id	sid	tid	token	lemma	pos
ma-12	1	1	2021	2021	NUM
ma-12	1	2	ada	ada	PROPN
ma-12	1	3	academica	academica	PROPN
ma-12	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-12	1	5	.	.	PUNCT
ma-12	2	1	j.	j.	PROPN
ma-12	2	2	math	math	PROPN
ma-12	2	3	.	.	PUNCT
ma-12	3	1	anal	anal	ADJ
ma-12	3	2	.	.	PUNCT
ma-12	4	1	1	1	NUM
ma-12	4	2	(	(	PUNCT
ma-12	4	3	2021	2021	NUM
ma-12	4	4	)	)	PUNCT
ma-12	4	5	45	45	NUM
ma-12	4	6	-	-	PUNCT
ma-12	4	7	67doi	67doi	NOUN
ma-12	4	8	:	:	PUNCT
ma-12	4	9	10.28924	10.28924	NUM
ma-12	4	10	/	/	SYM
ma-12	4	11	ada	ada	PROPN
ma-12	4	12	/	/	SYM
ma-12	4	13	ma.1.45	ma.1.45	ADJ
ma-12	4	14	convergence	convergence	NOUN
ma-12	4	15	of	of	ADP
ma-12	4	16	a	a	DET
ma-12	4	17	three	three	NUM
ma-12	4	18	-	-	PUNCT
ma-12	4	19	step	step	NOUN
ma-12	4	20	iteration	iteration	NOUN
ma-12	4	21	scheme	scheme	NOUN
ma-12	4	22	to	to	ADP
ma-12	4	23	the	the	DET
ma-12	4	24	common	common	ADJ
ma-12	4	25	fixed	fix	VERB
ma-12	4	26	points	point	NOUN
ma-12	4	27	of	of	ADP
ma-12	4	28	mixed	mixed	ADJ
ma-12	4	29	-	-	PUNCT
ma-12	4	30	type	type	NOUN
ma-12	4	31	total	total	NOUN
ma-12	4	32	asymtotically	asymtotically	ADV
ma-12	4	33	nonexpansive	nonexpansive	ADJ
ma-12	4	34	mappings	mapping	NOUN
ma-12	4	35	in	in	ADP
ma-12	4	36	uniformly	uniformly	ADV
ma-12	4	37	convex	convex	NOUN
ma-12	4	38	banach	banach	NOUN
ma-12	4	39	spaces	space	VERB
ma-12	4	40	imo	imo	PROPN
ma-12	4	41	kalu	kalu	PROPN
ma-12	4	42	agwu1,∗	agwu1,∗	PROPN
ma-12	4	43	,	,	PUNCT
ma-12	4	44	donatus	donatus	X
ma-12	4	45	ikechi	ikechi	PROPN
ma-12	4	46	igbokwe1	igbokwe1	NOUN
ma-12	4	47	,	,	PUNCT
ma-12	4	48	nathenial	nathenial	ADJ
ma-12	4	49	c.	c.	PROPN
ma-12	4	50	ukeje2	ukeje2	PROPN
ma-12	5	1	1department	1department	NUM
ma-12	5	2	of	of	ADP
ma-12	5	3	mathematics	mathematic	NOUN
ma-12	5	4	,	,	PUNCT
ma-12	5	5	micheal	micheal	NOUN
ma-12	5	6	okpara	okpara	NOUN
ma-12	5	7	university	university	PROPN
ma-12	5	8	of	of	ADP
ma-12	5	9	agriculture	agriculture	PROPN
ma-12	5	10	,	,	PUNCT
ma-12	5	11	umudike	umudike	NOUN
ma-12	5	12	,	,	PUNCT
ma-12	5	13	umuahia	umuahia	PROPN
ma-12	5	14	abia	abia	PROPN
ma-12	5	15	state	state	PROPN
ma-12	5	16	nigeria	nigeria	PROPN
ma-12	6	1	agwuimo@gmail.com	agwuimo@gmail.com	PROPN
ma-12	6	2	,	,	PUNCT
ma-12	6	3	igbokwedi@yahoo.com	igbokwedi@yahoo.com	PROPN
ma-12	7	1	2department	2department	NUM
ma-12	7	2	of	of	ADP
ma-12	7	3	mathematics	mathematic	NOUN
ma-12	7	4	and	and	CCONJ
ma-12	7	5	statistics	statistic	NOUN
ma-12	7	6	,	,	PUNCT
ma-12	7	7	university	university	NOUN
ma-12	7	8	of	of	ADP
ma-12	7	9	port	port	NOUN
ma-12	7	10	-	-	PUNCT
ma-12	7	11	harcourt	harcourt	NOUN
ma-12	7	12	,	,	PUNCT
ma-12	7	13	port	port	NOUN
ma-12	7	14	-	-	PUNCT
ma-12	7	15	harcourt	harcourt	NOUN
ma-12	7	16	rivers	river	NOUN
ma-12	7	17	state	state	PROPN
ma-12	7	18	nigeria	nigeria	PROPN
ma-12	7	19	ukejechukwuebuanathan@gmail.com	ukejechukwuebuanathan@gmail.com	PUNCT
ma-12	8	1	∗correspondence	∗correspondence	NOUN
ma-12	8	2	:	:	PUNCT
ma-12	8	3	agwuimo@gmail.com	agwuimo@gmail.com	X
ma-12	9	1	abstract	abstract	ADJ
ma-12	9	2	.	.	PUNCT
ma-12	10	1	we	we	PRON
ma-12	10	2	propose	propose	VERB
ma-12	10	3	a	a	DET
ma-12	10	4	three	three	NUM
ma-12	10	5	-	-	PUNCT
ma-12	10	6	step	step	NOUN
ma-12	10	7	iteration	iteration	NOUN
ma-12	10	8	scheme	scheme	NOUN
ma-12	10	9	of	of	ADP
ma-12	10	10	hybrid	hybrid	ADJ
ma-12	10	11	mixed	mixed	ADJ
ma-12	10	12	-	-	PUNCT
ma-12	10	13	type	type	NOUN
ma-12	10	14	for	for	ADP
ma-12	10	15	three	three	NUM
ma-12	10	16	total	total	ADJ
ma-12	10	17	asymptoti	asymptoti	NOUN
ma-12	10	18	-	-	PUNCT
ma-12	10	19	cally	cally	ADV
ma-12	10	20	nonexpansive	nonexpansive	ADJ
ma-12	10	21	self	self	NOUN
ma-12	10	22	mappings	mapping	NOUN
ma-12	10	23	and	and	CCONJ
ma-12	10	24	three	three	NUM
ma-12	10	25	total	total	ADJ
ma-12	10	26	asymptotically	asymptotically	ADV
ma-12	10	27	nonexpansive	nonexpansive	ADJ
ma-12	10	28	nonself	nonself	PROPN
ma-12	10	29	mappings	mapping	NOUN
ma-12	10	30	.	.	PUNCT
ma-12	11	1	inaddition	inaddition	NOUN
ma-12	11	2	,	,	PUNCT
ma-12	11	3	we	we	PRON
ma-12	11	4	establish	establish	VERB
ma-12	11	5	some	some	DET
ma-12	11	6	weak	weak	ADJ
ma-12	11	7	convergence	convergence	NOUN
ma-12	11	8	theorems	theorem	NOUN
ma-12	11	9	of	of	ADP
ma-12	11	10	the	the	DET
ma-12	11	11	scheme	scheme	NOUN
ma-12	11	12	to	to	ADP
ma-12	11	13	the	the	DET
ma-12	11	14	common	common	ADJ
ma-12	11	15	fixed	fix	VERB
ma-12	11	16	point	point	NOUN
ma-12	12	1	ofthe	ofthe	ADJ
ma-12	12	2	mappings	mapping	NOUN
ma-12	12	3	in	in	ADP
ma-12	12	4	uniformly	uniformly	ADV
ma-12	12	5	convex	convex	NOUN
ma-12	12	6	banach	banach	NOUN
ma-12	12	7	spaces	space	VERB
ma-12	12	8	.	.	PUNCT
ma-12	13	1	our	our	PRON
ma-12	13	2	results	result	NOUN
ma-12	13	3	extend	extend	VERB
ma-12	13	4	and	and	CCONJ
ma-12	13	5	generalize	generalize	VERB
ma-12	13	6	numerous	numerous	ADJ
ma-12	13	7	resultscurrently	resultscurrently	ADV
ma-12	13	8	in	in	ADP
ma-12	13	9	literature	literature	NOUN
ma-12	13	10	.	.	PUNCT
ma-12	14	1	1	1	X
ma-12	14	2	.	.	X
ma-12	14	3	introduction	introduction	NOUN
ma-12	14	4	let	let	VERB
ma-12	14	5	k	k	PROPN
ma-12	14	6	be	be	AUX
ma-12	14	7	a	a	DET
ma-12	14	8	nonempty	nonempty	ADJ
ma-12	14	9	subset	subset	NOUN
ma-12	14	10	of	of	ADP
ma-12	14	11	a	a	DET
ma-12	14	12	real	real	ADJ
ma-12	14	13	banach	banach	NOUN
ma-12	14	14	space	space	NOUN
ma-12	14	15	e.	e.	PROPN
ma-12	14	16	let	let	VERB
ma-12	14	17	t	t	NOUN
ma-12	14	18	:	:	PUNCT
ma-12	14	19	k	k	PROPN
ma-12	14	20	−→	−→	NOUN
ma-12	14	21	k	k	PROPN
ma-12	14	22	be	be	AUX
ma-12	14	23	a	a	DET
ma-12	14	24	nonlinear	nonlinear	ADJ
ma-12	14	25	mapping	mapping	NOUN
ma-12	14	26	,	,	PUNCT
ma-12	14	27	we	we	PRON
ma-12	14	28	denote	denote	VERB
ma-12	14	29	the	the	DET
ma-12	14	30	set	set	NOUN
ma-12	14	31	of	of	ADP
ma-12	14	32	all	all	DET
ma-12	14	33	fixed	fix	VERB
ma-12	14	34	points	point	NOUN
ma-12	14	35	of	of	ADP
ma-12	14	36	t	t	NOUN
ma-12	14	37	by	by	ADP
ma-12	14	38	f	f	PROPN
ma-12	14	39	(	(	PUNCT
ma-12	14	40	t	t	PROPN
ma-12	14	41	)	)	PUNCT
ma-12	14	42	.	.	PUNCT
ma-12	15	1	the	the	DET
ma-12	15	2	set	set	NOUN
ma-12	15	3	of	of	ADP
ma-12	15	4	common	common	ADJ
ma-12	15	5	fixed	fix	VERB
ma-12	15	6	points	point	NOUN
ma-12	15	7	of	of	ADP
ma-12	15	8	six	six	NUM
ma-12	15	9	mappings	mapping	NOUN
ma-12	15	10	s1	s1	NOUN
ma-12	15	11	,	,	PUNCT
ma-12	15	12	s2	s2	PROPN
ma-12	15	13	,	,	PUNCT
ma-12	15	14	s3	s3	PROPN
ma-12	15	15	,	,	PUNCT
