id	sid	tid	token	lemma	pos
iajs-3486	1	1	424	424	NUM
iajs-3486	1	2	this	this	DET
iajs-3486	1	3	work	work	NOUN
iajs-3486	1	4	is	be	AUX
iajs-3486	1	5	licensed	license	VERB
iajs-3486	1	6	under	under	ADP
iajs-3486	1	7	a	a	DET
iajs-3486	1	8	creative	creative	ADJ
iajs-3486	1	9	commons	common	NOUN
iajs-3486	1	10	attribution	attribution	NOUN
iajs-3486	1	11	4.0	4.0	NUM
iajs-3486	1	12	international	international	ADJ
iajs-3486	1	13	license	license	NOUN
iajs-3486	1	14	ihjpas.37	ihjpas.37	PROPN
iajs-3486	1	15	(	(	PUNCT
iajs-3486	1	16	2	2	NUM
iajs-3486	1	17	)	)	PUNCT
iajs-3486	1	18	2024	2024	NUM
iajs-3486	1	19	ibn	ibn	PROPN
iajs-3486	1	20	al	al	PROPN
iajs-3486	1	21	-	-	PUNCT
iajs-3486	1	22	haitham	haitham	PROPN
iajs-3486	1	23	journal	journal	PROPN
iajs-3486	1	24	for	for	ADP
iajs-3486	1	25	pure	pure	ADJ
iajs-3486	1	26	and	and	CCONJ
iajs-3486	1	27	applied	applied	ADJ
iajs-3486	1	28	sciences	sciences	PROPN
iajs-3486	1	29	journal	journal	PROPN
iajs-3486	1	30	homepage	homepage	NOUN
iajs-3486	1	31	:	:	PUNCT
iajs-3486	1	32	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	NOUN
iajs-3486	1	33	pissn	pissn	ADJ
iajs-3486	1	34	:	:	PUNCT
iajs-3486	1	35	1609	1609	NUM
iajs-3486	1	36	-	-	SYM
iajs-3486	1	37	4042	4042	NUM
iajs-3486	1	38	,	,	PUNCT
iajs-3486	1	39	eissn	eissn	NOUN
iajs-3486	1	40	:	:	PUNCT
iajs-3486	1	41	2521	2521	NUM
iajs-3486	1	42	-	-	SYM
iajs-3486	1	43	3407	3407	NUM
iajs-3486	1	44	wisam	wisam	NOUN
iajs-3486	1	45	mohammed	mohammed	PROPN
iajs-3486	1	46	mukhlif	mukhlif	VERB
iajs-3486	1	47	1	1	NUM
iajs-3486	1	48	*	*	PUNCT
iajs-3486	1	49	,	,	PUNCT
iajs-3486	1	50	sabah	sabah	PROPN
iajs-3486	1	51	hassan	hassan	PROPN
iajs-3486	1	52	malih2	malih2	PROPN
iajs-3486	1	53	and	and	CCONJ
iajs-3486	1	54	shrooq	shrooq	NOUN
iajs-3486	1	55	bahjat	bahjat	PROPN
iajs-3486	1	56	smeein3	smeein3	PROPN
iajs-3486	2	1	1department	1department	NUM
iajs-3486	2	2	of	of	ADP
iajs-3486	2	3	mathematics	mathematic	NOUN
iajs-3486	2	4	,	,	PUNCT
iajs-3486	2	5	college	college	NOUN
iajs-3486	2	6	of	of	ADP
iajs-3486	2	7	education	education	NOUN
iajs-3486	2	8	for	for	ADP
iajs-3486	2	9	pure	pure	ADJ
iajs-3486	2	10	science	science	NOUN
iajs-3486	2	11	(	(	PUNCT
iajs-3486	2	12	ibnalhaitham	ibnalhaitham	NOUN
iajs-3486	2	13	)	)	PUNCT
iajs-3486	2	14	,	,	PUNCT
iajs-3486	2	15	university	university	NOUN
iajs-3486	2	16	of	of	ADP
iajs-3486	2	17	baghdad	baghdad	PROPN
iajs-3486	2	18	,	,	PUNCT
iajs-3486	2	19	city	city	PROPN
iajs-3486	2	20	baghdad	baghdad	PROPN
iajs-3486	2	21	,	,	PUNCT
iajs-3486	2	22	iraq	iraq	PROPN
iajs-3486	2	23	.	.	PUNCT
iajs-3486	3	1	2department	2department	NUM
iajs-3486	3	2	of	of	ADP
iajs-3486	3	3	mathematics	mathematic	NOUN
iajs-3486	3	4	,	,	PUNCT
iajs-3486	3	5	college	college	NOUN
iajs-3486	3	6	of	of	ADP
iajs-3486	3	7	education	education	NOUN
iajs-3486	3	8	for	for	ADP
iajs-3486	3	9	pure	pure	ADJ
iajs-3486	3	10	science	science	NOUN
iajs-3486	3	11	(	(	PUNCT
iajs-3486	3	12	ibnalhaitham	ibnalhaitham	NOUN
iajs-3486	3	13	)	)	PUNCT
iajs-3486	3	14	,	,	PUNCT
iajs-3486	3	15	university	university	NOUN
iajs-3486	3	16	of	of	ADP
iajs-3486	3	17	baghdad	baghdad	PROPN
iajs-3486	3	18	,	,	PUNCT
iajs-3486	3	19	city	city	PROPN
iajs-3486	3	20	baghdad	baghdad	PROPN
iajs-3486	3	21	,	,	PUNCT
iajs-3486	3	22	iraq	iraq	PROPN
iajs-3486	3	23	.	.	PUNCT
iajs-3486	4	1	3information	3information	NUM
iajs-3486	4	2	department	department	NOUN
iajs-3486	4	3	,	,	PUNCT
iajs-3486	4	4	section	section	NOUN
iajs-3486	4	5	mathematics	mathematic	NOUN
iajs-3486	4	6	sultanate	sultanate	NOUN
iajs-3486	4	7	of	of	ADP
iajs-3486	4	8	oman	oman	NOUN
iajs-3486	4	9	.	.	PUNCT
iajs-3486	5	1	*	*	PUNCT
iajs-3486	5	2	corresponding	correspond	VERB
iajs-3486	5	3	author	author	NOUN
iajs-3486	5	4	.	.	PUNCT
iajs-3486	6	1	abstract	abstract	ADJ
iajs-3486	6	2	in	in	ADP
iajs-3486	6	3	this	this	DET
iajs-3486	6	4	paper	paper	NOUN
iajs-3486	6	5	,	,	PUNCT
iajs-3486	6	6	we	we	PRON
iajs-3486	6	7	introduce	introduce	VERB
iajs-3486	6	8	a	a	DET
iajs-3486	6	9	new	new	ADJ
iajs-3486	6	10	one	one	NUM
iajs-3486	6	11	-	-	PUNCT
iajs-3486	6	12	step	step	NOUN
iajs-3486	6	13	iteration	iteration	NOUN
iajs-3486	6	14	process	process	NOUN
iajs-3486	6	15	in	in	ADP
iajs-3486	6	16	banach	banach	NOUN
iajs-3486	6	17	space	space	NOUN
iajs-3486	6	18	and	and	CCONJ
iajs-3486	6	19	prove	prove	VERB
iajs-3486	6	20	the	the	DET
iajs-3486	6	21	existence	existence	NOUN
iajs-3486	6	22	of	of	ADP
iajs-3486	6	23	a	a	DET
iajs-3486	6	24	common	common	ADJ
iajs-3486	6	25	random	random	ADJ
iajs-3486	6	26	fixed	fix	VERB
iajs-3486	6	27	point	point	NOUN
iajs-3486	6	28	of	of	ADP
iajs-3486	6	29	three	three	NUM
iajs-3486	6	30	non	non	ADJ
iajs-3486	6	31	-	-	ADJ
iajs-3486	6	32	expansive	expansive	ADJ
iajs-3486	6	33	multivalued	multivalued	ADJ
iajs-3486	6	34	random	random	ADJ
iajs-3486	6	35	operators	operator	NOUN
iajs-3486	6	36	through	through	ADP
iajs-3486	6	37	strong	strong	ADJ
iajs-3486	6	38	and	and	CCONJ
iajs-3486	6	39	weak	weak	ADJ
iajs-3486	6	40	convergences	convergence	NOUN
iajs-3486	6	41	of	of	ADP
iajs-3486	6	42	an	an	DET
iajs-3486	6	43	iterative	iterative	NOUN
iajs-3486	6	44	process	process	NOUN
iajs-3486	6	45	.	.	PUNCT
iajs-3486	7	1	the	the	DET
iajs-3486	7	2	necessary	necessary	ADJ
iajs-3486	7	3	and	and	CCONJ
iajs-3486	7	4	sufficient	sufficient	ADJ
iajs-3486	7	5	condition	condition	NOUN
iajs-3486	7	6	for	for	ADP
iajs-3486	7	7	the	the	DET
iajs-3486	7	8	convergence	convergence	NOUN
iajs-3486	7	9	of	of	ADP
iajs-3486	7	10	a	a	DET
iajs-3486	7	11	sequence	sequence	NOUN
iajs-3486	7	12	of	of	ADP
iajs-3486	7	13	measurable	measurable	ADJ
iajs-3486	7	14	functions	function	NOUN
iajs-3486	7	15	to	to	ADP
iajs-3486	7	16	a	a	DET
iajs-3486	7	17	random	random	ADJ
iajs-3486	7	18	fixed	fix	VERB
iajs-3486	7	19	point	point	NOUN
iajs-3486	7	20	of	of	ADP
iajs-3486	7	21	non	non	ADJ
iajs-3486	7	22	-	-	ADJ
iajs-3486	7	23	expansive	expansive	ADJ
iajs-3486	7	24	multivalued	multivalued	ADJ
iajs-3486	7	25	random	random	ADJ
iajs-3486	7	26	operators	operator	NOUN
iajs-3486	7	27	in	in	ADP
iajs-3486	7	28	uniformly	uniformly	ADV
iajs-3486	7	29	convex	convex	NOUN
iajs-3486	7	30	banach	banach	NOUN
iajs-3486	7	31	spaces	space	NOUN
iajs-3486	7	32	is	be	AUX
iajs-3486	7	33	also	also	ADV
iajs-3486	7	34	established	establish	VERB
iajs-3486	7	35	.	.	PUNCT
iajs-3486	8	1	our	our	PRON
iajs-3486	8	2	random	random	ADJ
iajs-3486	8	3	iteration	iteration	NOUN
iajs-3486	8	4	scheme	scheme	NOUN
iajs-3486	8	5	includes	include	VERB
iajs-3486	8	6	new	new	ADJ
iajs-3486	8	7	random	random	ADJ
iajs-3486	8	8	multivalued	multivalue	VERB
iajs-3486	8	9	iterations	iteration	NOUN
iajs-3486	8	10	as	as	ADP
iajs-3486	8	11	special	special	ADJ
iajs-3486	8	12	cases	case	NOUN
iajs-3486	8	13	.	.	PUNCT
iajs-3486	9	1	the	the	DET
iajs-3486	9	2	results	result	NOUN
iajs-3486	9	3	obtained	obtain	VERB
iajs-3486	9	4	in	in	ADP
iajs-3486	9	5	this	this	DET
iajs-3486	9	6	paper	paper	NOUN
iajs-3486	9	7	are	be	AUX
iajs-3486	9	8	an	an	DET
iajs-3486	9	9	extension	extension	NOUN
iajs-3486	9	10	and	and	CCONJ
iajs-3486	9	11	refinement	refinement	NOUN
iajs-3486	9	12	of	of	ADP
iajs-3486	9	13	previously	previously	ADV
iajs-3486	9	14	known	know	VERB
iajs-3486	9	15	results	result	NOUN
iajs-3486	9	16	.	.	PUNCT
iajs-3486	10	1	a	a	DET
iajs-3486	10	2	new	new	ADJ
iajs-3486	10	3	random	random	ADJ
iajs-3486	10	4	iterative	iterative	NOUN
iajs-3486	10	5	scheme	scheme	NOUN
iajs-3486	10	6	for	for	ADP
iajs-3486	10	7	approximating	approximate	VERB
iajs-3486	10	8	random	random	ADJ
iajs-3486	10	9	common	common	ADJ
iajs-3486	10	10	fixed	fix	VERB
iajs-3486	10	11	points	point	NOUN
iajs-3486	10	12	of	of	ADP
iajs-3486	10	13	three	three	NUM
iajs-3486	10	14	random	random	ADJ
iajs-3486	10	15	non	non	ADJ
iajs-3486	10	16	-	-	ADJ
iajs-3486	10	17	expansive	expansive	ADJ
iajs-3486	10	18	multivalued	multivalued	ADJ
iajs-3486	10	19	random	random	ADJ
iajs-3486	10	20	operators	operator	NOUN
iajs-3486	10	21	is	be	AUX
iajs-3486	10	22	defined	define	VERB
iajs-3486	10	23	and	and	CCONJ
iajs-3486	10	24	we	we	PRON
iajs-3486	10	25	have	have	AUX
iajs-3486	10	26	proved	prove	VERB
iajs-3486	10	27	weak	weak	ADJ
iajs-3486	10	28	and	and	CCONJ
iajs-3486	10	29	strong	strong	ADJ
iajs-3486	10	30	convergence	convergence	NOUN
iajs-3486	10	31	theorems	theorem	NOUN
iajs-3486	10	32	in	in	ADP
iajs-3486	10	33	a	a	DET
iajs-3486	10	34	uniformly	uniformly	ADJ
iajs-3486	10	35	convex	convex	NOUN
iajs-3486	10	36	banach	banach	NOUN
iajs-3486	10	37	space	space	NOUN
iajs-3486	10	38	.	.	PUNCT
iajs-3486	11	1	keywords	keyword	NOUN
iajs-3486	11	2	:	:	PUNCT
iajs-3486	11	3	common	common	ADJ
iajs-3486	11	4	fixed	fix	VERB
iajs-3486	11	5	points	point	NOUN
iajs-3486	11	6	,	,	PUNCT
iajs-3486	11	7	random	random	ADJ
iajs-3486	11	8	operators	operator	NOUN
iajs-3486	11	9	,	,	PUNCT
iajs-3486	11	10	one	one	NUM
iajs-3486	11	11	-	-	PUNCT
iajs-3486	11	12	step	step	NOUN
iajs-3486	11	13	iteration	iteration	NOUN
iajs-3486	11	14	,	,	PUNCT
iajs-3486	11	15	banach	banach	NOUN
iajs-3486	11	16	spaces	space	VERB
iajs-3486	11	17	.	.	PUNCT
iajs-3486	12	1	1	1	X
iajs-3486	12	2	.	.	X
iajs-3486	12	3	introduction	introduction	NOUN
iajs-3486	12	4	the	the	DET
iajs-3486	12	5	random	random	ADJ
iajs-3486	12	6	fixed	fix	VERB
iajs-3486	12	7	point	point	NOUN
iajs-3486	12	8	theories	theory	NOUN
iajs-3486	12	9	are	be	AUX
iajs-3486	12	10	a	a	DET
iajs-3486	12	11	generalizations	generalization	NOUN
iajs-3486	12	12	of	of	ADP
iajs-3486	12	13	the	the	DET
iajs-3486	12	14	classical	classical	ADJ
iajs-3486	12	15	fixed	fix	VERB
iajs-3486	12	16	point	point	NOUN
iajs-3486	12	17	theories	theory	NOUN
iajs-3486	12	18	.	.	PUNCT
iajs-3486	13	1	in	in	ADP
iajs-3486	13	2	the	the	DET
iajs-3486	13	3	1950s	1950	NOUN
iajs-3486	13	4	the	the	DET
iajs-3486	13	5	probability	probability	NOUN
iajs-3486	13	6	school	school	NOUN
iajs-3486	13	7	in	in	ADP
iajs-3486	13	8	prague	prague	PROPN
iajs-3486	13	9	presented	present	VERB
iajs-3486	13	10	a	a	DET
iajs-3486	13	11	study	study	NOUN
iajs-3486	13	12	on	on	ADP
iajs-3486	13	13	random	random	ADJ
iajs-3486	13	14	fixed	fix	VERB
iajs-3486	13	15	point	point	NOUN
iajs-3486	13	16	theory	theory	NOUN
iajs-3486	13	17	[	[	X
iajs-3486	13	18	1].on	1].on	NUM
iajs-3486	13	19	other	other	ADJ
iajs-3486	13	20	hand,[2	hand,[2	PROPN
iajs-3486	13	21	]	]	PUNCT
iajs-3486	13	22	obtain	obtain	VERB
iajs-3486	13	23	common	common	ADJ
iajs-3486	13	24	random	random	ADJ
iajs-3486	13	25	fixed	fix	VERB
iajs-3486	13	26	point	point	NOUN
iajs-3486	13	27	of	of	ADP
iajs-3486	13	28	two	two	NUM
iajs-3486	13	29	multivalued	multivalue	VERB
iajs-3486	13	30	random	random	ADJ
iajs-3486	13	31	operator	operator	NOUN
iajs-3486	13	32	.recently	.recently	ADV
iajs-3486	13	33	,	,	PUNCT
iajs-3486	13	34	s.	s.	PROPN
iajs-3486	13	35	h.	h.	PROPN
iajs-3486	13	36	khan	khan	PROPN
iajs-3486	13	37	et	et	PROPN
iajs-3486	13	38	al.[3]introduce	al.[3]introduce	PROPN
iajs-3486	13	39	a	a	DET
iajs-3486	13	40	new	new	ADJ
iajs-3486	13	41	one	one	NUM
iajs-3486	13	42	-	-	PUNCT
iajs-3486	13	43	step	step	NOUN
iajs-3486	13	44	iterative	iterative	NOUN
iajs-3486	13	45	process	process	NOUN
iajs-3486	13	46	to	to	PART
iajs-3486	13	47	find	find	VERB
iajs-3486	13	48	the	the	DET
iajs-3486	13	49	common	common	ADJ
iajs-3486	13	50	random	random	ADJ
iajs-3486	13	51	fixed	fix	VERB
iajs-3486	13	52	point	point	NOUN
iajs-3486	13	53	of	of	ADP
iajs-3486	13	54	two	two	NUM
iajs-3486	13	55	multivalued	multivalued	ADJ
iajs-3486	13	56	non	non	ADJ
iajs-3486	13	57	-	-	ADJ
iajs-3486	13	58	expansive	expansive	ADJ
iajs-3486	13	59	random	random	ADJ
iajs-3486	13	60	operator	operator	NOUN
iajs-3486	13	61	.	.	PUNCT
iajs-3486	14	1	,	,	PUNCT
iajs-3486	14	2	the	the	DET
iajs-3486	14	3	nonlinear	nonlinear	ADJ
iajs-3486	14	4	random	random	ADJ
iajs-3486	14	5	systems	system	NOUN
iajs-3486	14	6	have	have	AUX
iajs-3486	14	7	appeared	appear	VERB
iajs-3486	14	8	in	in	ADP
iajs-3486	14	9	the	the	DET
iajs-3486	14	10	literature	literature	NOUN
iajs-3486	14	11	(	(	PUNCT
iajs-3486	14	12	see	see	VERB
iajs-3486	14	13	[	[	X
iajs-3486	14	14	4	4	NUM
iajs-3486	14	15	-	-	SYM
iajs-3486	14	16	17	17	NUM
iajs-3486	14	17	]	]	PUNCT
iajs-3486	14	18	)	)	PUNCT
iajs-3486	14	19	.	.	PUNCT
iajs-3486	15	1	let	let	VERB
iajs-3486	15	2	𝛷	𝛷	NOUN
iajs-3486	15	3	be	be	AUX
iajs-3486	15	4	a	a	DET
iajs-3486	15	5	separable	separable	ADJ
iajs-3486	15	6	banach	banach	NOUN
iajs-3486	15	7	space	space	NOUN
iajs-3486	15	8	.	.	PUNCT
iajs-3486	16	1	let𝛶	let𝛶	ADJ
iajs-3486	16	2	is	be	AUX
iajs-3486	16	3	subset	subset	VERB
iajs-3486	16	4	of	of	ADP
iajs-3486	16	5	𝛷is	𝛷is	PROPN
iajs-3486	16	6	called	call	VERB
iajs-3486	16	7	proximinal	proximinal	ADJ
iajs-3486	16	8	if	if	SCONJ
iajs-3486	16	9	∀𝑢	∀𝑢	DET
iajs-3486	16	10	∈	∈	PROPN
iajs-3486	16	11	𝛷	𝛷	PROPN
iajs-3486	16	12	,	,	PUNCT
iajs-3486	16	13	∃𝑘	∃𝑘	PROPN
iajs-3486	16	14	∈	∈	PROPN
iajs-3486	16	15	𝛶	𝛶	PROPN
iajs-3486	16	16	∋	∋	NOUN
iajs-3486	16	17	𝑑(𝑢	𝑑(𝑢	PROPN
iajs-3486	16	18	,	,	PUNCT
iajs-3486	16	19	𝑘	𝑘	NOUN
iajs-3486	16	20	)	)	PUNCT
iajs-3486	16	21	=	=	SYM
iajs-3486	16	22	inf{‖𝑢	inf{‖𝑢	PUNCT
iajs-3486	16	23	−	−	PROPN
iajs-3486	16	24	𝑣‖	𝑣‖	PROPN
iajs-3486	16	25	∶	∶	NOUN
iajs-3486	16	26	𝑣	𝑣	ADP
iajs-3486	16	27	∈	∈	NOUN
iajs-3486	16	28	𝛶	𝛶	PROPN
iajs-3486	16	29	}	}	PUNCT
iajs-3486	16	30	=	=	SYM
iajs-3486	16	31	𝑑(𝑢	𝑑(𝑢	ADJ
iajs-3486	16	32	,	,	PUNCT
iajs-3486	16	33	𝛶	𝛶	NOUN
iajs-3486	16	34	)	)	PUNCT
iajs-3486	16	35	.	.	PUNCT
iajs-3486	17	1	we	we	PRON
iajs-3486	17	2	denote	denote	VERB
iajs-3486	17	3	the	the	DET
iajs-3486	17	4	set	set	NOUN
iajs-3486	17	5	of	of	ADP
iajs-3486	17	6	all	all	DET
iajs-3486	17	7	subsets	subset	NOUN
iajs-3486	17	8	bounded	bound	VERB
iajs-3486	17	9	proximinal	proximinal	ADJ
iajs-3486	17	10	of𝛶	of𝛶	NOUN
iajs-3486	17	11	by	by	ADP
iajs-3486	17	12	𝛱(𝛶	𝛱(𝛶	NOUN
iajs-3486	17	13	)	)	PUNCT
iajs-3486	18	1	[	[	X
iajs-3486	18	2	18	18	NUM
iajs-3486	18	3	]	]	PUNCT
iajs-3486	18	4	.	.	PUNCT
iajs-3486	19	1	let	let	VERB
iajs-3486	19	2	𝐶𝐵(𝛶	𝐶𝐵(𝛶	NOUN
iajs-3486	19	3	)	)	PUNCT
iajs-3486	19	4	be	be	AUX
iajs-3486	19	5	be	be	AUX
iajs-3486	19	6	all	all	PRON
iajs-3486	19	7	closed	closed	ADJ
iajs-3486	19	8	boundedsubsets	boundedsubset	NOUN
iajs-3486	19	9	of	of	ADP
iajs-3486	19	10	𝛶.	𝛶.	PROPN
iajs-3486	19	11	and	and	CCONJ
iajs-3486	19	12	let	let	VERB
iajs-3486	19	13	his	his	PRON
iajs-3486	19	14	hausdorff	hausdorff	NOUN
iajs-3486	19	15	induced	induce	VERB
iajs-3486	19	16	by	by	ADP
iajs-3486	19	17	the	the	DET
iajs-3486	19	18	metric	metric	ADJ
iajs-3486	19	19	space	space	NOUN
iajs-3486	19	20	𝑑	𝑑	PROPN
iajs-3486	19	21	of	of	ADP
iajs-3486	19	22	𝛷	𝛷	PROPN
iajs-3486	19	23	,	,	PUNCT
iajs-3486	19	24	implies	imply	VERB
iajs-3486	19	25	h(𝜛	h(𝜛	PROPN
iajs-3486	19	26	,	,	PUNCT
iajs-3486	19	27	β	β	X
iajs-3486	19	28	)	)	PUNCT
iajs-3486	19	29	=	=	SYM
iajs-3486	19	30	𝑚𝑎𝑥{𝑠𝑢𝑝	𝑚𝑎𝑥{𝑠𝑢𝑝	ADJ
iajs-3486	19	31	𝑑(𝑢	𝑑(𝑢	ADJ
iajs-3486	19	32	,	,	PUNCT
iajs-3486	19	33	β)𝑢∈𝜛	β)𝑢∈𝜛	PROPN
iajs-3486	19	34	,	,	PUNCT
iajs-3486	19	35	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
iajs-3486	19	36	𝑑(𝑣	𝑑(𝑣	NOUN
iajs-3486	19	37	,	,	PUNCT
iajs-3486	19	38	𝜛	𝜛	NOUN
iajs-3486	19	39	)	)	PUNCT
iajs-3486	19	40	𝑣∈β	𝑣∈β	NOUN
iajs-3486	19	41	}	}	PUNCT
iajs-3486	19	42	for	for	ADP
iajs-3486	19	43	every	every	DET
iajs-3486	19	44	𝜛	𝜛	PROPN
iajs-3486	19	45	,	,	PUNCT
iajs-3486	19	46	β	β	PROPN
iajs-3486	19	47	∈	∈	PROPN
iajs-3486	19	48	𝐶𝐵(𝛶	𝐶𝐵(𝛶	NOUN
iajs-3486	19	49	)	)	PUNCT
iajs-3486	19	50	.	.	PUNCT
iajs-3486	20	1	a	a	DET
iajs-3486	20	2	multivalued	multivalue	VERB
iajs-3486	20	3	random	random	ADJ
iajs-3486	20	4	operator	operator	NOUN
iajs-3486	20	5	𝐵	𝐵	NOUN
iajs-3486	20	6	:	:	PUNCT
iajs-3486	20	7	𝛹	𝛹	PROPN
iajs-3486	20	8	×	×	NOUN
iajs-3486	20	9	𝛶	𝛶	PROPN
iajs-3486	20	10	→	→	SYM
iajs-3486	20	11	𝛱(𝛶)is	𝛱(𝛶)is	ADJ
iajs-3486	20	12	called	call	VERB
iajs-3486	20	13	contraction	contraction	NOUN
iajs-3486	20	14	if	if	SCONJ
iajs-3486	20	15	there	there	PRON
iajs-3486	20	16	is𝑘(ϱ	is𝑘(ϱ	NOUN
iajs-3486	20	17	)	)	PUNCT
iajs-3486	20	18	∈	∈	PROPN
iajs-3486	21	1	[	[	X
iajs-3486	21	2	0,1	0,1	NUM
iajs-3486	21	3	)	)	PUNCT
iajs-3486	21	4	and	and	CCONJ
iajs-3486	21	5	received	receive	VERB
iajs-3486	21	6	:	:	PUNCT
iajs-3486	21	7	13	13	NUM
iajs-3486	21	8	may	may	PROPN
iajs-3486	21	9	2023	2023	NUM
iajs-3486	21	10	accepted	accept	VERB
iajs-3486	21	11	:	:	PUNCT
iajs-3486	21	12	22	22	NUM
iajs-3486	21	13	june	june	PROPN
iajs-3486	21	14	2023	2023	NUM
iajs-3486	21	15	published	publish	VERB
iajs-3486	21	16	:	:	PUNCT
iajs-3486	21	17	20	20	NUM
iajs-3486	21	18	april	april	PROPN
iajs-3486	21	19	2024	2024	NUM
iajs-3486	21	20	common	common	ADJ
iajs-3486	21	21	fixed	fix	VERB
iajs-3486	21	22	points	point	NOUN
iajs-3486	21	23	of	of	ADP
iajs-3486	21	24	three	three	NUM
iajs-3486	21	25	multivalued	multivalue	VERB
iajs-3486	21	26	nonexpansive	nonexpansive	ADJ
iajs-3486	21	27	random	random	ADJ
iajs-3486	21	28	operators	operator	NOUN
iajs-3486	21	29	for	for	ADP
iajs-3486	21	30	one	one	NUM
iajs-3486	21	31	step	step	NOUN
iajs-3486	21	32	iterative	iterative	NOUN
iajs-3486	21	33	scheme	scheme	NOUN
iajs-3486	21	34	doi.org/10.30526/37.2.3486	doi.org/10.30526/37.2.3486	PROPN
iajs-3486	21	35	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3486	21	36	https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042	https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042	NOUN
