	128	9	t	t	PROPN
cana-5932	128	10	)	)	PUNCT
cana-5932	128	11	)	)	PUNCT
cana-5932	128	12	.	.	PUNCT
cana-5932	129	1	as	as	ADP
cana-5932	129	2	a	a	DET
cana-5932	129	3	consequence	consequence	NOUN
cana-5932	129	4	,	,	PUNCT
cana-5932	129	5	we	we	PRON
cana-5932	129	6	have	have	VERB
cana-5932	129	7	f	f	PROPN
cana-5932	129	8	(	(	PUNCT
cana-5932	129	9	m	m	PROPN
cana-5932	129	10	(	(	PUNCT
cana-5932	129	11	z	z	PROPN
cana-5932	129	12	,	,	PUNCT
cana-5932	129	13	u	u	PROPN
cana-5932	129	14	,	,	PUNCT
cana-5932	129	15	t	t	PROPN
cana-5932	129	16	)	)	PUNCT
cana-5932	129	17	)	)	PUNCT
cana-5932	130	1	>	>	X
cana-5932	131	1	f	f	X
cana-5932	131	2	(	(	PUNCT
cana-5932	131	3	m	m	PROPN
cana-5932	131	4	(	(	PUNCT
cana-5932	131	5	z	z	PROPN
cana-5932	131	6	,	,	PUNCT
cana-5932	131	7	u	u	PROPN
cana-5932	131	8	,	,	PUNCT
cana-5932	131	9	t	t	PROPN
cana-5932	131	10	)	)	PUNCT
cana-5932	131	11	)	)	PUNCT
cana-5932	131	12	,	,	PUNCT
cana-5932	131	13	this	this	PRON
cana-5932	131	14	imply	imply	VERB
cana-5932	131	15	,	,	PUNCT
cana-5932	131	16	m	m	VERB
cana-5932	131	17	(	(	PUNCT
cana-5932	131	18	z	z	PROPN
cana-5932	131	19	,	,	PUNCT
cana-5932	131	20	u	u	PROPN
cana-5932	131	21	,	,	PUNCT
cana-5932	131	22	t	t	PROPN
cana-5932	131	23	)	)	PUNCT
cana-5932	131	24	>	>	X
cana-5932	132	1	m	m	PROPN
cana-5932	132	2	(	(	PUNCT
cana-5932	132	3	z	z	PROPN
cana-5932	132	4	,	,	PUNCT
cana-5932	132	5	u	u	PROPN
cana-5932	132	6	,	,	PUNCT
cana-5932	132	7	t	t	PROPN
cana-5932	132	8	)	)	PUNCT
cana-5932	132	9	.	.	PUNCT
cana-5932	133	1	this	this	PRON
cana-5932	133	2	is	be	AUX
cana-5932	133	3	a	a	DET
cana-5932	133	4	contradiction	contradiction	NOUN
cana-5932	133	5	,	,	PUNCT
cana-5932	133	6	hence	hence	ADV
cana-5932	133	7	z	z	NOUN
cana-5932	133	8	=	=	PUNCT
cana-5932	133	9	u.	u.	NOUN
cana-5932	133	10	that	that	PRON
cana-5932	133	11	is	be	AUX
cana-5932	133	12	t	t	NOUN
cana-5932	133	13	has	have	VERB
cana-5932	133	14	a	a	DET
cana-5932	133	15	unique	unique	ADJ
cana-5932	133	16	fixed	fix	VERB
cana-5932	133	17	point	point	NOUN
cana-5932	133	18	in	in	ADP
cana-5932	133	19	x.	x.	PROPN
cana-5932	133	20	example	example	NOUN
cana-5932	134	1	1	1	X
cana-5932	134	2	.	.	PUNCT
cana-5932	134	3	let	let	VERB
cana-5932	134	4	x	x	PUNCT
cana-5932	134	5	=	=	SYM
cana-5932	134	6	r	r	NOUN
cana-5932	134	7	,	,	PUNCT
cana-5932	134	8	a	a	DET
cana-5932	134	9	∗	∗	NOUN
cana-5932	134	10	b	b	NOUN
cana-5932	134	11	=	=	SYM
cana-5932	134	12	min{a	min{a	PROPN
cana-5932	134	13	,	,	PUNCT
cana-5932	134	14	b	b	NOUN
cana-5932	134	15	}	}	PUNCT
cana-5932	134	16	for	for	ADP
cana-5932	134	17	all	all	DET
cana-5932	134	18	a	a	PRON
cana-5932	134	19	,	,	PUNCT
cana-5932	134	20	b	b	X
cana-5932	134	21	∈	∈	NOUN
cana-5932	135	1	i	i	PRON
cana-5932	135	2	and	and	CCONJ
cana-5932	135	3	(	(	PUNCT
cana-5932	135	4	,	,	PUNCT
cana-5932	135	5	,	,	PUNCT
cana-5932	135	6	)	)	PUNCT
cana-5932	136	1	|	|	ADV
cana-5932	136	2	|	|	ADV
cana-5932	136	3	t	t	INTJ
cana-5932	136	4	m	m	VERB
cana-5932	136	5	x	x	VERB
cana-5932	136	6	y	y	PROPN
cana-5932	136	7	t	t	PROPN
cana-5932	136	8	t	t	NOUN
cana-5932	137	1	x	x	PUNCT
cana-5932	137	2	y	y	PROPN
cana-5932	137	3	=	=	PUNCT
cana-5932	138	1	+	+	CCONJ
cana-5932	138	2	−	−	NOUN
cana-5932	138	3	for	for	ADP
cana-5932	138	4	all	all	DET
cana-5932	138	5	x	x	NOUN
cana-5932	139	1	,	,	PUNCT
cana-5932	139	2	y	y	PROPN
cana-5932	139	3	∈	∈	PROPN
cana-5932	139	4	x	x	X
cana-5932	139	5	and	and	CCONJ
cana-5932	139	6	t	t	X
cana-5932	139	7	>	>	X
cana-5932	139	8	0	0	X
cana-5932	139	9	.	.	PUNCT
cana-5932	139	10	subsequently	subsequently	ADV
cana-5932	139	11	,	,	PUNCT
cana-5932	139	12	(	(	PUNCT
cana-5932	139	13	x	x	X
cana-5932	139	14	,	,	PUNCT
cana-5932	139	15	m	m	PROPN
cana-5932	139	16	,	,	PUNCT
cana-5932	139	17	∗	∗	NOUN
cana-5932	139	18	)	)	PUNCT
cana-5932	139	19	is	be	AUX
cana-5932	139	20	a	a	DET
cana-5932	139	21	complete	complete	ADJ
cana-5932	139	22	fuzzy	fuzzy	ADJ
cana-5932	139	23	metric	metric	ADJ
cana-5932	139	24	space	space	NOUN
cana-5932	139	25	.	.	PUNCT
cana-5932	140	1	now	now	ADV
cana-5932	140	2	,	,	PUNCT
cana-5932	140	3	let	let	VERB
cana-5932	140	4	f	f	PROPN
cana-5932	140	5	∈	∈	PROPN
cana-5932	140	6	f	f	AUX
cana-5932	140	7	be	be	AUX
cana-5932	140	8	defined	define	VERB
cana-5932	140	9	as	as	ADP
cana-5932	140	10	2	2	NUM
cana-5932	140	11	1	1	NUM
cana-5932	140	12	(	(	PUNCT
cana-5932	140	13	)	)	PUNCT
cana-5932	141	1	(	(	PUNCT
cana-5932	141	2	1	1	X
cana-5932	141	3	)	)	PUNCT
cana-5932	141	4	p	p	NOUN
cana-5932	141	5	f	f	X
cana-5932	142	1	p	p	X
cana-5932	142	2	p	p	X
cana-5932	142	3	p	p	X
cana-5932	142	4	−	−	PROPN
cana-5932	142	5	=	=	SYM
cana-5932	142	6	−	−	PROPN
cana-5932	142	7	for	for	ADP
cana-5932	142	8	all	all	DET
cana-5932	142	9	p	p	NOUN
cana-5932	142	10	∈	∈	PROPN
cana-5932	142	11	(	(	PUNCT
cana-5932	142	12	0	0	NUM
cana-5932	142	13	,	,	PUNCT
cana-5932	142	14	1	1	NUM
cana-5932	142	15	)	)	PUNCT
cana-5932	142	16	,	,	PUNCT
cana-5932	142	17	and	and	CCONJ
cana-5932	142	18	let	let	VERB
cana-5932	142	19	h	h	PRON
cana-5932	142	20	∈	∈	PROPN
cana-5932	142	21	h	h	NOUN
cana-5932	142	22	defined	define	VERB
cana-5932	142	23	as	as	ADP
cana-5932	142	24	h(x	h(x	PROPN
cana-5932	142	25	,	,	PUNCT
cana-5932	142	26	y	y	NOUN
cana-5932	142	27	)	)	PUNCT
cana-5932	142	28	=	=	NOUN
cana-5932	142	29	|x	|x	NOUN
cana-5932	142	30	−	−	PROPN
cana-5932	142	31	y|	y|	NOUN
cana-5932	142	32	.	.	PUNCT
cana-5932	143	1	now	now	ADV
cana-5932	143	2	,	,	PUNCT
cana-5932	143	3	define	define	VERB
cana-5932	143	4	t	t	PROPN
cana-5932	143	5	:	:	PUNCT
cana-5932	143	6	x→x	x→x	NUM
cana-5932	143	7	,	,	PUNCT
cana-5932	143	8	(	(	PUNCT
cana-5932	143	9	)	)	PUNCT
cana-5932	143	10	4	4	NUM
cana-5932	143	11	x	x	SYM
cana-5932	143	12	t	t	NOUN
cana-5932	143	13	x	x	SYM
cana-5932	143	14	=	=	PUNCT
cana-5932	143	15	for	for	ADP
cana-5932	143	16	all	all	DET
cana-5932	143	17	x	x	SYM
cana-5932	143	18	∈	∈	PROPN
cana-5932	143	19	x.	x.	NOUN
cana-5932	143	20	figure	figure	NOUN
cana-5932	143	21	1	1	NUM
cana-5932	143	22	:	:	PUNCT
cana-5932	143	23	h	h	PROPN
cana-5932	143	24	–	–	PUNCT
cana-5932	143	25	f	f	X
cana-5932	143	26	–	–	PUNCT
cana-5932	143	27	contractive	contractive	ADJ
cana-5932	143	28	from	from	ADP
cana-5932	143	29	the	the	DET
cana-5932	143	30	graph	graph	NOUN
cana-5932	143	31	given	give	VERB
cana-5932	143	32	in	in	ADP
cana-5932	143	33	the	the	DET
cana-5932	143	34	figure	figure	NOUN
cana-5932	143	35	1	1	NUM
cana-5932	143	36	,	,	PUNCT
cana-5932	143	37	it	it	PRON
cana-5932	143	38	is	be	AUX
cana-5932	143	39	clear	clear	ADJ
cana-5932	143	40	that	that	SCONJ
cana-5932	143	41	,	,	PUNCT
cana-5932	143	42	for	for	ADP
cana-5932	143	43	any	any	DET
cana-5932	143	44	x	x	NOUN
cana-5932	143	45	,	,	PUNCT
cana-5932	143	46	y	y	PROPN
cana-5932	143	47	∈	∈	PROPN
cana-5932	143	48	x	x	X
cana-5932	143	49	and	and	CCONJ
cana-5932	143	50	t	t	X
cana-5932	143	51	>	>	X
cana-5932	143	52	0	0	PROPN
cana-5932	143	53	,	,	PUNCT
cana-5932	143	54	t	t	PROPN
cana-5932	143	55	is	be	AUX
cana-5932	143	56	a	a	DET
cana-5932	143	57	h	h	NOUN
cana-5932	143	58	–	–	PUNCT
cana-5932	143	59	f	f	X
cana-5932	143	60	–	–	PUNCT
cana-5932	143	61	contractive	contractive	ADJ
cana-5932	143	62	.	.	PUNCT
cana-5932	144	1	thus	thus	ADV
cana-5932	144	2	,	,	PUNCT
cana-5932	144	3	all	all	DET
cana-5932	144	4	the	the	DET
cana-5932	144	5	conditions	condition	NOUN
cana-5932	144	6	of	of	ADP
cana-5932	144	7	theorem	theorem	ADJ
cana-5932	144	8	1	1	NUM
cana-5932	144	9	are	be	AUX
cana-5932	144	10	satisfied	satisfied	ADJ
cana-5932	144	11	,	,	PUNCT
cana-5932	144	12	therefore	therefore	ADV
cana-5932	144	13	t	t	PROPN
cana-5932	144	14	has	have	VERB
cana-5932	144	15	a	a	DET
cana-5932	144	16	unique	unique	ADJ
cana-5932	144	17	fixed	fix	VERB
cana-5932	144	18	point	point	NOUN
cana-5932	144	19	communications	communication	NOUN