ma-12	15	16	t1	t1	PROPN
ma-12	15	17	,	,	PUNCT
ma-12	15	18	t2	t2	NOUN
ma-12	15	19	and	and	CCONJ
ma-12	15	20	t3	t3	PROPN
ma-12	15	21	will	will	AUX
ma-12	15	22	be	be	AUX
ma-12	15	23	denoted	denote	VERB
ma-12	15	24	by	by	ADP
ma-12	15	25	f	f	PROPN
ma-12	15	26	=	=	PRON
ma-12	15	27	∩3i=1(f	∩3i=1(f	X
ma-12	15	28	(	(	PUNCT
ma-12	15	29	ti	ti	NOUN
ma-12	15	30	)	)	PUNCT
ma-12	15	31	∩	∩	ADJ
ma-12	15	32	f	f	X
ma-12	15	33	(	(	PUNCT
ma-12	15	34	si	si	NOUN
ma-12	15	35	)	)	PUNCT
ma-12	15	36	)	)	PUNCT
ma-12	15	37	.	.	PUNCT
ma-12	16	1	definition	definition	NOUN
ma-12	16	2	1.1	1.1	NUM
ma-12	16	3	.	.	PUNCT
ma-12	17	1	a	a	DET
ma-12	17	2	mapping	mapping	NOUN
ma-12	17	3	t	t	NOUN
ma-12	17	4	:	:	PUNCT
ma-12	17	5	k	k	X
ma-12	17	6	−→	−→	NOUN
ma-12	17	7	k	k	PROPN
ma-12	17	8	is	be	AUX
ma-12	17	9	said	say	VERB
ma-12	17	10	to	to	ADP
ma-12	17	11	asymptotically	asymptotically	ADV
ma-12	17	12	nonexpansive	nonexpansive	ADJ
ma-12	18	1	[	[	X
ma-12	18	2	6	6	NUM
ma-12	18	3	]	]	PUNCT
ma-12	18	4	if	if	SCONJ
ma-12	18	5	there	there	PRON
ma-12	18	6	exists	exist	VERB
ma-12	18	7	a	a	DET
ma-12	18	8	sequence	sequence	NOUN
ma-12	18	9	{	{	PUNCT
ma-12	18	10	kn	kn	NOUN
ma-12	18	11	}	}	PUNCT
ma-12	18	12	in	in	ADP
ma-12	18	13	[	[	X
ma-12	18	14	1,∞	1,∞	NUM
ma-12	18	15	)	)	PUNCT
ma-12	18	16	with	with	ADP
ma-12	18	17	limn→∞	limn→∞	PROPN
ma-12	19	1	kn	kn	PROPN
ma-12	19	2	=	=	NOUN
ma-12	19	3	1	1	NUM
ma-12	19	4	such	such	ADJ
ma-12	19	5	that	that	SCONJ
ma-12	19	6	‖t	‖t	NOUN
ma-12	19	7	n(x)−	n(x)−	PROPN
ma-12	19	8	t	t	VERB
ma-12	19	9	n(y)‖	n(y)‖	PRON
ma-12	19	10	≤	≤	PROPN
ma-12	19	11	kn‖x	kn‖x	PROPN
ma-12	19	12	−	−	NUM
ma-12	20	1	y‖,∀x	y‖,∀x	NOUN
ma-12	20	2	,	,	PUNCT
ma-12	20	3	y	y	PROPN
ma-12	20	4	∈	∈	PROPN
ma-12	20	5	n	n	CCONJ
ma-12	20	6	(	(	PUNCT
ma-12	20	7	1.1	1.1	NUM
ma-12	20	8	)	)	PUNCT
ma-12	20	9	.	.	PUNCT
ma-12	21	1	ln	ln	ADJ
ma-12	21	2	1972	1972	NUM
ma-12	21	3	,	,	PUNCT
ma-12	21	4	the	the	DET
ma-12	21	5	class	class	NOUN
ma-12	21	6	of	of	ADP
ma-12	21	7	asymptotically	asymptotically	ADV
ma-12	21	8	nonexpansive	nonexpansive	ADJ
ma-12	21	9	mapping	mapping	NOUN
ma-12	21	10	was	be	AUX
ma-12	21	11	introduced	introduce	VERB
ma-12	21	12	by	by	ADP
ma-12	21	13	goebel	goebel	NOUN
ma-12	21	14	and	and	CCONJ
ma-12	21	15	kirk	kirk	NOUN
ma-12	22	1	[	[	X
ma-12	22	2	6].they	6].they	PRON
ma-12	22	3	proved	prove	VERB
ma-12	22	4	that	that	SCONJ
ma-12	22	5	if	if	SCONJ
ma-12	22	6	k	k	PROPN
ma-12	22	7	is	be	AUX
ma-12	22	8	a	a	DET
ma-12	22	9	nonempty	nonempty	ADV
ma-12	22	10	closed	close	VERB
ma-12	22	11	convex	convex	NOUN
ma-12	22	12	subset	subset	NOUN
ma-12	22	13	of	of	ADP
ma-12	22	14	a	a	DET
ma-12	22	15	uniformly	uniformly	ADV
ma-12	22	16	convex	convex	NOUN
ma-12	22	17	banach	banach	NOUN
ma-12	22	18	space	space	NOUN
ma-12	22	19	and	and	CCONJ
ma-12	22	20	t	t	PROPN
ma-12	22	21	is	be	AUX
ma-12	22	22	an	an	DET
ma-12	22	23	asymptotically	asymptotically	ADV
ma-12	22	24	nonexpansive	nonexpansive	ADJ
ma-12	22	25	mapping	mapping	NOUN
ma-12	22	26	of	of	ADP
ma-12	22	27	k	k	PROPN
ma-12	22	28	,	,	PUNCT
ma-12	22	29	then	then	ADV
ma-12	22	30	t	t	PROPN
ma-12	22	31	has	have	VERB
ma-12	22	32	a	a	DET
ma-12	22	33	fixed	fix	VERB
ma-12	22	34	point	point	NOUN
ma-12	22	35	.	.	PUNCT
ma-12	23	1	received	receive	VERB
ma-12	23	2	:	:	PUNCT
ma-12	23	3	29	29	NUM
ma-12	23	4	aug	aug	PROPN
ma-12	23	5	2021	2021	NUM
ma-12	23	6	.	.	PUNCT
ma-12	24	1	key	key	ADJ
ma-12	24	2	words	word	NOUN
ma-12	24	3	and	and	CCONJ
ma-12	24	4	phrases	phrase	NOUN
ma-12	24	5	.	.	PUNCT
ma-12	25	1	asymtotically	asymtotically	ADV
ma-12	25	2	nonexpansive	nonexpansive	ADJ
ma-12	25	3	mapping	mapping	NOUN
ma-12	25	4	;	;	PUNCT
ma-12	25	5	total	total	ADJ
ma-12	25	6	asymptotically	asymptotically	ADV
ma-12	25	7	nonexpansive	nonexpansive	ADJ
ma-12	25	8	nonself	nonself	PROPN
ma-12	25	9	mapping;hybrid	mapping;hybrid	PROPN
ma-12	25	10	mixed	mixed	ADJ
ma-12	25	11	type	type	NOUN
ma-12	25	12	iteration	iteration	NOUN
ma-12	25	13	scheme	scheme	NOUN
ma-12	25	14	;	;	PUNCT
ma-12	25	15	common	common	ADJ
ma-12	25	16	fixed	fix	VERB
ma-12	25	17	point	point	NOUN
ma-12	25	18	;	;	PUNCT
ma-12	25	19	uniformly	uniformly	ADV
ma-12	25	20	convex	convex	VERB
ma-12	25	21	banach	banach	NOUN
ma-12	25	22	space	space	NOUN
ma-12	25	23	;	;	PUNCT
ma-12	25	24	weak	weak	ADJ
ma-12	25	25	convergence.45	convergence.45	PROPN
ma-12	25	26	https://adac.ee	https://adac.ee	PROPN
ma-12	25	27	https://doi.org/10.28924/ada/ma.1.45	https://doi.org/10.28924/ada/ma.1.45	X
ma-12	25	28	eur	eur	NOUN
ma-12	25	29	.	.	PUNCT
ma-12	26	1	j.	j.	PROPN
ma-12	26	2	math	math	PROPN
ma-12	26	3	.	.	PUNCT
ma-12	27	1	anal	anal	ADJ
ma-12	27	2	.	.	PUNCT
ma-12	28	1	1	1	NUM
ma-12	28	2	(	(	PUNCT
ma-12	28	3	2021	2021	NUM
ma-12	28	4	)	)	PUNCT
ma-12	28	5	46	46	NUM
ma-12	28	6	definition	definition	NOUN
ma-12	28	7	1.2	1.2	NUM
ma-12	28	8	.	.	PUNCT
ma-12	29	1	a	a	DET
ma-12	29	2	mapping	mapping	NOUN
ma-12	29	3	t	t	NOUN
ma-12	29	4	is	be	AUX
ma-12	29	5	said	say	VERB
ma-12	29	6	to	to	PART
ma-12	29	7	be	be	AUX
ma-12	29	8	total	total	ADV
ma-12	29	9	asymptotically	asymptotically	ADV
ma-12	29	10	nonexpansive	nonexpansive	ADJ
ma-12	30	1	[	[	X
ma-12	30	2	1	1	X
ma-12	30	3	]	]	PUNCT
ma-12	30	4	if	if	SCONJ
ma-12	30	5	‖t	‖t	ADJ
ma-12	30	6	n(x)−	n(x)−	PROPN
ma-12	30	7	t	t	VERB
ma-12	30	8	n(y)‖	n(y)‖	PRON
ma-12	30	9	≤	≤	NUM
ma-12	30	10	‖x	‖x	PUNCT
ma-12	31	1	−	−	PROPN
ma-12	32	1	y‖+	y‖+	PROPN
ma-12	32	2	µnφ(‖x	µnφ(‖x	PROPN
ma-12	32	3	−	−	PROPN
ma-12	32	4	y‖	y‖	PROPN
ma-12	32	5	)	)	PUNCT
ma-12	33	1	+	+	CCONJ
ma-12	33	2	νn,∀x	νn,∀x	PROPN
ma-12	33	3	,	,	PUNCT
ma-12	33	4	y	y	PROPN
ma-12	33	5	∈	∈	PROPN
ma-12	33	6	k,∀n	k,∀n	PROPN
ma-12	33	7	∈	∈	PROPN
ma-12	33	8	n	n	CCONJ
ma-12	33	9	,	,	PUNCT
ma-12	33	10	(	(	PUNCT
ma-12	33	11	1.2	1.2	NUM
ma-12	33	12	)	)	PUNCT
ma-12	33	13	where	where	SCONJ
ma-12	33	14	{	{	PUNCT
ma-12	33	15	µn}and	µn}and	PUNCT
ma-12	33	16	{	{	PUNCT
ma-12	33	17	νn	νn	AUX
ma-12	33	18	}	}	PUNCT
ma-12	33	19	are	be	AUX
ma-12	33	20	nonnegative	nonnegative	ADJ
ma-12	33	21	real	real	ADJ
ma-12	33	22	sequences	sequence	NOUN
ma-12	33	23	such	such	ADJ
ma-12	33	24	that	that	SCONJ
ma-12	33	25	µn	µn	PROPN
ma-12	33	26	→	→	SYM
ma-12	33	27	0	0	NUM
ma-12	33	28	and	and	CCONJ
ma-12	33	29	νn	νn	ADJ
ma-12	33	30	→	→	SYM
ma-12	33	31	0	0	PUNCT
ma-12	33	32	as	as	ADP
ma-12	33	33	n	n	PROPN
ma-12	33	34	→∞	→∞	NOUN
ma-12	33	35	and	and	CCONJ
ma-12	33	36	φ	φ	PROPN
ma-12	33	37	is	be	AUX
ma-12	33	38	a	a	DET