iajs-3486	21	37	https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407	https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407	VERB
iajs-3486	21	38	https://orcid.org/0009-0009-8635-2210	https://orcid.org/0009-0009-8635-2210	NOUN
iajs-3486	21	39	mailto:wassoalraqy@gmail.com	mailto:wassoalraqy@gmail.com	PROPN
iajs-3486	22	1	https://orcid.org/0000-0002-1600-3631	https://orcid.org/0000-0002-1600-3631	PROPN
iajs-3486	22	2	mailto:sabah.h.m@ihcoedu.uobaghdad.edu.iq	mailto:sabah.h.m@ihcoedu.uobaghdad.edu.iq	VERB
iajs-3486	22	3	https://orcid.org/0009-0002-9351-4176	https://orcid.org/0009-0002-9351-4176	PROPN
iajs-3486	22	4	mailto:shrooq.bahjat@utas.edu.om	mailto:shrooq.bahjat@utas.edu.om	PROPN
iajs-3486	22	5	ihjpas.37	ihjpas.37	PROPN
iajs-3486	22	6	(	(	PUNCT
iajs-3486	22	7	2	2	NUM
iajs-3486	22	8	)	)	PUNCT
iajs-3486	22	9	2024	2024	NUM
iajs-3486	22	10	425	425	NUM
iajs-3486	22	11	for	for	ADP
iajs-3486	22	12	each	each	DET
iajs-3486	22	13	ϱ	ϱ	PROPN
iajs-3486	22	14	∈	∈	PROPN
iajs-3486	22	15	ψ	ψ	ADP
iajs-3486	22	16	∋	∋	NOUN
iajs-3486	22	17	h	h	NOUN
iajs-3486	22	18	(	(	PUNCT
iajs-3486	22	19	𝐵(ϱ	𝐵(ϱ	PROPN
iajs-3486	22	20	,	,	PUNCT
iajs-3486	22	21	𝑢	𝑢	NOUN
iajs-3486	22	22	)	)	PUNCT
iajs-3486	22	23	,	,	PUNCT
iajs-3486	22	24	𝐵(ϱ	𝐵(ϱ	PRON
iajs-3486	22	25	,	,	PUNCT
iajs-3486	22	26	𝑣	𝑣	NOUN
iajs-3486	22	27	)	)	PUNCT
iajs-3486	22	28	)	)	PUNCT
iajs-3486	23	1	≤	≤	NOUN
iajs-3486	23	2	𝑘(ϱ)‖𝑢	𝑘(ϱ)‖𝑢	VERB
iajs-3486	23	3	−	−	PROPN
iajs-3486	23	4	𝑣‖for	𝑣‖for	ADP
iajs-3486	23	5	all	all	DET
iajs-3486	23	6	𝑢	𝑢	NOUN
iajs-3486	23	7	,	,	PUNCT
iajs-3486	23	8	𝑣	𝑣	PROPN
iajs-3486	23	9	∈	∈	PROPN
iajs-3486	23	10	𝛶	𝛶	PROPN
iajs-3486	23	11	and	and	CCONJ
iajs-3486	23	12	𝐵	𝐵	PROPN
iajs-3486	23	13	is	be	AUX
iajs-3486	23	14	said	say	VERB
iajs-3486	23	15	to	to	PART
iajs-3486	23	16	be	be	AUX
iajs-3486	23	17	nonexpansive	nonexpansive	ADJ
iajs-3486	23	18	random	random	ADJ
iajs-3486	23	19	operator	operator	NOUN
iajs-3486	23	20	if	if	SCONJ
iajs-3486	23	21	for	for	ADP
iajs-3486	23	22	each	each	DET
iajs-3486	23	23	ϱ	ϱ	ADP
iajs-3486	23	24	∈	∈	PROPN
iajs-3486	23	25	ψ	ψ	X
iajs-3486	23	26	h	h	X
iajs-3486	23	27	(	(	PUNCT
iajs-3486	23	28	𝐵(ϱ	𝐵(ϱ	PROPN
iajs-3486	23	29	,	,	PUNCT
iajs-3486	23	30	𝑢	𝑢	NOUN
iajs-3486	23	31	)	)	PUNCT
iajs-3486	23	32	,	,	PUNCT
iajs-3486	23	33	𝐵(ϱ	𝐵(ϱ	PRON
iajs-3486	23	34	,	,	PUNCT
iajs-3486	23	35	𝑣	𝑣	NOUN
iajs-3486	23	36	)	)	PUNCT
iajs-3486	23	37	)	)	PUNCT
iajs-3486	24	1	≤	≤	NOUN
iajs-3486	24	2	‖𝑢	‖𝑢	NOUN
iajs-3486	24	3	−	−	ADP
iajs-3486	24	4	𝑣‖	𝑣‖	PROPN
iajs-3486	24	5	,	,	PUNCT
iajs-3486	24	6	a	a	DET
iajs-3486	24	7	point	point	NOUN
iajs-3486	24	8	ϑ(ϱ	ϑ(ϱ	NUM
iajs-3486	24	9	)	)	PUNCT
iajs-3486	24	10	is	be	AUX
iajs-3486	24	11	called	call	VERB
iajs-3486	24	12	random	random	ADJ
iajs-3486	24	13	fixed	fix	VERB
iajs-3486	24	14	point	point	NOUN
iajs-3486	24	15	of	of	ADP
iajs-3486	24	16	𝐵	𝐵	PRON
iajs-3486	24	17	if	if	SCONJ
iajs-3486	24	18	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	24	19	)	)	PUNCT
iajs-3486	24	20	∈	∈	PROPN
iajs-3486	24	21	𝐵(ϑ(ϱ	𝐵(ϑ(ϱ	PROPN
iajs-3486	24	22	)	)	PUNCT
iajs-3486	24	23	)	)	PUNCT
iajs-3486	24	24	.	.	PUNCT
iajs-3486	25	1	study	study	VERB
iajs-3486	25	2	the	the	DET
iajs-3486	25	3	fixed	fix	VERB
iajs-3486	25	4	point	point	NOUN
iajs-3486	25	5	of	of	ADP
iajs-3486	25	6	results	result	NOUN
iajs-3486	25	7	paper	paper	NOUN
iajs-3486	26	1	[	[	X
iajs-3486	26	2	22	22	NUM
iajs-3486	26	3	-	-	SYM
iajs-3486	26	4	30	30	NUM
iajs-3486	26	5	]	]	PUNCT
iajs-3486	26	6	under	under	ADP
iajs-3486	26	7	multivalued	multivalued	ADJ
iajs-3486	26	8	non	non	ADJ
iajs-3486	26	9	-	-	ADJ
iajs-3486	26	10	expansive	expansive	ADJ
iajs-3486	26	11	random	random	ADJ
iajs-3486	26	12	operator	operator	NOUN
iajs-3486	26	13	.	.	PUNCT
iajs-3486	27	1	we	we	PRON
iajs-3486	27	2	will	will	AUX
iajs-3486	27	3	give	give	VERB
iajs-3486	27	4	some	some	DET
iajs-3486	27	5	definition	definition	NOUN
iajs-3486	27	6	…	…	PUNCT
iajs-3486	27	7	definition	definition	NOUN
iajs-3486	27	8	1.1.[19	1.1.[19	NUM
iajs-3486	27	9	]	]	PUNCT
iajs-3486	27	10	:	:	PUNCT
iajs-3486	27	11	a	a	DET
iajs-3486	27	12	separable	separable	ADJ
iajs-3486	27	13	banach	banach	NOUN
iajs-3486	27	14	space	space	NOUN
iajs-3486	27	15	𝛷	𝛷	NOUN
iajs-3486	27	16	is	be	AUX
iajs-3486	27	17	said	say	VERB
iajs-3486	27	18	to	to	PART
iajs-3486	27	19	satisfy	satisfy	VERB
iajs-3486	27	20	opials	opial	NOUN
iajs-3486	27	21	condition	condition	NOUN
iajs-3486	27	22	if	if	SCONJ
iajs-3486	27	23	the	the	DET
iajs-3486	27	24	sequence	sequence	NOUN
iajs-3486	27	25	{	{	PUNCT
iajs-3486	27	26	𝑆𝑛	𝑆𝑛	NOUN
iajs-3486	27	27	}	}	PUNCT
iajs-3486	27	28	in	in	ADP
iajs-3486	27	29	𝛷	𝛷	PROPN
iajs-3486	27	30	,	,	PUNCT
iajs-3486	27	31	𝑆𝑛	𝑆𝑛	PROPN
iajs-3486	27	32	→	→	NOUN
iajs-3486	27	33	𝑢	𝑢	NOUN
iajs-3486	27	34	implies	imply	VERB
iajs-3486	27	35	that	that	SCONJ
iajs-3486	27	36	lim	lim	PROPN
iajs-3486	27	37	𝑛→∞	𝑛→∞	NUM
iajs-3486	27	38	𝑠𝑢𝑝‖𝑆𝑛	𝑠𝑢𝑝‖𝑆𝑛	PROPN
iajs-3486	27	39	−	−	PROPN
iajs-3486	27	40	𝑢‖	𝑢‖	PROPN
iajs-3486	27	41	<	<	X
iajs-3486	27	42	lim	lim	PROPN
iajs-3486	27	43	𝑛→∞	𝑛→∞	NUM
iajs-3486	27	44	𝑠𝑢𝑝‖𝑆𝑛	𝑠𝑢𝑝‖𝑆𝑛	PROPN
iajs-3486	27	45	−	−	X
iajs-3486	27	46	𝑣‖	𝑣‖	NOUN
iajs-3486	27	47	for	for	ADP
iajs-3486	27	48	all	all	PRON
iajs-3486	27	49	𝑣	𝑣	DET
iajs-3486	27	50	∈	∈	PROPN
iajs-3486	27	51	𝛷	𝛷	PROPN
iajs-3486	27	52	,	,	PUNCT
iajs-3486	27	53	𝑣	𝑣	DET
iajs-3486	27	54	≠	≠	PROPN
iajs-3486	27	55	𝑢.	𝑢.	NOUN
iajs-3486	27	56	definition	definition	NOUN
iajs-3486	27	57	1.2	1.2	NUM
iajs-3486	27	58	.	.	PUNCT
iajs-3486	28	1	let	let	VERB
iajs-3486	28	2	υ	υ	NOUN
iajs-3486	28	3	subset	subset	NOUN
iajs-3486	28	4	of	of	ADP
iajs-3486	28	5	𝛷	𝛷	PROPN
iajs-3486	28	6	,	,	PUNCT
iajs-3486	28	7	and	and	CCONJ
iajs-3486	28	8	ℛ	ℛ	PROPN
iajs-3486	28	9	,	,	PUNCT
iajs-3486	28	10	ζ	ζ	NOUN
iajs-3486	28	11	,	,	PUNCT
iajs-3486	28	12	𝐵	𝐵	NOUN
iajs-3486	28	13	∶	∶	NOUN
iajs-3486	28	14	ψ	ψ	X
iajs-3486	28	15	×	×	NOUN
iajs-3486	28	16	υ	υ	PROPN
iajs-3486	28	17	→	→	SYM
iajs-3486	28	18	𝛱(υ)three	𝛱(υ)three	NOUN
iajs-3486	28	19	multivalued	multivalue	VERB
iajs-3486	28	20	nonexpansive	nonexpansive	ADJ
iajs-3486	28	21	random	random	ADJ
iajs-3486	28	22	operator	operator	NOUN
iajs-3486	28	23	are	be	AUX
iajs-3486	28	24	satisfy	satisfy	VERB
iajs-3486	28	25	the	the	DET
iajs-3486	28	26	condition	condition	NOUN
iajs-3486	28	27	(	(	PUNCT
iajs-3486	28	28	𝐴′	𝐴′	PROPN
iajs-3486	28	29	)	)	PUNCT
iajs-3486	28	30	if	if	SCONJ
iajs-3486	28	31	∃a	∃a	NOUN
iajs-3486	28	32	non	non	ADJ
iajs-3486	28	33	-	-	ADJ
iajs-3486	28	34	decreasing	decrease	VERB
iajs-3486	28	35	function	function	NOUN
iajs-3486	28	36	𝜒	𝜒	NOUN
iajs-3486	28	37	:	:	PUNCT
iajs-3486	28	38	[	[	X
iajs-3486	28	39	0	0	NUM
iajs-3486	28	40	,	,	PUNCT
iajs-3486	28	41	∞	∞	PROPN
iajs-3486	28	42	)	)	PUNCT
iajs-3486	28	43	→	→	PUNCT
iajs-3486	29	1	[	[	X
iajs-3486	29	2	0	0	NUM
iajs-3486	29	3	,	,	PUNCT
iajs-3486	29	4	∞	∞	PROPN
iajs-3486	29	5	)	)	PUNCT
iajs-3486	29	6	with	with	ADP
iajs-3486	29	7	𝜒(0	𝜒(0	PROPN
iajs-3486	29	8	)	)	PUNCT
iajs-3486	29	9	=	=	SYM
iajs-3486	29	10	0	0	NUM
iajs-3486	29	11	,	,	PUNCT
iajs-3486	29	12	𝜒(𝑟	𝜒(𝑟	NOUN
iajs-3486	29	13	)	)	PUNCT
iajs-3486	29	14	>	>	X
iajs-3486	29	15	0	0	NUM
iajs-3486	29	16	,	,	PUNCT
iajs-3486	29	17	∀𝑟	∀𝑟	PROPN
iajs-3486	29	18	∈	∈	PROPN
iajs-3486	29	19	(	(	PUNCT
iajs-3486	29	20	0	0	NUM
iajs-3486	29	21	,	,	PUNCT
iajs-3486	29	22	∞	∞	NUM
iajs-3486	29	23	)	)	PUNCT
iajs-3486	29	24	∋	∋	NOUN
iajs-3486	29	25	either	either	CCONJ
iajs-3486	29	26	𝑑(𝑢(𝜚	𝑑(𝑢(𝜚	PROPN
iajs-3486	29	27	)	)	PUNCT
iajs-3486	29	28	,	,	PUNCT
iajs-3486	29	29	𝐵	𝐵	NOUN
iajs-3486	29	30	𝑢(𝜚	𝑢(𝜚	NOUN
iajs-3486	29	31	)	)	PUNCT
iajs-3486	29	32	)	)	PUNCT
iajs-3486	29	33	≥	≥	NOUN
iajs-3486	29	34	𝜒(𝑑(𝑢(𝜚	𝜒(𝑑(𝑢(𝜚	NUM
iajs-3486	29	35	)	)	PUNCT
iajs-3486	29	36	,	,	PUNCT
iajs-3486	29	37	𝐹))or	𝐹))or	VERB
iajs-3486	29	38	𝑑(𝑢(𝜚	𝑑(𝑢(𝜚	NOUN
iajs-3486	29	39	)	)	PUNCT
iajs-3486	29	40	,	,	PUNCT
iajs-3486	29	41	ζ𝑢(𝜚	ζ𝑢(𝜚	NOUN
iajs-3486	29	42	)	)	PUNCT
iajs-3486	29	43	)	)	PUNCT
iajs-3486	29	44	≥	≥	NOUN
iajs-3486	29	45	𝜒(𝑑(𝑢(𝜚	𝜒(𝑑(𝑢(𝜚	NUM
iajs-3486	29	46	)	)	PUNCT
iajs-3486	29	47	,	,	PUNCT
iajs-3486	29	48	𝐹))or	𝐹))or	VERB
iajs-3486	29	49	𝑑(𝑢(𝜚	𝑑(𝑢(𝜚	NOUN
iajs-3486	29	50	)	)	PUNCT
iajs-3486	29	51	,	,	PUNCT
iajs-3486	29	52	r𝑢(𝜚	r𝑢(𝜚	NUM
iajs-3486	29	53	)	)	PUNCT
iajs-3486	29	54	)	)	PUNCT
iajs-3486	30	1	≥	≥	NOUN
iajs-3486	30	2	𝜒(𝑑(𝑢(𝜚	𝜒(𝑑(𝑢(𝜚	NUM
iajs-3486	30	3	)	)	PUNCT
iajs-3486	30	4	,	,	PUNCT
iajs-3486	30	5	𝐹	𝐹	PROPN
iajs-3486	30	6	)	)	PUNCT
iajs-3486	30	7	)	)	PUNCT
iajs-3486	30	8	for	for	ADP
iajs-3486	30	9	all	all	DET
iajs-3486	30	10	𝑢(𝜚	𝑢(𝜚	NOUN
iajs-3486	30	11	)	)	PUNCT
iajs-3486	30	12	∈	∈	PROPN
iajs-3486	30	13	υ	υ	PRON
iajs-3486	30	14	definition	definition	NOUN
iajs-3486	30	15	1.3	1.3	NUM
iajs-3486	30	16	let	let	VERB
iajs-3486	30	17	𝛶	𝛶	PROPN
iajs-3486	30	18	be	be	AUX
iajs-3486	30	19	a	a	DET
iajs-3486	30	20	non	non	X
iajs-3486	30	21	empty	empty	ADJ
iajs-3486	30	22	subset	subset	NOUN
iajs-3486	30	23	of	of	ADP
iajs-3486	30	24	a	a	DET
iajs-3486	30	25	separable	separable	ADJ
iajs-3486	30	26	banach	banach	NOUN
iajs-3486	30	27	space	space	NOUN
iajs-3486	30	28	𝛷satisfying	𝛷satisfye	VERB
iajs-3486	30	29	opials	opial	NOUN
iajs-3486	30	30	condition	condition	NOUN
iajs-3486	30	31	and	and	CCONJ
iajs-3486	30	32	𝐵	𝐵	NOUN
iajs-3486	30	33	∶	∶	NOUN
iajs-3486	30	34	ψ	ψ	ADP
iajs-3486	30	35	×	×	PROPN
iajs-3486	30	36	𝛶	𝛶	PROPN
iajs-3486	30	37	→	→	SYM
iajs-3486	30	38	𝛱(𝛶	𝛱(𝛶	NOUN
iajs-3486	30	39	)	)	PUNCT
iajs-3486	30	40	be	be	AUX
iajs-3486	30	41	amultivalued	amultivalue	VERB
iajs-3486	30	42	random	random	ADJ
iajs-3486	30	43	operator	operator	NOUN
iajs-3486	30	44	is	be	AUX
iajs-3486	30	45	said	say	VERB
iajs-3486	30	46	to	to	PART
iajs-3486	30	47	be	be	AUX
iajs-3486	30	48	demiclosed	demiclose	VERB
iajs-3486	30	49	at	at	ADP
iajs-3486	30	50	𝜁(ϱ	𝜁(ϱ	NOUN
iajs-3486	30	51	)	)	PUNCT
iajs-3486	30	52	)	)	PUNCT
iajs-3486	31	1	if	if	SCONJ
iajs-3486	31	2	{	{	PUNCT
iajs-3486	31	3	𝑆𝑛(𝜚	𝑆𝑛(𝜚	NOUN
iajs-3486	31	4	)	)	PUNCT
iajs-3486	31	5	}	}	PUNCT
iajs-3486	31	6	and	and	CCONJ
iajs-3486	31	7	{	{	PUNCT
iajs-3486	31	8	ζ𝑛(𝜚	ζ𝑛(𝜚	ADV
iajs-3486	31	9	)	)	PUNCT
iajs-3486	31	10	}	}	PUNCT
iajs-3486	31	11	are	be	AUX
iajs-3486	31	12	two	two	NUM
iajs-3486	31	13	sequences	sequence	NOUN
iajs-3486	31	14	∋	∋	NOUN
iajs-3486	31	15	{	{	PUNCT
iajs-3486	31	16	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	31	17	)	)	PUNCT
iajs-3486	31	18	}	}	PUNCT
iajs-3486	31	19	converges	converge	VERB
iajs-3486	31	20	weak	weak	ADJ
iajs-3486	31	21	to	to	ADP
iajs-3486	31	22	𝑆(𝜚)also	𝑆(𝜚)also	PROPN
iajs-3486	31	23	{	{	PUNCT
iajs-3486	31	24	𝐵((ϱ	𝐵((ϱ	PROPN
iajs-3486	31	25	)	)	PUNCT
iajs-3486	31	26	,	,	PUNCT
iajs-3486	31	27	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	31	28	)	)	PUNCT
iajs-3486	31	29	}	}	PUNCT
iajs-3486	31	30	converges	converge	VERB
iajs-3486	31	31	to	to	ADP
iajs-3486	31	32	𝜁(𝜚	𝜁(𝜚	NOUN
iajs-3486	31	33	)	)	PUNCT
iajs-3486	31	34	imply	imply	VERB
iajs-3486	31	35	that	that	SCONJ
iajs-3486	31	36	𝑆(𝜚	𝑆(𝜚	X
iajs-3486	31	37	)	)	PUNCT
iajs-3486	31	38	∈	∈	PROPN
iajs-3486	31	39	𝛶	𝛶	PROPN
iajs-3486	31	40	and	and	CCONJ
iajs-3486	31	41	𝜁(𝜚	𝜁(𝜚	PROPN
iajs-3486	31	42	)	)	PUNCT
iajs-3486	31	43	𝜖	𝜖	PROPN
iajs-3486	31	44	𝐵(𝑆(𝜚	𝐵(𝑆(𝜚	PROPN
iajs-3486	31	45	)	)	PUNCT
iajs-3486	31	46	)	)	PUNCT
iajs-3486	31	47	for	for	ADP
iajs-3486	31	48	each	each	DET
iajs-3486	31	49	ϱ	ϱ	PROPN
iajs-3486	31	50	∈	∈	PROPN
iajs-3486	31	51	𝛹	𝛹	PROPN
iajs-3486	31	52	,	,	PUNCT
iajs-3486	31	53	then	then	ADV
iajs-3486	31	54	𝐼	𝐼	PROPN
iajs-3486	31	55	−	−	PROPN
iajs-3486	31	56	𝐵	𝐵	PROPN
iajs-3486	31	57	is	be	AUX
iajs-3486	31	58	dimiclosed	dimiclose	VERB
iajs-3486	31	59	with	with	ADP
iajs-3486	31	60	respect	respect	NOUN
iajs-3486	31	61	to	to	ADP
iajs-3486	31	62	0	0	NUM
iajs-3486	31	63	.	.	PUNCT
iajs-3486	32	1	lemma	lemma	PROPN
iajs-3486	32	2	1.4[20	1.4[20	PROPN
iajs-3486	32	3	]	]	PUNCT
iajs-3486	32	4	let	let	VERB
iajs-3486	32	5	{	{	PUNCT
iajs-3486	32	6	𝑆𝑛	𝑆𝑛	NOUN
iajs-3486	32	7	}	}	PUNCT
iajs-3486	32	8	,	,	PUNCT
iajs-3486	32	9	{	{	PUNCT
iajs-3486	32	10	𝜂𝑛	𝜂𝑛	NOUN
iajs-3486	32	11	)	)	PUNCT
iajs-3486	32	12	}	}	PUNCT
iajs-3486	32	13	,	,	PUNCT
iajs-3486	32	14	{	{	PUNCT
iajs-3486	32	15	𝜌𝑛	𝜌𝑛	NOUN
iajs-3486	32	16	}	}	PUNCT
iajs-3486	32	17	be	be	AUX
iajs-3486	32	18	a	a	DET
iajs-3486	32	19	sequence	sequence	NOUN
iajs-3486	32	20	in	in	ADP
iajs-3486	32	21	uniformly	uniformly	ADV
iajs-3486	32	22	convex	convex	VERB
iajs-3486	32	23	banach	banach	NOUN
iajs-3486	32	24	space	space	NOUN
iajs-3486	32	25	𝛷.	𝛷.	PROPN
iajs-3486	32	26	let	let	VERB
iajs-3486	32	27	{	{	PUNCT
iajs-3486	32	28	𝛼𝑛	𝛼𝑛	PROPN
iajs-3486	32	29	}	}	PUNCT
iajs-3486	32	30	,	,	PUNCT
iajs-3486	32	31	{	{	PUNCT
iajs-3486	32	32	𝜅𝑛	𝜅𝑛	ADP
iajs-3486	32	33	}	}	PUNCT
iajs-3486	32	34	,	,	PUNCT
iajs-3486	32	35	{	{	PUNCT
iajs-3486	32	36	𝜎𝑛	𝜎𝑛	NOUN
iajs-3486	32	37	}	}	PUNCT
iajs-3486	32	38	are	be	AUX
iajs-3486	32	39	sequence	sequence	NOUN
iajs-3486	32	40	in	in	ADP
iajs-3486	32	41	[	[	X
iajs-3486	32	42	0,1	0,1	NUM
iajs-3486	32	43	]	]	PUNCT
iajs-3486	32	44	with	with	ADP
iajs-3486	32	45	𝛼𝑛	𝛼𝑛	PROPN
iajs-3486	32	46	+	+	NUM
iajs-3486	32	47	𝜅𝑛	𝜅𝑛	ADP
iajs-3486	32	48	+	+	NOUN
iajs-3486	32	49	𝜎𝑛	𝜎𝑛	NOUN
iajs-3486	32	50	=	=	SYM
iajs-3486	32	51	1	1	NUM
iajs-3486	32	52	,	,	PUNCT
iajs-3486	32	53	lim	lim	NOUN
iajs-3486	32	54	𝑛→∞	𝑛→∞	NUM
iajs-3486	32	55	𝑠𝑢𝑝‖𝑆𝑛‖	𝑠𝑢𝑝‖𝑆𝑛‖	PROPN
iajs-3486	32	56	=	=	SYM
iajs-3486	32	57	𝑞	𝑞	PROPN
iajs-3486	32	58	,	,	PUNCT
iajs-3486	32	59	lim	lim	PROPN
iajs-3486	32	60	𝑛→∞	𝑛→∞	NUM
iajs-3486	32	61	𝑠𝑢𝑝‖𝜂𝑛‖	𝑠𝑢𝑝‖𝜂𝑛‖	PROPN
iajs-3486	32	62	=	=	SYM
iajs-3486	32	63	𝑞	𝑞	PROPN
iajs-3486	32	64	,	,	PUNCT
iajs-3486	32	65	lim	lim	PROPN
iajs-3486	32	66	𝑛→∞	𝑛→∞	NUM
iajs-3486	32	67	𝑠𝑢𝑝‖𝜌𝑛‖	𝑠𝑢𝑝‖𝜌𝑛‖	PROPN
iajs-3486	32	68	=	=	SYM
iajs-3486	32	69	𝑞	𝑞	NOUN
iajs-3486	32	70	and	and	CCONJ
iajs-3486	32	71	lim	lim	PROPN
iajs-3486	32	72	𝑛→∞	𝑛→∞	NUM
iajs-3486	32	73	‖𝛼𝑛𝑆𝑛	‖𝛼𝑛𝑆𝑛	PUNCT
iajs-3486	33	1	+	+	NUM
iajs-3486	33	2	𝜅𝑛𝜂𝑛	𝜅𝑛𝜂𝑛	ADJ
iajs-3486	33	3	+	+	CCONJ
iajs-3486	33	4	𝜎𝑛𝜌𝑛‖	𝜎𝑛𝜌𝑛‖	NUM
iajs-3486	33	5	=	=	SYM
iajs-3486	33	6	𝑞	𝑞	NOUN
iajs-3486	33	7	if	if	SCONJ
iajs-3486	33	8	lim	lim	PROPN
iajs-3486	33	9	𝑛→∞	𝑛→∞	NUM
iajs-3486	33	10	inf	inf	PROPN
iajs-3486	33	11	𝛼𝑛	𝛼𝑛	PROPN
iajs-3486	33	12	>	>	X
iajs-3486	33	13	0	0	PROPN
iajs-3486	33	14	,	,	PUNCT
iajs-3486	33	15	lim	lim	PROPN
iajs-3486	33	16	𝑛→∞	𝑛→∞	NUM
iajs-3486	33	17	inf	inf	PROPN
iajs-3486	33	18	𝜅𝑛	𝜅𝑛	ADP
iajs-3486	33	19	>	>	X
iajs-3486	33	20	0	0	PROPN
iajs-3486	33	21	,	,	PUNCT
iajs-3486	33	22	lim	lim	PROPN
iajs-3486	33	23	𝑛→∞	𝑛→∞	NUM
iajs-3486	33	24	inf	inf	PROPN
iajs-3486	33	25	𝜎𝑛	𝜎𝑛	ADP
iajs-3486	33	26	>	>	X
iajs-3486	33	27	0	0	PROPN
iajs-3486	33	28	,	,	PUNCT
iajs-3486	33	29	then	then	ADV
iajs-3486	33	30	lim	lim	PROPN
iajs-3486	33	31	𝑛→∞	𝑛→∞	NUM
iajs-3486	33	32	‖𝑆𝑛	‖𝑆𝑛	PUNCT
iajs-3486	33	33	−	−	NOUN
iajs-3486	33	34	𝜂𝑛‖	𝜂𝑛‖	NUM
iajs-3486	34	1	=	=	SYM
iajs-3486	34	2	lim	lim	PROPN
iajs-3486	34	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	34	4	‖𝑆𝑛	‖𝑆𝑛	PUNCT
iajs-3486	34	5	−	−	PUNCT
iajs-3486	34	6	𝜌𝑛‖	𝜌𝑛‖	NOUN
iajs-3486	34	7	=	=	SYM
iajs-3486	34	8	lim	lim	PROPN
iajs-3486	34	9	𝑛→∞	𝑛→∞	NUM
iajs-3486	34	10	‖𝜌𝑛	‖𝜌𝑛	PROPN
iajs-3486	34	11	−	−	PROPN
iajs-3486	34	12	𝜂𝑛‖	𝜂𝑛‖	PROPN
iajs-3486	34	13	=	=	SYM
iajs-3486	34	14	0	0	NUM
iajs-3486	34	15	.	.	NOUN
iajs-3486	34	16	2	2	NUM
iajs-3486	34	17	.	.	X
iajs-3486	34	18	main	main	ADJ
iajs-3486	34	19	results	result	NOUN
iajs-3486	34	20	idefinition2.1	idefinition2.1	NUM
iajs-3486	34	21	let	let	VERB
iajs-3486	34	22	𝛷	𝛷	NOUN
iajs-3486	34	23	be	be	AUX
iajs-3486	34	24	a	a	DET
iajs-3486	34	25	separable	separable	ADJ
iajs-3486	34	26	banach	banach	NOUN
iajs-3486	34	27	space	space	NOUN
iajs-3486	34	28	and	and	CCONJ
iajs-3486	34	29	υ	υ	PRON
iajs-3486	34	30	≠	≠	PROPN
iajs-3486	34	31	∅	∅	NOUN
iajs-3486	34	32	closed	close	VERB
iajs-3486	34	33	subset	subset	NOUN
iajs-3486	34	34	and	and	CCONJ
iajs-3486	34	35	convex	convex	PROPN
iajs-3486	34	36	.we	.we	PUNCT
iajs-3486	34	37	can	can	AUX
iajs-3486	34	38	be	be	AUX
iajs-3486	34	39	written	write	VERB
iajs-3486	34	40	𝐹	𝐹	PROPN
iajs-3486	35	1	=	=	SYM
iajs-3486	35	2	𝐹	𝐹	PROPN
iajs-3486	35	3	(	(	PUNCT
iajs-3486	35	4	𝐵	𝐵	NOUN
iajs-3486	35	5	)	)	PUNCT
iajs-3486	35	6	∩	∩	NOUN
iajs-3486	35	7	𝐹(ℛ	𝐹(ℛ	NUM
iajs-3486	35	8	)	)	PUNCT
iajs-3486	35	9	∩	∩	NOUN
iajs-3486	35	10	𝐹	𝐹	PROPN
iajs-3486	35	11	(	(	PUNCT
iajs-3486	35	12	ζ	ζ	NOUN
iajs-3486	35	13	)	)	PUNCT
iajs-3486	35	14	the	the	DET
iajs-3486	35	15	set	set	NOUN
iajs-3486	35	16	of	of	ADP
iajs-3486	35	17	all	all	DET
iajs-3486	35	18	common	common	ADJ
iajs-3486	35	19	random	random	ADJ
iajs-3486	35	20	fixed	fix	VERB
iajs-3486	35	21	point	point	NOUN
iajs-3486	35	22	of	of	ADP
iajs-3486	35	23	the	the	DET
iajs-3486	35	24	ℛ	ℛ	PROPN
iajs-3486	35	25	,	,	PUNCT
iajs-3486	35	26	ζ	ζ	NOUN
iajs-3486	35	27	,	,	PUNCT
iajs-3486	35	28	𝐵	𝐵	NOUN
iajs-3486	35	29	∶	∶	NOUN
iajs-3486	35	30	ψ	ψ	X
iajs-3486	35	31	×	×	NOUN
iajs-3486	35	32	υ	υ	NOUN
iajs-3486	35	33	→	→	SYM
iajs-3486	35	34	𝛱(υ	𝛱(υ	PROPN
iajs-3486	35	35	)	)	PUNCT
iajs-3486	35	36	be	be	VERB
iajs-3486	35	37	three	three	NUM
iajs-3486	35	38	multivalued	multivalued	ADJ
iajs-3486	35	39	non	non	ADJ
iajs-3486	35	40	-	-	ADJ
iajs-3486	35	41	expansive	expansive	ADJ
iajs-3486	35	42	random	random	ADJ
iajs-3486	35	43	operator	operator	NOUN
iajs-3486	35	44	with	with	ADP
iajs-3486	35	45	common	common	ADJ
iajs-3486	35	46	random	random	ADJ
iajs-3486	35	47	fixed	fix	VERB
iajs-3486	35	48	point	point	NOUN
iajs-3486	35	49	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	35	50	)	)	PUNCT