cana-5932	144	20	on	on	ADP
cana-5932	144	21	applied	apply	VERB
cana-5932	144	22	nonlinear	nonlinear	ADJ
cana-5932	144	23	analysis	analysis	NOUN
cana-5932	144	24	issn	issn	NOUN
cana-5932	144	25	:	:	PUNCT
cana-5932	144	26	1074	1074	NUM
cana-5932	144	27	-	-	PUNCT
cana-5932	144	28	133x	133x	NUM
cana-5932	144	29	vol	vol	VERB
cana-5932	144	30	32	32	NUM
cana-5932	144	31	no	no	NOUN
cana-5932	144	32	.	.	PUNCT
cana-5932	145	1	10s	10	NOUN
cana-5932	145	2	(	(	PUNCT
cana-5932	145	3	2025	2025	NUM
cana-5932	145	4	)	)	PUNCT
cana-5932	145	5	3131	3131	NUM
cana-5932	146	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	146	2	z	z	NOUN
cana-5932	146	3	=	=	SYM
cana-5932	146	4	0	0	X
cana-5932	146	5	.	.	PUNCT
cana-5932	146	6	to	to	PART
cana-5932	146	7	verify	verify	VERB
cana-5932	146	8	the	the	DET
cana-5932	146	9	above	above	ADJ
cana-5932	146	10	calculations	calculation	NOUN
cana-5932	146	11	,	,	PUNCT
cana-5932	146	12	the	the	DET
cana-5932	146	13	reader	reader	NOUN
cana-5932	146	14	is	be	AUX
cana-5932	146	15	referred	refer	VERB
cana-5932	146	16	to	to	ADP
cana-5932	146	17	the	the	DET
cana-5932	146	18	computational	computational	ADJ
cana-5932	146	19	data	datum	NOUN
cana-5932	146	20	for	for	ADP
cana-5932	146	21	t	t	NOUN
cana-5932	146	22	=	=	SYM
cana-5932	146	23	1	1	NUM
cana-5932	146	24	presented	present	VERB
cana-5932	146	25	to	to	ADP
cana-5932	146	26	the	the	DET
cana-5932	146	27	table	table	NOUN
cana-5932	146	28	shown	show	VERB
cana-5932	146	29	in	in	ADP
cana-5932	146	30	the	the	DET
cana-5932	146	31	below	below	ADJ
cana-5932	146	32	table	table	NOUN
cana-5932	146	33	2	2	NUM
cana-5932	146	34	.	.	PUNCT
cana-5932	146	35	table	table	NOUN
cana-5932	146	36	1	1	NUM
cana-5932	146	37	:	:	PUNCT
cana-5932	146	38	computation	computation	NOUN
cana-5932	146	39	of	of	ADP
cana-5932	146	40	fixed	fix	VERB
cana-5932	146	41	point	point	NOUN
cana-5932	146	42	for	for	ADP
cana-5932	146	43	t	t	NOUN
cana-5932	146	44	=	=	SYM
cana-5932	146	45	1	1	NUM
cana-5932	146	46	,	,	PUNCT
cana-5932	146	47	where	where	SCONJ
cana-5932	146	48	lhs	lhs	PROPN
cana-5932	146	49	=	=	PROPN
cana-5932	146	50	h(xn	h(xn	PROPN
cana-5932	146	51	,	,	PUNCT
cana-5932	146	52	xn+1	xn+1	NUM
cana-5932	146	53	)	)	PUNCT
cana-5932	147	1	+	+	CCONJ
cana-5932	147	2	f	f	X
cana-5932	147	3	(	(	PUNCT
cana-5932	147	4	m	m	PROPN
cana-5932	147	5	(	(	PUNCT
cana-5932	147	6	xn	xn	PROPN
cana-5932	147	7	,	,	PUNCT
cana-5932	147	8	xn+1	xn+1	PROPN
cana-5932	147	9	,	,	PUNCT
cana-5932	147	10	t	t	PROPN
cana-5932	147	11	)	)	PUNCT
cana-5932	147	12	)	)	PUNCT
cana-5932	147	13	.	.	PUNCT
cana-5932	148	1	however	however	ADV
cana-5932	148	2	,	,	PUNCT
cana-5932	148	3	varying	vary	VERB
cana-5932	148	4	the	the	DET
cana-5932	148	5	value	value	NOUN
cana-5932	148	6	of	of	ADP
cana-5932	148	7	t	t	PROPN
cana-5932	148	8	influences	influence	VERB
cana-5932	148	9	the	the	DET
cana-5932	148	10	number	number	NOUN
cana-5932	148	11	of	of	ADP
cana-5932	148	12	iterations	iteration	NOUN
cana-5932	148	13	required	require	VERB
cana-5932	148	14	to	to	PART
cana-5932	148	15	approach	approach	VERB
cana-5932	148	16	a	a	DET
cana-5932	148	17	fixed	fix	VERB
cana-5932	148	18	point	point	NOUN
cana-5932	148	19	.	.	PUNCT
cana-5932	149	1	as	as	SCONJ
cana-5932	149	2	illustrated	illustrate	VERB
cana-5932	149	3	in	in	ADP
cana-5932	149	4	figure	figure	NOUN
cana-5932	149	5	2	2	NUM
cana-5932	149	6	,	,	PUNCT
cana-5932	149	7	increasing	increase	VERB
cana-5932	149	8	the	the	DET
cana-5932	149	9	value	value	NOUN
cana-5932	149	10	of	of	ADP
cana-5932	149	11	t	t	PROPN
cana-5932	149	12	reduces	reduce	VERB
cana-5932	149	13	the	the	DET
cana-5932	149	14	number	number	NOUN
cana-5932	149	15	of	of	ADP
cana-5932	149	16	iterations	iteration	NOUN
cana-5932	149	17	required	require	VERB
cana-5932	149	18	to	to	PART
cana-5932	149	19	approximate	approximate	VERB
cana-5932	149	20	the	the	DET
cana-5932	149	21	fixed	fixed	ADJ
cana-5932	149	22	point	point	NOUN
cana-5932	149	23	x	x	PUNCT
cana-5932	149	24	=	=	SYM
cana-5932	149	25	0	0	X
cana-5932	149	26	.	.	PUNCT
cana-5932	149	27	figure	figure	NOUN
cana-5932	149	28	2	2	NUM
cana-5932	149	29	:	:	PUNCT
cana-5932	149	30	effect	effect	NOUN
cana-5932	149	31	of	of	ADP
cana-5932	149	32	t	t	PROPN
cana-5932	149	33	on	on	ADP
cana-5932	149	34	the	the	DET
cana-5932	149	35	number	number	NOUN
cana-5932	149	36	of	of	ADP
cana-5932	149	37	iterations	iteration	NOUN
cana-5932	149	38	n.	n.	NOUN
cana-5932	149	39	communications	communication	NOUN
cana-5932	149	40	on	on	ADP
cana-5932	149	41	applied	apply	VERB
cana-5932	149	42	nonlinear	nonlinear	ADJ
cana-5932	149	43	analysis	analysis	NOUN
cana-5932	149	44	issn	issn	NOUN
cana-5932	149	45	:	:	PUNCT
cana-5932	149	46	1074	1074	NUM
cana-5932	149	47	-	-	PUNCT
cana-5932	149	48	133x	133x	NUM
cana-5932	149	49	vol	vol	VERB
cana-5932	149	50	32	32	NUM
cana-5932	149	51	no	no	NOUN
cana-5932	149	52	.	.	PUNCT
cana-5932	150	1	10s	10	NOUN
cana-5932	150	2	(	(	PUNCT
cana-5932	150	3	2025	2025	NUM
cana-5932	150	4	)	)	PUNCT
cana-5932	150	5	3132	3132	NUM
cana-5932	150	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	150	7	example	example	NOUN
cana-5932	150	8	2	2	NUM
cana-5932	150	9	let	let	VERB
cana-5932	150	10	x	x	SYM
cana-5932	150	11	=	=	SYM
cana-5932	150	12	c[0	c[0	PROPN
cana-5932	150	13	,	,	PUNCT
cana-5932	150	14	1	1	NUM
cana-5932	150	15	]	]	PUNCT
cana-5932	150	16	be	be	AUX
cana-5932	150	17	a	a	DET
cana-5932	150	18	fuzzy	fuzzy	ADJ
cana-5932	150	19	metric	metric	ADJ
cana-5932	150	20	space	space	NOUN
cana-5932	150	21	with	with	ADP
cana-5932	150	22	a	a	DET
cana-5932	150	23	∗	∗	NOUN
cana-5932	150	24	b	b	NOUN
cana-5932	150	25	=	=	SYM
cana-5932	150	26	ab	ab	PROPN
cana-5932	150	27	for	for	ADP
cana-5932	150	28	all	all	DET
cana-5932	150	29	a	a	PRON
cana-5932	150	30	,	,	PUNCT
cana-5932	151	1	b	b	X
cana-5932	151	2	∈	∈	NOUN
cana-5932	151	3	i	i	PRON
cana-5932	151	4	and	and	CCONJ
cana-5932	151	5	fuzzy	fuzzy	ADJ
cana-5932	151	6	metric	metric	ADJ
cana-5932	151	7	||	||	NOUN
cana-5932	152	1	||	||	PROPN
cana-5932	153	1	(	(	PUNCT
cana-5932	153	2	,	,	PUNCT
cana-5932	153	3	,	,	PUNCT
cana-5932	153	4	)	)	PUNCT
cana-5932	153	5	exp	exp	NOUN
cana-5932	153	6	f	f	PROPN
cana-5932	154	1	g	g	PROPN
cana-5932	154	2	m	m	PROPN
cana-5932	154	3	f	f	PROPN
cana-5932	154	4	g	g	PROPN
cana-5932	154	5	t	t	PROPN
cana-5932	154	6	t	t	PROPN
cana-5932	154	7	−	−	VERB
cana-5932	154	8	−	−	NOUN
cana-5932	155	1			NOUN
cana-5932	155	2	=	=	SYM
cana-5932	155	3			PROPN
cana-5932	155	4			PROPN
cana-5932	156	1			ADJ
cana-5932	156	2			NOUN
cana-5932	156	3	for	for	ADP
cana-5932	156	4	all	all	DET
cana-5932	156	5	f	f	NOUN
cana-5932	156	6	,	,	PUNCT
cana-5932	156	7	g	g	PROPN
cana-5932	156	8	∈	∈	PROPN
cana-5932	156	9	x	x	X
cana-5932	156	10	and	and	CCONJ
cana-5932	156	11	t	t	X
cana-5932	156	12	>	>	X
cana-5932	156	13	0	0	X
cana-5932	156	14	.	.	PUNCT
cana-5932	157	1	it	it	PRON
cana-5932	157	2	is	be	AUX
cana-5932	157	3	clear	clear	ADJ
cana-5932	157	4	that	that	SCONJ
cana-5932	157	5	(	(	PUNCT
cana-5932	157	6	x	x	X
cana-5932	157	7	,	,	PUNCT
cana-5932	157	8	m	m	PROPN
cana-5932	157	9	,	,	PUNCT
cana-5932	157	10	∗	∗	NOUN
cana-5932	157	11	)	)	PUNCT
cana-5932	157	12	is	be	AUX
cana-5932	157	13	a	a	DET
cana-5932	157	14	fuzzy	fuzzy	ADJ
cana-5932	157	15	metric	metric	ADJ
cana-5932	157	16	space	space	NOUN
cana-5932	157	17	,	,	PUNCT
cana-5932	157	18	to	to	PART
cana-5932	157	19	verify	verify	VERB
cana-5932	157	20	its	its	PRON
cana-5932	157	21	completeness	completeness	NOUN
cana-5932	157	22	,	,	PUNCT
cana-5932	157	23	let	let	VERB
cana-5932	157	24	us	we	PRON
cana-5932	157	25	consider	consider	VERB
cana-5932	157	26	the	the	DET
cana-5932	157	27	sequence	sequence	NOUN
cana-5932	157	28	of	of	ADP