ma-12	33	39	strictly	strictly	ADV
ma-12	33	40	increasing	increase	VERB
ma-12	33	41	continuous	continuous	ADJ
ma-12	33	42	function	function	NOUN
ma-12	33	43	φ	φ	NOUN
ma-12	33	44	:	:	PUNCT
ma-12	34	1	[	[	X
ma-12	34	2	0,∞)→	0,∞)→	NOUN
ma-12	34	3	[	[	X
ma-12	34	4	0,∞	0,∞	NOUN
ma-12	34	5	)	)	PUNCT
ma-12	34	6	with	with	ADP
ma-12	34	7	φ(0	φ(0	ADJ
ma-12	34	8	)	)	PUNCT
ma-12	34	9	=	=	SYM
ma-12	34	10	0	0	X
ma-12	34	11	.	.	PUNCT
ma-12	34	12	from	from	ADP
ma-12	34	13	the	the	DET
ma-12	34	14	above	above	ADJ
ma-12	34	15	definitions	definition	NOUN
ma-12	34	16	,	,	PUNCT
ma-12	34	17	we	we	PRON
ma-12	34	18	see	see	VERB
ma-12	34	19	that	that	SCONJ
ma-12	34	20	the	the	DET
ma-12	34	21	class	class	NOUN
ma-12	34	22	of	of	ADP
ma-12	34	23	total	total	ADJ
ma-12	34	24	asymptotically	asymptotically	ADV
ma-12	34	25	nonexpansive	nonexpansive	ADJ
ma-12	34	26	mappingsincludes	mappingsinclude	VERB
ma-12	34	27	the	the	DET
ma-12	34	28	class	class	NOUN
ma-12	34	29	of	of	ADP
ma-12	34	30	asymptotically	asymptotically	ADV
ma-12	34	31	nonexpansive	nonexpansive	ADJ
ma-12	34	32	mapping	mapping	NOUN
ma-12	34	33	as	as	ADP
ma-12	34	34	a	a	DET
ma-12	34	35	special	special	ADJ
ma-12	34	36	case	case	NOUN
ma-12	34	37	;	;	PUNCT
ma-12	34	38	see	see	VERB
ma-12	34	39	[	[	X
ma-12	34	40	4	4	X
ma-12	34	41	]	]	PUNCT
ma-12	34	42	for	for	ADP
ma-12	34	43	moredetails	moredetail	NOUN
ma-12	34	44	.	.	PUNCT
ma-12	35	1	each	each	DET
ma-12	35	2	asymptotically	asymptotically	ADV
ma-12	35	3	nonexpansive	nonexpansive	ADJ
ma-12	35	4	mapping	mapping	NOUN
ma-12	35	5	is	be	AUX
ma-12	35	6	total	total	ADJ
ma-12	35	7	asymptotically	asymptotically	ADV
ma-12	35	8	nonexpansive	nonexpansive	ADJ
ma-12	35	9	mappingwith	mappingwith	ADJ
ma-12	35	10	νn	νn	NOUN
ma-12	35	11	=	=	SYM
ma-12	35	12	0	0	NUM
ma-12	35	13	,	,	PUNCT
ma-12	35	14	µn	µn	PROPN
ma-12	35	15	=	=	SYM
ma-12	35	16	kn	kn	PROPN
ma-12	35	17	−	−	PROPN
ma-12	35	18	1	1	NUM
ma-12	35	19	f	f	PROPN
ma-12	35	20	or	or	CCONJ
ma-12	35	21	al	al	PROPN
ma-12	35	22	l	l	PROPN
ma-12	35	23	n	n	X
ma-12	35	24	≥	≥	NUM
ma-12	35	25	1	1	NUM
ma-12	35	26	,	,	PUNCT
ma-12	35	27	φ(t	φ(t	PROPN
ma-12	35	28	)	)	PUNCT
ma-12	35	29	=	=	SYM
ma-12	35	30	t	t	PROPN
ma-12	35	31	,	,	PUNCT
ma-12	35	32	t	t	PROPN
ma-12	35	33	≥	≥	PROPN
ma-12	35	34	0	0	NUM
ma-12	35	35	.	.	PUNCT
ma-12	36	1	definition	definition	NOUN
ma-12	36	2	1.3	1.3	NUM
ma-12	36	3	.	.	PUNCT
ma-12	37	1	a	a	DET
ma-12	37	2	subset	subset	NOUN
ma-12	37	3	k	k	NOUN
ma-12	37	4	of	of	ADP
ma-12	37	5	a	a	DET
ma-12	37	6	banach	banach	NOUN
ma-12	37	7	space	space	NOUN
ma-12	37	8	e	e	NOUN
ma-12	37	9	is	be	AUX
ma-12	37	10	said	say	VERB
ma-12	37	11	to	to	PART
ma-12	37	12	be	be	AUX
ma-12	37	13	a	a	DET
ma-12	37	14	retract	retract	NOUN
ma-12	37	15	of	of	ADP
ma-12	37	16	e	e	NOUN
ma-12	37	17	if	if	SCONJ
ma-12	37	18	there	there	PRON
ma-12	37	19	exists	exist	VERB
ma-12	37	20	a	a	DET
ma-12	37	21	continuous	continuous	ADJ
ma-12	37	22	mapping	mapping	NOUN
ma-12	37	23	p	p	NOUN
ma-12	37	24	:	:	PUNCT
ma-12	37	25	e	e	X
ma-12	37	26	−→	−→	NOUN
ma-12	37	27	k	k	PROPN
ma-12	37	28	(	(	PUNCT
ma-12	37	29	cal	cal	PROPN
ma-12	37	30	led	lead	VERB
ma-12	37	31	retraction	retraction	NOUN
ma-12	37	32	)	)	PUNCT
ma-12	38	1	such	such	ADJ
ma-12	38	2	that	that	SCONJ
ma-12	38	3	p	p	X
ma-12	38	4	(	(	PUNCT
ma-12	38	5	x	x	NOUN
ma-12	38	6	)	)	PUNCT
ma-12	38	7	=	=	SYM
ma-12	38	8	x	x	PUNCT
ma-12	38	9	for	for	ADP
ma-12	38	10	all	all	DET
ma-12	38	11	x	x	SYM
ma-12	38	12	∈	∈	PROPN
ma-12	38	13	k.	k.	NOUN
ma-12	39	1	if	if	SCONJ
ma-12	39	2	,	,	PUNCT
ma-12	39	3	in	in	ADP
ma-12	39	4	addition	addition	NOUN
ma-12	39	5	p	p	NOUN
ma-12	39	6	is	be	AUX
ma-12	39	7	nonexpansive	nonexpansive	ADJ
ma-12	39	8	,	,	PUNCT
ma-12	39	9	then	then	ADV
ma-12	39	10	p	p	NOUN
ma-12	39	11	is	be	AUX
ma-12	39	12	said	say	VERB
ma-12	39	13	to	to	PART
ma-12	39	14	be	be	AUX
ma-12	39	15	nonexpansive	nonexpansive	ADJ
ma-12	39	16	retraction	retraction	NOUN
ma-12	39	17	of	of	ADP
ma-12	39	18	e.	e.	PROPN
ma-12	39	19	if	if	SCONJ
ma-12	39	20	p	p	X
ma-12	39	21	:	:	PUNCT
ma-12	39	22	e	e	X
ma-12	39	23	−→	−→	NOUN
ma-12	39	24	k	k	PROPN
ma-12	39	25	is	be	AUX
ma-12	39	26	a	a	DET
ma-12	39	27	retraction	retraction	NOUN
ma-12	39	28	,	,	PUNCT
ma-12	39	29	then	then	ADV
ma-12	39	30	p	p	X
ma-12	39	31	2	2	NUM
ma-12	39	32	=	=	SYM
ma-12	39	33	p.	p.	NOUN
ma-12	39	34	a	a	DET
ma-12	39	35	retract	retract	NOUN
ma-12	39	36	of	of	ADP
ma-12	39	37	a	a	DET
ma-12	39	38	hausdorff	hausdorff	NOUN
ma-12	39	39	space	space	NOUN
ma-12	39	40	must	must	AUX
ma-12	39	41	be	be	AUX
ma-12	39	42	a	a	DET
ma-12	39	43	closed	closed	ADJ
ma-12	39	44	subset	subset	NOUN
ma-12	39	45	.	.	PUNCT
ma-12	40	1	every	every	DET
ma-12	40	2	closed	close	VERB
ma-12	40	3	convex	convex	NOUN
ma-12	40	4	subset	subset	NOUN
ma-12	40	5	of	of	ADP
ma-12	40	6	a	a	DET
ma-12	40	7	uniformly	uniformly	ADJ
ma-12	40	8	convex	convex	NOUN
ma-12	40	9	banach	banach	NOUN
ma-12	40	10	space	space	NOUN
ma-12	40	11	is	be	AUX
ma-12	40	12	a	a	DET
ma-12	40	13	retract	retract	NOUN
ma-12	40	14	.	.	PUNCT
ma-12	41	1	in	in	ADP
ma-12	41	2	2012	2012	NUM
ma-12	41	3	,	,	PUNCT
ma-12	41	4	yolacan	yolacan	NOUN
ma-12	41	5	and	and	CCONJ
ma-12	41	6	kiziltune	kiziltune	PROPN
ma-12	42	1	[	[	X
ma-12	42	2	18	18	NUM
ma-12	42	3	]	]	PUNCT
ma-12	42	4	defined	define	VERB
ma-12	42	5	the	the	DET
ma-12	42	6	following	following	NOUN
ma-12	42	7	:	:	PUNCT
ma-12	42	8	definition	definition	NOUN
ma-12	42	9	1.4	1.4	NUM
ma-12	42	10	.	.	PUNCT
ma-12	43	1	let	let	VERB
ma-12	43	2	k	k	PRON
ma-12	43	3	be	be	AUX
ma-12	43	4	a	a	DET
ma-12	43	5	nonempty	nonempty	ADJ
ma-12	43	6	and	and	CCONJ
ma-12	43	7	closed	closed	ADJ
ma-12	43	8	convex	convex	NOUN
ma-12	43	9	subset	subset	NOUN
ma-12	43	10	of	of	ADP
ma-12	43	11	a	a	DET
ma-12	43	12	banach	banach	NOUN
ma-12	43	13	space	space	NOUN
ma-12	43	14	e.	e.	PROPN
ma-12	43	15	a	a	DET
ma-12	43	16	nonself	nonself	PROPN
ma-12	43	17	mapping	mapping	NOUN
ma-12	43	18	t	t	NOUN
ma-12	43	19	:	:	PUNCT
ma-12	43	20	k	k	X
ma-12	43	21	→	→	PUNCT
ma-12	43	22	e	e	PROPN
ma-12	43	23	is	be	AUX
ma-12	43	24	said	say	VERB
ma-12	43	25	to	to	PART
ma-12	43	26	be	be	AUX
ma-12	43	27	total	total	ADJ
ma-12	43	28	asymptotically	asymptotically	ADV
ma-12	43	29	nonexpansive	nonexpansive	ADJ
ma-12	43	30	mapping	mapping	NOUN
ma-12	43	31	if	if	SCONJ
ma-12	43	32	there	there	PRON
ma-12	43	33	exist	exist	VERB
ma-12	43	34	sequences	sequence	NOUN
ma-12	43	35	k(1)n	k(1)n	X
