id	sid	tid	token	lemma	pos
iajs-3828	1	1	376	376	NUM
iajs-3828	1	2	©	©	ADP
iajs-3828	1	3	2025	2025	NUM
iajs-3828	1	4	the	the	DET
iajs-3828	1	5	author(s	author(s	NOUN
iajs-3828	1	6	)	)	PUNCT
iajs-3828	1	7	.	.	PUNCT
iajs-3828	2	1	published	publish	VERB
iajs-3828	2	2	by	by	ADP
iajs-3828	2	3	college	college	NOUN
iajs-3828	2	4	of	of	ADP
iajs-3828	2	5	education	education	NOUN
iajs-3828	2	6	for	for	ADP
iajs-3828	2	7	pure	pure	ADJ
iajs-3828	2	8	science	science	NOUN
iajs-3828	2	9	(	(	PUNCT
iajs-3828	2	10	ibn	ibn	PROPN
iajs-3828	2	11	al	al	PROPN
iajs-3828	2	12	-	-	PUNCT
iajs-3828	2	13	haitham	haitham	PROPN
iajs-3828	2	14	)	)	PUNCT
iajs-3828	2	15	,	,	PUNCT
iajs-3828	2	16	university	university	NOUN
iajs-3828	2	17	of	of	ADP
iajs-3828	2	18	baghdad	baghdad	PROPN
iajs-3828	2	19	.	.	PUNCT
iajs-3828	3	1	this	this	PRON
iajs-3828	3	2	is	be	AUX
iajs-3828	3	3	an	an	DET
iajs-3828	3	4	open	open	ADJ
iajs-3828	3	5	-	-	PUNCT
iajs-3828	3	6	access	access	NOUN
iajs-3828	3	7	article	article	NOUN
iajs-3828	3	8	distributed	distribute	VERB
iajs-3828	3	9	under	under	ADP
iajs-3828	3	10	the	the	DET
iajs-3828	3	11	terms	term	NOUN
iajs-3828	3	12	of	of	ADP
iajs-3828	3	13	the	the	DET
iajs-3828	3	14	creative	creative	ADJ
iajs-3828	3	15	commons	common	NOUN
iajs-3828	3	16	attribution	attribution	NOUN
iajs-3828	3	17	4.0	4.0	NUM
iajs-3828	3	18	international	international	ADJ
iajs-3828	3	19	license	license	NOUN
iajs-3828	3	20	fixed	fix	VERB
iajs-3828	3	21	point	point	NOUN
iajs-3828	3	22	results	result	NOUN
iajs-3828	3	23	and	and	CCONJ
iajs-3828	3	24	application	application	NOUN
iajs-3828	3	25	of	of	ADP
iajs-3828	3	26	cyclic	cyclic	ADJ
iajs-3828	3	27	contractive	contractive	ADJ
iajs-3828	3	28	maps	map	NOUN
iajs-3828	3	29	in	in	ADP
iajs-3828	3	30	bmetric	bmetric	ADJ
iajs-3828	3	31	spaces	space	NOUN
iajs-3828	3	32	abbas	abbas	PROPN
iajs-3828	3	33	karim	karim	PROPN
iajs-3828	3	34	nahi1	nahi1	PROPN
iajs-3828	3	35	*	*	PUNCT
iajs-3828	3	36	and	and	CCONJ
iajs-3828	3	37	salwa	salwa	PROPN
iajs-3828	3	38	salman	salman	PROPN
iajs-3828	3	39	abed2	abed2	PROPN
iajs-3828	4	1	1,2department	1,2department	NUM
iajs-3828	4	2	of	of	ADP
iajs-3828	4	3	mathematics	mathematic	NOUN
iajs-3828	4	4	,	,	PUNCT
iajs-3828	4	5	college	college	NOUN
iajs-3828	4	6	of	of	ADP
iajs-3828	4	7	education	education	NOUN
iajs-3828	4	8	for	for	ADP
iajs-3828	4	9	pure	pure	ADJ
iajs-3828	4	10	science	science	NOUN
iajs-3828	4	11	(	(	PUNCT
iajs-3828	4	12	ibn	ibn	PROPN
iajs-3828	4	13	al	al	PROPN
iajs-3828	4	14	-	-	PUNCT
iajs-3828	4	15	haitham	haitham	PROPN
iajs-3828	4	16	)	)	PUNCT
iajs-3828	4	17	,	,	PUNCT
iajs-3828	4	18	university	university	NOUN
iajs-3828	4	19	of	of	ADP
iajs-3828	4	20	baghdad	baghdad	PROPN
iajs-3828	4	21	,	,	PUNCT
iajs-3828	4	22	baghdad	baghdad	PROPN
iajs-3828	4	23	,	,	PUNCT
iajs-3828	4	24	iraq	iraq	PROPN
iajs-3828	4	25	.	.	PUNCT
iajs-3828	5	1	*	*	PUNCT
iajs-3828	5	2	corresponding	correspond	VERB
iajs-3828	5	3	author	author	NOUN
iajs-3828	5	4	.	.	PUNCT
iajs-3828	6	1	received:10	received:10	PROPN
iajs-3828	7	1	november	november	PROPN
iajs-3828	7	2	2023	2023	NUM
iajs-3828	7	3	accepted:16	accepted:16	X
iajs-3828	7	4	january	january	PROPN
iajs-3828	7	5	2024	2024	NUM
iajs-3828	7	6	published	publish	VERB
iajs-3828	7	7	:	:	PUNCT
iajs-3828	7	8	20	20	NUM
iajs-3828	7	9	april	april	PROPN
iajs-3828	7	10	2025	2025	NUM
iajs-3828	7	11	abstract	abstract	ADJ
iajs-3828	7	12	one	one	NUM
iajs-3828	7	13	of	of	ADP
iajs-3828	7	14	the	the	DET
iajs-3828	7	15	generalizations	generalization	NOUN
iajs-3828	7	16	of	of	ADP
iajs-3828	7	17	the	the	DET
iajs-3828	7	18	usual	usual	ADJ
iajs-3828	7	19	metric	metric	ADJ
iajs-3828	7	20	function	function	NOUN
iajs-3828	7	21	is	be	AUX
iajs-3828	7	22	the	the	DET
iajs-3828	7	23	b	b	NOUN
iajs-3828	7	24	-	-	PUNCT
iajs-3828	7	25	metric	metric	ADJ
iajs-3828	7	26	,	,	PUNCT
iajs-3828	7	27	which	which	PRON
iajs-3828	7	28	provides	provide	VERB
iajs-3828	7	29	researchers	researcher	NOUN
iajs-3828	7	30	with	with	ADP
iajs-3828	7	31	a	a	DET
iajs-3828	7	32	broader	broad	ADJ
iajs-3828	7	33	field	field	NOUN
iajs-3828	7	34	for	for	ADP
iajs-3828	7	35	deriving	derive	VERB
iajs-3828	7	36	numerous	numerous	ADJ
iajs-3828	7	37	results	result	NOUN
iajs-3828	7	38	and	and	CCONJ
iajs-3828	7	39	applications	application	NOUN
iajs-3828	7	40	related	relate	VERB
iajs-3828	7	41	to	to	ADP
iajs-3828	7	42	fixed	fix	VERB
iajs-3828	7	43	point	point	NOUN
iajs-3828	7	44	theory	theory	NOUN
iajs-3828	7	45	.	.	PUNCT
iajs-3828	8	1	the	the	DET
iajs-3828	8	2	aim	aim	NOUN
iajs-3828	8	3	of	of	ADP
iajs-3828	8	4	this	this	DET
iajs-3828	8	5	paper	paper	NOUN
iajs-3828	8	6	is	be	AUX
iajs-3828	8	7	to	to	PART
iajs-3828	8	8	develop	develop	VERB
iajs-3828	8	9	three	three	NUM
iajs-3828	8	10	new	new	ADJ
iajs-3828	8	11	fixed	fix	VERB
iajs-3828	8	12	point	point	NOUN
iajs-3828	8	13	principles	principle	NOUN
iajs-3828	8	14	in	in	ADP
iajs-3828	8	15	the	the	DET
iajs-3828	8	16	complete	complete	ADJ
iajs-3828	8	17	b	b	X
iajs-3828	8	18	-	-	PUNCT
iajs-3828	8	19	metric	metric	ADJ
iajs-3828	8	20	space	space	NOUN
iajs-3828	8	21	(	(	PUNCT
iajs-3828	8	22	£	£	NOUN
iajs-3828	8	23	,	,	PUNCT
iajs-3828	8	24	𝜌	𝜌	ADP
iajs-3828	8	25	)	)	PUNCT
iajs-3828	8	26	when	when	SCONJ
iajs-3828	8	27	𝜌	𝜌	PRON
iajs-3828	8	28	is	be	AUX
iajs-3828	8	29	a	a	DET
iajs-3828	8	30	continuous	continuous	ADJ
iajs-3828	8	31	function	function	NOUN
iajs-3828	8	32	in	in	ADP
iajs-3828	8	33	two	two	NUM
iajs-3828	8	34	variables	variable	NOUN
iajs-3828	8	35	.	.	PUNCT
iajs-3828	9	1	here	here	ADV
iajs-3828	9	2	there	there	PRON
iajs-3828	9	3	are	be	VERB
iajs-3828	9	4	three	three	NUM
iajs-3828	9	5	directions	direction	NOUN
iajs-3828	9	6	to	to	PART
iajs-3828	9	7	prove	prove	VERB
iajs-3828	9	8	the	the	DET
iajs-3828	9	9	existence	existence	NOUN
iajs-3828	9	10	and	and	CCONJ
iajs-3828	9	11	uniqueness	uniqueness	NOUN
iajs-3828	9	12	of	of	ADP
iajs-3828	9	13	fixed	fix	VERB
iajs-3828	9	14	points	point	NOUN
iajs-3828	9	15	.	.	PUNCT
iajs-3828	10	1	first	first	ADV
iajs-3828	10	2	,	,	PUNCT
iajs-3828	10	3	we	we	PRON
iajs-3828	10	4	derive	derive	VERB
iajs-3828	10	5	a	a	DET
iajs-3828	10	6	result	result	NOUN
iajs-3828	10	7	in	in	ADP
iajs-3828	10	8	terms	term	NOUN
iajs-3828	10	9	of	of	ADP
iajs-3828	10	10	branciari	branciari	NOUN
iajs-3828	10	11	’s	’s	PART
iajs-3828	10	12	theorem	theorem	NOUN
iajs-3828	10	13	by	by	ADP
iajs-3828	10	14	combining	combine	VERB
iajs-3828	10	15	integral	integral	ADJ
iajs-3828	10	16	contractive	contractive	ADJ
iajs-3828	10	17	conditions	condition	NOUN
iajs-3828	10	18	with	with	ADP
iajs-3828	10	19	the	the	DET
iajs-3828	10	20	notion	notion	NOUN
iajs-3828	10	21	of	of	ADP
iajs-3828	10	22	a	a	DET
iajs-3828	10	23	cyclic	cyclic	ADJ
iajs-3828	10	24	map	map	NOUN
iajs-3828	10	25	.	.	PUNCT
iajs-3828	11	1	second	second	ADJ
iajs-3828	11	2	,	,	PUNCT
iajs-3828	11	3	we	we	PRON
iajs-3828	11	4	apply	apply	VERB
iajs-3828	11	5	the	the	DET
iajs-3828	11	6	notion	notion	NOUN
iajs-3828	11	7	of	of	ADP
iajs-3828	11	8	cyclic	cyclic	ADJ
iajs-3828	11	9	representation	representation	NOUN
iajs-3828	11	10	to	to	ADP
iajs-3828	11	11	maps	map	NOUN
iajs-3828	11	12	satisfying	satisfy	VERB
iajs-3828	11	13	general	general	ADJ
iajs-3828	11	14	weak	weak	ADJ
iajs-3828	11	15	conditions	condition	NOUN
iajs-3828	11	16	,	,	PUNCT
iajs-3828	11	17	including	include	VERB
iajs-3828	11	18	a	a	DET
iajs-3828	11	19	changing	change	VERB
iajs-3828	11	20	distance	distance	NOUN
iajs-3828	11	21	function	function	NOUN
iajs-3828	11	22	,	,	PUNCT
iajs-3828	11	23	to	to	PART
iajs-3828	11	24	simulate	simulate	VERB
iajs-3828	11	25	the	the	DET
iajs-3828	11	26	content	content	NOUN
iajs-3828	11	27	of	of	ADP
iajs-3828	11	28	boyd	boyd	PROPN
iajs-3828	11	29	and	and	CCONJ
iajs-3828	11	30	wong	wong	PROPN
iajs-3828	11	31	's	's	PART
iajs-3828	11	32	theorem	theorem	NOUN
iajs-3828	11	33	.	.	PUNCT
iajs-3828	12	1	using	use	VERB
iajs-3828	12	2	this	this	DET
iajs-3828	12	3	result	result	NOUN
iajs-3828	12	4	,	,	PUNCT
iajs-3828	12	5	an	an	DET
iajs-3828	12	6	application	application	NOUN
iajs-3828	12	7	to	to	ADP
iajs-3828	12	8	the	the	DET
iajs-3828	12	9	existence	existence	NOUN
iajs-3828	12	10	and	and	CCONJ
iajs-3828	12	11	uniqueness	uniqueness	NOUN
iajs-3828	12	12	of	of	ADP
iajs-3828	12	13	the	the	DET
iajs-3828	12	14	solution	solution	NOUN
iajs-3828	12	15	of	of	ADP
iajs-3828	12	16	an	an	DET
iajs-3828	12	17	integral	integral	ADJ
iajs-3828	12	18	equation	equation	NOUN
iajs-3828	12	19	is	be	AUX
iajs-3828	12	20	given	give	VERB
iajs-3828	12	21	.	.	PUNCT
iajs-3828	13	1	finally	finally	ADV
iajs-3828	13	2	,	,	PUNCT
iajs-3828	13	3	an	an	DET
iajs-3828	13	4	implicit	implicit	ADJ
iajs-3828	13	5	relation	relation	NOUN
iajs-3828	13	6	with	with	ADP
iajs-3828	13	7	a	a	DET
iajs-3828	13	8	changing	change	VERB
iajs-3828	13	9	distance	distance	NOUN
iajs-3828	13	10	function	function	NOUN
iajs-3828	13	11	is	be	AUX
iajs-3828	13	12	used	use	VERB
iajs-3828	13	13	to	to	PART
iajs-3828	13	14	construct	construct	VERB
iajs-3828	13	15	a	a	DET
iajs-3828	13	16	cyclic	cyclic	ADJ
iajs-3828	13	17	contractive	contractive	ADJ
iajs-3828	13	18	map	map	NOUN
iajs-3828	13	19	.	.	PUNCT
iajs-3828	14	1	some	some	DET
iajs-3828	14	2	examples	example	NOUN
iajs-3828	14	3	are	be	AUX
iajs-3828	14	4	also	also	ADV
iajs-3828	14	5	presented	present	VERB
iajs-3828	14	6	to	to	PART
iajs-3828	14	7	analyze	analyze	VERB
iajs-3828	14	8	and	and	CCONJ
iajs-3828	14	9	illustrate	illustrate	VERB
iajs-3828	14	10	the	the	DET
iajs-3828	14	11	main	main	ADJ
iajs-3828	14	12	results	result	NOUN
iajs-3828	14	13	.	.	PUNCT
iajs-3828	15	1	keywords	keyword	NOUN
iajs-3828	15	2	:	:	PUNCT
iajs-3828	15	3	alternating	alternate	VERB
iajs-3828	15	4	distance	distance	NOUN
iajs-3828	15	5	functions	function	NOUN
iajs-3828	15	6	,	,	PUNCT
iajs-3828	15	7	complete	complete	ADJ
iajs-3828	15	8	b	b	X
iajs-3828	15	9	-	-	ADJ
iajs-3828	15	10	metric	metric	ADJ
iajs-3828	15	11	spaces	space	NOUN
iajs-3828	15	12	,	,	PUNCT
iajs-3828	15	13	contractive	contractive	ADJ
iajs-3828	15	14	conditions	condition	NOUN
iajs-3828	15	15	,	,	PUNCT
iajs-3828	15	16	cyclic	cyclic	ADJ
iajs-3828	15	17	representation	representation	NOUN
iajs-3828	15	18	,	,	PUNCT
iajs-3828	15	19	fixed	fix	VERB
iajs-3828	15	20	points	point	NOUN
iajs-3828	15	21	.	.	PUNCT
iajs-3828	16	1	1	1	X
iajs-3828	16	2	.	.	X
iajs-3828	16	3	introduction	introduction	NOUN
iajs-3828	16	4	in	in	ADP
iajs-3828	16	5	2003	2003	NUM
iajs-3828	16	6	,	,	PUNCT
iajs-3828	16	7	cyclic	cyclic	ADJ
iajs-3828	16	8	contraction	contraction	NOUN
iajs-3828	16	9	was	be	AUX
iajs-3828	16	10	first	first	ADV
iajs-3828	16	11	introduced	introduce	VERB
iajs-3828	16	12	by	by	ADP
iajs-3828	16	13	kirk	kirk	PROPN
iajs-3828	16	14	et	et	PROPN
iajs-3828	16	15	al	al	PROPN
iajs-3828	16	16	.	.	PROPN
iajs-3828	17	1	(	(	PUNCT
iajs-3828	17	2	1	1	NUM
iajs-3828	17	3	)	)	PUNCT
iajs-3828	17	4	,	,	PUNCT
iajs-3828	17	5	who	who	PRON
iajs-3828	17	6	introduced	introduce	VERB
iajs-3828	17	7	results	result	NOUN
iajs-3828	17	8	dealing	deal	VERB
iajs-3828	17	9	with	with	ADP
iajs-3828	17	10	mappings	mapping	NOUN
iajs-3828	17	11	of	of	ADP
iajs-3828	17	12	the	the	DET
iajs-3828	17	13	type	type	NOUN
iajs-3828	18	1	𝑓	𝑓	PRON
iajs-3828	18	2	:	:	PUNCT
iajs-3828	18	3	£	£	SYM
iajs-3828	18	4	𝑖	𝑖	NOUN
iajs-3828	18	5	→	→	SYM
iajs-3828	18	6	£	£	SYM
iajs-3828	18	7	𝑖+1	𝑖+1	NUM
iajs-3828	18	8	,	,	PUNCT
iajs-3828	18	9	𝑖	𝑖	SYM
iajs-3828	18	10	=	=	SYM
iajs-3828	18	11	1	1	NUM
iajs-3828	18	12	,	,	PUNCT
iajs-3828	18	13	2	2	NUM
iajs-3828	18	14	,	,	PUNCT
iajs-3828	18	15	·	·	PUNCT
iajs-3828	18	16	·	·	PUNCT
iajs-3828	18	17	·	·	PUNCT
iajs-3828	18	18	,	,	PUNCT
iajs-3828	18	19	𝑝	𝑝	NOUN
iajs-3828	18	20	+	+	NOUN
iajs-3828	18	21	1	1	NUM
iajs-3828	18	22	,	,	PUNCT
iajs-3828	18	23	with	with	ADP
iajs-3828	18	24	£	£	SYM
iajs-3828	18	25	𝑖	𝑖	NOUN
iajs-3828	18	26	=	=	PUNCT
iajs-3828	18	27	£	£	SYM
iajs-3828	18	28	𝑖+1	𝑖+1	NUM
iajs-3828	18	29	,	,	PUNCT
iajs-3828	18	30	where	where	SCONJ
iajs-3828	18	31	the	the	DET
iajs-3828	18	32	contractive	contractive	ADJ
iajs-3828	18	33	hypotheses	hypothesis	NOUN
iajs-3828	18	34	are	be	AUX
iajs-3828	18	35	restricted	restrict	VERB
iajs-3828	18	36	to	to	ADP
iajs-3828	18	37	pairs	pair	NOUN
iajs-3828	18	38	(	(	PUNCT
iajs-3828	18	39	𝑎	𝑎	X
iajs-3828	18	40	,	,	PUNCT
iajs-3828	18	41	𝑏	𝑏	NOUN
iajs-3828	18	42	)	)	PUNCT
iajs-3828	18	43	∈	∈	NOUN
iajs-3828	18	44	£	£	SYM
iajs-3828	18	45	𝑖	𝑖	NOUN
iajs-3828	18	46	×	×	NOUN
iajs-3828	18	47	£	£	SYM
iajs-3828	18	48	𝑖+1	𝑖+1	NUM
iajs-3828	18	49	.	.	PUNCT
iajs-3828	18	50	extensions	extension	NOUN
iajs-3828	18	51	of	of	ADP
iajs-3828	18	52	the	the	DET
iajs-3828	18	53	banach	banach	NOUN
iajs-3828	18	54	’s	’s	NOUN
iajs-3828	18	55	theorem	theorem	NOUN
iajs-3828	18	56	and	and	CCONJ
iajs-3828	18	57	an	an	DET
iajs-3828	18	58	extension	extension	NOUN
iajs-3828	18	59	of	of	ADP
iajs-3828	18	60	the	the	DET
iajs-3828	18	61	caristi	caristi	PROPN
iajs-3828	18	62	theorem	theorem	PROPN
iajs-3828	18	63	were	be	AUX
iajs-3828	18	64	proved	prove	VERB
iajs-3828	18	65	.	.	PUNCT
iajs-3828	19	1	in	in	ADP
iajs-3828	19	2	addition	addition	NOUN
iajs-3828	19	3	,	,	PUNCT
iajs-3828	19	4	results	result	NOUN
iajs-3828	19	5	related	relate	VERB
iajs-3828	19	6	to	to	ADP
iajs-3828	19	7	non	non	ADJ
iajs-3828	19	8	-	-	ADJ
iajs-3828	19	9	expansive	expansive	ADJ
iajs-3828	19	10	mappings	mapping	NOUN
iajs-3828	19	11	in	in	ADP
iajs-3828	19	12	a	a	DET
iajs-3828	19	13	banach	banach	NOUN
iajs-3828	19	14	space	space	NOUN
iajs-3828	19	15	were	be	AUX
iajs-3828	19	16	included	include	VERB
iajs-3828	19	17	.	.	PUNCT
iajs-3828	20	1	several	several	ADJ
iajs-3828	20	2	authors	author	NOUN
iajs-3828	20	3	have	have	AUX
iajs-3828	20	4	contributed	contribute	VERB
iajs-3828	20	5	to	to	PART
iajs-3828	20	6	research	research	VERB
iajs-3828	20	7	on	on	ADP
iajs-3828	20	8	fixed	fix	VERB
iajs-3828	20	9	points	point	NOUN
iajs-3828	20	10	in	in	ADP
iajs-3828	20	11	various	various	ADJ
iajs-3828	20	12	cases	case	NOUN
iajs-3828	20	13	for	for	ADP
iajs-3828	20	14	cyclic	cyclic	ADJ
iajs-3828	20	15	contractions	contraction	NOUN
iajs-3828	20	16	(	(	PUNCT
iajs-3828	20	17	29	29	NUM
iajs-3828	20	18	)	)	PUNCT
iajs-3828	20	19	.	.	PUNCT
iajs-3828	21	1	backhtin	backhtin	NOUN
iajs-3828	21	2	(	(	PUNCT
iajs-3828	21	3	10	10	NUM
iajs-3828	21	4	)	)	PUNCT
iajs-3828	21	5	presented	present	VERB
iajs-3828	21	6	a	a	DET
iajs-3828	21	7	definition	definition	NOUN
iajs-3828	21	8	of	of	ADP
iajs-3828	21	9	b	b	NOUN
iajs-3828	21	10	-	-	PUNCT
iajs-3828	21	11	metric	metric	ADJ
iajs-3828	21	12	by	by	ADP
iajs-3828	21	13	replacing	replace	VERB
iajs-3828	21	14	the	the	DET
iajs-3828	21	15	triangle	triangle	NOUN
iajs-3828	21	16	inequality	inequality	NOUN
iajs-3828	21	17	as	as	ADP
iajs-3828	21	18	the	the	DET
iajs-3828	21	19	following	follow	VERB
iajs-3828	21	20	definition1.1	definition1.1	NOUN
iajs-3828	21	21	:	:	PUNCT
iajs-3828	21	22	let	let	VERB
iajs-3828	21	23	£	£	PART
iajs-3828	21	24	be	be	AUX
iajs-3828	21	25	a	a	DET
iajs-3828	21	26	nonempty	nonempty	ADV
iajs-3828	21	27	set	set	VERB
iajs-3828	21	28	and	and	CCONJ
iajs-3828	21	29	𝑞	𝑞	PRON
iajs-3828	21	30	≥	≥	PROPN
iajs-3828	21	31	1	1	NUM
iajs-3828	21	32	.	.	PUNCT
iajs-3828	22	1	a	a	DET
iajs-3828	22	2	function	function	NOUN
iajs-3828	22	3	𝜌	𝜌	ADP
iajs-3828	22	4	∶	∶	NOUN
iajs-3828	22	5	£	£	SYM
iajs-3828	22	6	×	×	NOUN
iajs-3828	22	7	£	£	NOUN
iajs-3828	22	8	→	→	PUNCT
iajs-3828	22	9	𝑅+	𝑅+	PROPN
iajs-3828	22	10	is	be	AUX
iajs-3828	22	11	said	say	VERB
iajs-3828	22	12	to	to	PART
iajs-3828	22	13	be	be	AUX
iajs-3828	22	14	a	a	DET
iajs-3828	22	15	b	b	NOUN
iajs-3828	22	16	-	-	NOUN
iajs-3828	22	17	metric	metric	ADJ
iajs-3828	22	18	on	on	ADP
iajs-3828	22	19	£	£	SYM
iajs-3828	22	20	if	if	SCONJ
iajs-3828	22	21	(	(	PUNCT
iajs-3828	22	22	10	10	NUM
iajs-3828	22	23	):	):	SYM
iajs-3828	22	24	1	1	NUM
iajs-3828	22	25	.	.	X
iajs-3828	23	1	𝜌(𝑎	𝜌(𝑎	NUM
iajs-3828	23	2	,	,	PUNCT
iajs-3828	23	3	𝑏	𝑏	NOUN
iajs-3828	23	4	)	)	PUNCT
iajs-3828	23	5	=	=	SYM
iajs-3828	23	6	0	0	PUNCT
iajs-3828	24	1	if	if	SCONJ
iajs-3828	24	2	and	and	CCONJ
iajs-3828	24	3	only	only	ADV
iajs-3828	24	4	if	if	SCONJ
iajs-3828	24	5	𝑎	𝑎	PROPN
iajs-3828	24	6	=	=	SYM
iajs-3828	24	7	𝑏	𝑏	NOUN
iajs-3828	24	8	;	;	PUNCT
iajs-3828	24	9	doi.org/10.30526/38.2.3828	doi.org/10.30526/38.2.3828	NOUN
iajs-3828	24	10	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3828	24	11	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3828	24	12	mailto	mailto	PROPN
iajs-3828	24	13	:	:	PUNCT
iajs-3828	24	14	https://orcid.org/0009	https://orcid.org/0009	NOUN
iajs-3828	24	15	-	-	SYM
iajs-3828	24	16	0006	0006	NUM
iajs-3828	24	17	-	-	SYM
iajs-3828	24	18	3158	3158	NUM
iajs-3828	24	19	-	-	PUNCT
iajs-3828	24	20	1348	1348	NUM
iajs-3828	24	21	mailto	mailto	NOUN
iajs-3828	24	22	:	:	PUNCT
iajs-3828	24	23	https://orcid.org/0000	https://orcid.org/0000	X
iajs-3828	24	24	-	-	PUNCT
iajs-3828	24	25	0002	0002	NUM
iajs-3828	24	26	-	-	PUNCT
iajs-3828	24	27	0581	0581	NUM
iajs-3828	24	28	-	-	PUNCT
iajs-3828	24	29	253x	253x	NOUN
iajs-3828	24	30	mailto:salwa.s.a@ihcoedu.uobaghdad.edu.iq	mailto:salwa.s.a@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-3828	24	31	mailto:abbas.nahi2103m@ihcoedu.uobaghdad.edu.iq	mailto:abbas.nahi2103m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-3828	24	32	ihjpas	ihjpa	NOUN
iajs-3828	24	33	.	.	PUNCT
iajs-3828	25	1	2025,38(2	2025,38(2	PROPN
iajs-3828	25	2	)	)	PUNCT
iajs-3828	25	3	377	377	NUM
iajs-3828	25	4	2	2	NUM
iajs-3828	25	5	.	.	PUNCT
iajs-3828	26	1	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	26	2	,	,	PUNCT
iajs-3828	26	3	𝑏	𝑏	NOUN
iajs-3828	26	4	)	)	PUNCT
iajs-3828	26	5	=	=	SYM
iajs-3828	26	6	𝜌(𝑏	𝜌(𝑏	NOUN
iajs-3828	26	7	,	,	PUNCT
iajs-3828	26	8	𝑎	𝑎	NOUN
iajs-3828	26	9	)	)	PUNCT
iajs-3828	26	10	for	for	ADP
iajs-3828	26	11	all	all	DET
iajs-3828	26	12	𝑎	𝑎	NOUN
iajs-3828	26	13	,	,	PUNCT
iajs-3828	26	14	𝑏	𝑏	PROPN
iajs-3828	26	15	∈	∈	PROPN
iajs-3828	26	16	£	£	NOUN
iajs-3828	26	17	;	;	PUNCT
iajs-3828	26	18	3	3	X
iajs-3828	26	19	.	.	X
iajs-3828	27	1	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	27	2	,	,	PUNCT
iajs-3828	27	3	𝑏	𝑏	NOUN
iajs-3828	27	4	)	)	PUNCT
iajs-3828	27	5	≤	≤	NOUN
iajs-3828	28	1	𝑞(𝜌(𝑎	𝑞(𝜌(𝑎	ADJ
iajs-3828	28	2	,	,	PUNCT
iajs-3828	28	3	𝑐	𝑐	NOUN
iajs-3828	28	4	)	)	PUNCT
iajs-3828	28	5	+	+	NUM
iajs-3828	28	6	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	28	7	,	,	PUNCT
iajs-3828	28	8	𝑏	𝑏	NOUN
iajs-3828	28	9	)	)	PUNCT
iajs-3828	28	10	)	)	PUNCT
iajs-3828	28	11	for	for	ADP
iajs-3828	28	12	all	all	DET
iajs-3828	28	13	𝑎	𝑎	PROPN
iajs-3828	28	14	,	,	PUNCT
iajs-3828	28	15	𝑏	𝑏	NOUN
iajs-3828	28	16	,	,	PUNCT
iajs-3828	28	17	𝑐	𝑐	PROPN
iajs-3828	28	18	∈	∈	PROPN
iajs-3828	28	19	£	£	PROPN
iajs-3828	28	20	.	.	PUNCT
iajs-3828	29	1	the	the	DET
iajs-3828	29	2	pair	pair	NOUN
iajs-3828	29	3	(	(	PUNCT
iajs-3828	29	4	£	£	NOUN
iajs-3828	29	5	,	,	PUNCT
iajs-3828	29	6	𝜌	𝜌	X
iajs-3828	29	7	)	)	PUNCT
iajs-3828	29	8	is	be	AUX
iajs-3828	29	9	called	call	VERB
iajs-3828	29	10	a	a	DET
iajs-3828	29	11	b	b	NOUN
iajs-3828	29	12	-	-	PUNCT
iajs-3828	29	13	metric	metric	ADJ
iajs-3828	29	14	space	space	NOUN
iajs-3828	29	15	.	.	PUNCT
iajs-3828	30	1	definition	definition	NOUN
iajs-3828	30	2	1.2	1.2	NUM
iajs-3828	30	3	:	:	PUNCT
iajs-3828	30	4	let	let	VERB
iajs-3828	30	5	(	(	PUNCT
iajs-3828	30	6	£	£	NOUN
iajs-3828	30	7	,	,	PUNCT
iajs-3828	30	8	𝜌	𝜌	AUX
iajs-3828	30	9	)	)	PUNCT
iajs-3828	30	10	be	be	VERB
iajs-3828	30	11	a	a	DET
iajs-3828	30	12	b	b	NOUN
iajs-3828	30	13	-	-	PUNCT
iajs-3828	30	14	metric	metric	ADJ
iajs-3828	30	15	space	space	NOUN
iajs-3828	30	16	,	,	PUNCT
iajs-3828	30	17	𝑎	𝑎	PROPN
iajs-3828	30	18	∈	∈	PROPN
iajs-3828	30	19	£	£	SYM
iajs-3828	30	20	and	and	CCONJ
iajs-3828	30	21	(	(	PUNCT
iajs-3828	30	22	𝑎𝑛	𝑎𝑛	NOUN
iajs-3828	30	23	)	)	PUNCT
iajs-3828	30	24	be	be	VERB
iajs-3828	30	25	a	a	DET
iajs-3828	30	26	sequence	sequence	NOUN
iajs-3828	30	27	in	in	ADP
iajs-3828	30	28	£	£	PROPN
iajs-3828	30	29	.	.	PUNCT
iajs-3828	31	1	then	then	ADV
iajs-3828	31	2	1	1	X
iajs-3828	31	3	.	.	PUNCT
iajs-3828	31	4	(	(	PUNCT
iajs-3828	31	5	𝑎𝑛	𝑎𝑛	NOUN
iajs-3828	31	6	)	)	PUNCT
iajs-3828	31	7	converges	converge	VERB
iajs-3828	31	8	to	to	ADP
iajs-3828	31	9	𝑎	𝑎	NOUN
iajs-3828	31	10	if	if	NOUN
iajs-3828	32	1	and	and	CCONJ
iajs-3828	32	2	only	only	ADV
iajs-3828	32	3	if	if	SCONJ
iajs-3828	32	4	lim	lim	PROPN
iajs-3828	32	5	𝑛→∞	𝑛→∞	NUM
iajs-3828	32	6	𝜌(𝑎𝑛	𝜌(𝑎𝑛	PROPN
iajs-3828	32	7	,	,	PUNCT
iajs-3828	32	8	𝑎	𝑎	NOUN
iajs-3828	32	9	)	)	PUNCT
iajs-3828	32	10	=	=	SYM
iajs-3828	32	11	0	0	X
iajs-3828	32	12	.	.	PUNCT
iajs-3828	33	1	we	we	PRON
iajs-3828	33	2	denote	denote	VERB
iajs-3828	33	3	this	this	PRON
iajs-3828	33	4	by	by	ADP
iajs-3828	33	5	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
iajs-3828	33	6	𝑛→∞	𝑛→∞	NUM
iajs-3828	33	7	𝑎𝑛	𝑎𝑛	NOUN
iajs-3828	33	8	=	=	SYM
iajs-3828	33	9	𝑎	𝑎	PROPN
iajs-3828	33	10	or	or	CCONJ
iajs-3828	33	11	𝑎𝑛	𝑎𝑛	PRON
iajs-3828	33	12	→	→	SYM
iajs-3828	33	13	𝑎	𝑎	X
iajs-3828	33	14	(	(	PUNCT
iajs-3828	33	15	as	as	ADP
iajs-3828	33	16	𝑛	𝑛	PROPN
iajs-3828	33	17	→	→	SYM
iajs-3828	33	18	∞	∞	NUM
iajs-3828	33	19	)	)	PUNCT
iajs-3828	33	20	.	.	PUNCT
iajs-3828	34	1	2	2	X
iajs-3828	34	2	.	.	X
iajs-3828	34	3	(	(	PUNCT
iajs-3828	34	4	𝑎𝑛	𝑎𝑛	PROPN
iajs-3828	34	5	)	)	PUNCT
iajs-3828	34	6	is	be	AUX
iajs-3828	34	7	cauchy	cauchy	ADJ
iajs-3828	34	8	if	if	SCONJ
iajs-3828	34	9	and	and	CCONJ
iajs-3828	34	10	only	only	ADV
iajs-3828	34	11	if	if	SCONJ
iajs-3828	34	12	lim	lim	PROPN
iajs-3828	34	13	𝑛,𝑚→∞	𝑛,𝑚→∞	PRON
iajs-3828	34	14	𝜌(𝑎𝑛	𝜌(𝑎𝑛	PROPN
iajs-3828	34	15	,	,	PUNCT
iajs-3828	34	16	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	34	17	)	)	PUNCT
iajs-3828	34	18	=	=	SYM
iajs-3828	35	1	0	0	NUM
iajs-3828	35	2	.	.	NOUN
iajs-3828	36	1	3	3	X
iajs-3828	36	2	.	.	X
iajs-3828	36	3	(	(	PUNCT
iajs-3828	36	4	£	£	NOUN
iajs-3828	36	5	,	,	PUNCT
iajs-3828	36	6	𝜌	𝜌	X
iajs-3828	36	7	)	)	PUNCT
iajs-3828	36	8	is	be	AUX
iajs-3828	36	9	complete	complete	ADJ
iajs-3828	36	10	if	if	SCONJ
iajs-3828	36	11	and	and	CCONJ
iajs-3828	36	12	only	only	ADV
iajs-3828	36	13	if	if	SCONJ
iajs-3828	36	14	every	every	DET
iajs-3828	36	15	cauchy	cauchy	ADJ
iajs-3828	36	16	sequence	sequence	NOUN
iajs-3828	36	17	in	in	ADP
iajs-3828	36	18	£	£	PROPN
iajs-3828	36	19	is	be	AUX
iajs-3828	36	20	convergent	convergent	NOUN
iajs-3828	36	21	.	.	PUNCT
iajs-3828	37	1	remark	remark	VERB
iajs-3828	37	2	1.3	1.3	NUM
iajs-3828	37	3	:	:	PUNCT
iajs-3828	37	4	in	in	ADP
iajs-3828	37	5	a	a	DET
iajs-3828	37	6	b	b	NOUN
iajs-3828	37	7	-	-	PUNCT
iajs-3828	37	8	metric	metric	ADJ
iajs-3828	37	9	space	space	NOUN
iajs-3828	37	10	(	(	PUNCT
iajs-3828	37	11	£	£	NOUN
iajs-3828	37	12	,	,	PUNCT
iajs-3828	37	13	𝜌	𝜌	ADP
iajs-3828	37	14	)	)	PUNCT
iajs-3828	37	15	,	,	PUNCT
iajs-3828	37	16	the	the	DET
iajs-3828	37	17	following	follow	VERB
iajs-3828	37	18	assertions	assertion	NOUN
iajs-3828	37	19	hold(11,12	hold(11,12	NOUN
iajs-3828	37	20	):	):	PUNCT
iajs-3828	37	21	1	1	X
iajs-3828	37	22	.	.	X
iajs-3828	38	1	a	a	DET
iajs-3828	38	2	convergent	convergent	NOUN
iajs-3828	38	3	sequence	sequence	NOUN
iajs-3828	38	4	has	have	VERB
iajs-3828	38	5	a	a	DET
iajs-3828	38	6	unique	unique	ADJ
iajs-3828	38	7	limit	limit	NOUN
iajs-3828	38	8	.	.	PUNCT
iajs-3828	39	1	2	2	X
iajs-3828	39	2	.	.	X
iajs-3828	39	3	each	each	DET
iajs-3828	39	4	convergent	convergent	NOUN
iajs-3828	39	5	sequence	sequence	NOUN
iajs-3828	39	6	is	be	AUX
iajs-3828	39	7	cauchy	cauchy	NOUN
iajs-3828	39	8	.	.	PUNCT
iajs-3828	40	1	3	3	X
iajs-3828	40	2	.	.	X
iajs-3828	40	3	in	in	ADP
iajs-3828	40	4	general	general	ADJ
iajs-3828	40	5	,	,	PUNCT
iajs-3828	40	6	a	a	DET
iajs-3828	40	7	b	b	X
iajs-3828	40	8	-	-	ADJ
iajs-3828	40	9	metric	metric	ADJ
iajs-3828	40	10	is	be	AUX
iajs-3828	40	11	not	not	PART
iajs-3828	40	12	continuous	continuous	ADJ
iajs-3828	40	13	.	.	PUNCT
iajs-3828	41	1	as	as	ADP
iajs-3828	41	2	in	in	ADP
iajs-3828	41	3	the	the	DET
iajs-3828	41	4	usual	usual	ADJ
iajs-3828	41	5	metric	metric	ADJ
iajs-3828	41	6	space	space	NOUN
iajs-3828	41	7	)	)	PUNCT
iajs-3828	41	8	1	1	NUM
iajs-3828	41	9	(	(	PUNCT
iajs-3828	41	10	,	,	PUNCT
iajs-3828	41	11	we	we	PRON
iajs-3828	41	12	reform	reform	VERB
iajs-3828	41	13	the	the	DET
iajs-3828	41	14	following	follow	VERB
iajs-3828	41	15	definition	definition	NOUN
iajs-3828	41	16	:	:	PUNCT
iajs-3828	41	17	definition1.4	definition1.4	NOUN
iajs-3828	41	18	:	:	PUNCT
iajs-3828	41	19	let	let	VERB
iajs-3828	41	20	{	{	PUNCT
iajs-3828	41	21	£	£	SYM
iajs-3828	41	22	𝑖}𝑖=1	𝑖}𝑖=1	NUM
iajs-3828	41	23	𝑛	𝑛	PRON
iajs-3828	41	24	be	be	AUX
iajs-3828	41	25	a	a	DET
iajs-3828	41	26	nonempty	nonempty	ADJ
iajs-3828	41	27	subsets	subset	NOUN
iajs-3828	41	28	of	of	ADP
iajs-3828	41	29	a	a	DET
iajs-3828	41	30	b	b	NOUN
iajs-3828	41	31	-	-	PUNCT
iajs-3828	41	32	metric	metric	ADJ
iajs-3828	41	33	space	space	NOUN
iajs-3828	41	34	(	(	PUNCT
iajs-3828	41	35	£	£	NOUN
iajs-3828	41	36	,	,	PUNCT
iajs-3828	41	37	𝜌	𝜌	ADP
iajs-3828	41	38	)	)	PUNCT
iajs-3828	41	39	,	,	PUNCT
iajs-3828	42	1	£	£	NOUN
iajs-3828	42	2	=	=	SYM
iajs-3828	42	3	⋃	⋃	ADP
iajs-3828	42	4	£	£	SYM
iajs-3828	42	5	𝑖	𝑖	SYM
iajs-3828	42	6	𝑛	𝑛	PRON
iajs-3828	42	7	𝑖=1	𝑖=1	PROPN
iajs-3828	42	8	and	and	CCONJ
iajs-3828	42	9	𝑓	𝑓	X
iajs-3828	42	10	:	:	PUNCT
iajs-3828	42	11	£	£	NOUN
iajs-3828	42	12	→	→	SYM
iajs-3828	42	13	£	£	NOUN
iajs-3828	42	14	such	such	ADJ
iajs-3828	42	15	that	that	DET
iajs-3828	42	16	1	1	NUM
iajs-3828	42	17	)	)	PUNCT
iajs-3828	42	18	𝑓(£1	𝑓(£1	PROPN
iajs-3828	42	19	)	)	PUNCT
iajs-3828	42	20	⊂	⊂	PROPN
iajs-3828	42	21	£	£	SYM
iajs-3828	42	22	2	2	NUM
iajs-3828	42	23	,	,	PUNCT
iajs-3828	42	24	…	…	PUNCT
iajs-3828	42	25	,	,	PUNCT
iajs-3828	42	26	𝑓(£𝑛−1	𝑓(£𝑛−1	PROPN
iajs-3828	42	27	)	)	PUNCT
iajs-3828	42	28	⊂	⊂	PROPN
iajs-3828	42	29	£	£	SYM
iajs-3828	42	30	𝑛	𝑛	NOUN
iajs-3828	42	31	,	,	PUNCT
iajs-3828	42	32	𝑓(£𝑛	𝑓(£𝑛	NOUN
iajs-3828	42	33	)	)	PUNCT
iajs-3828	42	34	⊂	⊂	PUNCT
iajs-3828	43	1	£	£	SYM
iajs-3828	43	2	1	1	NUM
iajs-3828	43	3	for	for	ADP
iajs-3828	43	4	1	1	NUM
iajs-3828	43	5	≤	≤	NUM
iajs-3828	43	6	𝑖	𝑖	SYM
iajs-3828	43	7	≤	≤	NOUN
iajs-3828	43	8	𝑛	𝑛	NOUN
iajs-3828	43	9	;	;	PUNCT
iajs-3828	43	10	2	2	X
iajs-3828	43	11	)	)	PUNCT
iajs-3828	43	12	∃𝑘	∃𝑘	PROPN
iajs-3828	43	13	∈	∈	PROPN
iajs-3828	43	14	(	(	PUNCT
iajs-3828	43	15	0,1	0,1	NOUN
iajs-3828	43	16	)	)	PUNCT
iajs-3828	43	17	such	such	ADJ
iajs-3828	43	18	that	that	SCONJ
iajs-3828	43	19	𝜌(𝑓𝑎	𝜌(𝑓𝑎	NOUN
iajs-3828	43	20	,	,	PUNCT
iajs-3828	43	21	𝑓𝑏	𝑓𝑏	PROPN
iajs-3828	43	22	)	)	PUNCT
iajs-3828	43	23	≤	≤	NOUN
iajs-3828	43	24	𝑘	𝑘	PRON
iajs-3828	43	25	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	43	26	,	,	PUNCT
iajs-3828	43	27	𝑏	𝑏	NOUN
iajs-3828	43	28	)	)	PUNCT
iajs-3828	43	29	∀𝑎	∀𝑎	PROPN
iajs-3828	43	30	∈	∈	PROPN
iajs-3828	44	1	£	£	SYM
iajs-3828	44	2	𝑖	𝑖	NOUN
iajs-3828	44	3	,	,	PUNCT
iajs-3828	44	4	𝑏	𝑏	PROPN
iajs-3828	44	5	∈	∈	NOUN
iajs-3828	44	6	£	£	SYM
iajs-3828	44	7	𝑖+1	𝑖+1	NUM
iajs-3828	44	8	for	for	ADP
iajs-3828	44	9	1	1	NUM
iajs-3828	44	10	≤	≤	NUM
iajs-3828	44	11	𝑖	𝑖	SYM
iajs-3828	44	12	≤	≤	NOUN
iajs-3828	44	13	𝑛	𝑛	PRON
iajs-3828	45	1	then	then	ADV
iajs-3828	45	2	𝑓	𝑓	PRON
iajs-3828	45	3	is	be	AUX
iajs-3828	45	4	the	the	DET
iajs-3828	45	5	cyclic	cyclic	ADJ
iajs-3828	45	6	contraction	contraction	NOUN
iajs-3828	45	7	map	map	NOUN
iajs-3828	45	8	and	and	CCONJ
iajs-3828	45	9	£	£	NOUN
iajs-3828	45	10	is	be	AUX
iajs-3828	45	11	cyclic	cyclic	ADJ
iajs-3828	45	12	representation	representation	NOUN
iajs-3828	45	13	w.r.t	w.r.t	NOUN
iajs-3828	45	14	.	.	PUNCT
iajs-3828	46	1	,	,	PUNCT
iajs-3828	46	2	𝑓.	𝑓.	NOUN
iajs-3828	46	3	in	in	ADP
iajs-3828	46	4	the	the	DET
iajs-3828	46	5	field	field	NOUN
iajs-3828	46	6	of	of	ADP
iajs-3828	46	7	fixed	fix	VERB
iajs-3828	46	8	point	point	NOUN
iajs-3828	46	9	theory	theory	NOUN
iajs-3828	46	10	for	for	ADP
iajs-3828	46	11	cyclicity	cyclicity	NOUN
iajs-3828	46	12	,	,	PUNCT
iajs-3828	46	13	see	see	VERB
iajs-3828	46	14	(	(	PUNCT
iajs-3828	46	15	13	13	NUM
iajs-3828	46	16	-	-	SYM
iajs-3828	46	17	20	20	NUM
iajs-3828	46	18	)	)	PUNCT
iajs-3828	46	19	example1.5	example1.5	PROPN
iajs-3828	46	20	:	:	PUNCT
iajs-3828	46	21	let	let	VERB
iajs-3828	46	22	£	£	PRON
iajs-3828	46	23	=	=	PUNCT
iajs-3828	47	1	[	[	X
iajs-3828	47	2	−1,1	−1,1	X
iajs-3828	47	3	]	]	X
iajs-3828	47	4	,	,	PUNCT
iajs-3828	47	5	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	47	6	,	,	PUNCT
iajs-3828	47	7	𝑏	𝑏	NOUN
iajs-3828	47	8	)	)	PUNCT
iajs-3828	47	9	=	=	SYM
iajs-3828	47	10	|𝑎	|𝑎	X
iajs-3828	47	11	−	−	PROPN
iajs-3828	47	12	𝑏|2	𝑏|2	PROPN
iajs-3828	47	13	is	be	AUX
iajs-3828	47	14	b	b	NOUN
iajs-3828	47	15	-	-	ADJ
iajs-3828	47	16	metric	metric	ADJ
iajs-3828	47	17	with	with	ADP
iajs-3828	47	18	𝑠	𝑠	PROPN
iajs-3828	47	19	=	=	SYM
iajs-3828	47	20	2	2	NUM
iajs-3828	47	21	and	and	CCONJ
iajs-3828	47	22	£	£	SYM
iajs-3828	47	23	1	1	NUM
iajs-3828	47	24	=	=	SYM
iajs-3828	48	1	[	[	X
iajs-3828	48	2	−1,0	−1,0	X
iajs-3828	48	3	]	]	PUNCT
iajs-3828	48	4	,	,	PUNCT
iajs-3828	48	5	£	£	SYM
iajs-3828	48	6	2	2	NUM
iajs-3828	48	7	=	=	SYM
iajs-3828	49	1	[	[	X
iajs-3828	49	2	0,1	0,1	NUM
iajs-3828	49	3	]	]	PUNCT
iajs-3828	49	4	,	,	PUNCT
iajs-3828	49	5	£	£	SYM
iajs-3828	49	6	3	3	NUM
iajs-3828	49	7	=	=	SYM
iajs-3828	50	1	[	[	X
iajs-3828	50	2	−1,0	−1,0	X
iajs-3828	50	3	]	]	PUNCT
iajs-3828	50	4	,	,	PUNCT
iajs-3828	50	5	£	£	SYM
iajs-3828	50	6	4	4	NUM
iajs-3828	50	7	=	=	SYM
iajs-3828	51	1	[	[	X
iajs-3828	51	2	0,1	0,1	NUM
iajs-3828	51	3	]	]	PUNCT
iajs-3828	51	4	,	,	PUNCT
iajs-3828	51	5	£	£	SYM
iajs-3828	51	6	5	5	NUM
iajs-3828	51	7	=	=	SYM
iajs-3828	52	1	[	[	X
iajs-3828	52	2	−1,0	−1,0	X
iajs-3828	52	3	]	]	PUNCT
iajs-3828	52	4	,	,	PUNCT
iajs-3828	52	5	£	£	SYM
iajs-3828	52	6	6	6	NUM
iajs-3828	52	7	=	=	SYM
iajs-3828	53	1	[	[	X
iajs-3828	53	2	0,1	0,1	NUM
iajs-3828	53	3	]	]	PUNCT
iajs-3828	53	4	.	.	PUNCT
iajs-3828	54	1	so	so	ADV
iajs-3828	54	2	,	,	PUNCT
iajs-3828	54	3	£	£	NOUN
iajs-3828	54	4	=	=	PUNCT
iajs-3828	54	5	⋃	⋃	ADP
iajs-3828	54	6	£	£	SYM
iajs-3828	54	7	𝑖	𝑖	ADP
iajs-3828	54	8	6	6	NUM
iajs-3828	54	9	𝑖=1	𝑖=1	PUNCT
iajs-3828	54	10	.	.	PUNCT
iajs-3828	54	11	define	define	VERB
iajs-3828	54	12	𝑓:⋃	𝑓:⋃	ADP
iajs-3828	54	13	£	£	SYM
iajs-3828	54	14	𝑖	𝑖	ADP
iajs-3828	54	15	6	6	NUM
iajs-3828	54	16	𝑖=1	𝑖=1	PUNCT
iajs-3828	54	17	→	→	SYM
iajs-3828	54	18	⋃	⋃	ADP
iajs-3828	54	19	£	£	SYM
iajs-3828	54	20	𝑖	𝑖	ADP
iajs-3828	54	21	6	6	NUM
iajs-3828	54	22	𝑖=1	𝑖=1	PUNCT
iajs-3828	54	23	such	such	ADJ
iajs-3828	54	24	that	that	PRON
iajs-3828	54	25	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	55	1	=	=	PUNCT
iajs-3828	55	2	−	−	PROPN
iajs-3828	55	3	𝑎	𝑎	PROPN
iajs-3828	55	4	2+𝑎	2+𝑎	NUM
iajs-3828	55	5	,	,	PUNCT
iajs-3828	55	6	∀𝑎	∀𝑎	PROPN
iajs-3828	55	7	∈	∈	PROPN
iajs-3828	56	1	⋃	⋃	ADP
iajs-3828	56	2	£	£	SYM
iajs-3828	56	3	𝑖	𝑖	ADP
iajs-3828	56	4	6	6	NUM
iajs-3828	56	5	𝑖=1	𝑖=1	PUNCT
iajs-3828	56	6	.	.	PUNCT
iajs-3828	57	1	here	here	ADV
iajs-3828	57	2	£	£	NOUN
iajs-3828	57	3	is	be	AUX
iajs-3828	57	4	cyclic	cyclic	ADJ
iajs-3828	57	5	representation	representation	NOUN
iajs-3828	57	6	w.r.t	w.r.t	NOUN
iajs-3828	57	7	.	.	PUNCT
iajs-3828	58	1	,𝑓.	,𝑓.	PROPN
iajs-3828	58	2	(	(	PUNCT
iajs-3828	58	3	𝑓(£1	𝑓(£1	PROPN
iajs-3828	58	4	)	)	PUNCT
iajs-3828	58	5	⊂	⊂	PROPN
iajs-3828	58	6	£	£	SYM
iajs-3828	58	7	2	2	NUM
iajs-3828	58	8	,	,	PUNCT
iajs-3828	58	9	𝑓(£2	𝑓(£2	PROPN
iajs-3828	58	10	)	)	PUNCT
iajs-3828	58	11	⊂	⊂	PROPN
iajs-3828	58	12	£	£	SYM
iajs-3828	58	13	3	3	NUM
iajs-3828	58	14	,	,	PUNCT
iajs-3828	58	15	𝑓(£3	𝑓(£3	NOUN
iajs-3828	58	16	)	)	PUNCT
iajs-3828	58	17	⊂	⊂	PUNCT
iajs-3828	59	1	£	£	SYM
iajs-3828	59	2	4	4	NUM
iajs-3828	59	3	,	,	PUNCT
iajs-3828	59	4	and	and	CCONJ
iajs-3828	59	5	𝑓(£4	𝑓(£4	NUM
iajs-3828	59	6	)	)	PUNCT
iajs-3828	60	1	⊂	⊂	PROPN
iajs-3828	60	2	£	£	SYM
iajs-3828	60	3	5	5	NUM
iajs-3828	60	4	,	,	PUNCT
iajs-3828	60	5	𝑓(£5	𝑓(£5	PROPN
iajs-3828	60	6	)	)	PUNCT
iajs-3828	60	7	⊂	⊂	PUNCT
iajs-3828	61	1	£	£	SYM
iajs-3828	61	2	6	6	NUM
iajs-3828	61	3	,	,	PUNCT
iajs-3828	61	4	𝑓(£6	𝑓(£6	X
iajs-3828	61	5	)	)	PUNCT
iajs-3828	61	6	⊂	⊂	PUNCT
iajs-3828	61	7	£	£	SYM
iajs-3828	61	8	1	1	NUM
iajs-3828	61	9	)	)	PUNCT
iajs-3828	61	10	and	and	CCONJ
iajs-3828	61	11	𝑓is	𝑓is	DET
iajs-3828	61	12	cyclic	cyclic	ADJ
iajs-3828	61	13	contraction	contraction	NOUN
iajs-3828	61	14	with	with	ADP
iajs-3828	61	15	constant	constant	ADJ
iajs-3828	62	1	0	0	PUNCT
iajs-3828	62	2	<	<	X
iajs-3828	62	3	𝑘	𝑘	X
iajs-3828	62	4	=	=	SYM
iajs-3828	62	5	1	1	NUM
iajs-3828	62	6	2	2	NUM
iajs-3828	62	7	,	,	PUNCT
iajs-3828	62	8	where	where	SCONJ
iajs-3828	62	9	𝑎	𝑎	DET
iajs-3828	62	10	∈	∈	NOUN
iajs-3828	62	11	£	£	SYM
iajs-3828	62	12	𝑖	𝑖	SYM
iajs-3828	62	13	𝑏	𝑏	NOUN
iajs-3828	62	14	∈	∈	PROPN
iajs-3828	62	15	£	£	SYM
iajs-3828	62	16	𝑖+1	𝑖+1	NUM
iajs-3828	62	17	.	.	PUNCT
iajs-3828	63	1	definition	definition	NOUN
iajs-3828	63	2	1.6	1.6	NUM
iajs-3828	63	3	:	:	PUNCT
iajs-3828	63	4	a	a	DET
iajs-3828	63	5	point	point	NOUN
iajs-3828	63	6	𝑎	𝑎	NOUN
iajs-3828	63	7	is	be	AUX
iajs-3828	63	8	called	call	VERB
iajs-3828	63	9	a	a	DET
iajs-3828	63	10	fixed	fix	VERB
iajs-3828	63	11	point	point	NOUN
iajs-3828	63	12	of	of	ADP
iajs-3828	63	13	a	a	DET
iajs-3828	63	14	map	map	NOUN
iajs-3828	63	15	𝑓	𝑓	NOUN
iajs-3828	63	16	:	:	PUNCT
iajs-3828	63	17	£	£	NOUN
iajs-3828	63	18	→	→	SYM
iajs-3828	63	19	£	£	NOUN
iajs-3828	63	20	if	if	SCONJ
iajs-3828	63	21	𝑓(𝑎	𝑓(𝑎	ADJ
iajs-3828	63	22	)	)	PUNCT
iajs-3828	63	23	=	=	SYM
iajs-3828	64	1	𝑎	𝑎	X
iajs-3828	64	2	(	(	PUNCT
iajs-3828	64	3	21	21	NUM
iajs-3828	64	4	)	)	PUNCT
iajs-3828	64	5	.	.	PUNCT
iajs-3828	65	1	an	an	DET
iajs-3828	65	2	example	example	NOUN
iajs-3828	65	3	,	,	PUNCT
iajs-3828	65	4	𝑎	𝑎	PROPN
iajs-3828	65	5	=	=	SYM
iajs-3828	65	6	0	0	NUM
iajs-3828	65	7	is	be	AUX
iajs-3828	65	8	a	a	DET
iajs-3828	65	9	fixed	fix	VERB
iajs-3828	65	10	point	point	NOUN
iajs-3828	65	11	of	of	ADP
iajs-3828	65	12	𝑓	𝑓	PRON
iajs-3828	65	13	in	in	ADP
iajs-3828	65	14	the	the	DET
iajs-3828	65	15	previous	previous	ADJ
iajs-3828	65	16	example	example	NOUN
iajs-3828	65	17	.	.	PUNCT
iajs-3828	66	1	this	this	DET
iajs-3828	66	2	work	work	NOUN
iajs-3828	66	3	includes	include	VERB
iajs-3828	66	4	four	four	NUM
iajs-3828	66	5	main	main	ADJ
iajs-3828	66	6	fixed	fix	VERB
iajs-3828	66	7	point	point	NOUN
iajs-3828	66	8	theorems	theorem	NOUN
iajs-3828	66	9	based	base	VERB
iajs-3828	66	10	on	on	ADP
iajs-3828	66	11	different	different	ADJ
iajs-3828	66	12	cyclic	cyclic	ADJ
iajs-3828	66	13	contractive	contractive	ADJ
iajs-3828	66	14	conditions	condition	NOUN
iajs-3828	66	15	.	.	PUNCT
iajs-3828	67	1	here	here	ADV
iajs-3828	67	2	,	,	PUNCT
iajs-3828	67	3	(	(	PUNCT
iajs-3828	67	4	£	£	SYM
iajs-3828	67	5	,	,	PUNCT
iajs-3828	67	6	ρ	ρ	NOUN
iajs-3828	67	7	)	)	PUNCT
iajs-3828	67	8	denote	denote	NOUN
iajs-3828	67	9	to	to	PART
iajs-3828	67	10	complete	complete	VERB
iajs-3828	67	11	b	b	X
iajs-3828	67	12	-	-	PUNCT
iajs-3828	67	13	metric	metric	ADJ
iajs-3828	67	14	space	space	NOUN
iajs-3828	67	15	where	where	SCONJ
iajs-3828	67	16	ρ	ρ	PROPN
iajs-3828	67	17	is	be	AUX
iajs-3828	67	18	continuous	continuous	ADJ
iajs-3828	67	19	.	.	PUNCT
iajs-3828	68	1	2	2	X
iajs-3828	68	2	.	.	X
iajs-3828	68	3	materials	material	NOUN
iajs-3828	68	4	and	and	CCONJ
iajs-3828	68	5	methods	method	NOUN
iajs-3828	68	6	in	in	ADP
iajs-3828	68	7	this	this	DET
iajs-3828	68	8	section	section	NOUN
iajs-3828	68	9	,	,	PUNCT
iajs-3828	68	10	there	there	PRON
iajs-3828	68	11	are	be	VERB
iajs-3828	68	12	three	three	NUM
iajs-3828	68	13	axes	axis	NOUN
iajs-3828	68	14	;	;	PUNCT
iajs-3828	68	15	the	the	DET
iajs-3828	68	16	first	first	ADJ
iajs-3828	68	17	one	one	NOUN
iajs-3828	68	18	is	be	AUX
iajs-3828	68	19	fixed	fix	VERB
iajs-3828	68	20	point	point	NOUN
iajs-3828	68	21	for	for	ADP
iajs-3828	68	22	cyclic	cyclic	ADJ
iajs-3828	68	23	contractive	contractive	ADJ
iajs-3828	68	24	maps	map	NOUN
iajs-3828	68	25	with	with	ADP
iajs-3828	68	26	integral	integral	ADJ
iajs-3828	68	27	condition	condition	NOUN
iajs-3828	68	28	not	not	PART
iajs-3828	68	29	that	that	SCONJ