cana-5932	157	29	functions	function	NOUN
cana-5932	157	30	1	1	NUM
cana-5932	157	31	(	(	PUNCT
cana-5932	157	32	)	)	PUNCT
cana-5932	157	33	sin	sin	NOUN
cana-5932	157	34	(	(	PUNCT
cana-5932	157	35	)	)	PUNCT
cana-5932	157	36	nf	nf	NOUN
cana-5932	157	37	x	x	NOUN
cana-5932	157	38	x	x	PUNCT
cana-5932	157	39	x	x	SYM
cana-5932	157	40	x	x	SYM
cana-5932	157	41	n	n	X
cana-5932	157	42			ADJ
cana-5932	157	43			NOUN
cana-5932	158	1			PROPN
cana-5932	159	1	=	=	PUNCT
cana-5932	160	1	+	+	CCONJ
cana-5932	160	2			PROPN
cana-5932	160	3			INTJ
cana-5932	160	4			PROPN
cana-5932	160	5			PROPN
cana-5932	160	6	and	and	CCONJ
cana-5932	160	7	a	a	DET
cana-5932	160	8	function	function	NOUN
cana-5932	160	9	f	f	X
cana-5932	160	10	(	(	PUNCT
cana-5932	160	11	x	x	X
cana-5932	160	12	)	)	PUNCT
cana-5932	160	13	=	=	PUNCT
cana-5932	161	1	x	x	X
cana-5932	161	2	in	in	ADP
cana-5932	161	3	x.	x.	NOUN
cana-5932	161	4	figure	figure	NOUN
cana-5932	161	5	3	3	NUM
cana-5932	161	6	:	:	PUNCT
cana-5932	161	7	convergence	convergence	NOUN
cana-5932	161	8	of	of	ADP
cana-5932	161	9	m	m	PROPN
cana-5932	161	10	(	(	PUNCT
cana-5932	161	11	fn(x	fn(x	NOUN
cana-5932	161	12	)	)	PUNCT
cana-5932	161	13	,	,	PUNCT
cana-5932	161	14	f	f	PROPN
cana-5932	161	15	(	(	PUNCT
cana-5932	161	16	x	x	NOUN
cana-5932	161	17	)	)	PUNCT
cana-5932	161	18	,	,	PUNCT
cana-5932	161	19	t	t	PROPN
cana-5932	161	20	)	)	PUNCT
cana-5932	161	21	as	as	ADP
cana-5932	161	22	n	n	PROPN
cana-5932	161	23	→	→	SYM
cana-5932	161	24	∞	∞	PROPN
cana-5932	161	25	for	for	ADP
cana-5932	161	26	t	t	NOUN
cana-5932	161	27	=	=	SYM
cana-5932	161	28	1	1	NUM
cana-5932	161	29	.	.	PUNCT
cana-5932	161	30	from	from	ADP
cana-5932	161	31	the	the	DET
cana-5932	161	32	graph	graph	NOUN
cana-5932	161	33	shown	show	VERB
cana-5932	161	34	in	in	ADP
cana-5932	161	35	figure	figure	NOUN
cana-5932	161	36	3	3	NUM
cana-5932	161	37	,	,	PUNCT
cana-5932	161	38	it	it	PRON
cana-5932	161	39	is	be	AUX
cana-5932	161	40	clear	clear	ADJ
cana-5932	161	41	that	that	SCONJ
cana-5932	161	42	(	(	PUNCT
cana-5932	161	43	x	x	X
cana-5932	161	44	,	,	PUNCT
cana-5932	161	45	m	m	PROPN
cana-5932	161	46	,	,	PUNCT
cana-5932	161	47	∗	∗	NOUN
cana-5932	161	48	)	)	PUNCT
cana-5932	161	49	is	be	AUX
cana-5932	161	50	a	a	DET
cana-5932	161	51	complete	complete	ADJ
cana-5932	161	52	fuzzy	fuzzy	ADJ
cana-5932	161	53	metric	metric	ADJ
cana-5932	161	54	space	space	NOUN
cana-5932	161	55	.	.	PUNCT
cana-5932	162	1	now	now	ADV
cana-5932	162	2	,	,	PUNCT
cana-5932	162	3	for	for	ADP
cana-5932	162	4	any	any	DET
cana-5932	162	5	g	g	PROPN
cana-5932	162	6	∈	∈	PROPN
cana-5932	162	7	x	x	PUNCT
cana-5932	162	8	define	define	VERB
cana-5932	162	9	1	1	NUM
cana-5932	162	10	(	(	PUNCT
cana-5932	162	11	,	,	PUNCT
cana-5932	162	12	)	)	PUNCT
cana-5932	162	13	||	||	NOUN
cana-5932	163	1	||	||	NOUN
cana-5932	164	1	2	2	NUM
cana-5932	164	2	h	h	NOUN
cana-5932	164	3	f	f	NOUN
cana-5932	165	1	g	g	PROPN
cana-5932	165	2	f	f	PROPN
cana-5932	165	3	g	g	NOUN
cana-5932	165	4	=	=	PROPN
cana-5932	165	5	−	−	PROPN
cana-5932	165	6	and	and	CCONJ
cana-5932	165	7	for	for	ADP
cana-5932	165	8	any	any	DET
cana-5932	165	9	q	q	NOUN
cana-5932	165	10	∈	∈	PROPN
cana-5932	165	11	(	(	PUNCT
cana-5932	165	12	0	0	NUM
cana-5932	165	13	,	,	PUNCT
cana-5932	165	14	1	1	X
cana-5932	165	15	)	)	PUNCT
cana-5932	165	16	define	define	NOUN
cana-5932	165	17	(	(	PUNCT
cana-5932	165	18	)	)	PUNCT
cana-5932	165	19	ln	ln	INTJ
cana-5932	165	20	.	.	PUNCT
cana-5932	166	1	1	1	NUM
cana-5932	166	2	p	p	NOUN
cana-5932	166	3	f	f	X
cana-5932	166	4	p	p	X
cana-5932	166	5	p	p	PROPN
cana-5932	166	6			NOUN
cana-5932	166	7			NOUN
cana-5932	166	8	=	=	SYM
cana-5932	166	9			PROPN
cana-5932	167	1			INTJ
cana-5932	167	2	−	−	NUM
cana-5932	167	3			NOUN
cana-5932	167	4	it	it	PRON
cana-5932	167	5	is	be	AUX
cana-5932	167	6	easy	easy	ADJ
cana-5932	167	7	to	to	PART
cana-5932	167	8	see	see	VERB
cana-5932	167	9	that	that	PRON
cana-5932	167	10	,	,	PUNCT
cana-5932	167	11	for	for	ADP
cana-5932	167	12	any	any	DET
cana-5932	167	13	x	x	SYM
cana-5932	167	14	∈	∈	PROPN
cana-5932	167	15	i	i	PRON
cana-5932	167	16	,	,	PUNCT
cana-5932	167	17	h(f	h(f	PROPN
cana-5932	167	18	,	,	PUNCT
cana-5932	167	19	f	f	PROPN
cana-5932	167	20	)	)	PUNCT
cana-5932	168	1	=	=	SYM
cana-5932	168	2	0	0	NUM
cana-5932	168	3	and	and	CCONJ
cana-5932	168	4	a	a	DET
cana-5932	168	5	sequence	sequence	NOUN
cana-5932	168	6	of	of	ADP
cana-5932	168	7	functions	function	NOUN
cana-5932	168	8	{	{	PUNCT
cana-5932	168	9	fn	fn	NOUN
cana-5932	168	10	}	}	PUNCT
cana-5932	168	11	⊂	⊂	PROPN
cana-5932	168	12	x	x	X
cana-5932	168	13	,	,	PUNCT
cana-5932	168	14	1lim	1lim	NUM
cana-5932	168	15	(	(	PUNCT
cana-5932	168	16	,	,	PUNCT
cana-5932	168	17	)	)	PUNCT
cana-5932	168	18	0.n	0.n	NUM
cana-5932	169	1	n	n	CCONJ
cana-5932	169	2	n	n	NOUN
cana-5932	169	3	h	h	NOUN
cana-5932	169	4	f	f	PROPN
cana-5932	169	5	f	f	PROPN
cana-5932	169	6	+	+	PROPN
cana-5932	169	7	→	→	PUNCT
cana-5932	169	8	=	=	PUNCT
cana-5932	169	9	also	also	ADV
cana-5932	169	10	,	,	PUNCT
cana-5932	169	11	we	we	PRON
cana-5932	169	12	have	have	VERB
cana-5932	169	13	0	0	NUM
cana-5932	169	14	lim	lim	NOUN
cana-5932	169	15	(	(	PUNCT
cana-5932	169	16	)	)	PUNCT
cana-5932	169	17	,	,	PUNCT
cana-5932	170	1	p	p	NOUN
cana-5932	170	2	f	f	X
cana-5932	170	3	p	p	X
cana-5932	170	4	+	+	PROPN
cana-5932	170	5	→	→	SYM
cana-5932	170	6	=	=	SYM
cana-5932	170	7	−	−	NOUN
cana-5932	170	8	and	and	CCONJ
cana-5932	170	9	1	1	NUM
cana-5932	170	10	lim	lim	NOUN
cana-5932	170	11	(	(	PUNCT
cana-5932	170	12	)	)	PUNCT
cana-5932	170	13	.	.	PUNCT
cana-5932	171	1	p	p	X
cana-5932	171	2	f	f	X
cana-5932	171	3	p	p	NOUN
cana-5932	171	4	−→	−→	NOUN
cana-5932	171	5	=	=	SYM
cana-5932	171	6			NOUN
cana-5932	171	7	now	now	ADV
cana-5932	171	8	,	,	PUNCT
cana-5932	171	9	define	define	VERB
cana-5932	171	10	t	t	NOUN
cana-5932	171	11	:	:	PUNCT
cana-5932	171	12	x	x	PUNCT
cana-5932	171	13	×	×	NOUN
cana-5932	171	14	x	x	PUNCT
cana-5932	171	15	→	→	SYM
cana-5932	171	16	x	x	SYM
cana-5932	171	17	,	,	PUNCT
cana-5932	171	18	1	1	NUM
cana-5932	171	19	0	0	NUM
cana-5932	171	20	(	(	PUNCT
cana-5932	171	21	)	)	PUNCT
cana-5932	171	22	(	(	PUNCT
cana-5932	171	23	)	)	PUNCT
cana-5932	171	24	(	(	PUNCT
cana-5932	171	25	)	)	PUNCT
cana-5932	171	26	t	t	NOUN
cana-5932	171	27	f	f	X
cana-5932	171	28	x	x	PROPN
cana-5932	171	29	xyf	xyf	PROPN
cana-5932	171	30	y	y	PROPN
cana-5932	171	31	dy=	dy=	NOUN
cana-5932	171	32			PUNCT
cana-5932	171	33	(	(	PUNCT
cana-5932	171	34	8)	8)	NUM
cana-5932	171	35	for	for	ADP
cana-5932	171	36	all	all	DET
cana-5932	171	37	f	f	NOUN
cana-5932	171	38	,	,	PUNCT
cana-5932	171	39	g	g	PROPN
cana-5932	171	40	∈	∈	PROPN
cana-5932	171	41	x	x	X
cana-5932	171	42	and	and	CCONJ
cana-5932	171	43	x	x	NOUN
cana-5932	171	44	,	,	PUNCT
cana-5932	171	45	y	y	PROPN
cana-5932	171	46	∈	∈	PROPN
cana-5932	171	47	i.	i.	NOUN
cana-5932	171	48	let	let	VERB
cana-5932	171	49	g(x	g(x	NOUN
cana-5932	171	50	)	)	PUNCT
cana-5932	172	1	=	=	SYM
cana-5932	172	2	sin	sin	NOUN
cana-5932	172	3	x	x	PUNCT
cana-5932	172	4	in	in	ADP
cana-5932	172	5	x	x	NOUN
cana-5932	172	6	,	,	PUNCT
cana-5932	172	7	then	then	ADV
cana-5932	172	8	we	we	PRON
cana-5932	172	9	have	have	VERB
cana-5932	172	10	,	,	PUNCT
cana-5932	173	1	1	1	NUM
cana-5932	173	2	1	1	NUM
cana-5932	173	3	2	2	NUM
cana-5932	173	4	0	0	NUM
cana-5932	173	5	0	0	NUM
cana-5932	173	6	(	(	PUNCT
cana-5932	173	7	)	)	PUNCT
cana-5932	173	8	(	(	PUNCT
cana-5932	173	9	)	)	PUNCT
cana-5932	173	10	(	(	PUNCT
cana-5932	173	11	)	)	PUNCT
cana-5932	173	12	,	,	PUNCT
cana-5932	173	13	3	3	NUM
cana-5932	173	14	x	x	SYM
cana-5932	173	15	t	t	NOUN
cana-5932	173	16	f	f	PROPN
cana-5932	173	17	x	x	SYM
cana-5932	173	18	xyf	xyf	PROPN
cana-5932	173	19	y	y	PROPN
cana-5932	173	20	dy	dy	VERB
cana-5932	173	21	xy	xy	PROPN