iajs-3486	35	51	.	.	PUNCT
iajs-3486	36	1	the	the	DET
iajs-3486	36	2	iterations	iteration	NOUN
iajs-3486	36	3	are	be	AUX
iajs-3486	36	4	as	as	SCONJ
iajs-3486	36	5	follows	follow	VERB
iajs-3486	36	6	:	:	PUNCT
iajs-3486	36	7	𝑆0(𝜚)𝜖𝛶	𝑆0(𝜚)𝜖𝛶	VERB
iajs-3486	36	8	𝑆𝑛+1(𝜚	𝑆𝑛+1(𝜚	NOUN
iajs-3486	36	9	)	)	PUNCT
iajs-3486	36	10	=	=	SYM
iajs-3486	36	11	𝛼𝑛𝜂𝑛(ϱ	𝛼𝑛𝜂𝑛(ϱ	NOUN
iajs-3486	36	12	)	)	PUNCT
iajs-3486	36	13	+	+	CCONJ
iajs-3486	36	14	𝜅𝑛𝜌𝑛(𝜚	𝜅𝑛𝜌𝑛(𝜚	NOUN
iajs-3486	36	15	)	)	PUNCT
iajs-3486	36	16	+	+	X
iajs-3486	37	1	𝜎𝑛𝜉𝑛(𝜚	𝜎𝑛𝜉𝑛(𝜚	NUM
iajs-3486	37	2	)	)	PUNCT
iajs-3486	38	1	,	,	PUNCT
iajs-3486	38	2	𝑛	𝑛	PRON
iajs-3486	38	3	∈	∈	NOUN
iajs-3486	38	4	𝑁	𝑁	PROPN
iajs-3486	38	5	(	(	PUNCT
iajs-3486	38	6	2.1	2.1	NUM
iajs-3486	38	7	)	)	PUNCT
iajs-3486	38	8	where	where	SCONJ
iajs-3486	38	9	𝜂𝑛(𝜚	𝜂𝑛(𝜚	PUNCT
iajs-3486	38	10	)	)	PUNCT
iajs-3486	38	11	∈	∈	PROPN
iajs-3486	38	12	𝐵(𝑆𝑛(ϱ	𝐵(𝑆𝑛(ϱ	PROPN
iajs-3486	38	13	)	)	PUNCT
iajs-3486	38	14	)	)	PUNCT
iajs-3486	38	15	and	and	CCONJ
iajs-3486	38	16	𝜌𝑛(𝜚	𝜌𝑛(𝜚	NOUN
iajs-3486	38	17	)	)	PUNCT
iajs-3486	38	18	∈	∈	PROPN
iajs-3486	38	19	ζ(𝑆𝑛(ϱ	ζ(𝑆𝑛(ϱ	NOUN
iajs-3486	38	20	)	)	PUNCT
iajs-3486	38	21	)	)	PUNCT
iajs-3486	38	22	such	such	ADJ
iajs-3486	38	23	that	that	SCONJ
iajs-3486	38	24	‖𝜂𝑛(𝜚	‖𝜂𝑛(𝜚	NOUN
iajs-3486	38	25	)	)	PUNCT
iajs-3486	38	26	−	−	PROPN
iajs-3486	38	27	𝜂𝑛+1(𝜚)‖	𝜂𝑛+1(𝜚)‖	X
iajs-3486	38	28	≤	≤	PROPN
iajs-3486	38	29	η	η	PROPN
iajs-3486	38	30	(	(	PUNCT
iajs-3486	38	31	𝐵(𝑆𝑛(ϱ	𝐵(𝑆𝑛(ϱ	PROPN
iajs-3486	38	32	)	)	PUNCT
iajs-3486	38	33	)	)	PUNCT
iajs-3486	38	34	,	,	PUNCT
iajs-3486	38	35	𝐵(𝑆𝑛+1(ϱ	𝐵(𝑆𝑛+1(ϱ	NOUN
iajs-3486	38	36	)	)	PUNCT
iajs-3486	38	37	)	)	PUNCT
iajs-3486	38	38	)	)	PUNCT
iajs-3486	39	1	+	+	CCONJ
iajs-3486	39	2	𝜇𝑛and	𝜇𝑛and	ADJ
iajs-3486	39	3	‖𝜌𝑛(𝜚	‖𝜌𝑛(𝜚	NUM
iajs-3486	39	4	)	)	PUNCT
iajs-3486	40	1	−	−	ADP
iajs-3486	40	2	𝜌𝑛+1(𝜚)‖	𝜌𝑛+1(𝜚)‖	VERB
iajs-3486	40	3	≤	≤	PROPN
iajs-3486	40	4	η	η	PROPN
iajs-3486	40	5	(	(	PUNCT
iajs-3486	40	6	ζ(𝑆𝑛(ϱ	ζ(𝑆𝑛(ϱ	NUM
iajs-3486	40	7	)	)	PUNCT
iajs-3486	40	8	)	)	PUNCT
iajs-3486	40	9	,	,	PUNCT
iajs-3486	40	10	ζ(𝑆𝑛+1(ϱ	ζ(𝑆𝑛+1(ϱ	NUM
iajs-3486	40	11	)	)	PUNCT
iajs-3486	40	12	)	)	PUNCT
iajs-3486	40	13	)	)	PUNCT
iajs-3486	41	1	+	+	CCONJ
iajs-3486	41	2	𝜇𝑛	𝜇𝑛	NOUN
iajs-3486	41	3	and	and	CCONJ
iajs-3486	41	4	𝜉𝑛(𝜚	𝜉𝑛(𝜚	NOUN
iajs-3486	41	5	)	)	PUNCT
iajs-3486	41	6	∈	∈	PROPN
iajs-3486	41	7	𝑅(𝑆𝑛(ϱ))‖𝜉𝑛(𝜚	𝑅(𝑆𝑛(ϱ))‖𝜉𝑛(𝜚	NOUN
iajs-3486	41	8	)	)	PUNCT
iajs-3486	41	9	−	−	PROPN
iajs-3486	42	1	𝜉𝑛+1(𝜚)‖	𝜉𝑛+1(𝜚)‖	PROPN
iajs-3486	42	2	≤	≤	NUM
iajs-3486	42	3	η	η	PROPN
iajs-3486	42	4	(	(	PUNCT
iajs-3486	42	5	ℛ(𝑆𝑛(ϱ	ℛ(𝑆𝑛(ϱ	NOUN
iajs-3486	42	6	)	)	PUNCT
iajs-3486	42	7	)	)	PUNCT
iajs-3486	42	8	,	,	PUNCT
iajs-3486	42	9	ℛ(𝑆𝑛+1(ϱ	ℛ(𝑆𝑛+1(ϱ	PROPN
iajs-3486	42	10	)	)	PUNCT
iajs-3486	42	11	)	)	PUNCT
iajs-3486	42	12	)	)	PUNCT
iajs-3486	43	1	+	+	CCONJ
iajs-3486	43	2	𝜇𝑛	𝜇𝑛	NOUN
iajs-3486	43	3	and	and	CCONJ
iajs-3486	43	4	{	{	PUNCT
iajs-3486	43	5	𝛼𝑛	𝛼𝑛	PROPN
iajs-3486	43	6	}	}	PUNCT
iajs-3486	43	7	,	,	PUNCT
iajs-3486	43	8	{	{	PUNCT
iajs-3486	43	9	𝜅𝑛	𝜅𝑛	ADP
iajs-3486	43	10	}	}	PUNCT
iajs-3486	43	11	,	,	PUNCT
iajs-3486	43	12	{	{	PUNCT
iajs-3486	43	13	𝜎𝑛	𝜎𝑛	X
iajs-3486	43	14	}	}	PUNCT
iajs-3486	43	15	are	be	AUX
iajs-3486	43	16	sequence	sequence	NOUN
iajs-3486	43	17	in	in	ADP
iajs-3486	43	18	(	(	PUNCT
iajs-3486	43	19	0,1	0,1	NOUN
iajs-3486	43	20	)	)	PUNCT
iajs-3486	43	21	satisfying	satisfy	VERB
iajs-3486	43	22	𝛼𝑛	𝛼𝑛	ADP
iajs-3486	43	23	+	+	NUM
iajs-3486	43	24	𝜅𝑛	𝜅𝑛	ADP
iajs-3486	43	25	+	+	NOUN
iajs-3486	43	26	𝜎𝑛	𝜎𝑛	NOUN
iajs-3486	43	27	=	=	SYM
iajs-3486	43	28	1	1	X
iajs-3486	43	29	.	.	PUNCT
iajs-3486	44	1	lemma	lemma	PROPN
iajs-3486	44	2	2.2	2.2	NUM
iajs-3486	44	3	let	let	VERB
iajs-3486	44	4	𝛷,υ	𝛷,υ	ADJ
iajs-3486	44	5	≠	≠	PROPN
iajs-3486	44	6	∅	∅	NOUN
iajs-3486	44	7	,	,	PUNCT
iajs-3486	44	8	and	and	CCONJ
iajs-3486	44	9	ℛ	ℛ	PROPN
iajs-3486	44	10	,	,	PUNCT
iajs-3486	44	11	ζ	ζ	NOUN
iajs-3486	44	12	,	,	PUNCT
iajs-3486	44	13	𝐵	𝐵	NOUN
iajs-3486	44	14	∶	∶	NOUN
iajs-3486	44	15	ψ	ψ	X
iajs-3486	44	16	×	×	NOUN
iajs-3486	44	17	υ	υ	NOUN
iajs-3486	44	18	→	→	SYM
iajs-3486	44	19	𝛱(υ	𝛱(υ	NOUN
iajs-3486	44	20	)	)	PUNCT
iajs-3486	44	21	,	,	PUNCT
iajs-3486	44	22	let	let	VERB
iajs-3486	44	23	{	{	PUNCT
iajs-3486	44	24	𝑆𝑛(𝜚)}issequence	𝑆𝑛(𝜚)}issequence	PROPN
iajs-3486	44	25	defined	define	VERB
iajs-3486	44	26	in	in	ADP
iajs-3486	44	27	2.1	2.1	NUM
iajs-3486	44	28	.	.	PUNCT
iajs-3486	45	1	if	if	SCONJ
iajs-3486	45	2	𝐹(𝐵	𝐹(𝐵	NOUN
iajs-3486	45	3	)	)	PUNCT
iajs-3486	45	4	≠	≠	PROPN
iajs-3486	45	5	∅	∅	NOUN
iajs-3486	45	6	and	and	CCONJ
iajs-3486	45	7	𝐵ϑ(ϱ	𝐵ϑ(ϱ	NOUN
iajs-3486	45	8	)	)	PUNCT
iajs-3486	45	9	=	=	SYM
iajs-3486	45	10	ℛϑ(ϱ	ℛϑ(ϱ	ADJ
iajs-3486	45	11	)	)	PUNCT
iajs-3486	45	12	=	=	PUNCT
iajs-3486	45	13	ζϑ(ϱ	ζϑ(ϱ	X
iajs-3486	45	14	)	)	PUNCT
iajs-3486	45	15	=	=	SYM
iajs-3486	45	16	{	{	PUNCT
iajs-3486	45	17	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	45	18	)	)	PUNCT
iajs-3486	45	19	}	}	PUNCT
iajs-3486	45	20	for	for	ADP
iajs-3486	45	21	any	any	DET
iajs-3486	45	22	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	45	23	)	)	PUNCT
iajs-3486	45	24	∈	∈	PROPN
iajs-3486	45	25	𝐹	𝐹	PROPN
iajs-3486	45	26	then	then	ADV
iajs-3486	45	27	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	45	28	𝑛→∞	𝑛→∞	PUNCT
iajs-3486	45	29	 	 	SPACE
iajs-3486	45	30	∥∥𝑥𝑛	∥∥𝑥𝑛	PUNCT
iajs-3486	46	1	−	−	PROPN
iajs-3486	47	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	47	2	exists	exist	VERB
iajs-3486	47	3	∀ϑ(ϱ	∀ϑ(ϱ	PROPN
iajs-3486	47	4	)	)	PUNCT
iajs-3486	47	5	∈	∈	PROPN
iajs-3486	47	6	𝐹.	𝐹.	PROPN
iajs-3486	47	7	ihjpas.37	ihjpas.37	PROPN
iajs-3486	47	8	(	(	PUNCT
iajs-3486	47	9	2	2	NUM
iajs-3486	47	10	)	)	PUNCT
iajs-3486	47	11	2024	2024	NUM
iajs-3486	47	12	426	426	NUM
iajs-3486	47	13	proof	proof	NOUN
iajs-3486	47	14	.	.	PUNCT
iajs-3486	48	1	let𝐹	let𝐹	ADJ
iajs-3486	48	2	≠	≠	PROPN
iajs-3486	48	3	∅.	∅.	ADV
iajs-3486	48	4	let	let	VERB
iajs-3486	48	5	ϑ(ϱ	ϑ(ϱ	NOUN
iajs-3486	48	6	)	)	PUNCT
iajs-3486	48	7	∈	∈	PROPN
iajs-3486	48	8	𝐹	𝐹	PROPN
iajs-3486	48	9	.	.	PUNCT
iajs-3486	49	1	then	then	ADV
iajs-3486	49	2	∥∥𝑆𝑛+1(𝜚	∥∥𝑆𝑛+1(𝜚	PROPN
iajs-3486	49	3	)	)	PUNCT
iajs-3486	49	4	−	−	PROPN
iajs-3486	50	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	50	2	=	=	PUNCT
iajs-3486	50	3	∥∥𝛼𝑛𝜂𝑛(𝜚	∥∥𝛼𝑛𝜂𝑛(𝜚	X
iajs-3486	50	4	)	)	PUNCT
iajs-3486	51	1	+	+	CCONJ
iajs-3486	51	2	𝜅𝑛𝜌𝑛(𝜚	𝜅𝑛𝜌𝑛(𝜚	NOUN
iajs-3486	51	3	)	)	PUNCT
iajs-3486	51	4	+	+	CCONJ
iajs-3486	51	5	𝜎𝑛𝜉𝑛(ϱ	𝜎𝑛𝜉𝑛(ϱ	X
iajs-3486	51	6	)	)	PUNCT
iajs-3486	51	7	−	−	PROPN
iajs-3486	52	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	52	2	=	=	SYM
iajs-3486	52	3	∥∥𝛼𝑛(𝜂𝑛(𝜚	∥∥𝛼𝑛(𝜂𝑛(𝜚	PROPN
iajs-3486	52	4	)	)	PUNCT
iajs-3486	52	5	−	−	PROPN
iajs-3486	52	6	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	52	7	)	)	PUNCT
iajs-3486	52	8	)	)	PUNCT
iajs-3486	53	1	+	+	PUNCT
iajs-3486	53	2	𝜅𝑛(𝜌𝑛(𝜚	𝜅𝑛(𝜌𝑛(𝜚	X
iajs-3486	53	3	)	)	PUNCT
iajs-3486	53	4	−	−	PROPN
iajs-3486	53	5	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	53	6	)	)	PUNCT
iajs-3486	53	7	)	)	PUNCT
iajs-3486	54	1	+	+	CCONJ
iajs-3486	54	2	𝜎𝑛(𝜉𝑛(ϱ	𝜎𝑛(𝜉𝑛(ϱ	NUM
iajs-3486	54	3	)	)	PUNCT
iajs-3486	54	4	−	−	PROPN
iajs-3486	54	5	ϑ(ϱ))∥∥	ϑ(ϱ))∥∥	PROPN
iajs-3486	54	6	⩽	⩽	NOUN
iajs-3486	54	7	𝛼𝑛∥∥𝜂𝑛(𝜚	𝛼𝑛∥∥𝜂𝑛(𝜚	PROPN
iajs-3486	54	8	)	)	PUNCT
iajs-3486	54	9	−	−	NOUN
iajs-3486	55	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	55	2	+	+	CCONJ
iajs-3486	55	3	𝜅𝑛∥∥𝜌𝑛(𝜚	𝜅𝑛∥∥𝜌𝑛(𝜚	NOUN
iajs-3486	55	4	)	)	PUNCT
iajs-3486	55	5	−	−	PUNCT
iajs-3486	56	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	56	2	+	+	CCONJ
iajs-3486	56	3	𝜎𝑛∥∥𝜉𝑛(ϱ	𝜎𝑛∥∥𝜉𝑛(ϱ	NOUN
iajs-3486	56	4	)	)	PUNCT
iajs-3486	56	5	−	−	PROPN
iajs-3486	56	6	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	56	7	⩽	⩽	PROPN
iajs-3486	56	8	𝛼𝑛𝑑	𝛼𝑛𝑑	PROPN
iajs-3486	56	9	(	(	PUNCT
iajs-3486	56	10	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	56	11	)	)	PUNCT
iajs-3486	56	12	,	,	PUNCT
iajs-3486	56	13	𝐵(ϑ(ϱ	𝐵(ϑ(ϱ	PROPN
iajs-3486	56	14	)	)	PUNCT
iajs-3486	56	15	)	)	PUNCT
iajs-3486	56	16	)	)	PUNCT
iajs-3486	57	1	+	+	NUM
iajs-3486	57	2	𝜅𝑛𝑑	𝜅𝑛𝑑	NOUN
iajs-3486	57	3	(	(	PUNCT
iajs-3486	57	4	𝜌𝑛(𝜚	𝜌𝑛(𝜚	NOUN
iajs-3486	57	5	)	)	PUNCT
iajs-3486	57	6	,	,	PUNCT
iajs-3486	57	7	ζ(ϑ(ϱ	ζ(ϑ(ϱ	PROPN
iajs-3486	57	8	)	)	PUNCT
iajs-3486	57	9	)	)	PUNCT
iajs-3486	57	10	)	)	PUNCT
iajs-3486	58	1	+	+	CCONJ
iajs-3486	58	2	𝜎𝑛𝑑	𝜎𝑛𝑑	NOUN
iajs-3486	58	3	(	(	PUNCT
iajs-3486	58	4	𝜉𝑛(ϱ	𝜉𝑛(ϱ	NOUN
iajs-3486	58	5	)	)	PUNCT
iajs-3486	58	6	,	,	PUNCT
iajs-3486	58	7	𝑅(ϑ(ϱ	𝑅(ϑ(ϱ	X
iajs-3486	58	8	)	)	PUNCT
iajs-3486	58	9	)	)	PUNCT
iajs-3486	58	10	)	)	PUNCT
iajs-3486	58	11	⩽	⩽	PROPN
iajs-3486	58	12	𝛼𝑛𝐻(𝐵𝑆𝑛(𝜚	𝛼𝑛𝐻(𝐵𝑆𝑛(𝜚	NUM
iajs-3486	58	13	)	)	PUNCT
iajs-3486	58	14	,	,	PUNCT
iajs-3486	58	15	𝐵ϑ(ϱ	𝐵ϑ(ϱ	NOUN
iajs-3486	58	16	)	)	PUNCT
iajs-3486	58	17	)	)	PUNCT
iajs-3486	59	1	+	+	CCONJ
iajs-3486	59	2	𝜅𝑛𝐻	𝜅𝑛𝐻	NOUN
iajs-3486	59	3	(	(	PUNCT
iajs-3486	59	4	ζ	ζ	NOUN
iajs-3486	59	5	(	(	PUNCT
iajs-3486	59	6	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	59	7	)	)	PUNCT
iajs-3486	59	8	)	)	PUNCT
iajs-3486	59	9	,	,	PUNCT
iajs-3486	59	10	ζ(ϑ(ϱ	ζ(ϑ(ϱ	PROPN
iajs-3486	59	11	)	)	PUNCT
iajs-3486	59	12	)	)	PUNCT
iajs-3486	59	13	)	)	PUNCT
iajs-3486	60	1	+	+	VERB
iajs-3486	60	2	𝜎𝑛𝐻	𝜎𝑛𝐻	NOUN
iajs-3486	60	3	(	(	PUNCT
iajs-3486	60	4	𝑅(𝑆𝑛(𝜚	𝑅(𝑆𝑛(𝜚	NUM
iajs-3486	60	5	)	)	PUNCT
iajs-3486	60	6	)	)	PUNCT
iajs-3486	60	7	,	,	PUNCT
iajs-3486	60	8	𝑅(ϑ(ϱ	𝑅(ϑ(ϱ	X
iajs-3486	60	9	)	)	PUNCT
iajs-3486	60	10	)	)	PUNCT
iajs-3486	60	11	)	)	PUNCT
iajs-3486	60	12	⩽	⩽	NOUN
iajs-3486	60	13	𝛼𝑛∥∥𝑆𝑛(𝜚	𝛼𝑛∥∥𝑆𝑛(𝜚	NOUN
iajs-3486	60	14	)	)	PUNCT
iajs-3486	60	15	−	−	PROPN
iajs-3486	61	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	61	2	+	+	CCONJ
iajs-3486	61	3	𝜅𝑛∥∥𝑆𝑛(𝜚	𝜅𝑛∥∥𝑆𝑛(𝜚	NOUN
iajs-3486	61	4	)	)	PUNCT
iajs-3486	61	5	−	−	PROPN
iajs-3486	62	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	62	2	+	+	CCONJ
iajs-3486	62	3	𝜎𝑛∥∥𝑆𝑛(𝜚	𝜎𝑛∥∥𝑆𝑛(𝜚	NOUN
iajs-3486	62	4	)	)	PUNCT
iajs-3486	62	5	−	−	PROPN
iajs-3486	62	6	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	62	7	=	=	SYM
iajs-3486	62	8	∥∥𝑆𝑛(𝜚	∥∥𝑆𝑛(𝜚	X
iajs-3486	62	9	)	)	PUNCT
iajs-3486	62	10	−	−	PROPN
iajs-3486	63	1	ϑ(ϱ)∥∥.	ϑ(ϱ)∥∥.	PROPN
iajs-3486	63	2	(	(	PUNCT
iajs-3486	63	3	2.2	2.2	NUM
iajs-3486	63	4	)	)	PUNCT
iajs-3486	63	5	thus	thus	ADV
iajs-3486	63	6	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	63	7	𝑛→∞	𝑛→∞	NUM
iajs-3486	63	8	 	 	SPACE
iajs-3486	63	9	∥∥𝑆𝑛(𝜚	∥∥𝑆𝑛(𝜚	NOUN
iajs-3486	63	10	)	)	PUNCT
iajs-3486	63	11	−	−	PROPN
iajs-3486	64	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	64	2	exists	exist	VERB
iajs-3486	64	3	for	for	ADP
iajs-3486	64	4	all	all	DET
iajs-3486	64	5	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	64	6	)	)	PUNCT
iajs-3486	64	7	∈	∈	PROPN
iajs-3486	64	8	𝐹	𝐹	PROPN
iajs-3486	64	9	.	.	PUNCT
iajs-3486	65	1	definition	definition	NOUN
iajs-3486	65	2	2.3	2.3	NUM
iajs-3486	65	3	letℛ	letℛ	NOUN
iajs-3486	65	4	,	,	PUNCT
iajs-3486	65	5	ζ	ζ	NOUN
iajs-3486	65	6	,	,	PUNCT
iajs-3486	66	1	𝐵	𝐵	PROPN
iajs-3486	66	2	∶	∶	NOUN
iajs-3486	66	3	ψ	ψ	X
iajs-3486	66	4	×	×	NOUN
iajs-3486	66	5	υ	υ	NOUN
iajs-3486	66	6	→	→	SYM
iajs-3486	66	7	𝛱(υ	𝛱(υ	NOUN
iajs-3486	66	8	)	)	PUNCT
iajs-3486	66	9	are	be	AUX
iajs-3486	66	10	said	say	VERB
iajs-3486	66	11	to	to	PART
iajs-3486	66	12	satisfy	satisfy	VERB
iajs-3486	66	13	condition	condition	NOUN
iajs-3486	66	14	(	(	PUNCT
iajs-3486	66	15	𝐶′	𝐶′	CCONJ
iajs-3486	66	16	)	)	PUNCT
iajs-3486	66	17	if	if	SCONJ
iajs-3486	66	18	𝑑	𝑑	PROPN
iajs-3486	66	19	(	(	PUNCT
iajs-3486	66	20	(	(	PUNCT
iajs-3486	66	21	𝜚	𝜚	NOUN
iajs-3486	66	22	,	,	PUNCT
iajs-3486	66	23	𝑢	𝑢	PART
iajs-3486	66	24	)	)	PUNCT
iajs-3486	66	25	,	,	PUNCT
iajs-3486	66	26	𝑣(𝜚	𝑣(𝜚	PROPN
iajs-3486	66	27	)	)	PUNCT
iajs-3486	66	28	)	)	PUNCT
iajs-3486	66	29	≤	≤	PROPN
iajs-3486	66	30	𝑑(𝑤(𝜚	𝑑(𝑤(𝜚	PROPN
iajs-3486	66	31	)	)	PUNCT
iajs-3486	66	32	,	,	PUNCT
iajs-3486	66	33	𝑣(𝜚	𝑣(𝜚	PROPN
iajs-3486	66	34	)	)	PUNCT
iajs-3486	66	35	)	)	PUNCT
iajs-3486	66	36	for	for	ADP
iajs-3486	66	37	𝑣(𝜚	𝑣(𝜚	PROPN
iajs-3486	66	38	)	)	PUNCT
iajs-3486	66	39	∈	∈	PROPN
iajs-3486	66	40	ζ(𝜚	ζ(𝜚	NOUN
iajs-3486	66	41	,	,	PUNCT
iajs-3486	66	42	𝑢	𝑢	NOUN
iajs-3486	66	43	)	)	PUNCT
iajs-3486	66	44	,	,	PUNCT
iajs-3486	66	45	𝑤(𝜚	𝑤(𝜚	NUM
iajs-3486	66	46	)	)	PUNCT
iajs-3486	66	47	∈	∈	PROPN
iajs-3486	66	48	ℛ(𝜚	ℛ(𝜚	PROPN
iajs-3486	66	49	,	,	PUNCT
iajs-3486	66	50	𝑢	𝑢	NOUN
iajs-3486	66	51	)	)	PUNCT
iajs-3486	66	52	.	.	PUNCT
iajs-3486	67	1	lemma2.4.let	lemma2.4.let	NOUN
iajs-3486	67	2	𝛷and	𝛷and	PROPN
iajs-3486	67	3	υandlet	υandlet	NOUN
iajs-3486	67	4	ℛ	ℛ	PROPN
iajs-3486	67	5	,	,	PUNCT
iajs-3486	67	6	ζ	ζ	NOUN
iajs-3486	67	7	,	,	PUNCT
iajs-3486	67	8	𝐵	𝐵	NOUN
iajs-3486	67	9	∶	∶	NOUN
iajs-3486	67	10	ψ	ψ	X
iajs-3486	67	11	×	×	NOUN
iajs-3486	67	12	υ	υ	X
iajs-3486	67	13	→	→	PUNCT
iajs-3486	67	14	𝛱(υ)satisfying	𝛱(υ)satisfying	NOUN
iajs-3486	67	15	conditionn(𝐶′	conditionn(𝐶′	NOUN
iajs-3486	67	16	)	)	PUNCT
iajs-3486	67	17	and	and	CCONJ
iajs-3486	67	18	{	{	PUNCT
iajs-3486	67	19	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	67	20	)	)	PUNCT
iajs-3486	67	21	}	}	PUNCT
iajs-3486	67	22	be	be	AUX
iajs-3486	67	23	sequence	sequence	NOUN
iajs-3486	67	24	defined	define	VERB
iajs-3486	67	25	in	in	ADP
iajs-3486	67	26	(	(	PUNCT
iajs-3486	67	27	2.1	2.1	NUM
iajs-3486	67	28	)	)	PUNCT
iajs-3486	67	29	.	.	PUNCT
iajs-3486	68	1	if	if	SCONJ
iajs-3486	68	2	𝐹	𝐹	PROPN
iajs-3486	68	3	≠	≠	PROPN
iajs-3486	68	4	∅	∅	NOUN
iajs-3486	68	5	and	and	CCONJ
iajs-3486	68	6	𝐵ϑ(ϱ	𝐵ϑ(ϱ	NOUN
iajs-3486	68	7	)	)	PUNCT
iajs-3486	68	8	=	=	SYM
iajs-3486	68	9	ℛϑ(ϱ	ℛϑ(ϱ	ADJ
iajs-3486	68	10	)	)	PUNCT
iajs-3486	68	11	=	=	PUNCT
iajs-3486	68	12	ζϑ(ϱ	ζϑ(ϱ	X
iajs-3486	68	13	)	)	PUNCT
iajs-3486	68	14	=	=	SYM
iajs-3486	68	15	{	{	PUNCT
iajs-3486	68	16	ϑ(ϱ)}for	ϑ(ϱ)}for	ADP
iajs-3486	68	17	any	any	DET
iajs-3486	68	18	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	68	19	)	)	PUNCT
iajs-3486	68	20	∈	∈	PROPN
iajs-3486	68	21	𝐹	𝐹	PROPN
iajs-3486	68	22	then	then	ADV
iajs-3486	68	23	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	68	24	𝚤→∞	𝚤→∞	NUM
iajs-3486	68	25	 	 	SPACE
iajs-3486	68	26	𝑑	𝑑	PROPN
iajs-3486	68	27	(	(	PUNCT
iajs-3486	68	28	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	68	29	)	)	PUNCT
iajs-3486	68	30	,	,	PUNCT
iajs-3486	68	31	𝐵(𝑆𝑛(𝜚	𝐵(𝑆𝑛(𝜚	NOUN
iajs-3486	68	32	)	)	PUNCT
iajs-3486	68	33	)	)	PUNCT
iajs-3486	68	34	)	)	PUNCT
iajs-3486	69	1	=	=	PRON
iajs-3486	69	2	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	69	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	69	4	 	 	SPACE
iajs-3486	69	5	𝑑	𝑑	PROPN
iajs-3486	69	6	(	(	PUNCT
iajs-3486	69	7	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	69	8	)	)	PUNCT
iajs-3486	69	9	,	,	PUNCT
iajs-3486	69	10	ζ(𝑆𝑛(𝜚	ζ(𝑆𝑛(𝜚	NOUN
iajs-3486	69	11	)	)	PUNCT
iajs-3486	69	12	)	)	PUNCT
iajs-3486	69	13	)	)	PUNCT
iajs-3486	70	1	=	=	SYM
iajs-3486	70	2	0	0	PUNCT
iajs-3486	71	1	=	=	NUM
iajs-3486	71	2	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	71	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	71	4	 	 	SPACE
iajs-3486	72	1	𝑑	𝑑	PROPN
iajs-3486	72	2	(	(	PUNCT
iajs-3486	72	3	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	72	4	)	)	PUNCT
iajs-3486	72	5	,	,	PUNCT
iajs-3486	72	6	ℛ(𝑆𝑛(𝜚	ℛ(𝑆𝑛(𝜚	NOUN
iajs-3486	72	7	)	)	PUNCT
iajs-3486	72	8	)	)	PUNCT
iajs-3486	72	9	)	)	PUNCT
iajs-3486	73	1	proof	proof	NOUN
iajs-3486	73	2	.	.	PUNCT
iajs-3486	74	1	by	by	ADP
iajs-3486	74	2	lemma	lemma	PROPN
iajs-3486	74	3	(	(	PUNCT
iajs-3486	74	4	2.2	2.2	NUM
iajs-3486	74	5	)	)	PUNCT
iajs-3486	74	6	,	,	PUNCT
iajs-3486	74	7	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	74	8	𝑛→∞	𝑛→∞	NUM
iajs-3486	74	9	 	 	SPACE
iajs-3486	74	10	∥∥𝑆𝑛(𝜚	∥∥𝑆𝑛(𝜚	NOUN
iajs-3486	74	11	)	)	PUNCT
iajs-3486	74	12	−	−	NOUN
iajs-3486	74	13	ϑ(ϱ)∥∥exists	ϑ(ϱ)∥∥exist	NOUN
iajs-3486	74	14	.	.	PUNCT
iajs-3486	75	1	we	we	PRON
iajs-3486	75	2	suppose	suppose	VERB
iajs-3486	75	3	that	that	SCONJ
iajs-3486	75	4	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	75	5	𝑛→∞	𝑛→∞	NUM
iajs-3486	75	6	 	 	SPACE
iajs-3486	75	7	∥∥𝑆𝑛(𝜚	∥∥𝑆𝑛(𝜚	NOUN
iajs-3486	75	8	)	)	PUNCT
iajs-3486	75	9	−	−	PROPN
iajs-3486	76	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	76	2	=	=	SYM
iajs-3486	76	3	𝑐	𝑐	PROPN
iajs-3486	76	4	for	for	ADP
iajs-3486	76	5	𝑐	𝑐	PROPN
iajs-3486	76	6	≥	≥	NUM
iajs-3486	76	7	0	0	NUM
iajs-3486	76	8	.	.	PUNCT
iajs-3486	77	1	also	also	ADV
iajs-3486	77	2	have𝐵	have𝐵	PROPN
iajs-3486	77	3	,	,	PUNCT
iajs-3486	77	4	s	s	PROPN
iajs-3486	77	5	,	,	PUNCT
iajs-3486	77	6	ℛ	ℛ	ADJ
iajs-3486	77	7	arenonexpansive	arenonexpansive	ADJ
iajs-3486	77	8	random	random	ADJ
iajs-3486	77	9	operator	operator	NOUN
iajs-3486	77	10	and	and	CCONJ
iajs-3486	77	11	𝐹	𝐹	PROPN
iajs-3486	77	12	≠	≠	PROPN
iajs-3486	77	13	∅	∅	NOUN
iajs-3486	77	14	,	,	PUNCT
iajs-3486	77	15	we	we	PRON
iajs-3486	77	16	have∥∥𝜂𝑛(𝜚	have∥∥𝜂𝑛(𝜚	ADJ
iajs-3486	77	17	)	)	PUNCT
iajs-3486	77	18	−	−	PROPN
iajs-3486	78	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	78	2	=	=	SYM
iajs-3486	78	3	𝑑	𝑑	PROPN
iajs-3486	78	4	(	(	PUNCT
iajs-3486	78	5	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	78	6	)	)	PUNCT
iajs-3486	78	7	,	,	PUNCT
iajs-3486	78	8	ζ(ϑ(ϱ	ζ(ϑ(ϱ	PROPN
iajs-3486	78	9	)	)	PUNCT
iajs-3486	78	10	)	)	PUNCT
iajs-3486	78	11	)	)	PUNCT
iajs-3486	79	1	⩽	⩽	NOUN
iajs-3486	79	2	𝐻	𝐻	PROPN
iajs-3486	79	3	(	(	PUNCT
iajs-3486	79	4	𝐵(𝑆𝑛(𝜚	𝐵(𝑆𝑛(𝜚	PROPN
iajs-3486	79	5	)	)	PUNCT
iajs-3486	79	6	)	)	PUNCT
iajs-3486	79	7	,	,	PUNCT
iajs-3486	79	8	𝐵(ϑ(ϱ	𝐵(ϑ(ϱ	PROPN
iajs-3486	79	9	)	)	PUNCT
iajs-3486	79	10	)	)	PUNCT
iajs-3486	79	11	)	)	PUNCT
iajs-3486	79	12	⩽	⩽	ADJ