ma-12	43	36	and	and	CCONJ
ma-12	43	37	k(2)n	k(2)n	PROPN
ma-12	43	38	in	in	ADP
ma-12	43	39	[	[	X
ma-12	43	40	0,∞	0,∞	NOUN
ma-12	43	41	)	)	PUNCT
ma-12	43	42	with	with	ADP
ma-12	43	43	k(1)n	k(1)n	PROPN
ma-12	43	44	→	→	SYM
ma-12	43	45	0	0	NUM
ma-12	43	46	and	and	CCONJ
ma-12	43	47	k(2)n	k(2)n	PROPN
ma-12	43	48	→	→	SYM
ma-12	43	49	0	0	PROPN
ma-12	43	50	as	as	ADP
ma-12	43	51	n	n	X
ma-12	43	52	→∞	→∞	PROPN
ma-12	43	53	and	and	CCONJ
ma-12	43	54	a	a	DET
ma-12	43	55	strictly	strictly	ADV
ma-12	43	56	increasing	increase	VERB
ma-12	43	57	function	function	NOUN
ma-12	43	58	φ	φ	NOUN
ma-12	43	59	:	:	PUNCT
ma-12	44	1	[	[	X
ma-12	44	2	0,∞)→	0,∞)→	NOUN
ma-12	44	3	[	[	X
ma-12	44	4	0,∞	0,∞	NOUN
ma-12	44	5	)	)	PUNCT
ma-12	44	6	with	with	ADP
ma-12	44	7	φ(0	φ(0	ADJ
ma-12	44	8	)	)	PUNCT
ma-12	44	9	=	=	SYM
ma-12	44	10	0	0	NUM
ma-12	44	11	such	such	ADJ
ma-12	44	12	that	that	DET
ma-12	44	13	‖t	‖t	NOUN
ma-12	44	14	(	(	PUNCT
ma-12	44	15	pt	pt	PROPN
ma-12	44	16	)	)	PUNCT
ma-12	44	17	n−1(x)−	n−1(x)−	PROPN
ma-12	44	18	t	t	PROPN
ma-12	44	19	(	(	PUNCT
ma-12	44	20	pt	pt	INTJ
ma-12	44	21	)	)	PUNCT
ma-12	44	22	n−1(y)‖	n−1(y)‖	PROPN
ma-12	44	23	≤	≤	ADV
ma-12	44	24	‖x	‖x	PUNCT
ma-12	45	1	−	−	PROPN
ma-12	45	2	y‖+	y‖+	PROPN
ma-12	45	3	k1(n)φ(‖x	k1(n)φ(‖x	PROPN
ma-12	45	4	−	−	PROPN
ma-12	45	5	y‖	y‖	PROPN
ma-12	45	6	)	)	PUNCT
ma-12	46	1	+	+	CCONJ
ma-12	46	2	k	k	X
ma-12	46	3	(	(	PUNCT
ma-12	46	4	2	2	NUM
ma-12	46	5	)	)	PUNCT
ma-12	46	6	n	n	NOUN
ma-12	46	7	,	,	PUNCT
ma-12	46	8	∀x	∀x	X
ma-12	46	9	,	,	PUNCT
ma-12	46	10	y	y	PROPN
ma-12	46	11	∈	∈	PROPN
ma-12	46	12	k	k	PROPN
ma-12	46	13	,	,	PUNCT
ma-12	46	14	n	n	PROPN
ma-12	46	15	∈	∈	PROPN
ma-12	46	16	n.	n.	NOUN
ma-12	46	17	(	(	PUNCT
ma-12	46	18	1.3	1.3	NUM
ma-12	46	19	)	)	PUNCT
ma-12	46	20	chidume	chidume	VERB
ma-12	46	21	et	et	PROPN
ma-12	46	22	al	al	PROPN
ma-12	46	23	.	.	PUNCT
ma-12	47	1	[	[	X
ma-12	47	2	3	3	NUM
ma-12	47	3	]	]	PUNCT
ma-12	47	4	studied	study	VERB
ma-12	47	5	the	the	DET
ma-12	47	6	following	following	ADJ
ma-12	47	7	iterative	iterative	NOUN
ma-12	47	8	scheme	scheme	NOUN
ma-12	47	9	in	in	ADP
ma-12	47	10	2004	2004	NUM
ma-12	47	11	:	:	PUNCT
ma-12	48	1	x1	x1	X
ma-12	48	2	=	=	PUNCT
ma-12	48	3	x	x	SYM
ma-12	48	4	∈	∈	PROPN
ma-12	48	5	k	k	X
ma-12	48	6	xn+1	xn+1	PROPN
ma-12	48	7	=	=	SYM
ma-12	48	8	p	p	X
ma-12	48	9	(	(	PUNCT
ma-12	48	10	αnt	αnt	X
ma-12	48	11	(	(	PUNCT
ma-12	48	12	pt	pt	INTJ
ma-12	48	13	)	)	PUNCT
ma-12	48	14	n−1xn	n−1xn	NOUN
ma-12	48	15	+	+	CCONJ
ma-12	48	16	(	(	PUNCT
ma-12	48	17	1−	1−	NUM
ma-12	48	18	αn)xn	αn)xn	NOUN
ma-12	48	19	)	)	PUNCT
ma-12	48	20	,	,	PUNCT
ma-12	48	21	n	n	X
ma-12	48	22	≥	≥	NOUN
ma-12	48	23	1	1	NUM
ma-12	48	24	,	,	PUNCT
ma-12	48	25	(	(	PUNCT
ma-12	48	26	1.4	1.4	NUM
ma-12	48	27	)	)	PUNCT
ma-12	48	28	where	where	SCONJ
ma-12	48	29	{	{	PUNCT
ma-12	48	30	αn	αn	NOUN
ma-12	48	31	}	}	PUNCT
ma-12	48	32	is	be	AUX
ma-12	48	33	a	a	DET
ma-12	48	34	sequence	sequence	NOUN
ma-12	48	35	in	in	ADP
ma-12	48	36	(	(	PUNCT
ma-12	48	37	0	0	NUM
ma-12	48	38	,	,	PUNCT
ma-12	48	39	1	1	NUM
ma-12	48	40	)	)	PUNCT
ma-12	48	41	,	,	PUNCT
ma-12	48	42	k	k	PROPN
ma-12	48	43	is	be	AUX
ma-12	48	44	a	a	DET
ma-12	48	45	nonempty	nonempty	ADV
ma-12	48	46	closed	close	VERB
ma-12	48	47	convex	convex	NOUN
ma-12	48	48	subset	subset	NOUN
ma-12	48	49	of	of	ADP
ma-12	48	50	of	of	ADP
ma-12	48	51	a	a	DET
ma-12	48	52	real	real	ADJ
ma-12	48	53	uniformlyconvex	uniformlyconvex	NOUN
ma-12	48	54	banach	banach	NOUN
ma-12	48	55	space	space	NOUN
ma-12	48	56	e	e	NOUN
ma-12	48	57	,	,	PUNCT
ma-12	48	58	p	p	PRON
ma-12	48	59	is	be	AUX
ma-12	48	60	a	a	DET
ma-12	48	61	nonexpansive	nonexpansive	ADJ
ma-12	48	62	retraction	retraction	NOUN
ma-12	48	63	of	of	ADP
ma-12	48	64	e	e	PROPN
ma-12	48	65	onto	onto	ADP
ma-12	48	66	k	k	PROPN
ma-12	48	67	,	,	PUNCT
ma-12	48	68	and	and	CCONJ
ma-12	48	69	proved	prove	VERB
ma-12	48	70	some	some	DET
ma-12	48	71	strong	strong	ADJ
ma-12	48	72	andweak	andweak	NOUN
ma-12	48	73	convergence	convergence	NOUN
ma-12	48	74	theorems	theorem	NOUN
ma-12	48	75	for	for	ADP
ma-12	48	76	asymptotically	asymptotically	ADV
ma-12	48	77	nonexpansive	nonexpansive	ADJ
ma-12	48	78	nonself	nonself	PROPN
ma-12	48	79	mappings	mapping	NOUN
ma-12	48	80	in	in	ADP
ma-12	48	81	the	the	DET
ma-12	48	82	intermediatesense	intermediatesense	NOUN
ma-12	48	83	in	in	ADP
ma-12	48	84	the	the	DET
ma-12	48	85	framework	framework	NOUN
ma-12	48	86	of	of	ADP
ma-12	48	87	uniformly	uniformly	ADV
ma-12	48	88	convex	convex	VERB
ma-12	48	89	banach	banach	NOUN
ma-12	48	90	spaces.ln	spaces.ln	NUM
ma-12	48	91	2006	2006	NUM
ma-12	48	92	,	,	PUNCT
ma-12	48	93	wang	wang	PROPN
ma-12	49	1	[	[	X
ma-12	49	2	17	17	NUM
ma-12	49	3	]	]	PUNCT
ma-12	49	4	generalised	generalise	VERB
ma-12	49	5	the	the	DET
ma-12	49	6	iteration	iteration	NOUN
ma-12	49	7	process	process	NOUN
ma-12	49	8	(	(	PUNCT
ma-12	49	9	1.4	1.4	NUM
ma-12	49	10	)	)	PUNCT
ma-12	49	11	as	as	SCONJ
ma-12	49	12	follows	follow	VERB
ma-12	49	13	:	:	PUNCT
ma-12	49	14	x1	x1	PROPN
ma-12	49	15	=	=	PUNCT
ma-12	49	16	x	x	SYM
ma-12	49	17	∈	∈	PROPN
ma-12	49	18	k	k	PROPN
ma-12	49	19	,	,	PUNCT
ma-12	49	20	xn+1	xn+1	PROPN
ma-12	50	1	=	=	SYM
ma-12	50	2	p	p	X
ma-12	50	3	(	(	PUNCT
ma-12	50	4	(	(	PUNCT
ma-12	50	5	1−	1−	NUM
ma-12	50	6	αn)xn	αn)xn	PROPN
ma-12	50	7	+	+	PUNCT
ma-12	50	8	αnt1(pt1	αnt1(pt1	ADJ
ma-12	50	9	)	)	PUNCT
ma-12	50	10	n−1yn	n−1yn	PROPN
ma-12	50	11	)	)	PUNCT
ma-12	50	12	,	,	PUNCT
ma-12	50	13	yn	yn	X
ma-12	51	1	=	=	PUNCT
ma-12	51	2	p	p	X
ma-12	51	3	(	(	PUNCT
ma-12	51	4	(	(	PUNCT
ma-12	51	5	1−	1−	NUM
ma-12	51	6	βn)xn	βn)xn	PUNCT
ma-12	51	7	+	+	CCONJ
ma-12	51	8	βnt2(pt2	βnt2(pt2	NOUN
ma-12	51	9	)	)	PUNCT
ma-12	51	10	n−1xn	n−1xn	NOUN
ma-12	51	11	)	)	PUNCT
ma-12	51	12	,	,	PUNCT
ma-12	51	13	n	n	PRON
ma-12	51	14	≥	≥	NOUN
ma-12	51	15	1	1	NUM
ma-12	51	16	,	,	PUNCT
ma-12	51	17	(	(	PUNCT
ma-12	51	18	1.5	1.5	NUM
ma-12	51	19	)	)	PUNCT
ma-12	51	20	eur	eur	PROPN
ma-12	51	21	.	.	PUNCT
ma-12	52	1	j.	j.	PROPN
ma-12	52	2	math	math	PROPN
ma-12	52	3	.	.	PUNCT
ma-12	53	1	anal	anal	ADJ
ma-12	53	2	.	.	PUNCT
ma-12	54	1	1	1	NUM
ma-12	54	2	(	(	PUNCT
ma-12	54	3	2021	2021	NUM
ma-12	54	4	)	)	PUNCT
ma-12	55	1	47where	47where	NUM
ma-12	55	2	t1	t1	NOUN
ma-12	55	3	,	,	PUNCT
ma-12	55	4	t2	t2	PROPN
ma-12	55	5	:	:	PUNCT
ma-12	55	6	k	k	X
ma-12	55	7	−→	−→	NOUN
ma-12	55	8	e	e	NOUN
ma-12	55	9	are	be	AUX
ma-12	55	10	two	two	NUM
ma-12	55	11	asymptotically	asymptotically	ADV