cana-5932	174	1	dy=	dy=	NOUN
cana-5932	174	2	=	=	PUNCT
cana-5932	175	1	=	=	NOUN
cana-5932	175	2			NOUN
cana-5932	175	3			SYM
cana-5932	175	4	1	1	NUM
cana-5932	175	5	1	1	NUM
cana-5932	175	6	0	0	NUM
cana-5932	175	7	0	0	NUM
cana-5932	175	8	(	(	PUNCT
cana-5932	175	9	)	)	PUNCT
cana-5932	175	10	(	(	PUNCT
cana-5932	175	11	)	)	PUNCT
cana-5932	175	12	(	(	PUNCT
cana-5932	175	13	)	)	PUNCT
cana-5932	175	14	sin	sin	NOUN
cana-5932	175	15	(	(	PUNCT
cana-5932	175	16	sin1	sin1	PROPN
cana-5932	175	17	cos1).t	cos1).t	NOUN
cana-5932	175	18	g	g	NOUN
cana-5932	175	19	x	x	X
cana-5932	175	20	xyg	xyg	NOUN
cana-5932	175	21	y	y	NOUN
cana-5932	175	22	dy	dy	NOUN
cana-5932	175	23	xy	xy	PROPN
cana-5932	175	24	y	y	PROPN
cana-5932	175	25	dy	dy	X
cana-5932	175	26	x=	x=	PUNCT
cana-5932	176	1	=	=	PUNCT
cana-5932	176	2	=	=	PUNCT
cana-5932	176	3	−	−	NOUN
cana-5932	176	4			PUNCT
cana-5932	176	5	and	and	CCONJ
cana-5932	176	6	1	1	NUM
cana-5932	176	7	||	||	NOUN
cana-5932	176	8	||	||	NOUN
cana-5932	177	1	sup	sup	NOUN
cana-5932	177	2	|	|	ADV
cana-5932	177	3	(	(	PUNCT
cana-5932	177	4	)	)	PUNCT
cana-5932	177	5	(	(	PUNCT
cana-5932	177	6	)	)	PUNCT
cana-5932	177	7	(	(	PUNCT
cana-5932	177	8	)	)	PUNCT
cana-5932	177	9	(	(	PUNCT
cana-5932	177	10	)	)	PUNCT
cana-5932	177	11	|	|	ADV
cana-5932	177	12	sin1	sin1	PROPN
cana-5932	177	13	cos1	cos1	PROPN
cana-5932	177	14	0.0322	0.0322	NUM
cana-5932	177	15	.	.	PUNCT
cana-5932	178	1	3x	3x	NUM
cana-5932	179	1	i	i	PRON
cana-5932	179	2	f	f	PROPN
cana-5932	180	1	g	g	PROPN
cana-5932	180	2	t	t	PROPN
cana-5932	180	3	f	f	X
cana-5932	180	4	x	x	SYM
cana-5932	180	5	t	t	PROPN
cana-5932	180	6	g	g	PROPN
cana-5932	180	7	x	x	PROPN
cana-5932	180	8			NOUN
cana-5932	180	9	−	−	PROPN
cana-5932	180	10	=	=	SYM
cana-5932	181	1	−	−	PROPN
cana-5932	181	2	=	=	SYM
cana-5932	182	1	−	−	PROPN
cana-5932	182	2	+	+	CCONJ
cana-5932	182	3			NOUN
cana-5932	182	4	for	for	ADP
cana-5932	182	5	t	t	NOUN
cana-5932	182	6	=	=	SYM
cana-5932	182	7	1	1	NUM
cana-5932	182	8	,	,	PUNCT
cana-5932	182	9	we	we	PRON
cana-5932	182	10	have	have	VERB
cana-5932	182	11	(	(	PUNCT
cana-5932	182	12	)	)	PUNCT
cana-5932	182	13	(	(	PUNCT
cana-5932	182	14	)	)	PUNCT
cana-5932	182	15	(	(	PUNCT
cana-5932	182	16	)	)	PUNCT
cana-5932	182	17	(	(	PUNCT
cana-5932	182	18	)	)	PUNCT
cana-5932	182	19	(	(	PUNCT
cana-5932	182	20	,	,	PUNCT
cana-5932	182	21	,	,	PUNCT
cana-5932	182	22	)	)	PUNCT
cana-5932	182	23	exp	exp	NOUN
cana-5932	183	1	||	||	NOUN
cana-5932	183	2	||	||	NOUN
cana-5932	184	1	0.8534	0.8534	NUM
cana-5932	184	2	,	,	PUNCT
cana-5932	184	3	(	(	PUNCT
cana-5932	184	4	(	(	PUNCT
cana-5932	184	5	)	)	PUNCT
cana-5932	184	6	,	,	PUNCT
cana-5932	184	7	(	(	PUNCT
cana-5932	184	8	)	)	PUNCT
cana-5932	184	9	,	,	PUNCT
cana-5932	184	10	)	)	PUNCT
cana-5932	184	11	exp	exp	NOUN
cana-5932	184	12	||	||	NOUN
cana-5932	184	13	(	(	PUNCT
cana-5932	184	14	)	)	PUNCT
cana-5932	184	15	(	(	PUNCT
cana-5932	184	16	)	)	PUNCT
cana-5932	184	17	||	||	X
cana-5932	185	1	0.9683	0.9683	NUM
cana-5932	185	2	,	,	PUNCT
cana-5932	185	3	1	1	NUM
cana-5932	185	4	(	(	PUNCT
cana-5932	185	5	)	)	PUNCT
cana-5932	185	6	,	,	PUNCT
cana-5932	185	7	(	(	PUNCT
cana-5932	185	8	)	)	PUNCT
cana-5932	185	9	||	||	NOUN
cana-5932	186	1	||	||	NUM
cana-5932	187	1	0.07925	0.07925	NUM
cana-5932	187	2	,	,	PUNCT
cana-5932	187	3	2	2	NUM
cana-5932	187	4	1	1	NUM
cana-5932	187	5	(	(	PUNCT
cana-5932	187	6	)	)	PUNCT
cana-5932	187	7	(	(	PUNCT
cana-5932	187	8	)	)	PUNCT
cana-5932	187	9	,	,	PUNCT
cana-5932	187	10	(	(	PUNCT
cana-5932	187	11	)	)	PUNCT
cana-5932	187	12	(	(	PUNCT
cana-5932	187	13	)	)	PUNCT
cana-5932	187	14	||	||	NOUN
cana-5932	188	1	(	(	PUNCT
cana-5932	188	2	)	)	PUNCT
cana-5932	188	3	(	(	PUNCT
cana-5932	188	4	)	)	PUNCT
cana-5932	188	5	||	||	X
cana-5932	189	1	0.0161	0.0161	NUM
cana-5932	189	2	.	.	PUNCT
cana-5932	190	1	2	2	NUM
cana-5932	190	2	m	m	NOUN
cana-5932	190	3	f	f	NOUN
cana-5932	190	4	g	g	PROPN
cana-5932	190	5	t	t	PROPN
cana-5932	191	1	f	f	PROPN
cana-5932	191	2	g	g	PROPN
cana-5932	191	3	m	m	PROPN
cana-5932	191	4	t	t	PROPN
cana-5932	192	1	f	f	PROPN
cana-5932	192	2	t	t	PROPN
cana-5932	192	3	g	g	PROPN
cana-5932	192	4	t	t	PROPN
cana-5932	192	5	t	t	PROPN
cana-5932	193	1	f	f	PROPN
cana-5932	193	2	t	t	PROPN
cana-5932	193	3	g	g	PROPN
cana-5932	193	4	h	h	NOUN
cana-5932	194	1	f	f	NOUN
cana-5932	194	2	x	x	X
cana-5932	195	1	g	g	NOUN
cana-5932	195	2	x	x	X
cana-5932	195	3	f	f	PROPN
cana-5932	195	4	g	g	PROPN
cana-5932	195	5	h	h	PROPN
cana-5932	196	1	t	t	PROPN
cana-5932	196	2	f	f	PROPN
cana-5932	196	3	x	x	X
cana-5932	197	1	t	t	PROPN
cana-5932	197	2	g	g	PROPN
cana-5932	197	3	x	x	PROPN
cana-5932	197	4	t	t	PROPN
cana-5932	197	5	f	f	PROPN
cana-5932	197	6	t	t	PROPN
cana-5932	197	7	g	g	NOUN
cana-5932	197	8			NOUN
cana-5932	197	9			VERB
cana-5932	197	10			NOUN
cana-5932	197	11			NOUN
cana-5932	197	12	=	=	SYM
cana-5932	197	13	−	−	NOUN
cana-5932	197	14	−	−	PROPN
cana-5932	197	15			PUNCT
cana-5932	197	16	=	=	SYM
cana-5932	197	17	−	−	NOUN
cana-5932	197	18	−	−	NOUN
cana-5932	197	19			PUNCT
cana-5932	197	20	=	=	SYM
cana-5932	197	21	−	−	NOUN
cana-5932	198	1	=	=	PUNCT
cana-5932	198	2	=	=	PUNCT
cana-5932	199	1	−	−	NOUN
cana-5932	200	1	=	=	NOUN
cana-5932	200	2	communications	communication	NOUN
cana-5932	200	3	on	on	ADP
cana-5932	200	4	applied	apply	VERB
cana-5932	200	5	nonlinear	nonlinear	ADJ
cana-5932	200	6	analysis	analysis	NOUN
cana-5932	200	7	issn	issn	NOUN
cana-5932	200	8	:	:	PUNCT
cana-5932	200	9	1074	1074	NUM
cana-5932	200	10	-	-	PUNCT
cana-5932	200	11	133x	133x	NUM
cana-5932	200	12	vol	vol	VERB
cana-5932	200	13	32	32	NUM
cana-5932	200	14	no	no	NOUN
cana-5932	200	15	.	.	PUNCT
cana-5932	201	1	10s	10	NOUN
cana-5932	201	2	(	(	PUNCT
cana-5932	201	3	2025	2025	NUM
cana-5932	201	4	)	)	PUNCT
cana-5932	201	5	3133	3133	NUM
cana-5932	201	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	202	1	now	now	ADV
cana-5932	202	2	,	,	PUNCT
cana-5932	202	3	from	from	ADP
cana-5932	202	4	contraction	contraction	NOUN
cana-5932	202	5	(	(	PUNCT
cana-5932	202	6	1	1	NUM
cana-5932	202	7	)	)	PUNCT
cana-5932	202	8	,	,	PUNCT
cana-5932	202	9	we	we	PRON
cana-5932	202	10	obtain	obtain	VERB
cana-5932	202	11	f	f	X
cana-5932	202	12	(	(	PUNCT
cana-5932	202	13	m	m	PROPN
cana-5932	202	14	(	(	PUNCT
cana-5932	202	15	t	t	PROPN
cana-5932	202	16	(	(	PUNCT
cana-5932	202	17	f	f	PROPN
cana-5932	202	18	)	)	PUNCT
cana-5932	202	19	(	(	PUNCT
cana-5932	202	20	x	x	X
cana-5932	202	21	)	)	PUNCT
cana-5932	202	22	,	,	PUNCT
cana-5932	202	23	t	t	PROPN
cana-5932	202	24	(	(	PUNCT
cana-5932	202	25	g)(x	g)(x	PROPN
cana-5932	202	26	)	)	PUNCT
cana-5932	202	27	,	,	PUNCT
cana-5932	202	28	t	t	PROPN
cana-5932	202	29	)	)	PUNCT
cana-5932	202	30	)	)	PUNCT
cana-5932	203	1	≈	≈	PROPN
cana-5932	203	2	3.42	3.42	NUM
cana-5932	203	3	>	>	PUNCT
cana-5932	203	4	1.84	1.84	NUM
cana-5932	204	1	≈	≈	PROPN
cana-5932	204	2	h(f	h(f	PROPN
cana-5932	204	3	(	(	PUNCT
cana-5932	204	4	x	x	NOUN
cana-5932	204	5	)	)	PUNCT
cana-5932	204	6	,	,	PUNCT
cana-5932	204	7	g(x	g(x	NOUN
cana-5932	204	8	)	)	PUNCT
cana-5932	204	9	)	)	PUNCT
cana-5932	205	1	+	+	CCONJ
cana-5932	205	2	f	f	X
cana-5932	205	3	(	(	PUNCT
cana-5932	205	4	m	m	PROPN
cana-5932	205	5	(	(	PUNCT
cana-5932	205	6	f	f	X
cana-5932	205	7	(	(	PUNCT
cana-5932	205	8	x	x	NOUN
cana-5932	205	9	)	)	PUNCT
cana-5932	205	10	,	,	PUNCT
cana-5932	205	11	g(x	g(x	NOUN
cana-5932	205	12	)	)	PUNCT
cana-5932	205	13	,	,	PUNCT
cana-5932	205	14	t	t	PROPN
cana-5932	205	15	)	)	PUNCT
cana-5932	205	16	)	)	PUNCT
cana-5932	205	17	.	.	PUNCT
cana-5932	206	1	hence	hence	ADV
cana-5932	206	2	,	,	PUNCT
cana-5932	206	3	it	it	PRON
cana-5932	206	4	can	can	AUX
cana-5932	206	5	be	be	AUX
cana-5932	206	6	concluded	conclude	VERB
cana-5932	206	7	that	that	SCONJ
cana-5932	206	8	for	for	ADP