iajs-3486	79	13	∥∥𝑆𝑛(𝜚	∥∥𝑆𝑛(𝜚	PROPN
iajs-3486	79	14	)	)	PUNCT
iajs-3486	79	15	−	−	PROPN
iajs-3486	80	1	ϑ(ϱ)∥∥.	ϑ(ϱ)∥∥.	NOUN
iajs-3486	80	2	take	take	VERB
iajs-3486	80	3	limsup	limsup	NOUN
iajs-3486	80	4	for	for	ADP
iajs-3486	80	5	both	both	DET
iajs-3486	80	6	side	side	NOUN
iajs-3486	80	7	,	,	PUNCT
iajs-3486	80	8	get	get	AUX
iajs-3486	80	9	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	80	10	 	 	SPACE
iajs-3486	80	11	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
iajs-3486	80	12	𝑛→∞	𝑛→∞	NUM
iajs-3486	80	13	 	 	SPACE
iajs-3486	80	14	∥∥𝜂𝑛(𝜚	∥∥𝜂𝑛(𝜚	NUM
iajs-3486	80	15	)	)	PUNCT
iajs-3486	80	16	−	−	PROPN
iajs-3486	81	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	81	2	⩽	⩽	PROPN
iajs-3486	81	3	𝑐	𝑐	PROPN
iajs-3486	81	4	similarly	similarly	ADV
iajs-3486	81	5	,	,	PUNCT
iajs-3486	81	6	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	81	7	 	 	SPACE
iajs-3486	81	8	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
iajs-3486	81	9	𝑛→∞	𝑛→∞	VERB
iajs-3486	81	10	 	 	SPACE
iajs-3486	81	11	∥∥𝜌𝑛(𝜚	∥∥𝜌𝑛(𝜚	NUM
iajs-3486	81	12	)	)	PUNCT
iajs-3486	81	13	−	−	PROPN
iajs-3486	82	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	82	2	⩽	⩽	PROPN
iajs-3486	82	3	𝑐	𝑐	PROPN
iajs-3486	82	4	and	and	CCONJ
iajs-3486	82	5	,	,	PUNCT
iajs-3486	82	6	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	82	7	 	 	SPACE
iajs-3486	82	8	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
iajs-3486	82	9	𝑛→∞	𝑛→∞	NUM
iajs-3486	82	10	 	 	SPACE
iajs-3486	82	11	∥∥𝜉𝑛(ϱ	∥∥𝜉𝑛(ϱ	NOUN
iajs-3486	82	12	)	)	PUNCT
iajs-3486	82	13	−	−	PROPN
iajs-3486	83	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	83	2	⩽	⩽	PROPN
iajs-3486	83	3	𝑐	𝑐	PROPN
iajs-3486	83	4	as	as	ADP
iajs-3486	83	5	,	,	PUNCT
iajs-3486	83	6	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	83	7	𝑛→∞	𝑛→∞	NUM
iajs-3486	83	8	 	 	SPACE
iajs-3486	83	9	∥∥𝑆𝑛+1(𝜚	∥∥𝑆𝑛+1(𝜚	PROPN
iajs-3486	83	10	)	)	PUNCT
iajs-3486	83	11	−	−	PROPN
iajs-3486	84	1	ϑ(ϱ)∥∥	ϑ(ϱ)∥∥	PROPN
iajs-3486	84	2	=	=	SYM
iajs-3486	84	3	𝑐	𝑐	PROPN
iajs-3486	84	4	that	that	PRON
iajs-3486	84	5	mean	mean	VERB
iajs-3486	84	6	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-3486	84	7	𝑛→∞	𝑛→∞	NUM
iajs-3486	84	8	 	 	SPACE
iajs-3486	84	9	∥∥𝛼𝑛(𝜂𝑛(𝜚	∥∥𝛼𝑛(𝜂𝑛(𝜚	PROPN
iajs-3486	84	10	)	)	PUNCT
iajs-3486	84	11	−	−	PROPN
iajs-3486	84	12	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	84	13	)	)	PUNCT
iajs-3486	84	14	)	)	PUNCT
iajs-3486	85	1	+	+	PUNCT
iajs-3486	85	2	𝜅𝑛(𝜌𝑛(𝜚	𝜅𝑛(𝜌𝑛(𝜚	X
iajs-3486	85	3	)	)	PUNCT
iajs-3486	85	4	−	−	PROPN
iajs-3486	85	5	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	85	6	)	)	PUNCT
iajs-3486	85	7	)	)	PUNCT
iajs-3486	86	1	+	+	CCONJ
iajs-3486	86	2	𝜎𝑛(𝜉𝑛(ϱ	𝜎𝑛(𝜉𝑛(ϱ	NUM
iajs-3486	86	3	)	)	PUNCT
iajs-3486	86	4	−	−	PROPN
iajs-3486	86	5	ϑ(ϱ))∥∥	ϑ(ϱ))∥∥	NOUN
iajs-3486	86	6	=	=	SYM
iajs-3486	86	7	𝑐	𝑐	PROPN
iajs-3486	86	8	applying	apply	VERB
iajs-3486	86	9	lemma	lemma	PROPN
iajs-3486	86	10	(	(	PUNCT
iajs-3486	86	11	1.4	1.4	NUM
iajs-3486	86	12	)	)	PUNCT
iajs-3486	86	13	,	,	PUNCT
iajs-3486	86	14	we	we	PRON
iajs-3486	86	15	get	get	VERB
iajs-3486	86	16	lim	lim	NOUN
iajs-3486	86	17	𝑛→∞	𝑛→∞	NUM
iajs-3486	86	18	‖𝜂𝑛(𝜚	‖𝜂𝑛(𝜚	NUM
iajs-3486	86	19	)	)	PUNCT
iajs-3486	86	20	−	−	PUNCT
iajs-3486	87	1	𝜌𝑛(𝜚)‖	𝜌𝑛(𝜚)‖	PUNCT
iajs-3486	87	2	=	=	PUNCT
iajs-3486	87	3	0	0	PROPN
iajs-3486	87	4	.	.	PUNCT
iajs-3486	88	1	but	but	CCONJ
iajs-3486	88	2	from	from	ADP
iajs-3486	88	3	the	the	DET
iajs-3486	88	4	condition	condition	NOUN
iajs-3486	88	5	(	(	PUNCT
iajs-3486	88	6	𝐶′	𝐶′	ADJ
iajs-3486	88	7	)	)	PUNCT
iajs-3486	88	8	we	we	PRON
iajs-3486	88	9	obtain	obtain	VERB
iajs-3486	88	10	that	that	SCONJ
iajs-3486	88	11	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	VERB
iajs-3486	88	12	)	)	PUNCT
iajs-3486	88	13	,	,	PUNCT
iajs-3486	88	14	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	88	15	)	)	PUNCT
iajs-3486	88	16	)	)	PUNCT
iajs-3486	88	17	≤	≤	NUM
iajs-3486	89	1	𝑑(𝜌𝑛(𝜚	𝑑(𝜌𝑛(𝜚	NOUN
iajs-3486	89	2	)	)	PUNCT
iajs-3486	89	3	,	,	PUNCT
iajs-3486	89	4	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	89	5	)	)	PUNCT
iajs-3486	89	6	)	)	PUNCT
iajs-3486	90	1	,	,	PUNCT
iajs-3486	90	2	lim	lim	PROPN
iajs-3486	90	3	sup	sup	NOUN
iajs-3486	90	4	𝑛→∞	𝑛→∞	NUM
iajs-3486	90	5	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	90	6	)	)	PUNCT
iajs-3486	90	7	,	,	PUNCT
iajs-3486	90	8	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	90	9	)	)	PUNCT
iajs-3486	90	10	)	)	PUNCT
iajs-3486	91	1	≤	≤	NUM
iajs-3486	91	2	0	0	NUM
iajs-3486	91	3	.	.	PUNCT
iajs-3486	92	1	that	that	PRON
iajs-3486	92	2	is	be	AUX
iajs-3486	92	3	,	,	PUNCT
iajs-3486	92	4	ihjpas.37	ihjpas.37	PROPN
iajs-3486	92	5	(	(	PUNCT
iajs-3486	92	6	2	2	NUM
iajs-3486	92	7	)	)	PUNCT
iajs-3486	92	8	2024	2024	NUM
iajs-3486	92	9	427	427	NUM
iajs-3486	92	10	lim	lim	NOUN
iajs-3486	92	11	𝑛→∞	𝑛→∞	NUM
iajs-3486	92	12	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	92	13	)	)	PUNCT
iajs-3486	92	14	−	−	PROPN
iajs-3486	93	1	𝜂𝑛(𝜚)‖	𝜂𝑛(𝜚)‖	NOUN
iajs-3486	93	2	=	=	NOUN
iajs-3486	93	3	0	0	PROPN
iajs-3486	93	4	.	.	PUNCT
iajs-3486	94	1	(	(	PUNCT
iajs-3486	94	2	2.3	2.3	NUM
iajs-3486	94	3	)	)	PUNCT
iajs-3486	94	4	also	also	ADV
iajs-3486	94	5	from	from	ADP
iajs-3486	94	6	lemma	lemma	PROPN
iajs-3486	94	7	(	(	PUNCT
iajs-3486	94	8	1.4	1.4	NUM
iajs-3486	94	9	)	)	PUNCT
iajs-3486	94	10	and	and	CCONJ
iajs-3486	94	11	(	(	PUNCT
iajs-3486	94	12	2.2	2.2	NUM
iajs-3486	94	13	)	)	PUNCT
iajs-3486	94	14	we	we	PRON
iajs-3486	94	15	obtain	obtain	VERB
iajs-3486	94	16	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	94	17	)	)	PUNCT
iajs-3486	94	18	−	−	NOUN
iajs-3486	95	1	𝜌𝑛(𝜚)‖	𝜌𝑛(𝜚)‖	ADP
iajs-3486	95	2	≤	≤	NUM
iajs-3486	95	3	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	95	4	)	)	PUNCT
iajs-3486	95	5	−	−	PROPN
iajs-3486	95	6	𝜂𝑛(𝜚)‖	𝜂𝑛(𝜚)‖	NOUN
iajs-3486	95	7	+	+	CCONJ
iajs-3486	95	8	‖𝜂𝑛(𝜚	‖𝜂𝑛(𝜚	NOUN
iajs-3486	95	9	)	)	PUNCT
iajs-3486	96	1	−	−	PUNCT
iajs-3486	96	2	𝜌𝑛(𝜚)‖	𝜌𝑛(𝜚)‖	NOUN
iajs-3486	96	3	implies	imply	VERB
iajs-3486	96	4	that	that	SCONJ
iajs-3486	96	5	lim	lim	PROPN
iajs-3486	96	6	𝑛→∞	𝑛→∞	NUM
iajs-3486	96	7	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	96	8	)	)	PUNCT
iajs-3486	96	9	−	−	NOUN
iajs-3486	96	10	𝜌𝑛(𝜚)‖	𝜌𝑛(𝜚)‖	PUNCT
iajs-3486	96	11	=	=	SYM
iajs-3486	96	12	0	0	PROPN
iajs-3486	96	13	.	.	PUNCT
iajs-3486	97	1	(	(	PUNCT
iajs-3486	97	2	2.4	2.4	NUM
iajs-3486	97	3	)	)	PUNCT
iajs-3486	97	4	also	also	ADV
iajs-3486	97	5	from	from	ADP
iajs-3486	97	6	lemma	lemma	PROPN
iajs-3486	97	7	(	(	PUNCT
iajs-3486	97	8	1.4	1.4	NUM
iajs-3486	97	9	)	)	PUNCT
iajs-3486	97	10	and	and	CCONJ
iajs-3486	97	11	(	(	PUNCT
iajs-3486	97	12	2.3	2.3	NUM
iajs-3486	97	13	)	)	PUNCT
iajs-3486	97	14	we	we	PRON
iajs-3486	97	15	have	have	VERB
iajs-3486	97	16	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	97	17	)	)	PUNCT
iajs-3486	98	1	−	−	PROPN
iajs-3486	98	2	𝜉𝑛(ϱ)‖	𝜉𝑛(ϱ)‖	PROPN
iajs-3486	98	3	≤	≤	PUNCT
iajs-3486	98	4	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	98	5	)	)	PUNCT
iajs-3486	98	6	−	−	NOUN
iajs-3486	98	7	𝜌𝑛(𝜚)‖	𝜌𝑛(𝜚)‖	PUNCT
iajs-3486	99	1	+	+	PUNCT
iajs-3486	99	2	‖𝜌𝑛(𝜚	‖𝜌𝑛(𝜚	NOUN
iajs-3486	99	3	)	)	PUNCT
iajs-3486	100	1	−	−	PUNCT
iajs-3486	100	2	𝜉𝑛(ϱ)‖	𝜉𝑛(ϱ)‖	PROPN
iajs-3486	100	3	implies	imply	VERB
iajs-3486	100	4	that	that	SCONJ
iajs-3486	100	5	lim	lim	PROPN
iajs-3486	100	6	𝑛→∞	𝑛→∞	NUM
iajs-3486	100	7	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	100	8	)	)	PUNCT
iajs-3486	100	9	−	−	PROPN
iajs-3486	100	10	𝜉𝑛(ϱ)‖	𝜉𝑛(ϱ)‖	PROPN
iajs-3486	100	11	=	=	PUNCT
iajs-3486	100	12	0	0	PUNCT
iajs-3486	100	13	.	.	PUNCT
iajs-3486	101	1	(	(	PUNCT
iajs-3486	101	2	2.5	2.5	NUM
iajs-3486	101	3	)	)	PUNCT
iajs-3486	101	4	now	now	ADV
iajs-3486	101	5	,	,	PUNCT
iajs-3486	101	6	we	we	PRON
iajs-3486	101	7	get	get	VERB
iajs-3486	101	8	𝑑	𝑑	PRON
iajs-3486	101	9	(	(	PUNCT
iajs-3486	101	10	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	101	11	)	)	PUNCT
iajs-3486	101	12	,	,	PUNCT
iajs-3486	101	13	𝐵(𝑆𝑛(𝜚	𝐵(𝑆𝑛(𝜚	NOUN
iajs-3486	101	14	)	)	PUNCT
iajs-3486	101	15	)	)	PUNCT
iajs-3486	101	16	)	)	PUNCT
iajs-3486	102	1	≤	≤	NOUN
iajs-3486	102	2	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	102	3	)	)	PUNCT
iajs-3486	102	4	,	,	PUNCT
iajs-3486	102	5	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	102	6	)	)	PUNCT
iajs-3486	102	7	)	)	PUNCT
iajs-3486	102	8	,	,	PUNCT
iajs-3486	102	9	also	also	ADV
iajs-3486	102	10	,	,	PUNCT
iajs-3486	102	11	𝑑	𝑑	PROPN
iajs-3486	102	12	(	(	PUNCT
iajs-3486	102	13	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	102	14	)	)	PUNCT
iajs-3486	102	15	,	,	PUNCT
iajs-3486	102	16	ζ(𝑆𝑛(𝜚	ζ(𝑆𝑛(𝜚	NOUN
iajs-3486	102	17	)	)	PUNCT
iajs-3486	102	18	)	)	PUNCT
iajs-3486	102	19	)	)	PUNCT
iajs-3486	102	20	≤	≤	NOUN
iajs-3486	102	21	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	102	22	)	)	PUNCT
iajs-3486	102	23	,	,	PUNCT
iajs-3486	102	24	𝜌𝑛(𝜚	𝜌𝑛(𝜚	NUM
iajs-3486	102	25	)	)	PUNCT
iajs-3486	102	26	)	)	PUNCT
iajs-3486	102	27	,	,	PUNCT
iajs-3486	102	28	and	and	CCONJ
iajs-3486	102	29	𝑑	𝑑	PROPN
iajs-3486	102	30	(	(	PUNCT
iajs-3486	102	31	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	102	32	)	)	PUNCT
iajs-3486	102	33	,	,	PUNCT
iajs-3486	102	34	ℛ(𝑆𝑛(𝜚	ℛ(𝑆𝑛(𝜚	NOUN
iajs-3486	102	35	)	)	PUNCT
iajs-3486	102	36	)	)	PUNCT
iajs-3486	102	37	)	)	PUNCT
iajs-3486	102	38	≤	≤	NOUN
iajs-3486	102	39	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	102	40	)	)	PUNCT
iajs-3486	102	41	,	,	PUNCT
iajs-3486	102	42	𝜉𝑛(ϱ	𝜉𝑛(ϱ	X
iajs-3486	102	43	)	)	PUNCT
iajs-3486	102	44	)	)	PUNCT
iajs-3486	102	45	,	,	PUNCT
iajs-3486	102	46	we	we	PRON
iajs-3486	102	47	gives	give	VERB
iajs-3486	102	48	𝑑	𝑑	PRON
iajs-3486	102	49	(	(	PUNCT
iajs-3486	102	50	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	102	51	)	)	PUNCT
iajs-3486	102	52	,	,	PUNCT
iajs-3486	102	53	𝐵(𝑆𝑛(𝜚	𝐵(𝑆𝑛(𝜚	NOUN
iajs-3486	102	54	)	)	PUNCT
iajs-3486	102	55	)	)	PUNCT
iajs-3486	102	56	)	)	PUNCT
iajs-3486	103	1	→	→	SYM
iajs-3486	103	2	0	0	NUM
iajs-3486	103	3	,	,	PUNCT
iajs-3486	103	4	𝑑	𝑑	PROPN
iajs-3486	103	5	(	(	PUNCT
iajs-3486	103	6	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	103	7	)	)	PUNCT
iajs-3486	103	8	,	,	PUNCT
iajs-3486	103	9	ζ(𝑆𝑛(𝜚	ζ(𝑆𝑛(𝜚	NOUN
iajs-3486	103	10	)	)	PUNCT
iajs-3486	103	11	)	)	PUNCT
iajs-3486	103	12	)	)	PUNCT
iajs-3486	103	13	→	→	SYM
iajs-3486	103	14	0	0	NUM
iajs-3486	103	15	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-3486	103	16	𝑑	𝑑	PROPN
iajs-3486	103	17	(	(	PUNCT
iajs-3486	103	18	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	103	19	)	)	PUNCT
iajs-3486	103	20	,	,	PUNCT
iajs-3486	103	21	ℛ(𝑆𝑛(𝜚	ℛ(𝑆𝑛(𝜚	NOUN
iajs-3486	103	22	)	)	PUNCT
iajs-3486	103	23	)	)	PUNCT
iajs-3486	103	24	)	)	PUNCT
iajs-3486	104	1	→	→	SYM
iajs-3486	104	2	0	0	NUM
iajs-3486	104	3	as	as	SCONJ
iajs-3486	104	4	n	n	PROPN
iajs-3486	104	5	→	→	SYM
iajs-3486	104	6	∞.	∞.	PROPN
iajs-3486	104	7	theorem	theorem	VERB
iajs-3486	104	8	2.5	2.5	NUM
iajs-3486	104	9	.	.	PUNCT
iajs-3486	105	1	let	let	VERB
iajs-3486	105	2	υ	υ	NOUN
iajs-3486	105	3	and	and	CCONJ
iajs-3486	105	4	let𝛷	let𝛷	NOUN
iajs-3486	105	5	satisfying	satisfy	VERB
iajs-3486	105	6	the	the	DET
iajs-3486	105	7	opial’scondition	opial’scondition	NOUN
iajs-3486	105	8	.	.	PUNCT
iajs-3486	106	1	and	and	CCONJ
iajs-3486	106	2	ℛ	ℛ	NOUN
iajs-3486	106	3	,	,	PUNCT
iajs-3486	106	4	ζ	ζ	NOUN
iajs-3486	106	5	,	,	PUNCT
iajs-3486	106	6	𝐵	𝐵	NOUN
iajs-3486	106	7	∶	∶	NOUN
iajs-3486	106	8	ψ	ψ	X
iajs-3486	106	9	×	×	NOUN
iajs-3486	106	10	υ	υ	PROPN
iajs-3486	106	11	→	→	PROPN
iajs-3486	106	12	𝛱(υ).if	𝛱(υ).if	PROPN
iajs-3486	106	13	𝐹	𝐹	PROPN
iajs-3486	106	14	≠	≠	PROPN
iajs-3486	106	15	∅	∅	NOUN
iajs-3486	106	16	and	and	CCONJ
iajs-3486	106	17	𝐵ϑ(ϱ	𝐵ϑ(ϱ	NOUN
iajs-3486	106	18	)	)	PUNCT
iajs-3486	106	19	=	=	SYM
iajs-3486	106	20	ℛϑ(ϱ	ℛϑ(ϱ	ADJ
iajs-3486	106	21	)	)	PUNCT
iajs-3486	106	22	=	=	PUNCT
iajs-3486	106	23	ζϑ(ϱ	ζϑ(ϱ	X
iajs-3486	106	24	)	)	PUNCT
iajs-3486	107	1	=	=	SYM
iajs-3486	107	2	{	{	PUNCT
iajs-3486	107	3	ϑ(ϱ)}for	ϑ(ϱ)}for	ADP
iajs-3486	107	4	any	any	DET
iajs-3486	107	5	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	107	6	)	)	PUNCT
iajs-3486	107	7	∈	∈	PROPN
iajs-3486	107	8	𝐹	𝐹	PROPN
iajs-3486	107	9	,	,	PUNCT
iajs-3486	107	10	𝐼	𝐼	ADP
iajs-3486	107	11	−	−	PROPN
iajs-3486	107	12	𝐵	𝐵	NOUN
iajs-3486	107	13	,	,	PUNCT
iajs-3486	107	14	𝐼	𝐼	PROPN
iajs-3486	107	15	−	−	PROPN
iajs-3486	107	16	ζand	ζand	NOUN
iajs-3486	108	1	𝐼	𝐼	SCONJ
iajs-3486	108	2	−	−	PROPN
iajs-3486	108	3	ℛare	ℛare	PROPN
iajs-3486	108	4	demi	demi	NOUN
iajs-3486	108	5	-	-	PUNCT
iajs-3486	108	6	closed	closed	ADJ
iajs-3486	108	7	to	to	ADP
iajs-3486	108	8	0	0	NUM
iajs-3486	108	9	,	,	PUNCT
iajs-3486	108	10	then	then	ADV
iajs-3486	108	11	{	{	PUNCT
iajs-3486	108	12	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	108	13	)	)	PUNCT
iajs-3486	108	14	}	}	PUNCT
iajs-3486	108	15	converges	converge	VERB
iajs-3486	108	16	to	to	ADP
iajs-3486	108	17	a	a	DET
iajs-3486	108	18	commonrandom	commonrandom	NOUN
iajs-3486	108	19	fixed	fix	VERB
iajs-3486	108	20	point	point	NOUN
iajs-3486	108	21	of	of	ADP
iajs-3486	108	22	𝐵	𝐵	PROPN
iajs-3486	108	23	,	,	PUNCT
iajs-3486	108	24	s	s	PROPN
iajs-3486	108	25	,	,	PUNCT
iajs-3486	108	26	r	r	NOUN
iajs-3486	108	27	and	and	CCONJ
iajs-3486	108	28	weakly	weakly	ADJ
iajs-3486	108	29	.	.	PUNCT
iajs-3486	109	1	proof	proof	NOUN
iajs-3486	109	2	:	:	PUNCT
iajs-3486	109	3	let	let	VERB
iajs-3486	109	4	ϑ(ϱ	ϑ(ϱ	NUM
iajs-3486	109	5	)	)	PUNCT
iajs-3486	109	6	∈	∈	PROPN
iajs-3486	109	7	𝐹.	𝐹.	PROPN
iajs-3486	109	8	by	by	ADP
iajs-3486	109	9	lemma	lemma	PROPN
iajs-3486	109	10	(	(	PUNCT
iajs-3486	109	11	2.2	2.2	NUM
iajs-3486	109	12	)	)	PUNCT
iajs-3486	109	13	,	,	PUNCT
iajs-3486	109	14	we	we	PRON
iajs-3486	109	15	have	have	VERB
iajs-3486	109	16	lim	lim	PROPN
iajs-3486	109	17	𝑛→∞	𝑛→∞	NUM
iajs-3486	109	18	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	109	19	)	)	PUNCT
iajs-3486	109	20	−	−	NOUN
iajs-3486	109	21	ϑ(ϱ)‖exists	ϑ(ϱ)‖exist	NOUN
iajs-3486	109	22	.	.	PUNCT
iajs-3486	110	1	we	we	PRON
iajs-3486	110	2	prove	prove	VERB
iajs-3486	110	3	that	that	SCONJ
iajs-3486	110	4	{	{	PUNCT
iajs-3486	110	5	𝑆𝑛(𝜚)}subsequentialhave	𝑆𝑛(𝜚)}subsequentialhave	VERB
iajs-3486	110	6	a	a	DET
iajs-3486	110	7	weak	weak	ADJ
iajs-3486	110	8	unique	unique	ADJ
iajs-3486	110	9	limit	limit	NOUN
iajs-3486	110	10	in	in	ADP
iajs-3486	110	11	f	f	PROPN
iajs-3486	110	12	.	.	PUNCT
iajs-3486	111	1	now	now	ADV
iajs-3486	111	2	,	,	PUNCT
iajs-3486	111	3	let	let	VERB
iajs-3486	111	4	𝜔1(ϱ	𝜔1(ϱ	SYM
iajs-3486	111	5	)	)	PUNCT
iajs-3486	111	6	,	,	PUNCT
iajs-3486	111	7	𝜔2(ϱ)𝑎𝑛𝑑	𝜔2(ϱ)𝑎𝑛𝑑	PUNCT
iajs-3486	111	8	𝜔3(ϱ	𝜔3(ϱ	X
iajs-3486	111	9	)	)	PUNCT
iajs-3486	111	10	be	be	AUX
iajs-3486	111	11	weak	weak	ADJ
iajs-3486	111	12	limits	limit	NOUN
iajs-3486	111	13	of	of	ADP
iajs-3486	111	14	the	the	DET
iajs-3486	111	15	subsequences	subsequence	NOUN
iajs-3486	111	16	{	{	PUNCT
iajs-3486	111	17	𝑆𝑛𝑖(𝜚	𝑆𝑛𝑖(𝜚	NOUN
iajs-3486	111	18	)	)	PUNCT
iajs-3486	111	19	}	}	PUNCT
iajs-3486	111	20	,	,	PUNCT
iajs-3486	111	21	{	{	PUNCT
iajs-3486	111	22	𝑆𝑛𝑗(𝜚	𝑆𝑛𝑗(𝜚	NOUN
iajs-3486	111	23	)	)	PUNCT
iajs-3486	111	24	}	}	PUNCT
iajs-3486	111	25	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3486	111	26	{	{	PUNCT
iajs-3486	111	27	𝑆𝑛𝑙(𝜚	𝑆𝑛𝑙(𝜚	NOUN
iajs-3486	111	28	)	)	PUNCT
iajs-3486	111	29	}	}	PUNCT
iajs-3486	111	30	𝑜𝑓	𝑜𝑓	ADP
iajs-3486	111	31	{	{	PUNCT
iajs-3486	111	32	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	111	33	)	)	PUNCT
iajs-3486	111	34	}	}	PUNCT
iajs-3486	111	35	,	,	PUNCT
iajs-3486	111	36	respect	respect	NOUN
iajs-3486	111	37	.	.	PUNCT
iajs-3486	112	1	by	by	ADP
iajs-3486	112	2	uselemma	uselemma	PROPN
iajs-3486	112	3	(	(	PUNCT
iajs-3486	112	4	2.4	2.4	NUM
iajs-3486	112	5	)	)	PUNCT
iajs-3486	112	6	,	,	PUNCT
iajs-3486	112	7	there	there	PRON
iajs-3486	112	8	is𝜂𝑛(𝜚	is𝜂𝑛(𝜚	PROPN
iajs-3486	112	9	)	)	PUNCT
iajs-3486	112	10	∈	∈	PROPN
iajs-3486	112	11	𝐵(𝑆𝑛(𝜚))such	𝐵(𝑆𝑛(𝜚))such	PUNCT
iajs-3486	112	12	that	that	SCONJ
iajs-3486	112	13	lim	lim	PROPN
iajs-3486	112	14	𝑛→∞	𝑛→∞	NUM
iajs-3486	112	15	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	112	16	)	)	PUNCT
iajs-3486	113	1	−	−	NOUN
iajs-3486	113	2	ϑ(ϱ)‖	ϑ(ϱ)‖	ADJ
iajs-3486	113	3	=	=	SYM
iajs-3486	113	4	0	0	PUNCT
iajs-3486	114	1	and	and	CCONJ
iajs-3486	114	2	𝐼	𝐼	PROPN
iajs-3486	114	3	−	−	PROPN
iajs-3486	114	4	𝐵is	𝐵is	PROPN
iajs-3486	114	5	demi	demi	NOUN
iajs-3486	114	6	-	-	PUNCT
iajs-3486	114	7	closedto	closedto	ADJ
iajs-3486	114	8	0	0	NUM
iajs-3486	114	9	,	,	PUNCT
iajs-3486	114	10	therefore	therefore	ADV
iajs-3486	114	11	we	we	PRON
iajs-3486	114	12	get𝜔1(ϱ	get𝜔1(ϱ	VERB
iajs-3486	114	13	)	)	PUNCT
iajs-3486	115	1	∈	∈	PROPN
iajs-3486	115	2	𝐵𝜔1(ϱ	𝐵𝜔1(ϱ	VERB
iajs-3486	115	3	)	)	PUNCT
iajs-3486	115	4	.	.	PUNCT
iajs-3486	116	1	similarly	similarly	ADV
iajs-3486	116	2	,	,	PUNCT
iajs-3486	116	3	𝜔1(ϱ	𝜔1(ϱ	SYM
iajs-3486	116	4	)	)	PUNCT
iajs-3486	116	5	∈	∈	PROPN
iajs-3486	116	6	ζ𝜔1(ϱ	ζ𝜔1(ϱ	PROPN
iajs-3486	116	7	)	)	PUNCT
iajs-3486	116	8	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-3486	116	9	𝜔1(ϱ	𝜔1(ϱ	SYM
iajs-3486	116	10	)	)	PUNCT
iajs-3486	116	11	∈	∈	PROPN
iajs-3486	117	1	ℛ𝜔1(ϱ).again	ℛ𝜔1(ϱ).again	NOUN
iajs-3486	117	2	in	in	ADP
iajs-3486	117	3	a	a	DET
iajs-3486	117	4	same	same	ADJ
iajs-3486	117	5	way	way	NOUN
iajs-3486	117	6	,	,	PUNCT
iajs-3486	117	7	we	we	PRON
iajs-3486	117	8	can	can	AUX
iajs-3486	117	9	prove	prove	VERB
iajs-3486	117	10	that	that	PRON
iajs-3486	117	11	𝜔1(ϱ	𝜔1(ϱ	SYM
iajs-3486	117	12	)	)	PUNCT
iajs-3486	117	13	,	,	PUNCT