ma-12	55	12	nonexpansive	nonexpansive	ADJ
ma-12	55	13	nonself	nonself	PROPN
ma-12	55	14	mappings	mapping	NOUN
ma-12	55	15	,	,	PUNCT
ma-12	55	16	{	{	PUNCT
ma-12	55	17	αn	αn	NOUN
ma-12	55	18	}	}	PUNCT
ma-12	55	19	and	and	CCONJ
ma-12	55	20	{	{	PUNCT
ma-12	55	21	βn	βn	ADJ
ma-12	55	22	}	}	PUNCT
ma-12	55	23	arereal	arereal	ADJ
ma-12	55	24	sequences	sequence	NOUN
ma-12	55	25	in	in	ADP
ma-12	55	26	[	[	X
ma-12	55	27	0	0	NUM
ma-12	55	28	,	,	PUNCT
ma-12	55	29	1	1	NUM
ma-12	55	30	)	)	PUNCT
ma-12	55	31	,	,	PUNCT
ma-12	55	32	and	and	CCONJ
ma-12	55	33	proved	prove	VERB
ma-12	55	34	some	some	DET
ma-12	55	35	weak	weak	ADJ
ma-12	55	36	and	and	CCONJ
ma-12	55	37	strong	strong	ADJ
ma-12	55	38	convergence	convergence	NOUN
ma-12	55	39	theorems	theorem	NOUN
ma-12	55	40	for	for	ADP
ma-12	55	41	asymptoticallynonexpansive	asymptoticallynonexpansive	ADJ
ma-12	55	42	nonself	nonself	PRON
ma-12	55	43	mappings.ln	mappings.ln	PROPN
ma-12	55	44	2012	2012	NUM
ma-12	55	45	,	,	PUNCT
ma-12	55	46	guo	guo	PROPN
ma-12	55	47	et	et	PROPN
ma-12	55	48	al	al	PROPN
ma-12	56	1	[	[	X
ma-12	56	2	8	8	NUM
ma-12	56	3	]	]	PUNCT
ma-12	56	4	generalised	generalise	VERB
ma-12	56	5	the	the	DET
ma-12	56	6	iteration	iteration	NOUN
ma-12	56	7	process	process	NOUN
ma-12	56	8	(	(	PUNCT
ma-12	56	9	1.5	1.5	NUM
ma-12	56	10	)	)	PUNCT
ma-12	56	11	as	as	SCONJ
ma-12	56	12	follows	follow	VERB
ma-12	56	13	:	:	PUNCT
ma-12	56	14	x1	x1	PROPN
ma-12	56	15	=	=	PUNCT
ma-12	56	16	x	x	SYM
ma-12	56	17	∈	∈	PROPN
ma-12	56	18	k	k	PROPN
ma-12	56	19	,	,	PUNCT
ma-12	56	20	xn+1	xn+1	PROPN
ma-12	57	1	=	=	SYM
ma-12	57	2	p	p	X
ma-12	57	3	(	(	PUNCT
ma-12	57	4	(	(	PUNCT
ma-12	57	5	1−	1−	NUM
ma-12	57	6	αn)sn1xn	αn)sn1xn	NOUN
ma-12	57	7	+	+	CCONJ
ma-12	57	8	αnt1(pt1	αnt1(pt1	ADJ
ma-12	57	9	)	)	PUNCT
ma-12	57	10	n−1yn	n−1yn	PROPN
ma-12	57	11	)	)	PUNCT
ma-12	57	12	,	,	PUNCT
ma-12	57	13	yn	yn	X
ma-12	57	14	=	=	PUNCT
ma-12	57	15	p	p	X
ma-12	57	16	(	(	PUNCT
ma-12	57	17	(	(	PUNCT
ma-12	57	18	1−	1−	NUM
ma-12	57	19	βn)sn2xn	βn)sn2xn	X
ma-12	57	20	+	+	CCONJ
ma-12	57	21	βnt2(pt2	βnt2(pt2	NOUN
ma-12	57	22	)	)	PUNCT
ma-12	57	23	n−1xn	n−1xn	NOUN
ma-12	57	24	)	)	PUNCT
ma-12	57	25	,	,	PUNCT
ma-12	57	26	n	n	PRON
ma-12	57	27	≥	≥	NOUN
ma-12	57	28	1	1	NUM
ma-12	57	29	,	,	PUNCT
ma-12	57	30	(	(	PUNCT
ma-12	57	31	1.6	1.6	NUM
ma-12	57	32	)	)	PUNCT
ma-12	57	33	where	where	SCONJ
ma-12	57	34	t1	t1	NOUN
ma-12	57	35	,	,	PUNCT
ma-12	57	36	t2	t2	NOUN
ma-12	57	37	:	:	PUNCT
ma-12	57	38	k	k	X
ma-12	57	39	−→	−→	NOUN
ma-12	57	40	e	e	NOUN
ma-12	57	41	are	be	AUX
ma-12	57	42	two	two	NUM
ma-12	57	43	asymptotically	asymptotically	ADV
ma-12	57	44	nonexpansive	nonexpansive	ADJ
ma-12	57	45	nonself	nonself	PROPN
ma-12	57	46	mappings	mapping	NOUN
ma-12	57	47	,	,	PUNCT
ma-12	57	48	s1	s1	NOUN
ma-12	57	49	,	,	PUNCT
ma-12	57	50	s2	s2	NOUN
ma-12	57	51	:	:	PUNCT
ma-12	57	52	k	k	X
ma-12	57	53	−→	−→	NOUN
ma-12	57	54	eare	eare	VERB
ma-12	57	55	two	two	NUM
ma-12	57	56	asymptotically	asymptotically	ADV
ma-12	57	57	nonexpansive	nonexpansive	ADJ
ma-12	57	58	self	self	NOUN
ma-12	57	59	mappings	mapping	NOUN
ma-12	57	60	and	and	CCONJ
ma-12	57	61	{	{	PUNCT
ma-12	57	62	αn	αn	NOUN
ma-12	57	63	}	}	PUNCT
ma-12	57	64	,	,	PUNCT
ma-12	57	65	{	{	PUNCT
ma-12	57	66	βn	βn	NOUN
ma-12	57	67	}	}	PUNCT
ma-12	57	68	are	be	AUX
ma-12	57	69	real	real	ADJ
ma-12	57	70	sequences	sequence	NOUN
ma-12	57	71	in	in	ADP
ma-12	57	72	[	[	X
ma-12	57	73	0	0	NUM
ma-12	57	74	,	,	PUNCT
ma-12	57	75	1),and	1),and	PRON
ma-12	57	76	proved	prove	VERB
ma-12	57	77	some	some	DET
ma-12	57	78	strong	strong	ADJ
ma-12	57	79	and	and	CCONJ
ma-12	57	80	weak	weak	ADJ
ma-12	57	81	convergence	convergence	NOUN
ma-12	57	82	theorems	theorem	NOUN
ma-12	57	83	for	for	ADP
ma-12	57	84	mixed	mixed	ADJ
ma-12	57	85	-	-	PUNCT
ma-12	57	86	type	type	NOUN
ma-12	57	87	asymptotically	asymptotically	ADV
ma-12	57	88	nonexpan	nonexpan	NOUN
ma-12	57	89	-	-	PUNCT
ma-12	57	90	sive	sive	ADJ
ma-12	57	91	mappings	mapping	NOUN
ma-12	57	92	.	.	PUNCT
ma-12	58	1	hybrid	hybrid	ADJ
ma-12	58	2	mixed	mix	VERB
ma-12	58	3	-	-	PUNCT
ma-12	58	4	type	type	NOUN
ma-12	58	5	iteration	iteration	NOUN
ma-12	58	6	schemelet	schemelet	NOUN
ma-12	58	7	e	e	PRON
ma-12	58	8	be	be	AUX
ma-12	58	9	a	a	DET
ma-12	58	10	real	real	ADV
ma-12	58	11	uniformly	uniformly	ADV
ma-12	58	12	convex	convex	NOUN
ma-12	58	13	banach	banach	NOUN
ma-12	58	14	space	space	NOUN
ma-12	58	15	,	,	PUNCT
ma-12	58	16	k	k	PROPN
ma-12	58	17	a	a	DET
ma-12	58	18	nonempty	nonempty	ADV
ma-12	58	19	closed	close	VERB
ma-12	58	20	convex	convex	NOUN
ma-12	58	21	subset	subset	NOUN
ma-12	58	22	of	of	ADP
ma-12	58	23	e	e	PROPN
ma-12	58	24	and	and	CCONJ
ma-12	58	25	p	p	X
ma-12	58	26	:	:	PUNCT
ma-12	58	27	e	e	X
ma-12	58	28	−→	−→	NOUN
ma-12	58	29	k	k	PROPN
ma-12	58	30	a	a	DET
ma-12	58	31	nonexpansive	nonexpansive	ADJ
ma-12	58	32	retraction	retraction	NOUN
ma-12	58	33	of	of	ADP
ma-12	58	34	e	e	PROPN
ma-12	58	35	onto	onto	ADP
ma-12	58	36	k.	k.	PROPN
ma-12	58	37	let	let	VERB
ma-12	58	38	s1	s1	NOUN
ma-12	58	39	,	,	PUNCT
ma-12	58	40	s2	s2	PROPN
ma-12	58	41	,	,	PUNCT
ma-12	58	42	s3	s3	PROPN
ma-12	58	43	:	:	PUNCT
ma-12	58	44	k	k	PROPN
ma-12	58	45	−→	−→	NOUN
ma-12	58	46	k	k	PROPN
ma-12	58	47	be	be	AUX
ma-12	58	48	three	three	NUM
ma-12	58	49	totalasymptotically	totalasymptotically	ADV
ma-12	58	50	nonexpansive	nonexpansive	ADJ
ma-12	58	51	self	self	NOUN
ma-12	58	52	mappings	mapping	NOUN
ma-12	58	53	and	and	CCONJ
ma-12	58	54	t1	t1	NOUN
ma-12	58	55	,	,	PUNCT
ma-12	58	56	t2	t2	NOUN
ma-12	58	57	,	,	PUNCT
ma-12	58	58	t3	t3	NOUN
ma-12	58	59	:	:	PUNCT
ma-12	58	60	k	k	X
ma-12	58	61	−→	−→	NOUN
ma-12	58	62	e	e	NOUN
ma-12	58	63	be	be	VERB
ma-12	58	64	three	three	NUM
ma-12	58	65	total	total	ADJ
ma-12	58	66	asymptoti	asymptoti	NOUN
ma-12	58	67	-	-	PUNCT
ma-12	58	68	cally	cally	ADV
ma-12	58	69	nonexpansive	nonexpansive	ADJ
ma-12	58	70	nonself	nonself	PROPN
ma-12	58	71	mappings	mapping	NOUN
ma-12	58	72	.	.	PUNCT
ma-12	59	1	then	then	ADV
ma-12	59	2	,	,	PUNCT
ma-12	59	3	the	the	DET
ma-12	59	4	hybrid	hybrid	ADJ
ma-12	59	5	iteration	iteration	NOUN
ma-12	59	6	scheme	scheme	NOUN
ma-12	59	7	for	for	ADP
ma-12	59	8	the	the	DET
ma-12	59	9	above	above	ADJ
ma-12	59	10	mentionedmappings	mentionedmapping	NOUN
ma-12	59	11	is	be	AUX
ma-12	59	12	as	as	ADP
ma-12	59	13	follows:	follows:	NOUN
ma-12	59	14	x1	x1	NOUN
ma-12	59	15	=	=	PUNCT
ma-12	59	16	x	x	SYM
ma-12	59	17	∈	∈	PROPN