iajs-3828	68	30	,	,	PUNCT
iajs-3828	68	31	a	a	DET
iajs-3828	68	32	lebesgue	lebesgue	NOUN
iajs-3828	68	33	-	-	PUNCT
iajs-3828	68	34	integrable	integrable	ADJ
iajs-3828	68	35	function	function	NOUN
iajs-3828	68	36	¥	¥	NOUN
iajs-3828	68	37	:	:	PUNCT
iajs-3828	69	1	[	[	X
iajs-3828	69	2	0,1	0,1	NUM
iajs-3828	69	3	)	)	PUNCT
iajs-3828	69	4	→	→	PUNCT
iajs-3828	70	1	[	[	X
iajs-3828	70	2	0,1	0,1	NUM
iajs-3828	70	3	)	)	PUNCT
iajs-3828	70	4	is	be	AUX
iajs-3828	70	5	called	call	VERB
iajs-3828	70	6	summable	summable	ADJ
iajs-3828	70	7	if	if	SCONJ
iajs-3828	70	8	∫¥(𝑟)𝑑𝑟	∫¥(𝑟)𝑑𝑟	ADJ
iajs-3828	70	9	<	<	X
iajs-3828	70	10	∞	∞	PROPN
iajs-3828	70	11	(	(	PUNCT
iajs-3828	70	12	22	22	NUM
iajs-3828	70	13	)	)	PUNCT
iajs-3828	70	14	.	.	PUNCT
iajs-3828	71	1	theorem	theorem	VERB
iajs-3828	71	2	2.1	2.1	NUM
iajs-3828	71	3	:	:	PUNCT
iajs-3828	71	4	let	let	VERB
iajs-3828	71	5	(	(	PUNCT
iajs-3828	71	6	£	£	NOUN
iajs-3828	71	7	,	,	PUNCT
iajs-3828	71	8	𝜌	𝜌	NOUN
iajs-3828	71	9	)	)	PUNCT
iajs-3828	71	10	be	be	VERB
iajs-3828	71	11	b	b	NUM
iajs-3828	71	12	-	-	PUNCT
iajs-3828	71	13	metric	metric	ADJ
iajs-3828	71	14	space	space	NOUN
iajs-3828	71	15	,	,	PUNCT
iajs-3828	71	16	𝑘	𝑘	PRON
iajs-3828	71	17	∈	∈	PROPN
iajs-3828	71	18	(	(	PUNCT
iajs-3828	71	19	0,1	0,1	NOUN
iajs-3828	71	20	)	)	PUNCT
iajs-3828	71	21	,	,	PUNCT
iajs-3828	71	22	and	and	CCONJ
iajs-3828	71	23	let	let	VERB
iajs-3828	71	24	𝑓	𝑓	X
iajs-3828	71	25	:	:	PUNCT
iajs-3828	71	26	£	£	NOUN
iajs-3828	71	27	→	→	SYM
iajs-3828	71	28	£	£	PUNCT
iajs-3828	71	29	be	be	AUX
iajs-3828	71	30	a	a	DET
iajs-3828	71	31	map	map	NOUN
iajs-3828	71	32	such	such	ADJ
iajs-3828	71	33	that	that	PRON
iajs-3828	71	34	for	for	ADP
iajs-3828	71	35	each	each	DET
iajs-3828	71	36	𝑎	𝑎	NOUN
iajs-3828	71	37	,	,	PUNCT
iajs-3828	71	38	𝑏	𝑏	PROPN
iajs-3828	71	39	∈	∈	PROPN
iajs-3828	71	40	£	£	PROPN
iajs-3828	71	41	∫	∫	NOUN
iajs-3828	71	42	¥	¥	PROPN
iajs-3828	71	43	(	(	PUNCT
iajs-3828	71	44	𝑟	𝑟	NOUN
iajs-3828	71	45	)	)	PUNCT
iajs-3828	71	46	𝜌(𝑓𝑎	𝜌(𝑓𝑎	NOUN
iajs-3828	71	47	,	,	PUNCT
iajs-3828	71	48	𝑓𝑏	𝑓𝑏	PROPN
iajs-3828	71	49	)	)	PUNCT
iajs-3828	71	50	0	0	NUM
iajs-3828	71	51	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	71	52	≤	≤	NOUN
iajs-3828	71	53	𝑘	𝑘	DET
iajs-3828	71	54	∫	∫	PROPN
iajs-3828	71	55	¥	¥	PROPN
iajs-3828	71	56	(	(	PUNCT
iajs-3828	71	57	𝑟)𝑑𝑟	𝑟)𝑑𝑟	PROPN
iajs-3828	71	58	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	71	59	,	,	PUNCT
iajs-3828	71	60	𝑏	𝑏	NOUN
iajs-3828	71	61	)	)	PUNCT
iajs-3828	71	62	0	0	NUM
iajs-3828	71	63	,	,	PUNCT
iajs-3828	71	64	∀𝑎	∀𝑎	PROPN
iajs-3828	71	65	∈	∈	PROPN
iajs-3828	71	66	£	£	AUX
iajs-3828	71	67	𝑖	𝑖	NOUN
iajs-3828	71	68	,	,	PUNCT
iajs-3828	71	69	𝑏	𝑏	PROPN
iajs-3828	71	70	∈	∈	PROPN
iajs-3828	71	71	£	£	SYM
iajs-3828	71	72	𝑖+1	𝑖+1	NUM
iajs-3828	71	73	(	(	PUNCT
iajs-3828	71	74	1	1	NUM
iajs-3828	71	75	)	)	PUNCT
iajs-3828	71	76	where	where	SCONJ
iajs-3828	71	77	,	,	PUNCT
iajs-3828	71	78	¥	¥	PROPN
iajs-3828	71	79	is	be	AUX
iajs-3828	71	80	summable	summable	ADJ
iajs-3828	71	81	on	on	ADP
iajs-3828	71	82	each	each	DET
iajs-3828	71	83	compact	compact	ADJ
iajs-3828	71	84	subset	subset	NOUN
iajs-3828	71	85	of	of	ADP
iajs-3828	71	86	[	[	X
iajs-3828	71	87	0,∞	0,∞	NOUN
iajs-3828	71	88	)	)	PUNCT
iajs-3828	71	89	,	,	PUNCT
iajs-3828	71	90	nonnegative	nonnegative	ADJ
iajs-3828	71	91	and	and	CCONJ
iajs-3828	71	92	for	for	ADP
iajs-3828	71	93	any	any	DET
iajs-3828	71	94	𝜀	𝜀	NOUN
iajs-3828	71	95	>	>	X
iajs-3828	71	96	0	0	PROPN
iajs-3828	71	97	,	,	PUNCT
iajs-3828	71	98	∫	∫	PROPN
iajs-3828	71	99	¥	¥	PROPN
iajs-3828	71	100	(	(	PUNCT
iajs-3828	71	101	𝑟	𝑟	NOUN
iajs-3828	71	102	)	)	PUNCT
iajs-3828	71	103	ε	ε	PROPN
iajs-3828	71	104	0	0	NUM
iajs-3828	71	105	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	71	106	>	>	X
iajs-3828	71	107	0	0	X
iajs-3828	71	108	.	.	PUNCT
iajs-3828	72	1	then	then	ADV
iajs-3828	72	2	∃	∃	PROPN
iajs-3828	72	3	!	!	PUNCT
iajs-3828	72	4	𝑐	𝑐	PROPN
iajs-3828	72	5	∈	∈	PROPN
iajs-3828	72	6	⋂	⋂	PROPN
iajs-3828	72	7	£	£	SYM
iajs-3828	72	8	𝑖	𝑖	SYM
iajs-3828	72	9	𝑛	𝑛	PRON
iajs-3828	72	10	𝑖=1	𝑖=1	PROPN
iajs-3828	72	11	,	,	PUNCT
iajs-3828	72	12	moreover	moreover	ADV
iajs-3828	72	13	,	,	PUNCT
iajs-3828	72	14	∀𝑎	∀𝑎	PROPN
iajs-3828	72	15	∈	∈	PROPN
iajs-3828	72	16	£	£	PROPN
iajs-3828	72	17	,	,	PUNCT
iajs-3828	72	18	lim	lim	NOUN
iajs-3828	72	19	𝑚→∞	𝑚→∞	NUM
iajs-3828	72	20	𝑓𝑚(𝑎	𝑓𝑚(𝑎	X
iajs-3828	72	21	)	)	PUNCT
iajs-3828	72	22	=	=	SYM
iajs-3828	73	1	𝑐.	𝑐.	NOUN
iajs-3828	73	2	proof	proof	NOUN
iajs-3828	73	3	:	:	PUNCT
iajs-3828	73	4	let	let	VERB
iajs-3828	73	5	𝑎0	𝑎0	VERB
iajs-3828	73	6	∈	∈	VERB
iajs-3828	73	7	£	£	NOUN
iajs-3828	73	8	=	=	PUNCT
iajs-3828	73	9	⋃	⋃	ADP
iajs-3828	73	10	£	£	SYM
iajs-3828	73	11	𝑖	𝑖	SYM
iajs-3828	73	12	𝑛	𝑛	PRON
iajs-3828	73	13	𝑖=1	𝑖=1	PUNCT
iajs-3828	73	14	and	and	CCONJ
iajs-3828	73	15	consider	consider	VERB
iajs-3828	73	16	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	73	17	=	=	SYM
iajs-3828	73	18	𝑓𝑎𝑚	𝑓𝑎𝑚	VERB
iajs-3828	73	19	for	for	ADP
iajs-3828	73	20	each	each	DET
iajs-3828	73	21	𝑚	𝑚	ADP
iajs-3828	73	22	∈	∈	PROPN
iajs-3828	73	23	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	73	24	}	}	PUNCT
iajs-3828	73	25	,	,	PUNCT
iajs-3828	73	26	so	so	SCONJ
iajs-3828	73	27	for	for	ADP
iajs-3828	73	28	any	any	DET
iajs-3828	73	29	𝑚	𝑚	PROPN
iajs-3828	73	30	∈	∈	PROPN
iajs-3828	73	31	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	73	32	}	}	PUNCT
iajs-3828	73	33	,	,	PUNCT
iajs-3828	73	34	∃𝑖𝑚	∃𝑖𝑚	NOUN
iajs-3828	73	35	∈	∈	PROPN
iajs-3828	73	36	{	{	PUNCT
iajs-3828	73	37	1,2	1,2	NUM
iajs-3828	73	38	,	,	PUNCT
iajs-3828	73	39	…	…	PUNCT
iajs-3828	73	40	,	,	PUNCT
iajs-3828	73	41	𝑛	𝑛	X
iajs-3828	73	42	}	}	PUNCT
iajs-3828	73	43	such	such	ADJ
iajs-3828	73	44	that	that	SCONJ
iajs-3828	73	45	𝑎𝑚	𝑎𝑚	PROPN
iajs-3828	73	46	∈	∈	PROPN
iajs-3828	73	47	£	£	NOUN
iajs-3828	73	48	𝑖𝑚	𝑖𝑚	NOUN
iajs-3828	73	49	and	and	CCONJ
iajs-3828	73	50	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	73	51	∈	∈	PROPN
iajs-3828	73	52	£	£	NOUN
iajs-3828	73	53	𝑖𝑚+1	𝑖𝑚+1	NOUN
iajs-3828	73	54	.	.	PUNCT
iajs-3828	74	1	if	if	SCONJ
iajs-3828	74	2	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	74	3	=	=	PUNCT
iajs-3828	74	4	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	74	5	+	+	PROPN
iajs-3828	74	6	1	1	NUM
iajs-3828	74	7	for	for	ADP
iajs-3828	74	8	some	some	DET
iajs-3828	74	9	𝑚0	𝑚0	NOUN
iajs-3828	74	10	then	then	ADV
iajs-3828	74	11	,	,	PUNCT
iajs-3828	74	12	since	since	SCONJ
iajs-3828	74	13	𝑎𝑚0	𝑎𝑚0	PROPN
iajs-3828	74	14	+	+	PROPN
iajs-3828	74	15	1	1	NUM
iajs-3828	74	16	=	=	NOUN
iajs-3828	74	17	𝑓𝑎𝑚0	𝑓𝑎𝑚0	NOUN
iajs-3828	74	18	=	=	PUNCT
iajs-3828	75	1	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	75	2	this	this	PRON
iajs-3828	75	3	mean	mean	VERB
iajs-3828	75	4	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	75	5	is	be	AUX
iajs-3828	75	6	a	a	DET
iajs-3828	75	7	fixed	fix	VERB
iajs-3828	75	8	point	point	NOUN
iajs-3828	75	9	ihjpas	ihjpa	NOUN
iajs-3828	75	10	.	.	PUNCT
iajs-3828	76	1	2025,38(2	2025,38(2	PROPN
iajs-3828	76	2	)	)	PUNCT
iajs-3828	76	3	378	378	NUM
iajs-3828	76	4	of	of	ADP
iajs-3828	76	5	𝑓.	𝑓.	NOUN
iajs-3828	76	6	thus	thus	ADV
iajs-3828	76	7	,	,	PUNCT
iajs-3828	76	8	assume	assume	VERB
iajs-3828	76	9	that	that	SCONJ
iajs-3828	76	10	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	76	11	≠	≠	NOUN
iajs-3828	76	12	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	76	13	for	for	ADP
iajs-3828	76	14	all	all	PRON
iajs-3828	76	15	𝑚	𝑚	ADP
iajs-3828	76	16	∈	∈	NOUN
iajs-3828	76	17	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	76	18	}	}	PUNCT
iajs-3828	76	19	.	.	PUNCT
iajs-3828	77	1	so	so	ADV
iajs-3828	77	2	,	,	PUNCT
iajs-3828	77	3	∫	∫	PROPN
iajs-3828	77	4	¥	¥	PROPN
iajs-3828	77	5	(	(	PUNCT
iajs-3828	77	6	𝑟	𝑟	NOUN
iajs-3828	77	7	)	)	PUNCT
iajs-3828	77	8	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	NOUN
iajs-3828	77	9	,	,	PUNCT
iajs-3828	77	10	𝑓𝑎𝑚+1	𝑓𝑎𝑚+1	ADJ
iajs-3828	77	11	)	)	PUNCT
iajs-3828	77	12	0	0	NUM
iajs-3828	77	13	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	77	14	≤	≤	NOUN
iajs-3828	77	15	𝑘	𝑘	PRON
iajs-3828	77	16	∫	∫	PROPN
iajs-3828	77	17	¥	¥	PROPN
iajs-3828	77	18	(	(	PUNCT
iajs-3828	77	19	𝑟	𝑟	NOUN
iajs-3828	77	20	)	)	PUNCT
iajs-3828	77	21	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	77	22	,	,	PUNCT
iajs-3828	77	23	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	77	24	)	)	PUNCT
iajs-3828	77	25	0	0	NUM
iajs-3828	78	1	𝑑𝑟.	𝑑𝑟.	X
iajs-3828	78	2	(	(	PUNCT
iajs-3828	78	3	2	2	NUM
iajs-3828	78	4	)	)	PUNCT
iajs-3828	78	5	by	by	ADP
iajs-3828	78	6	repeating	repeat	VERB
iajs-3828	78	7	the	the	DET
iajs-3828	78	8	inequality	inequality	NOUN
iajs-3828	78	9	(	(	PUNCT
iajs-3828	78	10	2	2	NUM
iajs-3828	78	11	)	)	PUNCT
iajs-3828	78	12	m	m	NOUN
iajs-3828	78	13	times	time	NOUN
iajs-3828	78	14	,	,	PUNCT
iajs-3828	78	15	it	it	PRON
iajs-3828	78	16	follows	follow	VERB
iajs-3828	78	17	directly	directly	ADV
iajs-3828	78	18	∫	∫	PROPN
iajs-3828	78	19	¥	¥	PROPN
iajs-3828	78	20	(	(	PUNCT
iajs-3828	78	21	𝑟	𝑟	NOUN
iajs-3828	78	22	)	)	PUNCT
iajs-3828	78	23	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	NOUN
iajs-3828	78	24	,	,	PUNCT
iajs-3828	78	25	𝑓𝑎𝑚+1	𝑓𝑎𝑚+1	ADJ
iajs-3828	78	26	)	)	PUNCT
iajs-3828	78	27	0	0	NUM
iajs-3828	78	28	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	78	29	≤	≤	NOUN
iajs-3828	79	1	𝑘	𝑘	DET
iajs-3828	79	2	∫	∫	PROPN
iajs-3828	79	3	¥	¥	PROPN
iajs-3828	79	4	(	(	PUNCT
iajs-3828	79	5	𝑟	𝑟	NOUN
iajs-3828	79	6	)	)	PUNCT
iajs-3828	79	7	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	79	8	,	,	PUNCT
iajs-3828	79	9	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	79	10	)	)	PUNCT
iajs-3828	79	11	0	0	NUM
iajs-3828	79	12	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	79	13	=	=	PRON
iajs-3828	79	14	𝑘𝑚	𝑘𝑚	NOUN
iajs-3828	79	15	∫	∫	PROPN
iajs-3828	79	16	¥	¥	PROPN
iajs-3828	79	17	(	(	PUNCT
iajs-3828	79	18	𝑟	𝑟	NOUN
iajs-3828	79	19	)	)	PUNCT
iajs-3828	79	20	𝜌(𝑎0	𝜌(𝑎0	ADJ
iajs-3828	79	21	,	,	PUNCT
iajs-3828	79	22	𝑓𝑎0	𝑓𝑎0	NOUN
iajs-3828	79	23	)	)	PUNCT
iajs-3828	79	24	0	0	NUM
iajs-3828	80	1	𝑑𝑟.	𝑑𝑟.	NOUN
iajs-3828	80	2	as	as	ADP
iajs-3828	80	3	consequence	consequence	NOUN
iajs-3828	80	4	,	,	PUNCT
iajs-3828	80	5	since	since	SCONJ
iajs-3828	80	6	𝑘	𝑘	PRON
iajs-3828	80	7	∈	∈	PROPN
iajs-3828	80	8	(	(	PUNCT
iajs-3828	80	9	0,1	0,1	NOUN
iajs-3828	80	10	)	)	PUNCT
iajs-3828	80	11	,	,	PUNCT
iajs-3828	80	12	obtaining	obtain	VERB
iajs-3828	80	13	a	a	DET
iajs-3828	80	14	monotone	monotone	NOUN
iajs-3828	80	15	decreasing	decrease	VERB
iajs-3828	80	16	sequence	sequence	NOUN
iajs-3828	80	17	(	(	PUNCT
iajs-3828	80	18	∫	∫	PROPN
iajs-3828	80	19	¥	¥	PROPN
iajs-3828	80	20	(	(	PUNCT
iajs-3828	80	21	𝑟	𝑟	NOUN
iajs-3828	80	22	)	)	PUNCT
iajs-3828	80	23	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	NOUN
iajs-3828	80	24	,	,	PUNCT
iajs-3828	80	25	𝑓𝑎𝑚+1	𝑓𝑎𝑚+1	ADJ
iajs-3828	80	26	)	)	PUNCT
iajs-3828	80	27	0	0	NUM
iajs-3828	80	28	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	80	29	)	)	PUNCT
iajs-3828	80	30	=	=	PUNCT
iajs-3828	81	1	(	(	PUNCT
iajs-3828	81	2	∫	∫	PROPN
iajs-3828	81	3	¥	¥	PROPN
iajs-3828	81	4	(	(	PUNCT
iajs-3828	81	5	𝑟	𝑟	NOUN
iajs-3828	81	6	)	)	PUNCT
iajs-3828	81	7	𝜌(𝑎𝑚+1	𝜌(𝑎𝑚+1	ADJ
iajs-3828	81	8	,	,	PUNCT
iajs-3828	81	9	𝑎𝑚+2	𝑎𝑚+2	NUM
iajs-3828	81	10	)	)	PUNCT
iajs-3828	81	11	0	0	NUM
iajs-3828	81	12	𝑑𝑟	𝑑𝑟	PROPN
iajs-3828	81	13	)	)	PUNCT
iajs-3828	81	14	which	which	PRON
iajs-3828	81	15	has	have	AUX
iajs-3828	81	16	lower	lower	ADV
iajs-3828	81	17	bound	bind	VERB
iajs-3828	81	18	is	be	AUX
iajs-3828	81	19	0	0	NUM
iajs-3828	81	20	.	.	PUNCT
iajs-3828	82	1	we	we	PRON
iajs-3828	82	2	have	have	VERB
iajs-3828	82	3	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	NOUN
iajs-3828	82	4	,	,	PUNCT
iajs-3828	82	5	𝑓𝑎𝑚+1	𝑓𝑎𝑚+1	ADJ
iajs-3828	82	6	)	)	PUNCT
iajs-3828	82	7	→	→	SYM
iajs-3828	82	8	0	0	NUM
iajs-3828	82	9	as𝑚	as𝑚	PROPN
iajs-3828	82	10	→	→	PUNCT
iajs-3828	82	11	∞.	∞.	PROPN
iajs-3828	82	12	by	by	ADP
iajs-3828	82	13	properties	property	NOUN
iajs-3828	82	14	of	of	ADP
iajs-3828	82	15	real	real	ADJ
iajs-3828	82	16	sequence	sequence	NOUN
iajs-3828	82	17	,	,	PUNCT
iajs-3828	82	18	(	(	PUNCT
iajs-3828	82	19	∫	∫	PROPN
iajs-3828	82	20	¥	¥	PROPN
iajs-3828	82	21	(	(	PUNCT
iajs-3828	82	22	𝑟	𝑟	NOUN
iajs-3828	82	23	)	)	PUNCT
iajs-3828	82	24	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	NOUN
iajs-3828	82	25	,	,	PUNCT
iajs-3828	82	26	𝑓𝑎𝑚+1	𝑓𝑎𝑚+1	ADJ
iajs-3828	82	27	)	)	PUNCT
iajs-3828	82	28	0	0	NUM
iajs-3828	82	29	𝑑𝑟	𝑑𝑟	PROPN
iajs-3828	82	30	)	)	PUNCT
iajs-3828	82	31	convergences	convergence	NOUN
iajs-3828	82	32	𝜀	𝜀	X
iajs-3828	82	33	≥	≥	X
iajs-3828	82	34	0	0	NUM
iajs-3828	82	35	such	such	ADJ
iajs-3828	82	36	that	that	SCONJ
iajs-3828	82	37	lim	lim	PROPN
iajs-3828	82	38	𝑚→∞	𝑚→∞	NUM
iajs-3828	82	39	∫	∫	PROPN
iajs-3828	82	40	¥	¥	PROPN
iajs-3828	82	41	(	(	PUNCT
iajs-3828	82	42	𝑟	𝑟	NOUN
iajs-3828	82	43	)	)	PUNCT
iajs-3828	82	44	𝜌(𝑎𝑚+1,𝑎𝑚+2	𝜌(𝑎𝑚+1,𝑎𝑚+2	X
iajs-3828	82	45	)	)	PUNCT
iajs-3828	82	46	0	0	NUM
iajs-3828	82	47	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	82	48	=	=	PUNCT
iajs-3828	82	49	𝜀.	𝜀.	NOUN
iajs-3828	82	50	suppose	suppose	VERB
iajs-3828	82	51	that	that	SCONJ
iajs-3828	82	52	𝜀	𝜀	VERB
iajs-3828	82	53	>	>	X
iajs-3828	82	54	0	0	NUM
iajs-3828	82	55	,	,	PUNCT
iajs-3828	82	56	it	it	PRON
iajs-3828	82	57	is	be	AUX
iajs-3828	82	58	enough	enough	ADJ
iajs-3828	82	59	to	to	PART
iajs-3828	82	60	assume	assume	VERB
iajs-3828	82	61	lim	lim	PROPN
iajs-3828	82	62	𝑚→∞	𝑚→∞	PROPN
iajs-3828	82	63	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
iajs-3828	82	64	𝜌(𝑎𝑚+1	𝜌(𝑎𝑚+1	NUM
iajs-3828	82	65	,	,	PUNCT
iajs-3828	82	66	𝑎𝑚+2	𝑎𝑚+2	X
iajs-3828	82	67	)	)	PUNCT
iajs-3828	82	68	=	=	SYM
iajs-3828	83	1	𝜀	𝜀	X
iajs-3828	83	2	>	>	X
iajs-3828	83	3	0	0	NUM
iajs-3828	83	4	.	.	PUNCT
iajs-3828	84	1	then	then	ADV
iajs-3828	84	2	there	there	PRON
iajs-3828	84	3	exists	exist	VERB
iajs-3828	84	4	a	a	DET
iajs-3828	84	5	𝑢𝜀𝜖𝑁	𝑢𝜀𝜖𝑁	NOUN
iajs-3828	84	6	and	and	CCONJ
iajs-3828	84	7	a	a	DET
iajs-3828	84	8	sequence	sequence	NOUN
iajs-3828	84	9	(	(	PUNCT
iajs-3828	84	10	𝑓𝑎𝑚𝑢	𝑓𝑎𝑚𝑢	NOUN
iajs-3828	84	11	)	)	PUNCT
iajs-3828	84	12	𝑢≥𝑢𝜀	𝑢≥𝑢𝜀	PROPN
iajs-3828	84	13	such	such	ADJ
iajs-3828	84	14	that	that	DET
iajs-3828	84	15	𝜌(𝑓𝑎𝑚𝑢	𝜌(𝑓𝑎𝑚𝑢	NOUN
iajs-3828	84	16	,	,	PUNCT
iajs-3828	84	17	𝑓𝑎𝑚𝑢+1	𝑓𝑎𝑚𝑢+1	NOUN
iajs-3828	84	18	)	)	PUNCT
iajs-3828	84	19	→	→	SYM
iajs-3828	84	20	𝜀	𝜀	X
iajs-3828	84	21	>	>	X
iajs-3828	84	22	0	0	PUNCT
iajs-3828	84	23	as	as	ADP
iajs-3828	84	24	𝑢	𝑢	X
iajs-3828	84	25	→	→	SYM
iajs-3828	84	26	∞	∞	PROPN
iajs-3828	84	27	and	and	CCONJ
iajs-3828	84	28	𝜌(𝑓𝑎𝑚𝑢	𝜌(𝑓𝑎𝑚𝑢	NOUN
iajs-3828	84	29	,	,	PUNCT
iajs-3828	84	30	𝑓𝑎𝑚𝑢+1	𝑓𝑎𝑚𝑢+1	NOUN
iajs-3828	84	31	)	)	PUNCT
iajs-3828	84	32	≥	≥	PROPN
iajs-3828	84	33	𝜀	𝜀	X
iajs-3828	84	34	2	2	NUM
iajs-3828	84	35	.	.	PUNCT
iajs-3828	85	1	for	for	ADP
iajs-3828	85	2	each	each	DET
iajs-3828	85	3	𝑢	𝑢	PROPN
iajs-3828	85	4	≥	≥	NOUN
iajs-3828	85	5	𝑢𝜀	𝑢𝜀	NOUN
iajs-3828	85	6	,	,	PUNCT
iajs-3828	85	7	by	by	ADP
iajs-3828	85	8	𝜀	𝜀	ADP
iajs-3828	85	9	2	2	NUM
iajs-3828	85	10	≤	≤	NOUN
iajs-3828	85	11	lim	lim	PROPN
iajs-3828	85	12	𝑚→∞	𝑚→∞	NUM
iajs-3828	85	13	𝜌(𝑓𝑎𝑚𝑢	𝜌(𝑓𝑎𝑚𝑢	NOUN
iajs-3828	85	14	,	,	PUNCT
iajs-3828	85	15	𝑓𝑎𝑚𝑢+1	𝑓𝑎𝑚𝑢+1	NOUN
iajs-3828	85	16	)	)	PUNCT
iajs-3828	86	1	=	=	VERB
iajs-3828	86	2	lim	lim	PROPN
iajs-3828	86	3	𝑚→∞	𝑚→∞	NUM
iajs-3828	86	4	sup	sup	NOUN
iajs-3828	86	5	𝜌(𝑓𝑎𝑚𝑢	𝜌(𝑓𝑎𝑚𝑢	NOUN
iajs-3828	86	6	,	,	PUNCT
iajs-3828	86	7	𝑓𝑎𝑚𝑢+1	𝑓𝑎𝑚𝑢+1	NOUN
iajs-3828	86	8	)	)	PUNCT
iajs-3828	86	9	=	=	SYM
iajs-3828	87	1	𝜀	𝜀	X
iajs-3828	87	2	>	>	X
iajs-3828	87	3	0	0	PROPN
iajs-3828	87	4	,	,	PUNCT
iajs-3828	87	5	this	this	PRON
iajs-3828	87	6	is	be	AUX
iajs-3828	87	7	true	true	ADJ
iajs-3828	87	8	only	only	ADV
iajs-3828	87	9	if	if	SCONJ
iajs-3828	87	10	𝜀	𝜀	X
iajs-3828	87	11	=	=	SYM
iajs-3828	87	12	0	0	PROPN
iajs-3828	87	13	.	.	PUNCT
iajs-3828	88	1	the	the	DET
iajs-3828	88	2	next	next	ADJ
iajs-3828	88	3	step	step	NOUN
iajs-3828	88	4	is	be	AUX
iajs-3828	88	5	proving	prove	VERB
iajs-3828	88	6	that	that	SCONJ
iajs-3828	88	7	for	for	ADP
iajs-3828	88	8	each	each	DET
iajs-3828	88	9	𝑎0	𝑎0	PROPN
iajs-3828	88	10	∈	∈	PROPN
iajs-3828	88	11	£	£	SYM
iajs-3828	88	12	,	,	PUNCT
iajs-3828	88	13	(	(	PUNCT
iajs-3828	88	14	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	88	15	)	)	PUNCT
iajs-3828	88	16	is	be	AUX
iajs-3828	88	17	a	a	DET
iajs-3828	88	18	cauchy	cauchy	ADJ
iajs-3828	88	19	sequence	sequence	NOUN
iajs-3828	88	20	.	.	PUNCT
iajs-3828	89	1	for	for	ADP
iajs-3828	89	2	𝑗	𝑗	INTJ
iajs-3828	89	3	>	>	X
iajs-3828	89	4	𝑛	𝑛	PRON
iajs-3828	89	5	define	define	VERB
iajs-3828	89	6	£	£	SYM
iajs-3828	89	7	𝑗	𝑗	NOUN
iajs-3828	89	8	=	=	NOUN
iajs-3828	89	9	£	£	SYM
iajs-3828	89	10	𝑖	𝑖	NOUN
iajs-3828	89	11	if	if	SCONJ
iajs-3828	89	12	𝑗	𝑗	ADJ
iajs-3828	89	13	=	=	SYM
iajs-3828	89	14	𝑖	𝑖	SYM
iajs-3828	89	15	mod	mod	ADJ
iajs-3828	89	16	𝑛.	𝑛.	NOUN
iajs-3828	89	17	claim	claim	VERB
iajs-3828	89	18	i	i	PRON
iajs-3828	89	19	:	:	PUNCT
iajs-3828	89	20	for	for	ADP
iajs-3828	89	21	all	all	DET
iajs-3828	89	22	𝜀	𝜀	NOUN
iajs-3828	89	23	>	>	X
iajs-3828	89	24	0	0	PUNCT
iajs-3828	89	25	there	there	PRON
iajs-3828	89	26	exist	exist	VERB
iajs-3828	89	27	𝑚	𝑚	ADP
iajs-3828	89	28	∈	∈	NOUN
iajs-3828	90	1	𝑁	𝑁	PROPN
iajs-3828	90	2	such	such	ADJ
iajs-3828	90	3	that	that	PRON
iajs-3828	90	4	for	for	ADP
iajs-3828	90	5	all	all	DET
iajs-3828	90	6	𝑗	𝑗	PROPN
iajs-3828	90	7	,	,	PUNCT
iajs-3828	90	8	𝑖	𝑖	X
iajs-3828	90	9	≥	≥	NOUN
iajs-3828	90	10	𝑚	𝑚	NOUN
iajs-3828	90	11	,	,	PUNCT
iajs-3828	90	12	𝑗	𝑗	INTJ
iajs-3828	90	13	−	−	PROPN
iajs-3828	90	14	𝑖	𝑖	SYM
iajs-3828	90	15	≡	≡	PROPN
iajs-3828	90	16	1(mod	1(mod	NUM
iajs-3828	90	17	𝑛	𝑛	NOUN
iajs-3828	90	18	)	)	PUNCT
iajs-3828	90	19	then	then	ADV
iajs-3828	90	20	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	90	21	,	,	PUNCT
iajs-3828	90	22	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	90	23	)	)	PUNCT
iajs-3828	90	24	<	<	X
iajs-3828	90	25	𝜀.	𝜀.	NOUN
iajs-3828	90	26	suppose	suppose	VERB
iajs-3828	90	27	that	that	SCONJ
iajs-3828	90	28	there	there	PRON
iajs-3828	90	29	exists	exist	VERB
iajs-3828	90	30	𝜀	𝜀	PROPN
iajs-3828	90	31	>	>	X
iajs-3828	90	32	0	0	NUM
iajs-3828	90	33	such	such	ADJ
iajs-3828	90	34	that	that	PRON
iajs-3828	90	35	for	for	ADP
iajs-3828	90	36	each	each	DET
iajs-3828	90	37	𝑚	𝑚	ADP
iajs-3828	90	38	∈	∈	PROPN
iajs-3828	90	39	𝑁	𝑁	PROPN
iajs-3828	90	40	,	,	PUNCT
iajs-3828	90	41	one	one	PRON
iajs-3828	90	42	can	can	AUX
iajs-3828	90	43	find	find	VERB
iajs-3828	90	44	𝑗	𝑗	PRON
iajs-3828	90	45	>	>	X
iajs-3828	90	46	𝑖	𝑖	X
iajs-3828	90	47	>	>	X
iajs-3828	90	48	𝑚	𝑚	X
iajs-3828	90	49	with	with	ADP
iajs-3828	90	50	𝑗	𝑗	INTJ
iajs-3828	90	51	−	−	PROPN
iajs-3828	90	52	𝑖	𝑖	SYM
iajs-3828	90	53	≡	≡	PROPN
iajs-3828	90	54	1(mod	1(mod	NUM
iajs-3828	90	55	𝑛	𝑛	X
iajs-3828	90	56	)	)	PUNCT
iajs-3828	90	57	satisfying	satisfy	VERB
iajs-3828	90	58	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	90	59	,	,	PUNCT
iajs-3828	90	60	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	90	61	)	)	PUNCT
iajs-3828	90	62	≥	≥	NOUN
iajs-3828	90	63	𝜀.	𝜀.	VERB
iajs-3828	90	64	clearly	clearly	ADV
iajs-3828	90	65	,	,	PUNCT
iajs-3828	90	66	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	90	67	,	,	PUNCT
iajs-3828	90	68	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	90	69	)	)	PUNCT
iajs-3828	90	70	<	<	X
iajs-3828	90	71	𝜀.	𝜀.	NOUN
iajs-3828	90	72	now	now	ADV
iajs-3828	90	73	,	,	PUNCT
iajs-3828	90	74	take	take	VERB
iajs-3828	90	75	𝑚	𝑚	ADP
iajs-3828	90	76	≥	≥	NUM
iajs-3828	90	77	2(mod	2(mod	NUM
iajs-3828	90	78	𝑛	𝑛	X
iajs-3828	90	79	)	)	PUNCT
iajs-3828	90	80	.	.	PUNCT
iajs-3828	91	1	then	then	ADV
iajs-3828	91	2	,	,	PUNCT
iajs-3828	91	3	corresponding	correspond	VERB
iajs-3828	91	4	to	to	ADP
iajs-3828	91	5	𝑖	𝑖	PRON
iajs-3828	91	6	≥	≥	NOUN
iajs-3828	91	7	𝑚	𝑚	ADP
iajs-3828	91	8	use	use	NOUN
iajs-3828	91	9	can	can	AUX
iajs-3828	91	10	choose	choose	VERB
iajs-3828	91	11	𝑗	𝑗	PRON
iajs-3828	91	12	in	in	ADP
iajs-3828	91	13	such	such	DET
iajs-3828	91	14	a	a	DET
iajs-3828	91	15	way	way	NOUN
iajs-3828	91	16	that	that	PRON
iajs-3828	91	17	it	it	PRON
iajs-3828	91	18	is	be	AUX
iajs-3828	91	19	the	the	DET
iajs-3828	91	20	smallest	small	ADJ
iajs-3828	91	21	integer	integer	NOUN
iajs-3828	91	22	with	with	ADP
iajs-3828	91	23	𝑗	𝑗	PROPN
iajs-3828	91	24	>	>	X
iajs-3828	91	25	𝑖	𝑖	PUNCT
iajs-3828	91	26	satisfying	satisfy	VERB
iajs-3828	91	27	𝑗	𝑗	INTJ
iajs-3828	91	28	−	−	PROPN
iajs-3828	91	29	𝑖	𝑖	SYM
iajs-3828	91	30	≡	≡	PROPN
iajs-3828	91	31	1(mod	1(mod	NUM
iajs-3828	91	32	𝑛	𝑛	NOUN
iajs-3828	91	33	)	)	PUNCT
iajs-3828	91	34	and	and	CCONJ
iajs-3828	91	35	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	91	36	,	,	PUNCT
iajs-3828	91	37	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	91	38	)	)	PUNCT
iajs-3828	91	39	≥	≥	NOUN
iajs-3828	91	40	𝜀.	𝜀.	VERB
iajs-3828	91	41	therefore	therefore	ADV
iajs-3828	91	42	,	,	PUNCT
iajs-3828	91	43	𝜌(𝑎𝑗−𝑛	𝜌(𝑎𝑗−𝑛	NUM
iajs-3828	91	44	,	,	PUNCT
iajs-3828	91	45	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	91	46	)	)	PUNCT
iajs-3828	91	47	≤	≤	NOUN
iajs-3828	91	48	𝜀.	𝜀.	NOUN
iajs-3828	91	49	by	by	ADP
iajs-3828	91	50	triangular	triangular	NOUN
iajs-3828	91	51	inequality	inequality	NOUN
iajs-3828	91	52	𝜀	𝜀	ADP
iajs-3828	91	53	≤	≤	ADJ
iajs-3828	91	54	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	91	55	,	,	PUNCT
iajs-3828	91	56	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	91	57	)	)	PUNCT
iajs-3828	91	58	≤	≤	NOUN
iajs-3828	91	59	𝑞(𝜌(𝑎𝑗	𝑞(𝜌(𝑎𝑗	NOUN
iajs-3828	91	60	,	,	PUNCT
iajs-3828	91	61	𝑎𝑖−𝑛	𝑎𝑖−𝑛	NOUN
iajs-3828	91	62	)	)	PUNCT
iajs-3828	92	1	+	+	CCONJ
iajs-3828	92	2	∑	∑	PUNCT
iajs-3828	92	3	𝜌(𝑎𝑖−𝑘	𝜌(𝑎𝑖−𝑘	VERB
iajs-3828	92	4	,	,	PUNCT
iajs-3828	92	5	𝑎𝑖−𝑘+1	𝑎𝑖−𝑘+1	VERB
iajs-3828	92	6	)	)	PUNCT
iajs-3828	92	7	𝑛	𝑛	DET
iajs-3828	92	8	𝑘=1	𝑘=1	NOUN
iajs-3828	92	9	)	)	PUNCT
iajs-3828	93	1	≤	≤	PROPN
iajs-3828	93	2	𝑞(∑	𝑞(∑	PROPN
iajs-3828	93	3	𝜌(𝑎𝑖−𝑘	𝜌(𝑎𝑖−𝑘	PROPN
iajs-3828	93	4	,	,	PUNCT
iajs-3828	93	5	𝑎𝑖−𝑘+1	𝑎𝑖−𝑘+1	VERB
iajs-3828	93	6	)	)	PUNCT
iajs-3828	93	7	𝑛	𝑛	DET
iajs-3828	93	8	𝑘=1	𝑘=1	NOUN
iajs-3828	93	9	)	)	PUNCT
iajs-3828	94	1	+	+	CCONJ
iajs-3828	94	2	𝑞	𝑞	X
iajs-3828	94	3	𝜀	𝜀	PROPN
iajs-3828	94	4	→	→	SYM
iajs-3828	94	5	𝑞	𝑞	X
iajs-3828	94	6	𝜀	𝜀	PROPN
iajs-3828	94	7	as	as	ADP
iajs-3828	94	8	𝑛	𝑛	PROPN
iajs-3828	94	9	→	→	SYM
iajs-3828	94	10	∞.	∞.	PROPN
iajs-3828	94	11	again	again	ADV
iajs-3828	94	12	,	,	PUNCT
iajs-3828	94	13	by	by	ADP
iajs-3828	94	14	triangular	triangular	NOUN
iajs-3828	94	15	inequality	inequality	NOUN
iajs-3828	94	16	𝜀	𝜀	ADP
iajs-3828	94	17	≤	≤	ADJ
iajs-3828	94	18	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	94	19	,	,	PUNCT
iajs-3828	94	20	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	94	21	)	)	PUNCT
iajs-3828	94	22	≤	≤	NOUN
iajs-3828	94	23	𝑞	𝑞	ADP
iajs-3828	94	24	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	94	25	,	,	PUNCT
iajs-3828	94	26	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	94	27	)	)	PUNCT
iajs-3828	94	28	+	+	CCONJ
iajs-3828	94	29	𝑞𝜌(𝑎𝑗+1	𝑞𝜌(𝑎𝑗+1	PROPN
iajs-3828	94	30	,	,	PUNCT
iajs-3828	94	31	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	94	32	)	)	PUNCT
iajs-3828	94	33	+	+	X
iajs-3828	94	34	𝑞𝜌(𝑎𝑖+1	𝑞𝜌(𝑎𝑖+1	PROPN
iajs-3828	94	35	,	,	PUNCT
iajs-3828	94	36	𝑎𝑖	𝑎𝑖	PROPN
iajs-3828	94	37	)	)	PUNCT
iajs-3828	94	38	→	→	PUNCT
iajs-3828	94	39	𝑞	𝑞	X
iajs-3828	94	40	ε	ε	PROPN
iajs-3828	94	41	as	as	ADP
iajs-3828	94	42	𝑗	𝑗	PROPN
iajs-3828	94	43	,	,	PUNCT
iajs-3828	94	44	𝑖	𝑖	SYM
iajs-3828	94	45	→	→	SYM
iajs-3828	94	46	∞	∞	PROPN
iajs-3828	94	47	,	,	PUNCT
iajs-3828	94	48	we	we	PRON
iajs-3828	94	49	get	get	VERB
iajs-3828	94	50	∫	∫	PROPN
iajs-3828	94	51	¥	¥	PROPN
iajs-3828	94	52	(	(	PUNCT
iajs-3828	94	53	𝑟	𝑟	NOUN
iajs-3828	94	54	)	)	PUNCT
iajs-3828	94	55	𝜌(𝑓𝑎𝑗+1,𝑓𝑎𝑖+1	𝜌(𝑓𝑎𝑗+1,𝑓𝑎𝑖+1	PROPN
iajs-3828	94	56	)	)	PUNCT
iajs-3828	94	57	0	0	NUM
iajs-3828	95	1	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	95	2	≤	≤	NOUN
iajs-3828	95	3	𝑘	𝑘	DET
iajs-3828	95	4	∫	∫	PROPN
iajs-3828	95	5	¥	¥	PROPN
iajs-3828	95	6	(	(	PUNCT
iajs-3828	95	7	𝑟	𝑟	NOUN
iajs-3828	95	8	)	)	PUNCT
iajs-3828	95	9	𝜌(𝑎𝑗+1,𝑎𝑖+1	𝜌(𝑎𝑗+1,𝑎𝑖+1	PROPN
iajs-3828	95	10	)	)	PUNCT
iajs-3828	95	11	0	0	NUM
iajs-3828	96	1	𝑑𝑟.	𝑑𝑟.	X
iajs-3828	96	2	(	(	PUNCT
iajs-3828	96	3	3	3	X
iajs-3828	96	4	)	)	PUNCT
iajs-3828	96	5	letting	let	VERB
iajs-3828	96	6	𝑗	𝑗	INTJ
iajs-3828	96	7	,	,	PUNCT
iajs-3828	96	8	𝑖	𝑖	SYM
iajs-3828	96	9	→	→	SYM
iajs-3828	96	10	∞	∞	PROPN
iajs-3828	96	11	implies	imply	VERB
iajs-3828	96	12	to	to	PART
iajs-3828	96	13	∫	∫	PROPN
iajs-3828	96	14	¥	¥	PROPN
iajs-3828	96	15	(	(	PUNCT
iajs-3828	96	16	𝑟	𝑟	NOUN
iajs-3828	96	17	)	)	PUNCT
iajs-3828	96	18	𝑞𝜀	𝑞𝜀	ADP
iajs-3828	96	19	0	0	NUM
iajs-3828	96	20	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	96	21	≤	≤	NOUN
iajs-3828	96	22	𝑘	𝑘	DET
iajs-3828	96	23	∫	∫	PROPN
iajs-3828	96	24	¥	¥	PROPN
iajs-3828	96	25	(	(	PUNCT
iajs-3828	96	26	𝑟	𝑟	NOUN
iajs-3828	96	27	)	)	PUNCT
iajs-3828	96	28	𝑞𝜀	𝑞𝜀	ADP
iajs-3828	96	29	0	0	NUM
iajs-3828	96	30	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	96	31	,	,	PUNCT
iajs-3828	96	32	which	which	PRON
iajs-3828	96	33	is	be	AUX
iajs-3828	96	34	a	a	DET
iajs-3828	96	35	contradiction	contradiction	NOUN
iajs-3828	96	36	.	.	PUNCT
iajs-3828	97	1	therefore	therefore	ADV
iajs-3828	97	2	,	,	PUNCT
iajs-3828	97	3	the	the	DET
iajs-3828	97	4	(	(	PUNCT
iajs-3828	97	5	claim	claim	NOUN
iajs-3828	97	6	i	i	NOUN
iajs-3828	97	7	)	)	PUNCT
iajs-3828	97	8	is	be	AUX
iajs-3828	97	9	proved	prove	VERB
iajs-3828	97	10	.	.	PUNCT
iajs-3828	98	1	now	now	ADV
iajs-3828	98	2	,	,	PUNCT
iajs-3828	98	3	to	to	PART
iajs-3828	98	4	prove	prove	VERB
iajs-3828	98	5	(	(	PUNCT
iajs-3828	98	6	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	98	7	)	)	PUNCT
iajs-3828	98	8	is	be	AUX
iajs-3828	98	9	cauchy	cauchy	ADJ
iajs-3828	98	10	sequence	sequence	NOUN
iajs-3828	98	11	in	in	ADP
iajs-3828	98	12	(	(	PUNCT
iajs-3828	98	13	£	£	NOUN
iajs-3828	98	14	,	,	PUNCT
iajs-3828	98	15	𝜌	𝜌	NOUN
iajs-3828	98	16	)	)	PUNCT
iajs-3828	98	17	.	.	PUNCT
iajs-3828	99	1	fix	fix	NOUN
iajs-3828	99	2	ε	ε	PROPN
iajs-3828	99	3	>	>	X
iajs-3828	99	4	0	0	NUM
iajs-3828	99	5	.	.	PUNCT
iajs-3828	100	1	by	by	ADP
iajs-3828	100	2	the	the	DET
iajs-3828	100	3	claim	claim	NOUN
iajs-3828	100	4	,	,	PUNCT
iajs-3828	100	5	∃𝑚0	∃𝑚0	NOUN
iajs-3828	100	6	such	such	ADJ
iajs-3828	100	7	that	that	SCONJ
iajs-3828	100	8	if	if	SCONJ
iajs-3828	100	9	𝑗	𝑗	PROPN
iajs-3828	100	10	,	,	PUNCT
iajs-3828	100	11	𝑖	𝑖	PRON
iajs-3828	100	12	≥	≥	NOUN
iajs-3828	100	13	𝑚0	𝑚0	NOUN
iajs-3828	100	14	with	with	ADP
iajs-3828	100	15	𝑗	𝑗	INTJ
iajs-3828	100	16	−	−	PROPN
iajs-3828	100	17	𝑖	𝑖	SYM
iajs-3828	100	18	≡	≡	PROPN
iajs-3828	100	19	1(mod	1(mod	NUM
iajs-3828	100	20	𝑛	𝑛	X
iajs-3828	100	21	)	)	PUNCT
iajs-3828	100	22	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	100	23	,	,	PUNCT
iajs-3828	100	24	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	100	25	)	)	PUNCT
iajs-3828	100	26	≤	≤	NUM
iajs-3828	100	27	ε	ε	PROPN
iajs-3828	100	28	𝑛	𝑛	PROPN
iajs-3828	100	29	.	.	PUNCT
iajs-3828	101	1	since	since	SCONJ
iajs-3828	101	2	lim	lim	PROPN
iajs-3828	101	3	𝑚→∞	𝑚→∞	NUM
iajs-3828	101	4	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	101	5	,	,	PUNCT
iajs-3828	101	6	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	101	7	)	)	PUNCT
iajs-3828	101	8	=	=	SYM
iajs-3828	101	9	0	0	NUM
iajs-3828	101	10	,	,	PUNCT
iajs-3828	101	11	∃𝑚1	∃𝑚1	NOUN
iajs-3828	101	12	∈	∈	NOUN
iajs-3828	102	1	𝑁	𝑁	PROPN
iajs-3828	102	2	such	such	ADJ
iajs-3828	102	3	that	that	SCONJ
iajs-3828	102	4	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	102	5	,	,	PUNCT
iajs-3828	102	6	𝑎𝑚+1	𝑎𝑚+1	NOUN
iajs-3828	102	7	)	)	PUNCT
iajs-3828	102	8	≤	≤	NOUN
iajs-3828	102	9	𝜀	𝜀	ADP
iajs-3828	102	10	𝑛	𝑛	PROPN
iajs-3828	102	11	,	,	PUNCT
iajs-3828	102	12	∀𝑚	∀𝑚	X
iajs-3828	102	13	≥	≥	NOUN
iajs-3828	102	14	𝑚1	𝑚1	NOUN
iajs-3828	102	15	suppose	suppose	VERB
iajs-3828	102	16	𝑐	𝑐	X
iajs-3828	102	17	,	,	PUNCT
iajs-3828	102	18	𝑣	𝑣	PRON
iajs-3828	102	19	≥	≥	NOUN
iajs-3828	102	20	𝑚𝑎𝑥{𝑚0	𝑚𝑎𝑥{𝑚0	X
iajs-3828	102	21	,	,	PUNCT
iajs-3828	102	22	𝑚1	𝑚1	NOUN
iajs-3828	102	23	}	}	PUNCT
iajs-3828	102	24	and	and	CCONJ
iajs-3828	103	1	𝑐	𝑐	PROPN
iajs-3828	103	2	>	>	X
iajs-3828	103	3	𝑣.	𝑣.	NOUN
iajs-3828	103	4	then	then	ADV
iajs-3828	103	5	there	there	PRON
iajs-3828	103	6	exists	exist	VERB
iajs-3828	103	7	ℎ	ℎ	PROPN
iajs-3828	103	8	∈	∈	PROPN
iajs-3828	103	9	{	{	PUNCT
iajs-3828	103	10	1,2	1,2	NUM
iajs-3828	103	11	,	,	PUNCT
iajs-3828	103	12	…	…	PUNCT
iajs-3828	103	13	.	.	PUNCT
iajs-3828	103	14	.	.	PUNCT
iajs-3828	104	1	,	,	PUNCT
iajs-3828	104	2	𝑛	𝑛	X
iajs-3828	104	3	}	}	PUNCT
iajs-3828	104	4	such	such	ADJ
iajs-3828	104	5	that	that	SCONJ
iajs-3828	104	6	𝑣	𝑣	PRON
iajs-3828	104	7	−	−	PROPN
iajs-3828	104	8	𝑐	𝑐	PROPN
iajs-3828	104	9	≡	≡	PROPN
iajs-3828	104	10	ℎ(mod	ℎ(mod	PROPN
iajs-3828	104	11	𝑛	𝑛	PROPN
iajs-3828	104	12	)	)	PUNCT
iajs-3828	104	13	.	.	PUNCT
iajs-3828	105	1	therefore	therefore	ADV
iajs-3828	105	2	,	,	PUNCT
iajs-3828	105	3	𝑣	𝑣	ADP
iajs-3828	105	4	−	−	PROPN
iajs-3828	105	5	𝑐	𝑐	PROPN
iajs-3828	105	6	+	+	NUM
iajs-3828	105	7	𝑟	𝑟	NUM
iajs-3828	105	8	≡	≡	PROPN
iajs-3828	105	9	1(mod	1(mod	NUM
iajs-3828	105	10	𝑛	𝑛	NOUN
iajs-3828	105	11	)	)	PUNCT
iajs-3828	105	12	for	for	ADP
iajs-3828	105	13	𝑟	𝑟	NOUN
iajs-3828	105	14	=	=	SYM
iajs-3828	105	15	𝑛	𝑛	DET
iajs-3828	105	16	−	−	NOUN
iajs-3828	105	17	ℎ	ℎ	X
iajs-3828	105	18	+	+	NOUN
iajs-3828	105	19	1	1	X
iajs-3828	105	20	.	.	PUNCT
iajs-3828	106	1	so	so	ADV
iajs-3828	106	2	,	,	PUNCT
iajs-3828	106	3	getting	get	VERB
iajs-3828	106	4	𝜌(𝑎𝑐	𝜌(𝑎𝑐	NUM
iajs-3828	106	5	,	,	PUNCT
iajs-3828	106	6	𝑎𝑣	𝑎𝑣	NOUN
iajs-3828	106	7	)	)	PUNCT
iajs-3828	106	8	≤	≤	NUM
iajs-3828	107	1	𝑞(𝜌(𝑎𝑐	𝑞(𝜌(𝑎𝑐	NOUN
iajs-3828	107	2	,	,	PUNCT
iajs-3828	107	3	𝑎𝑣+𝑟	𝑎𝑣+𝑟	PROPN
iajs-3828	107	4	)	)	PUNCT
iajs-3828	107	5	+	+	NUM
iajs-3828	107	6	𝜌(𝑎𝑣+𝑟	𝜌(𝑎𝑣+𝑟	NOUN
iajs-3828	107	7	,	,	PUNCT
iajs-3828	107	8	𝑎𝑣	𝑎𝑣	NOUN
iajs-3828	107	9	)	)	PUNCT
iajs-3828	107	10	)	)	PUNCT
iajs-3828	107	11	.	.	PUNCT
iajs-3828	108	1	≤	≤	NUM
iajs-3828	108	2	𝑞	𝑞	X
iajs-3828	108	3	𝜌(𝑎𝑐	𝜌(𝑎𝑐	PROPN
iajs-3828	108	4	,	,	PUNCT
iajs-3828	108	5	𝑎𝑣+𝑟	𝑎𝑣+𝑟	PROPN
iajs-3828	108	6	)	)	PUNCT
iajs-3828	108	7	+	+	CCONJ
iajs-3828	108	8	𝑞	𝑞	X
iajs-3828	108	9	2	2	NUM
iajs-3828	108	10	𝜌(𝑎𝑣+𝑟	𝜌(𝑎𝑣+𝑟	NOUN
iajs-3828	108	11	,	,	PUNCT
iajs-3828	108	12	𝑎𝑣+𝑟−1	𝑎𝑣+𝑟−1	ADJ
iajs-3828	108	13	)	)	PUNCT
iajs-3828	108	14	+	+	NUM
iajs-3828	108	15	𝑞	𝑞	PROPN
iajs-3828	108	16	3𝜌(𝑎𝑣+𝑟−1	3𝜌(𝑎𝑣+𝑟−1	NUM
iajs-3828	108	17	,	,	PUNCT
iajs-3828	108	18	𝑎𝑣+𝑟−2	𝑎𝑣+𝑟−2	PROPN
iajs-3828	108	19	)	)	PUNCT
iajs-3828	109	1	+	+	NUM
iajs-3828	109	2	⋯+	⋯+	NOUN
iajs-3828	109	3	𝑞	𝑞	NOUN
iajs-3828	109	4	𝑟𝜌(𝑎𝑣+1	𝑟𝜌(𝑎𝑣+1	NOUN
iajs-3828	109	5	,	,	PUNCT
iajs-3828	109	6	𝑎𝑣	𝑎𝑣	X
iajs-3828	109	7	)	)	PUNCT
iajs-3828	109	8	by	by	ADP
iajs-3828	109	9	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	109	10	,	,	PUNCT
iajs-3828	109	11	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	109	12	)	)	PUNCT
iajs-3828	109	13	≤	≤	NUM
iajs-3828	109	14	ε	ε	PROPN
iajs-3828	109	15	𝑛	𝑛	PROPN
iajs-3828	109	16	and	and	CCONJ
iajs-3828	109	17	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	109	18	,	,	PUNCT
iajs-3828	109	19	𝑎𝑚+1	𝑎𝑚+1	NOUN
iajs-3828	109	20	)	)	PUNCT
iajs-3828	109	21	≤	≤	NOUN
iajs-3828	109	22	𝜀	𝜀	ADP
iajs-3828	109	23	𝑛	𝑛	PROPN
iajs-3828	109	24	and	and	CCONJ
iajs-3828	109	25	from	from	ADP
iajs-3828	109	26	the	the	DET
iajs-3828	109	27	last	last	ADJ
iajs-3828	109	28	inequality	inequality	NOUN
iajs-3828	109	29	,	,	PUNCT
iajs-3828	109	30	𝜌(𝑎𝑐	𝜌(𝑎𝑐	NUM
iajs-3828	109	31	,	,	PUNCT
iajs-3828	109	32	𝑎𝑣	𝑎𝑣	NOUN
iajs-3828	109	33	)	)	PUNCT
iajs-3828	109	34	≤	≤	NOUN
iajs-3828	109	35	𝑞	𝑞	X
iajs-3828	109	36	𝜀	𝜀	PROPN
iajs-3828	109	37	𝑛	𝑛	PROPN
iajs-3828	109	38	+	+	NUM
iajs-3828	109	39	𝑞2	𝑞2	NOUN
iajs-3828	109	40	𝜀	𝜀	X
iajs-3828	109	41	𝑛	𝑛	PROPN
iajs-3828	109	42	+	+	CCONJ
iajs-3828	109	43	𝑞3	𝑞3	PROPN
iajs-3828	109	44	𝜀	𝜀	NOUN
iajs-3828	109	45	𝑛	𝑛	PRON
iajs-3828	109	46	+	+	NOUN
iajs-3828	109	47	⋯+	⋯+	NOUN
iajs-3828	109	48	𝑞𝑟	𝑞𝑟	ADP
iajs-3828	109	49	𝜀	𝜀	ADP
iajs-3828	109	50	𝑛	𝑛	VERB
iajs-3828	109	51	≤	≤	NUM
iajs-3828	109	52	𝑞	𝑞	PROPN
iajs-3828	109	53	𝜀	𝜀	X
iajs-3828	109	54	𝑛	𝑛	PROPN
iajs-3828	109	55	(	(	PUNCT
iajs-3828	109	56	1	1	NUM
iajs-3828	109	57	1−𝑞	1−𝑞	NUM
iajs-3828	109	58	)	)	PUNCT
iajs-3828	110	1	→	→	SYM
iajs-3828	110	2	0	0	NUM
iajs-3828	110	3	,	,	PUNCT
iajs-3828	110	4	as	as	ADP
iajs-3828	110	5	𝑛	𝑛	PROPN
iajs-3828	110	6	→	→	SYM
iajs-3828	110	7	∞	∞	PROPN
iajs-3828	110	8	this	this	PRON
iajs-3828	110	9	proves	prove	VERB
iajs-3828	110	10	that	that	SCONJ
iajs-3828	110	11	(	(	PUNCT
iajs-3828	110	12	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	110	13	)	)	PUNCT
iajs-3828	110	14	is	be	AUX
iajs-3828	110	15	the	the	DET
iajs-3828	110	16	cauchy	cauchy	ADJ
iajs-3828	110	17	sequence	sequence	NOUN
iajs-3828	110	18	.	.	PUNCT
iajs-3828	111	1	the	the	DET
iajs-3828	111	2	completeness	completeness	NOUN
iajs-3828	111	3	of	of	ADP
iajs-3828	111	4	(	(	PUNCT
iajs-3828	111	5	£	£	PROPN
iajs-3828	111	6	,	,	PUNCT
iajs-3828	111	7	𝜌	𝜌	X
iajs-3828	111	8	)	)	PUNCT
iajs-3828	111	9	implies	imply	VERB
iajs-3828	111	10	to	to	PART
iajs-3828	111	11	exist	exist	VERB
iajs-3828	111	12	𝑐	𝑐	PROPN
iajs-3828	111	13	∈	∈	PROPN
iajs-3828	111	14	ihjpas	ihjpa	NOUN
iajs-3828	111	15	.	.	PUNCT
iajs-3828	112	1	2025,38(2	2025,38(2	PROPN
iajs-3828	112	2	)	)	PUNCT
iajs-3828	112	3	379	379	NUM
iajs-3828	112	4	£	£	NOUN
iajs-3828	113	1	such	such	ADJ
iajs-3828	113	2	that	that	SCONJ
iajs-3828	113	3	lim	lim	PROPN
iajs-3828	113	4	𝑚→∞	𝑚→∞	PUNCT
iajs-3828	113	5	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	113	6	=	=	PUNCT
iajs-3828	113	7	𝑐.	𝑐.	NOUN
iajs-3828	113	8	to	to	PART
iajs-3828	113	9	prove	prove	VERB
iajs-3828	113	10	𝑐	𝑐	PROPN
iajs-3828	113	11	is	be	AUX
iajs-3828	113	12	a	a	DET
iajs-3828	113	13	fixed	fix	VERB
iajs-3828	113	14	point	point	NOUN
iajs-3828	113	15	for	for	ADP
iajs-3828	113	16	𝑓.	𝑓.	NOUN
iajs-3828	113	17	since	since	SCONJ
iajs-3828	113	18	£	£	NOUN
iajs-3828	113	19	=	=	SYM
iajs-3828	113	20	⋃	⋃	ADP
iajs-3828	113	21	£	£	SYM
iajs-3828	113	22	𝑖	𝑖	SYM
iajs-3828	113	23	𝑛	𝑛	PRON
iajs-3828	113	24	𝑖=1	𝑖=1	PROPN
iajs-3828	113	25	is	be	AUX
iajs-3828	113	26	a	a	DET
iajs-3828	113	27	cyclic	cyclic	ADJ
iajs-3828	113	28	representation	representation	NOUN
iajs-3828	113	29	of	of	ADP
iajs-3828	113	30	£	£	SYM
iajs-3828	113	31	w.r.t	w.r.t	NOUN
iajs-3828	113	32	.	.	PUNCT
iajs-3828	113	33	,	,	PUNCT
iajs-3828	113	34	𝑓	𝑓	X
iajs-3828	113	35	,	,	PUNCT
iajs-3828	113	36	the	the	DET
iajs-3828	113	37	sequence	sequence	NOUN
iajs-3828	113	38	(	(	PUNCT
iajs-3828	113	39	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	113	40	)	)	PUNCT
iajs-3828	113	41	has	have	AUX
iajs-3828	113	42	infinite	infinite	ADJ
iajs-3828	113	43	terms	term	NOUN
iajs-3828	113	44	in	in	ADP
iajs-3828	113	45	each	each	DET
iajs-3828	113	46	£	£	PROPN
iajs-3828	113	47	𝑖𝑚for	𝑖𝑚for	ADP
iajs-3828	113	48	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	113	49	∈	∈	PROPN
iajs-3828	113	50	{	{	PUNCT
iajs-3828	113	51	1,2	1,2	NUM
iajs-3828	113	52	,	,	PUNCT
iajs-3828	113	53	.	.	PUNCT
iajs-3828	113	54	.	.	PUNCT
iajs-3828	114	1	.	.	PUNCT
iajs-3828	115	1	,	,	PUNCT
iajs-3828	115	2	𝑛	𝑛	PROPN
iajs-3828	115	3	}	}	PUNCT
iajs-3828	115	4	.	.	PUNCT
iajs-3828	116	1	closeness	closeness	NOUN
iajs-3828	116	2	of	of	ADP
iajs-3828	116	3	£	£	SYM
iajs-3828	116	4	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	116	5	for	for	ADP
iajs-3828	116	6	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	116	7	∈	∈	PROPN
iajs-3828	116	8	{	{	PUNCT
iajs-3828	116	9	1,2	1,2	NUM
iajs-3828	116	10	,	,	PUNCT
iajs-3828	116	11	.	.	PUNCT
iajs-3828	116	12	.	.	PUNCT
iajs-3828	117	1	.	.	PUNCT
iajs-3828	118	1	,	,	PUNCT
iajs-3828	118	2	𝑛	𝑛	X
iajs-3828	118	3	}	}	PUNCT
iajs-3828	118	4	implies	imply	VERB
iajs-3828	118	5	to	to	ADP
iajs-3828	118	6	𝑐	𝑐	PROPN
iajs-3828	118	7	∈	∈	PROPN
iajs-3828	118	8	⋂	⋂	PROPN