cana-5932	206	9	any	any	DET
cana-5932	206	10	f	f	NOUN
cana-5932	206	11	,	,	PUNCT
cana-5932	206	12	g	g	PROPN
cana-5932	206	13	∈	∈	PROPN
cana-5932	206	14	x	x	PROPN
cana-5932	206	15	,	,	PUNCT
cana-5932	206	16	t	t	PROPN
cana-5932	206	17	is	be	AUX
cana-5932	206	18	a	a	DET
cana-5932	206	19	h	h	NOUN
cana-5932	206	20	–	–	PUNCT
cana-5932	206	21	f	f	X
cana-5932	206	22	–	–	PUNCT
cana-5932	206	23	contractive	contractive	ADJ
cana-5932	206	24	,	,	PUNCT
cana-5932	206	25	and	and	CCONJ
cana-5932	206	26	hence	hence	ADV
cana-5932	206	27	from	from	ADP
cana-5932	206	28	theorem	theorem	ADJ
cana-5932	206	29	1	1	NUM
cana-5932	206	30	,	,	PUNCT
cana-5932	206	31	t	t	PROPN
cana-5932	206	32	has	have	VERB
cana-5932	206	33	a	a	DET
cana-5932	206	34	unique	unique	ADJ
cana-5932	206	35	fixed	fix	VERB
cana-5932	206	36	point	point	NOUN
cana-5932	206	37	in	in	ADP
cana-5932	206	38	x.	x.	NOUN
cana-5932	206	39	let	let	VERB
cana-5932	206	40	us	we	PRON
cana-5932	206	41	consider	consider	VERB
cana-5932	206	42	h∗	h∗	NOUN
cana-5932	206	43	∈	∈	PROPN
cana-5932	206	44	x	x	PUNCT
cana-5932	206	45	is	be	AUX
cana-5932	206	46	a	a	DET
cana-5932	206	47	fixed	fix	VERB
cana-5932	206	48	point	point	NOUN
cana-5932	206	49	of	of	ADP
cana-5932	206	50	t	t	PROPN
cana-5932	206	51	,	,	PUNCT
cana-5932	206	52	that	that	ADV
cana-5932	206	53	is	is	ADV
cana-5932	206	54	,	,	PUNCT
cana-5932	206	55	t	t	PROPN
cana-5932	206	56	(	(	PUNCT
cana-5932	206	57	h∗)(x	h∗)(x	NOUN
cana-5932	206	58	)	)	PUNCT
cana-5932	206	59	=	=	SYM
cana-5932	206	60	h∗(x	h∗(x	NOUN
cana-5932	206	61	)	)	PUNCT
cana-5932	206	62	∀	∀	X
cana-5932	206	63	x	x	SYM
cana-5932	206	64	∈	∈	NOUN
cana-5932	206	65	i.	i.	NOUN
cana-5932	206	66	(	(	PUNCT
cana-5932	206	67	9	9	NUM
cana-5932	206	68	)	)	PUNCT
cana-5932	206	69	to	to	PART
cana-5932	206	70	find	find	VERB
cana-5932	206	71	h∗	h∗	NOUN
cana-5932	206	72	take	take	VERB
cana-5932	206	73	any	any	DET
cana-5932	206	74	function	function	NOUN
cana-5932	206	75	h	h	NOUN
cana-5932	206	76	from	from	ADP
cana-5932	206	77	x.	x.	NOUN
cana-5932	206	78	by	by	ADP
cana-5932	206	79	the	the	DET
cana-5932	206	80	definition	definition	NOUN
cana-5932	206	81	of	of	ADP
cana-5932	206	82	t	t	PROPN
cana-5932	206	83	in	in	ADP
cana-5932	206	84	the	the	DET
cana-5932	206	85	equation	equation	NOUN
cana-5932	206	86	(	(	PUNCT
cana-5932	206	87	8)	8)	NUM
cana-5932	206	88	,	,	PUNCT
cana-5932	206	89	we	we	PRON
cana-5932	206	90	construct	construct	VERB
cana-5932	206	91	a	a	DET
cana-5932	206	92	sequence	sequence	NOUN
cana-5932	206	93	of	of	ADP
cana-5932	206	94	functions	function	NOUN
cana-5932	206	95	as	as	SCONJ
cana-5932	206	96	follows	follow	VERB
cana-5932	206	97	;	;	PUNCT
cana-5932	206	98	1	1	NUM
cana-5932	206	99	1	1	NUM
cana-5932	206	100	1	1	NUM
cana-5932	206	101	0	0	NUM
cana-5932	206	102	0	0	NUM
cana-5932	206	103	1	1	NUM
cana-5932	206	104	1	1	NUM
cana-5932	206	105	1	1	NUM
cana-5932	206	106	1	1	NUM
cana-5932	206	107	1	1	NUM
cana-5932	206	108	2	2	NUM
cana-5932	206	109	0	0	NUM
cana-5932	206	110	0	0	NUM
cana-5932	206	111	1	1	NUM
cana-5932	206	112	1	1	NUM
cana-5932	206	113	2	2	NUM
cana-5932	206	114	2	2	NUM
cana-5932	206	115	2	2	NUM
cana-5932	206	116	32	32	NUM
cana-5932	206	117	0	0	NUM
cana-5932	206	118	0	0	NUM
cana-5932	206	119	1	1	NUM
cana-5932	206	120	1	1	NUM
cana-5932	206	121	3	3	NUM
cana-5932	206	122	3	3	NUM
cana-5932	206	123	3	3	NUM
cana-5932	206	124	3	3	NUM
cana-5932	206	125	0	0	NUM
cana-5932	206	126	0	0	NUM
cana-5932	206	127	(	(	PUNCT
cana-5932	206	128	)	)	PUNCT
cana-5932	206	129	(	(	PUNCT
cana-5932	206	130	)	)	PUNCT
cana-5932	206	131	(	(	PUNCT
cana-5932	206	132	)	)	PUNCT
cana-5932	206	133	(	(	PUNCT
cana-5932	206	134	)	)	PUNCT
cana-5932	206	135	(	(	PUNCT
cana-5932	206	136	)	)	PUNCT
cana-5932	206	137	(	(	PUNCT
cana-5932	206	138	)	)	PUNCT
cana-5932	206	139	(	(	PUNCT
cana-5932	206	140	)	)	PUNCT
cana-5932	206	141	(	(	PUNCT
cana-5932	206	142	)	)	PUNCT
cana-5932	206	143	(	(	PUNCT
cana-5932	206	144	)	)	PUNCT
cana-5932	206	145	(	(	PUNCT
cana-5932	206	146	)	)	PUNCT
cana-5932	206	147	3	3	NUM
cana-5932	206	148	(	(	PUNCT
cana-5932	206	149	)	)	PUNCT
cana-5932	206	150	(	(	PUNCT
cana-5932	206	151	)	)	PUNCT
cana-5932	206	152	(	(	PUNCT
cana-5932	206	153	)	)	PUNCT
cana-5932	206	154	(	(	PUNCT
cana-5932	206	155	)	)	PUNCT
cana-5932	206	156	(	(	PUNCT
cana-5932	206	157	)	)	PUNCT
cana-5932	206	158	3	3	NUM
cana-5932	206	159	(	(	PUNCT
cana-5932	206	160	)	)	PUNCT
cana-5932	206	161	(	(	PUNCT
cana-5932	206	162	)	)	PUNCT
cana-5932	206	163	(	(	PUNCT
cana-5932	206	164	)	)	PUNCT
cana-5932	206	165	(	(	PUNCT
cana-5932	206	166	)	)	PUNCT
cana-5932	206	167	3	3	NUM
cana-5932	206	168	t	t	NOUN
cana-5932	206	169	h	h	NOUN
cana-5932	206	170	x	x	X
cana-5932	206	171	xyh	xyh	PROPN
cana-5932	206	172	y	y	PROPN
cana-5932	206	173	dy	dy	NOUN
cana-5932	206	174	x	x	VERB
cana-5932	206	175	yh	yh	NOUN
cana-5932	206	176	y	y	PROPN
cana-5932	206	177	dy	dy	NOUN
cana-5932	206	178	cx	cx	PROPN
cana-5932	206	179	h	h	PROPN
cana-5932	207	1	x	x	PROPN
cana-5932	207	2	cx	cx	PROPN
cana-5932	208	1	t	t	NOUN
cana-5932	208	2	h	h	NOUN
cana-5932	208	3	x	x	X
cana-5932	208	4	xyh	xyh	PROPN
cana-5932	208	5	y	y	PROPN
cana-5932	208	6	dy	dy	NOUN
cana-5932	208	7	x	x	VERB
cana-5932	208	8	yh	yh	NOUN
cana-5932	208	9	y	y	PROPN
cana-5932	208	10	dy	dy	NOUN
cana-5932	208	11	h	h	NOUN
cana-5932	209	1	x	x	PROPN
cana-5932	209	2	cx	cx	PROPN
cana-5932	209	3	t	t	NOUN
cana-5932	209	4	h	h	NOUN
cana-5932	209	5	x	x	X
cana-5932	209	6	xyh	xyh	PROPN
cana-5932	209	7	y	y	PROPN
cana-5932	209	8	dy	dy	NOUN
cana-5932	209	9	x	x	VERB
cana-5932	209	10	yh	yh	NOUN
cana-5932	209	11	y	y	PROPN
cana-5932	209	12	dy	dy	NOUN
cana-5932	209	13	h	h	NOUN
cana-5932	210	1	x	x	PROPN
cana-5932	210	2	cx	cx	PROPN
cana-5932	210	3	t	t	NOUN
cana-5932	210	4	h	h	NOUN
cana-5932	210	5	x	x	X
cana-5932	210	6	xyh	xyh	PROPN
cana-5932	210	7	y	y	PROPN
cana-5932	210	8	dy	dy	NOUN
cana-5932	210	9	x	x	VERB
cana-5932	210	10	yh	yh	NOUN
cana-5932	210	11	y	y	PROPN
cana-5932	210	12	dy	dy	X
cana-5932	210	13			NOUN
cana-5932	210	14			NOUN
cana-5932	210	15	=	=	PUNCT
cana-5932	211	1	=	=	PUNCT
cana-5932	211	2	=	=	PUNCT
cana-5932	211	3	=	=	SYM
cana-5932	211	4			PROPN
cana-5932	211	5			NOUN
cana-5932	212	1			PROPN
cana-5932	213	1			ADJ
cana-5932	213	2			NOUN
cana-5932	213	3			VERB
cana-5932	213	4			NOUN
cana-5932	213	5	=	=	PUNCT
cana-5932	214	1	=	=	PUNCT
cana-5932	214	2	=	=	PUNCT
cana-5932	214	3	=	=	SYM
cana-5932	214	4			PROPN
cana-5932	214	5			NOUN
cana-5932	215	1			PROPN
cana-5932	216	1			ADJ
cana-5932	216	2			NOUN
cana-5932	216	3			VERB
cana-5932	216	4			NOUN
cana-5932	216	5	=	=	PUNCT
cana-5932	217	1	=	=	PUNCT
cana-5932	217	2	=	=	PUNCT
cana-5932	217	3	=	=	SYM
cana-5932	217	4			PROPN
cana-5932	217	5			NOUN
cana-5932	218	1			PROPN
cana-5932	219	1			ADJ
cana-5932	219	2			NOUN
cana-5932	219	3			VERB
cana-5932	219	4			NOUN
cana-5932	219	5	=	=	PUNCT
cana-5932	220	1	=	=	PUNCT
cana-5932	220	2	=	=	SYM
cana-5932	220	3			PROPN
cana-5932	220	4			NOUN
cana-5932	220	5			PROPN
cana-5932	221	1			ADJ
cana-5932	221	2			NOUN
cana-5932	221	3			PUNCT
cana-5932	221	4			X
cana-5932	221	5			X
cana-5932	222	1			X
cana-5932	222	2			X
cana-5932	223	1			NOUN
cana-5932	223	2			NOUN
cana-5932	224	1			SYM
cana-5932	225	1	4	4	NUM
cana-5932	225	2	1	1	NUM
cana-5932	225	3	1	1	NUM
cana-5932	225	4	1	1	NUM
cana-5932	225	5	0	0	NUM
cana-5932	225	6	0	0	NUM
cana-5932	225	7	(	(	PUNCT
cana-5932	225	8	)	)	PUNCT
cana-5932	225	9	.	.	PUNCT
cana-5932	225	10	.	.	PUNCT
cana-5932	225	11	.	.	PUNCT
cana-5932	226	1	(	(	PUNCT
cana-5932	226	2	)	)	PUNCT
cana-5932	226	3	(	(	PUNCT
cana-5932	226	4	)	)	PUNCT
cana-5932	226	5	(	(	PUNCT
cana-5932	226	6	)	)	PUNCT
cana-5932	226	7	(	(	PUNCT
cana-5932	226	8	)	)	PUNCT
cana-5932	226	9	(	(	PUNCT