ma-12	59	18	k	k	NOUN
ma-12	59	19	;	;	PUNCT
ma-12	59	20	xn+1	xn+1	X
ma-12	59	21	=	=	SYM
ma-12	59	22	p	p	X
ma-12	59	23	(	(	PUNCT
ma-12	59	24	(	(	PUNCT
ma-12	59	25	1−	1−	NUM
ma-12	59	26	αn)sn1xn	αn)sn1xn	NOUN
ma-12	59	27	+	+	CCONJ
ma-12	59	28	αnt1(pt1	αnt1(pt1	ADJ
ma-12	59	29	)	)	PUNCT
ma-12	59	30	n−1yn	n−1yn	PROPN
ma-12	59	31	)	)	PUNCT
ma-12	59	32	;	;	PUNCT
ma-12	60	1	yn	yn	PROPN
ma-12	60	2	=	=	PUNCT
ma-12	60	3	p	p	X
ma-12	60	4	(	(	PUNCT
ma-12	60	5	(	(	PUNCT
ma-12	60	6	1−	1−	NUM
ma-12	60	7	βn)sn2xn	βn)sn2xn	X
ma-12	60	8	+	+	CCONJ
ma-12	60	9	βnt2(pt2	βnt2(pt2	NOUN
ma-12	60	10	)	)	PUNCT
ma-12	60	11	n−1zn	n−1zn	NOUN
ma-12	60	12	)	)	PUNCT
ma-12	60	13	;	;	PUNCT
ma-12	60	14	zn	zn	X
ma-12	60	15	=	=	SYM
ma-12	60	16	p	p	X
ma-12	60	17	(	(	PUNCT
ma-12	60	18	(	(	PUNCT
ma-12	60	19	1−	1−	NUM
ma-12	60	20	γn)sn3xn	γn)sn3xn	X
ma-12	60	21	+	+	CCONJ
ma-12	60	22	γnt3(pt3	γnt3(pt3	X
ma-12	60	23	)	)	PUNCT
ma-12	60	24	n−1xn	n−1xn	NOUN
ma-12	60	25	)	)	PUNCT
ma-12	60	26	,	,	PUNCT
ma-12	60	27	(	(	PUNCT
ma-12	60	28	1.7	1.7	NUM
ma-12	60	29	)	)	PUNCT
ma-12	60	30	where	where	SCONJ
ma-12	60	31	{	{	PUNCT
ma-12	60	32	αn	αn	NOUN
ma-12	60	33	}	}	PUNCT
ma-12	60	34	,	,	PUNCT
ma-12	60	35	{	{	PUNCT
ma-12	60	36	βn	βn	NOUN
ma-12	60	37	}	}	PUNCT
ma-12	60	38	,	,	PUNCT
ma-12	60	39	and	and	CCONJ
ma-12	60	40	{	{	PUNCT
ma-12	60	41	γn	γn	NOUN
ma-12	60	42	}	}	PUNCT
ma-12	60	43	are	be	AUX
ma-12	60	44	real	real	ADJ
ma-12	60	45	sequences	sequence	NOUN
ma-12	60	46	in	in	ADP
ma-12	60	47	[	[	X
ma-12	60	48	0	0	NUM
ma-12	60	49	,	,	PUNCT
ma-12	60	50	1).the	1).the	DET
ma-12	60	51	aim	aim	NOUN
ma-12	60	52	of	of	ADP
ma-12	60	53	this	this	DET
ma-12	60	54	paper	paper	NOUN
ma-12	60	55	is	be	AUX
ma-12	60	56	to	to	PART
ma-12	60	57	study	study	VERB
ma-12	60	58	this	this	DET
ma-12	60	59	new	new	ADJ
ma-12	60	60	hybrid	hybrid	ADJ
ma-12	60	61	mixed	mixed	ADJ
ma-12	60	62	-	-	PUNCT
ma-12	60	63	type	type	NOUN
ma-12	60	64	iteration	iteration	NOUN
ma-12	60	65	scheme	scheme	NOUN
ma-12	60	66	(	(	PUNCT
ma-12	60	67	1.7	1.7	NUM
ma-12	60	68	)	)	PUNCT
ma-12	60	69	,	,	PUNCT
ma-12	60	70	prove	prove	VERB
ma-12	60	71	demi	demi	NOUN
ma-12	60	72	-	-	PUNCT
ma-12	60	73	closedness	closedness	ADJ
ma-12	60	74	principle	principle	NOUN
ma-12	60	75	for	for	ADP
ma-12	60	76	total	total	ADJ
ma-12	60	77	asymptotically	asymptotically	ADV
ma-12	60	78	nonexpansive	nonexpansive	ADJ
ma-12	60	79	nonself	nonself	PROPN
ma-12	60	80	map	map	NOUN
ma-12	60	81	and	and	CCONJ
ma-12	60	82	establish	establish	VERB
ma-12	60	83	some	some	DET
ma-12	60	84	conver	conver	NOUN
ma-12	60	85	-	-	PUNCT
ma-12	60	86	gence	gence	NOUN
ma-12	60	87	theorems	theorem	NOUN
ma-12	60	88	for	for	ADP
ma-12	60	89	mixed	mixed	ADJ
ma-12	60	90	-	-	PUNCT
ma-12	60	91	type	type	NOUN
ma-12	60	92	mappings	mapping	NOUN
ma-12	60	93	in	in	ADP
ma-12	60	94	the	the	DET
ma-12	60	95	setting	setting	NOUN
ma-12	60	96	of	of	ADP
ma-12	60	97	uniformly	uniformly	ADV
ma-12	60	98	convex	convex	NOUN
ma-12	60	99	banach	banach	NOUN
ma-12	60	100	spaces	space	VERB
ma-12	60	101	.	.	PUNCT
ma-12	61	1	2	2	X
ma-12	61	2	.	.	X
ma-12	61	3	preliminary	preliminary	ADJ
ma-12	61	4	for	for	ADP
ma-12	61	5	the	the	DET
ma-12	61	6	sake	sake	NOUN
ma-12	61	7	of	of	ADP
ma-12	61	8	convenience	convenience	NOUN
ma-12	61	9	,	,	PUNCT
ma-12	61	10	we	we	PRON
ma-12	61	11	restate	restate	VERB
ma-12	61	12	the	the	DET
ma-12	61	13	following	follow	VERB
ma-12	61	14	concepts	concept	NOUN
ma-12	61	15	and	and	CCONJ
ma-12	61	16	results	result	NOUN
ma-12	61	17	:	:	PUNCT
ma-12	61	18	let	let	VERB
ma-12	61	19	e	e	PRON
ma-12	61	20	be	be	AUX
ma-12	61	21	a	a	DET
ma-12	61	22	banach	banach	NOUN
ma-12	61	23	space	space	NOUN
ma-12	61	24	with	with	ADP
ma-12	61	25	its	its	PRON
ma-12	61	26	dimension	dimension	NOUN
ma-12	61	27	greater	great	ADJ
ma-12	61	28	than	than	ADP
ma-12	61	29	or	or	CCONJ
ma-12	61	30	equal	equal	ADJ
ma-12	61	31	to	to	ADP
ma-12	61	32	2	2	NUM
ma-12	61	33	.	.	PUNCT
ma-12	62	1	the	the	DET
ma-12	62	2	modulus	modulus	NOUN
ma-12	62	3	of	of	ADP
ma-12	62	4	convexityof	convexityof	PROPN
ma-12	62	5	e	e	PROPN
ma-12	62	6	is	be	AUX
ma-12	62	7	a	a	DET
ma-12	62	8	function	function	NOUN
ma-12	62	9	δe(ε	δe(ε	NUM
ma-12	62	10	)	)	PUNCT
ma-12	62	11	:	:	PUNCT
ma-12	62	12	(	(	PUNCT
ma-12	62	13	0	0	NUM
ma-12	62	14	,	,	PUNCT
ma-12	62	15	2	2	NUM
ma-12	62	16	]	]	X
ma-12	62	17	−→	−→	NOUN
ma-12	62	18	(	(	PUNCT
ma-12	62	19	0	0	NUM
ma-12	62	20	,	,	PUNCT
ma-12	62	21	2	2	NUM
ma-12	62	22	]	]	PUNCT
ma-12	62	23	defined	define	VERB
ma-12	62	24	by	by	ADP
ma-12	62	25	δe(ε	δe(ε	NOUN
ma-12	62	26	)	)	PUNCT
ma-12	62	27	=	=	VERB
ma-12	63	1	inf{1−	inf{1−	PROPN
ma-12	63	2	‖	‖	PROPN
ma-12	63	3	1	1	NUM
ma-12	63	4	2	2	NUM
ma-12	63	5	(	(	PUNCT
ma-12	63	6	x	x	X
ma-12	64	1	+	+	PUNCT
ma-12	64	2	y)‖	y)‖	NOUN
ma-12	64	3	:	:	PUNCT
ma-12	64	4	‖x‖	‖x‖	X
ma-12	64	5	=	=	SYM
ma-12	64	6	1	1	NUM
ma-12	64	7	,	,	PUNCT
ma-12	64	8	‖y‖	‖y‖	PROPN
ma-12	64	9	=	=	SYM
ma-12	64	10	1	1	NUM
ma-12	64	11	,	,	PUNCT
ma-12	64	12	ε	ε	PROPN
ma-12	64	13	=	=	SYM
ma-12	64	14	‖x	‖x	PROPN
ma-12	65	1	−	−	PROPN
ma-12	65	2	y‖	y‖	PROPN
ma-12	65	3	}	}	PUNCT
ma-12	65	4	.	.	PUNCT
ma-12	66	1	eur	eur	PROPN
ma-12	66	2	.	.	PUNCT
ma-12	67	1	j.	j.	PROPN
ma-12	67	2	math	math	PROPN
ma-12	67	3	.	.	PUNCT
ma-12	68	1	anal	anal	ADJ
ma-12	68	2	.	.	PUNCT
ma-12	69	1	1	1	NUM
ma-12	69	2	(	(	PUNCT
ma-12	69	3	2021	2021	NUM
ma-12	69	4	)	)	PUNCT
ma-12	69	5	48a	48a	NOUN
ma-12	69	6	banach	banach	NOUN
ma-12	69	7	space	space	NOUN
ma-12	69	8	e	e	NOUN
ma-12	69	9	is	be	AUX
ma-12	69	10	uniformly	uniformly	ADV
ma-12	69	11	convex	convex	ADJ
ma-12	69	12	if	if	SCONJ
ma-12	69	13	and	and	CCONJ
ma-12	69	14	if	if	SCONJ
ma-12	69	15	δe(ε	δe(ε	NOUN
ma-12	69	16	)	)	PUNCT
ma-12	69	17	>	>	X
ma-12	69	18	0	0	NUM
ma-12	69	19	,	,	PUNCT
ma-12	69	20	for	for	ADP
ma-12	69	21	all	all	DET
ma-12	69	22	ε	ε	PROPN
ma-12	69	23	∈	∈	PROPN
ma-12	69	24	(	(	PUNCT
ma-12	69	25	0	0	NUM
ma-12	69	26	,	,	PUNCT
ma-12	69	27	2].we	2].we	NUM
ma-12	69	28	recall	recall	VERB
ma-12	69	29	the	the	DET
ma-12	69	30	following	following	NOUN
ma-12	69	31	:	:	PUNCT
ma-12	69	32	definition	definition	NOUN
ma-12	69	33	2.1	2.1	NUM
ma-12	69	34	.	.	PUNCT
ma-12	70	1	(	(	PUNCT
ma-12	70	2	see	see	VERB
ma-12	70	3	[	[	X
ma-12	70	4	19	19	NUM
ma-12	70	5	]	]	X
ma-12	70	6	:	:	PUNCT
ma-12	70	7	let	let	VERB
ma-12	70	8	%	%	NOUN
ma-12	70	9	=	=	PRON
ma-12	70	10	{	{	PUNCT
ma-12	70	11	x	x	SYM
ma-12	70	12	∈	∈	PROPN
ma-12	70	13	e	e	NOUN