iajs-3828	118	9	£	£	SYM
iajs-3828	118	10	𝑖	𝑖	SYM
iajs-3828	118	11	𝑛	𝑛	PRON
iajs-3828	118	12	𝑖=1	𝑖=1	PROPN
iajs-3828	118	13	.	.	PUNCT
iajs-3828	119	1	suppose	suppose	VERB
iajs-3828	119	2	that	that	SCONJ
iajs-3828	119	3	𝑐	𝑐	PROPN
iajs-3828	119	4	∈	∈	PROPN
iajs-3828	119	5	£	£	SYM
iajs-3828	119	6	𝑖	𝑖	NOUN
iajs-3828	119	7	and	and	CCONJ
iajs-3828	119	8	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	119	9	∈	∈	PROPN
iajs-3828	119	10	£	£	SYM
iajs-3828	119	11	𝑖+1	𝑖+1	NUM
iajs-3828	119	12	.	.	PUNCT
iajs-3828	120	1	since	since	SCONJ
iajs-3828	120	2	(	(	PUNCT
iajs-3828	120	3	£	£	NOUN
iajs-3828	120	4	,	,	PUNCT
iajs-3828	120	5	𝜌	𝜌	X
iajs-3828	120	6	)	)	PUNCT
iajs-3828	120	7	is	be	AUX
iajs-3828	120	8	complete	complete	ADJ
iajs-3828	120	9	,	,	PUNCT
iajs-3828	120	10	there	there	PRON
iajs-3828	120	11	exists	exist	VERB
iajs-3828	120	12	appoint	appoint	VERB
iajs-3828	120	13	𝑐	𝑐	PROPN
iajs-3828	120	14	∈	∈	NOUN
iajs-3828	121	1	£	£	NOUN
iajs-3828	121	2	=	=	PUNCT
iajs-3828	121	3	⋃	⋃	ADP
iajs-3828	121	4	£	£	SYM
iajs-3828	121	5	𝑖	𝑖	SYM
iajs-3828	121	6	𝑛	𝑛	NOUN
iajs-3828	121	7	𝑖=1	𝑖=1	PUNCT
iajs-3828	121	8	such	such	ADJ
iajs-3828	121	9	that	that	PRON
iajs-3828	121	10	𝑐	𝑐	PROPN
iajs-3828	121	11	=	=	SYM
iajs-3828	121	12	lim	lim	PROPN
iajs-3828	121	13	𝑚→∞	𝑚→∞	NUM
iajs-3828	121	14	𝑓𝑎𝑚	𝑓𝑎𝑚	VERB
iajs-3828	121	15	0	0	NUM
iajs-3828	121	16	<	<	X
iajs-3828	121	17	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	121	18	,	,	PUNCT
iajs-3828	121	19	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	121	20	)	)	PUNCT
iajs-3828	121	21	≤	≤	NOUN
iajs-3828	121	22	𝑞	𝑞	PRON
iajs-3828	121	23	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	121	24	,	,	PUNCT
iajs-3828	121	25	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	121	26	)	)	PUNCT
iajs-3828	122	1	+	+	CCONJ
iajs-3828	122	2	𝑞	𝑞	X
iajs-3828	122	3	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	PROPN
iajs-3828	122	4	,	,	PUNCT
iajs-3828	122	5	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	122	6	)	)	PUNCT
iajs-3828	122	7	→	→	SYM
iajs-3828	122	8	0	0	NUM
iajs-3828	122	9	as	as	ADP
iajs-3828	122	10	𝑚	𝑚	PROPN
iajs-3828	122	11	→	→	SYM
iajs-3828	122	12	∞.	∞.	PROPN
iajs-3828	122	13	indeed	indeed	ADV
iajs-3828	122	14	both	both	DET
iajs-3828	122	15	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	122	16	,	,	PUNCT
iajs-3828	122	17	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	122	18	)	)	PUNCT
iajs-3828	122	19	and	and	CCONJ
iajs-3828	122	20	𝜌(𝑓𝑎𝑚𝑟	𝜌(𝑓𝑎𝑚𝑟	ADJ
iajs-3828	122	21	,	,	PUNCT
iajs-3828	122	22	𝑓𝑐	𝑓𝑐	PROPN
iajs-3828	122	23	)	)	PUNCT
iajs-3828	122	24	converge	converge	VERB
iajs-3828	122	25	to	to	ADP
iajs-3828	122	26	0	0	NUM
iajs-3828	122	27	as	as	ADP
iajs-3828	122	28	𝑚	𝑚	PROPN
iajs-3828	122	29	→	→	SYM
iajs-3828	122	30	∞	∞	PROPN
iajs-3828	122	31	,	,	PUNCT
iajs-3828	122	32	for	for	ADP
iajs-3828	122	33	the	the	DET
iajs-3828	122	34	first	first	ADJ
iajs-3828	122	35	one	one	NOUN
iajs-3828	122	36	it	it	PRON
iajs-3828	122	37	is	be	AUX
iajs-3828	122	38	obvious	obvious	ADJ
iajs-3828	122	39	,	,	PUNCT
iajs-3828	122	40	while	while	SCONJ
iajs-3828	122	41	for	for	ADP
iajs-3828	122	42	the	the	DET
iajs-3828	122	43	second	second	ADJ
iajs-3828	122	44	one	one	NOUN
iajs-3828	122	45	we	we	PRON
iajs-3828	122	46	have	have	VERB
iajs-3828	122	47	∫	∫	PROPN
iajs-3828	122	48	¥	¥	NUM
iajs-3828	122	49	(	(	PUNCT
iajs-3828	122	50	𝑟	𝑟	NOUN
iajs-3828	122	51	)	)	PUNCT
iajs-3828	122	52	𝜌(𝑓𝑎𝑚,𝑓𝑐	𝜌(𝑓𝑎𝑚,𝑓𝑐	NOUN
iajs-3828	122	53	)	)	PUNCT
iajs-3828	122	54	0	0	NUM
iajs-3828	122	55	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	122	56	≤	≤	NOUN
iajs-3828	122	57	𝑘	𝑘	DET
iajs-3828	122	58	∫	∫	PROPN
iajs-3828	122	59	¥	¥	PROPN
iajs-3828	122	60	(	(	PUNCT
iajs-3828	122	61	𝑟	𝑟	NOUN
iajs-3828	122	62	)	)	PUNCT
iajs-3828	122	63	𝜌(𝑎𝑚,𝑐	𝜌(𝑎𝑚,𝑐	NOUN
iajs-3828	122	64	)	)	PUNCT
iajs-3828	122	65	0	0	NUM
iajs-3828	123	1	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	123	2	→	→	SYM
iajs-3828	123	3	0	0	NUM
iajs-3828	123	4	as	as	ADP
iajs-3828	123	5	𝑚	𝑚	PROPN
iajs-3828	123	6	→	→	SYM
iajs-3828	123	7	∞.	∞.	PROPN
iajs-3828	123	8	now	now	ADV
iajs-3828	123	9	,	,	PUNCT
iajs-3828	123	10	if	if	SCONJ
iajs-3828	123	11	𝜌(𝑓𝑎𝑚	𝜌(𝑓𝑎𝑚	PROPN
iajs-3828	123	12	,	,	PUNCT
iajs-3828	123	13	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	123	14	)	)	PUNCT
iajs-3828	123	15	does	do	AUX
iajs-3828	123	16	not	not	PART
iajs-3828	123	17	converge	converge	VERB
iajs-3828	123	18	to	to	ADP
iajs-3828	123	19	0	0	NUM
iajs-3828	123	20	as	as	ADP
iajs-3828	123	21	𝑚	𝑚	PROPN
iajs-3828	123	22	→	→	SYM
iajs-3828	123	23	∞	∞	PROPN
iajs-3828	123	24	,	,	PUNCT
iajs-3828	123	25	then	then	ADV
iajs-3828	123	26	there	there	PRON
iajs-3828	123	27	exists	exist	VERB
iajs-3828	123	28	a	a	DET
iajs-3828	123	29	subsequence	subsequence	NOUN
iajs-3828	123	30	(	(	PUNCT
iajs-3828	123	31	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	123	32	)	)	PUNCT
iajs-3828	123	33	𝑟∈𝑁	𝑟∈𝑁	PROPN
iajs-3828	123	34	of	of	ADP
iajs-3828	123	35	(	(	PUNCT
iajs-3828	123	36	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	123	37	)	)	PUNCT
iajs-3828	123	38	with	with	ADP
iajs-3828	123	39	𝑎𝑚𝑟	𝑎𝑚𝑟	ADJ
iajs-3828	123	40	∈	∈	PROPN
iajs-3828	123	41	£	£	SYM
iajs-3828	123	42	𝑖−1	𝑖−1	NOUN
iajs-3828	123	43	such	such	ADJ
iajs-3828	123	44	that	that	SCONJ
iajs-3828	123	45	𝜌(𝑓𝑎𝑚𝑟	𝜌(𝑓𝑎𝑚𝑟	ADV
iajs-3828	123	46	,	,	PUNCT
iajs-3828	123	47	𝑓𝑐	𝑓𝑐	PROPN
iajs-3828	123	48	)	)	PUNCT
iajs-3828	123	49	≥	≥	NOUN
iajs-3828	123	50	𝜀	𝜀	NOUN
iajs-3828	123	51	for	for	ADP
iajs-3828	123	52	a	a	DET
iajs-3828	123	53	certain	certain	ADJ
iajs-3828	123	54	𝜀	𝜀	NOUN
iajs-3828	123	55	>	>	X
iajs-3828	123	56	0	0	NUM
iajs-3828	123	57	,	,	PUNCT
iajs-3828	123	58	we	we	PRON
iajs-3828	123	59	have	have	VERB
iajs-3828	123	60	the	the	DET
iajs-3828	123	61	following	follow	VERB
iajs-3828	123	62	contradiction	contradiction	NOUN
iajs-3828	123	63	0	0	PUNCT
iajs-3828	123	64	<	<	X
iajs-3828	123	65	∫	∫	PROPN
iajs-3828	123	66	¥	¥	PROPN
iajs-3828	123	67	(	(	PUNCT
iajs-3828	123	68	𝑟	𝑟	NOUN
iajs-3828	123	69	)	)	PUNCT
iajs-3828	123	70	𝜀	𝜀	PART
iajs-3828	123	71	0	0	NUM
iajs-3828	123	72	𝑑𝑟	𝑑𝑟	PROPN
iajs-3828	123	73	≤	≤	NUM
iajs-3828	123	74	∫	∫	PROPN
iajs-3828	123	75	¥	¥	PROPN
iajs-3828	123	76	(	(	PUNCT
iajs-3828	123	77	𝑟	𝑟	NOUN
iajs-3828	123	78	)	)	PUNCT
iajs-3828	123	79	𝜌(𝑓𝑎𝑚𝑟	𝜌(𝑓𝑎𝑚𝑟	ADV
iajs-3828	123	80	,	,	PUNCT
iajs-3828	123	81	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	123	82	)	)	PUNCT
iajs-3828	123	83	0	0	NUM
iajs-3828	123	84	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	123	85	→	→	SYM
iajs-3828	123	86	0	0	NUM
iajs-3828	123	87	as	as	ADP
iajs-3828	123	88	𝑟	𝑟	NOUN
iajs-3828	123	89	→	→	SYM
iajs-3828	123	90	∞.	∞.	PROPN
iajs-3828	123	91	this	this	PRON
iajs-3828	123	92	means	mean	VERB
iajs-3828	123	93	that	that	SCONJ
iajs-3828	123	94	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	123	95	,	,	PUNCT
iajs-3828	123	96	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	123	97	)	)	PUNCT
iajs-3828	123	98	≤	≤	NOUN
iajs-3828	123	99	0	0	PUNCT
iajs-3828	124	1	thus	thus	ADV
iajs-3828	124	2	,	,	PUNCT
iajs-3828	124	3	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	124	4	,	,	PUNCT
iajs-3828	124	5	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	124	6	)	)	PUNCT
iajs-3828	124	7	=	=	SYM
iajs-3828	124	8	0	0	NUM
iajs-3828	124	9	,	,	PUNCT
iajs-3828	124	10	𝑐	𝑐	PROPN
iajs-3828	124	11	is	be	AUX
iajs-3828	124	12	the	the	DET
iajs-3828	124	13	fixed	fix	VERB
iajs-3828	124	14	point	point	NOUN
iajs-3828	124	15	of	of	ADP
iajs-3828	124	16	𝑓.	𝑓.	NOUN
iajs-3828	124	17	for	for	ADP
iajs-3828	124	18	the	the	DET
iajs-3828	124	19	uniqueness	uniqueness	NOUN
iajs-3828	124	20	of	of	ADP
iajs-3828	124	21	fixed	fix	VERB
iajs-3828	124	22	point	point	NOUN
iajs-3828	124	23	𝑐.	𝑐.	NOUN
iajs-3828	124	24	assume	assume	VERB
iajs-3828	124	25	a	a	DET
iajs-3828	124	26	fixed	fixed	ADJ
iajs-3828	124	27	point	point	NOUN
iajs-3828	124	28	𝑤	𝑤	ADP
iajs-3828	124	29	of	of	ADP
iajs-3828	124	30	𝑓	𝑓	DET
iajs-3828	124	31	differs	differ	NOUN
iajs-3828	124	32	from	from	ADP
iajs-3828	124	33	𝑐	𝑐	PROPN
iajs-3828	124	34	,	,	PUNCT
iajs-3828	124	35	i.e.	i.e.	X
iajs-3828	124	36	,	,	PUNCT
iajs-3828	124	37	𝑓𝑤	𝑓𝑤	ADP
iajs-3828	124	38	=	=	NOUN
iajs-3828	124	39	𝑤.	𝑤.	NOUN
iajs-3828	124	40	the	the	DET
iajs-3828	124	41	cyclic	cyclic	ADJ
iajs-3828	124	42	character	character	NOUN
iajs-3828	124	43	of	of	ADP
iajs-3828	124	44	𝑓	𝑓	PRON
iajs-3828	124	45	and	and	CCONJ
iajs-3828	124	46	𝑐	𝑐	NOUN
iajs-3828	124	47	,	,	PUNCT
iajs-3828	124	48	𝑤	𝑤	ADP
iajs-3828	124	49	∈	∈	NOUN
iajs-3828	125	1	£	£	NOUN
iajs-3828	125	2	=	=	PUNCT
iajs-3828	125	3	⋃	⋃	ADP
iajs-3828	125	4	£	£	SYM
iajs-3828	125	5	𝑖	𝑖	SYM
iajs-3828	125	6	𝑛	𝑛	PRON
iajs-3828	125	7	𝑖=1	𝑖=1	PROPN
iajs-3828	125	8	are	be	AUX
iajs-3828	125	9	fixed	fix	VERB
iajs-3828	125	10	points	point	NOUN
iajs-3828	125	11	of	of	ADP
iajs-3828	125	12	𝑓	𝑓	DET
iajs-3828	125	13	implying	imply	VERB
iajs-3828	125	14	that	that	SCONJ
iajs-3828	125	15	𝑐	𝑐	NOUN
iajs-3828	125	16	,	,	PUNCT
iajs-3828	125	17	𝑤	𝑤	ADP
iajs-3828	125	18	∈	∈	PROPN
iajs-3828	125	19	⋂	⋂	PROPN
iajs-3828	125	20	£	£	SYM
iajs-3828	125	21	𝑖	𝑖	SYM
iajs-3828	125	22	𝑛	𝑛	PRON
iajs-3828	125	23	𝑖=1	𝑖=1	PROPN
iajs-3828	125	24	.	.	PUNCT
iajs-3828	126	1	by	by	ADP
iajs-3828	126	2	(	(	PUNCT
iajs-3828	126	3	1	1	NUM
iajs-3828	126	4	)	)	PUNCT
iajs-3828	126	5	,	,	PUNCT
iajs-3828	126	6	we	we	PRON
iajs-3828	126	7	obtain	obtain	VERB
iajs-3828	126	8	∫	∫	PROPN
iajs-3828	126	9	¥	¥	PROPN
iajs-3828	126	10	(	(	PUNCT
iajs-3828	126	11	𝑟	𝑟	NOUN
iajs-3828	126	12	)	)	PUNCT
iajs-3828	126	13	𝜌(𝑐,𝑤	𝜌(𝑐,𝑤	NUM
iajs-3828	126	14	)	)	PUNCT
iajs-3828	126	15	0	0	NUM
iajs-3828	127	1	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	127	2	=	=	SYM
iajs-3828	127	3	∫	∫	PROPN
iajs-3828	127	4	¥	¥	PROPN
iajs-3828	127	5	(	(	PUNCT
iajs-3828	127	6	𝑟	𝑟	NOUN
iajs-3828	127	7	)	)	PUNCT
iajs-3828	127	8	𝜌(𝑓𝑐,𝑓𝑤	𝜌(𝑓𝑐,𝑓𝑤	NUM
iajs-3828	127	9	)	)	PUNCT
iajs-3828	127	10	0	0	NUM
iajs-3828	127	11	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	127	12	≤	≤	NOUN
iajs-3828	127	13	𝑘	𝑘	PRON
iajs-3828	127	14	∫	∫	PROPN
iajs-3828	127	15	¥	¥	PROPN
iajs-3828	127	16	(	(	PUNCT
iajs-3828	127	17	𝑟	𝑟	NOUN
iajs-3828	127	18	)	)	PUNCT
iajs-3828	127	19	𝜌(𝑐,𝑤	𝜌(𝑐,𝑤	NUM
iajs-3828	127	20	)	)	PUNCT
iajs-3828	127	21	0	0	NUM
iajs-3828	127	22	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	127	23	,	,	PUNCT
iajs-3828	127	24	which	which	PRON
iajs-3828	127	25	is	be	AUX
iajs-3828	127	26	contradiction	contradiction	NOUN
iajs-3828	127	27	,	,	PUNCT
iajs-3828	127	28	consequently	consequently	ADV
iajs-3828	127	29	,	,	PUNCT
iajs-3828	127	30	𝑐	𝑐	NOUN
iajs-3828	127	31	=	=	PUNCT
iajs-3828	127	32	𝑤	𝑤	PROPN
iajs-3828	127	33	for	for	ADP
iajs-3828	127	34	each	each	DET
iajs-3828	127	35	𝑎	𝑎	PROPN
iajs-3828	127	36	∈	∈	PROPN
iajs-3828	127	37	£	£	SYM
iajs-3828	127	38	,	,	PUNCT
iajs-3828	127	39	lim	lim	NOUN
iajs-3828	127	40	𝑚→∞	𝑚→∞	ADV
iajs-3828	127	41	𝑓𝑚𝑎	𝑓𝑚𝑎	PROPN
iajs-3828	127	42	=	=	PUNCT
iajs-3828	127	43	𝑐.	𝑐.	NOUN
iajs-3828	127	44	example	example	NOUN
iajs-3828	127	45	2.2	2.2	NUM
iajs-3828	127	46	:	:	PUNCT
iajs-3828	127	47	let	let	VERB
iajs-3828	127	48	£	£	PRON
iajs-3828	127	49	=	=	PUNCT
iajs-3828	128	1	[	[	X
iajs-3828	128	2	−1	−1	NOUN
iajs-3828	128	3	,	,	PUNCT
iajs-3828	128	4	1	1	NUM
iajs-3828	128	5	]	]	PUNCT
iajs-3828	128	6	and	and	CCONJ
iajs-3828	128	7	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	128	8	,	,	PUNCT
iajs-3828	128	9	𝑏	𝑏	NOUN
iajs-3828	128	10	)	)	PUNCT
iajs-3828	128	11	=	=	SYM
iajs-3828	128	12	|𝑎	|𝑎	X
iajs-3828	129	1	−	−	PROPN
iajs-3828	129	2	𝑏|2	𝑏|2	PROPN
iajs-3828	129	3	is	be	AUX
iajs-3828	129	4	b	b	NOUN
iajs-3828	129	5	-	-	ADJ
iajs-3828	129	6	metric	metric	ADJ
iajs-3828	129	7	with	with	ADP
iajs-3828	129	8	𝑞	𝑞	PROPN
iajs-3828	129	9	=	=	PROPN
iajs-3828	129	10	2	2	X
iajs-3828	129	11	.	.	PUNCT
iajs-3828	129	12	suppose	suppose	VERB
iajs-3828	129	13	£	£	SYM
iajs-3828	129	14	1	1	NUM
iajs-3828	129	15	=	=	SYM
iajs-3828	130	1	[	[	X
iajs-3828	130	2	−1	−1	NOUN
iajs-3828	130	3	,	,	PUNCT
iajs-3828	130	4	0	0	NUM
iajs-3828	130	5	]	]	PUNCT
iajs-3828	130	6	,	,	PUNCT
iajs-3828	130	7	£	£	SYM
iajs-3828	130	8	2	2	NUM
iajs-3828	130	9	=	=	SYM
iajs-3828	131	1	[	[	X
iajs-3828	131	2	0	0	NUM
iajs-3828	131	3	,	,	PUNCT
iajs-3828	131	4	1	1	NUM
iajs-3828	131	5	]	]	PUNCT
iajs-3828	131	6	and	and	CCONJ
iajs-3828	131	7	£	£	NOUN
iajs-3828	131	8	=	=	SYM
iajs-3828	131	9	⋃	⋃	ADP
iajs-3828	131	10	£	£	SYM
iajs-3828	131	11	𝑖	𝑖	ADP
iajs-3828	131	12	2	2	NUM
iajs-3828	131	13	𝑖=1	𝑖=1	PUNCT
iajs-3828	131	14	.	.	PUNCT
iajs-3828	132	1	define	define	VERB
iajs-3828	132	2	𝑓:⋃	𝑓:⋃	ADP
iajs-3828	132	3	£	£	SYM
iajs-3828	132	4	𝑖	𝑖	ADP
iajs-3828	132	5	2	2	NUM
iajs-3828	132	6	𝑖=1	𝑖=1	PUNCT
iajs-3828	132	7	→	→	PUNCT
iajs-3828	132	8	⋃	⋃	ADP
iajs-3828	132	9	£	£	SYM
iajs-3828	132	10	𝑖	𝑖	ADP
iajs-3828	132	11	2	2	NUM
iajs-3828	132	12	𝑖=1	𝑖=1	PUNCT
iajs-3828	132	13	such	such	ADJ
iajs-3828	132	14	that	that	SCONJ
iajs-3828	132	15	(	(	PUNCT
iajs-3828	132	16	𝑎	𝑎	NOUN
iajs-3828	132	17	)	)	PUNCT
iajs-3828	132	18	=	=	NOUN
iajs-3828	133	1	−𝑎	−𝑎	VERB
iajs-3828	133	2	2	2	NUM
iajs-3828	133	3	∀𝑎.	∀𝑎.	ADJ
iajs-3828	133	4	so	so	ADV
iajs-3828	133	5	,	,	PUNCT
iajs-3828	133	6	𝑓(£1	𝑓(£1	PROPN
iajs-3828	133	7	)	)	PUNCT
iajs-3828	133	8	⊂	⊂	PROPN
iajs-3828	133	9	£	£	SYM
iajs-3828	133	10	2	2	NUM
iajs-3828	133	11	,	,	PUNCT
iajs-3828	133	12	𝑓(£2	𝑓(£2	PROPN
iajs-3828	133	13	)	)	PUNCT
iajs-3828	133	14	⊂	⊂	PROPN
iajs-3828	133	15	£	£	SYM
iajs-3828	133	16	1	1	NUM
iajs-3828	133	17	and	and	CCONJ
iajs-3828	133	18	𝑓	𝑓	PRON
iajs-3828	133	19	is	be	AUX
iajs-3828	133	20	contraction	contraction	NOUN
iajs-3828	133	21	of	of	ADP
iajs-3828	133	22	integral	integral	ADJ
iajs-3828	133	23	type	type	NOUN
iajs-3828	133	24	with	with	ADP
iajs-3828	133	25	constant	constant	ADJ
iajs-3828	133	26	𝑘	𝑘	X
iajs-3828	133	27	=	=	SYM
iajs-3828	133	28	1	1	NUM
iajs-3828	133	29	2	2	NUM
iajs-3828	133	30	∈	∈	NOUN
iajs-3828	133	31	(	(	PUNCT
iajs-3828	133	32	0	0	NUM
iajs-3828	133	33	,	,	PUNCT
iajs-3828	133	34	1	1	NUM
iajs-3828	133	35	)	)	PUNCT
iajs-3828	133	36	and	and	CCONJ
iajs-3828	133	37	¥	¥	NUM
iajs-3828	133	38	(	(	PUNCT
iajs-3828	133	39	𝑟	𝑟	NOUN
iajs-3828	133	40	)	)	PUNCT
iajs-3828	133	41	=	=	SYM
iajs-3828	133	42	𝑟	𝑟	NOUN
iajs-3828	133	43	3	3	NUM
iajs-3828	133	44	,	,	PUNCT
iajs-3828	133	45	for	for	ADP
iajs-3828	133	46	𝑎	𝑎	PROPN
iajs-3828	133	47	∈	∈	NOUN
iajs-3828	133	48	£	£	SYM
iajs-3828	133	49	1	1	NUM
iajs-3828	133	50	,	,	PUNCT
iajs-3828	133	51	𝑏	𝑏	PROPN
iajs-3828	133	52	∈	∈	NOUN
iajs-3828	133	53	£	£	SYM
iajs-3828	133	54	2	2	NUM
iajs-3828	133	55	∫	∫	NOUN
iajs-3828	133	56	𝑟	𝑟	NOUN
iajs-3828	133	57	3	3	NUM
iajs-3828	133	58	𝜌(𝑓𝑎,𝑓𝑏	𝜌(𝑓𝑎,𝑓𝑏	PROPN
iajs-3828	133	59	)	)	PUNCT
iajs-3828	133	60	0	0	NUM
iajs-3828	133	61	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	133	62	=	=	SYM
iajs-3828	133	63	∫	∫	PROPN
iajs-3828	134	1	𝑟	𝑟	NOUN
iajs-3828	134	2	3	3	NUM
iajs-3828	134	3	1	1	NUM
iajs-3828	134	4	2	2	NUM
iajs-3828	134	5	|𝑏−𝑎|2	|𝑏−𝑎|2	VERB
iajs-3828	134	6	0	0	NUM
iajs-3828	134	7	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	134	8	≤	≤	NUM
iajs-3828	134	9	1	1	NUM
iajs-3828	134	10	2	2	NUM
iajs-3828	134	11	∫	∫	NOUN
iajs-3828	134	12	𝑟	𝑟	NOUN
iajs-3828	134	13	3	3	NUM
iajs-3828	134	14	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	134	15	𝜌(𝑎,𝑏	𝜌(𝑎,𝑏	NOUN
iajs-3828	134	16	)	)	PUNCT
iajs-3828	134	17	0	0	NUM
iajs-3828	134	18	.	.	PUNCT
iajs-3828	135	1	hence	hence	ADV
iajs-3828	135	2	𝑓	𝑓	DET
iajs-3828	135	3	satisfies	satisfie	NOUN
iajs-3828	135	4	the	the	DET
iajs-3828	135	5	hypothesis	hypothesis	NOUN
iajs-3828	135	6	of	of	ADP
iajs-3828	135	7	(	(	PUNCT
iajs-3828	135	8	theorem	theorem	NOUN
iajs-3828	135	9	2.1	2.1	NUM
iajs-3828	135	10	)	)	PUNCT
iajs-3828	135	11	,	,	PUNCT
iajs-3828	135	12	which	which	PRON
iajs-3828	135	13	has	have	VERB
iajs-3828	135	14	a	a	DET
iajs-3828	135	15	unique	unique	ADJ
iajs-3828	135	16	fixed	fix	VERB
iajs-3828	135	17	point	point	NOUN
iajs-3828	135	18	at	at	ADP
iajs-3828	135	19	0	0	NUM
iajs-3828	135	20	.	.	PUNCT
iajs-3828	136	1	fixed	fix	VERB
iajs-3828	136	2	point	point	NOUN
iajs-3828	136	3	for	for	ADP
iajs-3828	136	4	general	general	ADJ
iajs-3828	136	5	cyclic(∅	cyclic(∅	PROPN
iajs-3828	136	6	−	−	PROPN
iajs-3828	136	7	𝜓	𝜓	NOUN
iajs-3828	136	8	)	)	PUNCT
iajs-3828	136	9	weak	weak	ADJ
iajs-3828	136	10	contractive	contractive	ADJ
iajs-3828	136	11	maps	map	NOUN
iajs-3828	136	12	recall	recall	VERB
iajs-3828	136	13	the	the	DET
iajs-3828	136	14	following	follow	VERB
iajs-3828	136	15	two	two	NUM
iajs-3828	136	16	definitions	definition	NOUN
iajs-3828	136	17	:	:	PUNCT
iajs-3828	136	18	definition	definition	NOUN
iajs-3828	136	19	2.3	2.3	NUM
iajs-3828	136	20	:	:	PUNCT
iajs-3828	136	21	let	let	VERB
iajs-3828	136	22	𝜓	𝜓	NOUN
iajs-3828	136	23	is	be	AUX
iajs-3828	136	24	function	function	NOUN
iajs-3828	136	25	where	where	SCONJ
iajs-3828	136	26	ψ	ψ	X
iajs-3828	136	27	:	:	PUNCT
iajs-3828	136	28	[	[	X
iajs-3828	136	29	0,∞	0,∞	NOUN
iajs-3828	136	30	)	)	PUNCT
iajs-3828	136	31	→	→	PUNCT
iajs-3828	137	1	[	[	X
iajs-3828	137	2	0,∞	0,∞	NOUN
iajs-3828	137	3	)	)	PUNCT
iajs-3828	137	4	is	be	AUX
iajs-3828	137	5	called	call	VERB
iajs-3828	137	6	altering	alter	VERB
iajs-3828	137	7	distance	distance	NOUN
iajs-3828	137	8	function	function	NOUN
iajs-3828	137	9	if	if	SCONJ
iajs-3828	137	10	satisfies(23	satisfies(23	NOUN
iajs-3828	137	11	)	)	PUNCT
iajs-3828	137	12	.	.	PUNCT
iajs-3828	138	1	1	1	X
iajs-3828	138	2	.	.	X
iajs-3828	138	3	𝜓	𝜓	PROPN
iajs-3828	138	4	is	be	AUX
iajs-3828	138	5	monotone	monotone	ADJ
iajs-3828	138	6	increasing	increase	VERB
iajs-3828	138	7	and	and	CCONJ
iajs-3828	138	8	lower	low	ADJ
iajs-3828	138	9	semi	semi	ADJ
iajs-3828	138	10	-	-	ADJ
iajs-3828	138	11	continuous	continuous	ADJ
iajs-3828	138	12	;	;	PUNCT
iajs-3828	138	13	2	2	NUM
iajs-3828	138	14	.	.	X
iajs-3828	138	15	𝜓(𝑟	𝜓(𝑟	NOUN
iajs-3828	138	16	)	)	PUNCT
iajs-3828	138	17	=	=	SYM
iajs-3828	138	18	0	0	PUNCT
iajs-3828	139	1	if	if	SCONJ
iajs-3828	139	2	and	and	CCONJ
iajs-3828	139	3	only	only	ADV
iajs-3828	139	4	if	if	SCONJ
iajs-3828	139	5	𝑟	𝑟	NOUN
iajs-3828	139	6	=	=	SYM
iajs-3828	139	7	0	0	X
iajs-3828	139	8	.	.	PUNCT
iajs-3828	140	1	as	as	ADP
iajs-3828	140	2	in	in	ADP
iajs-3828	140	3	usual	usual	ADJ
iajs-3828	140	4	metric	metric	ADJ
iajs-3828	140	5	spaces	space	NOUN
iajs-3828	140	6	)	)	PUNCT
iajs-3828	140	7	4	4	NUM
iajs-3828	140	8	(	(	PUNCT
iajs-3828	140	9	,	,	PUNCT
iajs-3828	140	10	below	below	ADV
iajs-3828	140	11	,	,	PUNCT
iajs-3828	140	12	we	we	PRON
iajs-3828	140	13	reform	reform	VERB
iajs-3828	140	14	many	many	ADJ
iajs-3828	140	15	concepts	concept	NOUN
iajs-3828	140	16	in	in	ADP
iajs-3828	140	17	a	a	DET
iajs-3828	140	18	b	b	NOUN
iajs-3828	140	19	-	-	PUNCT
iajs-3828	140	20	metric	metric	ADJ
iajs-3828	140	21	space	space	NOUN
iajs-3828	140	22	definition	definition	NOUN
iajs-3828	140	23	2.4	2.4	NUM
iajs-3828	140	24	:	:	PUNCT
iajs-3828	140	25	let	let	VERB
iajs-3828	140	26	(	(	PUNCT
iajs-3828	140	27	£	£	NOUN
iajs-3828	140	28	,	,	PUNCT
iajs-3828	140	29	𝜌	𝜌	AUX
iajs-3828	140	30	)	)	PUNCT
iajs-3828	140	31	be	be	VERB
iajs-3828	140	32	a	a	DET
iajs-3828	140	33	b	b	NOUN
iajs-3828	140	34	-	-	PUNCT
iajs-3828	140	35	metric	metric	ADJ
iajs-3828	140	36	space	space	NOUN
iajs-3828	140	37	,	,	PUNCT
iajs-3828	140	38	𝑛	𝑛	PRON
iajs-3828	140	39	a	a	DET
iajs-3828	140	40	positive	positive	ADJ
iajs-3828	140	41	integer	integer	NOUN
iajs-3828	140	42	£	£	SYM
iajs-3828	140	43	1	1	NUM
iajs-3828	140	44	,	,	PUNCT
iajs-3828	140	45	£	£	SYM
iajs-3828	140	46	2	2	NUM
iajs-3828	140	47	,	,	PUNCT
iajs-3828	140	48	…	…	PUNCT
iajs-3828	140	49	,	,	PUNCT
iajs-3828	140	50	£	£	SYM
iajs-3828	140	51	𝑛	𝑛	DET
iajs-3828	140	52	nonempty	nonempty	X
iajs-3828	140	53	closed	close	VERB
iajs-3828	140	54	subsets	subset	NOUN
iajs-3828	140	55	of	of	ADP
iajs-3828	140	56	£	£	NOUN
iajs-3828	140	57	and	and	CCONJ
iajs-3828	140	58	£	£	NOUN
iajs-3828	140	59	=	=	SYM
iajs-3828	141	1	⋃	⋃	ADP
iajs-3828	141	2	£	£	SYM
iajs-3828	141	3	𝑖	𝑖	SYM
iajs-3828	141	4	𝑛	𝑛	PRON
iajs-3828	141	5	𝑖=1	𝑖=1	PROPN
iajs-3828	141	6	.	.	PUNCT
iajs-3828	142	1	an	an	DET
iajs-3828	142	2	operator	operator	NOUN
iajs-3828	142	3	𝑓	𝑓	DET
iajs-3828	142	4	∶	∶	NOUN
iajs-3828	142	5	£	£	NOUN
iajs-3828	142	6	→	→	SYM
iajs-3828	142	7	£	£	NOUN
iajs-3828	142	8	is	be	AUX
iajs-3828	142	9	said	say	VERB
iajs-3828	142	10	to	to	PART
iajs-3828	142	11	be	be	AUX
iajs-3828	142	12	a	a	DET
iajs-3828	142	13	cyclic	cyclic	ADJ
iajs-3828	142	14	weak	weak	ADJ
iajs-3828	142	15	(	(	PUNCT
iajs-3828	142	16	∅	∅	NOUN
iajs-3828	142	17	−	−	NOUN
iajs-3828	142	18	𝜓)-contraction	𝜓)-contraction	PUNCT
iajs-3828	142	19	if	if	SCONJ
iajs-3828	142	20	1	1	NUM
iajs-3828	142	21	)	)	PUNCT
iajs-3828	143	1	£	£	NOUN
iajs-3828	143	2	=	=	PUNCT
iajs-3828	143	3	⋃	⋃	ADP
iajs-3828	143	4	£	£	SYM
iajs-3828	143	5	𝑖	𝑖	SYM
iajs-3828	143	6	𝑛	𝑛	PRON
iajs-3828	143	7	𝑖=1	𝑖=1	PROPN
iajs-3828	143	8	is	be	AUX
iajs-3828	143	9	a	a	DET
iajs-3828	143	10	cyclic	cyclic	ADJ
iajs-3828	143	11	representation	representation	NOUN
iajs-3828	143	12	of	of	ADP
iajs-3828	143	13	£	£	SYM
iajs-3828	143	14	w.r.t	w.r.t	NOUN
iajs-3828	143	15	.	.	PUNCT
iajs-3828	144	1	,	,	PUNCT
iajs-3828	144	2	𝑓.	𝑓.	NOUN
iajs-3828	144	3	2	2	NUM
iajs-3828	144	4	)	)	PUNCT
iajs-3828	144	5	∅(𝜌(𝑓𝑎	∅(𝜌(𝑓𝑎	PROPN
iajs-3828	144	6	,	,	PUNCT
iajs-3828	144	7	𝑓𝑏	𝑓𝑏	NOUN
iajs-3828	144	8	)	)	PUNCT
iajs-3828	144	9	)	)	PUNCT
iajs-3828	144	10	≤	≤	NUM
iajs-3828	145	1	∅(𝜌(𝑎	∅(𝜌(𝑎	NOUN
iajs-3828	145	2	,	,	PUNCT
iajs-3828	145	3	𝑏	𝑏	NOUN
iajs-3828	145	4	)	)	PUNCT
iajs-3828	145	5	)	)	PUNCT
iajs-3828	146	1	−	−	PROPN
iajs-3828	146	2	𝜓(𝜌(𝑎	𝜓(𝜌(𝑎	NOUN
iajs-3828	146	3	,	,	PUNCT
iajs-3828	146	4	𝑏	𝑏	NOUN
iajs-3828	146	5	)	)	PUNCT
iajs-3828	146	6	)	)	PUNCT
iajs-3828	146	7	,	,	PUNCT
iajs-3828	146	8	for	for	ADP
iajs-3828	146	9	any	any	DET
iajs-3828	146	10	𝑎	𝑎	PROPN
iajs-3828	146	11	∈	∈	NOUN
iajs-3828	146	12	£	£	SYM
iajs-3828	146	13	𝑖	𝑖	NOUN
iajs-3828	146	14	,	,	PUNCT
iajs-3828	146	15	𝑏	𝑏	PROPN
iajs-3828	146	16	∈	∈	PROPN
iajs-3828	146	17	£	£	SYM
iajs-3828	146	18	𝑖+1	𝑖+1	NUM
iajs-3828	146	19	,	,	PUNCT
iajs-3828	146	20	𝑖	𝑖	SYM
iajs-3828	146	21	=	=	SYM
iajs-3828	146	22	1	1	NUM
iajs-3828	146	23	,	,	PUNCT
iajs-3828	146	24	2	2	NUM
iajs-3828	146	25	,	,	PUNCT
iajs-3828	146	26	…	…	PUNCT
iajs-3828	146	27	,	,	PUNCT
iajs-3828	146	28	𝑛	𝑛	NOUN
iajs-3828	146	29	,	,	PUNCT
iajs-3828	146	30	where	where	SCONJ
iajs-3828	146	31	£	£	SYM
iajs-3828	146	32	𝑛+1	𝑛+1	PROPN
iajs-3828	146	33	=	=	SYM
iajs-3828	146	34	£	£	SYM
iajs-3828	146	35	1	1	NUM
iajs-3828	146	36	and	and	CCONJ
iajs-3828	146	37	∅,𝜓	∅,𝜓	NOUN
iajs-3828	146	38	:	:	PUNCT
iajs-3828	146	39	[	[	X
iajs-3828	146	40	0,∞	0,∞	NOUN
iajs-3828	146	41	)	)	PUNCT
iajs-3828	146	42	→	→	PUNCT
iajs-3828	147	1	[	[	X
iajs-3828	147	2	0,∞	0,∞	NUM
iajs-3828	147	3	)	)	PUNCT
iajs-3828	147	4	is	be	AUX
iajs-3828	147	5	a	a	DET
iajs-3828	147	6	non	non	ADJ
iajs-3828	147	7	-	-	ADJ
iajs-3828	147	8	decreasing	decrease	VERB
iajs-3828	147	9	and	and	CCONJ
iajs-3828	147	10	continuous	continuous	ADJ
iajs-3828	147	11	function	function	NOUN
iajs-3828	147	12	satisfying	satisfy	VERB
iajs-3828	147	13	∅(𝑟	∅(𝑟	NOUN
iajs-3828	147	14	)	)	PUNCT
iajs-3828	147	15	>	>	X
iajs-3828	147	16	0	0	NUM
iajs-3828	147	17	,	,	PUNCT
iajs-3828	147	18	𝜓(𝑟	𝜓(𝑟	NOUN
iajs-3828	147	19	)	)	PUNCT
iajs-3828	147	20	>	>	X
iajs-3828	147	21	0	0	NUM
iajs-3828	148	1	for	for	ADP
iajs-3828	148	2	𝑟	𝑟	DET
iajs-3828	148	3	∈	∈	PROPN
iajs-3828	148	4	(	(	PUNCT
iajs-3828	148	5	0,∞	0,∞	NOUN
iajs-3828	148	6	)	)	PUNCT
iajs-3828	148	7	and	and	CCONJ
iajs-3828	148	8	∅(0	∅(0	PROPN
iajs-3828	148	9	)	)	PUNCT
iajs-3828	148	10	=	=	SYM
iajs-3828	148	11	0	0	NUM
iajs-3828	148	12	,	,	PUNCT
iajs-3828	148	13	𝜓(0	𝜓(0	PROPN
iajs-3828	148	14	)	)	PUNCT
iajs-3828	148	15	=	=	SYM
iajs-3828	148	16	0	0	X
iajs-3828	148	17	.	.	PUNCT
iajs-3828	148	18	theorem	theorem	VERB
iajs-3828	148	19	2.5	2.5	NUM
iajs-3828	148	20	:	:	PUNCT
iajs-3828	148	21	let	let	VERB
iajs-3828	148	22	𝑓	𝑓	PRON
iajs-3828	148	23	be	be	AUX
iajs-3828	148	24	a	a	DET
iajs-3828	148	25	self	self	NOUN
iajs-3828	148	26	-	-	PUNCT
iajs-3828	148	27	map	map	NOUN
iajs-3828	148	28	of	of	ADP
iajs-3828	148	29	(	(	PUNCT
iajs-3828	148	30	£	£	PROPN
iajs-3828	148	31	,	,	PUNCT
iajs-3828	148	32	𝜌	𝜌	NOUN
iajs-3828	148	33	)	)	PUNCT
iajs-3828	148	34	satisfies	satisfy	VERB
iajs-3828	148	35	∅(𝜌(𝑓𝑎	∅(𝜌(𝑓𝑎	PROPN
iajs-3828	148	36	,	,	PUNCT
iajs-3828	148	37	𝑓𝑏	𝑓𝑏	PROPN
iajs-3828	148	38	)	)	PUNCT
iajs-3828	148	39	)	)	PUNCT
iajs-3828	149	1	≤	≤	NOUN
iajs-3828	149	2	∅(𝑀(𝑎	∅(𝑀(𝑎	NUM
iajs-3828	149	3	,	,	PUNCT
iajs-3828	149	4	𝑏	𝑏	NOUN
iajs-3828	149	5	)	)	PUNCT
iajs-3828	149	6	)	)	PUNCT
iajs-3828	150	1	−	−	PROPN
iajs-3828	150	2	𝜓(𝑁(𝑎	𝜓(𝑁(𝑎	PROPN
iajs-3828	150	3	,	,	PUNCT
iajs-3828	150	4	𝑏	𝑏	NOUN
iajs-3828	150	5	)	)	PUNCT
iajs-3828	150	6	)	)	PUNCT
iajs-3828	150	7	,	,	PUNCT
iajs-3828	150	8	∀𝑎	∀𝑎	PROPN
iajs-3828	150	9	∈	∈	PROPN
iajs-3828	150	10	£	£	AUX
iajs-3828	150	11	𝑖	𝑖	NOUN
iajs-3828	150	12	,	,	PUNCT
iajs-3828	150	13	𝑏	𝑏	PROPN
iajs-3828	150	14	∈	∈	PROPN
iajs-3828	150	15	£	£	SYM
iajs-3828	150	16	𝑖+1	𝑖+1	NUM
iajs-3828	150	17	(	(	PUNCT
iajs-3828	150	18	4	4	NUM
iajs-3828	150	19	)	)	PUNCT
iajs-3828	150	20	where	where	SCONJ
iajs-3828	150	21	ihjpas	ihjpas	PROPN
iajs-3828	150	22	.	.	PUNCT
iajs-3828	151	1	2025,38(2	2025,38(2	PROPN
iajs-3828	151	2	)	)	PUNCT
iajs-3828	151	3	380	380	NUM
iajs-3828	152	1	𝑀(𝑎	𝑀(𝑎	PROPN
iajs-3828	152	2	,	,	PUNCT
iajs-3828	152	3	𝑏	𝑏	NOUN
iajs-3828	152	4	)	)	PUNCT
iajs-3828	152	5	=	=	SYM
iajs-3828	152	6	𝑡	𝑡	PROPN
iajs-3828	152	7	max{𝜌(𝑎	max{𝜌(𝑎	PROPN
iajs-3828	152	8	,	,	PUNCT
iajs-3828	152	9	𝑏	𝑏	NOUN
iajs-3828	152	10	)	)	PUNCT
iajs-3828	152	11	,	,	PUNCT
iajs-3828	152	12	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	152	13	,	,	PUNCT
iajs-3828	152	14	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	152	15	)	)	PUNCT
iajs-3828	152	16	,	,	PUNCT
iajs-3828	152	17	𝜌(𝑏	𝜌(𝑏	NUM
iajs-3828	152	18	,	,	PUNCT
iajs-3828	152	19	𝑓𝑏	𝑓𝑏	NOUN
iajs-3828	152	20	)	)	PUNCT
iajs-3828	152	21	,	,	PUNCT
iajs-3828	152	22	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	152	23	,	,	PUNCT
iajs-3828	152	24	𝑓𝑏	𝑓𝑏	PROPN
iajs-3828	152	25	)	)	PUNCT
iajs-3828	152	26	,	,	PUNCT
iajs-3828	152	27	𝜌(𝑏	𝜌(𝑏	PRON
iajs-3828	152	28	,	,	PUNCT
iajs-3828	152	29	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	152	30	)	)	PUNCT
iajs-3828	152	31	}	}	PUNCT
iajs-3828	152	32	,	,	PUNCT
iajs-3828	152	33	and	and	CCONJ
iajs-3828	152	34	𝑁(𝑎	𝑁(𝑎	NOUN
iajs-3828	152	35	,	,	PUNCT
iajs-3828	152	36	𝑏	𝑏	NOUN
iajs-3828	152	37	)	)	PUNCT
iajs-3828	152	38	=	=	SYM
iajs-3828	153	1	𝑡	𝑡	PROPN
iajs-3828	153	2	min{𝜌(𝑎	min{𝜌(𝑎	PROPN
iajs-3828	153	3	,	,	PUNCT
iajs-3828	153	4	𝑏	𝑏	NOUN
iajs-3828	153	5	)	)	PUNCT
iajs-3828	153	6	,	,	PUNCT
iajs-3828	153	7	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	153	8	,	,	PUNCT
iajs-3828	153	9	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	153	10	)	)	PUNCT
iajs-3828	153	11	,	,	PUNCT
iajs-3828	153	12	𝜌(𝑏	𝜌(𝑏	NUM
iajs-3828	153	13	,	,	PUNCT
iajs-3828	153	14	𝑓𝑏	𝑓𝑏	NOUN
iajs-3828	153	15	)	)	PUNCT
iajs-3828	153	16	,	,	PUNCT
iajs-3828	153	17	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	153	18	,	,	PUNCT
iajs-3828	153	19	𝑓𝑏	𝑓𝑏	PROPN
iajs-3828	153	20	)	)	PUNCT
iajs-3828	153	21	,	,	PUNCT
iajs-3828	153	22	𝜌(𝑏	𝜌(𝑏	PRON
iajs-3828	153	23	,	,	PUNCT
iajs-3828	153	24	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	153	25	)	)	PUNCT
iajs-3828	153	26	}	}	PUNCT
iajs-3828	153	27	,	,	PUNCT
iajs-3828	153	28	where	where	SCONJ
iajs-3828	153	29	𝑡	𝑡	PROPN
iajs-3828	153	30	∈	∈	PROPN
iajs-3828	153	31	(	(	PUNCT
iajs-3828	153	32	0,1	0,1	NOUN
iajs-3828	153	33	)	)	PUNCT
iajs-3828	153	34	,	,	PUNCT
iajs-3828	153	35	and	and	CCONJ
iajs-3828	153	36	𝜓	𝜓	NOUN
iajs-3828	153	37	,	,	PUNCT
iajs-3828	153	38	∅	∅	NOUN
iajs-3828	153	39	:	:	PUNCT
iajs-3828	153	40	[	[	X
iajs-3828	153	41	0,∞	0,∞	NOUN
iajs-3828	153	42	)	)	PUNCT
iajs-3828	153	43	→	→	PUNCT
iajs-3828	154	1	[	[	X
iajs-3828	154	2	0,∞	0,∞	NOUN
iajs-3828	154	3	)	)	PUNCT
iajs-3828	154	4	are	be	AUX
iajs-3828	154	5	altering	alter	VERB
iajs-3828	154	6	distance	distance	NOUN
iajs-3828	154	7	functions	function	NOUN
iajs-3828	154	8	.	.	PUNCT
iajs-3828	155	1	then	then	ADV
iajs-3828	155	2	∃	∃	PROPN
iajs-3828	155	3	𝑐	𝑐	PROPN
iajs-3828	155	4	∈	∈	PROPN
iajs-3828	155	5	⋂	⋂	PROPN
iajs-3828	155	6	£	£	SYM
iajs-3828	155	7	𝑖	𝑖	SYM
iajs-3828	155	8	𝑛	𝑛	PRON
iajs-3828	155	9	𝑖=1	𝑖=1	PUNCT
iajs-3828	155	10	,	,	PUNCT
iajs-3828	155	11	𝑐	𝑐	PROPN
iajs-3828	155	12	is	be	AUX
iajs-3828	155	13	a	a	DET
iajs-3828	155	14	unique	unique	ADJ
iajs-3828	155	15	fixed	fix	VERB
iajs-3828	155	16	point	point	NOUN
iajs-3828	155	17	of	of	ADP
iajs-3828	155	18	𝑓.	𝑓.	NOUN
iajs-3828	155	19	proof	proof	NOUN
iajs-3828	155	20	:	:	PUNCT
iajs-3828	155	21	let	let	VERB
iajs-3828	155	22	𝑎0	𝑎0	VERB
iajs-3828	155	23	∈	∈	VERB
iajs-3828	155	24	⋃	⋃	ADP
iajs-3828	155	25	£	£	SYM
iajs-3828	155	26	𝑖	𝑖	SYM
iajs-3828	155	27	𝑛	𝑛	PRON
iajs-3828	155	28	𝑖=1	𝑖=1	PUNCT
iajs-3828	155	29	and	and	CCONJ
iajs-3828	155	30	consider	consider	VERB
iajs-3828	155	31	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	155	32	=	=	SYM
iajs-3828	155	33	𝑓𝑎𝑚	𝑓𝑎𝑚	VERB
iajs-3828	155	34	for	for	ADP
iajs-3828	155	35	each	each	DET
iajs-3828	155	36	𝑚	𝑚	ADP
iajs-3828	155	37	∈	∈	PROPN
iajs-3828	155	38	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	155	39	}	}	PUNCT
iajs-3828	155	40	,	,	PUNCT
iajs-3828	155	41	so	so	SCONJ
iajs-3828	155	42	for	for	ADP
iajs-3828	155	43	any	any	DET
iajs-3828	155	44	𝑚	𝑚	PROPN
iajs-3828	155	45	∈	∈	PROPN
iajs-3828	155	46	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	155	47	}	}	PUNCT
iajs-3828	155	48	,	,	PUNCT
iajs-3828	155	49	∃𝑖𝑚	∃𝑖𝑚	NOUN
iajs-3828	155	50	∈	∈	PROPN
iajs-3828	155	51	{	{	PUNCT
iajs-3828	155	52	1	1	NUM
iajs-3828	155	53	,	,	PUNCT
iajs-3828	155	54	2	2	NUM
iajs-3828	155	55	,	,	PUNCT
iajs-3828	155	56	…	…	PUNCT
iajs-3828	155	57	,	,	PUNCT
iajs-3828	155	58	𝑛	𝑛	X
iajs-3828	155	59	}	}	PUNCT
iajs-3828	155	60	such	such	ADJ
iajs-3828	155	61	that	that	SCONJ
iajs-3828	155	62	𝑎𝑚	𝑎𝑚	PROPN
iajs-3828	155	63	∈	∈	PROPN
iajs-3828	155	64	£	£	NOUN
iajs-3828	155	65	𝑖𝑚	𝑖𝑚	NOUN
iajs-3828	155	66	and	and	CCONJ
iajs-3828	155	67	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	155	68	∈	∈	PROPN
iajs-3828	155	69	£	£	NOUN
iajs-3828	155	70	𝑖𝑚+1	𝑖𝑚+1	NOUN
iajs-3828	155	71	.	.	PUNCT
iajs-3828	156	1	if	if	SCONJ
iajs-3828	156	2	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	156	3	=	=	PUNCT
iajs-3828	156	4	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	156	5	+	+	PROPN
iajs-3828	156	6	1	1	NUM
iajs-3828	156	7	for	for	ADP
iajs-3828	156	8	some	some	DET
iajs-3828	156	9	𝑚0	𝑚0	NOUN
iajs-3828	156	10	then	then	ADV
iajs-3828	156	11	,	,	PUNCT
iajs-3828	156	12	since	since	SCONJ
iajs-3828	156	13	𝑎𝑚0	𝑎𝑚0	PROPN
iajs-3828	156	14	+	+	PROPN
iajs-3828	156	15	1	1	NUM
iajs-3828	156	16	=	=	SYM
iajs-3828	156	17	𝑓𝑚0𝑎0	𝑓𝑚0𝑎0	NOUN
iajs-3828	156	18	=	=	SYM
iajs-3828	156	19	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	156	20	this	this	PRON
iajs-3828	156	21	means	mean	VERB
iajs-3828	156	22	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	156	23	is	be	AUX
iajs-3828	156	24	fixed	fix	VERB
iajs-3828	156	25	point	point	NOUN
iajs-3828	156	26	of	of	ADP
iajs-3828	156	27	𝑓.	𝑓.	NOUN
iajs-3828	156	28	thus	thus	ADV
iajs-3828	156	29	,	,	PUNCT
iajs-3828	156	30	assume	assume	VERB
iajs-3828	156	31	that	that	SCONJ
iajs-3828	156	32	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	156	33	≠	≠	NOUN
iajs-3828	156	34	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	156	35	for	for	ADP
iajs-3828	156	36	all	all	PRON
iajs-3828	156	37	𝑚	𝑚	ADP
iajs-3828	156	38	∈	∈	NOUN
iajs-3828	156	39	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	156	40	}	}	PUNCT
iajs-3828	156	41	.	.	PUNCT
iajs-3828	157	1	getting	get	VERB
iajs-3828	157	2	∅(𝜌(𝑎𝑚	∅(𝜌(𝑎𝑚	PROPN
iajs-3828	157	3	,	,	PUNCT
iajs-3828	157	4	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	157	5	)	)	PUNCT
iajs-3828	157	6	)	)	PUNCT
iajs-3828	158	1	≤	≤	NUM
iajs-3828	158	2	∅(𝑀(𝑎𝑚−1	∅(𝑀(𝑎𝑚−1	NOUN
iajs-3828	158	3	,	,	PUNCT
iajs-3828	158	4	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	158	5	)	)	PUNCT
iajs-3828	158	6	)	)	PUNCT
iajs-3828	159	1	−	−	ADP
iajs-3828	159	2	𝜓(𝑁(𝑎𝑚−1	𝜓(𝑁(𝑎𝑚−1	PROPN
iajs-3828	159	3	,	,	PUNCT
iajs-3828	159	4	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	5	)	)	PUNCT
iajs-3828	159	6	)	)	PUNCT
iajs-3828	159	7	,	,	PUNCT
iajs-3828	159	8	(	(	PUNCT
iajs-3828	159	9	5	5	X
iajs-3828	159	10	)	)	PUNCT
iajs-3828	159	11	where	where	SCONJ
iajs-3828	159	12	𝑀(𝑎𝑚−1	𝑀(𝑎𝑚−1	PROPN
iajs-3828	159	13	,	,	PUNCT
iajs-3828	159	14	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	15	)	)	PUNCT
iajs-3828	159	16	=	=	PUNCT
iajs-3828	159	17	𝑡	𝑡	PROPN
iajs-3828	159	18	max	max	PROPN
iajs-3828	159	19	{	{	PUNCT
iajs-3828	159	20	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	159	21	,	,	PUNCT
iajs-3828	159	22	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	23	)	)	PUNCT
iajs-3828	159	24	,	,	PUNCT
iajs-3828	159	25	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	159	26	,	,	PUNCT
iajs-3828	159	27	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	159	28	)	)	PUNCT
iajs-3828	159	29	,	,	PUNCT
iajs-3828	159	30	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	NUM
iajs-3828	159	31	,	,	PUNCT
iajs-3828	159	32	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	159	33	)	)	PUNCT
iajs-3828	159	34	,	,	PUNCT
iajs-3828	159	35	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	159	36	,	,	PUNCT
iajs-3828	159	37	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	38	)	)	PUNCT
iajs-3828	159	39	}	}	PUNCT
iajs-3828	159	40	,	,	PUNCT
iajs-3828	159	41	and	and	CCONJ
iajs-3828	159	42	𝑁(𝑎𝑚−1	𝑁(𝑎𝑚−1	PRON
iajs-3828	159	43	,	,	PUNCT
iajs-3828	159	44	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	45	)	)	PUNCT
iajs-3828	159	46	=	=	NOUN
iajs-3828	159	47	𝑡min	𝑡min	NOUN
iajs-3828	159	48	{	{	PUNCT
iajs-3828	159	49	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	159	50	,	,	PUNCT
iajs-3828	159	51	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	52	)	)	PUNCT
iajs-3828	159	53	,	,	PUNCT
iajs-3828	159	54	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	159	55	,	,	PUNCT
iajs-3828	159	56	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	159	57	)	)	PUNCT
iajs-3828	159	58	,	,	PUNCT
iajs-3828	159	59	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	NUM
iajs-3828	159	60	,	,	PUNCT
iajs-3828	159	61	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	159	62	)	)	PUNCT
iajs-3828	159	63	,	,	PUNCT
iajs-3828	159	64	0	0	NUM
iajs-3828	159	65	}	}	PUNCT
iajs-3828	159	66	=	=	SYM
iajs-3828	159	67	0	0	X
iajs-3828	159	68	.	.	PUNCT
iajs-3828	159	69	by	by	ADP
iajs-3828	159	70	triangular	triangular	NOUN
iajs-3828	159	71	inequality	inequality	NOUN
iajs-3828	159	72	,	,	PUNCT
iajs-3828	159	73	so	so	ADV
iajs-3828	159	74	𝑀(𝑎𝑚−1	𝑀(𝑎𝑚−1	PROPN
iajs-3828	159	75	,	,	PUNCT
iajs-3828	159	76	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	77	)	)	PUNCT
iajs-3828	159	78	=	=	PUNCT
iajs-3828	159	79	𝑡	𝑡	PROPN
iajs-3828	159	80	max	max	PROPN
iajs-3828	159	81	{	{	PUNCT
iajs-3828	159	82	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	159	83	,	,	PUNCT
iajs-3828	159	84	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	85	)	)	PUNCT
iajs-3828	159	86	,	,	PUNCT
iajs-3828	159	87	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	159	88	,	,	PUNCT
iajs-3828	159	89	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	159	90	)	)	PUNCT
iajs-3828	159	91	,	,	PUNCT
iajs-3828	159	92	𝑞	𝑞	PROPN
iajs-3828	159	93	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	159	94	,	,	PUNCT
iajs-3828	159	95	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	96	)	)	PUNCT
iajs-3828	159	97	+	+	CCONJ
iajs-3828	159	98	𝑞	𝑞	X
iajs-3828	159	99	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	159	100	,	,	PUNCT
iajs-3828	159	101	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	159	102	)	)	PUNCT