iajs-3486	117	14	𝜔2(ϱ	𝜔2(ϱ	PROPN
iajs-3486	117	15	)	)	PUNCT
iajs-3486	117	16	,	,	PUNCT
iajs-3486	117	17	𝜔3(ϱ	𝜔3(ϱ	X
iajs-3486	117	18	)	)	PUNCT
iajs-3486	117	19	∈	∈	PROPN
iajs-3486	117	20	𝐹.	𝐹.	PROPN
iajs-3486	117	21	now	now	ADV
iajs-3486	117	22	,	,	PUNCT
iajs-3486	117	23	we	we	PRON
iajs-3486	117	24	need	need	VERB
iajs-3486	117	25	to	to	PART
iajs-3486	117	26	prove	prove	VERB
iajs-3486	117	27	unique	unique	ADJ
iajs-3486	117	28	.	.	PUNCT
iajs-3486	118	1	for	for	ADP
iajs-3486	118	2	this	this	DET
iajs-3486	118	3	let𝜔1(ϱ	let𝜔1(ϱ	NUM
iajs-3486	118	4	)	)	PUNCT
iajs-3486	118	5	≠	≠	PROPN
iajs-3486	118	6	𝜔2(ϱ	𝜔2(ϱ	ADP
iajs-3486	118	7	)	)	PUNCT
iajs-3486	118	8	≠	≠	PROPN
iajs-3486	118	9	𝜔3(ϱ	𝜔3(ϱ	PROPN
iajs-3486	118	10	)	)	PUNCT
iajs-3486	118	11	.	.	PUNCT
iajs-3486	119	1	then	then	ADV
iajs-3486	119	2	by	by	ADP
iajs-3486	119	3	the	the	DET
iajs-3486	119	4	opial	opial	NOUN
iajs-3486	119	5	’s	’s	PART
iajs-3486	119	6	condition	condition	NOUN
iajs-3486	119	7	,	,	PUNCT
iajs-3486	119	8	lim	lim	PROPN
iajs-3486	119	9	𝑛→∞	𝑛→∞	NUM
iajs-3486	119	10	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	119	11	)	)	PUNCT
iajs-3486	119	12	−	−	NOUN
iajs-3486	119	13	𝜔1(ϱ)‖	𝜔1(ϱ)‖	NOUN
iajs-3486	120	1	=	=	SYM
iajs-3486	120	2	lim	lim	PROPN
iajs-3486	120	3	𝑛𝑖→∞	𝑛𝑖→∞	NUM
iajs-3486	120	4	‖𝑆𝑛𝑖(𝜚	‖𝑆𝑛𝑖(𝜚	NOUN
iajs-3486	120	5	)	)	PUNCT
iajs-3486	120	6	−	−	ADP
iajs-3486	120	7	𝜔1(ϱ)‖	𝜔1(ϱ)‖	PRON
iajs-3486	120	8	<	<	X
iajs-3486	120	9	lim	lim	PROPN
iajs-3486	120	10	𝑛𝑖→∞	𝑛𝑖→∞	NUM
iajs-3486	120	11	‖𝑆𝑛𝑖(𝜚	‖𝑆𝑛𝑖(𝜚	NOUN
iajs-3486	120	12	)	)	PUNCT
iajs-3486	120	13	−	−	PROPN
iajs-3486	120	14	𝜔2(ϱ)‖	𝜔2(ϱ)‖	VERB
iajs-3486	121	1	=	=	SYM
iajs-3486	121	2	lim	lim	NOUN
iajs-3486	121	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	121	4	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	121	5	)	)	PUNCT
iajs-3486	121	6	−	−	PROPN
iajs-3486	121	7	𝜔2(ϱ)‖	𝜔2(ϱ)‖	VERB
iajs-3486	122	1	=	=	SYM
iajs-3486	122	2	lim	lim	PROPN
iajs-3486	122	3	𝑛𝑗→∞	𝑛𝑗→∞	NOUN
iajs-3486	122	4	‖𝑆𝑛𝑗(𝜚	‖𝑆𝑛𝑗(𝜚	NOUN
iajs-3486	122	5	)	)	PUNCT
iajs-3486	122	6	−	−	PROPN
iajs-3486	122	7	𝜔2(ϱ)‖	𝜔2(ϱ)‖	VERB
iajs-3486	122	8	<	<	X
iajs-3486	122	9	lim	lim	PROPN
iajs-3486	122	10	𝑛𝑗→∞	𝑛𝑗→∞	PROPN
iajs-3486	122	11	‖𝑆𝑛𝑗(𝜚	‖𝑆𝑛𝑗(𝜚	NOUN
iajs-3486	122	12	)	)	PUNCT
iajs-3486	122	13	−	−	NOUN
iajs-3486	123	1	𝜔3(ϱ)‖	𝜔3(ϱ)‖	PUNCT
iajs-3486	124	1	=	=	NOUN
iajs-3486	124	2	lim	lim	NOUN
iajs-3486	124	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	124	4	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	124	5	)	)	PUNCT
iajs-3486	124	6	−	−	PROPN
iajs-3486	124	7	𝜔3(ϱ)‖	𝜔3(ϱ)‖	PUNCT
iajs-3486	125	1	ihjpas.37	ihjpas.37	NOUN
iajs-3486	125	2	(	(	PUNCT
iajs-3486	125	3	2	2	NUM
iajs-3486	125	4	)	)	PUNCT
iajs-3486	125	5	2024	2024	NUM
iajs-3486	125	6	428	428	NUM
iajs-3486	125	7	=	=	SYM
iajs-3486	125	8	lim	lim	PROPN
iajs-3486	125	9	𝑛𝑙→∞	𝑛𝑙→∞	NOUN
iajs-3486	125	10	‖𝑆𝑛𝑙(𝜚	‖𝑆𝑛𝑙(𝜚	PROPN
iajs-3486	125	11	)	)	PUNCT
iajs-3486	125	12	−	−	NOUN
iajs-3486	125	13	𝜔3(ϱ)‖	𝜔3(ϱ)‖	PUNCT
iajs-3486	126	1	<	<	X
iajs-3486	126	2	lim	lim	PROPN
iajs-3486	126	3	𝑛𝑙→∞	𝑛𝑙→∞	PROPN
iajs-3486	126	4	‖𝑆𝑛𝑙(𝜚	‖𝑆𝑛𝑙(𝜚	PROPN
iajs-3486	126	5	)	)	PUNCT
iajs-3486	126	6	−	−	ADP
iajs-3486	126	7	𝜔1(ϱ)‖	𝜔1(ϱ)‖	NOUN
iajs-3486	127	1	=	=	SYM
iajs-3486	127	2	lim	lim	PROPN
iajs-3486	127	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	127	4	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	127	5	)	)	PUNCT
iajs-3486	127	6	−	−	NOUN
iajs-3486	127	7	𝜔1(ϱ)‖	𝜔1(ϱ)‖	NOUN
iajs-3486	127	8	hence	hence	ADV
iajs-3486	127	9	,	,	PUNCT
iajs-3486	127	10	its	its	PRON
iajs-3486	127	11	a	a	DET
iajs-3486	127	12	contradiction	contradiction	NOUN
iajs-3486	127	13	.	.	PUNCT
iajs-3486	128	1	hence	hence	ADV
iajs-3486	128	2	{	{	PUNCT
iajs-3486	128	3	𝑆𝑛(𝜚)}converges	𝑆𝑛(𝜚)}converge	NOUN
iajs-3486	128	4	and	and	CCONJ
iajs-3486	128	5	weakly	weakly	ADJ
iajs-3486	128	6	in	in	ADP
iajs-3486	128	7	𝐹.	𝐹.	PROPN
iajs-3486	128	8	remark	remark	NOUN
iajs-3486	128	9	2.6let	2.6let	NUM
iajs-3486	128	10	υ	υ	NOUN
iajs-3486	128	11	and	and	CCONJ
iajs-3486	128	12	let𝛷	let𝛷	NOUN
iajs-3486	128	13	satisfying	satisfy	VERB
iajs-3486	128	14	the	the	DET
iajs-3486	128	15	opial’scondition	opial’scondition	NOUN
iajs-3486	128	16	and	and	CCONJ
iajs-3486	128	17	,	,	PUNCT
iajs-3486	128	18	ℛ	ℛ	PROPN
iajs-3486	128	19	,	,	PUNCT
iajs-3486	128	20	ζ	ζ	NOUN
iajs-3486	128	21	,	,	PUNCT
iajs-3486	128	22	𝐵	𝐵	NOUN
iajs-3486	128	23	∶	∶	NOUN
iajs-3486	128	24	ψ	ψ	X
iajs-3486	128	25	×	×	NOUN
iajs-3486	128	26	υ	υ	NOUN
iajs-3486	128	27	→	→	SYM
iajs-3486	128	28	𝛱(υ	𝛱(υ	NOUN
iajs-3486	128	29	)	)	PUNCT
iajs-3486	128	30	and	and	CCONJ
iajs-3486	128	31	{	{	PUNCT
iajs-3486	128	32	𝑆𝑛(𝜚)}be	𝑆𝑛(𝜚)}be	PROPN
iajs-3486	128	33	the	the	DET
iajs-3486	128	34	sequence	sequence	NOUN
iajs-3486	128	35	in	in	ADP
iajs-3486	128	36	(	(	PUNCT
iajs-3486	128	37	2.1	2.1	NUM
iajs-3486	128	38	)	)	PUNCT
iajs-3486	128	39	,	,	PUNCT
iajs-3486	128	40	.	.	PUNCT
iajs-3486	129	1	if	if	SCONJ
iajs-3486	129	2	𝐹	𝐹	PROPN
iajs-3486	129	3	≠	≠	PROPN
iajs-3486	129	4	∅	∅	NOUN
iajs-3486	129	5	and	and	CCONJ
iajs-3486	129	6	𝐵ϑ(ϱ	𝐵ϑ(ϱ	NOUN
iajs-3486	129	7	)	)	PUNCT
iajs-3486	129	8	=	=	SYM
iajs-3486	129	9	ℛϑ(ϱ	ℛϑ(ϱ	ADJ
iajs-3486	129	10	)	)	PUNCT
iajs-3486	129	11	=	=	PUNCT
iajs-3486	129	12	ζϑ(ϱ	ζϑ(ϱ	X
iajs-3486	129	13	)	)	PUNCT
iajs-3486	129	14	=	=	SYM
iajs-3486	130	1	{	{	PUNCT
iajs-3486	130	2	ϑ(ϱ)}then	ϑ(ϱ)}then	ADV
iajs-3486	130	3	{	{	PUNCT
iajs-3486	130	4	𝑆𝑛(𝜚)}converges	𝑆𝑛(𝜚)}converge	NOUN
iajs-3486	130	5	to	to	ADP
iajs-3486	130	6	a	a	DET
iajs-3486	130	7	common	common	ADJ
iajs-3486	130	8	-	-	PUNCT
iajs-3486	130	9	random	random	ADJ
iajs-3486	130	10	fixed	fix	VERB
iajs-3486	130	11	point	point	NOUN
iajs-3486	130	12	of	of	ADP
iajs-3486	130	13	𝐵	𝐵	PROPN
iajs-3486	130	14	,	,	PUNCT
iajs-3486	130	15	ζ,ℛ	ζ,ℛ	NOUN
iajs-3486	130	16	and	and	CCONJ
iajs-3486	130	17	weakly	weakly	ADJ
iajs-3486	130	18	.	.	PUNCT
iajs-3486	131	1	theorem	theorem	VERB
iajs-3486	131	2	2.7	2.7	NUM
iajs-3486	131	3	.	.	PUNCT
iajs-3486	132	1	let	let	VERB
iajs-3486	132	2	𝛷,υ	𝛷,υ	NUM
iajs-3486	132	3	,	,	PUNCT
iajs-3486	132	4	and	and	CCONJ
iajs-3486	132	5	{	{	PUNCT
iajs-3486	132	6	𝑆𝑛(𝜚	𝑆𝑛(𝜚	ADJ
iajs-3486	132	7	)	)	PUNCT
iajs-3486	132	8	}	}	PUNCT
iajs-3486	132	9	and𝐵	and𝐵	NOUN
iajs-3486	132	10	,	,	PUNCT
iajs-3486	132	11	ζ,,ℛ	ζ,,ℛ	PROPN
iajs-3486	132	12	defined	define	VERB
iajs-3486	132	13	in	in	ADP
iajs-3486	132	14	the	the	DET
iajs-3486	132	15	lema(2.4	lema(2.4	NOUN
iajs-3486	132	16	)	)	PUNCT
iajs-3486	132	17	.	.	PUNCT
iajs-3486	133	1	and	and	CCONJ
iajs-3486	133	2	𝐹	𝐹	PROPN
iajs-3486	133	3	≠	≠	PROPN
iajs-3486	133	4	∅and𝐵ϑ(ϱ	∅and𝐵ϑ(ϱ	ADJ
iajs-3486	133	5	)	)	PUNCT
iajs-3486	133	6	=	=	SYM
iajs-3486	133	7	ℛϑ(ϱ	ℛϑ(ϱ	ADJ
iajs-3486	133	8	)	)	PUNCT
iajs-3486	133	9	=	=	PUNCT
iajs-3486	133	10	ζϑ(ϱ	ζϑ(ϱ	X
iajs-3486	133	11	)	)	PUNCT
iajs-3486	133	12	=	=	SYM
iajs-3486	133	13	{	{	PUNCT
iajs-3486	133	14	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	133	15	)	)	PUNCT
iajs-3486	133	16	}	}	PUNCT
iajs-3486	133	17	for	for	ADP
iajs-3486	133	18	any	any	DET
iajs-3486	133	19	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	133	20	)	)	PUNCT
iajs-3486	133	21	∈	∈	PROPN
iajs-3486	133	22	𝐹	𝐹	PROPN
iajs-3486	133	23	,	,	PUNCT
iajs-3486	133	24	then	then	ADV
iajs-3486	133	25	{	{	PUNCT
iajs-3486	133	26	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	133	27	)	)	PUNCT
iajs-3486	133	28	}	}	PUNCT
iajs-3486	133	29	converges	converge	VERB
iajs-3486	133	30	to	to	ADP
iajs-3486	133	31	a	a	DET
iajs-3486	133	32	commonrandom	commonrandom	NOUN
iajs-3486	133	33	fixed	fix	VERB
iajs-3486	133	34	point	point	NOUN
iajs-3486	133	35	of	of	ADP
iajs-3486	133	36	𝐵	𝐵	PROPN
iajs-3486	133	37	,	,	PUNCT
iajs-3486	133	38	ℛ	ℛ	PROPN
iajs-3486	133	39	,	,	PUNCT
iajs-3486	133	40	ζandstronglyiff	ζandstronglyiff	PROPN
iajs-3486	133	41	lim	lim	PROPN
iajs-3486	133	42	𝑛→∞	𝑛→∞	NUM
iajs-3486	133	43	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3486	133	44	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	133	45	)	)	PUNCT
iajs-3486	133	46	,	,	PUNCT
iajs-3486	133	47	𝐹	𝐹	PROPN
iajs-3486	133	48	)	)	PUNCT
iajs-3486	133	49	=	=	SYM
iajs-3486	134	1	0	0	NUM
iajs-3486	134	2	proof	proof	NOUN
iajs-3486	134	3	.	.	PUNCT
iajs-3486	135	1	the	the	DET
iajs-3486	135	2	first	first	ADJ
iajs-3486	135	3	direction	direction	NOUN
iajs-3486	135	4	of	of	ADP
iajs-3486	135	5	the	the	DET
iajs-3486	135	6	proof	proof	NOUN
iajs-3486	135	7	is	be	AUX
iajs-3486	135	8	clear	clear	ADJ
iajs-3486	135	9	.	.	PUNCT
iajs-3486	136	1	conversely	conversely	ADV
iajs-3486	136	2	,	,	PUNCT
iajs-3486	136	3	let	let	VERB
iajs-3486	136	4	lim	lim	PROPN
iajs-3486	136	5	𝑛→∞	𝑛→∞	NUM
iajs-3486	136	6	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3486	136	7	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	136	8	)	)	PUNCT
iajs-3486	136	9	,	,	PUNCT
iajs-3486	136	10	𝐹	𝐹	PROPN
iajs-3486	136	11	)	)	PUNCT
iajs-3486	136	12	=	=	SYM
iajs-3486	137	1	0	0	X
iajs-3486	137	2	.	.	PUNCT
iajs-3486	137	3	by	by	ADP
iajs-3486	137	4	lemma(2.2	lemma(2.2	ADJ
iajs-3486	137	5	)	)	PUNCT
iajs-3486	137	6	,	,	PUNCT
iajs-3486	137	7	‖𝑆𝑛+1(𝜚	‖𝑆𝑛+1(𝜚	NOUN
iajs-3486	137	8	)	)	PUNCT
iajs-3486	137	9	−	−	VERB
iajs-3486	137	10	ϑ(ϱ)‖	ϑ(ϱ)‖	PROPN
iajs-3486	137	11	≤	≤	NUM
iajs-3486	137	12	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	137	13	)	)	PUNCT
iajs-3486	137	14	−	−	ADP
iajs-3486	138	1	ϑ(ϱ)‖.	ϑ(ϱ)‖.	VERB
iajs-3486	138	2	this	this	PRON
iajs-3486	138	3	gives	give	VERB
iajs-3486	138	4	𝑑(𝑆𝑛+1(𝜚	𝑑(𝑆𝑛+1(𝜚	NOUN
iajs-3486	138	5	)	)	PUNCT
iajs-3486	138	6	,	,	PUNCT
iajs-3486	138	7	𝐹	𝐹	PROPN
iajs-3486	138	8	)	)	PUNCT
iajs-3486	138	9	≤	≤	NOUN
iajs-3486	138	10	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	138	11	)	)	PUNCT
iajs-3486	138	12	,	,	PUNCT
iajs-3486	138	13	𝐹	𝐹	PROPN
iajs-3486	138	14	)	)	PUNCT
iajs-3486	138	15	,	,	PUNCT
iajs-3486	138	16	so	so	SCONJ
iajs-3486	138	17	that	that	SCONJ
iajs-3486	138	18	lim	lim	PROPN
iajs-3486	138	19	𝑛→∞	𝑛→∞	NUM
iajs-3486	138	20	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3486	138	21	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	138	22	)	)	PUNCT
iajs-3486	138	23	,	,	PUNCT
iajs-3486	138	24	𝐹)exists	𝐹)exist	NOUN
iajs-3486	138	25	.	.	PUNCT
iajs-3486	139	1	but	but	CCONJ
iajs-3486	139	2	,	,	PUNCT
iajs-3486	139	3	by	by	ADP
iajs-3486	139	4	use	use	NOUN
iajs-3486	139	5	hypothesis	hypothesis	NOUN
iajs-3486	139	6	,	,	PUNCT
iajs-3486	139	7	lim	lim	PROPN
iajs-3486	139	8	𝑛→∞	𝑛→∞	NUM
iajs-3486	139	9	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3486	139	10	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	139	11	)	)	PUNCT
iajs-3486	139	12	,	,	PUNCT
iajs-3486	139	13	𝐹	𝐹	PROPN
iajs-3486	139	14	)	)	PUNCT
iajs-3486	140	1	=	=	SYM
iajs-3486	140	2	0	0	X
iajs-3486	140	3	.	.	PUNCT
iajs-3486	141	1	therefore	therefore	ADV
iajs-3486	141	2	we	we	PRON
iajs-3486	141	3	must	must	AUX
iajs-3486	141	4	have	have	VERB
iajs-3486	141	5	lim	lim	PROPN
iajs-3486	141	6	𝑛→∞	𝑛→∞	NUM
iajs-3486	141	7	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3486	141	8	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	141	9	)	)	PUNCT
iajs-3486	141	10	,	,	PUNCT
iajs-3486	141	11	𝐹	𝐹	PROPN
iajs-3486	141	12	)	)	PUNCT
iajs-3486	141	13	=	=	SYM
iajs-3486	142	1	0	0	X
iajs-3486	142	2	.	.	PUNCT
iajs-3486	143	1	we	we	PRON
iajs-3486	143	2	need	need	VERB
iajs-3486	143	3	toprove	toprove	NOUN
iajs-3486	143	4	that	that	PRON
iajs-3486	143	5	{	{	PUNCT
iajs-3486	143	6	𝑆𝑛(𝜚)}is	𝑆𝑛(𝜚)}is	PROPN
iajs-3486	143	7	a	a	DET
iajs-3486	143	8	cauchy	cauchy	ADJ
iajs-3486	143	9	sequence	sequence	NOUN
iajs-3486	143	10	in	in	ADP
iajs-3486	143	11	υ	υ	PROPN
iajs-3486	143	12	.	.	PUNCT
iajs-3486	143	13	suppose	suppose	VERB
iajs-3486	143	14	ε	ε	PROPN
iajs-3486	143	15	>	>	PROPN
iajs-3486	143	16	0	0	PROPN
iajs-3486	143	17	.	.	PUNCT
iajs-3486	144	1	we	we	PRON
iajs-3486	144	2	have	have	VERB
iajs-3486	144	3	lim	lim	NOUN
iajs-3486	144	4	𝑛→∞	𝑛→∞	NUM
iajs-3486	144	5	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	144	6	)	)	PUNCT
iajs-3486	144	7	,	,	PUNCT
iajs-3486	144	8	𝐹	𝐹	PROPN
iajs-3486	144	9	)	)	PUNCT
iajs-3486	144	10	=	=	SYM
iajs-3486	144	11	0	0	NUM
iajs-3486	144	12	,	,	PUNCT
iajs-3486	144	13	thereis	thereis	NOUN
iajs-3486	144	14	a	a	DET
iajs-3486	144	15	constant	constant	ADJ
iajs-3486	144	16	𝑛0	𝑛0	VERB
iajs-3486	144	17	such	such	ADJ
iajs-3486	144	18	that	that	SCONJ
iajs-3486	144	19	∀𝑛	∀𝑛	NOUN
iajs-3486	144	20	≥	≥	PRON
iajs-3486	144	21	𝑛0	𝑛0	VERB
iajs-3486	144	22	,	,	PUNCT
iajs-3486	144	23	we	we	PRON
iajs-3486	144	24	have	have	VERB
iajs-3486	144	25	lim	lim	PROPN
iajs-3486	144	26	𝑛→∞	𝑛→∞	NUM
iajs-3486	144	27	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3486	144	28	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	144	29	)	)	PUNCT
iajs-3486	144	30	,	,	PUNCT
iajs-3486	144	31	𝐹	𝐹	PROPN
iajs-3486	144	32	)	)	PUNCT
iajs-3486	144	33	<	<	X
iajs-3486	144	34	𝜖	𝜖	X
iajs-3486	144	35	4	4	NUM
iajs-3486	144	36	in	in	ADP
iajs-3486	144	37	particular	particular	ADJ
iajs-3486	144	38	,	,	PUNCT
iajs-3486	144	39	inf{‖𝑆𝑛0(𝜚	inf{‖𝑆𝑛0(𝜚	NUM
iajs-3486	144	40	)	)	PUNCT
iajs-3486	144	41	−	−	NOUN
iajs-3486	144	42	ϑ(ϱ)‖	ϑ(ϱ)‖	DET
iajs-3486	144	43	∶	∶	NOUN
iajs-3486	145	1	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	145	2	)	)	PUNCT
iajs-3486	145	3	∈	∈	PROPN
iajs-3486	145	4	𝐹	𝐹	PROPN
iajs-3486	145	5	}	}	PUNCT
iajs-3486	145	6	<	<	X
iajs-3486	145	7	𝜖	𝜖	PROPN
iajs-3486	145	8	4	4	NUM
iajs-3486	145	9	.	.	PUNCT
iajs-3486	146	1	there	there	PRON
iajs-3486	146	2	must	must	AUX
iajs-3486	146	3	exist	exist	VERB
iajs-3486	146	4	a	a	DET
iajs-3486	146	5	ϑ(ϱ)∗	ϑ(ϱ)∗	NOUN
iajs-3486	146	6	∈	∈	PROPN
iajs-3486	146	7	𝐹such	𝐹such	PROPN
iajs-3486	146	8	that	that	SCONJ
iajs-3486	146	9	‖𝑆𝑛0(𝜚	‖𝑆𝑛0(𝜚	VERB
iajs-3486	146	10	)	)	PUNCT
iajs-3486	146	11	−	−	PROPN
iajs-3486	147	1	ϑ(ϱ)∗‖	ϑ(ϱ)∗‖	NOUN
iajs-3486	147	2	<	<	X
iajs-3486	147	3	𝜖	𝜖	PROPN
iajs-3486	147	4	2	2	NUM
iajs-3486	147	5	now	now	ADV
iajs-3486	147	6	for	for	ADP
iajs-3486	147	7	𝑚	𝑚	NOUN
iajs-3486	147	8	,	,	PUNCT
iajs-3486	147	9	𝑛	𝑛	DET
iajs-3486	147	10	≥	≥	NOUN
iajs-3486	147	11	𝑛0	𝑛0	VERB
iajs-3486	147	12	,	,	PUNCT
iajs-3486	147	13	we	we	PRON
iajs-3486	147	14	have	have	AUX
iajs-3486	147	15	‖𝑆𝑛+𝑚(𝜚	‖𝑆𝑛+𝑚(𝜚	VERB
iajs-3486	147	16	)	)	PUNCT
iajs-3486	148	1	−	−	NOUN
iajs-3486	148	2	𝑆𝑛(𝜚)‖	𝑆𝑛(𝜚)‖	NOUN
iajs-3486	148	3	≤	≤	NUM
iajs-3486	148	4	‖𝑆𝑛+𝑚(𝜚	‖𝑆𝑛+𝑚(𝜚	PROPN
iajs-3486	148	5	)	)	PUNCT
iajs-3486	148	6	−	−	PROPN
iajs-3486	149	1	ϑ(ϱ)∗‖	ϑ(ϱ)∗‖	NOUN
iajs-3486	149	2	+	+	CCONJ
iajs-3486	149	3	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	149	4	)	)	PUNCT
iajs-3486	149	5	−	−	PROPN
iajs-3486	149	6	ϑ(ϱ)∗‖	ϑ(ϱ)∗‖	NOUN
iajs-3486	149	7	≤	≤	NUM
iajs-3486	149	8	2‖𝑆𝑛0(𝜚	2‖𝑆𝑛0(𝜚	NUM
iajs-3486	149	9	)	)	PUNCT
iajs-3486	149	10	−	−	PROPN
iajs-3486	150	1	ϑ(ϱ)∗‖	ϑ(ϱ)∗‖	NOUN
iajs-3486	150	2	<	<	X
iajs-3486	150	3	2	2	NUM
iajs-3486	150	4	(	(	PUNCT
iajs-3486	150	5	𝜖	𝜖	PROPN
iajs-3486	150	6	2	2	NUM
iajs-3486	150	7	)	)	PUNCT
iajs-3486	150	8	=	=	PUNCT
iajs-3486	150	9	𝜖	𝜖	X
iajs-3486	150	10	hence	hence	ADV
iajs-3486	150	11	{	{	PUNCT
iajs-3486	150	12	𝑆𝑛(𝜚)}is	𝑆𝑛(𝜚)}is	PROPN
iajs-3486	150	13	a	a	DET
iajs-3486	150	14	cauchy	cauchy	ADJ
iajs-3486	150	15	sequence	sequence	NOUN
iajs-3486	150	16	inυof	inυof	PROPN
iajs-3486	150	17	𝛷	𝛷	PROPN
iajs-3486	150	18	,	,	PUNCT
iajs-3486	150	19	and	and	CCONJ
iajs-3486	150	20	therefore	therefore	ADV
iajs-3486	150	21	it	it	PRON
iajs-3486	150	22	must	must	AUX
iajs-3486	150	23	converge	converge	VERB
iajs-3486	150	24	in	in	ADP
iajs-3486	150	25	υ	υ	PROPN
iajs-3486	150	26	.	.	PUNCT
iajs-3486	151	1	let	let	VERB
iajs-3486	151	2	lim	lim	PROPN
iajs-3486	151	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	151	4	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	151	5	)	)	PUNCT
iajs-3486	151	6	=	=	SYM
iajs-3486	152	1	𝑞(ϱ	𝑞(ϱ	NOUN
iajs-3486	152	2	)	)	PUNCT
iajs-3486	152	3	now	now	ADV
iajs-3486	152	4	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	152	5	)	)	PUNCT
iajs-3486	152	6	,	,	PUNCT
iajs-3486	152	7	𝐵𝑞(ϱ	𝐵𝑞(ϱ	NOUN
iajs-3486	152	8	)	)	PUNCT
iajs-3486	152	9	)	)	PUNCT
iajs-3486	153	1	≤	≤	NUM
iajs-3486	153	2	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	153	3	)	)	PUNCT
iajs-3486	153	4	,	,	PUNCT
iajs-3486	153	5	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	153	6	)	)	PUNCT
iajs-3486	153	7	)	)	PUNCT
iajs-3486	154	1	+	+	CCONJ
iajs-3486	154	2	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	154	3	)	)	PUNCT
iajs-3486	154	4	,	,	PUNCT
iajs-3486	154	5	𝐵𝑆𝑛(𝜚	𝐵𝑆𝑛(𝜚	PROPN
iajs-3486	154	6	)	)	PUNCT
iajs-3486	154	7	)	)	PUNCT
iajs-3486	155	1	+	+	CCONJ
iajs-3486	155	2	𝐻(𝐵𝑆𝑛(𝜚	𝐻(𝐵𝑆𝑛(𝜚	X
iajs-3486	155	3	)	)	PUNCT
iajs-3486	155	4	,	,	PUNCT
iajs-3486	155	5	𝐵𝑞(ϱ	𝐵𝑞(ϱ	NOUN
iajs-3486	155	6	)	)	PUNCT
iajs-3486	155	7	)	)	PUNCT
iajs-3486	156	1	≤	≤	NUM
iajs-3486	156	2	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	156	3	)	)	PUNCT
iajs-3486	156	4	,	,	PUNCT
iajs-3486	156	5	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	156	6	)	)	PUNCT
iajs-3486	156	7	)	)	PUNCT
iajs-3486	157	1	+	+	CCONJ
iajs-3486	157	2	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	157	3	)	)	PUNCT
iajs-3486	157	4	,	,	PUNCT
iajs-3486	157	5	𝜂𝑛(𝜚	𝜂𝑛(𝜚	NOUN
iajs-3486	157	6	)	)	PUNCT
iajs-3486	157	7	)	)	PUNCT
iajs-3486	157	8	+	+	CCONJ
iajs-3486	157	9	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	157	10	)	)	PUNCT
iajs-3486	157	11	,	,	PUNCT
iajs-3486	157	12	𝑞(ϱ	𝑞(ϱ	NOUN
iajs-3486	157	13	)	)	PUNCT
iajs-3486	157	14	)	)	PUNCT