cana-5932	226	10	)	)	PUNCT
cana-5932	226	11	3	3	NUM
cana-5932	226	12	n	n	CCONJ
cana-5932	226	13	n	n	CCONJ
cana-5932	226	14	n	n	CCONJ
cana-5932	226	15	nn	nn	PROPN
cana-5932	226	16	h	h	NOUN
cana-5932	226	17	x	x	PROPN
cana-5932	227	1	cx	cx	PROPN
cana-5932	227	2	t	t	NOUN
cana-5932	227	3	h	h	NOUN
cana-5932	227	4	x	x	X
cana-5932	227	5	xyh	xyh	PROPN
cana-5932	227	6	y	y	PROPN
cana-5932	227	7	dy	dy	NOUN
cana-5932	227	8	x	x	VERB
cana-5932	227	9	yh	yh	NOUN
cana-5932	227	10	y	y	PROPN
cana-5932	227	11	dy	dy	NOUN
cana-5932	227	12	h	h	NOUN
cana-5932	227	13	x+	x+	PUNCT
cana-5932	228	1	=	=	PUNCT
cana-5932	228	2			NOUN
cana-5932	228	3			NOUN
cana-5932	228	4	=	=	PUNCT
cana-5932	228	5	=	=	PUNCT
cana-5932	229	1	=	=	PUNCT
cana-5932	229	2	=	=	SYM
cana-5932	229	3			PROPN
cana-5932	229	4			NOUN
cana-5932	229	5			PROPN
cana-5932	230	1			ADJ
cana-5932	230	2			NOUN
cana-5932	230	3			PUNCT
cana-5932	230	4			ADP
cana-5932	230	5	where	where	SCONJ
cana-5932	230	6	1	1	NUM
cana-5932	230	7	0	0	NUM
cana-5932	230	8	(	(	PUNCT
cana-5932	230	9	)	)	PUNCT
cana-5932	230	10	.c	.c	NOUN
cana-5932	231	1	yh	yh	NOUN
cana-5932	231	2	y	y	PROPN
cana-5932	231	3	dy=	dy=	NOUN
cana-5932	231	4			ADP
cana-5932	231	5	this	this	PRON
cana-5932	231	6	implies	imply	VERB
cana-5932	231	7	that	that	SCONJ
cana-5932	231	8	,	,	PUNCT
cana-5932	231	9	lim	lim	PROPN
cana-5932	231	10	(	(	PUNCT
cana-5932	231	11	)	)	PUNCT
cana-5932	231	12	lim	lim	PROPN
cana-5932	231	13	0	0	NUM
cana-5932	231	14	,	,	PUNCT
cana-5932	231	15	3	3	NUM
cana-5932	231	16	1	1	NUM
cana-5932	231	17	n	n	ADP
cana-5932	231	18	nn	nn	PROPN
cana-5932	231	19	n	n	CCONJ
cana-5932	231	20	cx	cx	NOUN
cana-5932	231	21	h	h	NOUN
cana-5932	231	22	x	x	PUNCT
cana-5932	231	23	→	→	PUNCT
cana-5932	231	24	→	→	PUNCT
cana-5932	231	25	=	=	NOUN
cana-5932	232	1	=	=	SYM
cana-5932	232	2	−	−	PROPN
cana-5932	232	3	and	and	CCONJ
cana-5932	232	4	(	(	PUNCT
cana-5932	232	5	)	)	PUNCT
cana-5932	232	6	1lim	1lim	NUM
cana-5932	232	7	(	(	PUNCT
cana-5932	232	8	)	)	PUNCT
cana-5932	232	9	(	(	PUNCT
cana-5932	232	10	)	)	PUNCT
cana-5932	232	11	lim	lim	PROPN
cana-5932	232	12	(	(	PUNCT
cana-5932	232	13	)	)	PUNCT
cana-5932	232	14	lim	lim	PROPN
cana-5932	232	15	(	(	PUNCT
cana-5932	232	16	)	)	PUNCT
cana-5932	232	17	lim	lim	PROPN
cana-5932	232	18	(	(	PUNCT
cana-5932	232	19	0	0	NUM
cana-5932	232	20	)	)	PUNCT
cana-5932	232	21	0	0	NUM
cana-5932	232	22	.	.	NOUN
cana-5932	232	23	3	3	NUM
cana-5932	232	24	n	n	CCONJ
cana-5932	232	25	n	n	CCONJ
cana-5932	232	26	n	n	CCONJ
cana-5932	232	27	n	n	CCONJ
cana-5932	232	28	n	n	CCONJ
cana-5932	232	29	nn	nn	PROPN
cana-5932	232	30	n	n	ADP
cana-5932	232	31	t	t	NOUN
cana-5932	232	32	h	h	NOUN
cana-5932	233	1	x	x	PUNCT
cana-5932	233	2	h	h	NOUN
cana-5932	233	3	x	x	X
cana-5932	233	4	cx	cx	PROPN
cana-5932	234	1	t	t	NOUN
cana-5932	234	2	h	h	NOUN
cana-5932	234	3	x	x	PROPN
cana-5932	234	4	t	t	PROPN
cana-5932	234	5	+	+	CCONJ
cana-5932	234	6	→	→	PUNCT
cana-5932	234	7	→	→	NUM
cana-5932	234	8	→	→	PUNCT
cana-5932	234	9	→	→	PUNCT
cana-5932	235	1	=	=	NOUN
cana-5932	235	2	=	=	PUNCT
cana-5932	235	3			NOUN
cana-5932	236	1	=	=	NOUN
cana-5932	236	2	hence	hence	ADV
cana-5932	236	3	from	from	ADP
cana-5932	236	4	the	the	DET
cana-5932	236	5	equation	equation	NOUN
cana-5932	236	6	(	(	PUNCT
cana-5932	236	7	9	9	NUM
cana-5932	236	8	)	)	PUNCT
cana-5932	236	9	,	,	PUNCT
cana-5932	236	10	the	the	DET
cana-5932	236	11	zero	zero	NUM
cana-5932	236	12	function	function	NOUN
cana-5932	236	13	h∗(x	h∗(x	NOUN
cana-5932	236	14	)	)	PUNCT
cana-5932	236	15	=	=	SYM
cana-5932	236	16	0	0	NUM
cana-5932	236	17	for	for	ADP
cana-5932	236	18	all	all	DET
cana-5932	236	19	x	x	SYM
cana-5932	236	20	∈	∈	PROPN
cana-5932	236	21	i	i	PRON
cana-5932	236	22	,	,	PUNCT
cana-5932	236	23	is	be	AUX
cana-5932	236	24	a	a	DET
cana-5932	236	25	unique	unique	ADJ
cana-5932	236	26	fixed	fix	VERB
cana-5932	236	27	point	point	NOUN
cana-5932	236	28	of	of	ADP
cana-5932	236	29	t.	t.	PROPN
cana-5932	236	30	remark	remark	NOUN
cana-5932	236	31	:	:	PUNCT
cana-5932	236	32	in	in	ADP
cana-5932	236	33	the	the	DET
cana-5932	236	34	above	above	ADJ
cana-5932	236	35	example	example	NOUN
cana-5932	236	36	2	2	NUM
cana-5932	236	37	,	,	PUNCT
cana-5932	236	38	consider	consider	VERB
cana-5932	236	39	the	the	DET
cana-5932	236	40	evaluation	evaluation	NOUN
cana-5932	236	41	x	x	PUNCT
cana-5932	237	1	=	=	PUNCT
cana-5932	237	2	0	0	X
cana-5932	237	3	.	.	PUNCT
cana-5932	238	1	we	we	PRON
cana-5932	238	2	observe	observe	VERB
cana-5932	238	3	that	that	SCONJ
cana-5932	238	4	1	1	NUM
cana-5932	238	5	0	0	NUM
cana-5932	238	6	(	(	PUNCT
cana-5932	238	7	)	)	PUNCT
cana-5932	238	8	(	(	PUNCT
cana-5932	238	9	0	0	NUM
cana-5932	238	10	)	)	PUNCT
cana-5932	238	11	0	0	NUM
cana-5932	238	12	(	(	PUNCT
cana-5932	238	13	)	)	PUNCT
cana-5932	238	14	0,t	0,t	PROPN
cana-5932	238	15	h	h	NOUN
cana-5932	238	16	yh	yh	NOUN
cana-5932	239	1	y	y	PROPN
cana-5932	239	2	dy=	dy=	NOUN
cana-5932	239	3	=	=	SYM
cana-5932	239	4			X
cana-5932	239	5	which	which	PRON
cana-5932	239	6	implies	imply	VERB
cana-5932	239	7	that	that	SCONJ
cana-5932	240	1	,	,	PUNCT
cana-5932	240	2	any	any	DET
cana-5932	240	3	function	function	NOUN
cana-5932	240	4	h	h	NOUN
cana-5932	240	5	∈	∈	NOUN
cana-5932	240	6	x	x	PUNCT
cana-5932	240	7	satisfying	satisfy	VERB
cana-5932	240	8	h(0	h(0	NOUN
cana-5932	240	9	)	)	PUNCT
cana-5932	241	1	=	=	SYM
cana-5932	241	2	0	0	PUNCT
cana-5932	241	3	also	also	ADV
cana-5932	241	4	fulfills	fulfill	VERB
cana-5932	241	5	the	the	DET
cana-5932	241	6	fixed	fix	VERB
cana-5932	241	7	point	point	NOUN
cana-5932	241	8	condition	condition	NOUN
cana-5932	241	9	at	at	ADP
cana-5932	241	10	the	the	DET
cana-5932	241	11	specific	specific	ADJ
cana-5932	241	12	point	point	NOUN
cana-5932	241	13	,	,	PUNCT
cana-5932	241	14	that	that	PRON
cana-5932	241	15	is	is	ADV
cana-5932	241	16	t	t	PROPN
cana-5932	241	17	(	(	PUNCT
cana-5932	241	18	h)(0	h)(0	ADJ
cana-5932	241	19	)	)	PUNCT
cana-5932	241	20	=	=	SYM
cana-5932	242	1	h(0	h(0	PROPN
cana-5932	242	2	)	)	PUNCT
cana-5932	242	3	.	.	PUNCT
cana-5932	243	1	for	for	ADP
cana-5932	243	2	instance	instance	NOUN
cana-5932	243	3	,	,	PUNCT
cana-5932	243	4	the	the	DET
cana-5932	243	5	non	non	ADJ
cana-5932	243	6	-	-	ADJ
cana-5932	243	7	trivial	trivial	ADJ
cana-5932	243	8	function	function	NOUN
cana-5932	243	9	h(x	h(x	PROPN
cana-5932	243	10	)	)	PUNCT
cana-5932	244	1	=	=	PUNCT
cana-5932	244	2	sin	sin	NOUN
cana-5932	244	3	x	x	X
cana-5932	244	4	∈	∈	PROPN
cana-5932	244	5	x	x	NOUN
cana-5932	244	6	,	,	PUNCT
cana-5932	244	7	satisfies	satisfy	VERB
cana-5932	244	8	this	this	DET
cana-5932	244	9	condition	condition	NOUN
cana-5932	244	10	since	since	SCONJ
cana-5932	244	11	h(0	h(0	PROPN
cana-5932	244	12	)	)	PUNCT
cana-5932	244	13	=	=	SYM
cana-5932	244	14	0	0	NUM
cana-5932	244	15	,	,	PUNCT
cana-5932	244	16	and	and	CCONJ
cana-5932	244	17	consequently	consequently	ADV
cana-5932	244	18	t	t	X
cana-5932	244	19	(	(	PUNCT
cana-5932	244	20	h)(0	h)(0	PROPN
cana-5932	244	21	)	)	PUNCT
cana-5932	244	22	=	=	SYM
cana-5932	244	23	0	0	PUNCT
cana-5932	244	24	=	=	SYM
cana-5932	244	25	h(0	h(0	PROPN
cana-5932	244	26	)	)	PUNCT
cana-5932	244	27	.	.	PUNCT
cana-5932	245	1	communications	communication	NOUN
cana-5932	245	2	on	on	ADP
cana-5932	245	3	applied	apply	VERB
cana-5932	245	4	nonlinear	nonlinear	ADJ
cana-5932	245	5	analysis	analysis	NOUN
cana-5932	245	6	issn	issn	NOUN
cana-5932	245	7	:	:	PUNCT
cana-5932	245	8	1074	1074	NUM
cana-5932	245	9	-	-	PUNCT
cana-5932	245	10	133x	133x	NUM
cana-5932	245	11	vol	vol	VERB
cana-5932	245	12	32	32	NUM
cana-5932	245	13	no	no	NOUN
cana-5932	245	14	.	.	PUNCT
cana-5932	246	1	10s	10	NOUN
cana-5932	246	2	(	(	PUNCT
cana-5932	246	3	2025	2025	NUM
cana-5932	246	4	)	)	PUNCT
cana-5932	246	5	3134	3134	NUM