iajs-3828	159	103	,	,	PUNCT
iajs-3828	159	104	0	0	NUM
iajs-3828	159	105	}	}	PUNCT
iajs-3828	159	106	,	,	PUNCT
iajs-3828	159	107	if	if	SCONJ
iajs-3828	159	108	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	NUM
iajs-3828	159	109	,	,	PUNCT
iajs-3828	159	110	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	159	111	)	)	PUNCT
iajs-3828	159	112	<	<	X
iajs-3828	160	1	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	160	2	,	,	PUNCT
iajs-3828	160	3	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	160	4	)	)	PUNCT
iajs-3828	160	5	,	,	PUNCT
iajs-3828	160	6	then	then	ADV
iajs-3828	160	7	𝑀(𝑎𝑚−1	𝑀(𝑎𝑚−1	NUM
iajs-3828	160	8	,	,	PUNCT
iajs-3828	160	9	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	160	10	)	)	PUNCT
iajs-3828	160	11	≤	≤	NUM
iajs-3828	161	1	𝑡	𝑡	PRON
iajs-3828	161	2	2	2	NUM
iajs-3828	161	3	𝑞	𝑞	ADP
iajs-3828	161	4	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	161	5	,	,	PUNCT
iajs-3828	161	6	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	161	7	)	)	PUNCT
iajs-3828	161	8	.	.	PUNCT
iajs-3828	162	1	and	and	CCONJ
iajs-3828	162	2	by	by	ADP
iajs-3828	162	3	(	(	PUNCT
iajs-3828	162	4	5	5	NUM
iajs-3828	162	5	)	)	PUNCT
iajs-3828	162	6	,	,	PUNCT
iajs-3828	162	7	we	we	PRON
iajs-3828	162	8	obtain	obtain	VERB
iajs-3828	162	9	∅(𝜌(𝑎𝑚	∅(𝜌(𝑎𝑚	PROPN
iajs-3828	162	10	,	,	PUNCT
iajs-3828	162	11	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	162	12	)	)	PUNCT
iajs-3828	162	13	)	)	PUNCT
iajs-3828	163	1	≤	≤	NUM
iajs-3828	163	2	∅(𝑡	∅(𝑡	NOUN
iajs-3828	163	3	2	2	NUM
iajs-3828	163	4	𝑞	𝑞	ADP
iajs-3828	163	5	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	163	6	,	,	PUNCT
iajs-3828	163	7	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	163	8	)	)	PUNCT
iajs-3828	163	9	)	)	PUNCT
iajs-3828	163	10	.	.	PUNCT
iajs-3828	164	1	since	since	SCONJ
iajs-3828	164	2	∅	∅	NOUN
iajs-3828	164	3	is	be	AUX
iajs-3828	164	4	altering	alter	VERB
iajs-3828	164	5	distance	distance	NOUN
iajs-3828	164	6	function	function	NOUN
iajs-3828	164	7	,	,	PUNCT
iajs-3828	164	8	subsequently	subsequently	ADV
iajs-3828	164	9	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	164	10	,	,	PUNCT
iajs-3828	164	11	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	164	12	)	)	PUNCT
iajs-3828	164	13	≤	≤	NUM
iajs-3828	164	14	𝑡	𝑡	PRON
iajs-3828	164	15	2	2	NUM
iajs-3828	164	16	𝑞	𝑞	ADP
iajs-3828	164	17	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	164	18	,	,	PUNCT
iajs-3828	164	19	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	164	20	)	)	PUNCT
iajs-3828	164	21	,	,	PUNCT
iajs-3828	164	22	this	this	PRON
iajs-3828	164	23	not	not	PART
iajs-3828	164	24	true	true	ADJ
iajs-3828	164	25	for	for	ADP
iajs-3828	164	26	all	all	PRON
iajs-3828	164	27	𝑡	𝑡	ADP
iajs-3828	164	28	∈	∈	PROPN
iajs-3828	164	29	(	(	PUNCT
iajs-3828	164	30	0,1	0,1	NOUN
iajs-3828	164	31	)	)	PUNCT
iajs-3828	164	32	,	,	PUNCT
iajs-3828	164	33	as	as	ADP
iajs-3828	164	34	result	result	NOUN
iajs-3828	164	35	𝑀(𝑎𝑚−1	𝑀(𝑎𝑚−1	PROPN
iajs-3828	164	36	,	,	PUNCT
iajs-3828	164	37	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	164	38	)	)	PUNCT
iajs-3828	164	39	=	=	PUNCT
iajs-3828	164	40	𝑡	𝑡	X
iajs-3828	164	41	2	2	NUM
iajs-3828	164	42	𝑞	𝑞	X
iajs-3828	164	43	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	164	44	,	,	PUNCT
iajs-3828	164	45	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	164	46	)	)	PUNCT
iajs-3828	164	47	and	and	CCONJ
iajs-3828	164	48	𝑁(𝑎𝑚−1	𝑁(𝑎𝑚−1	PRON
iajs-3828	164	49	,	,	PUNCT
iajs-3828	164	50	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	164	51	)	)	PUNCT
iajs-3828	164	52	=	=	SYM
iajs-3828	164	53	0	0	X
iajs-3828	164	54	.	.	PUNCT
iajs-3828	165	1	(	(	PUNCT
iajs-3828	165	2	6	6	NUM
iajs-3828	165	3	)	)	PUNCT
iajs-3828	165	4	now	now	ADV
iajs-3828	165	5	put	put	VERB
iajs-3828	165	6	(	(	PUNCT
iajs-3828	165	7	6	6	NUM
iajs-3828	165	8	)	)	PUNCT
iajs-3828	165	9	in	in	ADP
iajs-3828	165	10	(	(	PUNCT
iajs-3828	165	11	5	5	X
iajs-3828	165	12	)	)	PUNCT
iajs-3828	165	13	∅(𝜌(𝑎𝑚	∅(𝜌(𝑎𝑚	PROPN
iajs-3828	165	14	,	,	PUNCT
iajs-3828	165	15	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	165	16	)	)	PUNCT
iajs-3828	165	17	)	)	PUNCT
iajs-3828	166	1	≤	≤	NUM
iajs-3828	166	2	∅(𝑡	∅(𝑡	NOUN
iajs-3828	166	3	2	2	NUM
iajs-3828	166	4	𝑞	𝑞	X
iajs-3828	166	5	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	166	6	,	,	PUNCT
iajs-3828	166	7	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	166	8	)	)	PUNCT
iajs-3828	166	9	)	)	PUNCT
iajs-3828	166	10	.	.	PUNCT
iajs-3828	167	1	(	(	PUNCT
iajs-3828	167	2	7	7	X
iajs-3828	167	3	)	)	PUNCT
iajs-3828	167	4	since	since	SCONJ
iajs-3828	167	5	∅	∅	NOUN
iajs-3828	167	6	is	be	AUX
iajs-3828	167	7	altering	alter	VERB
iajs-3828	167	8	distance	distance	NOUN
iajs-3828	167	9	function	function	NOUN
iajs-3828	167	10	,	,	PUNCT
iajs-3828	167	11	as	as	ADP
iajs-3828	167	12	a	a	DET
iajs-3828	167	13	result	result	NOUN
iajs-3828	167	14	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	167	15	,	,	PUNCT
iajs-3828	167	16	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	167	17	)	)	PUNCT
iajs-3828	167	18	≤	≤	NUM
iajs-3828	168	1	𝑡	𝑡	SYM
iajs-3828	168	2	2	2	NUM
iajs-3828	168	3	𝑞	𝑞	X
iajs-3828	168	4	𝜌(𝑎𝑚−1	𝜌(𝑎𝑚−1	PROPN
iajs-3828	168	5	,	,	PUNCT
iajs-3828	168	6	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	168	7	)	)	PUNCT
iajs-3828	168	8	.	.	PUNCT
iajs-3828	169	1	thus	thus	ADV
iajs-3828	169	2	for	for	ADP
iajs-3828	169	3	all	all	PRON
iajs-3828	169	4	𝑚	𝑚	ADP
iajs-3828	169	5	∈	∈	NOUN
iajs-3828	169	6	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	169	7	}	}	PUNCT
iajs-3828	169	8	we	we	PRON
iajs-3828	169	9	have	have	VERB
iajs-3828	169	10	a	a	DET
iajs-3828	169	11	monotone	monotone	ADJ
iajs-3828	169	12	decreasing	decrease	VERB
iajs-3828	169	13	sequence	sequence	NOUN
iajs-3828	169	14	(	(	PUNCT
iajs-3828	169	15	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	169	16	,	,	PUNCT
iajs-3828	169	17	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	169	18	)	)	PUNCT
iajs-3828	169	19	)	)	PUNCT
iajs-3828	170	1	=	=	SYM
iajs-3828	170	2	(	(	PUNCT
iajs-3828	170	3	𝜌(𝑓𝑎𝑚−1	𝜌(𝑓𝑎𝑚−1	PROPN
iajs-3828	170	4	,	,	PUNCT
iajs-3828	170	5	𝑓𝑎𝑚	𝑓𝑎𝑚	NOUN
iajs-3828	170	6	)	)	PUNCT
iajs-3828	170	7	)	)	PUNCT
iajs-3828	170	8	.	.	PUNCT
iajs-3828	171	1	by	by	ADP
iajs-3828	171	2	properties	property	NOUN
iajs-3828	171	3	of	of	ADP
iajs-3828	171	4	real	real	ADJ
iajs-3828	171	5	sequence	sequence	NOUN
iajs-3828	171	6	,	,	PUNCT
iajs-3828	171	7	(	(	PUNCT
iajs-3828	171	8	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	171	9	,	,	PUNCT
iajs-3828	171	10	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	171	11	)	)	PUNCT
iajs-3828	171	12	)	)	PUNCT
iajs-3828	171	13	there	there	PRON
iajs-3828	171	14	exist	exist	VERB
iajs-3828	171	15	𝜀	𝜀	NOUN
iajs-3828	171	16	≥	≥	NOUN
iajs-3828	171	17	0	0	NUM
iajs-3828	171	18	such	such	ADJ
iajs-3828	171	19	that	that	SCONJ
iajs-3828	171	20	lim	lim	PROPN
iajs-3828	171	21	𝑚→∞	𝑚→∞	NUM
iajs-3828	171	22	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	171	23	,	,	PUNCT
iajs-3828	171	24	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	171	25	)	)	PUNCT
iajs-3828	171	26	=	=	VERB
iajs-3828	171	27	𝜀.	𝜀.	NOUN
iajs-3828	171	28	on	on	ADP
iajs-3828	171	29	letting	let	VERB
iajs-3828	171	30	𝑚	𝑚	X
iajs-3828	171	31	→	→	SYM
iajs-3828	171	32	∞	∞	NUM
iajs-3828	171	33	in	in	ADP
iajs-3828	171	34	(	(	PUNCT
iajs-3828	171	35	7	7	NUM
iajs-3828	171	36	)	)	PUNCT
iajs-3828	171	37	,	,	PUNCT
iajs-3828	171	38	obtaining	obtain	VERB
iajs-3828	171	39	∅(𝜀	∅(𝜀	NOUN
iajs-3828	171	40	)	)	PUNCT
iajs-3828	171	41	≤	≤	NUM
iajs-3828	171	42	∅(𝑡	∅(𝑡	NOUN
iajs-3828	171	43	2	2	NUM
iajs-3828	171	44	𝑞	𝑞	PART
iajs-3828	171	45	𝜀	𝜀	NOUN
iajs-3828	171	46	)	)	PUNCT
iajs-3828	171	47	.	.	PUNCT
iajs-3828	172	1	assume	assume	VERB
iajs-3828	172	2	that	that	SCONJ
iajs-3828	172	3	𝜀	𝜀	VERB
iajs-3828	172	4	≠	≠	PROPN
iajs-3828	172	5	0	0	NUM
iajs-3828	172	6	.	.	PUNCT
iajs-3828	173	1	so	so	ADV
iajs-3828	173	2	since	since	SCONJ
iajs-3828	173	3	∅	∅	NOUN
iajs-3828	173	4	is	be	AUX
iajs-3828	173	5	altering	alter	VERB
iajs-3828	173	6	distance	distance	NOUN
iajs-3828	173	7	function	function	NOUN
iajs-3828	173	8	,	,	PUNCT
iajs-3828	173	9	getting	get	VERB
iajs-3828	173	10	𝜀	𝜀	PRON
iajs-3828	173	11	≤	≤	NUM
iajs-3828	173	12	𝑡	𝑡	X
iajs-3828	173	13	2	2	NUM
iajs-3828	173	14	𝑞	𝑞	PART
iajs-3828	173	15	𝜀	𝜀	X
iajs-3828	173	16	<	<	X
iajs-3828	173	17	ε	ε	PROPN
iajs-3828	173	18	,	,	PUNCT
iajs-3828	173	19	which	which	PRON
iajs-3828	173	20	a	a	DET
iajs-3828	173	21	contradiction	contradiction	NOUN
iajs-3828	173	22	.	.	PUNCT
iajs-3828	174	1	hence	hence	ADV
iajs-3828	174	2	lim	lim	PROPN
iajs-3828	174	3	𝑚→∞	𝑚→∞	NUM
iajs-3828	174	4	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	174	5	,	,	PUNCT
iajs-3828	174	6	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	174	7	)	)	PUNCT
iajs-3828	174	8	=	=	SYM
iajs-3828	174	9	0	0	X
iajs-3828	174	10	.	.	PUNCT
iajs-3828	175	1	for	for	ADP
iajs-3828	175	2	𝑗	𝑗	INTJ
iajs-3828	175	3	>	>	X
iajs-3828	175	4	𝑛	𝑛	PRON
iajs-3828	175	5	define	define	VERB
iajs-3828	175	6	£	£	SYM
iajs-3828	175	7	𝑗	𝑗	NOUN
iajs-3828	175	8	=	=	NOUN
iajs-3828	175	9	£	£	SYM
iajs-3828	175	10	𝑖	𝑖	NOUN
iajs-3828	175	11	if	if	SCONJ
iajs-3828	175	12	𝑗	𝑗	ADJ
iajs-3828	175	13	=	=	SYM
iajs-3828	175	14	𝑖	𝑖	SYM
iajs-3828	175	15	mod	mod	ADJ
iajs-3828	175	16	𝑛.	𝑛.	NOUN
iajs-3828	175	17	claim	claim	VERB
iajs-3828	175	18	i	i	PRON
iajs-3828	175	19	:	:	PUNCT
iajs-3828	175	20	for	for	ADP
iajs-3828	175	21	all	all	DET
iajs-3828	175	22	𝜀	𝜀	NOUN
iajs-3828	175	23	>	>	X
iajs-3828	175	24	0	0	PUNCT
iajs-3828	175	25	there	there	PRON
iajs-3828	175	26	exist	exist	VERB
iajs-3828	175	27	𝑚	𝑚	ADP
iajs-3828	175	28	∈	∈	NOUN
iajs-3828	176	1	𝑁	𝑁	PROPN
iajs-3828	176	2	such	such	ADJ
iajs-3828	176	3	that	that	PRON
iajs-3828	176	4	for	for	ADP
iajs-3828	176	5	all	all	DET
iajs-3828	176	6	𝑗	𝑗	PROPN
iajs-3828	176	7	,	,	PUNCT
iajs-3828	176	8	𝑖	𝑖	X
iajs-3828	176	9	≥	≥	NOUN
iajs-3828	176	10	𝑚	𝑚	NOUN
iajs-3828	176	11	,	,	PUNCT
iajs-3828	176	12	𝑗	𝑗	INTJ
iajs-3828	176	13	−	−	PROPN
iajs-3828	176	14	𝑖	𝑖	SYM
iajs-3828	176	15	≡	≡	PROPN
iajs-3828	176	16	1(mod	1(mod	NUM
iajs-3828	176	17	𝑛	𝑛	NOUN
iajs-3828	176	18	)	)	PUNCT
iajs-3828	176	19	then	then	ADV
iajs-3828	176	20	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	176	21	,	,	PUNCT
iajs-3828	176	22	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	176	23	)	)	PUNCT
iajs-3828	176	24	<	<	X
iajs-3828	176	25	𝜀.	𝜀.	NOUN
iajs-3828	176	26	suppose	suppose	VERB
iajs-3828	176	27	that	that	SCONJ
iajs-3828	176	28	there	there	PRON
iajs-3828	176	29	exists	exist	VERB
iajs-3828	176	30	𝜀	𝜀	PROPN
iajs-3828	176	31	>	>	X
iajs-3828	176	32	0	0	NUM
iajs-3828	176	33	such	such	ADJ
iajs-3828	176	34	that	that	PRON
iajs-3828	176	35	for	for	ADP
iajs-3828	176	36	each	each	DET
iajs-3828	176	37	𝑚	𝑚	ADP
iajs-3828	176	38	∈	∈	NOUN
iajs-3828	176	39	𝑁	𝑁	NOUN
iajs-3828	176	40	we	we	PRON
iajs-3828	176	41	can	can	AUX
iajs-3828	176	42	find	find	VERB
iajs-3828	176	43	𝑗	𝑗	PRON
iajs-3828	176	44	>	>	X
iajs-3828	176	45	𝑖	𝑖	X
iajs-3828	176	46	>	>	X
iajs-3828	176	47	𝑚	𝑚	X
iajs-3828	176	48	with	with	ADP
iajs-3828	176	49	𝑗	𝑗	INTJ
iajs-3828	176	50	−	−	PROPN
iajs-3828	176	51	𝑖	𝑖	SYM
iajs-3828	176	52	≡	≡	PROPN
iajs-3828	176	53	1(mod	1(mod	NUM
iajs-3828	176	54	𝑛	𝑛	X
iajs-3828	176	55	)	)	PUNCT
iajs-3828	176	56	satisfying	satisfy	VERB
iajs-3828	176	57	𝜌(𝑎𝑖	𝜌(𝑎𝑖	PROPN
iajs-3828	176	58	,	,	PUNCT
iajs-3828	176	59	𝑎𝑗	𝑎𝑗	NOUN
iajs-3828	176	60	)	)	PUNCT
iajs-3828	176	61	≥	≥	NOUN
iajs-3828	176	62	𝜀.	𝜀.	VERB
iajs-3828	176	63	now	now	ADV
iajs-3828	176	64	,	,	PUNCT
iajs-3828	176	65	take	take	VERB
iajs-3828	176	66	𝑚	𝑚	ADP
iajs-3828	176	67	≥	≥	NUM
iajs-3828	176	68	2(mod	2(mod	NUM
iajs-3828	176	69	𝑛	𝑛	X
iajs-3828	176	70	)	)	PUNCT
iajs-3828	176	71	.	.	PUNCT
iajs-3828	177	1	then	then	ADV
iajs-3828	177	2	,	,	PUNCT
iajs-3828	177	3	corresponding	correspond	VERB
iajs-3828	177	4	to	to	ADP
iajs-3828	177	5	𝑖	𝑖	PRON
iajs-3828	177	6	≥	≥	NOUN
iajs-3828	177	7	𝑚	𝑚	ADP
iajs-3828	177	8	use	use	NOUN
iajs-3828	177	9	can	can	AUX
iajs-3828	177	10	choose	choose	VERB
iajs-3828	177	11	𝑗	𝑗	PRON
iajs-3828	177	12	in	in	ADP
iajs-3828	177	13	such	such	DET
iajs-3828	177	14	a	a	DET
iajs-3828	177	15	way	way	NOUN
iajs-3828	177	16	that	that	PRON
iajs-3828	177	17	it	it	PRON
iajs-3828	177	18	is	be	AUX
iajs-3828	177	19	the	the	DET
iajs-3828	177	20	smallest	small	ADJ
iajs-3828	177	21	integer	integer	NOUN
iajs-3828	177	22	with	with	ADP
iajs-3828	177	23	𝑗	𝑗	PROPN
iajs-3828	177	24	>	>	X
iajs-3828	177	25	𝑖	𝑖	PUNCT
iajs-3828	177	26	satisfying	satisfy	VERB
iajs-3828	177	27	𝑗	𝑗	INTJ
iajs-3828	177	28	−	−	PROPN
iajs-3828	177	29	𝑖	𝑖	SYM
iajs-3828	177	30	≡	≡	PROPN
iajs-3828	177	31	1(mod	1(mod	NUM
iajs-3828	177	32	𝑛	𝑛	NOUN
iajs-3828	177	33	)	)	PUNCT
iajs-3828	177	34	and	and	CCONJ
iajs-3828	177	35	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	177	36	,	,	PUNCT
iajs-3828	177	37	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	177	38	)	)	PUNCT
iajs-3828	177	39	≥	≥	NOUN
iajs-3828	177	40	𝜀.	𝜀.	VERB
iajs-3828	177	41	therefore	therefore	ADV
iajs-3828	177	42	,	,	PUNCT
iajs-3828	177	43	𝜌(𝑎𝑗−𝑛	𝜌(𝑎𝑗−𝑛	NUM
iajs-3828	177	44	,	,	PUNCT
iajs-3828	177	45	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	177	46	)	)	PUNCT
iajs-3828	177	47	≤	≤	NOUN
iajs-3828	177	48	𝜀.	𝜀.	NOUN
iajs-3828	177	49	by	by	ADP
iajs-3828	177	50	triangular	triangular	NOUN
iajs-3828	177	51	inequality	inequality	NOUN
iajs-3828	177	52	𝜀	𝜀	ADP
iajs-3828	177	53	≤	≤	ADJ
iajs-3828	177	54	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	177	55	,	,	PUNCT
iajs-3828	177	56	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	177	57	)	)	PUNCT
iajs-3828	177	58	≤	≤	NOUN
iajs-3828	177	59	𝑞	𝑞	ADP
iajs-3828	177	60	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	177	61	,	,	PUNCT
iajs-3828	177	62	𝑎𝑗−𝑛	𝑎𝑗−𝑛	NOUN
iajs-3828	177	63	)	)	PUNCT
iajs-3828	178	1	+	+	CCONJ
iajs-3828	178	2	𝑞	𝑞	SYM
iajs-3828	178	3	∑	∑	ADV
iajs-3828	178	4	𝜌(𝑎𝑖−𝑘	𝜌(𝑎𝑖−𝑘	NOUN
iajs-3828	178	5	,	,	PUNCT
iajs-3828	178	6	𝑎𝑖−𝑘+1	𝑎𝑖−𝑘+1	VERB
iajs-3828	178	7	)	)	PUNCT
iajs-3828	178	8	𝑛	𝑛	DET
iajs-3828	178	9	𝑘=1	𝑘=1	ADJ
iajs-3828	178	10	≤	≤	NUM
iajs-3828	178	11	𝑞	𝑞	X
iajs-3828	178	12	∑	∑	ADV
iajs-3828	178	13	𝜌(𝑎𝑖−𝑘	𝜌(𝑎𝑖−𝑘	VERB
iajs-3828	178	14	,	,	PUNCT
iajs-3828	178	15	𝑎𝑖−𝑘+1	𝑎𝑖−𝑘+1	VERB
iajs-3828	178	16	)	)	PUNCT
iajs-3828	179	1	𝑛	𝑛	DET
iajs-3828	179	2	𝑘=1	𝑘=1	ADJ
iajs-3828	179	3	+	+	NUM
iajs-3828	179	4	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	179	5	taking	take	VERB
iajs-3828	179	6	𝑛	𝑛	PRON
iajs-3828	179	7	→	→	SYM
iajs-3828	179	8	∞	∞	PROPN
iajs-3828	179	9	and	and	CCONJ
iajs-3828	179	10	since	since	SCONJ
iajs-3828	179	11	lim	lim	PROPN
iajs-3828	179	12	𝑚→∞	𝑚→∞	PUNCT
iajs-3828	179	13	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	179	14	,	,	PUNCT
iajs-3828	179	15	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	179	16	)	)	PUNCT
iajs-3828	179	17	=	=	SYM
iajs-3828	179	18	0	0	NUM
iajs-3828	179	19	,	,	PUNCT
iajs-3828	179	20	we	we	PRON
iajs-3828	179	21	obtain	obtain	VERB
iajs-3828	179	22	lim	lim	NOUN
iajs-3828	179	23	𝑖,𝑗→∞	𝑖,𝑗→∞	X
iajs-3828	180	1	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	180	2	,	,	PUNCT
iajs-3828	180	3	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	180	4	)	)	PUNCT
iajs-3828	180	5	=	=	VERB
iajs-3828	180	6	𝑞𝜀.	𝑞𝜀.	X
iajs-3828	180	7	again	again	ADV
iajs-3828	180	8	,	,	PUNCT
iajs-3828	180	9	by	by	ADP
iajs-3828	180	10	triangular	triangular	NOUN
iajs-3828	180	11	inequality	inequality	NOUN
iajs-3828	180	12	𝜀	𝜀	ADP
iajs-3828	180	13	≤	≤	ADJ
iajs-3828	180	14	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	180	15	,	,	PUNCT
iajs-3828	180	16	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	180	17	)	)	PUNCT
iajs-3828	180	18	≤	≤	NOUN
iajs-3828	180	19	2𝑞𝜌(𝑎𝑗+1	2𝑞𝜌(𝑎𝑗+1	NUM
iajs-3828	180	20	,	,	PUNCT
iajs-3828	180	21	𝑎𝑗	𝑎𝑗	NOUN
iajs-3828	180	22	)	)	PUNCT
iajs-3828	180	23	+	+	CCONJ
iajs-3828	180	24	𝑞𝜌(𝑎𝑗	𝑞𝜌(𝑎𝑗	PROPN
iajs-3828	180	25	,	,	PUNCT
iajs-3828	180	26	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	180	27	)	)	PUNCT
iajs-3828	180	28	+	+	CCONJ
iajs-3828	180	29	2𝑞𝜌(𝑎𝑖+1	2𝑞𝜌(𝑎𝑖+1	NUM
iajs-3828	180	30	,	,	PUNCT
iajs-3828	180	31	𝑎𝑖	𝑎𝑖	NUM
iajs-3828	180	32	)	)	PUNCT
iajs-3828	180	33	.	.	PUNCT
iajs-3828	181	1	letting	let	VERB
iajs-3828	181	2	𝑗	𝑗	INTJ
iajs-3828	181	3	,	,	PUNCT
iajs-3828	181	4	𝑖	𝑖	SYM
iajs-3828	181	5	→	→	SYM
iajs-3828	181	6	∞	∞	PROPN
iajs-3828	181	7	and	and	CCONJ
iajs-3828	181	8	since	since	SCONJ
iajs-3828	181	9	lim	lim	PROPN
iajs-3828	181	10	𝑚→∞	𝑚→∞	PUNCT
iajs-3828	181	11	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	181	12	,	,	PUNCT
iajs-3828	181	13	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	181	14	)	)	PUNCT
iajs-3828	181	15	=	=	SYM
iajs-3828	181	16	0	0	NUM
iajs-3828	181	17	,	,	PUNCT
iajs-3828	181	18	so	so	ADV
iajs-3828	181	19	lim	lim	PROPN
iajs-3828	181	20	𝑖,𝑗→∞	𝑖,𝑗→∞	X
iajs-3828	181	21	𝜌(𝑎𝑗+1	𝜌(𝑎𝑗+1	PROPN
iajs-3828	181	22	,	,	PUNCT
iajs-3828	181	23	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	181	24	)	)	PUNCT
iajs-3828	181	25	=	=	PUNCT
iajs-3828	182	1	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	182	2	ihjpas	ihjpas	PROPN
iajs-3828	182	3	.	.	PUNCT
iajs-3828	183	1	2025,38(2	2025,38(2	PROPN
iajs-3828	183	2	)	)	PUNCT
iajs-3828	183	3	381	381	NUM
iajs-3828	183	4	since	since	SCONJ
iajs-3828	183	5	𝑎𝑖	𝑎𝑖	ADV
iajs-3828	183	6	,	,	PUNCT
iajs-3828	183	7	𝑎𝑗	𝑎𝑗	PART
iajs-3828	183	8	belong	belong	VERB
iajs-3828	183	9	to	to	ADP
iajs-3828	183	10	different	different	ADJ
iajs-3828	183	11	sets	set	NOUN
iajs-3828	183	12	£	£	SYM
iajs-3828	183	13	𝑖	𝑖	NOUN
iajs-3828	183	14	and	and	CCONJ
iajs-3828	183	15	£	£	SYM
iajs-3828	183	16	𝑖+1	𝑖+1	NUM
iajs-3828	183	17	,	,	PUNCT
iajs-3828	183	18	and	and	CCONJ
iajs-3828	183	19	using	use	VERB
iajs-3828	183	20	(	(	PUNCT
iajs-3828	183	21	2.4	2.4	NUM
iajs-3828	183	22	)	)	PUNCT
iajs-3828	183	23	∅(𝜌(𝑓𝑎𝑗	∅(𝜌(𝑓𝑎𝑗	NOUN
iajs-3828	183	24	,	,	PUNCT
iajs-3828	183	25	𝑓𝑎𝑖	𝑓𝑎𝑖	ADJ
iajs-3828	183	26	)	)	PUNCT
iajs-3828	183	27	)	)	PUNCT
iajs-3828	183	28	≤	≤	NOUN
iajs-3828	183	29	∅	∅	NOUN
iajs-3828	183	30	(	(	PUNCT
iajs-3828	183	31	𝑀(𝑎𝑗	𝑀(𝑎𝑗	PROPN
iajs-3828	183	32	,	,	PUNCT
iajs-3828	183	33	𝑎𝑖	𝑎𝑖	PROPN
iajs-3828	183	34	)	)	PUNCT
iajs-3828	183	35	)	)	PUNCT
iajs-3828	183	36	−	−	PROPN
iajs-3828	183	37	𝜓	𝜓	PROPN
iajs-3828	183	38	(	(	PUNCT
iajs-3828	183	39	𝑁(𝑎𝑗	𝑁(𝑎𝑗	PROPN
iajs-3828	183	40	,	,	PUNCT
iajs-3828	183	41	𝑎𝑖	𝑎𝑖	PROPN
iajs-3828	183	42	)	)	PUNCT
iajs-3828	183	43	)	)	PUNCT
iajs-3828	183	44	,	,	PUNCT
iajs-3828	183	45	(	(	PUNCT
iajs-3828	183	46	8)	8)	NUM
iajs-3828	183	47	where	where	SCONJ
iajs-3828	183	48	𝑀(𝑎𝑗	𝑀(𝑎𝑗	PROPN
iajs-3828	183	49	,	,	PUNCT
iajs-3828	183	50	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	183	51	)	)	PUNCT
iajs-3828	183	52	=	=	SYM
iajs-3828	183	53	𝑡	𝑡	PROPN
iajs-3828	183	54	max	max	PROPN
iajs-3828	183	55	{	{	PUNCT
iajs-3828	183	56	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	183	57	,	,	PUNCT
iajs-3828	183	58	𝑎𝑖	𝑎𝑖	PROPN
iajs-3828	183	59	)	)	PUNCT
iajs-3828	183	60	,	,	PUNCT
iajs-3828	183	61	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	183	62	,	,	PUNCT
iajs-3828	183	63	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	183	64	)	)	PUNCT
iajs-3828	183	65	,	,	PUNCT
iajs-3828	183	66	𝜌(𝑎𝑖	𝜌(𝑎𝑖	PROPN
iajs-3828	183	67	,	,	PUNCT
iajs-3828	183	68	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	183	69	)	)	PUNCT
iajs-3828	183	70	,	,	PUNCT
iajs-3828	183	71	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	183	72	,	,	PUNCT
iajs-3828	183	73	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	183	74	)	)	PUNCT
iajs-3828	183	75	,	,	PUNCT
iajs-3828	183	76	𝜌(𝑎𝑖	𝜌(𝑎𝑖	PROPN
iajs-3828	183	77	,	,	PUNCT
iajs-3828	183	78	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	183	79	)	)	PUNCT
iajs-3828	183	80	}	}	PUNCT
iajs-3828	183	81	𝑁(𝑎𝑗	𝑁(𝑎𝑗	PROPN
iajs-3828	183	82	,	,	PUNCT
iajs-3828	183	83	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	183	84	)	)	PUNCT
iajs-3828	183	85	=	=	SYM
iajs-3828	183	86	𝑡	𝑡	PART
iajs-3828	183	87	min	min	PROPN
iajs-3828	183	88	{	{	PUNCT
iajs-3828	183	89	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	183	90	,	,	PUNCT
iajs-3828	183	91	𝑎𝑖	𝑎𝑖	PROPN
iajs-3828	183	92	)	)	PUNCT
iajs-3828	183	93	,	,	PUNCT
iajs-3828	183	94	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	183	95	,	,	PUNCT
iajs-3828	183	96	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	183	97	)	)	PUNCT
iajs-3828	183	98	,	,	PUNCT
iajs-3828	183	99	𝜌(𝑎𝑖	𝜌(𝑎𝑖	PROPN
iajs-3828	183	100	,	,	PUNCT
iajs-3828	183	101	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	183	102	)	)	PUNCT
iajs-3828	183	103	,	,	PUNCT
iajs-3828	183	104	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	183	105	,	,	PUNCT
iajs-3828	183	106	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	183	107	)	)	PUNCT
iajs-3828	183	108	,	,	PUNCT
iajs-3828	183	109	𝜌(𝑎𝑖	𝜌(𝑎𝑖	PROPN
iajs-3828	183	110	,	,	PUNCT
iajs-3828	183	111	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	183	112	)	)	PUNCT
iajs-3828	183	113	}	}	PUNCT
iajs-3828	183	114	.	.	PUNCT
iajs-3828	184	1	since	since	SCONJ
iajs-3828	184	2	lim	lim	PROPN
iajs-3828	184	3	𝑖,𝑗→∞	𝑖,𝑗→∞	X
iajs-3828	184	4	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	184	5	,	,	PUNCT
iajs-3828	184	6	𝑎𝑖	𝑎𝑖	PROPN
iajs-3828	184	7	)	)	PUNCT
iajs-3828	184	8	=	=	SYM
iajs-3828	184	9	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	184	10	,	,	PUNCT
iajs-3828	184	11	lim	lim	NOUN
iajs-3828	184	12	𝑖,𝑗→∞	𝑖,𝑗→∞	X
iajs-3828	184	13	𝜌(𝑎𝑗+1	𝜌(𝑎𝑗+1	PROPN
iajs-3828	184	14	,	,	PUNCT
iajs-3828	184	15	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	184	16	)	)	PUNCT
iajs-3828	184	17	=	=	SYM
iajs-3828	184	18	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	184	19	,	,	PUNCT
iajs-3828	184	20	and	and	CCONJ
iajs-3828	184	21	by	by	ADP
iajs-3828	184	22	the	the	DET
iajs-3828	184	23	triangle	triangle	NOUN
iajs-3828	184	24	inequality	inequality	NOUN
iajs-3828	184	25	𝜌(𝑎𝑖+1	𝜌(𝑎𝑖+1	SYM
iajs-3828	184	26	,	,	PUNCT
iajs-3828	184	27	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	184	28	)	)	PUNCT
iajs-3828	184	29	≤	≤	NUM
iajs-3828	184	30	𝑞𝜌(𝑎𝑖+1	𝑞𝜌(𝑎𝑖+1	PROPN
iajs-3828	184	31	,	,	PUNCT
iajs-3828	184	32	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	184	33	)	)	PUNCT
iajs-3828	185	1	+	+	NUM
iajs-3828	185	2	𝑞𝜌(𝑎𝑖	𝑞𝜌(𝑎𝑖	PROPN
iajs-3828	185	3	,	,	PUNCT
iajs-3828	185	4	𝑎𝑗	𝑎𝑗	NOUN
iajs-3828	185	5	)	)	PUNCT
iajs-3828	185	6	+	+	CCONJ
iajs-3828	185	7	𝑞𝜌(𝑎𝑗	𝑞𝜌(𝑎𝑗	PROPN
iajs-3828	185	8	,	,	PUNCT
iajs-3828	185	9	𝑎𝑗+1	𝑎𝑗+1	X
iajs-3828	185	10	)	)	PUNCT
iajs-3828	185	11	→	→	SYM
iajs-3828	185	12	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	185	13	as	as	ADP
iajs-3828	185	14	𝑖	𝑖	X
iajs-3828	185	15	,	,	PUNCT
iajs-3828	185	16	𝑗	𝑗	PROPN
iajs-3828	185	17	→	→	X
iajs-3828	185	18	∞.	∞.	PROPN
iajs-3828	185	19	again	again	ADV
iajs-3828	185	20	by	by	ADP
iajs-3828	185	21	the	the	DET
iajs-3828	185	22	triangle	triangle	NOUN
iajs-3828	185	23	inequality	inequality	NOUN
iajs-3828	185	24	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	185	25	,	,	PUNCT
iajs-3828	185	26	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	185	27	)	)	PUNCT
iajs-3828	185	28	≤	≤	NOUN
iajs-3828	185	29	𝑞𝜌(𝑎𝑗	𝑞𝜌(𝑎𝑗	PROPN
iajs-3828	185	30	,	,	PUNCT
iajs-3828	185	31	𝑎𝑗+1	𝑎𝑗+1	X
iajs-3828	185	32	)	)	PUNCT
iajs-3828	186	1	+	+	X
iajs-3828	186	2	𝑞𝜌(𝑎𝑗+1	𝑞𝜌(𝑎𝑗+1	PROPN
iajs-3828	186	3	,	,	PUNCT
iajs-3828	186	4	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	186	5	)	)	PUNCT
iajs-3828	186	6	→	→	SYM
iajs-3828	186	7	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	186	8	as	as	ADP
iajs-3828	186	9	𝑖	𝑖	X
iajs-3828	186	10	,	,	PUNCT
iajs-3828	186	11	𝑗	𝑗	PROPN
iajs-3828	186	12	→	→	X
iajs-3828	186	13	∞.	∞.	PROPN
iajs-3828	186	14	again	again	ADV
iajs-3828	186	15	by	by	ADP
iajs-3828	186	16	the	the	DET
iajs-3828	186	17	triangle	triangle	NOUN
iajs-3828	186	18	inequality	inequality	PROPN
iajs-3828	186	19	𝜌(𝑎𝑖	𝜌(𝑎𝑖	PROPN
iajs-3828	186	20	,	,	PUNCT
iajs-3828	186	21	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	186	22	)	)	PUNCT
iajs-3828	186	23	≤	≤	NOUN
iajs-3828	186	24	𝑞𝜌(𝑎𝑖	𝑞𝜌(𝑎𝑖	PROPN
iajs-3828	186	25	,	,	PUNCT
iajs-3828	186	26	𝑎𝑖+1	𝑎𝑖+1	NUM
iajs-3828	186	27	)	)	PUNCT
iajs-3828	186	28	+	+	X
iajs-3828	186	29	𝑞𝜌(𝑎𝑖+1	𝑞𝜌(𝑎𝑖+1	PROPN
iajs-3828	186	30	,	,	PUNCT
iajs-3828	186	31	𝑎𝑗+1	𝑎𝑗+1	NUM
iajs-3828	186	32	)	)	PUNCT
iajs-3828	186	33	→	→	SYM
iajs-3828	186	34	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	186	35	as	as	ADP
iajs-3828	186	36	𝑖	𝑖	X
iajs-3828	186	37	,	,	PUNCT
iajs-3828	186	38	𝑗	𝑗	PROPN
iajs-3828	186	39	→	→	SYM
iajs-3828	186	40	∞.	∞.	PROPN
iajs-3828	186	41	letting	let	VERB
iajs-3828	186	42	𝑖	𝑖	SYM
iajs-3828	186	43	,	,	PUNCT
iajs-3828	186	44	𝑗	𝑗	NOUN
iajs-3828	186	45	→	→	SYM
iajs-3828	186	46	∞	∞	PROPN
iajs-3828	186	47	,	,	PUNCT
iajs-3828	186	48	we	we	PRON
iajs-3828	186	49	have	have	AUX
iajs-3828	186	50	𝑀(𝑎𝑗	𝑀(𝑎𝑗	VERB
iajs-3828	186	51	,	,	PUNCT
iajs-3828	186	52	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	186	53	)	)	PUNCT
iajs-3828	186	54	=	=	SYM
iajs-3828	186	55	𝑡	𝑡	PROPN
iajs-3828	186	56	max	max	PROPN
iajs-3828	186	57	{	{	PUNCT
iajs-3828	186	58	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	186	59	,	,	PUNCT
iajs-3828	186	60	0,0	0,0	NOUN
iajs-3828	186	61	,	,	PUNCT
iajs-3828	186	62	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	186	63	,	,	PUNCT
iajs-3828	186	64	𝑞𝜀	𝑞𝜀	ADP
iajs-3828	186	65	}	}	PUNCT
iajs-3828	186	66	→	→	SYM
iajs-3828	186	67	𝑡𝑞𝜀	𝑡𝑞𝜀	NOUN
iajs-3828	186	68	and	and	CCONJ
iajs-3828	186	69	𝑁(𝑎𝑗	𝑁(𝑎𝑗	NOUN
iajs-3828	186	70	,	,	PUNCT
iajs-3828	186	71	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	186	72	)	)	PUNCT
iajs-3828	186	73	→	→	SYM
iajs-3828	186	74	0	0	NUM
iajs-3828	186	75	and	and	CCONJ
iajs-3828	186	76	by	by	ADP
iajs-3828	186	77	the	the	DET
iajs-3828	186	78	inequality	inequality	NOUN
iajs-3828	186	79	(	(	PUNCT
iajs-3828	186	80	8)	8)	NUM
iajs-3828	186	81	,	,	PUNCT
iajs-3828	186	82	we	we	PRON
iajs-3828	186	83	have	have	VERB
iajs-3828	186	84	∅(𝜀	∅(𝜀	NOUN
iajs-3828	186	85	)	)	PUNCT
iajs-3828	186	86	≤	≤	NOUN
iajs-3828	186	87	∅(𝑡𝑞𝜀	∅(𝑡𝑞𝜀	NOUN
iajs-3828	186	88	)	)	PUNCT
iajs-3828	186	89	−	−	PROPN
iajs-3828	187	1	𝜓(0	𝜓(0	PROPN
iajs-3828	187	2	)	)	PUNCT
iajs-3828	187	3	.	.	PUNCT
iajs-3828	188	1	since	since	SCONJ
iajs-3828	188	2	∅	∅	NOUN
iajs-3828	188	3	is	be	AUX
iajs-3828	188	4	altering	alter	VERB
iajs-3828	188	5	distance	distance	NOUN
iajs-3828	188	6	function	function	NOUN
iajs-3828	188	7	and	and	CCONJ
iajs-3828	188	8	𝑡	𝑡	NOUN
iajs-3828	188	9	∈	∈	PROPN
iajs-3828	188	10	(	(	PUNCT
iajs-3828	188	11	0,1	0,1	NUM
iajs-3828	188	12	)	)	PUNCT
iajs-3828	188	13	,	,	PUNCT
iajs-3828	188	14	then	then	ADV
iajs-3828	188	15	𝜀	𝜀	PROPN
iajs-3828	188	16	≤	≤	NUM
iajs-3828	188	17	𝑡𝑞𝜀	𝑡𝑞𝜀	NOUN
iajs-3828	188	18	which	which	PRON
iajs-3828	188	19	a	a	DET
iajs-3828	188	20	contradiction	contradiction	NOUN
iajs-3828	188	21	.	.	PUNCT
iajs-3828	189	1	therefore	therefore	ADV
iajs-3828	189	2	,	,	PUNCT
iajs-3828	189	3	the	the	DET
iajs-3828	189	4	claim	claim	NOUN
iajs-3828	189	5	(	(	PUNCT
iajs-3828	189	6	i	i	NOUN
iajs-3828	189	7	)	)	PUNCT
iajs-3828	189	8	is	be	AUX
iajs-3828	189	9	held	hold	VERB
iajs-3828	189	10	.	.	PUNCT
iajs-3828	190	1	now	now	ADV
iajs-3828	190	2	,	,	PUNCT
iajs-3828	190	3	we	we	PRON
iajs-3828	190	4	will	will	AUX
iajs-3828	190	5	prove	prove	VERB
iajs-3828	190	6	(	(	PUNCT
iajs-3828	190	7	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	190	8	)	)	PUNCT
iajs-3828	190	9	is	be	AUX
iajs-3828	190	10	a	a	DET
iajs-3828	190	11	cauchy	cauchy	ADJ
iajs-3828	190	12	sequence	sequence	NOUN
iajs-3828	190	13	in	in	ADP
iajs-3828	190	14	(	(	PUNCT
iajs-3828	190	15	£	£	NOUN
iajs-3828	190	16	,	,	PUNCT
iajs-3828	190	17	𝜌	𝜌	NOUN
iajs-3828	190	18	)	)	PUNCT
iajs-3828	190	19	.	.	PUNCT
iajs-3828	191	1	fix	fix	NOUN
iajs-3828	191	2	ε	ε	PROPN
iajs-3828	191	3	>	>	X
iajs-3828	191	4	0	0	NUM
iajs-3828	191	5	.	.	PUNCT
iajs-3828	192	1	by	by	ADP
iajs-3828	192	2	the	the	DET
iajs-3828	192	3	claim	claim	NOUN
iajs-3828	192	4	,	,	PUNCT
iajs-3828	192	5	∃𝑚0	∃𝑚0	NOUN
iajs-3828	192	6	such	such	ADJ
iajs-3828	192	7	that	that	SCONJ
iajs-3828	192	8	if	if	SCONJ
iajs-3828	192	9	𝑗	𝑗	PROPN
iajs-3828	192	10	,	,	PUNCT
iajs-3828	192	11	𝑖	𝑖	PRON
iajs-3828	192	12	≥	≥	NOUN
iajs-3828	192	13	𝑚0	𝑚0	NOUN
iajs-3828	192	14	with	with	ADP
iajs-3828	192	15	𝑗	𝑗	INTJ
iajs-3828	192	16	−	−	PROPN
iajs-3828	192	17	𝑖	𝑖	SYM
iajs-3828	192	18	≡	≡	PROPN
iajs-3828	192	19	1(mod𝑛	1(mod𝑛	NUM
iajs-3828	192	20	)	)	PUNCT
iajs-3828	192	21	such	such	ADJ
iajs-3828	192	22	that	that	SCONJ
iajs-3828	192	23	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	192	24	,	,	PUNCT
iajs-3828	192	25	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	192	26	)	)	PUNCT
iajs-3828	192	27	≤	≤	NUM
iajs-3828	192	28	ε	ε	PROPN
iajs-3828	192	29	2	2	NUM
iajs-3828	192	30	.	.	PUNCT
iajs-3828	193	1	since	since	SCONJ
iajs-3828	193	2	lim	lim	PROPN
iajs-3828	193	3	𝑚→∞	𝑚→∞	PUNCT
iajs-3828	193	4	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	193	5	,	,	PUNCT
iajs-3828	193	6	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	193	7	)	)	PUNCT
iajs-3828	193	8	=	=	SYM
iajs-3828	193	9	0	0	NUM
iajs-3828	193	10	,	,	PUNCT
iajs-3828	193	11	also	also	ADV
iajs-3828	193	12	∃𝑚1	∃𝑚1	NOUN
iajs-3828	193	13	∈	∈	NOUN
iajs-3828	194	1	𝑁	𝑁	ADP
iajs-3828	194	2	such	such	ADJ
iajs-3828	194	3	that𝜌(𝑎𝑚	that𝜌(𝑎𝑚	NOUN
iajs-3828	194	4	,	,	PUNCT
iajs-3828	194	5	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	194	6	)	)	PUNCT
iajs-3828	194	7	≤	≤	NOUN
iajs-3828	194	8	𝜀	𝜀	ADP
iajs-3828	194	9	2𝑛	2𝑛	PROPN
iajs-3828	194	10	,	,	PUNCT
iajs-3828	194	11	∀𝑚	∀𝑚	PROPN
iajs-3828	194	12	≥	≥	NOUN
iajs-3828	194	13	𝑚1	𝑚1	NOUN
iajs-3828	194	14	.	.	PUNCT
iajs-3828	194	15	suppose	suppose	VERB
iajs-3828	194	16	𝑐	𝑐	X
iajs-3828	194	17	,	,	PUNCT
iajs-3828	194	18	𝑣	𝑣	DET
iajs-3828	194	19	≥	≥	NOUN
iajs-3828	194	20	max{𝑚0,𝑚1	max{𝑚0,𝑚1	NOUN
iajs-3828	194	21	}	}	PUNCT
iajs-3828	194	22	and	and	CCONJ
iajs-3828	194	23	𝑐	𝑐	PROPN
iajs-3828	194	24	>	>	X
iajs-3828	194	25	𝑣.	𝑣.	NOUN
iajs-3828	194	26	then	then	ADV
iajs-3828	194	27	there	there	PRON
iajs-3828	194	28	exists	exist	VERB
iajs-3828	194	29	ℎ	ℎ	PROPN
iajs-3828	194	30	∈	∈	PROPN
iajs-3828	194	31	{	{	PUNCT
iajs-3828	194	32	1	1	NUM
iajs-3828	194	33	,	,	PUNCT
iajs-3828	194	34	2	2	NUM
iajs-3828	194	35	,	,	PUNCT
iajs-3828	194	36	…	…	PUNCT
iajs-3828	194	37	.	.	PUNCT
iajs-3828	194	38	.	.	PUNCT
iajs-3828	195	1	,	,	PUNCT
iajs-3828	195	2	𝑛	𝑛	X
iajs-3828	195	3	}	}	PUNCT
iajs-3828	195	4	such	such	ADJ
iajs-3828	195	5	that	that	SCONJ
iajs-3828	195	6	𝑣	𝑣	PRON
iajs-3828	195	7	−	−	PROPN
iajs-3828	195	8	𝑐	𝑐	PROPN
iajs-3828	195	9	≡	≡	PROPN
iajs-3828	195	10	ℎ(mod	ℎ(mod	PROPN
iajs-3828	195	11	𝑛	𝑛	PROPN
iajs-3828	195	12	)	)	PUNCT
iajs-3828	195	13	.	.	PUNCT
iajs-3828	196	1	therefore	therefore	ADV
iajs-3828	196	2	,	,	PUNCT
iajs-3828	196	3	𝑣	𝑣	ADP
iajs-3828	196	4	−	−	PROPN
iajs-3828	196	5	𝑐	𝑐	PROPN
iajs-3828	196	6	+	+	NUM
iajs-3828	196	7	𝑟	𝑟	NUM
iajs-3828	196	8	≡	≡	PROPN
iajs-3828	196	9	1(mod	1(mod	NUM
iajs-3828	196	10	𝑛	𝑛	NOUN
iajs-3828	196	11	)	)	PUNCT
iajs-3828	196	12	for	for	ADP
iajs-3828	196	13	𝑟	𝑟	NOUN
iajs-3828	196	14	=	=	SYM
iajs-3828	196	15	𝑛	𝑛	DET
iajs-3828	196	16	−	−	NOUN
iajs-3828	196	17	ℎ	ℎ	X
iajs-3828	196	18	+	+	NOUN
iajs-3828	196	19	1	1	X
iajs-3828	196	20	.	.	PUNCT
iajs-3828	197	1	so	so	ADV
iajs-3828	197	2	,	,	PUNCT
iajs-3828	197	3	we	we	PRON
iajs-3828	197	4	have	have	VERB
iajs-3828	197	5	𝜌(𝑎𝑐	𝜌(𝑎𝑐	NUM
iajs-3828	197	6	,	,	PUNCT
iajs-3828	197	7	𝑎𝑣	𝑎𝑣	NOUN
iajs-3828	197	8	)	)	PUNCT
iajs-3828	197	9	≤	≤	X
iajs-3828	197	10	𝑞	𝑞	X
iajs-3828	197	11	𝜌(𝑎𝑐	𝜌(𝑎𝑐	NUM
iajs-3828	197	12	,	,	PUNCT
iajs-3828	197	13	𝑎𝑣+𝑟	𝑎𝑣+𝑟	PROPN
iajs-3828	197	14	)	)	PUNCT
iajs-3828	197	15	+	+	CCONJ
iajs-3828	197	16	𝑞	𝑞	X
iajs-3828	197	17	2	2	NUM
iajs-3828	197	18	𝜌(𝑎𝑣+𝑟	𝜌(𝑎𝑣+𝑟	NOUN
iajs-3828	197	19	,	,	PUNCT
iajs-3828	197	20	𝑎𝑣+𝑟−1	𝑎𝑣+𝑟−1	ADJ
iajs-3828	197	21	)	)	PUNCT
iajs-3828	197	22	+	+	NUM
iajs-3828	197	23	𝑞	𝑞	PROPN
iajs-3828	197	24	3𝜌(𝑎𝑣+𝑟−1	3𝜌(𝑎𝑣+𝑟−1	NUM
iajs-3828	197	25	,	,	PUNCT
iajs-3828	197	26	𝑎𝑣+𝑟−2	𝑎𝑣+𝑟−2	PROPN
iajs-3828	197	27	)	)	PUNCT
iajs-3828	197	28	+	+	PUNCT
iajs-3828	198	1	+	+	NUM
iajs-3828	198	2	𝑞𝑟𝜌(𝑎𝑣+1	𝑞𝑟𝜌(𝑎𝑣+1	NOUN
iajs-3828	198	3	,	,	PUNCT
iajs-3828	198	4	𝑎𝑣	𝑎𝑣	X
iajs-3828	198	5	)	)	PUNCT
iajs-3828	198	6	(	(	PUNCT
iajs-3828	198	7	9	9	NUM
iajs-3828	198	8	)	)	PUNCT
iajs-3828	198	9	by	by	ADP
iajs-3828	198	10	𝜌(𝑎𝑗	𝜌(𝑎𝑗	PROPN
iajs-3828	198	11	,	,	PUNCT
iajs-3828	198	12	𝑎𝑖	𝑎𝑖	CCONJ
iajs-3828	198	13	)	)	PUNCT
iajs-3828	198	14	≤	≤	NUM
iajs-3828	198	15	ε	ε	PROPN
iajs-3828	198	16	2	2	NUM
iajs-3828	198	17	and	and	CCONJ
iajs-3828	198	18	𝜌(𝑎𝑚	𝜌(𝑎𝑚	PROPN
iajs-3828	198	19	,	,	PUNCT
iajs-3828	198	20	𝑎𝑚+1	𝑎𝑚+1	NOUN
iajs-3828	198	21	)	)	PUNCT
iajs-3828	198	22	≤	≤	NOUN
iajs-3828	198	23	𝜀	𝜀	ADP
iajs-3828	198	24	2𝑛	2𝑛	PROPN
iajs-3828	198	25	and	and	CCONJ
iajs-3828	198	26	from	from	ADP
iajs-3828	198	27	(	(	PUNCT
iajs-3828	198	28	9	9	NUM
iajs-3828	198	29	)	)	PUNCT
iajs-3828	198	30	,	,	PUNCT
iajs-3828	198	31	𝜌(𝑎𝑐	𝜌(𝑎𝑐	NUM
iajs-3828	198	32	,	,	PUNCT
iajs-3828	198	33	𝑎𝑣	𝑎𝑣	NOUN
iajs-3828	198	34	)	)	PUNCT
iajs-3828	198	35	≤	≤	NOUN
iajs-3828	198	36	𝑞	𝑞	X
iajs-3828	198	37	𝜀	𝜀	PROPN
iajs-3828	198	38	𝑛	𝑛	PROPN
iajs-3828	198	39	(	(	PUNCT
iajs-3828	198	40	1	1	NUM
iajs-3828	198	41	1−𝑞	1−𝑞	NUM
iajs-3828	198	42	)	)	PUNCT
iajs-3828	199	1	→	→	SYM
iajs-3828	199	2	0	0	NUM
iajs-3828	199	3	,	,	PUNCT
iajs-3828	199	4	as	as	ADP
iajs-3828	199	5	𝑛	𝑛	PROPN
iajs-3828	199	6	→	→	SYM
iajs-3828	199	7	∞	∞	PROPN
iajs-3828	199	8	this	this	PRON
iajs-3828	199	9	proves	prove	VERB
iajs-3828	199	10	that	that	SCONJ
iajs-3828	199	11	(	(	PUNCT
iajs-3828	199	12	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	199	13	)	)	PUNCT
iajs-3828	199	14	is	be	AUX
iajs-3828	199	15	a	a	DET
iajs-3828	199	16	cauchy	cauchy	ADJ
iajs-3828	199	17	sequence	sequence	NOUN
iajs-3828	199	18	.	.	PUNCT
iajs-3828	200	1	the	the	DET
iajs-3828	200	2	completeness	completeness	NOUN
iajs-3828	200	3	of	of	ADP
iajs-3828	200	4	(	(	PUNCT
iajs-3828	200	5	£	£	PROPN
iajs-3828	200	6	,	,	PUNCT
iajs-3828	200	7	𝜌	𝜌	X
iajs-3828	200	8	)	)	PUNCT
iajs-3828	200	9	implies	imply	VERB
iajs-3828	200	10	to	to	PART
iajs-3828	200	11	exists	exist	VERB
iajs-3828	200	12	𝑐	𝑐	PROPN
iajs-3828	200	13	∈	∈	PROPN
iajs-3828	200	14	£	£	NOUN
iajs-3828	200	15	such	such	ADJ
iajs-3828	200	16	that	that	SCONJ
iajs-3828	200	17	lim	lim	PROPN
iajs-3828	200	18	𝑚→∞	𝑚→∞	PUNCT
iajs-3828	200	19	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	200	20	=	=	X
iajs-3828	200	21	𝑐.	𝑐.	ADV
iajs-3828	200	22	now	now	ADV
iajs-3828	200	23	,	,	PUNCT
iajs-3828	200	24	to	to	PART
iajs-3828	200	25	prove	prove	VERB
iajs-3828	200	26	𝑐	𝑐	PROPN
iajs-3828	200	27	is	be	AUX
iajs-3828	200	28	a	a	DET
iajs-3828	200	29	fixed	fix	VERB
iajs-3828	200	30	point	point	NOUN
iajs-3828	200	31	for	for	ADP
iajs-3828	200	32	𝑓.	𝑓.	NOUN
iajs-3828	200	33	since	since	SCONJ
iajs-3828	200	34	£	£	NOUN
iajs-3828	200	35	=	=	SYM
iajs-3828	200	36	⋃	⋃	ADP
iajs-3828	200	37	£	£	SYM
iajs-3828	200	38	𝑖	𝑖	SYM
iajs-3828	200	39	𝑛	𝑛	PRON
iajs-3828	200	40	𝑖=1	𝑖=1	PROPN
iajs-3828	200	41	is	be	AUX
iajs-3828	200	42	a	a	DET
iajs-3828	200	43	cyclic	cyclic	ADJ
iajs-3828	200	44	representation	representation	NOUN
iajs-3828	200	45	of	of	ADP
iajs-3828	200	46	£	£	SYM
iajs-3828	200	47	w.r.t	w.r.t	NOUN
iajs-3828	200	48	.	.	PUNCT
iajs-3828	201	1	,	,	PUNCT
iajs-3828	201	2	𝑓	𝑓	X
iajs-3828	201	3	,	,	PUNCT
iajs-3828	201	4	the	the	DET
iajs-3828	201	5	sequence	sequence	NOUN
iajs-3828	201	6	(	(	PUNCT
iajs-3828	201	7	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	201	8	)	)	PUNCT
iajs-3828	201	9	has	have	AUX
iajs-3828	201	10	infinite	infinite	ADJ
iajs-3828	201	11	terms	term	NOUN
iajs-3828	201	12	in	in	ADP
iajs-3828	201	13	each	each	DET
iajs-3828	201	14	£	£	NOUN
iajs-3828	201	15	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	201	16	for	for	ADP