iajs-3486	157	15	→	→	SYM
iajs-3486	157	16	0	0	NUM
iajs-3486	157	17	𝑎𝑠	𝑎𝑠	PROPN
iajs-3486	157	18	𝑛	𝑛	PROPN
iajs-3486	157	19	→	→	SYM
iajs-3486	157	20	∞	∞	PROPN
iajs-3486	157	21	,	,	PUNCT
iajs-3486	157	22	gives	give	VERB
iajs-3486	157	23	that	that	DET
iajs-3486	157	24	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NOUN
iajs-3486	157	25	)	)	PUNCT
iajs-3486	157	26	,	,	PUNCT
iajs-3486	157	27	𝐵𝑞(ϱ	𝐵𝑞(ϱ	NOUN
iajs-3486	157	28	)	)	PUNCT
iajs-3486	157	29	)	)	PUNCT
iajs-3486	158	1	=	=	SYM
iajs-3486	158	2	0	0	NUM
iajs-3486	158	3	which	which	PRON
iajs-3486	158	4	implies	imply	VERB
iajs-3486	158	5	that	that	SCONJ
iajs-3486	158	6	𝑞(ϱ	𝑞(ϱ	NOUN
iajs-3486	158	7	)	)	PUNCT
iajs-3486	158	8	∈	∈	PROPN
iajs-3486	158	9	𝐵𝑞(ϱ	𝐵𝑞(ϱ	NOUN
iajs-3486	158	10	)	)	PUNCT
iajs-3486	158	11	.	.	PUNCT
iajs-3486	159	1	similarly	similarly	ADV
iajs-3486	159	2	,	,	PUNCT
iajs-3486	159	3	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	PROPN
iajs-3486	159	4	)	)	PUNCT
iajs-3486	159	5	,	,	PUNCT
iajs-3486	159	6	𝑆𝑞(ϱ	𝑆𝑞(ϱ	NOUN
iajs-3486	159	7	)	)	PUNCT
iajs-3486	159	8	)	)	PUNCT
iajs-3486	159	9	≤	≤	NUM
iajs-3486	159	10	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	159	11	)	)	PUNCT
iajs-3486	159	12	,	,	PUNCT
iajs-3486	159	13	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	159	14	)	)	PUNCT
iajs-3486	159	15	)	)	PUNCT
iajs-3486	160	1	+	+	CCONJ
iajs-3486	160	2	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	160	3	)	)	PUNCT
iajs-3486	160	4	,	,	PUNCT
iajs-3486	160	5	ζ𝑆𝑛(𝜚	ζ𝑆𝑛(𝜚	NOUN
iajs-3486	160	6	)	)	PUNCT
iajs-3486	160	7	)	)	PUNCT
iajs-3486	161	1	+	+	CCONJ
iajs-3486	161	2	𝐻(ζ𝑆𝑛(𝜚	𝐻(ζ𝑆𝑛(𝜚	NOUN
iajs-3486	161	3	)	)	PUNCT
iajs-3486	161	4	,	,	PUNCT
iajs-3486	161	5	ζ𝑞(ϱ	ζ𝑞(ϱ	NOUN
iajs-3486	161	6	)	)	PUNCT
iajs-3486	161	7	)	)	PUNCT
iajs-3486	161	8	≤	≤	NUM
iajs-3486	161	9	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	161	10	)	)	PUNCT
iajs-3486	161	11	,	,	PUNCT
iajs-3486	161	12	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	161	13	)	)	PUNCT
iajs-3486	161	14	)	)	PUNCT
iajs-3486	162	1	+	+	CCONJ
iajs-3486	162	2	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	162	3	)	)	PUNCT
iajs-3486	162	4	,	,	PUNCT
iajs-3486	162	5	𝜌𝑛(𝜚	𝜌𝑛(𝜚	NUM
iajs-3486	162	6	)	)	PUNCT
iajs-3486	162	7	)	)	PUNCT
iajs-3486	162	8	+	+	CCONJ
iajs-3486	162	9	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	162	10	)	)	PUNCT
iajs-3486	162	11	,	,	PUNCT
iajs-3486	162	12	𝑞(ϱ	𝑞(ϱ	NOUN
iajs-3486	162	13	)	)	PUNCT
iajs-3486	162	14	)	)	PUNCT
iajs-3486	162	15	→	→	SYM
iajs-3486	162	16	0	0	NUM
iajs-3486	162	17	𝑎𝑠	𝑎𝑠	PROPN
iajs-3486	162	18	𝑛	𝑛	PROPN
iajs-3486	162	19	→	→	SYM
iajs-3486	162	20	∞	∞	PROPN
iajs-3486	162	21	,	,	PUNCT
iajs-3486	162	22	ihjpas.37	ihjpas.37	PROPN
iajs-3486	162	23	(	(	PUNCT
iajs-3486	162	24	2	2	NUM
iajs-3486	162	25	)	)	PUNCT
iajs-3486	162	26	2024	2024	NUM
iajs-3486	162	27	429	429	NUM
iajs-3486	162	28	gives	give	VERB
iajs-3486	162	29	that	that	PRON
iajs-3486	162	30	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NOUN
iajs-3486	162	31	)	)	PUNCT
iajs-3486	162	32	,	,	PUNCT
iajs-3486	162	33	ζ𝑞(ϱ	ζ𝑞(ϱ	NOUN
iajs-3486	162	34	)	)	PUNCT
iajs-3486	162	35	)	)	PUNCT
iajs-3486	163	1	=	=	SYM
iajs-3486	163	2	0	0	NUM
iajs-3486	163	3	which	which	PRON
iajs-3486	163	4	implies	imply	VERB
iajs-3486	163	5	that	that	SCONJ
iajs-3486	163	6	𝑞(ϱ	𝑞(ϱ	NOUN
iajs-3486	163	7	)	)	PUNCT
iajs-3486	163	8	∈	∈	PROPN
iajs-3486	163	9	ζ𝑞(ϱ).similarly	ζ𝑞(ϱ).similarly	ADV
iajs-3486	163	10	,	,	PUNCT
iajs-3486	163	11	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	PROPN
iajs-3486	163	12	)	)	PUNCT
iajs-3486	163	13	,	,	PUNCT
iajs-3486	163	14	𝑅𝑞(ϱ	𝑅𝑞(ϱ	NOUN
iajs-3486	163	15	)	)	PUNCT
iajs-3486	163	16	)	)	PUNCT
iajs-3486	163	17	≤	≤	NUM
iajs-3486	163	18	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	163	19	)	)	PUNCT
iajs-3486	163	20	,	,	PUNCT
iajs-3486	163	21	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	163	22	)	)	PUNCT
iajs-3486	163	23	)	)	PUNCT
iajs-3486	164	1	+	+	CCONJ
iajs-3486	164	2	𝑑	𝑑	PROPN
iajs-3486	164	3	(	(	PUNCT
iajs-3486	164	4	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	164	5	)	)	PUNCT
iajs-3486	164	6	,	,	PUNCT
iajs-3486	164	7	ℛ𝑆𝑛(𝜚	ℛ𝑆𝑛(𝜚	NUM
iajs-3486	164	8	)	)	PUNCT
iajs-3486	164	9	)	)	PUNCT
iajs-3486	164	10	+	+	CCONJ
iajs-3486	164	11	𝐻(ℛ𝑆𝑛(𝜚	𝐻(ℛ𝑆𝑛(𝜚	NOUN
iajs-3486	164	12	)	)	PUNCT
iajs-3486	164	13	,	,	PUNCT
iajs-3486	164	14	ℛ𝑞(ϱ	ℛ𝑞(ϱ	NOUN
iajs-3486	164	15	)	)	PUNCT
iajs-3486	164	16	)	)	PUNCT
iajs-3486	165	1	≤	≤	NUM
iajs-3486	165	2	𝑑(𝑞(ϱ	𝑑(𝑞(ϱ	NUM
iajs-3486	165	3	)	)	PUNCT
iajs-3486	165	4	,	,	PUNCT
iajs-3486	165	5	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	165	6	)	)	PUNCT
iajs-3486	165	7	)	)	PUNCT
iajs-3486	166	1	+	+	CCONJ
iajs-3486	166	2	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	166	3	)	)	PUNCT
iajs-3486	166	4	,	,	PUNCT
iajs-3486	166	5	𝜌𝑛(𝜚	𝜌𝑛(𝜚	NUM
iajs-3486	166	6	)	)	PUNCT
iajs-3486	166	7	)	)	PUNCT
iajs-3486	166	8	+	+	CCONJ
iajs-3486	166	9	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	166	10	)	)	PUNCT
iajs-3486	166	11	,	,	PUNCT
iajs-3486	166	12	𝑞(ϱ	𝑞(ϱ	NOUN
iajs-3486	166	13	)	)	PUNCT
iajs-3486	166	14	)	)	PUNCT
iajs-3486	166	15	→	→	SYM
iajs-3486	166	16	0	0	NUM
iajs-3486	166	17	𝑎𝑠	𝑎𝑠	PROPN
iajs-3486	166	18	𝑛	𝑛	PROPN
iajs-3486	166	19	→	→	SYM
iajs-3486	166	20	∞	∞	PROPN
iajs-3486	166	21	,	,	PUNCT
iajs-3486	166	22	implies	imply	VERB
iajs-3486	166	23	that𝑞(ϱ	that𝑞(ϱ	PROPN
iajs-3486	166	24	)	)	PUNCT
iajs-3486	166	25	∈	∈	PROPN
iajs-3486	166	26	ℛ𝑞(ϱ	ℛ𝑞(ϱ	NOUN
iajs-3486	166	27	)	)	PUNCT
iajs-3486	166	28	.	.	PUNCT
iajs-3486	167	1	consequently	consequently	ADV
iajs-3486	167	2	,	,	PUNCT
iajs-3486	167	3	𝑞(ϱ	𝑞(ϱ	PROPN
iajs-3486	167	4	)	)	PUNCT
iajs-3486	167	5	∈	∈	PROPN
iajs-3486	167	6	𝐹.	𝐹.	PROPN
iajs-3486	167	7	nowwe	nowwe	NOUN
iajs-3486	167	8	can	can	AUX
iajs-3486	167	9	use	use	VERB
iajs-3486	167	10	the	the	DET
iajs-3486	167	11	conditional	conditional	ADJ
iajs-3486	167	12	(	(	PUNCT
iajs-3486	167	13	𝐴	𝐴	PROPN
iajs-3486	167	14	`	`	PUNCT
iajs-3486	167	15	)	)	PUNCT
iajs-3486	167	16	to	to	PART
iajs-3486	167	17	find	find	VERB
iajs-3486	167	18	the	the	DET
iajs-3486	167	19	strong	strong	ADJ
iajs-3486	167	20	converge	converge	NOUN
iajs-3486	167	21	of	of	ADP
iajs-3486	167	22	{	{	PUNCT
iajs-3486	167	23	𝑆𝑛(𝜚)}define	𝑆𝑛(𝜚)}define	PROPN
iajs-3486	167	24	in(2.1	in(2.1	PROPN
iajs-3486	167	25	)	)	PUNCT
iajs-3486	167	26	.	.	PUNCT
iajs-3486	168	1	,	,	PUNCT
iajs-3486	168	2	we	we	PRON
iajs-3486	168	3	assume	assume	VERB
iajs-3486	168	4	that	that	SCONJ
iajs-3486	168	5	ℛ	ℛ	PROPN
iajs-3486	168	6	,	,	PUNCT
iajs-3486	168	7	ζ	ζ	NOUN
iajs-3486	168	8	,	,	PUNCT
iajs-3486	168	9	𝐵	𝐵	NOUN
iajs-3486	168	10	∶	∶	NOUN
iajs-3486	168	11	ψ	ψ	X
iajs-3486	168	12	×	×	NOUN
iajs-3486	168	13	υ	υ	NOUN
iajs-3486	168	14	→	→	SYM
iajs-3486	168	15	𝛱(υ)satisfycondition	𝛱(υ)satisfycondition	NOUN
iajs-3486	168	16	(	(	PUNCT
iajs-3486	168	17	𝐴	𝐴	PROPN
iajs-3486	168	18	`	`	PUNCT
iajs-3486	168	19	)	)	PUNCT
iajs-3486	168	20	.	.	PUNCT
iajs-3486	169	1	theorem	theorem	NOUN
iajs-3486	169	2	2.8	2.8	NUM
iajs-3486	169	3	.	.	PUNCT
iajs-3486	170	1	let	let	VERB
iajs-3486	170	2	υ	υ	NOUN
iajs-3486	170	3	,	,	PUNCT
iajs-3486	170	4	𝛷	𝛷	NOUN
iajs-3486	170	5	and	and	CCONJ
iajs-3486	170	6	,	,	PUNCT
iajs-3486	170	7	the	the	DET
iajs-3486	170	8	sequence{𝑆𝑛(𝜚	sequence{𝑆𝑛(𝜚	PROPN
iajs-3486	170	9	)	)	PUNCT
iajs-3486	170	10	}	}	PUNCT
iajs-3486	170	11	beas	bea	NOUN
iajs-3486	170	12	in	in	ADP
iajs-3486	170	13	lemma	lemma	PROPN
iajs-3486	170	14	(	(	PUNCT
iajs-3486	170	15	2.4	2.4	NUM
iajs-3486	170	16	)	)	PUNCT
iajs-3486	170	17	.	.	PUNCT
iajs-3486	171	1	let	let	VERB
iajs-3486	171	2	ℛ	ℛ	NOUN
iajs-3486	171	3	,	,	PUNCT
iajs-3486	171	4	ζ	ζ	NOUN
iajs-3486	171	5	,	,	PUNCT
iajs-3486	171	6	𝐵	𝐵	NOUN
iajs-3486	171	7	∶	∶	NOUN
iajs-3486	171	8	ψ	ψ	X
iajs-3486	171	9	×	×	NOUN
iajs-3486	171	10	υ	υ	NOUN
iajs-3486	171	11	→	→	PUNCT
iajs-3486	171	12	𝛱(υ)satisfying	𝛱(υ)satisfying	NOUN
iajs-3486	171	13	condition	condition	NOUN
iajs-3486	171	14	(	(	PUNCT
iajs-3486	171	15	𝐴	𝐴	PROPN
iajs-3486	171	16	`	`	PUNCT
iajs-3486	171	17	)	)	PUNCT
iajs-3486	171	18	.	.	PUNCT
iajs-3486	172	1	if	if	SCONJ
iajs-3486	172	2	𝐹	𝐹	PROPN
iajs-3486	172	3	≠	≠	PROPN
iajs-3486	172	4	∅	∅	NOUN
iajs-3486	172	5	and	and	CCONJ
iajs-3486	172	6	𝐵ϑ(ϱ	𝐵ϑ(ϱ	NOUN
iajs-3486	172	7	)	)	PUNCT
iajs-3486	172	8	=	=	SYM
iajs-3486	172	9	ℛϑ(ϱ	ℛϑ(ϱ	ADJ
iajs-3486	172	10	)	)	PUNCT
iajs-3486	172	11	=	=	PUNCT
iajs-3486	172	12	ζϑ(ϱ	ζϑ(ϱ	X
iajs-3486	172	13	)	)	PUNCT
iajs-3486	172	14	=	=	SYM
iajs-3486	172	15	{	{	PUNCT
iajs-3486	172	16	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	172	17	)	)	PUNCT
iajs-3486	172	18	}	}	PUNCT
iajs-3486	172	19	for	for	ADP
iajs-3486	172	20	any	any	DET
iajs-3486	172	21	ϑ(ϱ	ϑ(ϱ	PROPN
iajs-3486	172	22	)	)	PUNCT
iajs-3486	172	23	∈	∈	PROPN
iajs-3486	172	24	𝐹	𝐹	PROPN
iajs-3486	172	25	then	then	ADV
iajs-3486	172	26	{	{	PUNCT
iajs-3486	172	27	𝑆𝑛(𝜚	𝑆𝑛(𝜚	PROPN
iajs-3486	172	28	)	)	PUNCT
iajs-3486	172	29	}	}	PUNCT
iajs-3486	172	30	converges	converge	VERB
iajs-3486	172	31	to	to	ADP
iajs-3486	172	32	a	a	DET
iajs-3486	172	33	commonfixed	commonfixed	NOUN
iajs-3486	172	34	point	point	NOUN
iajs-3486	172	35	of	of	ADP
iajs-3486	172	36	𝐵	𝐵	PROPN
iajs-3486	172	37	,	,	PUNCT
iajs-3486	172	38	ζ,ℛ	ζ,ℛ	NOUN
iajs-3486	172	39	and	and	CCONJ
iajs-3486	172	40	strongly	strongly	ADV
iajs-3486	172	41	.	.	PUNCT
iajs-3486	173	1	proof	proof	NOUN
iajs-3486	173	2	:	:	PUNCT
iajs-3486	173	3	since	since	SCONJ
iajs-3486	173	4	by	by	ADP
iajs-3486	173	5	used	used	ADJ
iajs-3486	173	6	lemma(2.4	lemma(2.4	NOUN
iajs-3486	173	7	)	)	PUNCT
iajs-3486	173	8	,	,	PUNCT
iajs-3486	173	9	have	have	VERB
iajs-3486	173	10	lim	lim	PROPN
iajs-3486	173	11	𝑛→∞	𝑛→∞	NUM
iajs-3486	173	12	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	PROPN
iajs-3486	173	13	)	)	PUNCT
iajs-3486	173	14	−	−	PROPN
iajs-3486	173	15	𝐹‖exists	𝐹‖exist	VERB
iajs-3486	173	16	∀ϑ(ϱ	∀ϑ(ϱ	PROPN
iajs-3486	173	17	)	)	PUNCT
iajs-3486	173	18	∈	∈	PROPN
iajs-3486	173	19	𝐹.	𝐹.	PROPN
iajs-3486	173	20	let	let	VERB
iajs-3486	173	21	𝑐	𝑐	PROPN
iajs-3486	173	22	≥	≥	VERB
iajs-3486	173	23	0	0	NUM
iajs-3486	173	24	.	.	PUNCT
iajs-3486	174	1	if	if	SCONJ
iajs-3486	174	2	𝑐	𝑐	PROPN
iajs-3486	174	3	=	=	SYM
iajs-3486	174	4	0	0	NUM
iajs-3486	174	5	,	,	PUNCT
iajs-3486	174	6	clear	clear	ADJ
iajs-3486	174	7	that	that	SCONJ
iajs-3486	174	8	.	.	PUNCT
iajs-3486	175	1	let𝑐	let𝑐	VERB
iajs-3486	175	2	>	>	X
iajs-3486	175	3	0	0	X
iajs-3486	175	4	.	.	PUNCT
iajs-3486	176	1	now	now	ADV
iajs-3486	176	2	‖𝑆𝑛+1(𝜚	‖𝑆𝑛+1(𝜚	VERB
iajs-3486	176	3	)	)	PUNCT
iajs-3486	177	1	−	−	VERB
iajs-3486	177	2	ϑ(ϱ)‖	ϑ(ϱ)‖	PROPN
iajs-3486	177	3	≤	≤	NUM
iajs-3486	177	4	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	177	5	)	)	PUNCT
iajs-3486	177	6	−	−	PROPN
iajs-3486	177	7	ϑ(ϱ)‖gives	ϑ(ϱ)‖give	VERB
iajs-3486	177	8	inf	inf	PROPN
iajs-3486	177	9	ϑ(ϱ)∈𝐹	ϑ(ϱ)∈𝐹	NOUN
iajs-3486	177	10	‖𝑆𝑛+1(𝜚	‖𝑆𝑛+1(𝜚	NOUN
iajs-3486	177	11	)	)	PUNCT
iajs-3486	177	12	−	−	VERB
iajs-3486	177	13	ϑ(ϱ)‖	ϑ(ϱ)‖	PROPN
iajs-3486	177	14	≤	≤	NUM
iajs-3486	177	15	inf	inf	PROPN
iajs-3486	177	16	ϑ(ϱ)∈𝐹	ϑ(ϱ)∈𝐹	NOUN
iajs-3486	177	17	‖𝑆𝑛(𝜚	‖𝑆𝑛(𝜚	NOUN
iajs-3486	177	18	)	)	PUNCT
iajs-3486	177	19	−	−	PROPN
iajs-3486	178	1	ϑ(ϱ)‖which	ϑ(ϱ)‖which	PROPN
iajs-3486	178	2	implies	imply	VERB
iajs-3486	178	3	that	that	PRON
iajs-3486	178	4	𝑑(𝑆𝑛+1(𝜚	𝑑(𝑆𝑛+1(𝜚	NOUN
iajs-3486	178	5	)	)	PUNCT
iajs-3486	178	6	,	,	PUNCT
iajs-3486	178	7	𝐹	𝐹	PROPN
iajs-3486	178	8	)	)	PUNCT
iajs-3486	178	9	≤	≤	NOUN
iajs-3486	178	10	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	178	11	)	)	PUNCT
iajs-3486	178	12	,	,	PUNCT
iajs-3486	178	13	𝐹	𝐹	PROPN
iajs-3486	178	14	)	)	PUNCT
iajs-3486	178	15	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-3486	179	1	so	so	ADV
iajs-3486	179	2	lim	lim	PROPN
iajs-3486	179	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	179	4	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	179	5	)	)	PUNCT
iajs-3486	179	6	,	,	PUNCT
iajs-3486	179	7	𝐹	𝐹	PROPN
iajs-3486	179	8	)	)	PUNCT
iajs-3486	179	9	exists	exist	VERB
iajs-3486	179	10	.	.	PUNCT
iajs-3486	180	1	by	by	ADP
iajs-3486	180	2	condition	condition	NOUN
iajs-3486	180	3	(	(	PUNCT
iajs-3486	180	4	𝐴`)either	𝐴`)either	NOUN
iajs-3486	180	5	lim	lim	PROPN
iajs-3486	180	6	𝑛→∞	𝑛→∞	NUM
iajs-3486	180	7	𝜒(𝑑(𝑆𝑛(𝜚	𝜒(𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	180	8	)	)	PUNCT
iajs-3486	180	9	,	,	PUNCT
iajs-3486	180	10	𝐹	𝐹	PROPN
iajs-3486	180	11	)	)	PUNCT
iajs-3486	180	12	)	)	PUNCT
iajs-3486	180	13	≤	≤	PROPN
iajs-3486	180	14	lim	lim	NOUN
iajs-3486	180	15	𝑛→∞	𝑛→∞	NUM
iajs-3486	180	16	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	180	17	)	)	PUNCT
iajs-3486	180	18	,	,	PUNCT
iajs-3486	180	19	𝐵𝑆𝑛(𝜚	𝐵𝑆𝑛(𝜚	PROPN
iajs-3486	180	20	)	)	PUNCT
iajs-3486	180	21	)	)	PUNCT
iajs-3486	181	1	=	=	SYM
iajs-3486	181	2	0	0	NUM
iajs-3486	181	3	or	or	CCONJ
iajs-3486	181	4	lim	lim	PROPN
iajs-3486	181	5	𝑛→∞	𝑛→∞	NUM
iajs-3486	181	6	𝜒(𝑑(𝑆𝑛(𝜚	𝜒(𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	181	7	)	)	PUNCT
iajs-3486	181	8	,	,	PUNCT
iajs-3486	181	9	𝐹	𝐹	PROPN
iajs-3486	181	10	)	)	PUNCT
iajs-3486	181	11	)	)	PUNCT
iajs-3486	182	1	≤	≤	PROPN
iajs-3486	182	2	lim	lim	NOUN
iajs-3486	182	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	182	4	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	182	5	)	)	PUNCT
iajs-3486	182	6	,	,	PUNCT
iajs-3486	182	7	ζ𝑆𝑛(𝜚	ζ𝑆𝑛(𝜚	NOUN
iajs-3486	182	8	)	)	PUNCT
iajs-3486	182	9	)	)	PUNCT
iajs-3486	183	1	=	=	SYM
iajs-3486	183	2	0	0	NUM
iajs-3486	183	3	or	or	CCONJ
iajs-3486	183	4	lim	lim	PROPN
iajs-3486	183	5	𝑛→∞	𝑛→∞	NUM
iajs-3486	183	6	𝜒(𝑑(𝑆𝑛(𝜚	𝜒(𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	183	7	)	)	PUNCT
iajs-3486	183	8	,	,	PUNCT
iajs-3486	183	9	𝐹	𝐹	PROPN
iajs-3486	183	10	)	)	PUNCT
iajs-3486	183	11	)	)	PUNCT
iajs-3486	184	1	≤	≤	PROPN
iajs-3486	184	2	lim	lim	NOUN
iajs-3486	184	3	𝑛→∞	𝑛→∞	NUM
iajs-3486	184	4	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	184	5	)	)	PUNCT
iajs-3486	184	6	,	,	PUNCT
iajs-3486	184	7	𝑅𝑆𝑛(𝜚	𝑅𝑆𝑛(𝜚	NOUN
iajs-3486	184	8	)	)	PUNCT
iajs-3486	184	9	)	)	PUNCT
iajs-3486	185	1	=	=	SYM
iajs-3486	185	2	0	0	NUM
iajs-3486	186	1	in	in	ADP
iajs-3486	186	2	both	both	CCONJ
iajs-3486	186	3	the	the	DET
iajs-3486	186	4	cases	case	NOUN
iajs-3486	186	5	,	,	PUNCT
iajs-3486	186	6	lim	lim	NOUN
iajs-3486	186	7	𝑛→∞	𝑛→∞	NUM
iajs-3486	186	8	𝜒(𝑑(𝑆𝑛(𝜚	𝜒(𝑑(𝑆𝑛(𝜚	NOUN
iajs-3486	186	9	)	)	PUNCT
iajs-3486	186	10	,	,	PUNCT
iajs-3486	186	11	𝐹	𝐹	PROPN
iajs-3486	186	12	)	)	PUNCT
iajs-3486	186	13	)	)	PUNCT
iajs-3486	186	14	=	=	SYM
iajs-3486	186	15	0	0	PUNCT
iajs-3486	186	16	since	since	SCONJ
iajs-3486	186	17	𝜒	𝜒	NOUN
iajs-3486	186	18	is	be	AUX
iajs-3486	186	19	a	a	DET
iajs-3486	186	20	non	non	ADJ
iajs-3486	186	21	-	-	ADJ
iajs-3486	186	22	decreasing	decrease	VERB
iajs-3486	186	23	function	function	NOUN
iajs-3486	186	24	where𝜒(0	where𝜒(0	NOUN
iajs-3486	186	25	)	)	PUNCT
iajs-3486	187	1	=	=	SYM
iajs-3486	187	2	0	0	NUM
iajs-3486	187	3	,	,	PUNCT
iajs-3486	187	4	lim	lim	NOUN
iajs-3486	187	5	𝑛→∞	𝑛→∞	NUM
iajs-3486	187	6	𝑑(𝑆𝑛(𝜚	𝑑(𝑆𝑛(𝜚	NUM
iajs-3486	187	7	)	)	PUNCT
iajs-3486	187	8	,	,	PUNCT
iajs-3486	187	9	𝐹	𝐹	PROPN
iajs-3486	187	10	)	)	PUNCT
iajs-3486	187	11	=	=	SYM
iajs-3486	187	12	0	0	NUM
iajs-3486	187	13	.	.	PUNCT
iajs-3486	188	1	5	5	X
iajs-3486	188	2	.	.	X
iajs-3486	188	3	conclusion	conclusion	VERB
iajs-3486	188	4	the	the	DET
iajs-3486	188	5	main	main	ADJ
iajs-3486	188	6	idea	idea	NOUN
iajs-3486	188	7	of	of	ADP
iajs-3486	188	8	the	the	DET
iajs-3486	188	9	this	this	DET
iajs-3486	188	10	paper	paper	NOUN
iajs-3486	188	11	is	be	AUX
iajs-3486	188	12	that	that	SCONJ
iajs-3486	188	13	we	we	PRON
iajs-3486	188	14	have	have	VERB
iajs-3486	188	15	to	to	PART
iajs-3486	188	16	found	find	VERB
iajs-3486	188	17	a	a	DET
iajs-3486	188	18	new	new	ADJ
iajs-3486	188	19	iterations	iteration	NOUN
iajs-3486	188	20	by	by	ADP
iajs-3486	188	21	one	one	NUM
iajs-3486	188	22	step	step	NOUN
iajs-3486	188	23	to	to	ADP
iajs-3486	188	24	approximation	approximation	NOUN
iajs-3486	188	25	for	for	ADP
iajs-3486	188	26	common	common	ADJ
iajs-3486	188	27	random	random	ADJ
iajs-3486	188	28	fixed	fix	VERB
iajs-3486	188	29	point	point	NOUN
iajs-3486	188	30	from	from	ADP
iajs-3486	188	31	three	three	NUM
iajs-3486	188	32	multi	multi	ADJ
iajs-3486	188	33	-	-	ADJ
iajs-3486	188	34	valued	value	VERB
iajs-3486	188	35	non	non	ADJ
iajs-3486	188	36	-	-	ADJ
iajs-3486	188	37	expansive	expansive	ADJ
iajs-3486	188	38	random	random	ADJ
iajs-3486	188	39	operator	operator	NOUN
iajs-3486	188	40	and	and	CCONJ
iajs-3486	188	41	obtain	obtain	VERB
iajs-3486	188	42	,	,	PUNCT
iajs-3486	188	43	as	as	ADV
iajs-3486	188	44	well	well	ADV
iajs-3486	188	45	as	as	ADP
iajs-3486	188	46	the	the	DET
iajs-3486	188	47	theories	theory	NOUN
iajs-3486	188	48	of	of	ADP
iajs-3486	188	49	strong	strong	ADJ
iajs-3486	188	50	and	and	CCONJ
iajs-3486	188	51	weak	weak	ADJ
iajs-3486	188	52	convergence	convergence	NOUN
iajs-3486	188	53	.	.	PUNCT
iajs-3486	189	1	acknowledgment	acknowledgment	NOUN
iajs-3486	189	2	the	the	DET
iajs-3486	189	3	authors	author	NOUN
iajs-3486	189	4	greatly	greatly	ADV
iajs-3486	189	5	appreciate	appreciate	VERB
iajs-3486	189	6	the	the	DET
iajs-3486	189	7	referees	referee	NOUN
iajs-3486	189	8	for	for	ADP
iajs-3486	189	9	their	their	PRON
iajs-3486	189	10	comments	comment	NOUN
iajs-3486	189	11	and	and	CCONJ
iajs-3486	189	12	suggestions	suggestion	NOUN
iajs-3486	189	13	for	for	ADP
iajs-3486	189	14	improving	improve	VERB