cana-5932	246	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5932	246	7	however	however	ADV
cana-5932	246	8	,	,	PUNCT
cana-5932	246	9	the	the	DET
cana-5932	246	10	point	point	NOUN
cana-5932	246	11	-	-	PUNCT
cana-5932	246	12	wise	wise	ADJ
cana-5932	246	13	existence	existence	NOUN
cana-5932	246	14	of	of	ADP
cana-5932	246	15	the	the	DET
cana-5932	246	16	fixed	fix	VERB
cana-5932	246	17	point	point	NOUN
cana-5932	246	18	condition	condition	NOUN
cana-5932	246	19	does	do	AUX
cana-5932	246	20	not	not	PART
cana-5932	246	21	ensure	ensure	VERB
cana-5932	246	22	that	that	SCONJ
cana-5932	246	23	t	t	PROPN
cana-5932	246	24	admits	admit	VERB
cana-5932	246	25	a	a	DET
cana-5932	246	26	global	global	ADJ
cana-5932	246	27	fixed	fix	VERB
cana-5932	246	28	point	point	NOUN
cana-5932	246	29	on	on	ADP
cana-5932	246	30	x.	x.	NOUN
cana-5932	246	31	this	this	PRON
cana-5932	246	32	highlights	highlight	VERB
cana-5932	246	33	the	the	DET
cana-5932	246	34	limitations	limitation	NOUN
cana-5932	246	35	of	of	ADP
cana-5932	246	36	point	point	NOUN
cana-5932	246	37	-	-	PUNCT
cana-5932	246	38	wise	wise	ADJ
cana-5932	246	39	analysis	analysis	NOUN
cana-5932	246	40	and	and	CCONJ
cana-5932	246	41	motivates	motivate	VERB
cana-5932	246	42	the	the	DET
cana-5932	246	43	use	use	NOUN
cana-5932	246	44	of	of	ADP
cana-5932	246	45	an	an	DET
cana-5932	246	46	iterative	iterative	NOUN
cana-5932	246	47	contraction	contraction	NOUN
cana-5932	246	48	framework	framework	NOUN
cana-5932	246	49	,	,	PUNCT
cana-5932	246	50	where	where	SCONJ
cana-5932	246	51	the	the	DET
cana-5932	246	52	sequence	sequence	NOUN
cana-5932	246	53	{	{	PUNCT
cana-5932	246	54	tn(f	tn(f	NUM
cana-5932	246	55	)	)	PUNCT
cana-5932	246	56	}	}	PUNCT
cana-5932	246	57	converges	converge	VERB
cana-5932	246	58	uniformly	uniformly	ADV
cana-5932	246	59	to	to	ADP
cana-5932	246	60	a	a	DET
cana-5932	246	61	unique	unique	ADJ
cana-5932	246	62	fixed	fix	VERB
cana-5932	246	63	point	point	NOUN
cana-5932	246	64	in	in	ADP
cana-5932	246	65	x	x	PRON
cana-5932	246	66	,	,	PUNCT
cana-5932	246	67	providing	provide	VERB
cana-5932	246	68	a	a	DET
cana-5932	246	69	more	more	ADV
cana-5932	246	70	rigorous	rigorous	ADJ
cana-5932	246	71	foundation	foundation	NOUN
cana-5932	246	72	for	for	ADP
cana-5932	246	73	the	the	DET
cana-5932	246	74	existence	existence	NOUN
cana-5932	246	75	and	and	CCONJ
cana-5932	246	76	uniqueness	uniqueness	NOUN
cana-5932	246	77	results	result	NOUN
cana-5932	246	78	.	.	PUNCT
cana-5932	247	1	definition	definition	NOUN
cana-5932	247	2	5	5	NUM
cana-5932	247	3	.	.	PUNCT
cana-5932	248	1	let	let	AUX
cana-5932	248	2	(	(	PUNCT
cana-5932	248	3	x	x	X
cana-5932	248	4	,	,	PUNCT
cana-5932	248	5	m	m	PROPN
cana-5932	248	6	,	,	PUNCT
cana-5932	248	7	∗	∗	NOUN
cana-5932	248	8	)	)	PUNCT
cana-5932	248	9	be	be	VERB
cana-5932	248	10	a	a	DET
cana-5932	248	11	complete	complete	ADJ
cana-5932	248	12	fuzzy	fuzzy	ADJ
cana-5932	248	13	metric	metric	ADJ
cana-5932	248	14	space	space	NOUN
cana-5932	248	15	.	.	PUNCT
cana-5932	249	1	for	for	ADP
cana-5932	249	2	any	any	DET
cana-5932	249	3	f	f	PROPN
cana-5932	249	4	∈	∈	PROPN
cana-5932	249	5	f	f	PROPN
cana-5932	249	6	,	,	PUNCT
cana-5932	249	7	a	a	DET
cana-5932	249	8	self	self	NOUN
cana-5932	249	9	mapping	mapping	NOUN
cana-5932	249	10	t	t	NOUN
cana-5932	249	11	:x	:x	PUNCT
cana-5932	249	12	→x	→x	PUNCT
cana-5932	249	13	is	be	AUX
cana-5932	249	14	said	say	VERB
cana-5932	249	15	to	to	PART
cana-5932	249	16	be	be	AUX
cana-5932	249	17	iterated	iterate	VERB
cana-5932	249	18	h	h	NOUN
cana-5932	249	19	–	–	PUNCT
cana-5932	249	20	f	f	NOUN
cana-5932	249	21	–	–	PUNCT
cana-5932	249	22	contractive	contractive	ADJ
cana-5932	249	23	if	if	SCONJ
cana-5932	249	24	there	there	PRON
cana-5932	249	25	exists	exist	VERB
cana-5932	249	26	a	a	DET
cana-5932	249	27	function	function	NOUN
cana-5932	249	28	h	h	NOUN
cana-5932	249	29	∈	∈	PROPN
cana-5932	249	30	h	h	NOUN
cana-5932	249	31	and	and	CCONJ
cana-5932	249	32	n	n	CCONJ
cana-5932	249	33	∈	∈	PROPN
cana-5932	250	1	n	n	PRON
cana-5932	250	2	such	such	ADJ
cana-5932	250	3	that	that	SCONJ
cana-5932	250	4	f	f	PROPN
cana-5932	250	5	(	(	PUNCT
cana-5932	250	6	m	m	PROPN
cana-5932	250	7	(	(	PUNCT
cana-5932	250	8	tnx	tnx	NOUN
cana-5932	250	9	,	,	PUNCT
cana-5932	250	10	tny	tny	PROPN
cana-5932	250	11	,	,	PUNCT
cana-5932	250	12	t	t	PROPN
cana-5932	250	13	)	)	PUNCT
cana-5932	250	14	)	)	PUNCT
cana-5932	250	15	≥	≥	PROPN
cana-5932	250	16	h(x	h(x	PROPN
cana-5932	250	17	,	,	PUNCT
cana-5932	250	18	y	y	PROPN
cana-5932	250	19	)	)	PUNCT
cana-5932	251	1	+	+	CCONJ
cana-5932	251	2	f	f	X
cana-5932	251	3	(	(	PUNCT
cana-5932	251	4	m	m	PROPN
cana-5932	251	5	(	(	PUNCT
cana-5932	251	6	x	x	NOUN
cana-5932	251	7	,	,	PUNCT
cana-5932	251	8	y	y	PROPN
cana-5932	251	9	,	,	PUNCT
cana-5932	251	10	t	t	PROPN
cana-5932	251	11	)	)	PUNCT
cana-5932	251	12	)	)	PUNCT
cana-5932	251	13	(	(	PUNCT
cana-5932	251	14	10	10	NUM
cana-5932	251	15	)	)	PUNCT
cana-5932	251	16	for	for	ADP
cana-5932	251	17	all	all	DET
cana-5932	251	18	x	x	NOUN
cana-5932	251	19	,	,	PUNCT
cana-5932	251	20	y	y	PROPN
cana-5932	251	21	∈	∈	PROPN
cana-5932	251	22	x	x	X
cana-5932	251	23	and	and	CCONJ
cana-5932	251	24	n	n	PRON
cana-5932	251	25	≥	≥	NOUN
cana-5932	251	26	n.	n.	PROPN
cana-5932	251	27	corollary	corollary	NOUN
cana-5932	251	28	1	1	NUM
cana-5932	251	29	.	.	PUNCT
cana-5932	252	1	let	let	AUX
cana-5932	252	2	(	(	PUNCT
cana-5932	252	3	x	x	X
cana-5932	252	4	,	,	PUNCT
cana-5932	252	5	m	m	PROPN
cana-5932	252	6	,	,	PUNCT
cana-5932	252	7	∗	∗	NOUN
cana-5932	252	8	)	)	PUNCT
cana-5932	252	9	be	be	VERB
cana-5932	252	10	a	a	DET
cana-5932	252	11	complete	complete	ADJ
cana-5932	252	12	fuzzy	fuzzy	ADJ
cana-5932	252	13	metric	metric	ADJ
cana-5932	252	14	space	space	NOUN
cana-5932	252	15	.	.	PUNCT
cana-5932	253	1	if	if	SCONJ
cana-5932	253	2	t	t	NOUN
cana-5932	253	3	:	:	PUNCT
cana-5932	253	4	x	x	X
cana-5932	253	5	→	→	PUNCT
cana-5932	253	6	x	x	X
cana-5932	253	7	is	be	AUX
cana-5932	253	8	an	an	DET
cana-5932	253	9	iterated	iterated	ADJ
cana-5932	253	10	h	h	NOUN
cana-5932	253	11	–	–	PUNCT
cana-5932	253	12	f	f	X
cana-5932	253	13	–	–	PUNCT
cana-5932	253	14	contractive	contractive	ADJ
cana-5932	253	15	mapping	mapping	NOUN
cana-5932	253	16	,	,	PUNCT
cana-5932	253	17	then	then	ADV
cana-5932	253	18	t	t	PROPN
cana-5932	253	19	has	have	VERB
cana-5932	253	20	a	a	DET
cana-5932	253	21	unique	unique	ADJ
cana-5932	253	22	fixed	fix	VERB
cana-5932	253	23	point	point	NOUN
cana-5932	253	24	in	in	ADP
cana-5932	253	25	x.	x.	NOUN
cana-5932	253	26	4	4	NUM
cana-5932	253	27	.	.	PUNCT
cana-5932	254	1	conclusion	conclusion	NOUN
cana-5932	254	2	we	we	PRON
cana-5932	254	3	introduced	introduce	VERB
cana-5932	254	4	the	the	DET
cana-5932	254	5	concept	concept	NOUN
cana-5932	254	6	of	of	ADP
cana-5932	254	7	h	h	NOUN
cana-5932	254	8	–	–	PUNCT
cana-5932	254	9	f	f	X
cana-5932	254	10	–	–	PUNCT
cana-5932	254	11	contractive	contractive	ADJ
cana-5932	254	12	mappings	mapping	NOUN
cana-5932	254	13	and	and	CCONJ
cana-5932	254	14	proved	prove	VERB
cana-5932	254	15	the	the	DET
cana-5932	254	16	fixed	fix	VERB
cana-5932	254	17	point	point	NOUN
cana-5932	254	18	theorem	theorem	VERB
cana-5932	254	19	in	in	ADP
cana-5932	254	20	complete	complete	ADJ
cana-5932	254	21	fuzzy	fuzzy	ADJ
cana-5932	254	22	metric	metric	ADJ
cana-5932	254	23	spaces	space	NOUN
cana-5932	254	24	.	.	PUNCT
cana-5932	255	1	the	the	DET
cana-5932	255	2	result	result	NOUN
cana-5932	255	3	generalizes	generalize	VERB
cana-5932	255	4	the	the	DET
cana-5932	255	5	existing	exist	VERB
cana-5932	255	6	work	work	NOUN
cana-5932	255	7	and	and	CCONJ
cana-5932	255	8	provides	provide	VERB
cana-5932	255	9	a	a	DET
cana-5932	255	10	versatile	versatile	ADJ
cana-5932	255	11	framework	framework	NOUN