iajs-3828	201	17	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	201	18	∈	∈	PROPN
iajs-3828	201	19	{	{	PUNCT
iajs-3828	201	20	1	1	NUM
iajs-3828	201	21	,	,	PUNCT
iajs-3828	201	22	2	2	NUM
iajs-3828	201	23	,	,	PUNCT
iajs-3828	201	24	.	.	PUNCT
iajs-3828	201	25	.	.	PUNCT
iajs-3828	202	1	.	.	PUNCT
iajs-3828	203	1	,	,	PUNCT
iajs-3828	203	2	𝑛	𝑛	PROPN
iajs-3828	203	3	}	}	PUNCT
iajs-3828	203	4	.	.	PUNCT
iajs-3828	204	1	closeness	closeness	NOUN
iajs-3828	204	2	of	of	ADP
iajs-3828	204	3	£	£	SYM
iajs-3828	204	4	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	204	5	for	for	ADP
iajs-3828	204	6	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	204	7	∈	∈	PROPN
iajs-3828	204	8	{	{	PUNCT
iajs-3828	204	9	1	1	NUM
iajs-3828	204	10	,	,	PUNCT
iajs-3828	204	11	2	2	NUM
iajs-3828	204	12	,	,	PUNCT
iajs-3828	204	13	.	.	PUNCT
iajs-3828	204	14	.	.	PUNCT
iajs-3828	205	1	.	.	PUNCT
iajs-3828	206	1	,	,	PUNCT
iajs-3828	206	2	𝑛	𝑛	X
iajs-3828	206	3	}	}	PUNCT
iajs-3828	206	4	implies	imply	VERB
iajs-3828	206	5	to	to	ADP
iajs-3828	206	6	𝑐	𝑐	PROPN
iajs-3828	206	7	∈	∈	PROPN
iajs-3828	206	8	⋂	⋂	PROPN
iajs-3828	206	9	£	£	SYM
iajs-3828	206	10	𝑖	𝑖	SYM
iajs-3828	206	11	𝑛	𝑛	PRON
iajs-3828	206	12	𝑖=1	𝑖=1	PROPN
iajs-3828	206	13	.	.	PUNCT
iajs-3828	207	1	suppose	suppose	VERB
iajs-3828	207	2	that	that	SCONJ
iajs-3828	207	3	𝑐	𝑐	PROPN
iajs-3828	207	4	∈	∈	PROPN
iajs-3828	207	5	£	£	SYM
iajs-3828	207	6	𝑖	𝑖	NOUN
iajs-3828	207	7	and	and	CCONJ
iajs-3828	207	8	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	207	9	∈	∈	PROPN
iajs-3828	207	10	£	£	SYM
iajs-3828	207	11	𝑖+1	𝑖+1	NUM
iajs-3828	207	12	and	and	CCONJ
iajs-3828	207	13	take	take	VERB
iajs-3828	207	14	a	a	DET
iajs-3828	207	15	subsequence	subsequence	NOUN
iajs-3828	207	16	(	(	PUNCT
iajs-3828	207	17	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	207	18	)	)	PUNCT
iajs-3828	207	19	𝑟∈𝑁	𝑟∈𝑁	PROPN
iajs-3828	207	20	of	of	ADP
iajs-3828	207	21	(	(	PUNCT
iajs-3828	207	22	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	207	23	)	)	PUNCT
iajs-3828	207	24	with	with	ADP
iajs-3828	207	25	𝑎𝑚𝑟	𝑎𝑚𝑟	ADJ
iajs-3828	207	26	∈	∈	PROPN
iajs-3828	208	1	£	£	SYM
iajs-3828	208	2	𝑖−1	𝑖−1	PROPN
iajs-3828	209	1	and	and	CCONJ
iajs-3828	209	2	take	take	VERB
iajs-3828	209	3	𝑎	𝑎	NOUN
iajs-3828	209	4	=	=	SYM
iajs-3828	209	5	𝑐	𝑐	NOUN
iajs-3828	209	6	,	,	PUNCT
iajs-3828	209	7	𝑏	𝑏	NOUN
iajs-3828	209	8	=	=	PUNCT
iajs-3828	209	9	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	209	10	in	in	ADP
iajs-3828	209	11	(	(	PUNCT
iajs-3828	209	12	4	4	NUM
iajs-3828	209	13	)	)	PUNCT
iajs-3828	209	14	∅(𝜌(𝑓𝑐	∅(𝜌(𝑓𝑐	PROPN
iajs-3828	209	15	,	,	PUNCT
iajs-3828	209	16	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	209	17	)	)	PUNCT
iajs-3828	209	18	)	)	PUNCT
iajs-3828	210	1	≤	≤	NUM
iajs-3828	210	2	∅(𝑀(𝑎𝑚𝑟	∅(𝑀(𝑎𝑚𝑟	PROPN
iajs-3828	210	3	,	,	PUNCT
iajs-3828	210	4	𝑐	𝑐	NOUN
iajs-3828	210	5	)	)	PUNCT
iajs-3828	210	6	)	)	PUNCT
iajs-3828	210	7	−	−	PROPN
iajs-3828	210	8	𝜓	𝜓	NOUN
iajs-3828	210	9	(	(	PUNCT
iajs-3828	210	10	𝑁(𝑎𝑚𝑟	𝑁(𝑎𝑚𝑟	ADV
iajs-3828	210	11	,	,	PUNCT
iajs-3828	210	12	𝑐	𝑐	NOUN
iajs-3828	210	13	)	)	PUNCT
iajs-3828	210	14	)	)	PUNCT
iajs-3828	210	15	(	(	PUNCT
iajs-3828	210	16	10	10	NUM
iajs-3828	210	17	)	)	PUNCT
iajs-3828	210	18	where	where	SCONJ
iajs-3828	210	19	𝑀(𝑎𝑚𝑟	𝑀(𝑎𝑚𝑟	ADJ
iajs-3828	210	20	,	,	PUNCT
iajs-3828	210	21	𝑐	𝑐	NOUN
iajs-3828	210	22	)	)	PUNCT
iajs-3828	210	23	=	=	SYM
iajs-3828	210	24	𝑡	𝑡	PROPN
iajs-3828	210	25	max	max	PROPN
iajs-3828	210	26	{	{	PUNCT
iajs-3828	210	27	𝜌(𝑎𝑚𝑟	𝜌(𝑎𝑚𝑟	PROPN
iajs-3828	210	28	,	,	PUNCT
iajs-3828	210	29	𝑐	𝑐	NOUN
iajs-3828	210	30	)	)	PUNCT
iajs-3828	210	31	,	,	PUNCT
iajs-3828	210	32	𝜌(𝑎𝑚𝑟	𝜌(𝑎𝑚𝑟	ADJ
iajs-3828	210	33	,	,	PUNCT
iajs-3828	210	34	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	210	35	)	)	PUNCT
iajs-3828	210	36	,	,	PUNCT
iajs-3828	210	37	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	210	38	,	,	PUNCT
iajs-3828	210	39	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	210	40	)	)	PUNCT
iajs-3828	210	41	,	,	PUNCT
iajs-3828	210	42	𝜌(𝑎𝑚𝑟	𝜌(𝑎𝑚𝑟	ADJ
iajs-3828	210	43	,	,	PUNCT
iajs-3828	210	44	𝑓𝑐	𝑓𝑐	PROPN
iajs-3828	210	45	)	)	PUNCT
iajs-3828	210	46	,	,	PUNCT
iajs-3828	210	47	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	210	48	,	,	PUNCT
iajs-3828	210	49	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	210	50	)	)	PUNCT
iajs-3828	210	51	}	}	PUNCT
iajs-3828	210	52	,	,	PUNCT
iajs-3828	210	53	𝑁(𝑎𝑚𝑟	𝑁(𝑎𝑚𝑟	ADV
iajs-3828	210	54	,	,	PUNCT
iajs-3828	210	55	𝑐	𝑐	NOUN
iajs-3828	210	56	)	)	PUNCT
iajs-3828	210	57	=	=	SYM
iajs-3828	210	58	𝑡	𝑡	PART
iajs-3828	210	59	min	min	NOUN
iajs-3828	210	60	{	{	PUNCT
iajs-3828	210	61	𝜌(𝑎𝑚𝑟	𝜌(𝑎𝑚𝑟	PROPN
iajs-3828	210	62	,	,	PUNCT
iajs-3828	210	63	𝑐	𝑐	NOUN
iajs-3828	210	64	)	)	PUNCT
iajs-3828	210	65	,	,	PUNCT
iajs-3828	210	66	𝜌(𝑎𝑚𝑟	𝜌(𝑎𝑚𝑟	ADJ
iajs-3828	210	67	,	,	PUNCT
iajs-3828	210	68	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	210	69	)	)	PUNCT
iajs-3828	210	70	,	,	PUNCT
iajs-3828	210	71	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	210	72	,	,	PUNCT
iajs-3828	210	73	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	210	74	)	)	PUNCT
iajs-3828	210	75	,	,	PUNCT
iajs-3828	210	76	𝜌(𝑎𝑚𝑟	𝜌(𝑎𝑚𝑟	ADJ
iajs-3828	210	77	,	,	PUNCT
iajs-3828	210	78	𝑓𝑐	𝑓𝑐	PROPN
iajs-3828	210	79	)	)	PUNCT
iajs-3828	210	80	,	,	PUNCT
iajs-3828	210	81	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	210	82	,	,	PUNCT
iajs-3828	210	83	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	210	84	)	)	PUNCT
iajs-3828	210	85	}	}	PUNCT
iajs-3828	210	86	.	.	PUNCT
iajs-3828	211	1	taking	take	VERB
iajs-3828	211	2	𝑟	𝑟	NOUN
iajs-3828	211	3	→	→	SYM
iajs-3828	211	4	∞	∞	PROPN
iajs-3828	211	5	,	,	PUNCT
iajs-3828	211	6	hence	hence	ADV
iajs-3828	211	7	𝑀(𝑐	𝑀(𝑐	NOUN
iajs-3828	211	8	,	,	PUNCT
iajs-3828	211	9	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	211	10	)	)	PUNCT
iajs-3828	212	1	=	=	SYM
iajs-3828	212	2	𝑡	𝑡	PROPN
iajs-3828	212	3	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	212	4	,	,	PUNCT
iajs-3828	212	5	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	212	6	)	)	PUNCT
iajs-3828	212	7	.	.	PUNCT
iajs-3828	213	1	and	and	CCONJ
iajs-3828	213	2	(	(	PUNCT
iajs-3828	213	3	𝑐	𝑐	NOUN
iajs-3828	213	4	,	,	PUNCT
iajs-3828	213	5	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	213	6	)	)	PUNCT
iajs-3828	213	7	=	=	SYM
iajs-3828	213	8	0	0	NUM
iajs-3828	213	9	,	,	PUNCT
iajs-3828	213	10	using	use	VERB
iajs-3828	213	11	(	(	PUNCT
iajs-3828	213	12	10	10	NUM
iajs-3828	213	13	)	)	PUNCT
iajs-3828	213	14	subsequently	subsequently	ADV
iajs-3828	213	15	∅(𝜌(𝑓𝑐	∅(𝜌(𝑓𝑐	PROPN
iajs-3828	213	16	,	,	PUNCT
iajs-3828	213	17	𝑐	𝑐	NOUN
iajs-3828	213	18	)	)	PUNCT
iajs-3828	213	19	)	)	PUNCT
iajs-3828	214	1	≤	≤	NUM
iajs-3828	214	2	∅(𝑡	∅(𝑡	NOUN
iajs-3828	214	3	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	214	4	,	,	PUNCT
iajs-3828	214	5	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	214	6	)	)	PUNCT
iajs-3828	214	7	)	)	PUNCT
iajs-3828	215	1	−	−	PROPN
iajs-3828	215	2	𝜓(0	𝜓(0	PROPN
iajs-3828	215	3	)	)	PUNCT
iajs-3828	216	1	≤	≤	NUM
iajs-3828	217	1	∅(𝑡	∅(𝑡	NOUN
iajs-3828	217	2	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	217	3	,	,	PUNCT
iajs-3828	217	4	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	217	5	)	)	PUNCT
iajs-3828	217	6	)	)	PUNCT
iajs-3828	217	7	.	.	PUNCT
iajs-3828	218	1	since	since	SCONJ
iajs-3828	218	2	∅	∅	NOUN
iajs-3828	218	3	is	be	AUX
iajs-3828	218	4	altering	alter	VERB
iajs-3828	218	5	distance	distance	NOUN
iajs-3828	218	6	function	function	NOUN
iajs-3828	218	7	,	,	PUNCT
iajs-3828	218	8	then	then	ADV
iajs-3828	218	9	𝜌(𝑓𝑐	𝜌(𝑓𝑐	PROPN
iajs-3828	218	10	,	,	PUNCT
iajs-3828	218	11	𝑐	𝑐	NOUN
iajs-3828	218	12	)	)	PUNCT
iajs-3828	218	13	<	<	X
iajs-3828	218	14	𝑡	𝑡	PROPN
iajs-3828	218	15	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	218	16	,	,	PUNCT
iajs-3828	218	17	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	218	18	)	)	PUNCT
iajs-3828	218	19	,	,	PUNCT
iajs-3828	218	20	which	which	PRON
iajs-3828	218	21	a	a	DET
iajs-3828	218	22	contradiction	contradiction	NOUN
iajs-3828	218	23	because	because	SCONJ
iajs-3828	218	24	𝑡	𝑡	PROPN
iajs-3828	218	25	∈	∈	PROPN
iajs-3828	218	26	(	(	PUNCT
iajs-3828	218	27	0,1	0,1	NOUN
iajs-3828	218	28	)	)	PUNCT
iajs-3828	218	29	,	,	PUNCT
iajs-3828	218	30	hence	hence	ADV
iajs-3828	218	31	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	218	32	=	=	PUNCT
iajs-3828	218	33	𝑐.	𝑐.	NOUN
iajs-3828	218	34	thus	thus	ADV
iajs-3828	218	35	𝑐	𝑐	PROPN
iajs-3828	218	36	is	be	AUX
iajs-3828	218	37	a	a	DET
iajs-3828	218	38	fixed	fix	VERB
iajs-3828	218	39	point	point	NOUN
iajs-3828	218	40	of	of	ADP
iajs-3828	218	41	𝑓.	𝑓.	NOUN
iajs-3828	218	42	for	for	ADP
iajs-3828	218	43	the	the	DET
iajs-3828	218	44	uniqueness	uniqueness	NOUN
iajs-3828	218	45	,	,	PUNCT
iajs-3828	218	46	suppose	suppose	VERB
iajs-3828	218	47	that	that	SCONJ
iajs-3828	218	48	there	there	PRON
iajs-3828	218	49	are	be	VERB
iajs-3828	218	50	two	two	NUM
iajs-3828	218	51	distinct	distinct	ADJ
iajs-3828	218	52	points	point	NOUN
iajs-3828	218	53	𝑐	𝑐	NOUN
iajs-3828	218	54	,	,	PUNCT
iajs-3828	218	55	𝑤	𝑤	X
iajs-3828	218	56	with	with	ADP
iajs-3828	218	57	𝑐	𝑐	PROPN
iajs-3828	218	58	and	and	CCONJ
iajs-3828	218	59	𝑤	𝑤	ADP
iajs-3828	218	60	fixed	fix	VERB
iajs-3828	218	61	points	point	NOUN
iajs-3828	218	62	of	of	ADP
iajs-3828	218	63	𝑓.	𝑓.	NOUN
iajs-3828	218	64	the	the	DET
iajs-3828	218	65	cyclic	cyclic	ADJ
iajs-3828	218	66	character	character	NOUN
iajs-3828	218	67	of	of	ADP
iajs-3828	218	68	𝑓	𝑓	PRON
iajs-3828	218	69	and	and	CCONJ
iajs-3828	218	70	the	the	DET
iajs-3828	218	71	fact	fact	NOUN
iajs-3828	218	72	that	that	SCONJ
iajs-3828	218	73	𝑐	𝑐	X
iajs-3828	218	74	,	,	PUNCT
iajs-3828	218	75	𝑤	𝑤	ADP
iajs-3828	218	76	∈	∈	NOUN
iajs-3828	219	1	£	£	NOUN
iajs-3828	219	2	=	=	PUNCT
iajs-3828	219	3	⋃	⋃	ADP
iajs-3828	219	4	£	£	SYM
iajs-3828	219	5	𝑖	𝑖	SYM
iajs-3828	219	6	𝑛	𝑛	PRON
iajs-3828	219	7	𝑖=1	𝑖=1	PROPN
iajs-3828	219	8	are	be	AUX
iajs-3828	219	9	fixed	fix	VERB
iajs-3828	219	10	points	point	NOUN
iajs-3828	219	11	of	of	ADP
iajs-3828	219	12	𝑓	𝑓	DET
iajs-3828	219	13	imply	imply	NOUN
iajs-3828	219	14	ihjpas	ihjpa	NOUN
iajs-3828	219	15	.	.	PUNCT
iajs-3828	220	1	2025,38(2	2025,38(2	PROPN
iajs-3828	220	2	)	)	PUNCT
iajs-3828	220	3	382	382	NUM
iajs-3828	221	1	that	that	PRON
iajs-3828	221	2	𝑐	𝑐	VERB
iajs-3828	221	3	,	,	PUNCT
iajs-3828	221	4	𝑤	𝑤	ADP
iajs-3828	221	5	∈	∈	PROPN
iajs-3828	221	6	⋂	⋂	PROPN
iajs-3828	221	7	£	£	SYM
iajs-3828	221	8	𝑖	𝑖	SYM
iajs-3828	221	9	𝑛	𝑛	PRON
iajs-3828	221	10	𝑖=1	𝑖=1	PROPN
iajs-3828	221	11	.	.	PUNCT
iajs-3828	222	1	using	use	VERB
iajs-3828	222	2	(	(	PUNCT
iajs-3828	222	3	4	4	NUM
iajs-3828	222	4	)	)	PUNCT
iajs-3828	222	5	,	,	PUNCT
iajs-3828	222	6	we	we	PRON
iajs-3828	222	7	can	can	AUX
iajs-3828	222	8	obtain	obtain	VERB
iajs-3828	222	9	∅(𝜌(𝑓𝑐	∅(𝜌(𝑓𝑐	PROPN
iajs-3828	222	10	,	,	PUNCT
iajs-3828	222	11	𝑓𝑤	𝑓𝑤	NOUN
iajs-3828	222	12	)	)	PUNCT
iajs-3828	222	13	)	)	PUNCT
iajs-3828	222	14	≤	≤	ADV
iajs-3828	222	15	∅(𝑀(𝑐	∅(𝑀(𝑐	ADV
iajs-3828	222	16	,	,	PUNCT
iajs-3828	222	17	𝑤	𝑤	ADP
iajs-3828	222	18	)	)	PUNCT
iajs-3828	222	19	)	)	PUNCT
iajs-3828	223	1	−	−	PROPN
iajs-3828	223	2	𝜓(𝑁(𝑐	𝜓(𝑁(𝑐	PROPN
iajs-3828	223	3	,	,	PUNCT
iajs-3828	223	4	𝑤	𝑤	ADP
iajs-3828	223	5	)	)	PUNCT
iajs-3828	223	6	)	)	PUNCT
iajs-3828	223	7	,	,	PUNCT
iajs-3828	223	8	where	where	SCONJ
iajs-3828	223	9	,	,	PUNCT
iajs-3828	223	10	𝑀(𝑐,𝑤	𝑀(𝑐,𝑤	PUNCT
iajs-3828	223	11	)	)	PUNCT
iajs-3828	223	12	=	=	SYM
iajs-3828	223	13	𝑡	𝑡	PROPN
iajs-3828	223	14	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	223	15	,	,	PUNCT
iajs-3828	223	16	𝑤	𝑤	ADP
iajs-3828	223	17	)	)	PUNCT
iajs-3828	223	18	and	and	CCONJ
iajs-3828	223	19	𝑁(𝑐	𝑁(𝑐	NOUN
iajs-3828	223	20	,	,	PUNCT
iajs-3828	223	21	𝑤	𝑤	NOUN
iajs-3828	223	22	)	)	PUNCT
iajs-3828	223	23	}	}	PUNCT
iajs-3828	223	24	=	=	SYM
iajs-3828	223	25	0	0	X
iajs-3828	223	26	.	.	PUNCT
iajs-3828	223	27	hence	hence	ADV
iajs-3828	223	28	∅(𝜌(𝑐	∅(𝜌(𝑐	NOUN
iajs-3828	223	29	,	,	PUNCT
iajs-3828	223	30	𝑤	𝑤	ADP
iajs-3828	223	31	)	)	PUNCT
iajs-3828	223	32	)	)	PUNCT
iajs-3828	224	1	=	=	PUNCT
iajs-3828	225	1	∅(𝜌(𝑓𝑐	∅(𝜌(𝑓𝑐	PROPN
iajs-3828	225	2	,	,	PUNCT
iajs-3828	225	3	𝑓𝑤	𝑓𝑤	NOUN
iajs-3828	225	4	)	)	PUNCT
iajs-3828	225	5	)	)	PUNCT
iajs-3828	225	6	≤	≤	NUM
iajs-3828	225	7	∅(𝑡	∅(𝑡	PROPN
iajs-3828	225	8	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	225	9	,	,	PUNCT
iajs-3828	225	10	𝑤	𝑤	ADP
iajs-3828	225	11	)	)	PUNCT
iajs-3828	225	12	)	)	PUNCT
iajs-3828	225	13	.	.	PUNCT
iajs-3828	226	1	and	and	CCONJ
iajs-3828	226	2	since	since	SCONJ
iajs-3828	226	3	∅	∅	NOUN
iajs-3828	226	4	is	be	AUX
iajs-3828	226	5	altering	alter	VERB
iajs-3828	226	6	distance	distance	NOUN
iajs-3828	226	7	function	function	NOUN
iajs-3828	226	8	,	,	PUNCT
iajs-3828	226	9	we	we	PRON
iajs-3828	226	10	obtain	obtain	VERB
iajs-3828	226	11	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	226	12	,	,	PUNCT
iajs-3828	226	13	𝑤	𝑤	ADP
iajs-3828	226	14	)	)	PUNCT
iajs-3828	226	15	≤	≤	NOUN
iajs-3828	226	16	𝑞	𝑞	PRON
iajs-3828	226	17	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	226	18	,	,	PUNCT
iajs-3828	226	19	𝑤	𝑤	ADP
iajs-3828	226	20	)	)	PUNCT
iajs-3828	226	21	which	which	PRON
iajs-3828	226	22	a	a	PRON
iajs-3828	226	23	is	be	AUX
iajs-3828	226	24	contradiction	contradiction	NOUN
iajs-3828	226	25	since	since	SCONJ
iajs-3828	226	26	it	it	PRON
iajs-3828	226	27	is	be	AUX
iajs-3828	226	28	not	not	PART
iajs-3828	226	29	true	true	ADJ
iajs-3828	226	30	for	for	ADP
iajs-3828	226	31	all	all	PRON
iajs-3828	226	32	𝑡	𝑡	ADP
iajs-3828	226	33	∈	∈	PROPN
iajs-3828	226	34	(	(	PUNCT
iajs-3828	226	35	0,1	0,1	NUM
iajs-3828	226	36	)	)	PUNCT
iajs-3828	226	37	,	,	PUNCT
iajs-3828	226	38	then	then	ADV
iajs-3828	226	39	𝑐	𝑐	NOUN
iajs-3828	226	40	=	=	NOUN
iajs-3828	226	41	𝑤.	𝑤.	NOUN
iajs-3828	226	42	hence	hence	ADV
iajs-3828	226	43	𝑓	𝑓	ADV
iajs-3828	226	44	has	have	VERB
iajs-3828	226	45	a	a	DET
iajs-3828	226	46	unique	unique	ADJ
iajs-3828	226	47	fixed	fix	VERB
iajs-3828	226	48	point	point	NOUN
iajs-3828	226	49	in	in	ADP
iajs-3828	226	50	£	£	PROPN
iajs-3828	226	51	and	and	CCONJ
iajs-3828	226	52	𝑐	𝑐	PROPN
iajs-3828	226	53	∈	∈	PROPN
iajs-3828	226	54	⋂	⋂	PROPN
iajs-3828	226	55	£	£	SYM
iajs-3828	226	56	𝑖	𝑖	SYM
iajs-3828	226	57	𝑛	𝑛	PRON
iajs-3828	226	58	𝑖=1	𝑖=1	PROPN
iajs-3828	226	59	.	.	PUNCT
iajs-3828	227	1	example	example	NOUN
iajs-3828	227	2	2.6	2.6	NUM
iajs-3828	227	3	:	:	PUNCT
iajs-3828	227	4	let	let	VERB
iajs-3828	227	5	£	£	NOUN
iajs-3828	227	6	=	=	SYM
iajs-3828	227	7	{	{	PUNCT
iajs-3828	227	8	6	6	NUM
iajs-3828	227	9	,	,	PUNCT
iajs-3828	227	10	7	7	NUM
iajs-3828	227	11	,	,	PUNCT
iajs-3828	227	12	8	8	NUM
iajs-3828	227	13	,	,	PUNCT
iajs-3828	227	14	9	9	NUM
iajs-3828	227	15	,	,	PUNCT
iajs-3828	227	16	10	10	NUM
iajs-3828	227	17	}	}	PUNCT
iajs-3828	227	18	with	with	ADP
iajs-3828	227	19	𝜌	𝜌	ADP
iajs-3828	227	20	:	:	PUNCT
iajs-3828	227	21	£	£	SYM
iajs-3828	227	22	×	×	NOUN
iajs-3828	227	23	£	£	NOUN
iajs-3828	227	24	→	→	SYM
iajs-3828	227	25	[	[	X
iajs-3828	227	26	0,∞	0,∞	NOUN
iajs-3828	227	27	)	)	PUNCT
iajs-3828	227	28	defined	define	VERB
iajs-3828	227	29	by	by	ADP
iajs-3828	227	30	𝜌(𝑎	𝜌(𝑎	PROPN
iajs-3828	227	31	,	,	PUNCT
iajs-3828	227	32	𝑏	𝑏	NOUN
iajs-3828	227	33	)	)	PUNCT
iajs-3828	227	34	=	=	NOUN
iajs-3828	227	35	{	{	PUNCT
iajs-3828	227	36	0	0	NUM
iajs-3828	227	37	,	,	PUNCT
iajs-3828	227	38	𝑖𝑓	𝑖𝑓	VERB
iajs-3828	228	1	𝑎	𝑎	X
iajs-3828	228	2	=	=	SYM
iajs-3828	228	3	𝑏	𝑏	PROPN
iajs-3828	228	4	6	6	NUM
iajs-3828	228	5	,	,	PUNCT
iajs-3828	228	6	𝑖𝑓	𝑖𝑓	ADP
iajs-3828	228	7	𝑎	𝑎	PRON
iajs-3828	228	8	≠	≠	PROPN
iajs-3828	228	9	𝑏	𝑏	PROPN
iajs-3828	228	10	,	,	PUNCT
iajs-3828	228	11	𝑎	𝑎	NOUN
iajs-3828	228	12	,	,	PUNCT
iajs-3828	228	13	𝑏	𝑏	PROPN
iajs-3828	228	14	∈	∈	PROPN
iajs-3828	228	15	{	{	PUNCT
iajs-3828	228	16	6,7,8,9	6,7,8,9	NUM
iajs-3828	228	17	}	}	SYM
iajs-3828	228	18	17	17	NUM
iajs-3828	228	19	,	,	PUNCT
iajs-3828	228	20	𝑖𝑓	𝑖𝑓	ADP
iajs-3828	228	21	𝑎	𝑎	NOUN
iajs-3828	228	22	,	,	PUNCT
iajs-3828	228	23	𝑏	𝑏	PROPN
iajs-3828	228	24	∈	∈	PROPN
iajs-3828	228	25	{	{	PUNCT
iajs-3828	228	26	9,10	9,10	NUM
iajs-3828	228	27	}	}	PUNCT
iajs-3828	228	28	and	and	CCONJ
iajs-3828	228	29	𝑎	𝑎	DET
iajs-3828	228	30	≠	≠	PROPN
iajs-3828	228	31	𝑏	𝑏	PROPN
iajs-3828	228	32	40	40	NUM
iajs-3828	228	33	,	,	PUNCT
iajs-3828	228	34	𝑖𝑓𝑎	𝑖𝑓𝑎	NOUN
iajs-3828	228	35	∈	∈	PROPN
iajs-3828	228	36	{	{	PUNCT
iajs-3828	228	37	6,7,8	6,7,8	NUM
iajs-3828	228	38	}	}	PUNCT
iajs-3828	228	39	and	and	CCONJ
iajs-3828	228	40	𝑏	𝑏	NOUN
iajs-3828	228	41	=	=	SYM
iajs-3828	228	42	10	10	NUM
iajs-3828	228	43	(	(	PUNCT
iajs-3828	228	44	or	or	CCONJ
iajs-3828	228	45	𝑏	𝑏	PRON
iajs-3828	228	46	∈	∈	PROPN
iajs-3828	228	47	{	{	PUNCT
iajs-3828	228	48	6,7,8	6,7,8	NUM
iajs-3828	228	49	}	}	PUNCT
iajs-3828	228	50	and	and	CCONJ
iajs-3828	228	51	𝑎	𝑎	X
iajs-3828	228	52	=	=	SYM
iajs-3828	228	53	10	10	NUM
iajs-3828	228	54	)	)	PUNCT
iajs-3828	228	55	.	.	PUNCT
iajs-3828	229	1	since	since	SCONJ
iajs-3828	229	2	all	all	DET
iajs-3828	229	3	cauchy	cauchy	ADJ
iajs-3828	229	4	sequences	sequence	NOUN
iajs-3828	229	5	in	in	ADP
iajs-3828	229	6	£	£	NOUN
iajs-3828	229	7	are	be	AUX
iajs-3828	229	8	constant	constant	ADJ
iajs-3828	229	9	.	.	PUNCT
iajs-3828	230	1	therefore	therefore	ADV
iajs-3828	230	2	,	,	PUNCT
iajs-3828	230	3	are	be	AUX
iajs-3828	230	4	convergent	convergent	ADJ
iajs-3828	230	5	.	.	PUNCT
iajs-3828	231	1	then	then	ADV
iajs-3828	231	2	(	(	PUNCT
iajs-3828	231	3	£	£	NOUN
iajs-3828	231	4	,	,	PUNCT
iajs-3828	231	5	𝜌	𝜌	X
iajs-3828	231	6	)	)	PUNCT
iajs-3828	231	7	is	be	AUX
iajs-3828	231	8	complete	complete	ADJ
iajs-3828	231	9	b	b	X
iajs-3828	231	10	-	-	PUNCT
iajs-3828	231	11	metric	metric	ADJ
iajs-3828	231	12	space	space	NOUN
iajs-3828	231	13	with	with	ADP
iajs-3828	231	14	𝑞	𝑞	PROPN
iajs-3828	231	15	=	=	PROPN
iajs-3828	231	16	2	2	NUM
iajs-3828	231	17	.	.	NOUN
iajs-3828	231	18	and	and	CCONJ
iajs-3828	231	19	£	£	SYM
iajs-3828	231	20	1	1	NUM
iajs-3828	231	21	=	=	SYM
iajs-3828	231	22	{	{	PUNCT
iajs-3828	231	23	6,8,10	6,8,10	NUM
iajs-3828	231	24	}	}	PUNCT
iajs-3828	231	25	and	and	CCONJ
iajs-3828	231	26	£	£	SYM
iajs-3828	231	27	2	2	NUM
iajs-3828	231	28	=	=	SYM
iajs-3828	231	29	{	{	PUNCT
iajs-3828	231	30	6,7,9	6,7,9	NUM
iajs-3828	231	31	}	}	PUNCT
iajs-3828	231	32	,	,	PUNCT
iajs-3828	231	33	£	£	NOUN
iajs-3828	231	34	=	=	PUNCT
iajs-3828	231	35	⋃	⋃	ADP
iajs-3828	231	36	£	£	SYM
iajs-3828	231	37	𝑖	𝑖	ADP
iajs-3828	231	38	2	2	NUM
iajs-3828	231	39	𝑖=1	𝑖=1	PUNCT
iajs-3828	231	40	.	.	PUNCT
iajs-3828	232	1	define	define	VERB
iajs-3828	232	2	𝑓:⋃	𝑓:⋃	ADP
iajs-3828	232	3	£	£	SYM
iajs-3828	232	4	𝑖	𝑖	ADP
iajs-3828	232	5	2	2	NUM
iajs-3828	232	6	𝑖=1	𝑖=1	PUNCT
iajs-3828	232	7	→	→	PUNCT
iajs-3828	232	8	⋃	⋃	ADP
iajs-3828	232	9	£	£	SYM
iajs-3828	232	10	𝑖	𝑖	ADP
iajs-3828	232	11	2	2	NUM
iajs-3828	232	12	𝑖=1	𝑖=1	PUNCT
iajs-3828	232	13	such	such	ADJ
iajs-3828	232	14	that	that	DET
iajs-3828	232	15	𝑓(𝑎	𝑓(𝑎	NOUN
iajs-3828	232	16	)	)	PUNCT
iajs-3828	232	17	=	=	SYM
iajs-3828	232	18	6	6	NUM
iajs-3828	232	19	and	and	CCONJ
iajs-3828	232	20	𝑖𝑓	𝑖𝑓	ADP
iajs-3828	232	21	𝑎	𝑎	X
iajs-3828	232	22	∈	∈	NOUN
iajs-3828	232	23	{	{	PUNCT
iajs-3828	232	24	6,7,8,9	6,7,8,9	NUM
iajs-3828	232	25	}	}	PUNCT
iajs-3828	232	26	and	and	CCONJ
iajs-3828	232	27	𝑓(10	𝑓(10	NOUN
iajs-3828	232	28	)	)	PUNCT
iajs-3828	232	29	=	=	SYM
iajs-3828	232	30	8	8	X
iajs-3828	232	31	.	.	PUNCT
iajs-3828	233	1	so	so	ADV
iajs-3828	233	2	,	,	PUNCT
iajs-3828	233	3	(	(	PUNCT
iajs-3828	233	4	£	£	SYM
iajs-3828	233	5	1	1	NUM
iajs-3828	233	6	)	)	PUNCT
iajs-3828	233	7	⊂	⊂	PUNCT
iajs-3828	234	1	£	£	SYM
iajs-3828	234	2	2,𝑓(£2	2,𝑓(£2	NOUN
iajs-3828	234	3	)	)	PUNCT
iajs-3828	235	1	⊂	⊂	PUNCT
iajs-3828	235	2	£	£	SYM
iajs-3828	235	3	1	1	NUM
iajs-3828	235	4	,	,	PUNCT
iajs-3828	235	5	for	for	ADP
iajs-3828	235	6	𝑎	𝑎	PROPN
iajs-3828	235	7	∈	∈	NOUN
iajs-3828	235	8	£	£	SYM
iajs-3828	235	9	1	1	NUM
iajs-3828	235	10	,	,	PUNCT
iajs-3828	235	11	𝑏	𝑏	PROPN
iajs-3828	235	12	∈	∈	PROPN
iajs-3828	235	13	£	£	SYM
iajs-3828	235	14	2	2	NUM
iajs-3828	235	15	,	,	PUNCT
iajs-3828	235	16	and	and	CCONJ
iajs-3828	235	17	take	take	VERB
iajs-3828	235	18	𝑡	𝑡	NOUN
iajs-3828	235	19	=	=	NOUN
iajs-3828	235	20	1	1	NUM
iajs-3828	235	21	4	4	NUM
iajs-3828	235	22	.	.	PUNCT
iajs-3828	236	1	let	let	VERB
iajs-3828	236	2	∅,𝜓	∅,𝜓	NOUN
iajs-3828	236	3	:	:	PUNCT
iajs-3828	236	4	[	[	X
iajs-3828	236	5	0,∞	0,∞	NOUN
iajs-3828	236	6	)	)	PUNCT
iajs-3828	236	7	→	→	PUNCT
iajs-3828	237	1	[	[	X
iajs-3828	237	2	0,∞	0,∞	NUM
iajs-3828	237	3	)	)	PUNCT
iajs-3828	237	4	such	such	ADJ
iajs-3828	237	5	that	that	DET
iajs-3828	237	6	∅(𝑟	∅(𝑟	NOUN
iajs-3828	237	7	)	)	PUNCT
iajs-3828	237	8	=	=	PUNCT
iajs-3828	237	9	𝑟	𝑟	NOUN
iajs-3828	237	10	4	4	NUM
iajs-3828	237	11	and	and	CCONJ
iajs-3828	237	12	𝜓(𝑟	𝜓(𝑟	NUM
iajs-3828	237	13	)	)	PUNCT
iajs-3828	237	14	=	=	SYM
iajs-3828	237	15	𝑟	𝑟	DET
iajs-3828	237	16	2	2	NUM
iajs-3828	237	17	.	.	PUNCT
iajs-3828	238	1	then	then	ADV
iajs-3828	238	2	∅	∅	NOUN
iajs-3828	238	3	and	and	CCONJ
iajs-3828	238	4	𝜓	𝜓	PROPN
iajs-3828	238	5	are	be	AUX
iajs-3828	238	6	altering	alter	VERB
iajs-3828	238	7	distance	distance	NOUN
iajs-3828	238	8	functions	function	NOUN
iajs-3828	238	9	.	.	PUNCT
iajs-3828	239	1	it	it	PRON
iajs-3828	239	2	is	be	AUX
iajs-3828	239	3	easy	easy	ADJ
iajs-3828	239	4	to	to	PART
iajs-3828	239	5	check	check	VERB
iajs-3828	239	6	condition	condition	NOUN
iajs-3828	239	7	(	(	PUNCT
iajs-3828	239	8	4	4	X
iajs-3828	239	9	)	)	PUNCT
iajs-3828	239	10	holds	hold	VERB
iajs-3828	239	11	with	with	ADP
iajs-3828	239	12	fixed	fix	VERB
iajs-3828	239	13	point	point	NOUN
iajs-3828	239	14	𝑎	𝑎	X
iajs-3828	239	15	=	=	SYM
iajs-3828	239	16	6	6	NUM
iajs-3828	239	17	.	.	PUNCT
iajs-3828	240	1	an	an	DET
iajs-3828	240	2	application	application	NOUN
iajs-3828	240	3	for	for	ADP
iajs-3828	240	4	solving	solve	VERB
iajs-3828	240	5	integral	integral	ADJ
iajs-3828	240	6	equations	equation	NOUN
iajs-3828	240	7	.	.	PUNCT
iajs-3828	241	1	consider	consider	VERB
iajs-3828	241	2	the	the	DET
iajs-3828	241	3	integral	integral	ADJ
iajs-3828	241	4	equation	equation	NOUN
iajs-3828	241	5	(	(	PUNCT
iajs-3828	241	6	24,25	24,25	NUM
iajs-3828	241	7	)	)	PUNCT
iajs-3828	241	8	𝑤(𝑡	𝑤(𝑡	NOUN
iajs-3828	241	9	)	)	PUNCT
iajs-3828	241	10	=	=	SYM
iajs-3828	241	11	∫	∫	PROPN
iajs-3828	241	12	𝑄(𝑡	𝑄(𝑡	PROPN
iajs-3828	241	13	,	,	PUNCT
iajs-3828	241	14	𝑟)𝑇(𝑟	𝑟)𝑇(𝑟	PROPN
iajs-3828	241	15	,	,	PUNCT
iajs-3828	241	16	𝑤(𝑟	𝑤(𝑟	NOUN
iajs-3828	241	17	)	)	PUNCT
iajs-3828	241	18	)	)	PUNCT
iajs-3828	242	1	𝐽	𝐽	NOUN
iajs-3828	242	2	0	0	NUM
iajs-3828	242	3	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	242	4	for	for	ADP
iajs-3828	242	5	all	all	DET
iajs-3828	242	6	𝑡	𝑡	ADP
iajs-3828	242	7	∈	∈	PROPN
iajs-3828	243	1	[	[	X
iajs-3828	243	2	0	0	NUM
iajs-3828	243	3	,	,	PUNCT
iajs-3828	243	4	𝐽	𝐽	PROPN
iajs-3828	243	5	]	]	X
iajs-3828	243	6	,	,	PUNCT
iajs-3828	243	7	(	(	PUNCT
iajs-3828	243	8	11	11	NUM
iajs-3828	243	9	)	)	PUNCT
iajs-3828	243	10	where	where	SCONJ
iajs-3828	243	11	𝐽	𝐽	PROPN
iajs-3828	243	12	>	>	X
iajs-3828	243	13	0	0	PROPN
iajs-3828	243	14	,	,	PUNCT
iajs-3828	243	15	𝑇	𝑇	PROPN
iajs-3828	243	16	:	:	PUNCT
iajs-3828	243	17	[	[	X
iajs-3828	243	18	0	0	NUM
iajs-3828	243	19	,	,	PUNCT
iajs-3828	243	20	𝐽	𝐽	PROPN
iajs-3828	243	21	]	]	X
iajs-3828	243	22	×	×	PROPN
iajs-3828	243	23	𝑅	𝑅	PROPN
iajs-3828	243	24	→	→	SYM
iajs-3828	243	25	𝑅	𝑅	PROPN
iajs-3828	243	26	and	and	CCONJ
iajs-3828	243	27	𝑄	𝑄	PROPN
iajs-3828	243	28	:	:	PUNCT
iajs-3828	244	1	[	[	X
iajs-3828	244	2	0	0	NUM
iajs-3828	244	3	,	,	PUNCT
iajs-3828	244	4	𝐽	𝐽	PROPN
iajs-3828	244	5	]	]	X
iajs-3828	244	6	×	×	NOUN
iajs-3828	245	1	[	[	X
iajs-3828	245	2	0	0	NUM
iajs-3828	245	3	,	,	PUNCT
iajs-3828	245	4	𝐽	𝐽	PROPN
iajs-3828	245	5	]	]	X
iajs-3828	245	6	→	→	X
iajs-3828	246	1	[	[	X
iajs-3828	246	2	0,∞	0,∞	NUM
iajs-3828	246	3	)	)	PUNCT
iajs-3828	246	4	are	be	AUX
iajs-3828	246	5	continuous	continuous	ADJ
iajs-3828	246	6	functions	function	NOUN
iajs-3828	246	7	.	.	PUNCT
iajs-3828	247	1	in	in	ADP
iajs-3828	247	2	this	this	DET
iajs-3828	247	3	section	section	NOUN
iajs-3828	247	4	,	,	PUNCT
iajs-3828	247	5	we	we	PRON
iajs-3828	247	6	look	look	VERB
iajs-3828	247	7	for	for	ADP
iajs-3828	247	8	a	a	DET
iajs-3828	247	9	nonnegative	nonnegative	ADJ
iajs-3828	247	10	solution	solution	NOUN
iajs-3828	247	11	to	to	ADP
iajs-3828	247	12	(	(	PUNCT
iajs-3828	247	13	11	11	NUM
iajs-3828	247	14	)	)	PUNCT
iajs-3828	247	15	in	in	ADP
iajs-3828	247	16	£	£	NOUN
iajs-3828	247	17	=	=	SYM
iajs-3828	247	18	𝐶([0	𝐶([0	PROPN
iajs-3828	247	19	,	,	PUNCT
iajs-3828	247	20	𝐽	𝐽	PROPN
iajs-3828	247	21	]	]	X
iajs-3828	247	22	,	,	PUNCT
iajs-3828	247	23	𝑅	𝑅	NOUN
iajs-3828	247	24	)	)	PUNCT
iajs-3828	247	25	by	by	ADP
iajs-3828	247	26	(	(	PUNCT
iajs-3828	247	27	theorem	theorem	NOUN
iajs-3828	247	28	2.5	2.5	NUM
iajs-3828	247	29	)	)	PUNCT
iajs-3828	247	30	.	.	PUNCT
iajs-3828	248	1	let	let	VERB
iajs-3828	248	2	£	£	NOUN
iajs-3828	248	3	=	=	SYM
iajs-3828	248	4	𝐶[0	𝐶[0	PROPN
iajs-3828	248	5	,	,	PUNCT
iajs-3828	248	6	𝐽	𝐽	PROPN
iajs-3828	248	7	]	]	PUNCT
iajs-3828	248	8	be	be	VERB
iajs-3828	248	9	the	the	DET
iajs-3828	248	10	set	set	NOUN
iajs-3828	248	11	of	of	ADP
iajs-3828	248	12	real	real	ADV
iajs-3828	248	13	valued	value	VERB
iajs-3828	248	14	continuous	continuous	ADJ
iajs-3828	248	15	functions	function	NOUN
iajs-3828	248	16	on	on	ADP
iajs-3828	248	17	[	[	X
iajs-3828	248	18	0	0	NUM
iajs-3828	248	19	,	,	PUNCT
iajs-3828	248	20	𝐽	𝐽	PROPN
iajs-3828	248	21	]	]	X
iajs-3828	248	22	,	,	PUNCT
iajs-3828	248	23	where	where	SCONJ
iajs-3828	248	24	[	[	X
iajs-3828	248	25	0	0	NUM
iajs-3828	248	26	,	,	PUNCT
iajs-3828	248	27	𝐽	𝐽	PROPN
iajs-3828	248	28	]	]	PUNCT
iajs-3828	248	29	is	be	AUX
iajs-3828	248	30	a	a	DET
iajs-3828	248	31	closed	closed	ADJ
iajs-3828	248	32	and	and	CCONJ
iajs-3828	248	33	bounded	bounded	ADJ
iajs-3828	248	34	interval	interval	NOUN
iajs-3828	248	35	in	in	ADP
iajs-3828	248	36	𝑅.	𝑅.	NOUN
iajs-3828	248	37	for	for	ADP
iajs-3828	248	38	𝑝	𝑝	NOUN
iajs-3828	248	39	>	>	SYM
iajs-3828	248	40	1	1	NUM
iajs-3828	248	41	,	,	PUNCT
iajs-3828	248	42	define	define	VERB
iajs-3828	248	43	𝜌	𝜌	ADP
iajs-3828	248	44	:	:	PUNCT
iajs-3828	248	45	[	[	X
iajs-3828	248	46	0	0	NUM
iajs-3828	248	47	,	,	PUNCT
iajs-3828	248	48	𝐽	𝐽	PROPN
iajs-3828	248	49	]	]	X
iajs-3828	248	50	×	×	NOUN
iajs-3828	249	1	[	[	X
iajs-3828	249	2	0	0	NUM
iajs-3828	249	3	,	,	PUNCT
iajs-3828	249	4	𝐽	𝐽	PROPN
iajs-3828	249	5	]	]	X
iajs-3828	249	6	→	→	SYM
iajs-3828	249	7	𝑅	𝑅	PROPN
iajs-3828	249	8	by	by	ADP
iajs-3828	249	9	𝜌(𝑤	𝜌(𝑤	PROPN
iajs-3828	249	10	,	,	PUNCT
iajs-3828	249	11	𝑣	𝑣	NOUN
iajs-3828	249	12	)	)	PUNCT
iajs-3828	249	13	=	=	SYM
iajs-3828	249	14	max	max	PROPN
iajs-3828	249	15	𝑡𝜖[0,𝐽	𝑡𝜖[0,𝐽	PROPN
iajs-3828	249	16	]	]	PUNCT
iajs-3828	249	17	|𝑤(𝑡	|𝑤(𝑡	PROPN
iajs-3828	249	18	)	)	PUNCT
iajs-3828	249	19	−	−	PROPN
iajs-3828	249	20	𝑣(𝑡)|𝑝	𝑣(𝑡)|𝑝	PROPN
iajs-3828	249	21	,	,	PUNCT
iajs-3828	249	22	for	for	ADP
iajs-3828	249	23	all	all	PRON
iajs-3828	249	24	𝑤	𝑤	ADP
iajs-3828	249	25	,	,	PUNCT
iajs-3828	249	26	𝑣	𝑣	PRON
iajs-3828	249	27	∈	∈	PROPN
iajs-3828	249	28	£	£	PROPN
iajs-3828	249	29	.	.	PUNCT
iajs-3828	250	1	therefore	therefore	ADV
iajs-3828	250	2	,	,	PUNCT
iajs-3828	250	3	(	(	PUNCT
iajs-3828	250	4	£	£	NOUN
iajs-3828	250	5	,	,	PUNCT
iajs-3828	250	6	𝜌)is	𝜌)is	PROPN
iajs-3828	250	7	a	a	DET
iajs-3828	250	8	complete	complete	ADJ
iajs-3828	250	9	b	b	X
iajs-3828	250	10	-	-	PUNCT
iajs-3828	250	11	metric	metric	ADJ
iajs-3828	250	12	space	space	NOUN
iajs-3828	250	13	with	with	ADP
iajs-3828	250	14	𝑞	𝑞	PROPN
iajs-3828	250	15	=	=	PROPN
iajs-3828	250	16	2𝑝−1	2𝑝−1	PROPN
iajs-3828	250	17	.	.	PUNCT
iajs-3828	251	1	let	let	VERB
iajs-3828	251	2	𝛼	𝛼	X
iajs-3828	251	3	,	,	PUNCT
iajs-3828	251	4	𝛽	𝛽	PROPN
iajs-3828	251	5	∈	∈	PROPN
iajs-3828	251	6	£	£	PROPN
iajs-3828	251	7	and	and	CCONJ
iajs-3828	251	8	𝛼0	𝛼0	PROPN
iajs-3828	251	9	,	,	PUNCT
iajs-3828	251	10	𝛽0	𝛽0	NOUN
iajs-3828	251	11	∈	∈	PROPN
iajs-3828	251	12	𝑅	𝑅	PROPN
iajs-3828	251	13	such	such	ADJ
iajs-3828	251	14	that	that	SCONJ
iajs-3828	251	15	𝛼0	𝛼0	PROPN
iajs-3828	251	16	≤	≤	PROPN
iajs-3828	251	17	α(𝑡	α(𝑡	PROPN
iajs-3828	251	18	)	)	PUNCT
iajs-3828	251	19	≤	≤	PUNCT
iajs-3828	252	1	𝛽(𝑡	𝛽(𝑡	PROPN
iajs-3828	252	2	)	)	PUNCT
iajs-3828	252	3	≤	≤	NOUN
iajs-3828	252	4	𝛽0	𝛽0	PROPN
iajs-3828	252	5	,	,	PUNCT
iajs-3828	252	6	∀	∀	NUM
iajs-3828	252	7	𝑡	𝑡	X
iajs-3828	252	8	∈	∈	PROPN
iajs-3828	253	1	[	[	X
iajs-3828	253	2	0	0	NUM
iajs-3828	253	3	,	,	PUNCT
iajs-3828	253	4	𝐽	𝐽	PROPN
iajs-3828	253	5	]	]	X
iajs-3828	253	6	.	.	PUNCT
iajs-3828	254	1	(	(	PUNCT
iajs-3828	254	2	12	12	NUM
iajs-3828	254	3	)	)	PUNCT
iajs-3828	254	4	suppose	suppose	VERB
iajs-3828	254	5	that	that	SCONJ
iajs-3828	254	6	for	for	ADP
iajs-3828	254	7	all	all	DET
iajs-3828	254	8	𝑡	𝑡	ADP
iajs-3828	254	9	∈	∈	PROPN
iajs-3828	254	10	[	[	X
iajs-3828	254	11	0	0	NUM
iajs-3828	254	12	,	,	PUNCT
iajs-3828	254	13	𝐽	𝐽	PROPN
iajs-3828	254	14	]	]	X
iajs-3828	254	15	,	,	PUNCT
iajs-3828	254	16	we	we	PRON
iajs-3828	254	17	have	have	VERB
iajs-3828	254	18	𝛼(𝑡	𝛼(𝑡	VERB
iajs-3828	254	19	)	)	PUNCT
iajs-3828	254	20	≤	≤	NUM
iajs-3828	254	21	∫	∫	PROPN
iajs-3828	254	22	𝑄(𝑡	𝑄(𝑡	PROPN
iajs-3828	254	23	,	,	PUNCT
iajs-3828	254	24	𝑟)𝑇(𝑟	𝑟)𝑇(𝑟	PROPN
iajs-3828	254	25	,	,	PUNCT
iajs-3828	254	26	𝛽(𝑟	𝛽(𝑟	NOUN
iajs-3828	254	27	)	)	PUNCT
iajs-3828	254	28	)	)	PUNCT
iajs-3828	255	1	𝐽	𝐽	PROPN
iajs-3828	255	2	0	0	NUM
iajs-3828	255	3	𝑑𝑟	𝑑𝑟	PROPN
iajs-3828	255	4	,	,	PUNCT
iajs-3828	255	5	(	(	PUNCT
iajs-3828	255	6	13	13	NUM
iajs-3828	255	7	)	)	PUNCT
iajs-3828	255	8	and	and	CCONJ
iajs-3828	255	9	𝛽(𝑡	𝛽(𝑡	PROPN
iajs-3828	255	10	)	)	PUNCT
iajs-3828	255	11	≥	≥	NOUN
iajs-3828	255	12	∫	∫	PROPN
iajs-3828	255	13	𝑄(𝑡	𝑄(𝑡	PROPN
iajs-3828	255	14	,	,	PUNCT
iajs-3828	255	15	𝑟)𝑇(𝑟	𝑟)𝑇(𝑟	PROPN
iajs-3828	255	16	,	,	PUNCT
iajs-3828	255	17	𝛼(𝑟	𝛼(𝑟	NUM
iajs-3828	255	18	)	)	PUNCT
iajs-3828	255	19	)	)	PUNCT
iajs-3828	256	1	𝐽	𝐽	PROPN
iajs-3828	256	2	0	0	NUM
iajs-3828	256	3	𝑑𝑟.	𝑑𝑟.	X
iajs-3828	256	4	(	(	PUNCT
iajs-3828	256	5	14	14	NUM
iajs-3828	256	6	)	)	PUNCT
iajs-3828	256	7	we	we	PRON
iajs-3828	256	8	suppose	suppose	VERB
iajs-3828	256	9	that	that	SCONJ
iajs-3828	256	10	∀	∀	VERB
iajs-3828	256	11	𝑟	𝑟	X
iajs-3828	256	12	∈	∈	NOUN
iajs-3828	257	1	[	[	X
iajs-3828	257	2	0	0	NUM
iajs-3828	257	3	,	,	PUNCT
iajs-3828	257	4	𝐽	𝐽	PROPN
iajs-3828	257	5	]	]	X
iajs-3828	257	6	,	,	PUNCT
iajs-3828	257	7	𝑇(𝑟	𝑇(𝑟	X
iajs-3828	257	8	,	,	PUNCT
iajs-3828	257	9	.	.	PUNCT
iajs-3828	257	10	)	)	PUNCT
iajs-3828	257	11	be	be	AUX
iajs-3828	257	12	a	a	DET
iajs-3828	257	13	decreasing	decrease	VERB
iajs-3828	257	14	function	function	NOUN
iajs-3828	257	15	,	,	PUNCT
iajs-3828	258	1	that	that	SCONJ
iajs-3828	258	2	𝑎	𝑎	X
iajs-3828	258	3	,	,	PUNCT
iajs-3828	258	4	𝑏	𝑏	PROPN
iajs-3828	258	5	∈	∈	PROPN
iajs-3828	258	6	𝑅	𝑅	PROPN
iajs-3828	258	7	,	,	PUNCT
iajs-3828	258	8	𝑎	𝑎	DET
iajs-3828	258	9	≥	≥	NOUN
iajs-3828	258	10	𝑏	𝑏	NOUN
iajs-3828	258	11	then	then	ADV
iajs-3828	258	12	𝑇(𝑟	𝑇(𝑟	ADJ
iajs-3828	258	13	,	,	PUNCT
iajs-3828	258	14	𝑎	𝑎	NOUN
iajs-3828	258	15	)	)	PUNCT
iajs-3828	258	16	≤	≤	NOUN
iajs-3828	258	17	𝑇(𝑟	𝑇(𝑟	NUM
iajs-3828	258	18	,	,	PUNCT
iajs-3828	258	19	𝑏	𝑏	NOUN
iajs-3828	258	20	)	)	PUNCT
iajs-3828	258	21	.	.	PUNCT
iajs-3828	259	1	(	(	PUNCT
iajs-3828	259	2	15	15	X
iajs-3828	259	3	)	)	PUNCT
iajs-3828	259	4	assume	assume	VERB
iajs-3828	259	5	that	that	SCONJ
iajs-3828	259	6	𝑘	𝑘	PROPN
iajs-3828	259	7	>	>	X
iajs-3828	259	8	0	0	NUM
iajs-3828	259	9	is	be	AUX
iajs-3828	259	10	such	such	ADJ
iajs-3828	259	11	that	that	SCONJ
iajs-3828	259	12	𝑘(max	𝑘(max	PROPN
iajs-3828	259	13	𝑡∈[0,𝐽	𝑡∈[0,𝐽	PROPN
iajs-3828	259	14	]	]	PUNCT
iajs-3828	259	15	∫	∫	PROPN
iajs-3828	259	16	𝑄(𝑡	𝑄(𝑡	X
iajs-3828	259	17	,	,	PUNCT
iajs-3828	259	18	𝑟	𝑟	NOUN
iajs-3828	259	19	)	)	PUNCT
iajs-3828	259	20	𝐽	𝐽	NOUN
iajs-3828	259	21	0	0	NUM
iajs-3828	259	22	𝑑𝑟	𝑑𝑟	PROPN
iajs-3828	259	23	)	)	PUNCT
iajs-3828	259	24	<	<	X
iajs-3828	259	25	1	1	NUM
iajs-3828	259	26	.	.	PUNCT
iajs-3828	260	1	(	(	PUNCT
iajs-3828	260	2	16	16	NUM
iajs-3828	260	3	)	)	PUNCT
iajs-3828	260	4	define	define	VERB
iajs-3828	260	5	a	a	DET
iajs-3828	260	6	map	map	NOUN
iajs-3828	260	7	𝑓	𝑓	NOUN
iajs-3828	260	8	:	:	PUNCT
iajs-3828	260	9	£	£	NOUN
iajs-3828	260	10	→	→	SYM
iajs-3828	260	11	£	£	NOUN
iajs-3828	260	12	by	by	ADP
iajs-3828	260	13	𝑓𝑤(𝑡	𝑓𝑤(𝑡	NUM
iajs-3828	260	14	)	)	PUNCT
iajs-3828	260	15	=	=	SYM
iajs-3828	261	1	∫	∫	PROPN
iajs-3828	261	2	𝑄(𝑡	𝑄(𝑡	PROPN
iajs-3828	261	3	,	,	PUNCT
iajs-3828	261	4	𝑟)𝑇(𝑟	𝑟)𝑇(𝑟	PROPN
iajs-3828	261	5	,	,	PUNCT
iajs-3828	261	6	𝑤(𝑟	𝑤(𝑟	NOUN
iajs-3828	261	7	)	)	PUNCT
iajs-3828	261	8	)	)	PUNCT
iajs-3828	262	1	𝐽	𝐽	PROPN
iajs-3828	262	2	0	0	NUM
iajs-3828	262	3	𝑑𝑟	𝑑𝑟	PROPN
iajs-3828	262	4	,	,	PUNCT
iajs-3828	262	5	for	for	ADP
iajs-3828	262	6	all	all	DET
iajs-3828	262	7	𝑡	𝑡	ADP
iajs-3828	262	8	∈	∈	PROPN
iajs-3828	263	1	[	[	X
iajs-3828	263	2	0	0	NUM
iajs-3828	263	3	,	,	PUNCT
iajs-3828	263	4	𝐽	𝐽	PROPN
iajs-3828	263	5	]	]	PUNCT
iajs-3828	263	6	.	.	PUNCT
iajs-3828	264	1	suppose	suppose	VERB
iajs-3828	264	2	that	that	SCONJ
iajs-3828	264	3	∀𝑟	∀𝑟	PROPN
iajs-3828	264	4	∈	∈	PROPN
iajs-3828	264	5	[	[	X
iajs-3828	264	6	0	0	NUM
iajs-3828	264	7	,	,	PUNCT
iajs-3828	264	8	𝐽	𝐽	PROPN
iajs-3828	264	9	]	]	PUNCT
iajs-3828	264	10	and	and	CCONJ
iajs-3828	264	11	𝑎	𝑎	PROPN
iajs-3828	264	12	,	,	PUNCT
iajs-3828	264	13	𝑏	𝑏	PROPN
iajs-3828	264	14	∈	∈	PROPN
iajs-3828	264	15	£	£	NOUN
iajs-3828	264	16	with	with	ADP
iajs-3828	264	17	(	(	PUNCT
iajs-3828	264	18	𝑎(𝑟	𝑎(𝑟	PROPN
iajs-3828	264	19	)	)	PUNCT
iajs-3828	264	20	≤	≤	NOUN
iajs-3828	264	21	𝛼0	𝛼0	PROPN
iajs-3828	264	22	and	and	CCONJ
iajs-3828	264	23	𝑏(𝑟	𝑏(𝑟	PROPN
iajs-3828	264	24	)	)	PUNCT
iajs-3828	264	25	≤	≤	NOUN
iajs-3828	264	26	𝛽0	𝛽0	PROPN
iajs-3828	264	27	)	)	PUNCT
iajs-3828	264	28	or	or	CCONJ
iajs-3828	264	29	vice	vice	NOUN
iajs-3828	264	30	versa	versa	ADV
iajs-3828	264	31	,	,	PUNCT
iajs-3828	264	32	0	0	NUM
iajs-3828	264	33	≤	≤	NOUN
iajs-3828	265	1	[	[	X
iajs-3828	265	2	𝑇(𝑞	𝑇(𝑞	NUM
iajs-3828	265	3	,	,	PUNCT
iajs-3828	265	4	𝑎(𝑟	𝑎(𝑟	PROPN
iajs-3828	265	5	)	)	PUNCT
iajs-3828	265	6	)	)	PUNCT
iajs-3828	266	1	−	−	PROPN
iajs-3828	266	2	𝑇(𝑟	𝑇(𝑟	NUM
iajs-3828	266	3	,	,	PUNCT
iajs-3828	266	4	𝑏(𝑟	𝑏(𝑟	PROPN
iajs-3828	266	5	)	)	PUNCT
iajs-3828	266	6	)	)	PUNCT
iajs-3828	266	7	]	]	PUNCT
iajs-3828	267	1	≤	≤	NOUN
iajs-3828	267	2	𝑘	𝑘	DET
iajs-3828	267	3	max	max	PROPN
iajs-3828	267	4	{	{	PUNCT
iajs-3828	267	5	|𝑎(𝑟	|𝑎(𝑟	PROPN
iajs-3828	267	6	)	)	PUNCT
iajs-3828	267	7	−	−	PROPN
iajs-3828	268	1	𝑏(𝑟)|𝑝	𝑏(𝑟)|𝑝	PROPN
iajs-3828	268	2	,	,	PUNCT
iajs-3828	268	3	0	0	NUM
iajs-3828	268	4	,	,	PUNCT
iajs-3828	268	5	|𝑏(𝑟	|𝑏(𝑟	NOUN
iajs-3828	268	6	)	)	PUNCT
iajs-3828	269	1	−	−	PROPN
iajs-3828	269	2	𝑓𝑏(𝑟)|𝑝	𝑓𝑏(𝑟)|𝑝	ADJ
iajs-3828	269	3	,	,	PUNCT
iajs-3828	269	4	|𝑎(𝑟	|𝑎(𝑟	PROPN
iajs-3828	269	5	)	)	PUNCT
iajs-3828	270	1	−	−	PROPN
iajs-3828	270	2	𝑓𝑏(𝑟)|𝑝	𝑓𝑏(𝑟)|𝑝	ADJ
iajs-3828	270	3	,	,	PUNCT
iajs-3828	270	4	|𝑏(𝑟	|𝑏(𝑟	PROPN
iajs-3828	270	5	)	)	PUNCT
iajs-3828	270	6	−	−	PROPN
iajs-3828	271	1	𝑎(𝑟)|𝑝	𝑎(𝑟)|𝑝	PROPN
iajs-3828	271	2	}	}	PUNCT
iajs-3828	271	3	)	)	PUNCT
iajs-3828	271	4	1	1	NUM
iajs-3828	271	5	𝑝	𝑝	PROPN
iajs-3828	271	6	(	(	PUNCT
iajs-3828	271	7	17	17	NUM
iajs-3828	271	8	)	)	PUNCT
iajs-3828	271	9	theorem	theorem	VERB
iajs-3828	271	10	2.7	2.7	NUM
iajs-3828	271	11	:	:	PUNCT
iajs-3828	271	12	under	under	ADP
iajs-3828	271	13	the	the	DET
iajs-3828	271	14	assumptions	assumption	NOUN
iajs-3828	271	15	(	(	PUNCT
iajs-3828	271	16	12)-(17	12)-(17	NUM
iajs-3828	271	17	)	)	PUNCT
iajs-3828	271	18	,	,	PUNCT
iajs-3828	271	19	the	the	DET
iajs-3828	271	20	integral	integral	ADJ
iajs-3828	271	21	equation	equation	NOUN
iajs-3828	271	22	(	(	PUNCT
iajs-3828	271	23	11	11	NUM
iajs-3828	271	24	)	)	PUNCT
iajs-3828	271	25	has	have	VERB
iajs-3828	271	26	a	a	DET
iajs-3828	271	27	solution	solution	NOUN
iajs-3828	271	28	in	in	ADP
iajs-3828	271	29	the	the	DET
iajs-3828	271	30	set	set	NOUN
iajs-3828	271	31	{	{	PUNCT
iajs-3828	271	32	𝑤	𝑤	ADP
iajs-3828	271	33	∈	∈	NOUN
iajs-3828	271	34	𝐶([0	𝐶([0	PROPN