iajs-3486	189	15	the	the	DET
iajs-3486	189	16	paper	paper	NOUN
iajs-3486	189	17	.	.	PUNCT
iajs-3486	190	1	conflict	conflict	NOUN
iajs-3486	190	2	of	of	ADP
iajs-3486	190	3	interest	interest	NOUN
iajs-3486	190	4	“	"	PUNCT
iajs-3486	190	5	conflict	conflict	NOUN
iajs-3486	190	6	of	of	ADP
iajs-3486	190	7	interest	interest	NOUN
iajs-3486	190	8	:	:	PUNCT
iajs-3486	190	9	the	the	DET
iajs-3486	190	10	authors	author	NOUN
iajs-3486	190	11	declare	declare	VERB
iajs-3486	190	12	that	that	SCONJ
iajs-3486	190	13	they	they	PRON
iajs-3486	190	14	have	have	VERB
iajs-3486	190	15	no	no	DET
iajs-3486	190	16	conflicts	conflict	NOUN
iajs-3486	190	17	of	of	ADP
iajs-3486	190	18	interest	interest	NOUN
iajs-3486	190	19	.	.	PUNCT
iajs-3486	190	20	”	"	PUNCT
iajs-3486	191	1	funding	funding	NOUN
iajs-3486	191	2	:	:	PUNCT
iajs-3486	191	3	none	none	NOUN
iajs-3486	191	4	.	.	PUNCT
iajs-3486	192	1	references	reference	NOUN
iajs-3486	192	2	1	1	NUM
iajs-3486	192	3	.	.	PUNCT
iajs-3486	193	1	hans	han	NOUN
iajs-3486	193	2	,	,	PUNCT
iajs-3486	193	3	o.	o.	INTJ
iajs-3486	193	4	,	,	PUNCT
iajs-3486	193	5	random	random	ADJ
iajs-3486	193	6	fixed	fix	VERB
iajs-3486	193	7	point	point	NOUN
iajs-3486	193	8	theorems	theorem	NOUN
iajs-3486	193	9	,	,	PUNCT
iajs-3486	193	10	in	in	ADP
iajs-3486	193	11	transactions	transaction	NOUN
iajs-3486	193	12	of	of	ADP
iajs-3486	193	13	the	the	DET
iajs-3486	193	14	first	first	ADJ
iajs-3486	193	15	prague	prague	PROPN
iajs-3486	193	16	conference	conference	NOUN
iajs-3486	193	17	on	on	ADP
iajs-3486	193	18	information	information	NOUN
iajs-3486	193	19	theory	theory	NOUN
iajs-3486	193	20	,	,	PUNCT
iajs-3486	193	21	statistical	statistical	ADJ
iajs-3486	193	22	decision	decision	NOUN
iajs-3486	193	23	functions	function	NOUN
iajs-3486	193	24	,	,	PUNCT
iajs-3486	193	25	random	random	ADJ
iajs-3486	193	26	process	process	NOUN
iajs-3486	193	27	,	,	PUNCT
iajs-3486	193	28	1957	1957	NUM
iajs-3486	193	29	,	,	PUNCT
iajs-3486	193	30	105–125	105–125	NUM
iajs-3486	193	31	.	.	PUNCT
iajs-3486	194	1	ihjpas.37	ihjpas.37	PROPN
iajs-3486	194	2	(	(	PUNCT
iajs-3486	194	3	2	2	NUM
iajs-3486	194	4	)	)	PUNCT
iajs-3486	194	5	2024	2024	NUM
iajs-3486	194	6	430	430	NUM
iajs-3486	194	7	https://doi.org/10.1016/b978-0-12-434160-9.50009-6	https://doi.org/10.1016/b978-0-12-434160-9.50009-6	NOUN
iajs-3486	194	8	2	2	NUM
iajs-3486	194	9	.	.	PUNCT
iajs-3486	195	1	huang	huang	PROPN
iajs-3486	195	2	,	,	PUNCT
iajs-3486	195	3	t.	t.	PROPN
iajs-3486	195	4	;	;	PUNCT
iajs-3486	195	5	rhoades	rhoade	NOUN
iajs-3486	195	6	,	,	PUNCT
iajs-3486	195	7	b.	b.	PROPN
iajs-3486	195	8	e.	e.	PROPN
iajs-3486	196	1	a	a	DET
iajs-3486	196	2	general	general	ADJ
iajs-3486	196	3	principle	principle	NOUN
iajs-3486	196	4	for	for	ADP
iajs-3486	196	5	ishikawa	ishikawa	PROPN
iajs-3486	196	6	iterations	iteration	NOUN
iajs-3486	196	7	for	for	ADP
iajs-3486	196	8	multi	multi	ADJ
iajs-3486	196	9	-	-	ADJ
iajs-3486	196	10	valued	value	VERB
iajs-3486	196	11	mappings	mapping	NOUN
iajs-3486	196	12	,	,	PUNCT
iajs-3486	196	13	indian	indian	PROPN
iajs-3486	196	14	j.	j.	PROPN
iajs-3486	196	15	math	math	PROPN
iajs-3486	196	16	.	.	PUNCT
iajs-3486	197	1	pure	pure	ADJ
iajs-3486	197	2	appl	appl	NOUN
iajs-3486	197	3	.	.	PUNCT
iajs-3486	198	1	1997	1997	NUM
iajs-3486	198	2	,	,	PUNCT
iajs-3486	198	3	1091–1098	1091–1098	NUM
iajs-3486	198	4	.	.	NOUN
iajs-3486	199	1	3	3	X
iajs-3486	199	2	.	.	X
iajs-3486	199	3	khan	khan	PROPN
iajs-3486	199	4	,	,	PUNCT
iajs-3486	199	5	s.	s.	PROPN
iajs-3486	199	6	h.	h.	PROPN
iajs-3486	199	7	;	;	PUNCT
iajs-3486	199	8	abbas	abbas	PROPN
iajs-3486	199	9	,	,	PUNCT
iajs-3486	199	10	m.	m.	NOUN
iajs-3486	199	11	;	;	PUNCT
iajs-3486	199	12	rhoades	rhoades	PROPN
iajs-3486	199	13	b.	b.	PROPN
iajs-3486	199	14	e.	e.	PROPN
iajs-3486	200	1	a	a	DET
iajs-3486	200	2	new	new	ADJ
iajs-3486	200	3	one	one	NUM
iajs-3486	200	4	-	-	PUNCT
iajs-3486	200	5	step	step	NOUN
iajs-3486	200	6	iterative	iterative	NOUN
iajs-3486	200	7	scheme	scheme	NOUN
iajs-3486	200	8	for	for	ADP
iajs-3486	200	9	approximating	approximate	VERB
iajs-3486	200	10	common	common	ADJ
iajs-3486	200	11	fixed	fix	VERB
iajs-3486	200	12	points	point	NOUN
iajs-3486	200	13	of	of	ADP
iajs-3486	200	14	two	two	NUM
iajs-3486	200	15	multivalued	multivalue	VERB
iajs-3486	200	16	nonexpansive	nonexpansive	ADJ
iajs-3486	200	17	mappings	mapping	NOUN
iajs-3486	200	18	,	,	PUNCT
iajs-3486	200	19	rend	rend	VERB
iajs-3486	200	20	.	.	PUNCT
iajs-3486	201	1	del	del	PROPN
iajs-3486	201	2	circ	circ	PROPN
iajs-3486	201	3	.	.	PUNCT
iajs-3486	202	1	mat	mat	PROPN
iajs-3486	202	2	.	.	PROPN
iajs-3486	202	3	di	di	PROPN
iajs-3486	202	4	palermo	palermo	PROPN
iajs-3486	202	5	.	.	PUNCT
iajs-3486	203	1	2010	2010	NUM
iajs-3486	203	2	,	,	PUNCT
iajs-3486	203	3	151	151	NUM
iajs-3486	203	4	–	–	PUNCT
iajs-3486	203	5	159	159	NUM
iajs-3486	203	6	.	.	PUNCT
iajs-3486	203	7	https://doi.org/10.1007/s12215-010-0012-4	https://doi.org/10.1007/s12215-010-0012-4	NUM
iajs-3486	203	8	4	4	NUM
iajs-3486	203	9	.	.	PUNCT
iajs-3486	203	10	engl	engl	PROPN
iajs-3486	203	11	,	,	PUNCT
iajs-3486	203	12	h.	h.	PROPN
iajs-3486	203	13	random	random	ADJ
iajs-3486	203	14	fixed	fix	VERB
iajs-3486	203	15	point	point	NOUN
iajs-3486	203	16	theorems	theorem	NOUN
iajs-3486	203	17	for	for	ADP
iajs-3486	203	18	multivalued	multivalued	ADJ
iajs-3486	203	19	mappings	mapping	NOUN
iajs-3486	203	20	,	,	PUNCT
iajs-3486	203	21	pacific	pacific	PROPN
iajs-3486	203	22	j.	j.	PROPN
iajs-3486	203	23	math	math	PROPN
iajs-3486	203	24	.	.	PUNCT
iajs-3486	204	1	1978	1978	NUM
iajs-3486	204	2	,	,	PUNCT
iajs-3486	204	3	351–360	351–360	NUM
iajs-3486	204	4	.	.	PUNCT
iajs-3486	205	1	https://doi.org/10.2140/pjm.1978.76.351	https://doi.org/10.2140/pjm.1978.76.351	NOUN
iajs-3486	205	2	5	5	X
iajs-3486	205	3	.	.	X
iajs-3486	205	4	engl	engl	PROPN
iajs-3486	205	5	,	,	PUNCT
iajs-3486	205	6	h.	h.	PROPN
iajs-3486	205	7	w.	w.	PROPN
iajs-3486	206	1	some	some	DET
iajs-3486	206	2	random	random	ADJ
iajs-3486	206	3	fixed	fix	VERB
iajs-3486	206	4	point	point	NOUN
iajs-3486	206	5	theorems	theorem	NOUN
iajs-3486	206	6	for	for	ADP
iajs-3486	206	7	strict	strict	ADJ
iajs-3486	206	8	contractions	contraction	NOUN
iajs-3486	206	9	and	and	CCONJ
iajs-3486	206	10	nonexpansive	nonexpansive	ADJ
iajs-3486	206	11	mappings	mapping	NOUN
iajs-3486	206	12	,	,	PUNCT
iajs-3486	206	13	nonlinear	nonlinear	ADJ
iajs-3486	206	14	anal	anal	NOUN
iajs-3486	206	15	.	.	PUNCT
iajs-3486	207	1	theory	theory	NOUN
iajs-3486	207	2	,	,	PUNCT
iajs-3486	207	3	methods	method	NOUN
iajs-3486	207	4	appl	appl	PROPN
iajs-3486	207	5	.	.	PROPN
iajs-3486	207	6	1978	1978	NUM
iajs-3486	207	7	,	,	PUNCT
iajs-3486	207	8	619–626	619–626	NUM
iajs-3486	207	9	.	.	PUNCT
iajs-3486	208	1	6	6	NUM
iajs-3486	208	2	.	.	NUM
iajs-3486	208	3	itoh	itoh	ADJ
iajs-3486	208	4	.s	.s	NOUN
iajs-3486	208	5	.	.	PUNCT
iajs-3486	209	1	,	,	PUNCT
iajs-3486	209	2	a	a	DET
iajs-3486	209	3	random	random	ADJ
iajs-3486	209	4	fixed	fix	VERB
iajs-3486	209	5	point	point	NOUN
iajs-3486	209	6	theorem	theorem	NOUN
iajs-3486	209	7	for	for	ADP
iajs-3486	209	8	a	a	DET
iajs-3486	209	9	multivalued	multivalued	ADJ
iajs-3486	209	10	contraction	contraction	NOUN
iajs-3486	209	11	mapping	mapping	NOUN
iajs-3486	209	12	,	,	PUNCT
iajs-3486	209	13	pacific	pacific	PROPN
iajs-3486	209	14	j.	j.	PROPN
iajs-3486	209	15	math	math	PROPN
iajs-3486	209	16	.	.	PUNCT
iajs-3486	210	1	1977	1977	NUM
iajs-3486	210	2	,	,	PUNCT
iajs-3486	210	3	85–90	85–90	NUM
iajs-3486	210	4	.	.	PUNCT
iajs-3486	211	1	https://doi.org/10.2140/pjm.1977.68.85	https://doi.org/10.2140/pjm.1977.68.85	ADP
iajs-3486	211	2	7	7	X
iajs-3486	211	3	.	.	PUNCT
iajs-3486	212	1	beg	beg	VERB
iajs-3486	212	2	,	,	PUNCT
iajs-3486	212	3	i.	i.	PROPN
iajs-3486	212	4	and	and	CCONJ
iajs-3486	212	5	shahzad	shahzad	PROPN
iajs-3486	212	6	,	,	PUNCT
iajs-3486	212	7	n.	n.	NOUN
iajs-3486	212	8	,	,	PUNCT
iajs-3486	212	9	random	random	ADJ
iajs-3486	212	10	fixed	fix	VERB
iajs-3486	212	11	point	point	NOUN
iajs-3486	212	12	theorems	theorem	NOUN
iajs-3486	212	13	for	for	ADP
iajs-3486	212	14	nonexpansive	nonexpansive	ADJ
iajs-3486	212	15	and	and	CCONJ
iajs-3486	212	16	contractive	contractive	ADJ
iajs-3486	212	17	-	-	PUNCT
iajs-3486	212	18	type	type	ADJ
iajs-3486	212	19	random	random	ADJ
iajs-3486	212	20	operators	operator	NOUN
iajs-3486	212	21	on	on	ADP
iajs-3486	212	22	banach	banach	NOUN
iajs-3486	212	23	spaces	space	NOUN
iajs-3486	212	24	,	,	PUNCT
iajs-3486	212	25	stoch	stoch	NOUN
iajs-3486	212	26	j.	j.	PROPN
iajs-3486	212	27	math.anal	math.anal	PROPN
iajs-3486	212	28	.	.	PROPN
iajs-3486	212	29	appl	appl	PROPN
iajs-3486	212	30	1994	1994	NUM
iajs-3486	212	31	,	,	PUNCT
iajs-3486	212	32	569–580	569–580	NUM
iajs-3486	212	33	.	.	PUNCT
iajs-3486	213	1	https://doi.org/10.1155/s1048953394000444	https://doi.org/10.1155/s1048953394000444	NUM
iajs-3486	213	2	8	8	NUM
iajs-3486	213	3	.	.	PUNCT
iajs-3486	214	1	papageorgiou	papageorgiou	NOUN
iajs-3486	214	2	,	,	PUNCT
iajs-3486	214	3	n.	n.	PROPN
iajs-3486	214	4	s.	s.	PROPN
iajs-3486	214	5	random	random	ADJ
iajs-3486	214	6	fixed	fix	VERB
iajs-3486	214	7	point	point	NOUN
iajs-3486	214	8	theorems	theorem	NOUN
iajs-3486	214	9	for	for	ADP
iajs-3486	214	10	measurable	measurable	ADJ
iajs-3486	214	11	multifunctions	multifunction	NOUN
iajs-3486	214	12	in	in	ADP
iajs-3486	214	13	banach	banach	NOUN
iajs-3486	214	14	spaces	space	NOUN
iajs-3486	214	15	,	,	PUNCT
iajs-3486	214	16	proc	proc	NOUN
iajs-3486	214	17	.	.	PUNCT
iajs-3486	215	1	am	be	AUX
iajs-3486	215	2	.	.	PUNCT
iajs-3486	216	1	math	math	NOUN
iajs-3486	216	2	.	.	PUNCT
iajs-3486	217	1	soc	soc	PROPN
iajs-3486	217	2	.	.	PUNCT
iajs-3486	218	1	1986	1986	NUM
iajs-3486	218	2	,	,	PUNCT
iajs-3486	218	3	507–514	507–514	NUM
iajs-3486	218	4	.	.	PUNCT
iajs-3486	219	1	https://doi.org/10.1090/s0002-9939-1986-0840638-3	https://doi.org/10.1090/s0002-9939-1986-0840638-3	PROPN
iajs-3486	219	2	9	9	NUM
iajs-3486	219	3	.	.	PUNCT
iajs-3486	219	4	sinacer	sinacer	PROPN
iajs-3486	219	5	,	,	PUNCT
iajs-3486	219	6	m.	m.	NOUN
iajs-3486	219	7	l.	l.	PROPN
iajs-3486	219	8	;	;	PUNCT
iajs-3486	219	9	nieto	nieto	PROPN
iajs-3486	219	10	j.	j.	PROPN
iajs-3486	219	11	j.	j.	PROPN
iajs-3486	219	12	;	;	PUNCT
iajs-3486	219	13	ouahab	ouahab	PROPN
iajs-3486	219	14	,	,	PUNCT
iajs-3486	219	15	a.	a.	NOUN
iajs-3486	219	16	random	random	ADJ
iajs-3486	219	17	fixed	fix	VERB
iajs-3486	219	18	point	point	NOUN
iajs-3486	219	19	theorem	theorem	VERB
iajs-3486	219	20	in	in	ADP
iajs-3486	219	21	generalized	generalized	ADJ
iajs-3486	219	22	banach	banach	NOUN
iajs-3486	219	23	space	space	NOUN
iajs-3486	219	24	and	and	CCONJ
iajs-3486	219	25	applications	application	NOUN
iajs-3486	219	26	,	,	PUNCT
iajs-3486	219	27	random	random	ADJ
iajs-3486	219	28	oper	oper	NOUN
iajs-3486	219	29	.	.	PROPN
iajs-3486	219	30	stoch	stoch	PROPN
iajs-3486	219	31	.	.	PUNCT
iajs-3486	220	1	equations	equation	NOUN
iajs-3486	220	2	.	.	PUNCT
iajs-3486	221	1	2016	2016	NUM
iajs-3486	221	2	,	,	PUNCT
iajs-3486	222	1	93–112	93–112	PROPN
iajs-3486	222	2	.	.	PUNCT
iajs-3486	223	1	http://dx.doi.org/10.1515/rose-20160007	http://dx.doi.org/10.1515/rose-20160007	PROPN
iajs-3486	223	2	10	10	NUM
iajs-3486	223	3	.	.	PUNCT
iajs-3486	224	1	dhage	dhage	NOUN
iajs-3486	224	2	,	,	PUNCT
iajs-3486	224	3	b.	b.	PROPN
iajs-3486	224	4	c.	c.	PROPN
iajs-3486	225	1	some	some	DET
iajs-3486	225	2	basic	basic	ADJ
iajs-3486	225	3	random	random	ADJ
iajs-3486	225	4	fixed	fix	VERB
iajs-3486	225	5	point	point	NOUN
iajs-3486	225	6	theorems	theorem	NOUN
iajs-3486	225	7	with	with	ADP
iajs-3486	225	8	ppf	ppf	PROPN
iajs-3486	225	9	dependence	dependence	NOUN
iajs-3486	225	10	and	and	CCONJ
iajs-3486	225	11	functional	functional	ADJ
iajs-3486	225	12	random	random	ADJ
iajs-3486	225	13	differential	differential	NOUN
iajs-3486	225	14	equations	equation	NOUN
iajs-3486	225	15	,	,	PUNCT
iajs-3486	225	16	differ	differ	VERB
iajs-3486	225	17	.	.	PUNCT
iajs-3486	226	1	equ	equ	PROPN
iajs-3486	226	2	.	.	PUNCT
iajs-3486	226	3	appl	appl	PROPN
iajs-3486	226	4	.	.	PROPN
iajs-3486	226	5	2012	2012	NUM
iajs-3486	226	6	,	,	PUNCT
iajs-3486	226	7	181–195	181–195	NUM
iajs-3486	226	8	.	.	PUNCT
iajs-3486	227	1	https://doi.org/10.7153/dea-04-11	https://doi.org/10.7153/dea-04-11	PROPN
iajs-3486	227	2	11	11	NUM
iajs-3486	227	3	.	.	PUNCT
iajs-3486	228	1	el	el	PROPN
iajs-3486	228	2	ghabi	ghabi	PROPN
iajs-3486	228	3	,	,	PUNCT
iajs-3486	228	4	a.	a.	NOUN
iajs-3486	228	5	random	random	ADJ
iajs-3486	228	6	fixed	fix	VERB
iajs-3486	228	7	point	point	NOUN
iajs-3486	228	8	theorems	theorem	NOUN
iajs-3486	228	9	with	with	ADP
iajs-3486	228	10	application	application	NOUN
iajs-3486	228	11	to	to	ADP
iajs-3486	228	12	random	random	ADJ
iajs-3486	228	13	differential	differential	ADJ
iajs-3486	228	14	equations	equation	NOUN
iajs-3486	228	15	in	in	ADP
iajs-3486	228	16	banach	banach	NOUN
iajs-3486	228	17	spaces	space	NOUN
iajs-3486	228	18	,	,	PUNCT
iajs-3486	228	19	23	23	NUM
iajs-3486	228	20	sep	sep	NOUN
iajs-3486	228	21	.	.	PROPN
iajs-3486	228	22	,	,	PUNCT
iajs-3486	228	23	inaug	inaug	PROPN
iajs-3486	228	24	.	.	PUNCT
iajs-3486	229	1	days	day	NOUN
iajs-3486	229	2	,	,	PUNCT
iajs-3486	229	3	54	54	NUM
iajs-3486	229	4	.	.	PUNCT
iajs-3486	230	1	https://doi.org/10.1155/2021/6648938	https://doi.org/10.1155/2021/6648938	ADJ
iajs-3486	230	2	12	12	NUM
iajs-3486	230	3	.	.	PUNCT
iajs-3486	231	1	beg	beg	INTJ
iajs-3486	231	2	,	,	PUNCT
iajs-3486	231	3	i.	i.	NOUN
iajs-3486	231	4	approximation	approximation	NOUN
iajs-3486	231	5	of	of	ADP
iajs-3486	231	6	random	random	ADJ
iajs-3486	231	7	fixed	fix	VERB
iajs-3486	231	8	points	point	NOUN
iajs-3486	231	9	in	in	ADP
iajs-3486	231	10	normed	normed	ADJ
iajs-3486	231	11	spaces	space	NOUN
iajs-3486	231	12	,	,	PUNCT
iajs-3486	231	13	nonlinear	nonlinear	ADJ
iajs-3486	231	14	anal	anal	NOUN
iajs-3486	231	15	.	.	PUNCT
iajs-3486	232	1	theory	theory	NOUN
iajs-3486	232	2	,	,	PUNCT
iajs-3486	232	3	methods	method	NOUN
iajs-3486	232	4	appl	appl	PROPN
iajs-3486	232	5	.	.	PROPN
iajs-3486	232	6	2002	2002	NUM
iajs-3486	232	7	,	,	PUNCT
iajs-3486	232	8	1363–1372	1363–1372	NUM
iajs-3486	232	9	.	.	PUNCT
iajs-3486	233	1	https://doi.org/10.1016/s0362-546x(01)00902-6	https://doi.org/10.1016/s0362-546x(01)00902-6	NOUN
iajs-3486	233	2	13	13	NUM
iajs-3486	233	3	.	.	PUNCT
iajs-3486	234	1	tan	tan	PROPN
iajs-3486	234	2	,	,	PUNCT
iajs-3486	234	3	k.-k	k.-k	PROPN
iajs-3486	234	4	.	.	PUNCT
iajs-3486	234	5	;	;	PUNCT
iajs-3486	235	1	yuan	yuan	NOUN
iajs-3486	235	2	,	,	PUNCT
iajs-3486	235	3	x.-z	x.-z	PROPN
iajs-3486	235	4	.	.	PUNCT
iajs-3486	236	1	random	random	ADJ
iajs-3486	236	2	fixed	fix	VERB
iajs-3486	236	3	point	point	NOUN
iajs-3486	236	4	theorems	theorem	NOUN
iajs-3486	236	5	and	and	CCONJ
iajs-3486	236	6	approximation	approximation	NOUN
iajs-3486	236	7	,	,	PUNCT
iajs-3486	236	8	stoch	stoch	NOUN
iajs-3486	236	9	.	.	PUNCT
iajs-3486	237	1	anal	anal	PROPN
iajs-3486	237	2	.	.	PUNCT
iajs-3486	237	3	appl	appl	PROPN
iajs-3486	237	4	.	.	PROPN
iajs-3486	237	5	,	,	PUNCT
iajs-3486	237	6	103	103	NUM
iajs-3486	237	7	–	–	SYM
iajs-3486	237	8	123	123	NUM
iajs-3486	237	9	.	.	PUNCT
iajs-3486	238	1	https://doi.org/10.1006/jmaa.1994.1256	https://doi.org/10.1006/jmaa.1994.1256	PROPN
iajs-3486	238	2	14	14	NUM
iajs-3486	238	3	.	.	PUNCT
iajs-3486	238	4	shahzad	shahzad	PROPN
iajs-3486	238	5	,	,	PUNCT
iajs-3486	238	6	n.	n.	PROPN
iajs-3486	238	7	random	random	ADJ
iajs-3486	238	8	fixed	fix	VERB
iajs-3486	238	9	point	point	NOUN
iajs-3486	238	10	theorems	theorem	NOUN
iajs-3486	238	11	for	for	ADP
iajs-3486	238	12	various	various	ADJ
iajs-3486	238	13	classes	class	NOUN
iajs-3486	238	14	of	of	ADP
iajs-3486	238	15	1	1	NUM
iajs-3486	238	16	-	-	PUNCT
iajs-3486	238	17	set	set	VERB
iajs-3486	238	18	-	-	PUNCT
iajs-3486	238	19	contractive	contractive	ADJ
iajs-3486	238	20	maps	map	NOUN
iajs-3486	238	21	in	in	ADP
iajs-3486	238	22	banach	banach	NOUN
iajs-3486	238	23	spaces	space	NOUN
iajs-3486	238	24	,	,	PUNCT
iajs-3486	238	25	j.	j.	PROPN
iajs-3486	238	26	math	math	PROPN
iajs-3486	238	27	.	.	PUNCT
iajs-3486	239	1	anal	anal	PROPN
iajs-3486	239	2	.	.	PUNCT
iajs-3486	239	3	appl	appl	PROPN
iajs-3486	239	4	.	.	PUNCT
iajs-3486	239	5	1996	1996	NUM
iajs-3486	239	6	,	,	PUNCT
iajs-3486	240	1	712–718	712–718	NUM
iajs-3486	240	2	.	.	PUNCT
iajs-3486	241	1	https://doi.org/10.1006/jmaa.1996.0407	https://doi.org/10.1006/jmaa.1996.0407	PROPN
iajs-3486	241	2	15	15	X
iajs-3486	241	3	.	.	PUNCT
iajs-3486	241	4	rashwan	rashwan	PROPN
iajs-3486	241	5	,	,	PUNCT
iajs-3486	241	6	r.	r.	PROPN
iajs-3486	241	7	a.	a.	PROPN
iajs-3486	241	8	;	;	PUNCT
iajs-3486	241	9	albaqeri	albaqeri	PROPN
iajs-3486	241	10	,	,	PUNCT
iajs-3486	241	11	d.	d.	PROPN
iajs-3486	241	12	m.	m.	PROPN
iajs-3486	241	13	a	a	DET
iajs-3486	241	14	common	common	ADJ
iajs-3486	241	15	random	random	ADJ
iajs-3486	241	16	fixed	fix	VERB
iajs-3486	241	17	point	point	NOUN
iajs-3486	241	18	theorem	theorem	NOUN
iajs-3486	241	19	and	and	CCONJ
iajs-3486	241	20	application	application	NOUN
iajs-3486	241	21	to	to	ADP
iajs-3486	241	22	random	random	ADJ
iajs-3486	241	23	integral	integral	ADJ
iajs-3486	241	24	equations	equation	NOUN
iajs-3486	241	25	,	,	PUNCT
iajs-3486	241	26	int	int	NOUN
iajs-3486	241	27	.	.	PUNCT
iajs-3486	242	1	j.math	j.math	PROPN
iajs-3486	242	2	.	.	PUNCT
iajs-3486	243	1	res	res	PROPN
iajs-3486	243	2	.	.	PUNCT
iajs-3486	243	3	appl	appl	PROPN