ma-12	70	14	:	:	PUNCT
ma-12	70	15	‖x‖	‖x‖	VERB
ma-12	70	16	=	=	SYM
ma-12	70	17	1	1	X
ma-12	70	18	}	}	PUNCT
ma-12	70	19	and	and	CCONJ
ma-12	70	20	let	let	VERB
ma-12	70	21	e	e	X
ma-12	70	22	?	?	PUNCT
ma-12	70	23	be	be	AUX
ma-12	70	24	the	the	DET
ma-12	70	25	dual	dual	ADJ
ma-12	70	26	of	of	ADP
ma-12	70	27	e.	e.	PROPN
ma-12	70	28	the	the	DET
ma-12	70	29	space	space	NOUN
ma-12	70	30	e	e	NOUN
ma-12	70	31	has	have	VERB
ma-12	70	32	gateaux	gateaux	ADV
ma-12	70	33	differentiable	differentiable	ADJ
ma-12	70	34	norm	norm	NOUN
ma-12	70	35	if	if	SCONJ
ma-12	70	36	limn→∞	limn→∞	PROPN
ma-12	70	37	‖x+ty‖−‖x‖	‖x+ty‖−‖x‖	PROPN
ma-12	70	38	t	t	PROPN
ma-12	70	39	exists	exist	VERB
ma-12	70	40	∀x	∀x	NUM
ma-12	70	41	,	,	PUNCT
ma-12	70	42	y	y	PROPN
ma-12	70	43	∈	∈	PROPN
ma-12	70	44	%	%	NOUN
ma-12	70	45	.	.	PUNCT
ma-12	71	1	definition	definition	NOUN
ma-12	71	2	2.2	2.2	NUM
ma-12	71	3	.	.	PUNCT
ma-12	72	1	(	(	PUNCT
ma-12	72	2	see	see	VERB
ma-12	72	3	[	[	X
ma-12	72	4	19	19	NUM
ma-12	72	5	]	]	X
ma-12	72	6	:	:	PUNCT
ma-12	72	7	the	the	DET
ma-12	72	8	space	space	NOUN
ma-12	72	9	e	e	NOUN
ma-12	72	10	has	have	VERB
ma-12	72	11	frechet	frechet	VERB
ma-12	72	12	differentiable	differentiable	ADJ
ma-12	72	13	norm	norm	NOUN
ma-12	72	14	[	[	X
ma-12	72	15	15	15	NUM
ma-12	72	16	]	]	X
ma-12	72	17	if	if	SCONJ
ma-12	72	18	for	for	ADP
ma-12	72	19	each	each	DET
ma-12	72	20	x	x	SYM
ma-12	72	21	∈	∈	PROPN
ma-12	72	22	%	%	NOUN
ma-12	72	23	,	,	PUNCT
ma-12	72	24	the	the	DET
ma-12	72	25	limit	limit	NOUN
ma-12	72	26	of	of	ADP
ma-12	72	27	the	the	DET
ma-12	72	28	norm	norm	NOUN
ma-12	72	29	above	above	ADP
ma-12	72	30	exists	exist	VERB
ma-12	72	31	and	and	CCONJ
ma-12	72	32	is	be	AUX
ma-12	72	33	attained	attain	VERB
ma-12	72	34	uniformly	uniformly	ADV
ma-12	72	35	for	for	ADP
ma-12	72	36	all	all	DET
ma-12	72	37	y	y	PROPN
ma-12	72	38	∈	∈	PROPN
ma-12	72	39	%	%	NOUN
ma-12	72	40	,	,	PUNCT
ma-12	72	41	and	and	CCONJ
ma-12	72	42	in	in	ADP
ma-12	72	43	this	this	DET
ma-12	72	44	case	case	NOUN
ma-12	72	45	,	,	PUNCT
ma-12	72	46	it	it	PRON
ma-12	72	47	is	be	AUX
ma-12	72	48	also	also	ADV
ma-12	72	49	well	well	ADV
ma-12	72	50	known	know	VERB
ma-12	72	51	that	that	SCONJ
ma-12	72	52	〈	〈	PROPN
ma-12	72	53	h	h	NOUN
ma-12	72	54	,	,	PUNCT
ma-12	72	55	j(x)〉+	j(x)〉+	NUM
ma-12	72	56	1	1	NUM
ma-12	72	57	2	2	NUM
ma-12	72	58	‖x‖2	‖x‖2	VERB
ma-12	72	59	≤	≤	NUM
ma-12	72	60	1	1	NUM
ma-12	72	61	2	2	NUM
ma-12	72	62	‖x	‖x	NOUN
ma-12	72	63	+	+	CCONJ
ma-12	72	64	h‖2	h‖2	NOUN
ma-12	72	65	≤	≤	PUNCT
ma-12	72	66	〈	〈	PROPN
ma-12	72	67	h	h	NOUN
ma-12	72	68	,	,	PUNCT
ma-12	72	69	j(x)〉+	j(x)〉+	NUM
ma-12	72	70	1	1	NUM
ma-12	72	71	2	2	NUM
ma-12	72	72	‖x‖2	‖x‖2	VERB
ma-12	72	73	+	+	NUM
ma-12	72	74	b(‖x‖	b(‖x‖	NOUN
ma-12	72	75	)	)	PUNCT
ma-12	72	76	,	,	PUNCT
ma-12	72	77	(	(	PUNCT
ma-12	72	78	2.1	2.1	NUM
ma-12	72	79	)	)	PUNCT
ma-12	72	80	∀x	∀x	NUM
ma-12	72	81	,	,	PUNCT
ma-12	72	82	y	y	PROPN
ma-12	72	83	∈	∈	PROPN
ma-12	72	84	e	e	NOUN
ma-12	72	85	,	,	PUNCT
ma-12	72	86	where	where	SCONJ
ma-12	72	87	j	j	PROPN
ma-12	72	88	is	be	AUX
ma-12	72	89	the	the	DET
ma-12	72	90	frechet	frechet	PROPN
ma-12	72	91	derivative	derivative	NOUN
ma-12	72	92	of	of	ADP
ma-12	72	93	the	the	DET
ma-12	72	94	functional	functional	ADJ
ma-12	72	95	12‖	12‖	PROPN
ma-12	72	96	·	·	PUNCT
ma-12	72	97	|2	|2	NUM
ma-12	72	98	at	at	ADP
ma-12	72	99	x	x	PROPN
ma-12	72	100	∈	∈	PROPN
ma-12	72	101	e	e	NOUN
ma-12	72	102	,	,	PUNCT
ma-12	72	103	〈	〈	PROPN
ma-12	72	104	·	·	SYM
ma-12	72	105	〉	〉	PROPN
ma-12	72	106	is	be	AUX
ma-12	72	107	the	the	DET
ma-12	72	108	pairing	pairing	NOUN
ma-12	72	109	between	between	ADP
ma-12	72	110	e	e	PROPN
ma-12	72	111	and	and	CCONJ
ma-12	72	112	e	e	NOUN
ma-12	72	113	?	?	PUNCT
ma-12	73	1	and	and	CCONJ
ma-12	73	2	b	b	NOUN
ma-12	73	3	is	be	AUX
ma-12	73	4	an	an	DET
ma-12	73	5	increasing	increase	VERB
ma-12	73	6	function	function	NOUN
ma-12	73	7	defined	define	VERB
ma-12	73	8	on	on	ADP
ma-12	73	9	[	[	X
ma-12	73	10	0,∞	0,∞	NOUN
ma-12	73	11	)	)	PUNCT
ma-12	73	12	such	such	ADJ
ma-12	73	13	that	that	DET
ma-12	73	14	limt→∞	limt→∞	PROPN
ma-12	73	15	b(t	b(t	PROPN
ma-12	73	16	)	)	PUNCT
ma-12	73	17	t	t	NOUN
ma-12	74	1	=	=	SYM
ma-12	74	2	0	0	X
ma-12	74	3	.	.	PUNCT
ma-12	75	1	definition	definition	NOUN
ma-12	75	2	2.3	2.3	NUM
ma-12	75	3	.	.	PUNCT
ma-12	76	1	:	:	PUNCT
ma-12	76	2	the	the	DET
ma-12	76	3	space	space	NOUN
ma-12	76	4	e	e	NOUN
ma-12	76	5	has	have	VERB
ma-12	76	6	opial	opial	ADJ
ma-12	76	7	condition	condition	NOUN
ma-12	76	8	[	[	X
ma-12	76	9	10	10	NUM
ma-12	76	10	]	]	X
ma-12	76	11	if	if	SCONJ
ma-12	76	12	for	for	ADP
ma-12	76	13	any	any	DET
ma-12	76	14	sequence	sequence	NOUN
ma-12	76	15	{	{	PUNCT
ma-12	76	16	xn	xn	NOUN
ma-12	76	17	}	}	PUNCT
ma-12	76	18	in	in	ADP
ma-12	76	19	e	e	NOUN
ma-12	76	20	,	,	PUNCT
ma-12	76	21	xn	xn	PROPN
ma-12	76	22	converges	converge	VERB
ma-12	76	23	to	to	ADP
ma-12	76	24	x	x	PUNCT
ma-12	76	25	weakly	weakly	ADV
ma-12	76	26	,	,	PUNCT
ma-12	76	27	then	then	ADV
ma-12	76	28	it	it	PRON
ma-12	76	29	follows	follow	VERB
ma-12	76	30	that	that	SCONJ
ma-12	76	31	lim	lim	PROPN
ma-12	76	32	supn→∞	supn→∞	PROPN
ma-12	76	33	‖xn	‖xn	PROPN
ma-12	76	34	−	−	PROPN
ma-12	76	35	x‖	x‖	PROPN
ma-12	76	36	<	<	X
ma-12	76	37	lim	lim	PROPN
ma-12	76	38	supn→∞	supn→∞	PROPN
ma-12	76	39	‖xn	‖xn	PROPN
ma-12	77	1	−	−	PROPN
ma-12	77	2	y‖	y‖	PROPN
ma-12	77	3	for	for	ADP
ma-12	77	4	all	all	DET
ma-12	77	5	y	y	PROPN
ma-12	77	6	∈	∈	PROPN
ma-12	77	7	e	e	NOUN
ma-12	77	8	with	with	ADP
ma-12	77	9	x	x	PROPN
ma-12	77	10	6=	6=	PROPN
ma-12	77	11	y	y	PROPN
ma-12	77	12	.	.	PUNCT
ma-12	78	1	examples	example	NOUN
ma-12	78	2	of	of	ADP
ma-12	78	3	banach	banach	NOUN
ma-12	78	4	spaces	space	NOUN
ma-12	78	5	satisfying	satisfy	VERB
ma-12	78	6	opial	opial	ADJ
ma-12	78	7	conditions	condition	NOUN
ma-12	78	8	are	be	AUX
ma-12	78	9	hilbert	hilbert	NOUN
ma-12	78	10	spaces	space	NOUN
ma-12	78	11	and	and	CCONJ
ma-12	78	12	all	all	DET
ma-12	78	13	spaces	space	VERB
ma-12	78	14	lp(1	lp(1	X
ma-12	78	15	<	<	X
ma-12	78	16	p	p	X
ma-12	78	17	<	<	X
ma-12	78	18	∞	∞	NUM
ma-12	78	19	)	)	PUNCT
ma-12	78	20	.	.	PUNCT
ma-12	79	1	on	on	ADP
ma-12	79	2	the	the	DET
ma-12	79	3	other	other	ADJ
ma-12	79	4	hand	hand	NOUN
ma-12	79	5	,	,	PUNCT
ma-12	79	6	lp[0	lp[0	PROPN
ma-12	79	7	,	,	PUNCT
ma-12	79	8	π	π	NOUN
ma-12	79	9	]	]	X
ma-12	79	10	with	with	ADP
ma-12	79	11	1	1	NUM
ma-12	79	12	<	<	X
ma-12	79	13	p	p	X
ma-12	79	14	6=	6=	NUM