iajs-3828	271	35	,	,	PUNCT
iajs-3828	271	36	𝐽	𝐽	PROPN
iajs-3828	271	37	]	]	PUNCT
iajs-3828	271	38	):	):	PUNCT
iajs-3828	271	39	𝛼	𝛼	X
iajs-3828	271	40	≤	≤	NUM
iajs-3828	271	41	𝑤	𝑤	ADP
iajs-3828	271	42	≤	≤	NUM
iajs-3828	271	43	𝛽	𝛽	NOUN
iajs-3828	271	44	}	}	PUNCT
iajs-3828	271	45	.	.	PUNCT
iajs-3828	272	1	ihjpas	ihjpas	PROPN
iajs-3828	272	2	.	.	PUNCT
iajs-3828	273	1	2025,38(2	2025,38(2	NOUN
iajs-3828	273	2	)	)	PUNCT
iajs-3828	273	3	383	383	NUM
iajs-3828	273	4	proof	proof	NOUN
iajs-3828	273	5	:	:	PUNCT
iajs-3828	273	6	we	we	PRON
iajs-3828	273	7	omit	omit	VERB
iajs-3828	273	8	the	the	DET
iajs-3828	273	9	details	detail	NOUN
iajs-3828	273	10	because	because	SCONJ
iajs-3828	273	11	the	the	DET
iajs-3828	273	12	proof	proof	NOUN
iajs-3828	273	13	steps	step	NOUN
iajs-3828	273	14	are	be	AUX
iajs-3828	273	15	classic	classic	ADJ
iajs-3828	273	16	with	with	ADP
iajs-3828	273	17	some	some	DET
iajs-3828	273	18	minor	minor	ADJ
iajs-3828	273	19	differences	difference	NOUN
iajs-3828	273	20	due	due	ADP
iajs-3828	273	21	to	to	ADP
iajs-3828	273	22	the	the	DET
iajs-3828	273	23	specificity	specificity	NOUN
iajs-3828	273	24	of	of	ADP
iajs-3828	273	25	the	the	DET
iajs-3828	273	26	b	b	NOUN
iajs-3828	273	27	-	-	PUNCT
iajs-3828	273	28	metric	metric	ADJ
iajs-3828	273	29	space	space	NOUN
iajs-3828	273	30	.	.	PUNCT
iajs-3828	274	1	fixed	fix	VERB
iajs-3828	274	2	points	point	NOUN
iajs-3828	274	3	by	by	ADP
iajs-3828	274	4	implicit	implicit	ADJ
iajs-3828	274	5	conditions	condition	NOUN
iajs-3828	274	6	the	the	DET
iajs-3828	274	7	following	follow	VERB
iajs-3828	274	8	list	list	NOUN
iajs-3828	274	9	of	of	ADP
iajs-3828	274	10	implicit	implicit	ADJ
iajs-3828	274	11	functions	function	NOUN
iajs-3828	274	12	under	under	ADP
iajs-3828	274	13	various	various	ADJ
iajs-3828	274	14	conditions	condition	NOUN
iajs-3828	274	15	(	(	PUNCT
iajs-3828	274	16	26	26	NUM
iajs-3828	274	17	)	)	PUNCT
iajs-3828	274	18	.	.	PUNCT
iajs-3828	275	1	let	let	VERB
iajs-3828	275	2	ω	ω	NUM
iajs-3828	275	3	be	be	AUX
iajs-3828	275	4	the	the	DET
iajs-3828	275	5	set	set	NOUN
iajs-3828	275	6	of	of	ADP
iajs-3828	275	7	all	all	DET
iajs-3828	275	8	real	real	ADJ
iajs-3828	275	9	continuous	continuous	ADJ
iajs-3828	275	10	functions	function	NOUN
iajs-3828	275	11	𝑀:𝑅+	𝑀:𝑅+	PROPN
iajs-3828	275	12	6	6	NUM
iajs-3828	275	13	→	→	SYM
iajs-3828	275	14	𝑅	𝑅	PROPN
iajs-3828	275	15	,	,	PUNCT
iajs-3828	275	16	satisfying	satisfy	VERB
iajs-3828	275	17	the	the	DET
iajs-3828	275	18	following	follow	VERB
iajs-3828	275	19	conditions	condition	NOUN
iajs-3828	275	20	:	:	PUNCT
iajs-3828	275	21	𝑀1	𝑀1	PROPN
iajs-3828	275	22	)	)	PUNCT
iajs-3828	275	23	is	be	AUX
iajs-3828	275	24	non	non	ADJ
iajs-3828	275	25	-	-	ADJ
iajs-3828	275	26	increasing	increase	VERB
iajs-3828	275	27	in	in	ADP
iajs-3828	275	28	variables	variable	NOUN
iajs-3828	275	29	𝑟2	𝑟2	NOUN
iajs-3828	275	30	,	,	PUNCT
iajs-3828	275	31	𝑟3	𝑟3	NOUN
iajs-3828	275	32	,	,	PUNCT
iajs-3828	275	33	𝑟4	𝑟4	PROPN
iajs-3828	275	34	,	,	PUNCT
iajs-3828	275	35	𝑟5	𝑟5	ADJ
iajs-3828	275	36	,	,	PUNCT
iajs-3828	275	37	𝑟6	𝑟6	NOUN
iajs-3828	275	38	𝑀2	𝑀2	PROPN
iajs-3828	275	39	)	)	PUNCT
iajs-3828	275	40	there	there	PRON
iajs-3828	275	41	exists	exist	VERB
iajs-3828	275	42	a	a	DET
iajs-3828	275	43	right	right	ADJ
iajs-3828	275	44	continuous	continuous	ADJ
iajs-3828	275	45	function	function	NOUN
iajs-3828	275	46	𝐴	𝐴	PROPN
iajs-3828	275	47	:	:	PUNCT
iajs-3828	276	1	[	[	X
iajs-3828	276	2	0,∞	0,∞	NOUN
iajs-3828	276	3	)	)	PUNCT
iajs-3828	276	4	→	→	PUNCT
iajs-3828	277	1	[	[	X
iajs-3828	277	2	0,∞	0,∞	NOUN
iajs-3828	277	3	)	)	PUNCT
iajs-3828	277	4	,	,	PUNCT
iajs-3828	277	5	𝐴(0	𝐴(0	X
iajs-3828	277	6	)	)	PUNCT
iajs-3828	277	7	=	=	SYM
iajs-3828	277	8	0,𝐴(𝑟	0,𝐴(𝑟	NOUN
iajs-3828	277	9	)	)	PUNCT
iajs-3828	277	10	<	<	X
iajs-3828	277	11	𝑟	𝑟	NOUN
iajs-3828	277	12	,	,	PUNCT
iajs-3828	277	13	for	for	ADP
iajs-3828	277	14	𝑟	𝑟	X
iajs-3828	277	15	>	>	PUNCT
iajs-3828	277	16	0	0	NUM
iajs-3828	277	17	such	such	ADJ
iajs-3828	277	18	that	that	PRON
iajs-3828	277	19	for	for	ADP
iajs-3828	277	20	𝑐	𝑐	PROPN
iajs-3828	277	21	≥	≥	X
iajs-3828	277	22	0	0	PUNCT
iajs-3828	277	23	𝑀(𝑐	𝑀(𝑐	NOUN
iajs-3828	277	24	,	,	PUNCT
iajs-3828	277	25	𝑢	𝑢	PROPN
iajs-3828	277	26	,	,	PUNCT
iajs-3828	277	27	𝑐	𝑐	NOUN
iajs-3828	277	28	,	,	PUNCT
iajs-3828	277	29	𝑢	𝑢	PROPN
iajs-3828	277	30	,	,	PUNCT
iajs-3828	277	31	0	0	NUM
iajs-3828	277	32	,	,	PUNCT
iajs-3828	277	33	𝑐	𝑐	NOUN
iajs-3828	277	34	+	+	NUM
iajs-3828	277	35	𝑢	𝑢	X
iajs-3828	277	36	)	)	PUNCT
iajs-3828	277	37	≤	≤	NOUN
iajs-3828	277	38	0	0	NUM
iajs-3828	277	39	or	or	CCONJ
iajs-3828	277	40	𝑀(𝑐	𝑀(𝑐	NOUN
iajs-3828	277	41	,	,	PUNCT
iajs-3828	277	42	𝑢	𝑢	NOUN
iajs-3828	277	43	,	,	PUNCT
iajs-3828	277	44	0,0	0,0	NOUN
iajs-3828	277	45	,	,	PUNCT
iajs-3828	277	46	𝑢	𝑢	NOUN
iajs-3828	277	47	,	,	PUNCT
iajs-3828	277	48	𝑢	𝑢	NOUN
iajs-3828	277	49	)	)	PUNCT
iajs-3828	277	50	≤	≤	NOUN
iajs-3828	277	51	0	0	NUM
iajs-3828	277	52	,	,	PUNCT
iajs-3828	277	53	implies	imply	VERB
iajs-3828	277	54	𝑐	𝑐	PROPN
iajs-3828	277	55	≤	≤	NUM
iajs-3828	277	56	𝐴(𝑢	𝐴(𝑢	NOUN
iajs-3828	277	57	)	)	PUNCT
iajs-3828	277	58	.	.	PUNCT
iajs-3828	278	1	𝑀3	𝑀3	PROPN
iajs-3828	278	2	)	)	PUNCT
iajs-3828	278	3	𝑀(𝑐	𝑀(𝑐	NOUN
iajs-3828	278	4	,	,	PUNCT
iajs-3828	278	5	0	0	NUM
iajs-3828	278	6	,	,	PUNCT
iajs-3828	278	7	𝑐	𝑐	NOUN
iajs-3828	278	8	,	,	PUNCT
iajs-3828	278	9	0,0	0,0	NOUN
iajs-3828	278	10	,	,	PUNCT
iajs-3828	278	11	𝑐	𝑐	NOUN
iajs-3828	278	12	)	)	PUNCT
iajs-3828	278	13	>	>	X
iajs-3828	278	14	0	0	PUNCT
iajs-3828	278	15	and	and	CCONJ
iajs-3828	278	16	𝑀(𝑐	𝑀(𝑐	NOUN
iajs-3828	278	17	,	,	PUNCT
iajs-3828	278	18	𝑐	𝑐	NOUN
iajs-3828	278	19	,	,	PUNCT
iajs-3828	278	20	0,0	0,0	NOUN
iajs-3828	278	21	,	,	PUNCT
iajs-3828	278	22	𝑐	𝑐	NOUN
iajs-3828	278	23	,	,	PUNCT
iajs-3828	278	24	𝑐	𝑐	NOUN
iajs-3828	278	25	)	)	PUNCT
iajs-3828	278	26	>	>	X
iajs-3828	278	27	0	0	NUM
iajs-3828	278	28	∀	∀	NOUN
iajs-3828	278	29	𝑐	𝑐	NOUN
iajs-3828	278	30	>	>	X
iajs-3828	278	31	0	0	X
iajs-3828	278	32	.	.	PUNCT
iajs-3828	279	1	also	also	ADV
iajs-3828	279	2	,	,	PUNCT
iajs-3828	279	3	let	let	VERB
iajs-3828	279	4	𝛹	𝛹	PRON
iajs-3828	279	5	≔	≔	VERB
iajs-3828	279	6	the	the	DET
iajs-3828	279	7	set	set	NOUN
iajs-3828	279	8	of	of	ADP
iajs-3828	279	9	functions	function	NOUN
iajs-3828	279	10	𝜓	𝜓	NOUN
iajs-3828	279	11	:	:	PUNCT
iajs-3828	279	12	[	[	X
iajs-3828	279	13	0,∞	0,∞	NOUN
iajs-3828	279	14	)	)	PUNCT
iajs-3828	279	15	→	→	PUNCT
iajs-3828	280	1	[	[	X
iajs-3828	280	2	0,∞	0,∞	NUM
iajs-3828	280	3	)	)	PUNCT
iajs-3828	280	4	such	such	ADJ
iajs-3828	280	5	that	that	SCONJ
iajs-3828	280	6	:	:	PUNCT
iajs-3828	280	7	i	i	X
iajs-3828	280	8	)	)	PUNCT
iajs-3828	280	9	𝜓	𝜓	PROPN
iajs-3828	280	10	is	be	AUX
iajs-3828	280	11	monotone	monotone	ADJ
iajs-3828	280	12	increasing	increase	VERB
iajs-3828	280	13	and	and	CCONJ
iajs-3828	280	14	continuous	continuous	ADJ
iajs-3828	280	15	;	;	PUNCT
iajs-3828	280	16	ii	ii	X
iajs-3828	280	17	)	)	PUNCT
iajs-3828	280	18	𝜓(𝑟	𝜓(𝑟	PROPN
iajs-3828	280	19	)	)	PUNCT
iajs-3828	280	20	=	=	SYM
iajs-3828	280	21	0	0	PUNCT
iajs-3828	281	1	if	if	SCONJ
iajs-3828	281	2	and	and	CCONJ
iajs-3828	281	3	only	only	ADV
iajs-3828	281	4	if	if	SCONJ
iajs-3828	281	5	𝑟	𝑟	NOUN
iajs-3828	281	6	=	=	SYM
iajs-3828	281	7	0	0	NUM
iajs-3828	281	8	;	;	PUNCT
iajs-3828	281	9	iii	iii	X
iajs-3828	281	10	)	)	PUNCT
iajs-3828	281	11	𝜓	𝜓	PROPN
iajs-3828	281	12	is	be	AUX
iajs-3828	281	13	subadditive	subadditive	ADJ
iajs-3828	281	14	,	,	PUNCT
iajs-3828	281	15	i.e.	i.e.	X
iajs-3828	281	16	,	,	PUNCT
iajs-3828	281	17	∀𝑟1	∀𝑟1	NOUN
iajs-3828	281	18	,	,	PUNCT
iajs-3828	281	19	𝑟2	𝑟2	NOUN
iajs-3828	281	20	∈	∈	NOUN
iajs-3828	281	21	[	[	X
iajs-3828	281	22	0,+∞),𝜓(𝑟1	0,+∞),𝜓(𝑟1	NUM
iajs-3828	281	23	+	+	NUM
iajs-3828	281	24	𝑟2	𝑟2	NOUN
iajs-3828	281	25	)	)	PUNCT
iajs-3828	281	26	=	=	SYM
iajs-3828	281	27	𝜓(𝑟1	𝜓(𝑟1	NOUN
iajs-3828	281	28	)	)	PUNCT
iajs-3828	281	29	+	+	CCONJ
iajs-3828	281	30	𝜓(𝑟2	𝜓(𝑟2	ADJ
iajs-3828	281	31	)	)	PUNCT
iajs-3828	281	32	.	.	PUNCT
iajs-3828	282	1	lemma	lemma	PROPN
iajs-3828	282	2	2.8	2.8	NUM
iajs-3828	282	3	:	:	PUNCT
iajs-3828	282	4	let	let	VERB
iajs-3828	282	5	𝐴	𝐴	PROPN
iajs-3828	282	6	:	:	PUNCT
iajs-3828	283	1	[	[	X
iajs-3828	283	2	0,∞	0,∞	NUM
iajs-3828	283	3	)	)	PUNCT
iajs-3828	283	4	→	→	PUNCT
iajs-3828	284	1	[	[	X
iajs-3828	284	2	0,∞	0,∞	X
iajs-3828	284	3	)	)	PUNCT
iajs-3828	284	4	be	be	VERB
iajs-3828	284	5	a	a	DET
iajs-3828	284	6	right	right	ADJ
iajs-3828	284	7	continuous	continuous	ADJ
iajs-3828	284	8	function	function	NOUN
iajs-3828	284	9	such	such	ADJ
iajs-3828	284	10	that	that	SCONJ
iajs-3828	284	11	𝐴(𝑟	𝐴(𝑟	NOUN
iajs-3828	284	12	)	)	PUNCT
iajs-3828	284	13	<	<	X
iajs-3828	284	14	𝑟	𝑟	NOUN
iajs-3828	284	15	,	,	PUNCT
iajs-3828	284	16	for	for	ADP
iajs-3828	284	17	𝑟	𝑟	X
iajs-3828	284	18	>	>	X
iajs-3828	284	19	0	0	NUM
iajs-3828	284	20	.	.	PUNCT
iajs-3828	285	1	(	(	PUNCT
iajs-3828	285	2	27	27	NUM
iajs-3828	285	3	(	(	PUNCT
iajs-3828	285	4	.	.	PUNCT
iajs-3828	286	1	then	then	ADV
iajs-3828	286	2	lim	lim	PROPN
iajs-3828	286	3	𝑚→∞	𝑚→∞	PUNCT
iajs-3828	286	4	𝐴𝑚(𝑟	𝐴𝑚(𝑟	PROPN
iajs-3828	286	5	)	)	PUNCT
iajs-3828	286	6	=	=	SYM
iajs-3828	286	7	0	0	NUM
iajs-3828	286	8	,	,	PUNCT
iajs-3828	286	9	where	where	SCONJ
iajs-3828	286	10	𝐴𝑚≔𝑚	𝐴𝑚≔𝑚	NUM
iajs-3828	286	11	times	time	NOUN
iajs-3828	286	12	repeated	repeat	VERB
iajs-3828	286	13	composition	composition	NOUN
iajs-3828	286	14	of	of	ADP
iajs-3828	286	15	𝐴.	𝐴.	PROPN
iajs-3828	286	16	theorem	theorem	NOUN
iajs-3828	286	17	2.9	2.9	NUM
iajs-3828	286	18	:	:	PUNCT
iajs-3828	286	19	if	if	SCONJ
iajs-3828	286	20	𝑀	𝑀	PROPN
iajs-3828	286	21	∈	∈	NOUN
iajs-3828	286	22	𝛺	𝛺	PROPN
iajs-3828	286	23	exists	exist	VERB
iajs-3828	286	24	and	and	CCONJ
iajs-3828	286	25	£	£	NOUN
iajs-3828	286	26	=	=	SYM
iajs-3828	286	27	⋃	⋃	ADP
iajs-3828	286	28	£	£	SYM
iajs-3828	286	29	𝑖	𝑖	SYM
iajs-3828	286	30	𝑛	𝑛	PRON
iajs-3828	286	31	𝑖=1	𝑖=1	PROPN
iajs-3828	286	32	is	be	AUX
iajs-3828	286	33	a	a	DET
iajs-3828	286	34	cyclic	cyclic	ADJ
iajs-3828	286	35	representation	representation	NOUN
iajs-3828	286	36	of	of	ADP
iajs-3828	286	37	£	£	SYM
iajs-3828	286	38	w.r.t	w.r.t	NOUN
iajs-3828	286	39	.	.	PUNCT
iajs-3828	286	40	,	,	PUNCT
iajs-3828	286	41	𝑓	𝑓	X
iajs-3828	286	42	:	:	PUNCT
iajs-3828	286	43	£	£	PROPN
iajs-3828	286	44	→	→	SYM
iajs-3828	286	45	£	£	PROPN
iajs-3828	286	46	.	.	PUNCT
iajs-3828	287	1	if	if	SCONJ
iajs-3828	287	2	for	for	ADP
iajs-3828	287	3	any	any	DET
iajs-3828	287	4	(	(	PUNCT
iajs-3828	287	5	𝑎	𝑎	NOUN
iajs-3828	287	6	,	,	PUNCT
iajs-3828	287	7	𝑏	𝑏	NOUN
iajs-3828	287	8	)	)	PUNCT
iajs-3828	287	9	∈	∈	NOUN
iajs-3828	287	10	£	£	SYM
iajs-3828	287	11	𝑖	𝑖	NOUN
iajs-3828	287	12	×	×	NOUN
iajs-3828	287	13	£	£	SYM
iajs-3828	287	14	𝑖+1	𝑖+1	NUM
iajs-3828	287	15	,	,	PUNCT
iajs-3828	287	16	𝑖	𝑖	NOUN
iajs-3828	287	17	=	=	SYM
iajs-3828	287	18	1,2	1,2	NUM
iajs-3828	287	19	,	,	PUNCT
iajs-3828	287	20	…	…	PUNCT
iajs-3828	287	21	,	,	PUNCT
iajs-3828	287	22	𝑛	𝑛	DET
iajs-3828	287	23	𝑀(𝜓(𝜌(𝑓𝑎	𝑀(𝜓(𝜌(𝑓𝑎	PROPN
iajs-3828	287	24	,	,	PUNCT
iajs-3828	287	25	𝑓𝑏	𝑓𝑏	NOUN
iajs-3828	287	26	)	)	PUNCT
iajs-3828	287	27	)	)	PUNCT
iajs-3828	287	28	,	,	PUNCT
iajs-3828	287	29	𝜓(𝜌(𝑎	𝜓(𝜌(𝑎	NOUN
iajs-3828	287	30	,	,	PUNCT
iajs-3828	287	31	𝑏	𝑏	NOUN
iajs-3828	287	32	)	)	PUNCT
iajs-3828	287	33	)	)	PUNCT
iajs-3828	287	34	,	,	PUNCT
iajs-3828	287	35	𝜓(𝜌(𝑎	𝜓(𝜌(𝑎	NOUN
iajs-3828	287	36	,	,	PUNCT
iajs-3828	287	37	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	287	38	)	)	PUNCT
iajs-3828	287	39	)	)	PUNCT
iajs-3828	287	40	,	,	PUNCT
iajs-3828	287	41	𝜓(𝜌(𝑏	𝜓(𝜌(𝑏	PROPN
iajs-3828	287	42	,	,	PUNCT
iajs-3828	287	43	𝑓𝑏	𝑓𝑏	NOUN
iajs-3828	287	44	)	)	PUNCT
iajs-3828	287	45	)	)	PUNCT
iajs-3828	287	46	,	,	PUNCT
iajs-3828	287	47	𝜓(𝜌(𝑎	𝜓(𝜌(𝑎	NOUN
iajs-3828	287	48	,	,	PUNCT
iajs-3828	287	49	𝑓𝑏	𝑓𝑏	NOUN
iajs-3828	287	50	)	)	PUNCT
iajs-3828	287	51	)	)	PUNCT
iajs-3828	287	52	,	,	PUNCT
iajs-3828	287	53	𝜓(𝜌(𝑏	𝜓(𝜌(𝑏	PROPN
iajs-3828	287	54	,	,	PUNCT
iajs-3828	287	55	𝑓𝑎	𝑓𝑎	PROPN
iajs-3828	287	56	)	)	PUNCT
iajs-3828	287	57	)	)	PUNCT
iajs-3828	287	58	)	)	PUNCT
iajs-3828	287	59	≤	≤	NUM
iajs-3828	287	60	0	0	NUM
iajs-3828	287	61	(	(	PUNCT
iajs-3828	287	62	18	18	NUM
iajs-3828	287	63	)	)	PUNCT
iajs-3828	287	64	,	,	PUNCT
iajs-3828	287	65	and	and	CCONJ
iajs-3828	287	66	𝜓	𝜓	ADP
iajs-3828	287	67	∈	∈	PROPN
iajs-3828	287	68	𝛹.	𝛹.	PROPN
iajs-3828	287	69	∃	∃	PROPN
iajs-3828	287	70	!	!	PUNCT
iajs-3828	288	1	𝑐	𝑐	PROPN
iajs-3828	288	2	∈	∈	PROPN
iajs-3828	288	3	⋂	⋂	PROPN
iajs-3828	288	4	£	£	SYM
iajs-3828	288	5	𝑖	𝑖	SYM
iajs-3828	288	6	𝑛	𝑛	PRON
iajs-3828	288	7	𝑖=1	𝑖=1	PUNCT
iajs-3828	288	8	,	,	PUNCT
iajs-3828	288	9	𝑐	𝑐	PROPN
iajs-3828	288	10	is	be	AUX
iajs-3828	288	11	a	a	DET
iajs-3828	288	12	unique	unique	ADJ
iajs-3828	288	13	fixed	fix	VERB
iajs-3828	288	14	point	point	NOUN
iajs-3828	288	15	.	.	PUNCT
iajs-3828	289	1	moreover	moreover	ADV
iajs-3828	289	2	,	,	PUNCT
iajs-3828	289	3	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3828	289	4	𝑚→∞	𝑚→∞	X
iajs-3828	289	5	𝑓𝑚(𝑎	𝑓𝑚(𝑎	X
iajs-3828	289	6	)	)	PUNCT
iajs-3828	290	1	=	=	SYM
iajs-3828	290	2	𝑐	𝑐	NOUN
iajs-3828	290	3	,	,	PUNCT
iajs-3828	290	4	for	for	ADP
iajs-3828	290	5	any	any	DET
iajs-3828	290	6	𝑎	𝑎	PROPN
iajs-3828	290	7	∈	∈	ADJ
iajs-3828	290	8	£	£	NOUN
iajs-3828	290	9	.	.	PUNCT
iajs-3828	291	1	proof	proof	NOUN
iajs-3828	291	2	:	:	PUNCT
iajs-3828	291	3	let	let	VERB
iajs-3828	291	4	𝑎0	𝑎0	VERB
iajs-3828	291	5	∈	∈	VERB
iajs-3828	291	6	⋃	⋃	ADP
iajs-3828	291	7	£	£	SYM
iajs-3828	291	8	𝑖	𝑖	SYM
iajs-3828	291	9	𝑛	𝑛	PRON
iajs-3828	291	10	𝑖=1	𝑖=1	PUNCT
iajs-3828	291	11	and	and	CCONJ
iajs-3828	291	12	𝑎𝑚	𝑎𝑚	ADP
iajs-3828	291	13	define	define	VERB
iajs-3828	291	14	by	by	ADP
iajs-3828	291	15	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	291	16	=	=	SYM
iajs-3828	291	17	𝑓𝑎𝑚.	𝑓𝑎𝑚.	NOUN
iajs-3828	291	18	so	so	ADV
iajs-3828	291	19	for	for	ADP
iajs-3828	291	20	𝑚	𝑚	PROPN
iajs-3828	291	21	≥	≥	NUM
iajs-3828	291	22	0	0	NUM
iajs-3828	291	23	,	,	PUNCT
iajs-3828	291	24	∃𝑖𝑚	∃𝑖𝑚	NOUN
iajs-3828	291	25	∈	∈	PROPN
iajs-3828	291	26	{	{	PUNCT
iajs-3828	291	27	1,2	1,2	NUM
iajs-3828	291	28	,	,	PUNCT
iajs-3828	291	29	…	…	PUNCT
iajs-3828	291	30	𝑛	𝑛	X
iajs-3828	291	31	}	}	PUNCT
iajs-3828	291	32	such	such	ADJ
iajs-3828	291	33	that	that	SCONJ
iajs-3828	291	34	𝑎𝑚−1	𝑎𝑚−1	PROPN
iajs-3828	291	35	∈	∈	PROPN
iajs-3828	291	36	£	£	AUX
iajs-3828	291	37	𝑖𝑚	𝑖𝑚	NOUN
iajs-3828	291	38	and	and	CCONJ
iajs-3828	291	39	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	291	40	∈	∈	PROPN
iajs-3828	291	41	£	£	NOUN
iajs-3828	291	42	𝑖𝑚+1	𝑖𝑚+1	NOUN
iajs-3828	291	43	.	.	PUNCT
iajs-3828	292	1	if	if	SCONJ
iajs-3828	292	2	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	292	3	=	=	PUNCT
iajs-3828	293	1	𝑎𝑚0−1for	𝑎𝑚0−1for	PROPN
iajs-3828	293	2	some	some	DET
iajs-3828	293	3	𝑚0	𝑚0	NOUN
iajs-3828	293	4	,	,	PUNCT
iajs-3828	293	5	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	293	6	=	=	SYM
iajs-3828	293	7	𝑓𝑎𝑚0−1	𝑓𝑎𝑚0−1	PROPN
iajs-3828	293	8	=	=	SYM
iajs-3828	293	9	𝑎𝑚0−1	𝑎𝑚0−1	PROPN
iajs-3828	293	10	then	then	ADV
iajs-3828	293	11	𝑎𝑚0	𝑎𝑚0	NOUN
iajs-3828	293	12	is	be	AUX
iajs-3828	293	13	fixed	fix	VERB
iajs-3828	293	14	point	point	NOUN
iajs-3828	293	15	of	of	ADP
iajs-3828	293	16	𝑓.	𝑓.	NOUN
iajs-3828	293	17	thus	thus	ADV
iajs-3828	293	18	,	,	PUNCT
iajs-3828	293	19	suppose	suppose	VERB
iajs-3828	293	20	that	that	SCONJ
iajs-3828	293	21	𝑎𝑚	𝑎𝑚	PROPN
iajs-3828	293	22	≠	≠	PROPN
iajs-3828	293	23	𝑎𝑚−1	𝑎𝑚−1	PROPN
iajs-3828	293	24	,	,	PUNCT
iajs-3828	293	25	for	for	ADP
iajs-3828	293	26	all	all	PRON
iajs-3828	293	27	𝑚	𝑚	ADP
iajs-3828	293	28	∈	∈	NOUN
iajs-3828	293	29	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	293	30	}	}	PUNCT
iajs-3828	293	31	.	.	PUNCT
iajs-3828	294	1	by	by	ADP
iajs-3828	294	2	using	use	VERB
iajs-3828	294	3	(	(	PUNCT
iajs-3828	294	4	2.18	2.18	NUM
iajs-3828	294	5	)	)	PUNCT
iajs-3828	294	6	therefore	therefore	ADV
iajs-3828	294	7	𝑀(𝜓(𝜌(𝑎𝑚+1	𝑀(𝜓(𝜌(𝑎𝑚+1	PROPN
iajs-3828	294	8	,	,	PUNCT
iajs-3828	294	9	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	294	10	)	)	PUNCT
iajs-3828	294	11	)	)	PUNCT
iajs-3828	294	12	,	,	PUNCT
iajs-3828	294	13	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	X
iajs-3828	294	14	,	,	PUNCT
iajs-3828	294	15	𝑎𝑚−1	𝑎𝑚−1	PROPN
iajs-3828	294	16	)	)	PUNCT
iajs-3828	294	17	)	)	PUNCT
iajs-3828	294	18	,	,	PUNCT
iajs-3828	294	19	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	X
iajs-3828	294	20	,	,	PUNCT
iajs-3828	294	21	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	294	22	)	)	PUNCT
iajs-3828	294	23	)	)	PUNCT
iajs-3828	294	24	,	,	PUNCT
iajs-3828	294	25	𝜓(𝜌(𝑎𝑚−1	𝜓(𝜌(𝑎𝑚−1	PROPN
iajs-3828	294	26	,	,	PUNCT
iajs-3828	294	27	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	294	28	)	)	PUNCT
iajs-3828	294	29	)	)	PUNCT
iajs-3828	294	30	,	,	PUNCT
iajs-3828	294	31	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	X
iajs-3828	294	32	,	,	PUNCT
iajs-3828	294	33	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	294	34	)	)	PUNCT
iajs-3828	294	35	)	)	PUNCT
iajs-3828	294	36	,	,	PUNCT
iajs-3828	294	37	𝜓(𝜌(𝑎𝑚−1	𝜓(𝜌(𝑎𝑚−1	PROPN
iajs-3828	294	38	,	,	PUNCT
iajs-3828	294	39	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	294	40	)	)	PUNCT
iajs-3828	294	41	)	)	PUNCT
iajs-3828	294	42	)	)	PUNCT
iajs-3828	295	1	≤	≤	ADV
iajs-3828	295	2	0	0	NUM
iajs-3828	295	3	.	.	PUNCT
iajs-3828	296	1	and	and	CCONJ
iajs-3828	296	2	since	since	SCONJ
iajs-3828	296	3	𝜓(𝑟	𝜓(𝑟	NUM
iajs-3828	296	4	)	)	PUNCT
iajs-3828	296	5	=	=	SYM
iajs-3828	296	6	0	0	PUNCT
iajs-3828	297	1	if	if	SCONJ
iajs-3828	297	2	and	and	CCONJ
iajs-3828	297	3	only	only	ADV
iajs-3828	297	4	if	if	SCONJ
iajs-3828	297	5	𝑟	𝑟	NOUN
iajs-3828	297	6	=	=	SYM
iajs-3828	297	7	0	0	NUM
iajs-3828	297	8	,	,	PUNCT
iajs-3828	297	9	also	also	ADV
iajs-3828	297	10	by	by	ADP
iajs-3828	297	11	using	use	VERB
iajs-3828	297	12	triangle	triangle	NOUN
iajs-3828	297	13	inequality	inequality	NOUN
iajs-3828	297	14	and	and	CCONJ
iajs-3828	297	15	since	since	SCONJ
iajs-3828	297	16	𝜓	𝜓	PROPN
iajs-3828	297	17	is	be	AUX
iajs-3828	297	18	subadditive	subadditive	ADJ
iajs-3828	297	19	,	,	PUNCT
iajs-3828	297	20	therefore	therefore	ADV
iajs-3828	297	21	𝑀(𝜓(𝜌(𝑎𝑚+1	𝑀(𝜓(𝜌(𝑎𝑚+1	PROPN
iajs-3828	297	22	,	,	PUNCT
iajs-3828	297	23	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	297	24	)	)	PUNCT
iajs-3828	297	25	)	)	PUNCT
iajs-3828	297	26	,	,	PUNCT
iajs-3828	297	27	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	X
iajs-3828	297	28	,	,	PUNCT
iajs-3828	297	29	𝑎𝑚−1	𝑎𝑚−1	PROPN
iajs-3828	297	30	)	)	PUNCT
iajs-3828	297	31	)	)	PUNCT
iajs-3828	297	32	,	,	PUNCT
iajs-3828	297	33	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	X
iajs-3828	297	34	,	,	PUNCT
iajs-3828	297	35	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	297	36	)	)	PUNCT
iajs-3828	297	37	)	)	PUNCT
iajs-3828	297	38	,	,	PUNCT
iajs-3828	297	39	𝜓(𝜌(𝑎𝑚−1	𝜓(𝜌(𝑎𝑚−1	PROPN
iajs-3828	297	40	,	,	PUNCT
iajs-3828	297	41	𝑎𝑚)),0	𝑎𝑚)),0	PROPN
iajs-3828	297	42	,	,	PUNCT
iajs-3828	297	43	𝜓(𝑞𝜌(𝑎𝑚−1	𝜓(𝑞𝜌(𝑎𝑚−1	PROPN
iajs-3828	297	44	,	,	PUNCT
iajs-3828	297	45	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	297	46	)	)	PUNCT
iajs-3828	297	47	)	)	PUNCT
iajs-3828	298	1	+	+	CCONJ
iajs-3828	298	2	𝜓(𝑞𝜌(𝑎𝑚	𝜓(𝑞𝜌(𝑎𝑚	NOUN
iajs-3828	298	3	,	,	PUNCT
iajs-3828	298	4	𝑎𝑚+1	𝑎𝑚+1	NUM
iajs-3828	298	5	)	)	PUNCT
iajs-3828	298	6	)	)	PUNCT
iajs-3828	298	7	)	)	PUNCT
iajs-3828	299	1	≤	≤	ADV
iajs-3828	299	2	0	0	NUM
iajs-3828	299	3	.	.	PUNCT
iajs-3828	300	1	and	and	CCONJ
iajs-3828	300	2	from	from	ADP
iajs-3828	300	3	𝑀2	𝑀2	PROPN
iajs-3828	300	4	,	,	PUNCT
iajs-3828	300	5	there	there	PRON
iajs-3828	300	6	exists	exist	VERB
iajs-3828	300	7	a	a	DET
iajs-3828	300	8	right	right	ADJ
iajs-3828	300	9	continuous	continuous	ADJ
iajs-3828	300	10	function	function	NOUN
iajs-3828	300	11	a	a	NOUN
iajs-3828	300	12	:	:	PUNCT
iajs-3828	300	13	[	[	X
iajs-3828	300	14	0,∞	0,∞	NOUN
iajs-3828	300	15	)	)	PUNCT
iajs-3828	300	16	→	→	PUNCT
iajs-3828	301	1	[	[	X
iajs-3828	301	2	0,∞	0,∞	NOUN
iajs-3828	301	3	)	)	PUNCT
iajs-3828	301	4	,	,	PUNCT
iajs-3828	301	5	a(0	a(0	PROPN
iajs-3828	301	6	)	)	PUNCT
iajs-3828	301	7	=	=	SYM
iajs-3828	301	8	0	0	NUM
iajs-3828	301	9	,	,	PUNCT
iajs-3828	301	10	a(𝑟	a(𝑟	NOUN
iajs-3828	301	11	)	)	PUNCT
iajs-3828	301	12	<	<	X
iajs-3828	301	13	𝑟	𝑟	NOUN
iajs-3828	301	14	,	,	PUNCT
iajs-3828	301	15	for	for	ADP
iajs-3828	301	16	𝑟	𝑟	X
iajs-3828	301	17	>	>	X
iajs-3828	301	18	0	0	NUM
iajs-3828	301	19	,	,	PUNCT
iajs-3828	301	20	such	such	ADJ
iajs-3828	301	21	that	that	SCONJ
iajs-3828	301	22	for	for	ADP
iajs-3828	301	23	all	all	PRON
iajs-3828	301	24	𝑚	𝑚	ADP
iajs-3828	301	25	∈	∈	NOUN
iajs-3828	301	26	𝑁⋃{0	𝑁⋃{0	ADP
iajs-3828	301	27	}	}	PUNCT
iajs-3828	301	28	𝜓(𝜌(𝑎𝑚+1	𝜓(𝜌(𝑎𝑚+1	VERB
iajs-3828	301	29	,	,	PUNCT
iajs-3828	301	30	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	301	31	)	)	PUNCT
iajs-3828	301	32	≤	≤	NOUN
iajs-3828	301	33	𝐴	𝐴	PROPN
iajs-3828	301	34	(	(	PUNCT
iajs-3828	301	35	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	PROPN
iajs-3828	301	36	,	,	PUNCT
iajs-3828	301	37	𝑎𝑚−1	𝑎𝑚−1	PROPN
iajs-3828	301	38	)	)	PUNCT
iajs-3828	301	39	)	)	PUNCT
iajs-3828	301	40	)	)	PUNCT
iajs-3828	301	41	.	.	PUNCT
iajs-3828	302	1	if	if	SCONJ
iajs-3828	302	2	we	we	PRON
iajs-3828	302	3	use	use	VERB
iajs-3828	302	4	this	this	DET
iajs-3828	302	5	procedure	procedure	NOUN
iajs-3828	302	6	,	,	PUNCT
iajs-3828	302	7	we	we	PRON
iajs-3828	302	8	get	get	AUX
iajs-3828	302	9	𝜓(𝜌(𝑎𝑚+1	𝜓(𝜌(𝑎𝑚+1	VERB
iajs-3828	302	10	,	,	PUNCT
iajs-3828	302	11	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	302	12	)	)	PUNCT
iajs-3828	302	13	≤	≤	NOUN
iajs-3828	302	14	𝐴	𝐴	PROPN
iajs-3828	302	15	(	(	PUNCT
iajs-3828	302	16	𝜓(𝜌(𝑎𝑚	𝜓(𝜌(𝑎𝑚	PROPN
iajs-3828	302	17	,	,	PUNCT
iajs-3828	302	18	𝑎𝑚−1	𝑎𝑚−1	PROPN
iajs-3828	302	19	)	)	PUNCT
iajs-3828	302	20	)	)	PUNCT
iajs-3828	302	21	)	)	PUNCT
iajs-3828	303	1	≤	≤	NOUN
iajs-3828	303	2	⋯	⋯	VERB
iajs-3828	303	3	≤	≤	NOUN
iajs-3828	303	4	a𝑚	a𝑚	ADP
iajs-3828	303	5	(	(	PUNCT
iajs-3828	303	6	𝜓(𝜌(𝑎1	𝜓(𝜌(𝑎1	ADJ
iajs-3828	303	7	,	,	PUNCT
iajs-3828	303	8	𝑎0	𝑎0	PROPN
iajs-3828	303	9	)	)	PUNCT
iajs-3828	303	10	)	)	PUNCT
iajs-3828	303	11	.	.	PUNCT
iajs-3828	304	1	(	(	PUNCT
iajs-3828	304	2	19	19	NUM
iajs-3828	304	3	)	)	PUNCT
iajs-3828	304	4	and	and	CCONJ
iajs-3828	304	5	by	by	ADP
iajs-3828	304	6	(	(	PUNCT
iajs-3828	304	7	lemma	lemma	PROPN
iajs-3828	304	8	8)	8)	NUM
iajs-3828	304	9	and	and	CCONJ
iajs-3828	304	10	continuity	continuity	NOUN
iajs-3828	304	11	of	of	ADP
iajs-3828	304	12	𝜓	𝜓	NOUN
iajs-3828	304	13	,	,	PUNCT
iajs-3828	304	14	subsequently	subsequently	ADV
iajs-3828	304	15	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3828	304	16	𝑚→∞	𝑚→∞	NUM
iajs-3828	304	17	𝜓	𝜓	PROPN
iajs-3828	304	18	(	(	PUNCT
iajs-3828	304	19	𝜌(𝑎𝑚+1	𝜌(𝑎𝑚+1	ADJ
iajs-3828	304	20	,	,	PUNCT
iajs-3828	304	21	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	304	22	)	)	PUNCT
iajs-3828	304	23	)	)	PUNCT
iajs-3828	305	1	=	=	SYM
iajs-3828	305	2	0	0	NUM
iajs-3828	305	3	=	=	SYM
iajs-3828	305	4	𝜓	𝜓	PROPN
iajs-3828	305	5	(	(	PUNCT
iajs-3828	305	6	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
iajs-3828	305	7	𝑚→∞	𝑚→∞	NUM
iajs-3828	305	8	𝜌(𝑎𝑚+1	𝜌(𝑎𝑚+1	ADJ
iajs-3828	305	9	,	,	PUNCT
iajs-3828	305	10	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	305	11	)	)	PUNCT
iajs-3828	305	12	)	)	PUNCT
iajs-3828	305	13	.	.	PUNCT
iajs-3828	306	1	since	since	SCONJ
iajs-3828	306	2	𝜓(𝑟	𝜓(𝑟	NUM
iajs-3828	306	3	)	)	PUNCT
iajs-3828	306	4	=	=	SYM
iajs-3828	306	5	0	0	PUNCT
iajs-3828	306	6	if	if	SCONJ
iajs-3828	306	7	and	and	CCONJ
iajs-3828	306	8	only	only	ADV
iajs-3828	306	9	if	if	SCONJ
iajs-3828	306	10	𝑟	𝑟	NOUN
iajs-3828	306	11	=	=	SYM
iajs-3828	306	12	0	0	NUM
iajs-3828	306	13	,	,	PUNCT
iajs-3828	306	14	therefore	therefore	ADV
iajs-3828	306	15	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3828	306	16	𝑚→∞	𝑚→∞	NUM
iajs-3828	306	17	𝜌(𝑎𝑚+1	𝜌(𝑎𝑚+1	ADJ
iajs-3828	306	18	,	,	PUNCT
iajs-3828	306	19	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	306	20	)	)	PUNCT
iajs-3828	306	21	=	=	SYM
iajs-3828	307	1	0	0	X
iajs-3828	307	2	.	.	PUNCT
iajs-3828	308	1	(	(	PUNCT
iajs-3828	308	2	20	20	NUM
iajs-3828	308	3	)	)	PUNCT
iajs-3828	308	4	to	to	PART
iajs-3828	308	5	prove	prove	VERB
iajs-3828	308	6	for	for	ADP
iajs-3828	308	7	each	each	DET
iajs-3828	308	8	𝑎0	𝑎0	PROPN
iajs-3828	308	9	∈	∈	PROPN
iajs-3828	308	10	£	£	SYM
iajs-3828	308	11	,	,	PUNCT
iajs-3828	308	12	(	(	PUNCT
iajs-3828	308	13	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	308	14	)	)	PUNCT
iajs-3828	308	15	is	be	AUX
iajs-3828	308	16	a	a	DET
iajs-3828	308	17	cauchy	cauchy	ADJ
iajs-3828	308	18	sequence	sequence	NOUN
iajs-3828	308	19	.	.	PUNCT
iajs-3828	309	1	assume	assume	VERB
iajs-3828	309	2	it	it	PRON
iajs-3828	309	3	is	be	AUX
iajs-3828	309	4	false	false	ADJ
iajs-3828	309	5	.	.	PUNCT
iajs-3828	310	1	then	then	ADV
iajs-3828	310	2	we	we	PRON
iajs-3828	310	3	can	can	AUX
iajs-3828	310	4	find	find	VERB
iajs-3828	310	5	a	a	DET
iajs-3828	310	6	𝜀	𝜀	X
iajs-3828	310	7	>	>	X
iajs-3828	310	8	0	0	PUNCT
iajs-3828	311	1	and	and	CCONJ
iajs-3828	311	2	{	{	PUNCT
iajs-3828	311	3	𝑝𝑟	𝑝𝑟	NOUN
iajs-3828	311	4	}	}	PUNCT
iajs-3828	311	5	,	,	PUNCT
iajs-3828	311	6	{	{	PUNCT
iajs-3828	311	7	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	311	8	}	}	PUNCT
iajs-3828	311	9	,	,	PUNCT
iajs-3828	311	10	𝑑𝑟	𝑑𝑟	VERB
iajs-3828	311	11	>	>	X
iajs-3828	311	12	𝑝𝑟	𝑝𝑟	PROPN
iajs-3828	311	13	≥	≥	NUM
iajs-3828	311	14	𝑟	𝑟	NOUN
iajs-3828	311	15	where	where	SCONJ
iajs-3828	311	16	{	{	PUNCT
iajs-3828	311	17	𝑝𝑟	𝑝𝑟	NOUN
iajs-3828	311	18	}	}	PUNCT
iajs-3828	311	19	,	,	PUNCT
iajs-3828	311	20	{	{	PUNCT
iajs-3828	311	21	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	311	22	}	}	PUNCT
iajs-3828	311	23	two	two	NUM
iajs-3828	311	24	subsequences	subsequence	NOUN
iajs-3828	311	25	of	of	ADP
iajs-3828	311	26	integers	integer	NOUN
iajs-3828	311	27	with	with	ADP
iajs-3828	311	28	𝜓(𝜌(𝑎𝑝𝑟	𝜓(𝜌(𝑎𝑝𝑟	NUM
iajs-3828	311	29	,	,	PUNCT
iajs-3828	311	30	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	311	31	)	)	PUNCT
iajs-3828	311	32	)	)	PUNCT
iajs-3828	311	33	≥	≥	PROPN
iajs-3828	311	34	𝜀	𝜀	NOUN
iajs-3828	311	35	for	for	ADP
iajs-3828	311	36	𝑛	𝑛	DET
iajs-3828	311	37	∈	∈	PROPN
iajs-3828	311	38	{	{	PUNCT
iajs-3828	311	39	1,2	1,2	NUM
iajs-3828	311	40	,	,	PUNCT
iajs-3828	311	41	…	…	PUNCT
iajs-3828	311	42	}	}	PUNCT
iajs-3828	311	43	.	.	PUNCT
iajs-3828	312	1	(	(	PUNCT
iajs-3828	312	2	21	21	NUM
iajs-3828	312	3	)	)	PUNCT
iajs-3828	312	4	we	we	PRON
iajs-3828	312	5	also	also	ADV
iajs-3828	312	6	assume	assume	VERB
iajs-3828	312	7	ihjpas	ihjpa	NOUN
iajs-3828	312	8	.	.	PUNCT
iajs-3828	313	1	2025,38(2	2025,38(2	PROPN
iajs-3828	313	2	)	)	PUNCT
iajs-3828	313	3	384	384	NUM
iajs-3828	313	4	𝜓(𝜌(𝑎𝑝𝑟	𝜓(𝜌(𝑎𝑝𝑟	NUM
iajs-3828	313	5	,	,	PUNCT
iajs-3828	313	6	𝑎𝑑𝑟−1	𝑎𝑑𝑟−1	PROPN
iajs-3828	313	7	)	)	PUNCT
iajs-3828	313	8	)	)	PUNCT
iajs-3828	313	9	<	<	X
iajs-3828	313	10	𝜀.	𝜀.	NOUN
iajs-3828	313	11	(	(	PUNCT
iajs-3828	313	12	22	22	NUM
iajs-3828	313	13	(	(	PUNCT
iajs-3828	313	14	by	by	ADP
iajs-3828	313	15	selecting	select	VERB
iajs-3828	313	16	𝑑𝑟	𝑑𝑟	NOUN
iajs-3828	313	17	to	to	PART
iajs-3828	313	18	be	be	AUX
iajs-3828	313	19	the	the	DET
iajs-3828	313	20	least	least	ADJ
iajs-3828	313	21	number	number	NOUN
iajs-3828	313	22	surpassing	surpass	VERB
iajs-3828	313	23	𝑝𝑟	𝑝𝑟	PROPN
iajs-3828	313	24	for	for	ADP
iajs-3828	313	25	which	which	DET
iajs-3828	313	26	inequality	inequality	NOUN
iajs-3828	313	27	(	(	PUNCT
iajs-3828	313	28	21	21	NUM
iajs-3828	313	29	)	)	PUNCT
iajs-3828	313	30	holds	hold	VERB
iajs-3828	313	31	,	,	PUNCT
iajs-3828	313	32	now	now	ADV
iajs-3828	313	33	by	by	ADP
iajs-3828	313	34	(	(	PUNCT
iajs-3828	313	35	19	19	NUM
iajs-3828	313	36	)	)	PUNCT
iajs-3828	313	37	and	and	CCONJ
iajs-3828	313	38	(	(	PUNCT
iajs-3828	313	39	21	21	NUM
iajs-3828	313	40	)	)	PUNCT
iajs-3828	313	41	,	,	PUNCT
iajs-3828	313	42	(	(	PUNCT
iajs-3828	313	43	22	22	NUM
iajs-3828	313	44	)	)	PUNCT
iajs-3828	313	45	,	,	PUNCT
iajs-3828	313	46	and	and	CCONJ
iajs-3828	313	47	since	since	SCONJ
iajs-3828	313	48	𝜓	𝜓	PROPN
iajs-3828	313	49	is	be	AUX
iajs-3828	313	50	subadditive	subadditive	ADJ
iajs-3828	313	51	,	,	PUNCT
iajs-3828	313	52	getting	get	VERB
iajs-3828	313	53	𝜀	𝜀	PRON
iajs-3828	313	54	≤	≤	ADJ
iajs-3828	313	55	𝜓(𝜌(𝑎𝑝𝑟	𝜓(𝜌(𝑎𝑝𝑟	NOUN
iajs-3828	313	56	,	,	PUNCT
iajs-3828	313	57	𝑎𝑑𝑟−1	𝑎𝑑𝑟−1	PROPN
iajs-3828	313	58	)	)	PUNCT
iajs-3828	313	59	)	)	PUNCT
iajs-3828	313	60	≤	≤	NUM
iajs-3828	313	61	𝜓(𝑞𝜌(𝑎𝑝𝑟	𝜓(𝑞𝜌(𝑎𝑝𝑟	NOUN
iajs-3828	313	62	,	,	PUNCT
iajs-3828	313	63	𝑎𝑑𝑟−1	𝑎𝑑𝑟−1	PROPN
iajs-3828	313	64	)	)	PUNCT
iajs-3828	313	65	)	)	PUNCT
iajs-3828	314	1	+	+	CCONJ
iajs-3828	314	2	𝜓(𝑞𝜌(𝑎𝑑𝑟−1	𝜓(𝑞𝜌(𝑎𝑑𝑟−1	NOUN
iajs-3828	314	3	,	,	PUNCT
iajs-3828	314	4	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	314	5	)	)	PUNCT
iajs-3828	314	6	)	)	PUNCT
iajs-3828	314	7	≤	≤	NOUN
iajs-3828	314	8	𝑞𝜀	𝑞𝜀	ADP
iajs-3828	314	9	+	+	X
iajs-3828	314	10	𝐴𝑑𝑟−1𝜓(𝑞𝜌(𝑎0	𝐴𝑑𝑟−1𝜓(𝑞𝜌(𝑎0	ADJ
iajs-3828	314	11	,	,	PUNCT
iajs-3828	314	12	𝑎1	𝑎1	NOUN
iajs-3828	314	13	)	)	PUNCT
iajs-3828	314	14	)	)	PUNCT
iajs-3828	314	15	.	.	PUNCT
iajs-3828	315	1	(	(	PUNCT
iajs-3828	315	2	23	23	NUM
iajs-3828	315	3	)	)	PUNCT
iajs-3828	315	4	and	and	CCONJ
iajs-3828	315	5	so	so	ADV
iajs-3828	315	6	lim	lim	PROPN
iajs-3828	315	7	𝑟→∞	𝑟→∞	NUM
iajs-3828	315	8	𝜓(𝜌(𝑎𝑑𝑟−1	𝜓(𝜌(𝑎𝑑𝑟−1	PROPN
iajs-3828	315	9	,	,	PUNCT
iajs-3828	315	10	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	315	11	)	)	PUNCT
iajs-3828	315	12	)	)	PUNCT
iajs-3828	315	13	=	=	SYM
iajs-3828	315	14	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	315	15	(	(	PUNCT
iajs-3828	315	16	24	24	NUM
iajs-3828	315	17	)	)	PUNCT
iajs-3828	315	18	on	on	ADP
iajs-3828	315	19	the	the	DET
iajs-3828	315	20	other	other	ADJ
iajs-3828	315	21	hand	hand	NOUN
iajs-3828	315	22	,	,	PUNCT
iajs-3828	315	23	∀𝑟	∀𝑟	PROPN
iajs-3828	315	24	,	,	PUNCT
iajs-3828	315	25	∃𝑖𝑟	∃𝑖𝑟	NOUN
iajs-3828	315	26	∈	∈	PROPN
iajs-3828	315	27	{	{	PUNCT
iajs-3828	315	28	1,2	1,2	NUM
iajs-3828	315	29	,	,	PUNCT
iajs-3828	315	30	…	…	PUNCT
iajs-3828	315	31	,	,	PUNCT
iajs-3828	315	32	𝑛	𝑛	X
iajs-3828	315	33	}	}	PUNCT
iajs-3828	315	34	such	such	ADJ
iajs-3828	315	35	that	that	SCONJ
iajs-3828	315	36	𝑑𝑟	𝑑𝑟	VERB
iajs-3828	315	37	−	−	PROPN
iajs-3828	315	38	𝑝𝑟	𝑝𝑟	PROPN
iajs-3828	316	1	+	+	CCONJ
iajs-3828	316	2	𝑖𝑟	𝑖𝑟	ADP
iajs-3828	316	3	≡	≡	PROPN
iajs-3828	316	4	1(mod	1(mod	NUM
iajs-3828	316	5	𝑛	𝑛	NOUN
iajs-3828	316	6	)	)	PUNCT
iajs-3828	316	7	.	.	PUNCT
iajs-3828	317	1	then	then	ADV
iajs-3828	317	2	𝑎𝑝𝑟−𝑖𝑟(for	𝑎𝑝𝑟−𝑖𝑟(for	ADP
iajs-3828	317	3	𝑟	𝑟	X
iajs-3828	317	4	large	large	ADJ
iajs-3828	317	5	enough	enough	ADV
iajs-3828	317	6	,	,	PUNCT
iajs-3828	317	7	𝑝𝑟	𝑝𝑟	PROPN
iajs-3828	317	8	>	>	X
iajs-3828	317	9	𝑖𝑟	𝑖𝑟	PROPN
iajs-3828	317	10	)	)	PUNCT
iajs-3828	317	11	and	and	CCONJ
iajs-3828	317	12	𝑎𝑑𝑟belong	𝑎𝑑𝑟belong	ADJ
iajs-3828	317	13	to	to	ADP
iajs-3828	317	14	different	different	ADJ
iajs-3828	317	15	sets	set	NOUN
iajs-3828	317	16	£	£	SYM
iajs-3828	317	17	𝑖	𝑖	NOUN
iajs-3828	317	18	and	and	CCONJ
iajs-3828	317	19	£	£	SYM
iajs-3828	317	20	𝑖+1	𝑖+1	NUM
iajs-3828	317	21	for	for	ADP
iajs-3828	317	22	𝑖	𝑖	PRON
iajs-3828	317	23	∈	∈	PROPN
iajs-3828	317	24	{	{	PUNCT
iajs-3828	317	25	1,2	1,2	NUM
iajs-3828	317	26	,	,	PUNCT
iajs-3828	317	27	…	…	PUNCT
iajs-3828	317	28	,	,	PUNCT
iajs-3828	317	29	𝑛	𝑛	NOUN
iajs-3828	317	30	}	}	PUNCT
iajs-3828	317	31	.	.	PUNCT
iajs-3828	318	1	by	by	ADP
iajs-3828	318	2	the	the	DET
iajs-3828	318	3	triangle	triangle	NOUN
iajs-3828	318	4	inequality	inequality	NOUN
iajs-3828	318	5	,	,	PUNCT
iajs-3828	318	6	also	also	ADV
iajs-3828	318	7	𝜓	𝜓	PROPN
iajs-3828	318	8	is	be	AUX
iajs-3828	318	9	subadditive	subadditive	ADJ
iajs-3828	318	10	,	,	PUNCT
iajs-3828	318	11	obtaining	obtain	VERB
iajs-3828	318	12	𝜓(𝜌(𝑎𝑑𝑟	𝜓(𝜌(𝑎𝑑𝑟	NOUN
iajs-3828	318	13	,	,	PUNCT
iajs-3828	318	14	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	318	15	)	)	PUNCT
iajs-3828	318	16	)	)	PUNCT
iajs-3828	318	17	≤	≤	NOUN
iajs-3828	319	1	𝜓(𝑞𝜌	𝜓(𝑞𝜌	X
iajs-3828	319	2	(	(	PUNCT
iajs-3828	319	3	𝑎𝑑𝑟	𝑎𝑑𝑟	ADJ
iajs-3828	319	4	,	,	PUNCT
iajs-3828	319	5	𝑎𝑑𝑟−𝑖𝑟	𝑎𝑑𝑟−𝑖𝑟	NUM
iajs-3828	319	6	)	)	PUNCT
iajs-3828	320	1	+	+	CCONJ
iajs-3828	320	2	𝑞𝜌(𝑎𝑑𝑟−𝑖𝑟	𝑞𝜌(𝑎𝑑𝑟−𝑖𝑟	PROPN
iajs-3828	320	3	,	,	PUNCT
iajs-3828	320	4	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	320	5	)	)	PUNCT
iajs-3828	320	6	)	)	PUNCT
iajs-3828	321	1	≤	≤	NOUN
iajs-3828	321	2	𝜓(𝑞𝜌(𝑎𝑑𝑟	𝜓(𝑞𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	321	3	,	,	PUNCT
iajs-3828	321	4	𝑎𝑑𝑟−𝑖𝑟	𝑎𝑑𝑟−𝑖𝑟	NUM
iajs-3828	321	5	)	)	PUNCT
iajs-3828	321	6	)	)	PUNCT
iajs-3828	322	1	+	+	CCONJ
iajs-3828	322	2	𝜓(𝑞𝜌(𝑎𝑑𝑟−𝑖𝑟	𝜓(𝑞𝜌(𝑎𝑑𝑟−𝑖𝑟	ADJ
iajs-3828	322	3	,	,	PUNCT
iajs-3828	322	4	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	322	5	)	)	PUNCT
iajs-3828	322	6	)	)	PUNCT
iajs-3828	322	7	.	.	PUNCT
iajs-3828	323	1	now	now	ADV
iajs-3828	323	2	taking	take	VERB
iajs-3828	323	3	𝑟	𝑟	NOUN
iajs-3828	323	4	→	→	SYM
iajs-3828	323	5	∞	∞	PROPN
iajs-3828	323	6	,	,	PUNCT
iajs-3828	323	7	by	by	ADP
iajs-3828	323	8	(	(	PUNCT
iajs-3828	323	9	19	19	NUM
iajs-3828	323	10	)	)	PUNCT
iajs-3828	323	11	and	and	CCONJ
iajs-3828	323	12	from	from	ADP
iajs-3828	323	13	(	(	PUNCT
iajs-3828	323	14	24	24	NUM
iajs-3828	323	15	)	)	PUNCT
iajs-3828	323	16	,	,	PUNCT
iajs-3828	323	17	as	as	ADP
iajs-3828	323	18	a	a	DET
iajs-3828	323	19	results	result	NOUN
iajs-3828	323	20	𝜓(𝜌(𝑎𝑝𝑟	𝜓(𝜌(𝑎𝑝𝑟	NUM
iajs-3828	323	21	,	,	PUNCT
iajs-3828	323	22	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	323	23	)	)	PUNCT
iajs-3828	323	24	)	)	PUNCT
iajs-3828	323	25	≤	≤	NUM
iajs-3828	323	26	𝐴	𝐴	PROPN
iajs-3828	323	27	𝑑𝑟−𝑖𝑟𝜓(𝑞𝜌(𝑎0	𝑑𝑟−𝑖𝑟𝜓(𝑞𝜌(𝑎0	NOUN
iajs-3828	323	28	,	,	PUNCT
iajs-3828	323	29	𝑎1	𝑎1	NOUN
iajs-3828	323	30	)	)	PUNCT
iajs-3828	323	31	)	)	PUNCT
iajs-3828	324	1	+	+	CCONJ
iajs-3828	324	2	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	324	3	and	and	CCONJ
iajs-3828	324	4	so	so	ADV
iajs-3828	324	5	,	,	PUNCT
iajs-3828	324	6	lim	lim	PROPN
iajs-3828	324	7	𝑟→∞	𝑟→∞	NUM
iajs-3828	324	8	𝜓(𝜌(𝑎𝑑𝑟	𝜓(𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	324	9	,	,	PUNCT
iajs-3828	324	10	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	324	11	)	)	PUNCT
iajs-3828	324	12	)	)	PUNCT
iajs-3828	325	1	=	=	SYM
iajs-3828	325	2	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	325	3	(	(	PUNCT
iajs-3828	325	4	25	25	NUM
iajs-3828	325	5	)	)	PUNCT
iajs-3828	325	6	by	by	ADP
iajs-3828	325	7	using	use	VERB
iajs-3828	325	8	(	(	PUNCT
iajs-3828	325	9	20	20	NUM
iajs-3828	325	10	)	)	PUNCT
iajs-3828	325	11	,	,	PUNCT
iajs-3828	325	12	so	so	ADV
iajs-3828	325	13	lim	lim	PROPN
iajs-3828	325	14	𝑟→∞	𝑟→∞	NUM
iajs-3828	325	15	𝜓(𝜌(𝑎𝑑𝑟+1	𝜓(𝜌(𝑎𝑑𝑟+1	ADV
iajs-3828	325	16	,	,	PUNCT
iajs-3828	325	17	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	325	18	)	)	PUNCT
iajs-3828	325	19	)	)	PUNCT
iajs-3828	326	1	=	=	SYM
iajs-3828	326	2	0	0	NUM
iajs-3828	326	3	,	,	PUNCT
iajs-3828	326	4	lim	lim	PROPN
iajs-3828	326	5	𝑟→∞	𝑟→∞	NUM
iajs-3828	326	6	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	326	7	,	,	PUNCT
iajs-3828	326	8	𝑎𝑝𝑟−𝑖𝑟	𝑎𝑝𝑟−𝑖𝑟	NOUN
iajs-3828	326	9	)	)	PUNCT
iajs-3828	326	10	)	)	PUNCT
iajs-3828	327	1	=	=	PUNCT
iajs-3828	327	2	0	0	X
iajs-3828	327	3	.	.	PUNCT
iajs-3828	328	1	(	(	PUNCT
iajs-3828	328	2	26	26	NUM
iajs-3828	328	3	)	)	PUNCT
iajs-3828	328	4	and	and	CCONJ
iajs-3828	328	5	by	by	ADP
iajs-3828	328	6	using	use	VERB
iajs-3828	328	7	the	the	DET
iajs-3828	328	8	triangle	triangle	NOUN
iajs-3828	328	9	inequality	inequality	NOUN
iajs-3828	328	10	,	,	PUNCT
iajs-3828	328	11	then	then	ADV
iajs-3828	328	12	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	328	13	,	,	PUNCT
iajs-3828	328	14	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	328	15	)	)	PUNCT
iajs-3828	328	16	)	)	PUNCT
iajs-3828	328	17	≤	≤	PUNCT
iajs-3828	329	1	𝜓(𝑞𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝑞𝜌(𝑎𝑝𝑟−𝑖𝑟	NOUN
iajs-3828	329	2	,	,	PUNCT
iajs-3828	329	3	𝑎𝑝𝑟	𝑎𝑝𝑟	ADJ
iajs-3828	329	4	)	)	PUNCT
iajs-3828	329	5	)	)	PUNCT
iajs-3828	330	1	+	+	PUNCT
iajs-3828	330	2	𝜓(𝑞𝜌	𝜓(𝑞𝜌	X
iajs-3828	330	3	(	(	PUNCT
iajs-3828	330	4	𝑎𝑝𝑟	𝑎𝑝𝑟	NOUN
iajs-3828	330	5	,	,	PUNCT