cana-5932	255	12	for	for	ADP
cana-5932	255	13	further	further	ADJ
cana-5932	255	14	developments	development	NOUN
cana-5932	255	15	in	in	ADP
cana-5932	255	16	fuzzy	fuzzy	ADJ
cana-5932	255	17	fixed	fix	VERB
cana-5932	255	18	point	point	NOUN
cana-5932	255	19	theory	theory	NOUN
cana-5932	255	20	.	.	PUNCT
cana-5932	256	1	future	future	ADJ
cana-5932	256	2	work	work	NOUN
cana-5932	256	3	may	may	AUX
cana-5932	256	4	involve	involve	VERB
cana-5932	256	5	exploring	explore	VERB
cana-5932	256	6	multivalued	multivalued	ADJ
cana-5932	256	7	mappings	mapping	NOUN
cana-5932	256	8	,	,	PUNCT
cana-5932	256	9	partial	partial	ADJ
cana-5932	256	10	fuzzy	fuzzy	ADJ
cana-5932	256	11	metrics	metric	NOUN
cana-5932	256	12	,	,	PUNCT
cana-5932	256	13	and	and	CCONJ
cana-5932	256	14	applications	application	NOUN
cana-5932	256	15	of	of	ADP
cana-5932	256	16	control	control	NOUN
cana-5932	256	17	theory	theory	NOUN
cana-5932	256	18	.	.	PUNCT
cana-5932	257	1	refrences	refrence	VERB
cana-5932	258	1	[	[	X
cana-5932	258	2	1	1	NUM
cana-5932	258	3	]	]	PUNCT
cana-5932	258	4	michalek	michalek	NOUN
cana-5932	258	5	kramosil	kramosil	PROPN
cana-5932	258	6	i.	i.	PROPN
cana-5932	258	7	fuzzy	fuzzy	PROPN
cana-5932	258	8	metric	metric	ADJ
cana-5932	258	9	and	and	CCONJ
cana-5932	258	10	statistical	statistical	ADJ
cana-5932	258	11	metric	metric	ADJ
cana-5932	258	12	spaces	space	NOUN
cana-5932	258	13	.	.	PUNCT
cana-5932	259	1	kybernetika	kybernetika	PROPN
cana-5932	259	2	,	,	PUNCT
cana-5932	259	3	11:326–334	11:326–334	PROPN
cana-5932	259	4	,	,	PUNCT
cana-5932	259	5	1975	1975	NUM
cana-5932	259	6	.	.	PUNCT
cana-5932	260	1	[	[	X
cana-5932	260	2	2	2	NUM
cana-5932	260	3	]	]	PUNCT
cana-5932	260	4	veeramani	veeramani	NOUN
cana-5932	260	5	p.	p.	PROPN
cana-5932	260	6	george	george	PROPN
cana-5932	260	7	a.	a.	PROPN
cana-5932	260	8	on	on	ADP
cana-5932	260	9	some	some	DET
cana-5932	260	10	results	result	NOUN
cana-5932	260	11	in	in	ADP
cana-5932	260	12	fuzzy	fuzzy	ADJ
cana-5932	260	13	metric	metric	ADJ
cana-5932	260	14	spaces	space	NOUN
cana-5932	260	15	.	.	PUNCT
cana-5932	261	1	fuzzy	fuzzy	ADJ
cana-5932	261	2	sets	set	NOUN
cana-5932	261	3	and	and	CCONJ
cana-5932	261	4	systems	system	NOUN
cana-5932	261	5	,	,	PUNCT
cana-5932	261	6	64:395–399	64:395–399	PROPN
cana-5932	261	7	,	,	PUNCT
cana-5932	261	8	1994	1994	NUM
cana-5932	261	9	.	.	PUNCT
cana-5932	262	1	[	[	X
cana-5932	262	2	3	3	X
cana-5932	262	3	]	]	X
cana-5932	262	4	grabiec	grabiec	PROPN
cana-5932	262	5	m.	m.	NOUN
cana-5932	262	6	fixed	fix	VERB
cana-5932	262	7	points	point	NOUN
cana-5932	262	8	in	in	ADP
cana-5932	262	9	fuzzy	fuzzy	ADJ
cana-5932	262	10	metric	metric	ADJ
cana-5932	262	11	spaces	space	NOUN
cana-5932	262	12	.	.	PUNCT
cana-5932	263	1	fuzzy	fuzzy	ADJ
cana-5932	263	2	sets	set	NOUN
cana-5932	263	3	and	and	CCONJ
cana-5932	263	4	systems	system	NOUN
cana-5932	263	5	,	,	PUNCT
cana-5932	263	6	27(3):385–389	27(3):385–389	NUM
cana-5932	263	7	,	,	PUNCT
cana-5932	263	8	1988	1988	NUM
cana-5932	263	9	.	.	PUNCT
cana-5932	264	1	[	[	X
cana-5932	264	2	4	4	NUM
cana-5932	264	3	]	]	PUNCT
cana-5932	264	4	dariusz	dariusz	NOUN
cana-5932	264	5	wardowski	wardowski	VERB
cana-5932	264	6	.	.	PUNCT
cana-5932	265	1	common	common	ADJ
cana-5932	265	2	fixed	fix	VERB
cana-5932	265	3	points	point	NOUN
cana-5932	265	4	of	of	ADP
cana-5932	265	5	chatterjea	chatterjea	ADJ
cana-5932	265	6	type	type	NOUN
cana-5932	265	7	fuzzy	fuzzy	ADJ
cana-5932	265	8	mappings	mapping	NOUN
cana-5932	265	9	on	on	ADP
cana-5932	265	10	closed	closed	ADJ
cana-5932	265	11	balls	ball	NOUN
cana-5932	265	12	.	.	PUNCT
cana-5932	266	1	neural	neural	ADJ
cana-5932	266	2	computing	computing	NOUN
cana-5932	266	3	and	and	CCONJ
cana-5932	266	4	applications	application	NOUN
cana-5932	266	5	,	,	PUNCT
cana-5932	266	6	21:313–317	21:313–317	NUM
cana-5932	266	7	,	,	PUNCT
cana-5932	266	8	2012	2012	NUM
cana-5932	266	9	.	.	PUNCT
cana-5932	267	1	[	[	X
cana-5932	267	2	5	5	X
cana-5932	267	3	]	]	PUNCT
cana-5932	267	4	t.	t.	NOUN
cana-5932	267	5	bag	bag	NOUN
cana-5932	267	6	.	.	PUNCT
cana-5932	268	1	fuzzy	fuzzy	ADJ
cana-5932	268	2	cone	cone	NOUN
cana-5932	268	3	metric	metric	ADJ
cana-5932	268	4	spaces	space	NOUN
cana-5932	268	5	and	and	CCONJ
cana-5932	268	6	fixed	fix	VERB
cana-5932	268	7	point	point	NOUN
cana-5932	268	8	theorems	theorem	NOUN
cana-5932	268	9	on	on	ADP
cana-5932	268	10	fuzzy	fuzzy	ADJ
cana-5932	268	11	tkannan	tkannan	NOUN
cana-5932	268	12	fuzzy	fuzzy	ADJ
cana-5932	268	13	tchatterjea	tchatterjea	PROPN
cana-5932	268	14	type	type	NOUN
cana-5932	268	15	contractive	contractive	ADJ
cana-5932	268	16	mappings	mapping	NOUN
cana-5932	268	17	.	.	PUNCT
cana-5932	269	1	fuzzy	fuzzy	ADJ
cana-5932	269	2	information	information	NOUN
cana-5932	269	3	and	and	CCONJ
cana-5932	269	4	engineering	engineering	NOUN
cana-5932	269	5	,	,	PUNCT
cana-5932	269	6	7(3):305–315	7(3):305–315	PROPN
cana-5932	269	7	,	,	PUNCT
cana-5932	269	8	2015	2015	NUM
cana-5932	269	9	.	.	PUNCT
cana-5932	270	1	[	[	X
cana-5932	270	2	6	6	NUM
cana-5932	270	3	]	]	PUNCT
cana-5932	270	4	salvador	salvador	PROPN
cana-5932	270	5	romaguera	romaguera	NOUN
cana-5932	270	6	.	.	PUNCT
cana-5932	271	1	a	a	DET
cana-5932	271	2	fixed	fix	VERB
cana-5932	271	3	point	point	NOUN
cana-5932	271	4	theorem	theorem	NOUN
cana-5932	271	5	of	of	ADP
cana-5932	271	6	kannan	kannan	PROPN
cana-5932	271	7	type	type	NOUN
cana-5932	271	8	that	that	PRON
cana-5932	271	9	characterizes	characterize	VERB
cana-5932	271	10	fuzzy	fuzzy	ADJ
cana-5932	271	11	metric	metric	ADJ
cana-5932	271	12	completeness	completeness	NOUN
cana-5932	271	13	.	.	PUNCT
cana-5932	272	1	filomat	filomat	NOUN
cana-5932	272	2	,	,	PUNCT
cana-5932	272	3	34(14):4811–4819	34(14):4811–4819	NUM
cana-5932	272	4	,	,	PUNCT
cana-5932	272	5	2020	2020	NUM
cana-5932	272	6	.	.	PUNCT
cana-5932	273	1	[	[	X
cana-5932	273	2	7	7	NUM
cana-5932	273	3	]	]	X
cana-5932	273	4	doˇsenovi´c	doˇsenovi´c	NOUN
cana-5932	273	5	.	.	PUNCT
cana-5932	274	1	t.	t.	NOUN
cana-5932	274	2	raki´c	raki´c	PROPN
cana-5932	274	3	.	.	PUNCT
cana-5932	275	1	d.	d.	PROPN
cana-5932	275	2	brdar	brdar	PROPN
cana-5932	275	3	m.	m.	PROPN
cana-5932	275	4	huang	huang	PROPN
cana-5932	275	5	h.	h.	PROPN
cana-5932	275	6	,	,	PUNCT
cana-5932	275	7	cari´c	cari´c	PROPN
cana-5932	275	8	.	.	PUNCT
cana-5932	276	1	b.	b.	PROPN
cana-5932	276	2	fixed	fix	VERB
cana-5932	276	3	-	-	PUNCT
cana-5932	276	4	point	point	NOUN
cana-5932	276	5	theorems	theorem	NOUN
cana-5932	276	6	in	in	ADP
cana-5932	276	7	fuzzy	fuzzy	ADJ
cana-5932	276	8	metric	metric	ADJ
cana-5932	276	9	spaces	space	NOUN
cana-5932	276	10	via	via	ADP
cana-5932	276	11	fuzzy	fuzzy	ADJ
cana-5932	276	12	f	f	NOUN
cana-5932	276	13	-	-	PUNCT
cana-5932	276	14	contraction	contraction	NOUN
cana-5932	276	15	.	.	PUNCT
cana-5932	277	1	mathematics	mathematic	NOUN
cana-5932	277	2	,	,	PUNCT
cana-5932	277	3	9(6):1	9(6):1	NUM
cana-5932	277	4	–	–	PUNCT
cana-5932	277	5	10	10	NUM
cana-5932	277	6	,	,	PUNCT
cana-5932	277	7	2021	2021	NUM
cana-5932	277	8	.	.	PUNCT
cana-5932	278	1	[	[	X
cana-5932	278	2	8	8	NUM
cana-5932	278	3	]	]	X
cana-5932	278	4	b.	b.	PROPN
cana-5932	278	5	schweizer	schweizer	PROPN
cana-5932	278	6	and	and	CCONJ
cana-5932	278	7	a.	a.	NOUN
cana-5932	278	8	sklar	sklar	PROPN
cana-5932	278	9	.	.	PUNCT
cana-5932	279	1	statistical	statistical	ADJ
cana-5932	279	2	metric	metric	ADJ
cana-5932	279	3	spaces	space	NOUN
cana-5932	279	4	,	,	PUNCT
cana-5932	279	5	.	.	PUNCT
cana-5932	280	1	pacific	pacific	PROPN
cana-5932	280	2	j.	j.	PROPN
cana-5932	280	3	math	math	PROPN
cana-5932	280	4	.	.	PUNCT
cana-5932	280	5	,	,	PUNCT
cana-5932	280	6	10(1):313–334	10(1):313–334	PROPN
cana-5932	280	7	,	,	PUNCT
cana-5932	280	8	1960	1960	NUM
cana-5932	280	9	.	.	PUNCT