iajs-3486	243	4	.	.	PROPN
iajs-3486	243	5	2014	2014	NUM
iajs-3486	243	6	.	.	PUNCT
iajs-3486	244	1	https://doi.org/10.14419/ijamr.v3i1.1690	https://doi.org/10.14419/ijamr.v3i1.1690	VERB
iajs-3486	244	2	16	16	NUM
iajs-3486	244	3	.	.	PUNCT
iajs-3486	245	1	plubtieng	plubtieng	PROPN
iajs-3486	245	2	,	,	PUNCT
iajs-3486	245	3	s.	s.	PROPN
iajs-3486	245	4	;	;	PUNCT
iajs-3486	245	5	kumam	kumam	PROPN
iajs-3486	245	6	,	,	PUNCT
iajs-3486	245	7	p.	p.	PROPN
iajs-3486	245	8	;	;	PUNCT
iajs-3486	245	9	wangkeeree	wangkeeree	PROPN
iajs-3486	245	10	,	,	PUNCT
iajs-3486	245	11	r.	r.	NOUN
iajs-3486	245	12	approximation	approximation	NOUN
iajs-3486	245	13	of	of	ADP
iajs-3486	245	14	a	a	DET
iajs-3486	245	15	common	common	ADJ
iajs-3486	245	16	random	random	ADJ
iajs-3486	245	17	fixed	fix	VERB
iajs-3486	245	18	point	point	NOUN
iajs-3486	245	19	for	for	ADP
iajs-3486	245	20	a	a	DET
iajs-3486	245	21	finite	finite	ADJ
iajs-3486	245	22	family	family	NOUN
iajs-3486	245	23	of	of	ADP
iajs-3486	245	24	random	random	ADJ
iajs-3486	245	25	operators	operator	NOUN
iajs-3486	245	26	,	,	PUNCT
iajs-3486	245	27	int	int	NOUN
iajs-3486	245	28	.	.	PUNCT
iajs-3486	246	1	j.	j.	PROPN
iajs-3486	246	2	math	math	PROPN
iajs-3486	246	3	.	.	PUNCT
iajs-3486	247	1	math	math	NOUN
iajs-3486	247	2	.	.	PUNCT
iajs-3486	248	1	sci	sci	PROPN
iajs-3486	248	2	.	.	PROPN
iajs-3486	248	3	2007	2007	NUM
iajs-3486	248	4	.	.	PUNCT
iajs-3486	249	1	https://doi.org/10.1155/2007%2f69626	https://doi.org/10.1155/2007%2f69626	PROPN
iajs-3486	249	2	17	17	NUM
iajs-3486	249	3	.	.	PUNCT
iajs-3486	250	1	beg	beg	PROPN
iajs-3486	250	2	,	,	PUNCT
iajs-3486	250	3	i.	i.	PROPN
iajs-3486	250	4	;	;	PUNCT
iajs-3486	250	5	abbas	abbas	PROPN
iajs-3486	250	6	,	,	PUNCT
iajs-3486	250	7	m.	m.	NOUN
iajs-3486	250	8	convergence	convergence	NOUN
iajs-3486	250	9	of	of	ADP
iajs-3486	250	10	iterative	iterative	ADJ
iajs-3486	250	11	algorithms	algorithm	NOUN
iajs-3486	250	12	to	to	ADP
iajs-3486	250	13	common	common	ADJ
iajs-3486	250	14	random	random	ADJ
iajs-3486	250	15	fixed	fix	VERB
iajs-3486	250	16	points	point	NOUN
iajs-3486	250	17	of	of	ADP
iajs-3486	250	18	random	random	ADJ
iajs-3486	250	19	operators	operator	NOUN
iajs-3486	250	20	,	,	PUNCT
iajs-3486	250	21	stoch	stoch	PROPN
iajs-3486	250	22	j.	j.	PROPN
iajs-3486	250	23	math	math	PROPN
iajs-3486	250	24	.	.	PUNCT
iajs-3486	251	1	anal	anal	PROPN
iajs-3486	251	2	.	.	PUNCT
iajs-3486	251	3	appl	appl	PROPN
iajs-3486	251	4	.	.	PUNCT
iajs-3486	252	1	2006	2006	NUM
iajs-3486	252	2	.	.	PUNCT
iajs-3486	253	1	https://doi.org/10.1155/jamsa%2f2006%2f89213	https://doi.org/10.1155/jamsa%2f2006%2f89213	PROPN
iajs-3486	253	2	18	18	NUM
iajs-3486	253	3	.	.	PUNCT
iajs-3486	254	1	nadler	nadler	PROPN
iajs-3486	254	2	jr	jr	PROPN
iajs-3486	254	3	,	,	PUNCT
iajs-3486	254	4	s.	s.	PROPN
iajs-3486	254	5	b.	b.	PROPN
iajs-3486	254	6	multi	multi	PROPN
iajs-3486	254	7	-	-	ADJ
iajs-3486	254	8	valued	value	VERB
iajs-3486	254	9	contraction	contraction	NOUN
iajs-3486	254	10	mappings	mapping	NOUN
iajs-3486	254	11	.	.	PUNCT
iajs-3486	254	12	,	,	PUNCT
iajs-3486	254	13	pacific	pacific	PROPN
iajs-3486	254	14	j.	j.	PROPN
iajs-3486	254	15	math	math	PROPN
iajs-3486	254	16	.	.	PUNCT
iajs-3486	255	1	1969	1969	NUM
iajs-3486	255	2	,	,	PUNCT
iajs-3486	255	3	475–488	475–488	NUM
iajs-3486	255	4	.	.	PUNCT
iajs-3486	256	1	https://doi.org/10.2140/pjm.1969.30.475	https://doi.org/10.2140/pjm.1969.30.475	X
iajs-3486	256	2	19	19	NUM
iajs-3486	256	3	.	.	PUNCT
iajs-3486	256	4	opial	opial	PROPN
iajs-3486	256	5	,	,	PUNCT
iajs-3486	256	6	z.	z.	PROPN
iajs-3486	256	7	weak	weak	ADJ
iajs-3486	256	8	convergence	convergence	NOUN
iajs-3486	256	9	of	of	ADP
iajs-3486	256	10	the	the	DET
iajs-3486	256	11	sequence	sequence	NOUN
iajs-3486	256	12	of	of	ADP
iajs-3486	256	13	successive	successive	ADJ
iajs-3486	256	14	approximations	approximation	NOUN
iajs-3486	256	15	for	for	ADP
iajs-3486	256	16	nonexpansive	nonexpansive	ADJ
iajs-3486	256	17	mappings	mapping	NOUN
iajs-3486	256	18	,	,	PUNCT
iajs-3486	256	19	bull	bull	NOUN
iajs-3486	256	20	.	.	PUNCT
iajs-3486	257	1	am	be	AUX
iajs-3486	257	2	.	.	PUNCT
iajs-3486	258	1	math	math	NOUN
iajs-3486	258	2	.	.	PUNCT
iajs-3486	259	1	soc	soc	PROPN
iajs-3486	259	2	.	.	PUNCT
iajs-3486	260	1	1967	1967	NUM
iajs-3486	260	2	,	,	PUNCT
iajs-3486	260	3	591–597	591–597	NUM
iajs-3486	260	4	.	.	PUNCT
iajs-3486	261	1	https://doi.org/10.1090/s0002-9904-1967-11761-0	https://doi.org/10.1090/s0002-9904-1967-11761-0	PROPN
iajs-3486	261	2	20	20	NUM
iajs-3486	261	3	.	.	PUNCT
iajs-3486	262	1	nilsrakoo	nilsrakoo	PROPN
iajs-3486	262	2	,	,	PUNCT
iajs-3486	262	3	w.	w.	PROPN
iajs-3486	262	4	;	;	PUNCT
iajs-3486	262	5	saejung	saejung	PROPN
iajs-3486	262	6	,	,	PUNCT
iajs-3486	262	7	s.	s.	PROPN
iajs-3486	262	8	,	,	PUNCT
iajs-3486	262	9	a	a	DET
iajs-3486	262	10	reconsideration	reconsideration	NOUN
iajs-3486	262	11	on	on	ADP
iajs-3486	262	12	convergence	convergence	NOUN
iajs-3486	262	13	of	of	ADP
iajs-3486	262	14	three	three	NUM
iajs-3486	262	15	-	-	PUNCT
iajs-3486	262	16	step	step	NOUN
iajs-3486	262	17	iterations	iteration	NOUN
iajs-3486	262	18	for	for	ADP
iajs-3486	262	19	asymptotically	asymptotically	ADV
iajs-3486	262	20	nonexpansive	nonexpansive	ADJ
iajs-3486	262	21	mappings	mapping	NOUN
iajs-3486	262	22	,	,	PUNCT
iajs-3486	262	23	math	math	PROPN
iajs-3486	262	24	.appl	.appl	PROPN
iajs-3486	262	25	.	.	PUNCT
iajs-3486	263	1	comput	comput	NOUN
iajs-3486	263	2	.	.	PUNCT
iajs-3486	264	1	2007	2007	NUM
iajs-3486	264	2	,	,	PUNCT
iajs-3486	264	3	1472–1478	1472–1478	NUM
iajs-3486	264	4	.	.	PUNCT
iajs-3486	265	1	https://doi.org/10.1016/j.amc.2007.02.026	https://doi.org/10.1016/j.amc.2007.02.026	PROPN
iajs-3486	265	2	21	21	NUM
iajs-3486	265	3	.	.	PUNCT
iajs-3486	266	1	malih	malih	PROPN
iajs-3486	266	2	,	,	PUNCT
iajs-3486	266	3	s.	s.	PROPN
iajs-3486	266	4	h.	h.	PROPN
iajs-3486	266	5	random	random	ADJ
iajs-3486	266	6	fixed	fix	VERB
iajs-3486	266	7	point	point	NOUN
iajs-3486	266	8	for	for	ADP
iajs-3486	266	9	random	random	ADJ
iajs-3486	266	10	fibonacci	fibonacci	NOUN
iajs-3486	266	11	noor	noor	PROPN
iajs-3486	266	12	iteration	iteration	PROPN
iajs-3486	266	13	scheme	scheme	PROPN
iajs-3486	266	14	j.	j.	PROPN
iajs-3486	266	15	math	math	PROPN
iajs-3486	266	16	.	.	PUNCT
iajs-3486	267	1	2021	2021	NUM
iajs-3486	267	2	,	,	PUNCT
iajs-3486	267	3	775	775	NUM
iajs-3486	267	4	–	–	PUNCT
iajs-3486	267	5	779	779	NUM
iajs-3486	267	6	.	.	PUNCT
iajs-3486	268	1	https://doi.org/10.1080/09720502.2021.1884392	https://doi.org/10.1080/09720502.2021.1884392	PROPN
iajs-3486	268	2	22	22	NUM
iajs-3486	268	3	.	.	PUNCT
iajs-3486	269	1	abed	abe	VERB
iajs-3486	269	2	,	,	PUNCT
iajs-3486	269	3	s.	s.	PROPN
iajs-3486	269	4	s.	s.	PROPN
iajs-3486	269	5	;	;	PUNCT
iajs-3486	269	6	hasan	hasan	PROPN
iajs-3486	269	7	,	,	PUNCT
iajs-3486	269	8	z.	z.	PROPN
iajs-3486	269	9	m.	m.	PROPN
iajs-3486	269	10	convergence	convergence	PROPN
iajs-3486	269	11	comparison	comparison	NOUN
iajs-3486	269	12	of	of	ADP
iajs-3486	269	13	two	two	NUM
iajs-3486	269	14	schemes	scheme	NOUN
iajs-3486	269	15	for	for	ADP
iajs-3486	269	16	common	common	ADJ
iajs-3486	269	17	fixed	fix	VERB
iajs-3486	269	18	points	point	NOUN
iajs-3486	269	19	with	with	ADP
iajs-3486	269	20	an	an	DET
iajs-3486	269	21	application	application	NOUN
iajs-3486	269	22	,	,	PUNCT
iajs-3486	269	23	ibn	ibn	PROPN
iajs-3486	269	24	al	al	PROPN
iajs-3486	269	25	-	-	PUNCT
iajs-3486	269	26	haitham	haitham	PROPN
iajs-3486	269	27	j.	j.	PROPN
iajs-3486	269	28	pure	pure	PROPN
iajs-3486	269	29	appl	appl	PROPN
iajs-3486	269	30	.	.	PUNCT
iajs-3486	270	1	sci	sci	PROPN
iajs-3486	270	2	.	.	PROPN
iajs-3486	270	3	2019	2019	NUM
iajs-3486	270	4	,	,	PUNCT
iajs-3486	270	5	81–92	81–92	NUM
iajs-3486	270	6	.	.	PUNCT
iajs-3486	271	1	https://doi.org/10.30526/32.2.2146	https://doi.org/10.30526/32.2.2146	PROPN
iajs-3486	271	2	23	23	NUM
iajs-3486	271	3	.	.	PUNCT
iajs-3486	271	4	abed	abe	VERB
iajs-3486	271	5	,	,	PUNCT
iajs-3486	271	6	s.	s.	PROPN
iajs-3486	271	7	s.	s.	PROPN
iajs-3486	271	8	;	;	PUNCT
iajs-3486	271	9	hasan	hasan	PROPN
iajs-3486	271	10	,	,	PUNCT
iajs-3486	271	11	z.	z.	PROPN
iajs-3486	271	12	m.	m.	PROPN
iajs-3486	271	13	common	common	ADJ
iajs-3486	271	14	fixed	fix	VERB
iajs-3486	271	15	point	point	NOUN
iajs-3486	271	16	of	of	ADP
iajs-3486	271	17	a	a	DET
iajs-3486	271	18	finite	finite	ADJ
iajs-3486	271	19	-	-	ADJ
iajs-3486	271	20	step	step	ADJ
iajs-3486	271	21	iteration	iteration	NOUN
iajs-3486	271	22	algorithm	algorithm	NOUN
iajs-3486	271	23	under	under	ADP
iajs-3486	271	24	total	total	ADJ
iajs-3486	271	25	ihjpas.37	ihjpas.37	PROPN
iajs-3486	271	26	(	(	PUNCT
iajs-3486	271	27	2	2	NUM
iajs-3486	271	28	)	)	PUNCT
iajs-3486	271	29	2024	2024	NUM
iajs-3486	271	30	431	431	NUM
iajs-3486	271	31	asymptotically	asymptotically	ADV
iajs-3486	271	32	quasi	quasi	ADJ
iajs-3486	271	33	-	-	ADJ
iajs-3486	271	34	nonexpansive	nonexpansive	ADJ
iajs-3486	271	35	maps	map	NOUN
iajs-3486	271	36	.	.	PUNCT
iajs-3486	272	1	baghdad	baghdad	PROPN
iajs-3486	272	2	sci	sci	PROPN
iajs-3486	272	3	.	.	PUNCT
iajs-3486	273	1	j.	j.	PROPN
iajs-3486	273	2	2019	2019	NUM
iajs-3486	273	3	,	,	PUNCT
iajs-3486	273	4	654–660	654–660	NUM
iajs-3486	273	5	.	.	PUNCT
iajs-3486	274	1	http://dx.doi.org/10.21123/bsj.2019.16.3.0654	http://dx.doi.org/10.21123/bsj.2019.16.3.0654	PROPN
iajs-3486	274	2	24	24	NUM
iajs-3486	274	3	.	.	PUNCT
iajs-3486	275	1	maibed	maibe	VERB
iajs-3486	275	2	,	,	PUNCT
iajs-3486	275	3	z.	z.	PROPN
iajs-3486	275	4	h.	h.	PROPN
iajs-3486	275	5	;	;	PUNCT
iajs-3486	275	6	thajil	thajil	X
iajs-3486	275	7	,	,	PUNCT
iajs-3486	275	8	a.	a.	PROPN
iajs-3486	275	9	q.	q.	PROPN
iajs-3486	275	10	,	,	PUNCT
iajs-3486	275	11	zenali	zenali	VERB
iajs-3486	275	12	iteration	iteration	NOUN
iajs-3486	275	13	method	method	NOUN
iajs-3486	275	14	for	for	ADP
iajs-3486	275	15	approximating	approximate	VERB
iajs-3486	275	16	fixed	fix	VERB
iajs-3486	275	17	point	point	NOUN
iajs-3486	275	18	of	of	ADP
iajs-3486	275	19	a	a	DET
iajs-3486	275	20	î	î	PROPN
iajs-3486	275	21	´	´	PROPN
iajs-3486	275	22	zaquasi	zaquasi	NOUN
iajs-3486	275	23	contractive	contractive	ADJ
iajs-3486	275	24	mappings	mapping	NOUN
iajs-3486	275	25	.	.	PUNCT
iajs-3486	276	1	ibn	ibn	PROPN
iajs-3486	276	2	al	al	PROPN
iajs-3486	276	3	-	-	PUNCT
iajs-3486	276	4	haitham	haitham	PROPN
iajs-3486	276	5	j.	j.	PROPN
iajs-3486	276	6	pure	pure	PROPN
iajs-3486	276	7	appl	appl	PROPN
iajs-3486	276	8	.	.	PUNCT
iajs-3486	277	1	sci	sci	PROPN
iajs-3486	277	2	.	.	PROPN
iajs-3486	277	3	2021	2021	NUM
iajs-3486	277	4	,	,	PUNCT
iajs-3486	277	5	78–92	78–92	NOUN
iajs-3486	277	6	.	.	PUNCT
iajs-3486	277	7	https://doi.org/10.30526/34.4.2705	https://doi.org/10.30526/34.4.2705	PROPN
iajs-3486	277	8	25	25	NUM
iajs-3486	277	9	.	.	PUNCT
iajs-3486	277	10	malih	malih	PROPN
iajs-3486	277	11	,	,	PUNCT
iajs-3486	277	12	s.	s.	PROPN
iajs-3486	277	13	h.	h.	PROPN
iajs-3486	277	14	common	common	ADJ
iajs-3486	277	15	fixed	fix	VERB
iajs-3486	277	16	point	point	NOUN
iajs-3486	277	17	for	for	ADP
iajs-3486	277	18	a	a	DET
iajs-3486	277	19	pair	pair	NOUN
iajs-3486	277	20	of	of	ADP
iajs-3486	277	21	asymptotically	asymptotically	ADV
iajs-3486	277	22	nonexpansive	nonexpansive	ADJ
iajs-3486	277	23	and	and	CCONJ
iajs-3486	277	24	multivalued	multivalued	ADJ
iajs-3486	277	25	mapping	mapping	NOUN
iajs-3486	277	26	under	under	ADP
iajs-3486	277	27	fibonacci	fibonacci	NOUN
iajs-3486	277	28	iteration	iteration	NOUN
iajs-3486	277	29	sequence	sequence	NOUN
iajs-3486	277	30	in	in	ADP
iajs-3486	277	31	cat	cat	NOUN
iajs-3486	277	32	(	(	PUNCT
iajs-3486	277	33	0	0	NUM
iajs-3486	277	34	)	)	PUNCT
iajs-3486	277	35	space	space	NOUN
iajs-3486	277	36	.	.	PUNCT
iajs-3486	278	1	in	in	ADP
iajs-3486	278	2	journal	journal	PROPN
iajs-3486	278	3	of	of	ADP
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iajs-3486	278	5	:	:	PUNCT
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iajs-3486	278	7	series	series	NOUN
iajs-3486	278	8	2021	2021	NUM
iajs-3486	278	9	.	.	PUNCT
iajs-3486	278	10	https://doi.org/10.1088/1742-6596/1897/1/012061	https://doi.org/10.1088/1742-6596/1897/1/012061	PROPN
iajs-3486	279	1	26	26	NUM
iajs-3486	279	2	.	.	PUNCT
iajs-3486	280	1	kadhim	kadhim	PROPN
iajs-3486	280	2	,	,	PUNCT
iajs-3486	280	3	a.	a.	PROPN
iajs-3486	280	4	j.	j.	PROPN
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iajs-3486	280	6	common	common	ADJ
iajs-3486	280	7	fixed	fix	VERB
iajs-3486	280	8	points	point	NOUN
iajs-3486	280	9	for	for	ADP
iajs-3486	280	10	total	total	ADJ
iajs-3486	280	11	asymptotically	asymptotically	ADV
iajs-3486	280	12	nonexpansive	nonexpansive	ADJ
iajs-3486	280	13	mapping	mapping	NOUN
iajs-3486	280	14	in	in	ADP
iajs-3486	280	15	cat	cat	NOUN
iajs-3486	280	16	(	(	PUNCT
iajs-3486	280	17	0	0	NUM
iajs-3486	280	18	)	)	PUNCT
iajs-3486	280	19	space	space	NOUN
iajs-3486	280	20	.	.	PUNCT
iajs-3486	281	1	baghdad	baghdad	PROPN
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iajs-3486	281	3	.	.	PUNCT
iajs-3486	282	1	j.2021	j.2021	PROPN
iajs-3486	282	2	.	.	PUNCT
iajs-3486	283	1	http://dx.doi.org/10.21123/bsj.2021.18.4.1286	http://dx.doi.org/10.21123/bsj.2021.18.4.1286	ADJ
iajs-3486	283	2	27	27	NUM
iajs-3486	283	3	.	.	PUNCT
iajs-3486	284	1	luaibi	luaibi	NOUN
iajs-3486	284	2	,	,	PUNCT
iajs-3486	284	3	h.	h.	PROPN
iajs-3486	284	4	h.	h.	PROPN
iajs-3486	284	5	;	;	PUNCT
iajs-3486	284	6	abed	abe	VERB
iajs-3486	284	7	,	,	PUNCT
iajs-3486	284	8	s.	s.	PROPN
iajs-3486	284	9	s.	s.	PROPN
iajs-3486	285	1	fixed	fix	VERB
iajs-3486	285	2	point	point	NOUN
iajs-3486	285	3	theorems	theorem	NOUN
iajs-3486	285	4	in	in	ADP
iajs-3486	285	5	general	general	ADJ
iajs-3486	285	6	metric	metric	ADJ
iajs-3486	285	7	space	space	NOUN
iajs-3486	285	8	with	with	ADP
iajs-3486	285	9	an	an	DET
iajs-3486	285	10	application	application	NOUN
iajs-3486	285	11	,	,	PUNCT
iajs-3486	285	12	baghdad	baghdad	PROPN
iajs-3486	285	13	sci	sci	PROPN
iajs-3486	285	14	j.	j.	PROPN
iajs-3486	285	15	2021	2021	NUM
iajs-3486	285	16	,	,	PUNCT
iajs-3486	285	17	812–815	812–815	NUM
iajs-3486	285	18	.	.	PUNCT
iajs-3486	286	1	https://doi.org/10.21123/bsj.2021.18.1%28suppl.%29.0812	https://doi.org/10.21123/bsj.2021.18.1%28suppl.%29.0812	PROPN
iajs-3486	286	2	28	28	NUM
iajs-3486	286	3	.	.	PUNCT
iajs-3486	287	1	ajeel	ajeel	PROPN
iajs-3486	287	2	,	,	PUNCT
iajs-3486	287	3	y.	y.	PROPN
iajs-3486	287	4	j.	j.	PROPN
iajs-3486	287	5	;	;	PUNCT
iajs-3486	287	6	kadhim	kadhim	PROPN
iajs-3486	287	7	,	,	PUNCT
iajs-3486	287	8	s.	s.	PROPN
iajs-3486	287	9	n.	n.	VERB
iajs-3486	287	10	some	some	DET
iajs-3486	287	11	common	common	ADJ
iajs-3486	287	12	fixed	fix	VERB
iajs-3486	287	13	points	point	NOUN
iajs-3486	287	14	theorems	theorem	NOUN
iajs-3486	287	15	of	of	ADP
iajs-3486	287	16	four	four	NUM
iajs-3486	287	17	weakly	weakly	ADV
iajs-3486	287	18	compatible	compatible	ADJ
iajs-3486	287	19	mappings	mapping	NOUN
iajs-3486	287	20	in	in	ADP
iajs-3486	287	21	metric	metric	ADJ
iajs-3486	287	22	spaces	space	NOUN
iajs-3486	287	23	,	,	PUNCT
iajs-3486	287	24	baghdad	baghdad	PROPN
iajs-3486	287	25	sci	sci	PROPN
iajs-3486	287	26	j.	j.	PROPN
iajs-3486	287	27	2021	2021	NUM
iajs-3486	287	28	,	,	PUNCT
iajs-3486	287	29	543–546	543–546	NUM
iajs-3486	287	30	.	.	PUNCT
iajs-3486	288	1	https://doi.org/10.21123/bsj.2021.18.3.0543	https://doi.org/10.21123/bsj.2021.18.3.0543	ADJ
iajs-3486	288	2	29	29	NUM
iajs-3486	288	3	.	.	PUNCT
iajs-3486	288	4	malih	malih	PROPN
iajs-3486	288	5	,	,	PUNCT
iajs-3486	288	6	s.	s.	PROPN
iajs-3486	288	7	h.	h.	PROPN
iajs-3486	288	8	,	,	PUNCT
iajs-3486	288	9	fixed	fix	VERB
iajs-3486	288	10	point	point	NOUN
iajs-3486	288	11	theorems	theorem	NOUN
iajs-3486	288	12	of	of	ADP
iajs-3486	288	13	modified	modify	VERB
iajs-3486	288	14	mann	mann	NOUN
iajs-3486	288	15	and	and	CCONJ
iajs-3486	288	16	ishikawa	ishikawa	PROPN
iajs-3486	288	17	iterations	iteration	NOUN
iajs-3486	288	18	,	,	PUNCT
iajs-3486	288	19	j.	j.	PROPN
iajs-3486	288	20	interdiscip	interdiscip	PROPN
iajs-3486	288	21	.	.	PUNCT
iajs-3486	289	1	math	math	NOUN
iajs-3486	289	2	.	.	PUNCT
iajs-3486	289	3	,	,	PUNCT
iajs-3486	289	4	2021	2021	NUM
iajs-3486	289	5	,	,	PUNCT
iajs-3486	289	6	1093–1097	1093–1097	NUM
iajs-3486	289	7	.	.	PUNCT
iajs-3486	290	1	https://doi.org/10.1080/09720502.2020.1790739	https://doi.org/10.1080/09720502.2020.1790739	PROPN
iajs-3486	290	2	30	30	NUM
iajs-3486	290	3	.	.	PUNCT
iajs-3486	291	1	karahan	karahan	PROPN
iajs-3486	291	2	,	,	PUNCT
iajs-3486	291	3	i.	i.	PROPN
iajs-3486	291	4	;	;	PUNCT
iajs-3486	291	5	ozdemir	ozdemir	PROPN
iajs-3486	291	6	,	,	PUNCT
iajs-3486	291	7	m.	m.	NOUN
iajs-3486	291	8	a	a	DET
iajs-3486	291	9	general	general	ADJ
iajs-3486	291	10	iterative	iterative	NOUN
iajs-3486	291	11	method	method	NOUN
iajs-3486	291	12	for	for	ADP
iajs-3486	291	13	approximation	approximation	NOUN
iajs-3486	291	14	of	of	ADP
iajs-3486	291	15	fixed	fix	VERB
iajs-3486	291	16	points	point	NOUN
iajs-3486	291	17	and	and	CCONJ
iajs-3486	291	18	their	their	PRON
iajs-3486	291	19	applications	application	NOUN
iajs-3486	291	20	.	.	PUNCT
iajs-3486	292	1	adv	adv	PROPN
iajs-3486	292	2	.	.	PUNCT
iajs-3486	292	3	fixed	fix	VERB
iajs-3486	292	4	point	point	NOUN
iajs-3486	292	5	theory	theory	NOUN
iajs-3486	292	6	.	.	PUNCT
iajs-3486	293	1	2013	2013	NUM
iajs-3486	293	2	,	,	PUNCT
iajs-3486	293	3	510–526	510–526	NUM
iajs-3486	293	4	.	.	PUNCT