iajs-3828	330	6	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	330	7	)	)	PUNCT
iajs-3828	330	8	)	)	PUNCT
iajs-3828	330	9	.	.	PUNCT
iajs-3828	331	1	letting	let	VERB
iajs-3828	331	2	𝑟	𝑟	NOUN
iajs-3828	331	3	→	→	SYM
iajs-3828	331	4	∞	∞	NUM
iajs-3828	331	5	in	in	ADP
iajs-3828	331	6	the	the	DET
iajs-3828	331	7	last	last	ADJ
iajs-3828	331	8	inequality	inequality	NOUN
iajs-3828	331	9	and	and	CCONJ
iajs-3828	331	10	using	use	VERB
iajs-3828	331	11	(	(	PUNCT
iajs-3828	331	12	2.19	2.19	NUM
iajs-3828	331	13	)	)	PUNCT
iajs-3828	331	14	and	and	CCONJ
iajs-3828	331	15	(	(	PUNCT
iajs-3828	331	16	2.25	2.25	NUM
iajs-3828	331	17	)	)	PUNCT
iajs-3828	331	18	,	,	PUNCT
iajs-3828	331	19	consequently	consequently	ADV
iajs-3828	331	20	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	PRON
iajs-3828	331	21	,	,	PUNCT
iajs-3828	331	22	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	331	23	)	)	PUNCT
iajs-3828	331	24	)	)	PUNCT
iajs-3828	332	1	≤	≤	PROPN
iajs-3828	332	2	𝐴𝑝𝑟−𝑖𝑟𝜓(𝑞𝜌(𝑎0	𝐴𝑝𝑟−𝑖𝑟𝜓(𝑞𝜌(𝑎0	PROPN
iajs-3828	332	3	,	,	PUNCT
iajs-3828	332	4	𝑎1	𝑎1	NOUN
iajs-3828	332	5	)	)	PUNCT
iajs-3828	332	6	)	)	PUNCT
iajs-3828	333	1	+	+	CCONJ
iajs-3828	333	2	𝑞𝜀	𝑞𝜀	ADP
iajs-3828	333	3	lim	lim	PROPN
iajs-3828	333	4	𝑟→∞	𝑟→∞	NUM
iajs-3828	333	5	𝜌	𝜌	X
iajs-3828	333	6	(	(	PUNCT
iajs-3828	333	7	𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜌(𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	333	8	,	,	PUNCT
iajs-3828	333	9	𝑎𝑑𝑟	𝑎𝑑𝑟	PROPN
iajs-3828	333	10	)	)	PUNCT
iajs-3828	333	11	)	)	PUNCT
iajs-3828	334	1	=	=	SYM
iajs-3828	334	2	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	334	3	(	(	PUNCT
iajs-3828	334	4	27	27	NUM
iajs-3828	334	5	)	)	PUNCT
iajs-3828	334	6	again	again	ADV
iajs-3828	334	7	,	,	PUNCT
iajs-3828	334	8	by	by	ADP
iajs-3828	334	9	using	use	VERB
iajs-3828	334	10	the	the	DET
iajs-3828	334	11	triangle	triangle	NOUN
iajs-3828	334	12	inequality	inequality	NOUN
iajs-3828	334	13	,	,	PUNCT
iajs-3828	334	14	obtaining	obtain	VERB
iajs-3828	334	15	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	PRON
iajs-3828	334	16	,	,	PUNCT
iajs-3828	334	17	𝑎𝑞𝑟+1	𝑎𝑞𝑟+1	NOUN
iajs-3828	334	18	)	)	PUNCT
iajs-3828	334	19	)	)	PUNCT
iajs-3828	334	20	≤	≤	NOUN
iajs-3828	335	1	𝜓(𝑞𝜌	𝜓(𝑞𝜌	PROPN
iajs-3828	335	2	(	(	PUNCT
iajs-3828	335	3	𝑎𝑝𝑟−𝑖𝑟	𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	335	4	,	,	PUNCT
iajs-3828	335	5	𝑎𝑑𝑟	𝑎𝑑𝑟	PROPN
iajs-3828	335	6	)	)	PUNCT
iajs-3828	335	7	)	)	PUNCT
iajs-3828	336	1	+	+	CCONJ
iajs-3828	336	2	𝜓(𝑞𝜌(𝑎𝑑𝑟	𝜓(𝑞𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	336	3	,	,	PUNCT
iajs-3828	336	4	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	NUM
iajs-3828	336	5	)	)	PUNCT
iajs-3828	336	6	)	)	PUNCT
iajs-3828	336	7	.	.	PUNCT
iajs-3828	337	1	letting	let	VERB
iajs-3828	337	2	𝑟	𝑟	NOUN
iajs-3828	337	3	→	→	SYM
iajs-3828	337	4	∞	∞	NUM
iajs-3828	337	5	in	in	ADP
iajs-3828	337	6	the	the	DET
iajs-3828	337	7	last	last	ADJ
iajs-3828	337	8	inequality	inequality	NOUN
iajs-3828	337	9	and	and	CCONJ
iajs-3828	337	10	using	use	VERB
iajs-3828	337	11	(	(	PUNCT
iajs-3828	337	12	26	26	NUM
iajs-3828	337	13	)	)	PUNCT
iajs-3828	337	14	and	and	CCONJ
iajs-3828	337	15	(	(	PUNCT
iajs-3828	337	16	27	27	NUM
iajs-3828	337	17	)	)	PUNCT
iajs-3828	337	18	,	,	PUNCT
iajs-3828	337	19	therefore	therefore	ADV
iajs-3828	337	20	lim	lim	PROPN
iajs-3828	337	21	𝑟→∞	𝑟→∞	NUM
iajs-3828	337	22	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	337	23	,	,	PUNCT
iajs-3828	337	24	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	PROPN
iajs-3828	337	25	)	)	PUNCT
iajs-3828	337	26	)	)	PUNCT
iajs-3828	338	1	=	=	SYM
iajs-3828	338	2	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	338	3	(	(	PUNCT
iajs-3828	338	4	28	28	NUM
iajs-3828	338	5	)	)	PUNCT
iajs-3828	338	6	and	and	CCONJ
iajs-3828	338	7	in	in	ADP
iajs-3828	338	8	the	the	DET
iajs-3828	338	9	same	same	ADJ
iajs-3828	338	10	way	way	NOUN
iajs-3828	338	11	,	,	PUNCT
iajs-3828	338	12	as	as	ADP
iajs-3828	338	13	a	a	DET
iajs-3828	338	14	results	result	NOUN
iajs-3828	338	15	𝜓(𝜌(𝑎𝑑𝑟	𝜓(𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	338	16	,	,	PUNCT
iajs-3828	338	17	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	338	18	)	)	PUNCT
iajs-3828	338	19	)	)	PUNCT
iajs-3828	338	20	≤	≤	NOUN
iajs-3828	338	21	𝜓(𝑞𝜌(𝑎𝑑𝑟	𝜓(𝑞𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	338	22	,	,	PUNCT
iajs-3828	338	23	𝑎𝑝𝑟−𝑖𝑟	𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	338	24	)	)	PUNCT
iajs-3828	338	25	)	)	PUNCT
iajs-3828	339	1	+	+	CCONJ
iajs-3828	339	2	𝜓(𝑞𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝑞𝜌(𝑎𝑝𝑟−𝑖𝑟	NOUN
iajs-3828	339	3	,	,	PUNCT
iajs-3828	339	4	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	339	5	)	)	PUNCT
iajs-3828	339	6	)	)	PUNCT
iajs-3828	339	7	.	.	PUNCT
iajs-3828	340	1	letting	let	VERB
iajs-3828	340	2	𝑟	𝑟	NOUN
iajs-3828	340	3	→	→	SYM
iajs-3828	340	4	∞	∞	PROPN
iajs-3828	340	5	and	and	CCONJ
iajs-3828	340	6	using	use	VERB
iajs-3828	340	7	(	(	PUNCT
iajs-3828	340	8	27	27	NUM
iajs-3828	340	9	)	)	PUNCT
iajs-3828	340	10	and	and	CCONJ
iajs-3828	340	11	(	(	PUNCT
iajs-3828	340	12	26	26	NUM
iajs-3828	340	13	)	)	PUNCT
iajs-3828	340	14	,	,	PUNCT
iajs-3828	340	15	getting	get	VERB
iajs-3828	340	16	lim	lim	PROPN
iajs-3828	340	17	𝑟→∞	𝑟→∞	NUM
iajs-3828	340	18	𝜓(𝜌(𝑎𝑑𝑟	𝜓(𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	340	19	,	,	PUNCT
iajs-3828	340	20	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	340	21	)	)	PUNCT
iajs-3828	340	22	)	)	PUNCT
iajs-3828	341	1	=	=	SYM
iajs-3828	341	2	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	341	3	(	(	PUNCT
iajs-3828	341	4	29	29	NUM
iajs-3828	341	5	)	)	PUNCT
iajs-3828	341	6	again	again	ADV
iajs-3828	341	7	,	,	PUNCT
iajs-3828	341	8	by	by	ADP
iajs-3828	341	9	using	use	VERB
iajs-3828	341	10	the	the	DET
iajs-3828	341	11	triangle	triangle	NOUN
iajs-3828	341	12	inequality	inequality	NOUN
iajs-3828	341	13	,	,	PUNCT
iajs-3828	341	14	therefore	therefore	ADV
iajs-3828	341	15	𝜓(𝜌	𝜓(𝜌	PROPN
iajs-3828	341	16	(	(	PUNCT
iajs-3828	341	17	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	341	18	,	,	PUNCT
iajs-3828	341	19	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	PROPN
iajs-3828	341	20	)	)	PUNCT
iajs-3828	341	21	)	)	PUNCT
iajs-3828	341	22	≤	≤	PUNCT
iajs-3828	342	1	𝜓(𝑞𝜌	𝜓(𝑞𝜌	X
iajs-3828	342	2	(	(	PUNCT
iajs-3828	342	3	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	342	4	,	,	PUNCT
iajs-3828	342	5	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	342	6	)	)	PUNCT
iajs-3828	342	7	)	)	PUNCT
iajs-3828	343	1	+	+	CCONJ
iajs-3828	343	2	𝜓(𝑞𝜌(𝑎𝑑𝑟	𝜓(𝑞𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	343	3	,	,	PUNCT
iajs-3828	343	4	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	NUM
iajs-3828	343	5	)	)	PUNCT
iajs-3828	343	6	)	)	PUNCT
iajs-3828	343	7	.	.	PUNCT
iajs-3828	344	1	letting	let	VERB
iajs-3828	344	2	𝑟	𝑟	NOUN
iajs-3828	344	3	→	→	SYM
iajs-3828	344	4	∞	∞	NUM
iajs-3828	344	5	in	in	ADP
iajs-3828	344	6	the	the	DET
iajs-3828	344	7	last	last	ADJ
iajs-3828	344	8	inequality	inequality	NOUN
iajs-3828	344	9	and	and	CCONJ
iajs-3828	344	10	using	use	VERB
iajs-3828	344	11	(	(	PUNCT
iajs-3828	344	12	2.26	2.26	NUM
iajs-3828	344	13	)	)	PUNCT
iajs-3828	344	14	and	and	CCONJ
iajs-3828	344	15	(	(	PUNCT
iajs-3828	344	16	2.29	2.29	NUM
iajs-3828	344	17	)	)	PUNCT
iajs-3828	344	18	,	,	PUNCT
iajs-3828	344	19	having	have	VERB
iajs-3828	344	20	lim	lim	PROPN
iajs-3828	344	21	𝑟→∞	𝑟→∞	NUM
iajs-3828	344	22	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	344	23	,	,	PUNCT
iajs-3828	344	24	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	NUM
iajs-3828	344	25	)	)	PUNCT
iajs-3828	344	26	)	)	PUNCT
iajs-3828	345	1	=	=	SYM
iajs-3828	345	2	𝑞𝜀.	𝑞𝜀.	PROPN
iajs-3828	345	3	(	(	PUNCT
iajs-3828	345	4	30	30	NUM
iajs-3828	345	5	)	)	PUNCT
iajs-3828	345	6	using	use	VERB
iajs-3828	345	7	(	(	PUNCT
iajs-3828	345	8	2.18	2.18	NUM
iajs-3828	345	9	)	)	PUNCT
iajs-3828	345	10	for	for	ADP
iajs-3828	345	11	𝑎	𝑎	PROPN
iajs-3828	345	12	=	=	SYM
iajs-3828	345	13	𝑎𝑝𝑟−𝑖𝑟	𝑎𝑝𝑟−𝑖𝑟	NOUN
iajs-3828	345	14	and	and	CCONJ
iajs-3828	345	15	𝑏	𝑏	NOUN
iajs-3828	345	16	=	=	PUNCT
iajs-3828	345	17	𝑎𝑑𝑟	𝑎𝑑𝑟	ADJ
iajs-3828	345	18	,	,	PUNCT
iajs-3828	345	19	subsequently	subsequently	ADV
iajs-3828	345	20	𝑀(𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1	𝑀(𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟+1	NUM
iajs-3828	345	21	,	,	PUNCT
iajs-3828	345	22	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	NUM
iajs-3828	345	23	)	)	PUNCT
iajs-3828	345	24	)	)	PUNCT
iajs-3828	345	25	,	,	PUNCT
iajs-3828	345	26	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	345	27	,	,	PUNCT
iajs-3828	345	28	𝑎𝑑𝑟	𝑎𝑑𝑟	NOUN
iajs-3828	345	29	)	)	PUNCT
iajs-3828	345	30	)	)	PUNCT
iajs-3828	345	31	,	,	PUNCT
iajs-3828	345	32	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	𝜓(𝜌(𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	345	33	,	,	PUNCT
iajs-3828	345	34	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	345	35	)	)	PUNCT
iajs-3828	345	36	)	)	PUNCT
iajs-3828	345	37	,	,	PUNCT
iajs-3828	345	38	ihjpas	ihjpas	PROPN
iajs-3828	345	39	.	.	PUNCT
iajs-3828	346	1	2025,38(2	2025,38(2	PROPN
iajs-3828	346	2	)	)	PUNCT
iajs-3828	346	3	385	385	NUM
iajs-3828	346	4	𝜓(𝜌(𝑎𝑑𝑟	𝜓(𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	346	5	,	,	PUNCT
iajs-3828	346	6	𝑎𝑑𝑟+1	𝑎𝑑𝑟+1	PROPN
iajs-3828	346	7	)	)	PUNCT
iajs-3828	346	8	)	)	PUNCT
iajs-3828	346	9	,	,	PUNCT
iajs-3828	346	10	𝜓(𝜌(𝑎𝑑𝑟+1	𝜓(𝜌(𝑎𝑑𝑟+1	AUX
iajs-3828	346	11	,	,	PUNCT
iajs-3828	346	12	𝑎𝑝𝑟−𝑖𝑟	𝑎𝑝𝑟−𝑖𝑟	PROPN
iajs-3828	346	13	)	)	PUNCT
iajs-3828	346	14	)	)	PUNCT
iajs-3828	346	15	,	,	PUNCT
iajs-3828	346	16	𝜓(𝜌(𝑎𝑑𝑟	𝜓(𝜌(𝑎𝑑𝑟	PROPN
iajs-3828	346	17	,	,	PUNCT
iajs-3828	346	18	𝑎𝑝𝑟−𝑖𝑟+1	𝑎𝑝𝑟−𝑖𝑟+1	PROPN
iajs-3828	346	19	)	)	PUNCT
iajs-3828	346	20	)	)	PUNCT
iajs-3828	346	21	)	)	PUNCT
iajs-3828	347	1	≤	≤	ADV
iajs-3828	347	2	0	0	X
iajs-3828	347	3	.	.	PUNCT
iajs-3828	348	1	letting	let	VERB
iajs-3828	348	2	𝑟	𝑟	NOUN
iajs-3828	348	3	→	→	SYM
iajs-3828	348	4	∞	∞	PROPN
iajs-3828	348	5	,	,	PUNCT
iajs-3828	348	6	and	and	CCONJ
iajs-3828	348	7	using	use	VERB
iajs-3828	348	8	(	(	PUNCT
iajs-3828	348	9	30	30	NUM
iajs-3828	348	10	)	)	PUNCT
iajs-3828	348	11	,	,	PUNCT
iajs-3828	348	12	(	(	PUNCT
iajs-3828	348	13	27	27	NUM
iajs-3828	348	14	)	)	PUNCT
iajs-3828	348	15	,	,	PUNCT
iajs-3828	348	16	(	(	PUNCT
iajs-3828	348	17	26	26	NUM
iajs-3828	348	18	)	)	PUNCT
iajs-3828	348	19	,	,	PUNCT
iajs-3828	348	20	(	(	PUNCT
iajs-3828	348	21	28	28	NUM
iajs-3828	348	22	)	)	PUNCT
iajs-3828	348	23	,	,	PUNCT
iajs-3828	348	24	(	(	PUNCT
iajs-3828	348	25	29	29	NUM
iajs-3828	348	26	)	)	PUNCT
iajs-3828	348	27	,	,	PUNCT
iajs-3828	348	28	then	then	ADV
iajs-3828	348	29	,	,	PUNCT
iajs-3828	348	30	by	by	ADP
iajs-3828	348	31	continuity	continuity	NOUN
iajs-3828	348	32	of	of	ADP
iajs-3828	348	33	𝑀	𝑀	PROPN
iajs-3828	348	34	and	and	CCONJ
iajs-3828	348	35	𝜓(𝑟	𝜓(𝑟	NUM
iajs-3828	348	36	)	)	PUNCT
iajs-3828	348	37	=	=	SYM
iajs-3828	348	38	0	0	PUNCT
iajs-3828	349	1	if	if	SCONJ
iajs-3828	349	2	and	and	CCONJ
iajs-3828	349	3	only	only	ADV
iajs-3828	349	4	if	if	SCONJ
iajs-3828	349	5	𝑟	𝑟	NOUN
iajs-3828	349	6	=	=	SYM
iajs-3828	349	7	0	0	NUM
iajs-3828	349	8	,	,	PUNCT
iajs-3828	349	9	𝑀(𝑞𝜀	𝑀(𝑞𝜀	NOUN
iajs-3828	349	10	,	,	PUNCT
iajs-3828	349	11	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	349	12	,	,	PUNCT
iajs-3828	349	13	0,0	0,0	NOUN
iajs-3828	349	14	,	,	PUNCT
iajs-3828	349	15	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	349	16	,	,	PUNCT
iajs-3828	349	17	𝑞𝜀	𝑞𝜀	NOUN
iajs-3828	349	18	)	)	PUNCT
iajs-3828	349	19	≤	≤	NOUN
iajs-3828	349	20	0	0	NUM
iajs-3828	349	21	,	,	PUNCT
iajs-3828	349	22	a	a	DET
iajs-3828	349	23	contradiction	contradiction	NOUN
iajs-3828	349	24	with	with	ADP
iajs-3828	349	25	𝑀3	𝑀3	NOUN
iajs-3828	349	26	.	.	PUNCT
iajs-3828	350	1	thus	thus	ADV
iajs-3828	350	2	,	,	PUNCT
iajs-3828	350	3	(	(	PUNCT
iajs-3828	350	4	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	350	5	)	)	PUNCT
iajs-3828	350	6	is	be	AUX
iajs-3828	350	7	cauchy	cauchy	ADJ
iajs-3828	350	8	sequence	sequence	NOUN
iajs-3828	350	9	in	in	ADP
iajs-3828	350	10	(	(	PUNCT
iajs-3828	350	11	£	£	NOUN
iajs-3828	350	12	,	,	PUNCT
iajs-3828	350	13	𝜌	𝜌	ADP
iajs-3828	350	14	)	)	PUNCT
iajs-3828	350	15	.	.	PUNCT
iajs-3828	351	1	now	now	ADV
iajs-3828	351	2	to	to	PART
iajs-3828	351	3	prove	prove	VERB
iajs-3828	351	4	that	that	SCONJ
iajs-3828	351	5	𝑐	𝑐	PROPN
iajs-3828	351	6	is	be	AUX
iajs-3828	351	7	fixed	fix	VERB
iajs-3828	351	8	point	point	NOUN
iajs-3828	351	9	of	of	ADP
iajs-3828	351	10	𝑓.	𝑓.	NOUN
iajs-3828	351	11	in	in	ADP
iajs-3828	351	12	fact	fact	NOUN
iajs-3828	351	13	𝑓𝑎𝑚	𝑓𝑎𝑚	NOUN
iajs-3828	351	14	→	→	SYM
iajs-3828	351	15	𝑐	𝑐	PROPN
iajs-3828	351	16	and	and	CCONJ
iajs-3828	351	17	since	since	SCONJ
iajs-3828	351	18	£	£	NOUN
iajs-3828	351	19	=	=	SYM
iajs-3828	351	20	⋃	⋃	ADP
iajs-3828	351	21	£	£	SYM
iajs-3828	351	22	𝑖	𝑖	SYM
iajs-3828	351	23	𝑛	𝑛	PRON
iajs-3828	351	24	𝑖=1	𝑖=1	PROPN
iajs-3828	351	25	is	be	AUX
iajs-3828	351	26	cyclic	cyclic	ADJ
iajs-3828	351	27	representation	representation	NOUN
iajs-3828	351	28	of	of	ADP
iajs-3828	351	29	£	£	SYM
iajs-3828	351	30	w.r.t	w.r.t	NOUN
iajs-3828	351	31	.	.	PUNCT
iajs-3828	351	32	,	,	PUNCT
iajs-3828	351	33	𝑓	𝑓	X
iajs-3828	351	34	,	,	PUNCT
iajs-3828	351	35	the	the	DET
iajs-3828	351	36	sequence	sequence	NOUN
iajs-3828	351	37	(	(	PUNCT
iajs-3828	351	38	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	351	39	)	)	PUNCT
iajs-3828	351	40	has	have	AUX
iajs-3828	351	41	infinite	infinite	ADJ
iajs-3828	351	42	terms	term	NOUN
iajs-3828	351	43	in	in	ADP
iajs-3828	351	44	each	each	DET
iajs-3828	351	45	£	£	PROPN
iajs-3828	351	46	𝑖𝑚for	𝑖𝑚for	ADP
iajs-3828	351	47	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	351	48	∈	∈	PROPN
iajs-3828	351	49	{	{	PUNCT
iajs-3828	351	50	1,2	1,2	NUM
iajs-3828	351	51	,	,	PUNCT
iajs-3828	351	52	…	…	PUNCT
iajs-3828	351	53	,	,	PUNCT
iajs-3828	351	54	𝑛	𝑛	NOUN
iajs-3828	351	55	}	}	PUNCT
iajs-3828	351	56	.	.	PUNCT
iajs-3828	352	1	considering	consider	VERB
iajs-3828	352	2	that	that	SCONJ
iajs-3828	352	3	£	£	SYM
iajs-3828	352	4	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	352	5	is	be	AUX
iajs-3828	352	6	closed	close	VERB
iajs-3828	352	7	for	for	ADP
iajs-3828	352	8	𝑖𝑚	𝑖𝑚	PROPN
iajs-3828	352	9	∈	∈	PROPN
iajs-3828	352	10	{	{	PUNCT
iajs-3828	352	11	1,2	1,2	NUM
iajs-3828	352	12	,	,	PUNCT
iajs-3828	352	13	…	…	PUNCT
iajs-3828	352	14	,	,	PUNCT
iajs-3828	352	15	𝑛	𝑛	X
iajs-3828	352	16	}	}	PUNCT
iajs-3828	352	17	we	we	PRON
iajs-3828	352	18	have	have	VERB
iajs-3828	352	19	𝑐	𝑐	PROPN
iajs-3828	352	20	∈	∈	PROPN
iajs-3828	352	21	⋂	⋂	PROPN
iajs-3828	352	22	£	£	SYM
iajs-3828	352	23	𝑖	𝑖	SYM
iajs-3828	352	24	𝑛	𝑛	PRON
iajs-3828	352	25	𝑖=1	𝑖=1	PROPN
iajs-3828	352	26	.	.	PUNCT
iajs-3828	353	1	suppose	suppose	VERB
iajs-3828	353	2	that	that	SCONJ
iajs-3828	353	3	𝑐	𝑐	PROPN
iajs-3828	353	4	∈	∈	PROPN
iajs-3828	353	5	£	£	SYM
iajs-3828	353	6	𝑖	𝑖	NOUN
iajs-3828	353	7	and	and	CCONJ
iajs-3828	353	8	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	353	9	∈	∈	PROPN
iajs-3828	353	10	£	£	SYM
iajs-3828	353	11	𝑖+1	𝑖+1	NUM
iajs-3828	353	12	,	,	PUNCT
iajs-3828	353	13	and	and	CCONJ
iajs-3828	353	14	take	take	VERB
iajs-3828	353	15	a	a	DET
iajs-3828	353	16	subsequence	subsequence	NOUN
iajs-3828	353	17	(	(	PUNCT
iajs-3828	353	18	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	353	19	)	)	PUNCT
iajs-3828	353	20	𝑟∈𝑁	𝑟∈𝑁	PROPN
iajs-3828	353	21	of	of	ADP
iajs-3828	353	22	(	(	PUNCT
iajs-3828	353	23	𝑎𝑚	𝑎𝑚	NOUN
iajs-3828	353	24	)	)	PUNCT
iajs-3828	353	25	with	with	ADP
iajs-3828	353	26	𝑎𝑚𝑟	𝑎𝑚𝑟	ADJ
iajs-3828	353	27	∈	∈	PROPN
iajs-3828	353	28	£	£	PRON
iajs-3828	353	29	𝑖−1	𝑖−1	PROPN
iajs-3828	353	30	,	,	PUNCT
iajs-3828	353	31	using	use	VERB
iajs-3828	353	32	(	(	PUNCT
iajs-3828	353	33	2.18	2.18	NUM
iajs-3828	353	34	)	)	PUNCT
iajs-3828	353	35	,	,	PUNCT
iajs-3828	353	36	take	take	VERB
iajs-3828	353	37	𝑎	𝑎	NOUN
iajs-3828	353	38	=	=	SYM
iajs-3828	353	39	𝑐	𝑐	PROPN
iajs-3828	353	40	and	and	CCONJ
iajs-3828	353	41	𝑏	𝑏	NOUN
iajs-3828	353	42	=	=	PUNCT
iajs-3828	353	43	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	353	44	,	,	PUNCT
iajs-3828	353	45	as	as	SCONJ
iajs-3828	353	46	result	result	VERB
iajs-3828	353	47	𝑀(𝜓(𝜌(𝑓𝑐	𝑀(𝜓(𝜌(𝑓𝑐	PROPN
iajs-3828	353	48	,	,	PUNCT
iajs-3828	353	49	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	353	50	)	)	PUNCT
iajs-3828	353	51	)	)	PUNCT
iajs-3828	353	52	,	,	PUNCT
iajs-3828	353	53	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	353	54	,	,	PUNCT
iajs-3828	353	55	𝑎𝑚𝑟	𝑎𝑚𝑟	ADV
iajs-3828	353	56	)	)	PUNCT
iajs-3828	353	57	)	)	PUNCT
iajs-3828	353	58	,	,	PUNCT
iajs-3828	353	59	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	353	60	,	,	PUNCT
iajs-3828	353	61	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	353	62	)	)	PUNCT
iajs-3828	353	63	)	)	PUNCT
iajs-3828	353	64	,	,	PUNCT
iajs-3828	353	65	𝜓(𝜌(𝑎𝑚𝑟	𝜓(𝜌(𝑎𝑚𝑟	ADJ
iajs-3828	353	66	,	,	PUNCT
iajs-3828	353	67	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	353	68	)	)	PUNCT
iajs-3828	353	69	)	)	PUNCT
iajs-3828	353	70	,	,	PUNCT
iajs-3828	353	71	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	353	72	,	,	PUNCT
iajs-3828	353	73	𝑓𝑎𝑚𝑟	𝑓𝑎𝑚𝑟	NOUN
iajs-3828	353	74	)	)	PUNCT
iajs-3828	353	75	)	)	PUNCT
iajs-3828	353	76	,	,	PUNCT
iajs-3828	353	77	𝜓(𝜌(𝑎𝑚𝑟	𝜓(𝜌(𝑎𝑚𝑟	NUM
iajs-3828	353	78	,	,	PUNCT
iajs-3828	353	79	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	353	80	)	)	PUNCT
iajs-3828	353	81	)	)	PUNCT
iajs-3828	353	82	)	)	PUNCT
iajs-3828	354	1	≤	≤	ADV
iajs-3828	354	2	0	0	X
iajs-3828	354	3	.	.	PUNCT
iajs-3828	355	1	taking	take	VERB
iajs-3828	355	2	𝑟	𝑟	NOUN
iajs-3828	355	3	→	→	SYM
iajs-3828	355	4	∞	∞	PROPN
iajs-3828	355	5	,	,	PUNCT
iajs-3828	355	6	hence	hence	ADV
iajs-3828	355	7	𝑀(𝜓(𝜌(𝑓𝑐	𝑀(𝜓(𝜌(𝑓𝑐	PROPN
iajs-3828	355	8	,	,	PUNCT
iajs-3828	355	9	𝑐	𝑐	NOUN
iajs-3828	355	10	)	)	PUNCT
iajs-3828	355	11	)	)	PUNCT
iajs-3828	355	12	,	,	PUNCT
iajs-3828	355	13	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	355	14	,	,	PUNCT
iajs-3828	355	15	𝑐	𝑐	NOUN
iajs-3828	355	16	)	)	PUNCT
iajs-3828	355	17	)	)	PUNCT
iajs-3828	355	18	,	,	PUNCT
iajs-3828	355	19	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	355	20	,	,	PUNCT
iajs-3828	355	21	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	355	22	)	)	PUNCT
iajs-3828	355	23	)	)	PUNCT
iajs-3828	355	24	,	,	PUNCT
iajs-3828	355	25	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	355	26	,	,	PUNCT
iajs-3828	355	27	𝑐	𝑐	NOUN
iajs-3828	355	28	)	)	PUNCT
iajs-3828	355	29	)	)	PUNCT
iajs-3828	355	30	,	,	PUNCT
iajs-3828	355	31	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	355	32	,	,	PUNCT
iajs-3828	355	33	𝑐	𝑐	NOUN
iajs-3828	355	34	)	)	PUNCT
iajs-3828	355	35	)	)	PUNCT
iajs-3828	355	36	,	,	PUNCT
iajs-3828	355	37	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	355	38	,	,	PUNCT
iajs-3828	355	39	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	355	40	)	)	PUNCT
iajs-3828	355	41	)	)	PUNCT
iajs-3828	355	42	)	)	PUNCT
iajs-3828	356	1	≤	≤	ADV
iajs-3828	356	2	0	0	NUM
iajs-3828	356	3	,	,	PUNCT
iajs-3828	356	4	and	and	CCONJ
iajs-3828	356	5	𝜓(𝑟	𝜓(𝑟	NUM
iajs-3828	356	6	)	)	PUNCT
iajs-3828	356	7	=	=	SYM
iajs-3828	356	8	0	0	PUNCT
iajs-3828	357	1	if	if	SCONJ
iajs-3828	357	2	and	and	CCONJ
iajs-3828	357	3	only	only	ADV
iajs-3828	357	4	if	if	SCONJ
iajs-3828	357	5	𝑟	𝑟	NOUN
iajs-3828	357	6	=	=	SYM
iajs-3828	357	7	0	0	NUM
iajs-3828	357	8	,	,	PUNCT
iajs-3828	357	9	then	then	ADV
iajs-3828	357	10	𝑀(𝜓(𝜌(𝑓𝑐	𝑀(𝜓(𝜌(𝑓𝑐	PROPN
iajs-3828	357	11	,	,	PUNCT
iajs-3828	357	12	𝑐)),0	𝑐)),0	PROPN
iajs-3828	357	13	,	,	PUNCT
iajs-3828	357	14	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	357	15	,	,	PUNCT
iajs-3828	357	16	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	357	17	)	)	PUNCT
iajs-3828	357	18	)	)	PUNCT
iajs-3828	357	19	,	,	PUNCT
iajs-3828	357	20	0,0	0,0	NOUN
iajs-3828	357	21	,	,	PUNCT
iajs-3828	357	22	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	357	23	,	,	PUNCT
iajs-3828	357	24	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	357	25	)	)	PUNCT
iajs-3828	357	26	)	)	PUNCT
iajs-3828	357	27	)	)	PUNCT
iajs-3828	358	1	≤	≤	ADV
iajs-3828	358	2	0	0	NUM
iajs-3828	358	3	,	,	PUNCT
iajs-3828	358	4	which	which	PRON
iajs-3828	358	5	is	be	AUX
iajs-3828	358	6	contradiction	contradiction	NOUN
iajs-3828	358	7	to	to	ADP
iajs-3828	358	8	𝑀3	𝑀3	NOUN
iajs-3828	358	9	.	.	PUNCT
iajs-3828	359	1	thus	thus	ADV
iajs-3828	359	2	𝜓(𝜌(𝑓𝑐	𝜓(𝜌(𝑓𝑐	NOUN
iajs-3828	359	3	,	,	PUNCT
iajs-3828	359	4	𝑐	𝑐	NOUN
iajs-3828	359	5	)	)	PUNCT
iajs-3828	359	6	)	)	PUNCT
iajs-3828	360	1	=	=	SYM
iajs-3828	360	2	0	0	PUNCT
iajs-3828	360	3	then	then	ADV
iajs-3828	360	4	𝜌(𝑓𝑐	𝜌(𝑓𝑐	NUM
iajs-3828	360	5	,	,	PUNCT
iajs-3828	360	6	𝑐	𝑐	NOUN
iajs-3828	360	7	)	)	PUNCT
iajs-3828	360	8	=	=	SYM
iajs-3828	360	9	0	0	NUM
iajs-3828	360	10	,	,	PUNCT
iajs-3828	360	11	then	then	ADV
iajs-3828	360	12	𝑓𝑐	𝑓𝑐	ADP
iajs-3828	360	13	=	=	PUNCT
iajs-3828	360	14	𝑐.	𝑐.	NOUN
iajs-3828	360	15	we	we	PRON
iajs-3828	360	16	obtain	obtain	VERB
iajs-3828	360	17	𝑐	𝑐	PROPN
iajs-3828	360	18	is	be	AUX
iajs-3828	360	19	a	a	DET
iajs-3828	360	20	fixed	fix	VERB
iajs-3828	360	21	point	point	NOUN
iajs-3828	360	22	of	of	ADP
iajs-3828	360	23	𝑓.	𝑓.	NOUN
iajs-3828	360	24	suppose	suppose	VERB
iajs-3828	360	25	that	that	SCONJ
iajs-3828	360	26	there	there	PRON
iajs-3828	360	27	are	be	VERB
iajs-3828	360	28	two	two	NUM
iajs-3828	360	29	distinct	distinct	ADJ
iajs-3828	360	30	points	point	NOUN
iajs-3828	360	31	𝑐	𝑐	NOUN
iajs-3828	360	32	,	,	PUNCT
iajs-3828	360	33	𝑤	𝑤	X
iajs-3828	360	34	with	with	ADP
iajs-3828	360	35	𝑐	𝑐	PROPN
iajs-3828	360	36	and	and	CCONJ
iajs-3828	360	37	𝑤	𝑤	PROPN
iajs-3828	360	38	are	be	AUX
iajs-3828	360	39	two	two	NUM
iajs-3828	360	40	fixed	fix	VERB
iajs-3828	360	41	points	point	NOUN
iajs-3828	360	42	of	of	ADP
iajs-3828	360	43	𝑓.	𝑓.	NOUN
iajs-3828	360	44	the	the	DET
iajs-3828	360	45	cyclic	cyclic	ADJ
iajs-3828	360	46	nature	nature	NOUN
iajs-3828	360	47	of	of	ADP
iajs-3828	360	48	𝑓	𝑓	PRON
iajs-3828	360	49	,	,	PUNCT
iajs-3828	360	50	also	also	ADV
iajs-3828	360	51	,	,	PUNCT
iajs-3828	360	52	the	the	DET
iajs-3828	360	53	fact	fact	NOUN
iajs-3828	360	54	𝑐	𝑐	NOUN
iajs-3828	360	55	,	,	PUNCT
iajs-3828	360	56	𝑤	𝑤	ADP
iajs-3828	360	57	∈	∈	NOUN
iajs-3828	361	1	£	£	NOUN
iajs-3828	361	2	=	=	PUNCT
iajs-3828	361	3	⋃	⋃	ADP
iajs-3828	361	4	£	£	SYM
iajs-3828	361	5	𝑖	𝑖	SYM
iajs-3828	361	6	𝑛	𝑛	PRON
iajs-3828	361	7	𝑖=1	𝑖=1	PROPN
iajs-3828	361	8	are	be	AUX
iajs-3828	361	9	fixed	fix	VERB
iajs-3828	361	10	points	point	NOUN
iajs-3828	361	11	of	of	ADP
iajs-3828	361	12	𝑓	𝑓	DET
iajs-3828	361	13	imply	imply	NOUN
iajs-3828	362	1	that	that	SCONJ
iajs-3828	362	2	𝑐	𝑐	NOUN
iajs-3828	362	3	,	,	PUNCT
iajs-3828	362	4	𝑤	𝑤	ADP
iajs-3828	362	5	∈	∈	PROPN
iajs-3828	362	6	⋂	⋂	PROPN
iajs-3828	362	7	£	£	SYM
iajs-3828	362	8	𝑖	𝑖	SYM
iajs-3828	362	9	𝑛	𝑛	PRON
iajs-3828	362	10	𝑖=1	𝑖=1	PUNCT
iajs-3828	362	11	and	and	CCONJ
iajs-3828	362	12	by	by	ADP
iajs-3828	362	13	(	(	PUNCT
iajs-3828	362	14	18	18	NUM
iajs-3828	362	15	)	)	PUNCT
iajs-3828	362	16	,	,	PUNCT
iajs-3828	362	17	obtaining	obtain	VERB
iajs-3828	362	18	𝑀(𝜓(𝜌(𝑓𝑐	𝑀(𝜓(𝜌(𝑓𝑐	PROPN
iajs-3828	362	19	,	,	PUNCT
iajs-3828	362	20	𝑓𝑤)),𝜓(𝜌(𝑐	𝑓𝑤)),𝜓(𝜌(𝑐	NOUN
iajs-3828	362	21	,	,	PUNCT
iajs-3828	362	22	𝑤	𝑤	NOUN
iajs-3828	362	23	)	)	PUNCT
iajs-3828	362	24	)	)	PUNCT
iajs-3828	362	25	,	,	PUNCT
iajs-3828	362	26	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	362	27	,	,	PUNCT
iajs-3828	362	28	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	362	29	)	)	PUNCT
iajs-3828	362	30	)	)	PUNCT
iajs-3828	362	31	,	,	PUNCT
iajs-3828	362	32	𝜓(𝜌(𝑤	𝜓(𝜌(𝑤	PROPN
iajs-3828	362	33	,	,	PUNCT
iajs-3828	362	34	𝑓𝑤)),𝜓(𝜌(𝑐	𝑓𝑤)),𝜓(𝜌(𝑐	NOUN
iajs-3828	362	35	,	,	PUNCT
iajs-3828	362	36	𝑓𝑤)),𝜓(𝜌(𝑤	𝑓𝑤)),𝜓(𝜌(𝑤	NOUN
iajs-3828	362	37	,	,	PUNCT
iajs-3828	362	38	𝑓𝑐	𝑓𝑐	NOUN
iajs-3828	362	39	)	)	PUNCT
iajs-3828	362	40	)	)	PUNCT
iajs-3828	362	41	)	)	PUNCT
iajs-3828	363	1	≤	≤	ADV
iajs-3828	363	2	0	0	X
iajs-3828	363	3	.	.	PUNCT
iajs-3828	364	1	since	since	SCONJ
iajs-3828	364	2	𝑐	𝑐	PROPN
iajs-3828	364	3	,	,	PUNCT
iajs-3828	364	4	𝑤	𝑤	PROPN
iajs-3828	364	5	are	be	AUX
iajs-3828	364	6	fixed	fix	VERB
iajs-3828	364	7	points	point	NOUN
iajs-3828	364	8	of	of	ADP
iajs-3828	364	9	𝑓	𝑓	PRON
iajs-3828	364	10	,	,	PUNCT
iajs-3828	364	11	𝜓(𝑟	𝜓(𝑟	NOUN
iajs-3828	364	12	)	)	PUNCT
iajs-3828	364	13	=	=	SYM
iajs-3828	365	1	0	0	PUNCT
iajs-3828	366	1	if	if	SCONJ
iajs-3828	366	2	and	and	CCONJ
iajs-3828	366	3	only	only	ADV
iajs-3828	366	4	if	if	SCONJ
iajs-3828	366	5	𝑟	𝑟	NOUN
iajs-3828	366	6	=	=	SYM
iajs-3828	366	7	0	0	NUM
iajs-3828	366	8	,	,	PUNCT
iajs-3828	366	9	we	we	PRON
iajs-3828	366	10	get	get	VERB
iajs-3828	366	11	𝑀(𝜓(𝜌(𝑐	𝑀(𝜓(𝜌(𝑐	PROPN
iajs-3828	366	12	,	,	PUNCT
iajs-3828	366	13	𝑤	𝑤	NOUN
iajs-3828	366	14	)	)	PUNCT
iajs-3828	366	15	)	)	PUNCT
iajs-3828	366	16	,	,	PUNCT
iajs-3828	366	17	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	366	18	,	,	PUNCT
iajs-3828	366	19	𝑤)),0,0	𝑤)),0,0	NOUN
iajs-3828	366	20	,	,	PUNCT
iajs-3828	366	21	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	366	22	,	,	PUNCT
iajs-3828	366	23	𝑓𝑤)),𝜓(𝜌(𝑤	𝑓𝑤)),𝜓(𝜌(𝑤	NOUN
iajs-3828	366	24	,	,	PUNCT
iajs-3828	366	25	𝑐	𝑐	NOUN
iajs-3828	366	26	)	)	PUNCT
iajs-3828	366	27	)	)	PUNCT
iajs-3828	366	28	)	)	PUNCT
iajs-3828	367	1	≤	≤	ADV
iajs-3828	367	2	0	0	NUM
iajs-3828	367	3	,	,	PUNCT
iajs-3828	367	4	which	which	PRON
iajs-3828	367	5	is	be	AUX
iajs-3828	367	6	a	a	DET
iajs-3828	367	7	contradiction	contradiction	NOUN
iajs-3828	367	8	to	to	ADP
iajs-3828	367	9	𝑀3	𝑀3	NOUN
iajs-3828	367	10	,	,	PUNCT
iajs-3828	367	11	then	then	ADV
iajs-3828	367	12	𝜓(𝜌(𝑐	𝜓(𝜌(𝑐	NOUN
iajs-3828	367	13	,	,	PUNCT
iajs-3828	367	14	𝑤	𝑤	NOUN
iajs-3828	367	15	)	)	PUNCT
iajs-3828	367	16	)	)	PUNCT
iajs-3828	368	1	=	=	SYM
iajs-3828	368	2	0	0	NUM
iajs-3828	368	3	,	,	PUNCT
iajs-3828	368	4	hence	hence	ADV
iajs-3828	368	5	𝜌(𝑐	𝜌(𝑐	NOUN
iajs-3828	368	6	,	,	PUNCT
iajs-3828	368	7	𝑤	𝑤	ADP
iajs-3828	368	8	)	)	PUNCT
iajs-3828	368	9	=	=	SYM
iajs-3828	368	10	0	0	NUM
iajs-3828	368	11	,	,	PUNCT
iajs-3828	368	12	that	that	ADV
iajs-3828	368	13	is	is	ADV
iajs-3828	368	14	,	,	PUNCT
iajs-3828	368	15	𝑐	𝑐	NOUN
iajs-3828	368	16	=	=	NOUN
iajs-3828	368	17	𝑤.	𝑤.	NOUN
iajs-3828	368	18	hence	hence	ADV
iajs-3828	368	19	𝑓	𝑓	ADV
iajs-3828	368	20	has	have	VERB
iajs-3828	368	21	a	a	DET
iajs-3828	368	22	unique	unique	ADJ
iajs-3828	368	23	fixed	fix	VERB
iajs-3828	368	24	point	point	NOUN
iajs-3828	368	25	in	in	ADP
iajs-3828	368	26	£	£	PROPN
iajs-3828	368	27	and	and	CCONJ
iajs-3828	368	28	𝑐	𝑐	PROPN
iajs-3828	368	29	∈	∈	PROPN
iajs-3828	368	30	⋂	⋂	PROPN
iajs-3828	368	31	£	£	SYM
iajs-3828	368	32	𝑖	𝑖	SYM
iajs-3828	368	33	𝑛	𝑛	PRON
iajs-3828	368	34	𝑖=1	𝑖=1	PROPN
iajs-3828	368	35	.	.	PUNCT
iajs-3828	369	1	remark	remark	PROPN
iajs-3828	369	2	2.10	2.10	NUM
iajs-3828	369	3	:	:	PUNCT
iajs-3828	369	4	it	it	PRON
iajs-3828	369	5	is	be	AUX
iajs-3828	369	6	worth	worth	ADJ
iajs-3828	369	7	noting	note	VERB
iajs-3828	369	8	,	,	PUNCT
iajs-3828	369	9	it	it	PRON
iajs-3828	369	10	is	be	AUX
iajs-3828	369	11	worth	worth	ADJ
iajs-3828	369	12	noting	note	VERB
iajs-3828	369	13	that	that	SCONJ
iajs-3828	369	14	we	we	PRON
iajs-3828	369	15	can	can	AUX
iajs-3828	369	16	obtain	obtain	VERB
iajs-3828	369	17	good	good	ADJ
iajs-3828	369	18	results	result	NOUN
iajs-3828	369	19	by	by	ADP
iajs-3828	369	20	including	include	VERB
iajs-3828	369	21	the	the	DET
iajs-3828	369	22	concept	concept	NOUN
iajs-3828	369	23	of	of	ADP
iajs-3828	369	24	cyclicity	cyclicity	NOUN
iajs-3828	369	25	in	in	ADP
iajs-3828	369	26	cases	case	NOUN
iajs-3828	369	27	of	of	ADP
iajs-3828	369	28	)	)	PUNCT
iajs-3828	369	29	28,29	28,29	PROPN
iajs-3828	369	30	(	(	PUNCT
iajs-3828	369	31	.	.	PUNCT
iajs-3828	370	1	3	3	X
iajs-3828	370	2	.	.	X
iajs-3828	370	3	discussion	discussion	NOUN
iajs-3828	370	4	this	this	DET
iajs-3828	370	5	work	work	NOUN
iajs-3828	370	6	is	be	AUX
iajs-3828	370	7	classified	classify	VERB
iajs-3828	370	8	within	within	ADP
iajs-3828	370	9	the	the	DET
iajs-3828	370	10	field	field	NOUN
iajs-3828	370	11	depending	depend	VERB
iajs-3828	370	12	on	on	ADP
iajs-3828	370	13	the	the	DET
iajs-3828	370	14	classification	classification	NOUN
iajs-3828	370	15	2010	2010	NUM
iajs-3828	370	16	msc	msc	NOUN
iajs-3828	370	17	:	:	PUNCT
iajs-3828	370	18	47h09	47h09	NUM
iajs-3828	370	19	,	,	PUNCT
iajs-3828	370	20	47h10	47h10	NUM
iajs-3828	370	21	.	.	PUNCT
iajs-3828	371	1	our	our	PRON
iajs-3828	371	2	study	study	NOUN
iajs-3828	371	3	of	of	ADP
iajs-3828	371	4	this	this	DET
iajs-3828	371	5	topic	topic	NOUN
iajs-3828	371	6	is	be	AUX
iajs-3828	371	7	the	the	DET
iajs-3828	371	8	first	first	ADJ
iajs-3828	371	9	in	in	ADP
iajs-3828	371	10	iraq	iraq	PROPN
iajs-3828	371	11	(	(	PUNCT
iajs-3828	371	12	to	to	ADP
iajs-3828	371	13	the	the	DET
iajs-3828	371	14	best	good	ADJ
iajs-3828	371	15	of	of	ADP
iajs-3828	371	16	our	our	PRON
iajs-3828	371	17	knowledge	knowledge	NOUN
iajs-3828	371	18	)	)	PUNCT
iajs-3828	371	19	in	in	ADP
iajs-3828	371	20	the	the	DET
iajs-3828	371	21	field	field	NOUN
iajs-3828	371	22	of	of	ADP
iajs-3828	371	23	fixed	fix	VERB
iajs-3828	371	24	points	point	NOUN
iajs-3828	371	25	for	for	ADP
iajs-3828	371	26	cyclic	cyclic	ADJ
iajs-3828	371	27	maps	map	NOUN
iajs-3828	371	28	,	,	PUNCT
iajs-3828	371	29	and	and	CCONJ
iajs-3828	371	30	it	it	PRON
iajs-3828	371	31	is	be	AUX
iajs-3828	371	32	taken	take	VERB
iajs-3828	371	33	from	from	ADP
iajs-3828	371	34	a	a	DET
iajs-3828	371	35	master	master	NOUN
iajs-3828	371	36	’s	’s	PART
iajs-3828	371	37	thesis	thesis	NOUN
iajs-3828	371	38	by	by	ADP
iajs-3828	371	39	researcher	researcher	PROPN
iajs-3828	371	40	abbas	abbas	PROPN
iajs-3828	371	41	karim	karim	PROPN
iajs-3828	371	42	nahi	nahi	PROPN
iajs-3828	371	43	.	.	PUNCT
iajs-3828	372	1	the	the	DET
iajs-3828	372	2	paper	paper	NOUN
iajs-3828	372	3	included	include	VERB
iajs-3828	372	4	new	new	ADJ
iajs-3828	372	5	results	result	NOUN
iajs-3828	372	6	in	in	ADP
iajs-3828	372	7	the	the	DET
iajs-3828	372	8	field	field	NOUN
iajs-3828	372	9	of	of	ADP
iajs-3828	372	10	integral	integral	ADJ
iajs-3828	372	11	contractions	contraction	NOUN
iajs-3828	372	12	in	in	ADP
iajs-3828	372	13	the	the	DET
iajs-3828	372	14	b	b	NOUN
iajs-3828	372	15	-	-	PUNCT
iajs-3828	372	16	metric	metric	ADJ
iajs-3828	372	17	spaces	space	NOUN
iajs-3828	372	18	of	of	ADP
iajs-3828	372	19	the	the	DET
iajs-3828	372	20	cyclic	cyclic	ADJ
iajs-3828	372	21	type	type	NOUN
iajs-3828	372	22	,	,	PUNCT
iajs-3828	372	23	as	as	ADV
iajs-3828	372	24	well	well	ADV
iajs-3828	372	25	as	as	ADP
iajs-3828	372	26	new	new	ADJ
iajs-3828	372	27	generalizations	generalization	NOUN
iajs-3828	372	28	of	of	ADP
iajs-3828	372	29	the	the	DET
iajs-3828	372	30	results	result	NOUN
iajs-3828	372	31	of	of	ADP
iajs-3828	372	32	other	other	ADJ
iajs-3828	372	33	researchers	researcher	NOUN
iajs-3828	372	34	in	in	ADP
iajs-3828	372	35	the	the	DET
iajs-3828	372	36	case	case	NOUN
iajs-3828	372	37	of	of	ADP
iajs-3828	372	38	the	the	DET
iajs-3828	372	39	b	b	NOUN
iajs-3828	372	40	-	-	PUNCT
iajs-3828	372	41	metric	metric	ADJ
iajs-3828	372	42	space	space	NOUN
iajs-3828	372	43	.	.	PUNCT
iajs-3828	373	1	in	in	ADP
iajs-3828	373	2	the	the	DET
iajs-3828	373	3	future	future	NOUN
iajs-3828	373	4	,	,	PUNCT
iajs-3828	373	5	we	we	PRON
iajs-3828	373	6	would	would	AUX
iajs-3828	373	7	like	like	VERB
iajs-3828	373	8	to	to	PART
iajs-3828	373	9	study	study	VERB
iajs-3828	373	10	the	the	DET
iajs-3828	373	11	results	result	NOUN
iajs-3828	373	12	in	in	ADP
iajs-3828	373	13	)	)	PUNCT
iajs-3828	373	14	30	30	NUM
iajs-3828	373	15	(	(	PUNCT
iajs-3828	373	16	in	in	ADP
iajs-3828	373	17	the	the	DET
iajs-3828	373	18	case	case	NOUN
iajs-3828	373	19	of	of	ADP
iajs-3828	373	20	cyclic	cyclic	ADJ
iajs-3828	373	21	maps	map	NOUN
iajs-3828	373	22	.	.	PUNCT
iajs-3828	374	1	4	4	X
iajs-3828	374	2	.	.	X
iajs-3828	374	3	conclusions	conclusion	NOUN
iajs-3828	374	4	in	in	ADP
iajs-3828	374	5	this	this	DET
iajs-3828	374	6	paper	paper	NOUN
iajs-3828	374	7	,	,	PUNCT
iajs-3828	374	8	new	new	ADJ
iajs-3828	374	9	theorems	theorem	NOUN
iajs-3828	374	10	were	be	AUX
iajs-3828	374	11	established	establish	VERB
iajs-3828	374	12	to	to	PART
iajs-3828	374	13	find	find	VERB
iajs-3828	374	14	fixed	fixed	ADJ
iajs-3828	374	15	points	point	NOUN
iajs-3828	374	16	.	.	PUNCT
iajs-3828	375	1	first	first	ADV
iajs-3828	375	2	,	,	PUNCT
iajs-3828	375	3	by	by	ADP
iajs-3828	375	4	merging	merge	VERB
iajs-3828	375	5	integral	integral	ADJ
iajs-3828	375	6	contractive	contractive	ADJ
iajs-3828	375	7	conditions	condition	NOUN
iajs-3828	375	8	with	with	ADP
iajs-3828	375	9	the	the	DET
iajs-3828	375	10	concept	concept	NOUN
iajs-3828	375	11	of	of	ADP
iajs-3828	375	12	cyclic	cyclic	ADJ
iajs-3828	375	13	map	map	NOUN
iajs-3828	375	14	,	,	PUNCT
iajs-3828	375	15	and	and	CCONJ
iajs-3828	375	16	second	second	ADJ
iajs-3828	375	17	,	,	PUNCT
iajs-3828	375	18	by	by	ADP
iajs-3828	375	19	applying	apply	VERB
iajs-3828	375	20	the	the	DET
iajs-3828	375	21	concept	concept	NOUN
iajs-3828	375	22	of	of	ADP
iajs-3828	375	23	cyclic	cyclic	ADJ
iajs-3828	375	24	representation	representation	NOUN
iajs-3828	375	25	with	with	ADP
iajs-3828	375	26	respect	respect	NOUN
iajs-3828	375	27	to	to	ADP
iajs-3828	375	28	maps	map	NOUN
iajs-3828	375	29	satisfying	satisfy	VERB
iajs-3828	375	30	general	general	ADJ
iajs-3828	375	31	weak	weak	ADJ
iajs-3828	375	32	conditions	condition	NOUN
iajs-3828	375	33	,	,	PUNCT
iajs-3828	375	34	including	include	VERB
iajs-3828	375	35	a	a	DET
iajs-3828	375	36	changing	change	VERB
iajs-3828	375	37	distance	distance	NOUN
iajs-3828	375	38	function	function	NOUN
iajs-3828	375	39	.	.	PUNCT
iajs-3828	376	1	and	and	CCONJ
iajs-3828	376	2	third	third	ADV
iajs-3828	376	3	,	,	PUNCT
iajs-3828	376	4	by	by	ADP
iajs-3828	376	5	merging	merge	VERB
iajs-3828	376	6	the	the	DET
iajs-3828	376	7	changing	change	VERB
iajs-3828	376	8	distance	distance	NOUN
iajs-3828	376	9	function	function	NOUN
iajs-3828	376	10	with	with	ADP
iajs-3828	376	11	the	the	DET
iajs-3828	376	12	sub	sub	NOUN
iajs-3828	376	13	additive	additive	NOUN
iajs-3828	376	14	to	to	PART
iajs-3828	376	15	find	find	VERB
iajs-3828	376	16	theorems	theorem	NOUN
iajs-3828	376	17	in	in	ADP
iajs-3828	376	18	the	the	DET
iajs-3828	376	19	field	field	NOUN
iajs-3828	376	20	of	of	ADP
iajs-3828	376	21	fixed	fix	VERB
iajs-3828	376	22	points	point	NOUN
iajs-3828	376	23	through	through	ADP
iajs-3828	376	24	the	the	DET
iajs-3828	376	25	concept	concept	NOUN
iajs-3828	376	26	of	of	ADP
iajs-3828	376	27	cyclic	cyclic	ADJ
iajs-3828	376	28	representation	representation	NOUN
iajs-3828	376	29	for	for	ADP
iajs-3828	376	30	maps	map	NOUN
iajs-3828	376	31	satisfying	satisfy	VERB
iajs-3828	376	32	an	an	DET
iajs-3828	376	33	implicit	implicit	ADJ
iajs-3828	376	34	relation	relation	NOUN
iajs-3828	376	35	including	include	VERB
iajs-3828	376	36	a	a	DET
iajs-3828	376	37	changing	change	VERB
iajs-3828	376	38	distance	distance	NOUN
iajs-3828	376	39	function	function	NOUN
iajs-3828	376	40	.	.	PUNCT
iajs-3828	377	1	ihjpas	ihjpas	PROPN
iajs-3828	377	2	.	.	PUNCT
iajs-3828	378	1	2025,38(2	2025,38(2	NOUN
iajs-3828	378	2	)	)	PUNCT
iajs-3828	378	3	386	386	NUM
iajs-3828	378	4	acknowledgments	acknowledgment	NOUN
iajs-3828	378	5	i	i	PRON
iajs-3828	378	6	would	would	AUX
iajs-3828	378	7	like	like	VERB
iajs-3828	378	8	to	to	PART
iajs-3828	378	9	thank	thank	VERB
iajs-3828	378	10	the	the	DET
iajs-3828	378	11	previous	previous	ADJ
iajs-3828	378	12	researchers	researcher	NOUN
iajs-3828	378	13	whose	whose	DET
iajs-3828	378	14	results	result	NOUN
iajs-3828	378	15	have	have	AUX
iajs-3828	378	16	played	play	VERB
iajs-3828	378	17	an	an	DET
iajs-3828	378	18	important	important	ADJ
iajs-3828	378	19	role	role	NOUN
iajs-3828	378	20	in	in	ADP
iajs-3828	378	21	obtaining	obtain	VERB
iajs-3828	378	22	our	our	PRON
iajs-3828	378	23	current	current	ADJ
iajs-3828	378	24	results	result	NOUN
iajs-3828	378	25	.	.	PUNCT
iajs-3828	379	1	especially	especially	ADV
iajs-3828	379	2	professors	professor	NOUN
iajs-3828	379	3	w.	w.	PROPN
iajs-3828	379	4	a.	a.	PROPN
iajs-3828	379	5	kirk	kirk	PROPN
iajs-3828	379	6	,	,	PUNCT
iajs-3828	379	7	p.s	p.s	PROPN
iajs-3828	379	8	.	.	PROPN
iajs-3828	379	9	srinivasan	srinivasan	PROPN
iajs-3828	379	10	and	and	CCONJ
iajs-3828	379	11	p.	p.	PROPN
iajs-3828	379	12	veeramani	veeramani	PROPN
iajs-3828	379	13	.	.	PUNCT
iajs-3828	380	1	conflict	conflict	NOUN
iajs-3828	380	2	of	of	ADP
iajs-3828	380	3	interest	interest	NOUN
iajs-3828	380	4	the	the	DET
iajs-3828	380	5	authors	author	NOUN
iajs-3828	380	6	declare	declare	VERB
iajs-3828	380	7	no	no	DET
iajs-3828	380	8	conflict	conflict	NOUN
iajs-3828	380	9	of	of	ADP
iajs-3828	380	10	interest	interest	NOUN
iajs-3828	380	11	.	.	PUNCT
iajs-3828	381	1	funding	fund	VERB
iajs-3828	381	2	this	this	DET
iajs-3828	381	3	research	research	NOUN
iajs-3828	381	4	received	receive	VERB
iajs-3828	381	5	no	no	DET
iajs-3828	381	6	external	external	ADJ
iajs-3828	381	7	funding	funding	NOUN
iajs-3828	381	8	.	.	PUNCT
iajs-3828	382	1	references	reference	NOUN
iajs-3828	382	2	1	1	NUM
iajs-3828	382	3	.	.	PUNCT
iajs-3828	382	4	kirk	kirk	PROPN
iajs-3828	382	5	wa	wa	PROPN
iajs-3828	382	6	,	,	PUNCT
iajs-3828	382	7	srinivasan	srinivasan	PROPN
iajs-3828	382	8	ps	ps	PROPN
iajs-3828	382	9	,	,	PUNCT
iajs-3828	382	10	veeramani	veeramani	NOUN
iajs-3828	383	1	p.	p.	PROPN
iajs-3828	383	2	fixed	fix	VERB
iajs-3828	383	3	points	point	NOUN
iajs-3828	383	4	for	for	ADP
iajs-3828	383	5	mappings	mapping	NOUN
iajs-3828	383	6	satisfying	satisfy	VERB
iajs-3828	383	7	cyclical	cyclical	ADJ
iajs-3828	383	8	contractive	contractive	ADJ
iajs-3828	383	9	conditions	condition	NOUN
iajs-3828	383	10	.	.	PUNCT
iajs-3828	384	1	fixed	fix	VERB
iajs-3828	384	2	point	point	NOUN
iajs-3828	384	3	theory	theory	NOUN
iajs-3828	384	4	.	.	PUNCT
iajs-3828	385	1	2003;4(1):79	2003;4(1):79	NUM
iajs-3828	385	2	-	-	SYM
iajs-3828	385	3	89	89	NUM
iajs-3828	385	4	.	.	PUNCT
iajs-3828	386	1	2	2	NUM
iajs-3828	386	2	.	.	X
iajs-3828	386	3	asem	asem	PROPN
iajs-3828	386	4	v	v	PROPN
iajs-3828	386	5	,	,	PUNCT
iajs-3828	386	6	singh	singh	PROPN
iajs-3828	386	7	ym	ym	PROPN
iajs-3828	386	8	,	,	PUNCT
iajs-3828	386	9	khan	khan	PROPN
iajs-3828	386	10	ms	ms	PROPN
iajs-3828	386	11	,	,	PUNCT
iajs-3828	386	12	sessa	sessa	PROPN
iajs-3828	386	13	s.	s.	PROPN
iajs-3828	386	14	on	on	ADP
iajs-3828	386	15	(	(	PUNCT
iajs-3828	386	16	α	α	X
iajs-3828	386	17	,	,	PUNCT
iajs-3828	386	18	p)-cyclic	p)-cyclic	ADJ
iajs-3828	386	19	contractions	contraction	NOUN
iajs-3828	386	20	and	and	CCONJ
iajs-3828	386	21	related	relate	VERB
iajs-3828	386	22	fixed	fix	VERB
iajs-3828	386	23	point	point	NOUN
iajs-3828	386	24	theorems	theorem	NOUN
iajs-3828	386	25	.	.	PUNCT
iajs-3828	386	26	symmetry	symmetry	PROPN
iajs-3828	386	27	.	.	PUNCT
iajs-3828	387	1	2023;15(10):1826	2023;15(10):1826	NUM
iajs-3828	387	2	.	.	NOUN
iajs-3828	388	1	http://doi.org/10.3390/sym15101826	http://doi.org/10.3390/sym15101826	PRON
iajs-3828	388	2	3	3	X
iajs-3828	388	3	.	.	PUNCT
iajs-3828	388	4	konar	konar	PROPN
iajs-3828	388	5	p	p	PROPN
iajs-3828	388	6	,	,	PUNCT
iajs-3828	388	7	bhandari	bhandari	PROPN
iajs-3828	388	8	sk	sk	PROPN
iajs-3828	388	9	,	,	PUNCT
iajs-3828	388	10	chandok	chandok	PROPN
iajs-3828	388	11	s	s	PROPN
iajs-3828	388	12	,	,	PUNCT
iajs-3828	388	13	mukheimer	mukheimer	PROPN
iajs-3828	388	14	a.	a.	NOUN
iajs-3828	388	15	multivalued	multivalue	VERB
iajs-3828	388	16	weak	weak	ADJ
iajs-3828	388	17	cyclic	cyclic	ADJ
iajs-3828	388	18	δ	δ	PROPN
iajs-3828	388	19	-	-	PUNCT
iajs-3828	388	20	contraction	contraction	NOUN
iajs-3828	388	21	mappings	mapping	NOUN
iajs-3828	388	22	.	.	PUNCT
iajs-3828	389	1	journal	journal	PROPN
iajs-3828	389	2	of	of	ADP
iajs-3828	389	3	inequalities	inequality	NOUN
iajs-3828	389	4	and	and	CCONJ
iajs-3828	389	5	applications	application	NOUN
iajs-3828	389	6	.	.	PUNCT
iajs-3828	390	1	2020;2020(1):111	2020;2020(1):111	X
iajs-3828	390	2	.	.	PUNCT
iajs-3828	391	1	http://doi.org/10.1186/s13660-020-02373-1	http://doi.org/10.1186/s13660-020-02373-1	NUM
iajs-3828	391	2	4	4	X
iajs-3828	391	3	.	.	X
iajs-3828	391	4	karapinar	karapinar	PROPN
iajs-3828	391	5	e	e	PROPN
iajs-3828	391	6	,	,	PUNCT
iajs-3828	391	7	sadarangani	sadarangani	PROPN
iajs-3828	391	8	k.	k.	PROPN
iajs-3828	391	9	fixed	fix	VERB
iajs-3828	391	10	point	point	NOUN
iajs-3828	391	11	theory	theory	NOUN
iajs-3828	391	12	for	for	ADP
iajs-3828	391	13	cyclic	cyclic	ADJ
iajs-3828	391	14	(	(	PUNCT
iajs-3828	391	15	ϕ-ψ)-contractions	ϕ-ψ)-contraction	NOUN
iajs-3828	391	16	.	.	PUNCT
iajs-3828	392	1	fixed	fix	VERB
iajs-3828	392	2	point	point	NOUN
iajs-3828	392	3	theory	theory	NOUN
iajs-3828	392	4	and	and	CCONJ
iajs-3828	392	5	applications	application	NOUN
iajs-3828	392	6	.	.	PUNCT
iajs-3828	393	1	2011;2011(1):1	2011;2011(1):1	NUM
iajs-3828	393	2	-	-	SYM
iajs-3828	393	3	8	8	NUM
iajs-3828	393	4	.	.	PUNCT
iajs-3828	394	1	http://doi.org/10.1186/1687-1812-2011-1	http://doi.org/10.1186/1687-1812-2011-1	NOUN
iajs-3828	394	2	5	5	NUM
iajs-3828	394	3	.	.	PUNCT
iajs-3828	394	4	hussain	hussain	PROPN
iajs-3828	394	5	n	n	CCONJ
iajs-3828	394	6	,	,	PUNCT
iajs-3828	394	7	parvaneh	parvaneh	PROPN
iajs-3828	394	8	v	v	NOUN
iajs-3828	394	9	,	,	PUNCT
iajs-3828	394	10	roshan	roshan	PROPN
iajs-3828	394	11	jr	jr	PROPN
iajs-3828	394	12	,	,	PUNCT
iajs-3828	394	13	kadelburg	kadelburg	PROPN
iajs-3828	394	14	z.	z.	PROPN
iajs-3828	394	15	fixed	fix	VERB
iajs-3828	394	16	points	point	NOUN
iajs-3828	394	17	of	of	ADP
iajs-3828	394	18	cyclic	cyclic	ADJ
iajs-3828	394	19	weakly	weakly	ADJ
iajs-3828	394	20	(	(	PUNCT
iajs-3828	394	21	ψ	ψ	NOUN
iajs-3828	394	22	,	,	PUNCT
iajs-3828	394	23	φ	φ	PROPN
iajs-3828	394	24	,	,	PUNCT
iajs-3828	394	25	l	l	PROPN
iajs-3828	394	26	,	,	PUNCT
iajs-3828	394	27	a	a	DET
iajs-3828	394	28	,	,	PUNCT
iajs-3828	394	29	b)contractive	b)contractive	ADJ
iajs-3828	394	30	mappings	mapping	NOUN
iajs-3828	394	31	in	in	ADP
iajs-3828	394	32	ordered	order	VERB
iajs-3828	394	33	b	b	X
iajs-3828	394	34	-	-	ADJ
iajs-3828	394	35	metric	metric	ADJ
iajs-3828	394	36	spaces	space	NOUN
iajs-3828	394	37	with	with	ADP
iajs-3828	394	38	applications	application	NOUN
iajs-3828	394	39	.	.	PUNCT
iajs-3828	395	1	fixed	fix	VERB
iajs-3828	395	2	point	point	NOUN
iajs-3828	395	3	theory	theory	NOUN
iajs-3828	395	4	and	and	CCONJ
iajs-3828	395	5	applications	application	NOUN
iajs-3828	395	6	.	.	PUNCT
iajs-3828	396	1	2013;2013(1):1	2013;2013(1):1	NUM
iajs-3828	396	2	-	-	SYM
iajs-3828	396	3	18	18	NUM
iajs-3828	396	4	.	.	PUNCT
iajs-3828	397	1	http://doi.org/10.1186/1687-1812-2013-1	http://doi.org/10.1186/1687-1812-2013-1	PROPN
iajs-3828	397	2	6	6	NUM
iajs-3828	397	3	.	.	PUNCT
iajs-3828	398	1	hiranmoy	hiranmoy	NOUN
iajs-3828	398	2	g	g	PROPN
iajs-3828	398	3	,	,	PUNCT
iajs-3828	398	4	karapınar	karapınar	PROPN
iajs-3828	398	5	e	e	NOUN
iajs-3828	398	6	,	,	PUNCT
iajs-3828	398	7	kanta	kanta	PROPN
iajs-3828	398	8	dey	dey	PROPN
iajs-3828	398	9	l.	l.	PROPN
iajs-3828	398	10	best	good	ADJ
iajs-3828	398	11	proximity	proximity	NOUN
iajs-3828	398	12	point	point	NOUN
iajs-3828	398	13	results	result	NOUN
iajs-3828	398	14	for	for	ADP
iajs-3828	398	15	contractive	contractive	ADJ
iajs-3828	398	16	and	and	CCONJ
iajs-3828	398	17	cyclic	cyclic	ADJ
iajs-3828	398	18	contractive	contractive	ADJ
iajs-3828	398	19	type	type	NOUN
iajs-3828	398	20	mappings	mapping	NOUN
iajs-3828	398	21	.	.	PUNCT
iajs-3828	399	1	numerical	numerical	ADJ
iajs-3828	399	2	functional	functional	ADJ
iajs-3828	399	3	analysis	analysis	NOUN
iajs-3828	399	4	and	and	CCONJ
iajs-3828	399	5	optimization	optimization	NOUN
iajs-3828	399	6	.	.	PUNCT
iajs-3828	400	1	2021;42(7):849864	2021;42(7):849864	NUM
iajs-3828	400	2	.	.	PUNCT
iajs-3828	400	3	http://doi.org/10.1080/01630563.2021.1929578	http://doi.org/10.1080/01630563.2021.1929578	PROPN
iajs-3828	401	1	7	7	X
iajs-3828	401	2	.	.	PUNCT
iajs-3828	401	3	chaipunya	chaipunya	PROPN
iajs-3828	401	4	p	p	PROPN
iajs-3828	401	5	,	,	PUNCT
iajs-3828	401	6	cho	cho	PROPN
iajs-3828	401	7	yj	yj	PROPN
iajs-3828	401	8	,	,	PUNCT
iajs-3828	401	9	sintunavarat	sintunavarat	PROPN
iajs-3828	401	10	w	w	PROPN
iajs-3828	401	11	,	,	PUNCT
iajs-3828	401	12	kumam	kumam	PROPN
iajs-3828	401	13	p.	p.	PROPN
iajs-3828	401	14	fixed	fix	VERB
iajs-3828	401	15	point	point	NOUN
iajs-3828	401	16	and	and	CCONJ
iajs-3828	401	17	common	common	ADJ
iajs-3828	401	18	fixed	fix	VERB
iajs-3828	401	19	point	point	NOUN
iajs-3828	401	20	theorems	theorem	NOUN
iajs-3828	401	21	for	for	ADP
iajs-3828	401	22	cyclic	cyclic	ADJ
iajs-3828	401	23	quasi	quasi	NOUN
iajs-3828	401	24	-	-	NOUN
iajs-3828	401	25	contractions	contraction	NOUN
iajs-3828	401	26	in	in	ADP
iajs-3828	401	27	metric	metric	ADJ
iajs-3828	401	28	and	and	CCONJ
iajs-3828	401	29	ultrametric	ultrametric	ADJ
iajs-3828	401	30	spaces	space	NOUN
iajs-3828	401	31	.	.	PUNCT
iajs-3828	402	1	advances	advance	NOUN
iajs-3828	402	2	in	in	ADP
iajs-3828	402	3	pure	pure	ADJ
iajs-3828	402	4	mathematics	mathematic	NOUN
iajs-3828	402	5	.	.	PUNCT
iajs-3828	403	1	2012;2(6):401	2012;2(6):401	NUM
iajs-3828	403	2	-	-	SYM
iajs-3828	403	3	407	407	NUM
iajs-3828	403	4	.	.	PUNCT
iajs-3828	404	1	http://doi.org/10.4236/apm.2012.26059	http://doi.org/10.4236/apm.2012.26059	PROPN
iajs-3828	404	2	8	8	NUM
iajs-3828	404	3	.	.	PUNCT
iajs-3828	404	4	abou	abou	PROPN
iajs-3828	404	5	bakr	bakr	PROPN
iajs-3828	404	6	,	,	PUNCT
iajs-3828	404	7	sahar	sahar	PROPN
iajs-3828	404	8	ma	ma	PROPN
iajs-3828	404	9	.	.	PROPN
iajs-3828	404	10	cyclic	cyclic	PROPN
iajs-3828	404	11	g‐ω	g‐ω	PROPN
iajs-3828	404	12	-	-	PUNCT
iajs-3828	404	13	weak	weak	ADJ
iajs-3828	404	14	contraction	contraction	NOUN
iajs-3828	404	15	-	-	PUNCT
iajs-3828	404	16	weak	weak	ADJ
iajs-3828	404	17	nonexpansive	nonexpansive	ADJ
iajs-3828	404	18	mappings	mapping	NOUN
iajs-3828	404	19	and	and	CCONJ
iajs-3828	404	20	some	some	DET
iajs-3828	404	21	fixed	fix	VERB
iajs-3828	404	22	point	point	NOUN
iajs-3828	404	23	theorems	theorem	NOUN
iajs-3828	404	24	in	in	ADP
iajs-3828	404	25	metric	metric	ADJ
iajs-3828	404	26	spaces	space	NOUN
iajs-3828	404	27	.	.	PUNCT
iajs-3828	405	1	abstract	abstract	ADJ
iajs-3828	405	2	and	and	CCONJ
iajs-3828	405	3	applied	apply	VERB
iajs-3828	405	4	analysis	analysis	NOUN
iajs-3828	405	5	.	.	PUNCT
iajs-3828	406	1	2021;2021:1	2021;2021:1	NUM
iajs-3828	406	2	-	-	PUNCT
iajs-3828	406	3	9	9	NUM
iajs-3828	406	4	.	.	NOUN
iajs-3828	406	5	9	9	NUM
iajs-3828	406	6	.	.	X
iajs-3828	406	7	karapınar	karapınar	PROPN
iajs-3828	406	8	e	e	PROPN
iajs-3828	406	9	,	,	PUNCT
iajs-3828	406	10	romaguera	romaguera	NOUN
iajs-3828	406	11	s	s	PROPN
iajs-3828	406	12	,	,	PUNCT
iajs-3828	406	13	taş	taş	PROPN
iajs-3828	406	14	k.	k.	X
iajs-3828	406	15	fixed	fix	VERB
iajs-3828	406	16	points	point	NOUN
iajs-3828	406	17	for	for	ADP
iajs-3828	406	18	cyclic	cyclic	ADJ
iajs-3828	406	19	orbital	orbital	ADJ
iajs-3828	406	20	generalized	generalized	ADJ
iajs-3828	406	21	contractions	contraction	NOUN
iajs-3828	406	22	on	on	ADP
iajs-3828	406	23	complete	complete	ADJ
iajs-3828	406	24	metric	metric	ADJ
iajs-3828	406	25	spaces	space	NOUN
iajs-3828	406	26	.	.	PUNCT
iajs-3828	407	1	open	open	ADJ
iajs-3828	407	2	mathematics	mathematic	NOUN
iajs-3828	407	3	.	.	PUNCT
iajs-3828	408	1	2013;11(3):552	2013;11(3):552	NUM
iajs-3828	408	2	-	-	SYM
iajs-3828	408	3	560	560	NUM
iajs-3828	408	4	.	.	PUNCT
iajs-3828	409	1	http://doi.org/10.2478/s11533013-0245-5	http://doi.org/10.2478/s11533013-0245-5	NOUN
iajs-3828	409	2	10	10	NUM
iajs-3828	409	3	.	.	PUNCT
iajs-3828	410	1	bakhtin	bakhtin	PROPN
iajs-3828	410	2	i.	i.	PROPN
iajs-3828	410	3	the	the	DET
iajs-3828	410	4	contraction	contraction	NOUN
iajs-3828	410	5	mapping	map	VERB
iajs-3828	410	6	principle	principle	NOUN
iajs-3828	410	7	in	in	ADP
iajs-3828	410	8	quasimetric	quasimetric	ADJ
iajs-3828	410	9	spaces	space	NOUN
iajs-3828	410	10	.	.	PUNCT
iajs-3828	411	1	functional	functional	ADJ
iajs-3828	411	2	analysis	analysis	NOUN
iajs-3828	411	3	.	.	PUNCT
iajs-3828	412	1	1989;30:26	1989;30:26	NUM
iajs-3828	412	2	-	-	SYM
iajs-3828	412	3	37	37	NUM
iajs-3828	412	4	.	.	PUNCT
iajs-3828	413	1	11	11	NUM
iajs-3828	413	2	.	.	PUNCT
iajs-3828	414	1	al	al	PROPN
iajs-3828	414	2	-	-	PUNCT
iajs-3828	414	3	bundi	bundi	PROPN
iajs-3828	414	4	ss	ss	PROPN
iajs-3828	414	5	.	.	PROPN
iajs-3828	414	6	iterated	iterate	VERB
iajs-3828	414	7	function	function	NOUN
iajs-3828	414	8	system	system	NOUN
iajs-3828	414	9	in	in	ADP
iajs-3828	414	10	metric	metric	ADJ
iajs-3828	414	11	spaces	space	NOUN
iajs-3828	414	12	.	.	PUNCT
iajs-3828	415	1	bol	bol	NOUN
iajs-3828	415	2	soc	soc	PROPN
iajs-3828	415	3	paran	paran	PROPN
iajs-3828	415	4	mat	mat	PROPN
iajs-3828	415	5	.	.	PUNCT
iajs-3828	415	6	2022;40:1	2022;40:1	NUM
iajs-3828	415	7	-	-	SYM
iajs-3828	415	8	10	10	NUM
iajs-3828	415	9	.	.	PUNCT
iajs-3828	415	10	12	12	NUM
iajs-3828	415	11	.	.	PUNCT
iajs-3828	416	1	abed	abed	PROPN
iajs-3828	416	2	ss	ss	PROPN
iajs-3828	416	3	.	.	PUNCT
iajs-3828	417	1	fixed	fix	VERB
iajs-3828	417	2	point	point	NOUN
iajs-3828	417	3	principles	principle	NOUN
iajs-3828	417	4	in	in	ADP
iajs-3828	417	5	general	general	ADJ
iajs-3828	417	6	b	b	X
iajs-3828	417	7	-	-	PUNCT
iajs-3828	417	8	metric	metric	ADJ
iajs-3828	417	9	spaces	space	NOUN
iajs-3828	417	10	and	and	CCONJ
iajs-3828	417	11	b	b	X
iajs-3828	417	12	-	-	PUNCT
iajs-3828	417	13	menger	menger	NOUN
iajs-3828	417	14	probabilistic	probabilistic	ADJ
iajs-3828	417	15	spaces	space	NOUN
iajs-3828	417	16	.	.	PUNCT
iajs-3828	418	1	journal	journal	PROPN
iajs-3828	418	2	of	of	ADP
iajs-3828	418	3	al	al	PROPN
iajs-3828	418	4	-	-	PUNCT
iajs-3828	418	5	qadisiyah	qadisiyah	NOUN
iajs-3828	418	6	for	for	ADP
iajs-3828	418	7	computer	computer	NOUN
iajs-3828	418	8	science	science	NOUN
iajs-3828	418	9	and	and	CCONJ
iajs-3828	418	10	mathematics	mathematic	NOUN
iajs-3828	418	11	.	.	PUNCT
iajs-3828	419	1	2018;10(2):42	2018;10(2):42	NUM
iajs-3828	419	2	.	.	PROPN
iajs-3828	419	3	13	13	NUM
iajs-3828	419	4	.	.	PUNCT
iajs-3828	420	1	karapınar	karapınar	PROPN
iajs-3828	420	2	e.	e.	PROPN
iajs-3828	420	3	fixed	fixed	PROPN
iajs-3828	420	4	point	point	NOUN
iajs-3828	420	5	theory	theory	NOUN
iajs-3828	420	6	for	for	ADP
iajs-3828	420	7	cyclic	cyclic	ADJ
iajs-3828	420	8	weak	weak	ADJ
iajs-3828	420	9	ϕ-contraction	ϕ-contraction	NOUN
iajs-3828	420	10	.	.	PUNCT
iajs-3828	421	1	applied	apply	VERB
iajs-3828	421	2	mathematics	mathematics	NOUN
iajs-3828	421	3	letters	letter	NOUN
iajs-3828	421	4	.	.	PUNCT
iajs-3828	422	1	2011;24(6):822	2011;24(6):822	NUM
iajs-3828	422	2	-	-	PUNCT
iajs-3828	422	3	825	825	NUM
iajs-3828	422	4	.	.	PUNCT
iajs-3828	423	1	http://doi.org/10.1016/j.aml.2011.01.021	http://doi.org/10.1016/j.aml.2011.01.021	PROPN
iajs-3828	423	2	14	14	NUM
iajs-3828	423	3	.	.	PUNCT
iajs-3828	424	1	hammad	hammad	PROPN
iajs-3828	424	2	ha	ha	PROPN
iajs-3828	424	3	,	,	PUNCT
iajs-3828	424	4	de	de	X
iajs-3828	424	5	la	la	PROPN
iajs-3828	424	6	sen	sen	PROPN
iajs-3828	424	7	m.	m.	NOUN
iajs-3828	424	8	solution	solution	NOUN
iajs-3828	424	9	of	of	ADP
iajs-3828	424	10	nonlinear	nonlinear	ADJ
iajs-3828	424	11	integral	integral	ADJ
iajs-3828	424	12	equation	equation	NOUN
iajs-3828	424	13	via	via	ADP
iajs-3828	424	14	fixed	fix	VERB
iajs-3828	424	15	point	point	NOUN
iajs-3828	424	16	of	of	ADP
iajs-3828	424	17	cyclic	cyclic	ADJ
iajs-3828	424	18	α	α	PRON
iajs-3828	424	19	l	l	NOUN
iajs-3828	424	20	ψrational	ψrational	ADJ
iajs-3828	424	21	contraction	contraction	NOUN
iajs-3828	424	22	mappings	mapping	NOUN
iajs-3828	424	23	in	in	ADP
iajs-3828	424	24	metric	metric	ADJ
iajs-3828	424	25	-	-	PUNCT
iajs-3828	424	26	like	like	ADJ
iajs-3828	424	27	spaces	space	NOUN
iajs-3828	424	28	.	.	PUNCT
iajs-3828	425	1	bulletin	bulletin	NOUN
iajs-3828	425	2	of	of	ADP
iajs-3828	425	3	the	the	DET
iajs-3828	425	4	brazilian	brazilian	ADJ
iajs-3828	425	5	mathematical	mathematical	ADJ
iajs-3828	425	6	society	society	NOUN
iajs-3828	425	7	.	.	PUNCT
iajs-3828	426	1	2020;51(1):81	2020;51(1):81	NUM
iajs-3828	426	2	-	-	SYM
iajs-3828	426	3	105	105	NUM
iajs-3828	426	4	.	.	PUNCT
iajs-3828	427	1	http://doi.org/10.1007/s00574-019-00162-1	http://doi.org/10.1007/s00574-019-00162-1	NUM
iajs-3828	427	2	15	15	NUM
iajs-3828	427	3	.	.	PUNCT
iajs-3828	428	1	jabbar	jabbar	PROPN
iajs-3828	428	2	ha	ha	INTJ
iajs-3828	428	3	,	,	PUNCT
iajs-3828	428	4	kadhim	kadhim	PROPN
iajs-3828	428	5	sn	sn	PROPN
iajs-3828	428	6	,	,	PUNCT
iajs-3828	428	7	abed	abe	VERB
iajs-3828	428	8	ss	ss	PROPN
iajs-3828	428	9	.	.	PUNCT
iajs-3828	429	1	best	good	ADJ
iajs-3828	429	2	approximation	approximation	NOUN
iajs-3828	429	3	in	in	ADP
iajs-3828	429	4	b	b	NOUN
iajs-3828	429	5	-	-	PUNCT
iajs-3828	429	6	modular	modular	ADJ
iajs-3828	429	7	spaces	space	NOUN
iajs-3828	429	8	.	.	PUNCT
iajs-3828	430	1	baghdad	baghdad	PROPN
iajs-3828	430	2	science	science	PROPN
iajs-3828	430	3	journal	journal	PROPN
iajs-3828	430	4	.	.	PUNCT
iajs-3828	431	1	2023	2023	NUM
iajs-3828	431	2	.	.	PUNCT
iajs-3828	432	1	http://doi.org/10.21123/bsj.2023.8230	http://doi.org/10.21123/bsj.2023.8230	PROPN
iajs-3828	432	2	16	16	NUM
iajs-3828	432	3	.	.	PUNCT
iajs-3828	433	1	qawaqneh	qawaqneh	PROPN
iajs-3828	433	2	h	h	PROPN
iajs-3828	433	3	,	,	PUNCT
iajs-3828	433	4	noorani	noorani	PROPN
iajs-3828	433	5	m	m	PROPN
iajs-3828	433	6	,	,	PUNCT
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iajs-3828	433	8	w	w	PROPN
iajs-3828	433	9	,	,	PUNCT
iajs-3828	433	10	alsamir	alsamir	VERB
iajs-3828	433	11	h.	h.	PROPN
iajs-3828	433	12	common	common	ADJ
iajs-3828	433	13	fixed	fix	VERB
iajs-3828	433	14	point	point	NOUN
iajs-3828	433	15	theorems	theorem	NOUN
iajs-3828	433	16	for	for	ADP
iajs-3828	433	17	generalized	generalized	ADJ
iajs-3828	433	18	geraghty	geraghty	PROPN
iajs-3828	433	19	(	(	PUNCT
iajs-3828	433	20	α	α	X
iajs-3828	433	21	,	,	PUNCT
iajs-3828	433	22	ψ	ψ	SYM
iajs-3828	433	23	,	,	PUNCT
iajs-3828	433	24	ϕ)-quasi	ϕ)-quasi	PUNCT
iajs-3828	433	25	contraction	contraction	NOUN
iajs-3828	433	26	type	type	NOUN
iajs-3828	433	27	mapping	mapping	NOUN
iajs-3828	433	28	in	in	ADP
iajs-3828	433	29	partially	partially	ADV
iajs-3828	433	30	ordered	order	VERB
iajs-3828	433	31	metric	metric	ADJ
iajs-3828	433	32	-	-	PUNCT
iajs-3828	433	33	like	like	ADJ
iajs-3828	433	34	spaces	space	NOUN
iajs-3828	433	35	.	.	PUNCT
iajs-3828	434	1	axioms	axiom	NOUN
iajs-3828	434	2	.	.	PUNCT
iajs-3828	435	1	2018;7(4):74	2018;7(4):74	X
iajs-3828	435	2	.	.	PUNCT
iajs-3828	436	1	http://doi.org/10.3390/axioms7040074	http://doi.org/10.3390/axioms7040074	PROPN
iajs-3828	436	2	http://doi.org/10.3390/sym15101826	http://doi.org/10.3390/sym15101826	VERB
iajs-3828	436	3	http://doi.org/10.1186/s13660-020-02373-1	http://doi.org/10.1186/s13660-020-02373-1	NUM
iajs-3828	436	4	http://doi.org/10.1186/1687-1812-2011-1	http://doi.org/10.1186/1687-1812-2011-1	ADJ
iajs-3828	436	5	http://doi.org/10.1186/1687-1812-2013-1	http://doi.org/10.1186/1687-1812-2013-1	NOUN
iajs-3828	436	6	http://doi.org/10.1080/01630563.2021.1929578	http://doi.org/10.1080/01630563.2021.1929578	PROPN
iajs-3828	437	1	http://doi.org/10.4236/apm.2012.26059	http://doi.org/10.4236/apm.2012.26059	PROPN
iajs-3828	437	2	http://doi.org/10.2478/s11533-013-0245-5	http://doi.org/10.2478/s11533-013-0245-5	PROPN
iajs-3828	437	3	http://doi.org/10.2478/s11533-013-0245-5	http://doi.org/10.2478/s11533-013-0245-5	NUM
iajs-3828	438	1	http://doi.org/10.1016/j.aml.2011.01.021	http://doi.org/10.1016/j.aml.2011.01.021	PROPN
iajs-3828	438	2	http://doi.org/10.1007/s00574-019-00162-1	http://doi.org/10.1007/s00574-019-00162-1	PROPN
iajs-3828	439	1	http://doi.org/10.21123/bsj.2023.8230	http://doi.org/10.21123/bsj.2023.8230	PROPN
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iajs-3828	439	3	ihjpas	ihjpas	PROPN
iajs-3828	439	4	.	.	PUNCT
iajs-3828	440	1	2025,38(2	2025,38(2	PROPN
iajs-3828	440	2	)	)	PUNCT
iajs-3828	440	3	387	387	NUM
iajs-3828	440	4	17	17	NUM
iajs-3828	440	5	.	.	PUNCT
iajs-3828	441	1	mustafa	mustafa	PROPN
iajs-3828	441	2	z	z	PROPN
iajs-3828	441	3	,	,	PUNCT
iajs-3828	441	4	mustafa	mustafa	PROPN
iajs-3828	441	5	z	z	PROPN
iajs-3828	441	6	,	,	PUNCT
iajs-3828	441	7	khan	khan	PROPN
iajs-3828	441	8	su	su	PROPN
iajs-3828	441	9	,	,	PUNCT
iajs-3828	441	10	jaradat	jaradat	PROPN
iajs-3828	441	11	mmm	mmm	PROPN
iajs-3828	441	12	,	,	PUNCT
iajs-3828	441	13	arshad	arshad	PROPN
iajs-3828	441	14	m	m	PROPN
iajs-3828	441	15	,	,	PUNCT
iajs-3828	441	16	jaradat	jaradat	PROPN
iajs-3828	441	17	hm	hm	INTJ
iajs-3828	441	18	.	.	PUNCT
iajs-3828	441	19	fixed	fix	VERB
iajs-3828	441	20	point	point	NOUN
iajs-3828	441	21	results	result	NOUN
iajs-3828	441	22	of	of	ADP
iajs-3828	441	23	frational	frational	ADJ
iajs-3828	441	24	cyclic	cyclic	ADJ
iajs-3828	441	25	contractive	contractive	ADJ
iajs-3828	441	26	mappings	mapping	NOUN
iajs-3828	441	27	on	on	ADP
iajs-3828	441	28	0	0	NUM
iajs-3828	441	29	-	-	PUNCT
iajs-3828	441	30	complete	complete	ADJ
iajs-3828	441	31	partial	partial	ADJ
iajs-3828	441	32	metric	metric	ADJ
iajs-3828	441	33	spaces	space	NOUN
iajs-3828	441	34	.	.	PUNCT
iajs-3828	442	1	forum	forum	PROPN
iajs-3828	442	2	.	.	PUNCT
iajs-3828	443	1	2018;40:394	2018;40:394	NUM
iajs-3828	443	2	-	-	PUNCT
iajs-3828	443	3	409	409	NUM
iajs-3828	443	4	.	.	NOUN
iajs-3828	444	1	18	18	NUM
iajs-3828	444	2	.	.	PUNCT
iajs-3828	444	3	păcurar	păcurar	PROPN
iajs-3828	444	4	m	m	PROPN
iajs-3828	444	5	,	,	PUNCT
iajs-3828	444	6	rus	rus	PROPN
iajs-3828	444	7	ia	ia	PROPN
iajs-3828	444	8	.	.	PROPN
iajs-3828	444	9	fixed	fix	VERB
iajs-3828	444	10	point	point	NOUN
iajs-3828	444	11	theory	theory	NOUN
iajs-3828	444	12	of	of	ADP
iajs-3828	444	13	cyclic	cyclic	ADJ
iajs-3828	444	14	operators	operator	NOUN
iajs-3828	444	15	.	.	PUNCT
iajs-3828	445	1	journal	journal	NOUN
iajs-3828	445	2	of	of	ADP
iajs-3828	445	3	fixed	fix	VERB
iajs-3828	445	4	point	point	NOUN
iajs-3828	445	5	theory	theory	NOUN
iajs-3828	445	6	and	and	CCONJ
iajs-3828	445	7	applications	application	NOUN
iajs-3828	445	8	.	.	PUNCT
iajs-3828	446	1	2022;24(4):79	2022;24(4):79	X
iajs-3828	446	2	.	.	NOUN
iajs-3828	446	3	19	19	NUM
iajs-3828	446	4	.	.	NOUN
iajs-3828	446	5	theivaraman	theivaraman	PROPN
iajs-3828	446	6	r	r	PROPN
iajs-3828	446	7	,	,	PUNCT
iajs-3828	446	8	srinivasan	srinivasan	NOUN
iajs-3828	446	9	p	p	NOUN
iajs-3828	446	10	,	,	PUNCT
iajs-3828	446	11	radenovic	radenovic	PROPN
iajs-3828	446	12	s	s	PROPN
iajs-3828	446	13	,	,	PUNCT
iajs-3828	446	14	park	park	PROPN
iajs-3828	446	15	c.	c.	PROPN
iajs-3828	446	16	new	new	PROPN
iajs-3828	446	17	approximate	approximate	ADJ
iajs-3828	446	18	fixed	fix	VERB
iajs-3828	446	19	point	point	NOUN
iajs-3828	446	20	results	result	NOUN
iajs-3828	446	21	for	for	ADP
iajs-3828	446	22	various	various	ADJ
iajs-3828	446	23	cyclic	cyclic	ADJ
iajs-3828	446	24	contraction	contraction	NOUN
iajs-3828	446	25	operators	operator	NOUN
iajs-3828	446	26	on	on	ADP
iajs-3828	446	27	e	e	ADJ
iajs-3828	446	28	-	-	ADJ
iajs-3828	446	29	metric	metric	ADJ
iajs-3828	446	30	spaces	space	NOUN
iajs-3828	446	31	.	.	PUNCT
iajs-3828	447	1	journal	journal	NOUN
iajs-3828	447	2	of	of	ADP
iajs-3828	447	3	the	the	DET
iajs-3828	447	4	korean	korean	ADJ
iajs-3828	447	5	society	society	NOUN
iajs-3828	447	6	for	for	ADP
iajs-3828	447	7	industrial	industrial	ADJ
iajs-3828	447	8	and	and	CCONJ
iajs-3828	447	9	applied	applied	ADJ
iajs-3828	447	10	mathematics	mathematic	NOUN
iajs-3828	447	11	.	.	PUNCT
iajs-3828	448	1	2023;27(3):160	2023;27(3):160	NUM
iajs-3828	448	2	-	-	SYM
iajs-3828	448	3	179	179	NUM
iajs-3828	448	4	.	.	PUNCT
iajs-3828	449	1	20	20	NUM
iajs-3828	449	2	.	.	PUNCT
iajs-3828	450	1	kumari	kumari	PROPN
iajs-3828	450	2	ps	ps	PROPN
iajs-3828	450	3	,	,	PUNCT
iajs-3828	450	4	panthi	panthi	PROPN
iajs-3828	450	5	d.	d.	PROPN
iajs-3828	450	6	cyclic	cyclic	PROPN
iajs-3828	450	7	contractions	contraction	NOUN
iajs-3828	450	8	and	and	CCONJ
iajs-3828	450	9	fixed	fix	VERB
iajs-3828	450	10	point	point	NOUN
iajs-3828	450	11	theorems	theorem	NOUN
iajs-3828	450	12	on	on	ADP
iajs-3828	450	13	various	various	ADJ
iajs-3828	450	14	generating	generate	VERB
iajs-3828	450	15	spaces	space	NOUN
iajs-3828	450	16	.	.	PUNCT
iajs-3828	451	1	fixed	fix	VERB
iajs-3828	451	2	point	point	NOUN
iajs-3828	451	3	theory	theory	NOUN
iajs-3828	451	4	and	and	CCONJ
iajs-3828	451	5	applications	application	NOUN
iajs-3828	451	6	.	.	PUNCT
iajs-3828	452	1	2015;2015(1):1	2015;2015(1):1	NUM
iajs-3828	452	2	-	-	SYM
iajs-3828	452	3	17	17	NUM
iajs-3828	452	4	.	.	PUNCT
iajs-3828	453	1	http://doi.org/10.1186/s13663015-0365-7	http://doi.org/10.1186/s13663015-0365-7	PROPN
iajs-3828	453	2	21	21	NUM
iajs-3828	453	3	.	.	PUNCT
iajs-3828	454	1	hussein	hussein	PROPN
iajs-3828	454	2	luaibi	luaibi	PROPN
iajs-3828	454	3	h	h	PROPN
iajs-3828	454	4	,	,	PUNCT
iajs-3828	454	5	abed	abe	VERB
iajs-3828	454	6	ss	ss	PROPN
iajs-3828	454	7	.	.	PUNCT
iajs-3828	455	1	fixed	fix	VERB
iajs-3828	455	2	point	point	NOUN
iajs-3828	455	3	theorems	theorem	NOUN
iajs-3828	455	4	in	in	ADP
iajs-3828	455	5	general	general	ADJ
iajs-3828	455	6	metric	metric	ADJ
iajs-3828	455	7	space	space	NOUN
iajs-3828	455	8	with	with	ADP
iajs-3828	455	9	an	an	DET
iajs-3828	455	10	application	application	NOUN
iajs-3828	455	11	.	.	PUNCT
iajs-3828	456	1	baghdad	baghdad	PROPN
iajs-3828	456	2	science	science	PROPN
iajs-3828	456	3	journal	journal	PROPN
iajs-3828	456	4	.	.	PUNCT
iajs-3828	457	1	2021;18(1	2021;18(1	NUM
iajs-3828	457	2	suppl.):0812	suppl.):0812	NOUN
iajs-3828	457	3	.	.	PROPN
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iajs-3828	457	5	.	.	PUNCT
iajs-3828	458	1	branciari	branciari	PROPN
iajs-3828	458	2	a.	a.	PROPN
iajs-3828	458	3	a	a	DET
iajs-3828	458	4	fixed	fix	VERB
iajs-3828	458	5	point	point	NOUN
iajs-3828	458	6	theorem	theorem	NOUN
iajs-3828	458	7	for	for	ADP
iajs-3828	458	8	mappings	mapping	NOUN
iajs-3828	458	9	satisfying	satisfy	VERB
iajs-3828	458	10	a	a	DET
iajs-3828	458	11	general	general	ADJ
iajs-3828	458	12	contractive	contractive	ADJ
iajs-3828	458	13	condition	condition	NOUN
iajs-3828	458	14	of	of	ADP
iajs-3828	458	15	integral	integral	ADJ
iajs-3828	458	16	type	type	NOUN
iajs-3828	458	17	.	.	PUNCT
iajs-3828	459	1	international	international	ADJ
iajs-3828	459	2	journal	journal	PROPN
iajs-3828	459	3	of	of	ADP
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iajs-3828	459	5	and	and	CCONJ
iajs-3828	459	6	mathematical	mathematical	ADJ
iajs-3828	459	7	sciences	science	NOUN
iajs-3828	459	8	.	.	PUNCT
iajs-3828	460	1	2002;29:531536	2002;29:531536	NUM
iajs-3828	460	2	.	.	PUNCT
iajs-3828	461	1	http://doi.org/10.1155/s0161171202007524	http://doi.org/10.1155/s0161171202007524	NUM
iajs-3828	461	2	23	23	NUM
iajs-3828	461	3	.	.	PUNCT
iajs-3828	462	1	khan	khan	PROPN
iajs-3828	462	2	ms	ms	PROPN
iajs-3828	462	3	,	,	PUNCT
iajs-3828	462	4	swaleh	swaleh	PROPN
iajs-3828	462	5	m	m	PROPN
iajs-3828	462	6	,	,	PUNCT
iajs-3828	462	7	sessa	sessa	PROPN
iajs-3828	462	8	s.	s.	PROPN
iajs-3828	462	9	fixed	fix	VERB
iajs-3828	462	10	point	point	NOUN
iajs-3828	462	11	theorems	theorem	NOUN
iajs-3828	462	12	by	by	ADP
iajs-3828	462	13	altering	alter	VERB
iajs-3828	462	14	distances	distance	NOUN
iajs-3828	462	15	between	between	ADP
iajs-3828	462	16	the	the	DET
iajs-3828	462	17	points	point	NOUN
iajs-3828	462	18	.	.	PUNCT
iajs-3828	463	1	bulletin	bulletin	NOUN
iajs-3828	463	2	of	of	ADP
iajs-3828	463	3	the	the	DET
iajs-3828	463	4	australian	australian	ADJ
iajs-3828	463	5	mathematical	mathematical	ADJ
iajs-3828	463	6	society	society	NOUN
iajs-3828	463	7	.	.	PUNCT
iajs-3828	464	1	1984;30(1):19	1984;30(1):19	X
iajs-3828	464	2	.	.	PUNCT
iajs-3828	465	1	http://doi.org/10.1017/s0004972700001659	http://doi.org/10.1017/s0004972700001659	NUM
iajs-3828	465	2	24	24	NUM
iajs-3828	465	3	.	.	PUNCT
iajs-3828	466	1	nieto	nieto	PROPN
iajs-3828	466	2	jj	jj	PROPN
iajs-3828	466	3	,	,	PUNCT
iajs-3828	466	4	rodríguez	rodríguez	NOUN
iajs-3828	466	5	-	-	PUNCT
iajs-3828	466	6	lópez	lópez	PROPN
iajs-3828	466	7	r.	r.	PROPN
iajs-3828	466	8	contractive	contractive	PROPN
iajs-3828	466	9	mapping	mapping	NOUN
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iajs-3828	466	11	in	in	ADP
iajs-3828	466	12	partially	partially	ADV
iajs-3828	466	13	ordered	order	VERB
iajs-3828	466	14	sets	set	NOUN
iajs-3828	466	15	and	and	CCONJ
iajs-3828	466	16	applications	application	NOUN
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iajs-3828	466	20	equations	equation	NOUN
iajs-3828	466	21	.	.	PUNCT
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iajs-3828	467	2	.	.	PUNCT
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iajs-3828	468	2	.	.	PUNCT
iajs-3828	469	1	http://doi.org/10.1007/s11083-005-9018-5	http://doi.org/10.1007/s11083-005-9018-5	NUM
iajs-3828	469	2	25	25	NUM
iajs-3828	469	3	.	.	PUNCT
iajs-3828	470	1	rabaiah	rabaiah	PROPN
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iajs-3828	470	3	,	,	PUNCT
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iajs-3828	470	6	,	,	PUNCT
iajs-3828	470	7	shatanawi	shatanawi	ADJ
iajs-3828	470	8	w.	w.	PROPN
iajs-3828	470	9	common	common	ADJ
iajs-3828	470	10	fixed	fix	VERB
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iajs-3828	470	12	results	result	NOUN
iajs-3828	470	13	for	for	ADP
iajs-3828	470	14	mappings	mapping	NOUN
iajs-3828	470	15	under	under	ADP
iajs-3828	470	16	nonlinear	nonlinear	ADJ
iajs-3828	470	17	contraction	contraction	NOUN
iajs-3828	470	18	of	of	ADP
iajs-3828	470	19	cyclic	cyclic	ADJ
iajs-3828	470	20	form	form	NOUN
iajs-3828	470	21	in	in	ADP
iajs-3828	470	22	b	b	NOUN
iajs-3828	470	23	-	-	ADJ
iajs-3828	470	24	metric	metric	ADJ
iajs-3828	470	25	spaces	space	NOUN
iajs-3828	470	26	.	.	PUNCT
iajs-3828	471	1	nonlinear	nonlinear	ADJ
iajs-3828	471	2	functional	functional	ADJ
iajs-3828	471	3	analysis	analysis	NOUN
iajs-3828	471	4	and	and	CCONJ
iajs-3828	471	5	applications	application	NOUN
iajs-3828	471	6	.	.	PUNCT
iajs-3828	472	1	2021;26(2):289	2021;26(2):289	NUM
iajs-3828	472	2	-	-	SYM
iajs-3828	472	3	301	301	NUM
iajs-3828	472	4	.	.	NOUN
iajs-3828	472	5	26	26	NUM
iajs-3828	472	6	.	.	PUNCT
iajs-3828	472	7	nashine	nashine	PROPN
iajs-3828	472	8	hk	hk	PROPN
iajs-3828	472	9	,	,	PUNCT
iajs-3828	472	10	kadelburg	kadelburg	PROPN
iajs-3828	472	11	z	z	PROPN
iajs-3828	472	12	,	,	PUNCT
iajs-3828	472	13	kumam	kumam	PROPN
iajs-3828	472	14	p.	p.	NOUN
iajs-3828	472	15	implicit	implicit	ADJ
iajs-3828	472	16	-	-	PUNCT
iajs-3828	472	17	relation	relation	NOUN
iajs-3828	472	18	-	-	PUNCT
iajs-3828	472	19	type	type	NOUN
iajs-3828	472	20	cyclic	cyclic	ADJ
iajs-3828	472	21	contractive	contractive	ADJ
iajs-3828	472	22	mappings	mapping	NOUN
iajs-3828	472	23	and	and	CCONJ
iajs-3828	472	24	applications	application	NOUN
iajs-3828	472	25	to	to	ADP
iajs-3828	472	26	integral	integral	ADJ
iajs-3828	472	27	equations	equation	NOUN
iajs-3828	472	28	.	.	PUNCT
iajs-3828	473	1	abstract	abstract	ADJ
iajs-3828	473	2	and	and	CCONJ
iajs-3828	473	3	applied	apply	VERB
iajs-3828	473	4	analysis	analysis	NOUN
iajs-3828	473	5	.	.	PUNCT
iajs-3828	474	1	2012;2012:1	2012;2012:1	NUM
iajs-3828	474	2	-	-	SYM
iajs-3828	474	3	15	15	NUM
iajs-3828	474	4	.	.	NOUN
iajs-3828	475	1	27	27	NUM
iajs-3828	475	2	.	.	PUNCT
iajs-3828	476	1	matkowski	matkowski	PROPN
iajs-3828	476	2	j.	j.	PROPN
iajs-3828	476	3	fixed	fixed	PROPN
iajs-3828	476	4	point	point	NOUN
iajs-3828	476	5	theorems	theorem	NOUN
iajs-3828	476	6	for	for	ADP
iajs-3828	476	7	mappings	mapping	NOUN
iajs-3828	476	8	with	with	ADP
iajs-3828	476	9	a	a	DET
iajs-3828	476	10	contractive	contractive	ADJ
iajs-3828	476	11	iterate	iterate	NOUN
iajs-3828	476	12	at	at	ADP
iajs-3828	476	13	a	a	DET
iajs-3828	476	14	point	point	NOUN
iajs-3828	476	15	.	.	PUNCT
iajs-3828	477	1	proceedings	proceeding	NOUN
iajs-3828	477	2	of	of	ADP
iajs-3828	477	3	the	the	DET
iajs-3828	477	4	american	american	PROPN
iajs-3828	477	5	mathematical	mathematical	PROPN
iajs-3828	477	6	society	society	NOUN
iajs-3828	477	7	.	.	PUNCT
iajs-3828	478	1	1977;62(2):344	1977;62(2):344	PROPN
iajs-3828	478	2	-	-	PUNCT
iajs-3828	478	3	348	348	NUM
iajs-3828	478	4	.	.	PUNCT
iajs-3828	479	1	http://doi.org/10.2307/2041931	http://doi.org/10.2307/2041931	NUM
iajs-3828	479	2	28	28	NUM
iajs-3828	479	3	.	.	PUNCT
iajs-3828	480	1	abed	abed	PROPN
iajs-3828	480	2	ss	ss	PROPN
iajs-3828	480	3	,	,	PUNCT
iajs-3828	480	4	abed	abe	VERB
iajs-3828	480	5	an	an	DET
iajs-3828	480	6	.	.	PUNCT
iajs-3828	480	7	convergence	convergence	NOUN
iajs-3828	480	8	and	and	CCONJ
iajs-3828	480	9	stability	stability	NOUN
iajs-3828	480	10	of	of	ADP
iajs-3828	480	11	iterative	iterative	ADJ
iajs-3828	480	12	scheme	scheme	NOUN
iajs-3828	480	13	for	for	ADP
iajs-3828	480	14	a	a	DET
iajs-3828	480	15	monotone	monotone	ADJ
iajs-3828	480	16	total	total	NOUN
iajs-3828	480	17	asymptotically	asymptotically	ADV
iajs-3828	480	18	non	non	ADJ
iajs-3828	480	19	-	-	ADJ
iajs-3828	480	20	expansive	expansive	ADJ
iajs-3828	480	21	mapping	mapping	NOUN
iajs-3828	480	22	.	.	PUNCT
iajs-3828	481	1	iraqi	iraqi	ADJ
iajs-3828	481	2	journal	journal	PROPN
iajs-3828	481	3	of	of	ADP
iajs-3828	481	4	science	science	NOUN
iajs-3828	481	5	.	.	PUNCT
iajs-3828	482	1	2022;63(1):241	2022;63(1):241	NUM
iajs-3828	482	2	-	-	PUNCT
iajs-3828	482	3	250	250	NUM
iajs-3828	482	4	.	.	PUNCT
iajs-3828	482	5	29	29	NUM
iajs-3828	482	6	.	.	PUNCT
iajs-3828	482	7	abed	abe	VERB
iajs-3828	482	8	ss	ss	PROPN
iajs-3828	482	9	,	,	PUNCT
iajs-3828	482	10	taresh	taresh	PROPN
iajs-3828	482	11	ns	ns	INTJ
iajs-3828	482	12	.	.	PROPN
iajs-3828	483	1	on	on	ADP
iajs-3828	483	2	stability	stability	NOUN
iajs-3828	483	3	of	of	ADP
iajs-3828	483	4	iterative	iterative	ADJ
iajs-3828	483	5	sequences	sequence	NOUN
iajs-3828	483	6	with	with	ADP
iajs-3828	483	7	error	error	NOUN
iajs-3828	483	8	.	.	PUNCT
iajs-3828	484	1	mathematics	mathematic	NOUN
iajs-3828	484	2	.	.	PUNCT
iajs-3828	485	1	2019;7(8):765	2019;7(8):765	NUM
iajs-3828	485	2	.	.	PUNCT
iajs-3828	486	1	http://doi.org/10.3390/math7080765	http://doi.org/10.3390/math7080765	PROPN
iajs-3828	486	2	30	30	NUM
iajs-3828	486	3	.	.	PUNCT
iajs-3828	487	1	monje	monje	PROPN
iajs-3828	487	2	zam	zam	PROPN
iajs-3828	487	3	,	,	PUNCT
iajs-3828	487	4	ahmed	ahmed	PROPN
iajs-3828	487	5	ba	ba	PROPN
iajs-3828	487	6	.	.	PUNCT
iajs-3828	488	1	a	a	DET
iajs-3828	488	2	study	study	NOUN
iajs-3828	488	3	of	of	ADP
iajs-3828	488	4	stability	stability	NOUN
iajs-3828	488	5	of	of	ADP
iajs-3828	488	6	first	first	ADJ
iajs-3828	488	7	-	-	PUNCT
iajs-3828	488	8	order	order	NOUN
iajs-3828	488	9	delay	delay	NOUN
iajs-3828	488	10	differential	differential	ADJ
iajs-3828	488	11	equations	equation	NOUN
iajs-3828	488	12	using	use	VERB
iajs-3828	488	13	fixed	fix	VERB
iajs-3828	488	14	point	point	NOUN
iajs-3828	488	15	theorem	theorem	ADJ
iajs-3828	488	16	banach	banach	NOUN
iajs-3828	488	17	.	.	PUNCT
iajs-3828	489	1	iraqi	iraqi	ADJ
iajs-3828	489	2	journal	journal	PROPN
iajs-3828	489	3	of	of	ADP
iajs-3828	489	4	science	science	NOUN
iajs-3828	489	5	.	.	PUNCT
iajs-3828	490	1	2019;60(12):2719	2019;60(12):2719	X
iajs-3828	490	2	-	-	SYM
iajs-3828	490	3	2724	2724	NUM
iajs-3828	490	4	http://doi.org/10.1186/s13663-015-0365-7	http://doi.org/10.1186/s13663-015-0365-7	NUM
iajs-3828	490	5	http://doi.org/10.1186/s13663-015-0365-7	http://doi.org/10.1186/s13663-015-0365-7	NUM
iajs-3828	491	1	http://doi.org/10.1155/s0161171202007524	http://doi.org/10.1155/s0161171202007524	NUM
iajs-3828	491	2	http://doi.org/10.1017/s0004972700001659	http://doi.org/10.1017/s0004972700001659	NUM
iajs-3828	491	3	http://doi.org/10.1007/s11083-005-9018-5	http://doi.org/10.1007/s11083-005-9018-5	NUM
iajs-3828	492	1	http://doi.org/10.2307/2041931	http://doi.org/10.2307/2041931	CCONJ
iajs-3828	492	2	http://doi.org/10.3390/math7080765	http://doi.org/10.3390/math7080765	PROPN