ma-12	79	15	2	2	NUM
ma-12	79	16	fails	fail	VERB
ma-12	79	17	to	to	PART
ma-12	79	18	satify	satify	VERB
ma-12	79	19	opial	opial	ADJ
ma-12	79	20	condition	condition	NOUN
ma-12	79	21	.	.	PUNCT
ma-12	80	1	definition	definition	NOUN
ma-12	80	2	2.4	2.4	NUM
ma-12	80	3	.	.	PUNCT
ma-12	81	1	:	:	PUNCT
ma-12	81	2	a	a	DET
ma-12	81	3	mapping	mapping	NOUN
ma-12	81	4	t	t	NOUN
ma-12	81	5	:	:	PUNCT
ma-12	82	1	k	k	X
ma-12	82	2	−→	−→	NOUN
ma-12	82	3	k	k	PROPN
ma-12	82	4	is	be	AUX
ma-12	82	5	said	say	VERB
ma-12	82	6	to	to	PART
ma-12	82	7	be	be	AUX
ma-12	82	8	demiclosed	demiclose	VERB
ma-12	82	9	at	at	ADP
ma-12	82	10	0	0	NUM
ma-12	82	11	,	,	PUNCT
ma-12	82	12	if	if	SCONJ
ma-12	82	13	for	for	ADP
ma-12	82	14	any	any	DET
ma-12	82	15	sequence	sequence	NOUN
ma-12	82	16	{	{	PUNCT
ma-12	82	17	xn	xn	NOUN
ma-12	82	18	}	}	PUNCT
ma-12	82	19	in	in	ADP
ma-12	82	20	k	k	PROPN
ma-12	82	21	,	,	PUNCT
ma-12	82	22	the	the	DET
ma-12	82	23	condition	condition	NOUN
ma-12	82	24	that	that	PRON
ma-12	82	25	xn	xn	PROPN
ma-12	82	26	converges	converge	VERB
ma-12	82	27	weakly	weakly	ADV
ma-12	82	28	to	to	ADP
ma-12	82	29	x	x	SYM
ma-12	82	30	∈	∈	PROPN
ma-12	82	31	k	k	NOUN
ma-12	82	32	and	and	CCONJ
ma-12	82	33	txn	txn	PROPN
ma-12	82	34	converges	converge	VERB
ma-12	82	35	strongly	strongly	ADV
ma-12	82	36	to	to	ADP
ma-12	82	37	0	0	NUM
ma-12	82	38	implies	imply	VERB
ma-12	82	39	tx	tx	PROPN
ma-12	82	40	=	=	SYM
ma-12	82	41	0	0	PROPN
ma-12	82	42	.	.	PUNCT
ma-12	83	1	definition	definition	NOUN
ma-12	83	2	2.5	2.5	NUM
ma-12	83	3	.	.	PUNCT
ma-12	84	1	:	:	PUNCT
ma-12	84	2	a	a	DET
ma-12	84	3	banach	banach	NOUN
ma-12	84	4	space	space	NOUN
ma-12	84	5	has	have	VERB
ma-12	84	6	the	the	DET
ma-12	84	7	kadec	kadec	NOUN
ma-12	84	8	-	-	PUNCT
ma-12	84	9	klec	klec	NOUN
ma-12	84	10	property	property	NOUN
ma-12	84	11	[	[	X
ma-12	84	12	14	14	NUM
ma-12	84	13	]	]	X
ma-12	84	14	if	if	SCONJ
ma-12	84	15	for	for	ADP
ma-12	84	16	every	every	DET
ma-12	84	17	sequence	sequence	NOUN
ma-12	84	18	xn	xn	PROPN
ma-12	84	19	in	in	ADP
ma-12	84	20	e	e	PROPN
ma-12	84	21	,	,	PUNCT
ma-12	84	22	xn	xn	PROPN
ma-12	85	1	→	→	SYM
ma-12	85	2	x	x	SYM
ma-12	85	3	weakly	weakly	ADJ
ma-12	85	4	and	and	CCONJ
ma-12	85	5	‖xn‖	‖xn‖	ADJ
ma-12	85	6	→	→	SYM
ma-12	85	7	‖x‖	‖x‖	PROPN
ma-12	85	8	,	,	PUNCT
ma-12	85	9	then	then	ADV
ma-12	85	10	it	it	PRON
ma-12	85	11	follows	follow	VERB
ma-12	85	12	that	that	SCONJ
ma-12	85	13	‖xn	‖xn	PROPN
ma-12	85	14	−	−	PROPN
ma-12	85	15	x‖	x‖	PROPN
ma-12	85	16	→	→	SYM
ma-12	85	17	0	0	PROPN
ma-12	85	18	.	.	PUNCT
ma-12	86	1	next	next	ADV
ma-12	86	2	,	,	PUNCT
ma-12	86	3	we	we	PRON
ma-12	86	4	state	state	VERB
ma-12	86	5	the	the	DET
ma-12	86	6	following	follow	VERB
ma-12	86	7	useful	useful	ADJ
ma-12	86	8	lemmas	lemma	NOUN
ma-12	86	9	which	which	PRON
ma-12	86	10	will	will	AUX
ma-12	86	11	be	be	AUX
ma-12	86	12	needed	need	VERB
ma-12	86	13	in	in	ADP
ma-12	86	14	order	order	NOUN
ma-12	86	15	to	to	PART
ma-12	86	16	prove	prove	VERB
ma-12	86	17	our	our	PRON
ma-12	86	18	mainresults	mainresult	NOUN
ma-12	86	19	.	.	PUNCT
ma-12	87	1	lemma	lemma	PROPN
ma-12	87	2	2.1	2.1	NUM
ma-12	87	3	.	.	PUNCT
ma-12	88	1	(	(	PUNCT
ma-12	88	2	see	see	VERB
ma-12	88	3	[	[	X
ma-12	88	4	16	16	NUM
ma-12	88	5	]	]	PUNCT
ma-12	88	6	):	):	PUNCT
ma-12	88	7	let	let	VERB
ma-12	88	8	{	{	PUNCT
ma-12	88	9	αn}∞n=1	αn}∞n=1	NUM
ma-12	88	10	,	,	PUNCT
ma-12	88	11	{	{	PUNCT
ma-12	88	12	βn}∞n=1	βn}∞n=1	PUNCT
ma-12	88	13	and	and	CCONJ
ma-12	88	14	{	{	PUNCT
ma-12	88	15	γn}∞n=1	γn}∞n=1	PUNCT
ma-12	88	16	be	be	AUX
ma-12	88	17	sequences	sequence	NOUN
ma-12	88	18	of	of	ADP
ma-12	88	19	nonnegative	nonnegative	ADJ
ma-12	88	20	numbers	number	NOUN
ma-12	88	21	satisfying	satisfy	VERB
ma-12	88	22	the	the	DET
ma-12	88	23	inequality	inequality	NOUN
ma-12	88	24	:	:	PUNCT
ma-12	88	25	αn+1	αn+1	NUM
ma-12	88	26	≤	≤	NUM
ma-12	88	27	(	(	PUNCT
ma-12	88	28	1	1	NUM
ma-12	88	29	+	+	CCONJ
ma-12	88	30	βn)αn	βn)αn	PUNCT
ma-12	89	1	+	+	CCONJ
ma-12	89	2	γn,∀n	γn,∀n	PROPN
ma-12	89	3	≥	≥	NOUN
ma-12	89	4	1	1	NUM
ma-12	89	5	.	.	PUNCT
ma-12	89	6	(	(	PUNCT
ma-12	89	7	2.2	2.2	NUM
ma-12	89	8	)	)	PUNCT
ma-12	89	9	if	if	SCONJ
ma-12	89	10	∑∞	∑∞	NOUN
ma-12	89	11	n=1	n=1	PRON
ma-12	89	12	βn	βn	VERB
ma-12	89	13	<	<	X
ma-12	89	14	∞	∞	NUM
ma-12	89	15	and	and	CCONJ
ma-12	89	16	∑∞	∑∞	NOUN
ma-12	89	17	n=1	n=1	PROPN
ma-12	89	18	γn	γn	ADP
ma-12	89	19	<	<	PROPN
ma-12	89	20	∞	∞	PROPN
ma-12	89	21	,	,	PUNCT
ma-12	89	22	then(1	then(1	PROPN
ma-12	89	23	)	)	PUNCT
ma-12	89	24	limn→∞	limn→∞	PROPN
ma-12	89	25	αn	αn	NOUN
ma-12	89	26	exists(2	exists(2	NOUN
ma-12	89	27	)	)	PUNCT
ma-12	89	28	ln	ln	ADV
ma-12	89	29	particular	particular	ADJ
ma-12	89	30	,	,	PUNCT
ma-12	89	31	if	if	SCONJ
ma-12	89	32	{	{	PUNCT
ma-12	89	33	αn}∞n=1	αn}∞n=1	NUM
ma-12	89	34	has	have	VERB
ma-12	89	35	a	a	DET
ma-12	89	36	subsequence	subsequence	NOUN
ma-12	89	37	which	which	PRON
ma-12	89	38	converges	converge	VERB
ma-12	89	39	strongly	strongly	ADV
ma-12	89	40	to	to	ADP
ma-12	89	41	0	0	NUM
ma-12	89	42	,	,	PUNCT
ma-12	89	43	then	then	ADV
ma-12	89	44	limn→∞	limn→∞	VERB
ma-12	89	45	αn	αn	NOUN
ma-12	89	46	=	=	SYM
ma-12	89	47	0	0	PROPN
ma-12	89	48	.	.	X
ma-12	89	49	eur	eur	PROPN
ma-12	89	50	.	.	PUNCT
ma-12	90	1	j.	j.	PROPN
ma-12	90	2	math	math	PROPN
ma-12	90	3	.	.	PUNCT
ma-12	91	1	anal	anal	ADJ
ma-12	91	2	.	.	PUNCT
ma-12	92	1	1	1	NUM
ma-12	92	2	(	(	PUNCT
ma-12	92	3	2021	2021	NUM
ma-12	92	4	)	)	PUNCT
ma-12	92	5	49	49	NUM
ma-12	93	1	lemma	lemma	PROPN
ma-12	93	2	2.2	2.2	NUM
ma-12	93	3	.	.	PUNCT
ma-12	94	1	(	(	PUNCT
ma-12	94	2	see	see	VERB
ma-12	94	3	[	[	X
ma-12	94	4	14	14	NUM
ma-12	94	5	]	]	PUNCT
ma-12	94	6	):	):	PUNCT
ma-12	94	7	let	let	VERB
ma-12	94	8	e	e	PRON
ma-12	94	9	be	be	AUX
ma-12	94	10	a	a	DET
ma-12	94	11	uniformly	uniformly	ADV
ma-12	94	12	convex	convex	NOUN
ma-12	94	13	banach	banach	NOUN
ma-12	94	14	space	space	NOUN
ma-12	94	15	and	and	CCONJ
ma-12	94	16	0	0	NUM
ma-12	94	17	<	<	X
ma-12	94	18	p	p	X
ma-12	94	19	≤	≤	NUM
ma-12	94	20	tn	tn	NOUN
ma-12	94	21	≤	≤	NOUN
ma-12	94	22	q	q	NOUN
ma-12	94	23	<	<	X
ma-12	94	24	1	1	NUM
ma-12	94	25	for	for	ADP
ma-12	94	26	each	each	DET
ma-12	94	27	n	n	PRON
ma-12	94	28	≥	≥	NOUN
ma-12	94	29	1	1	NUM
ma-12	94	30	.	.	PUNCT
ma-12	94	31	suppose	suppose	VERB
ma-12	94	32	that	that	SCONJ
ma-12	94	33	{	{	PUNCT
ma-12	94	34	xn	xn	X
ma-12	94	35	}	}	PUNCT
ma-12	94	36	and	and	CCONJ
ma-12	94	37	{	{	PUNCT
ma-12	94	38	yn	yn	NOUN
ma-12	94	39	}	}	PUNCT
ma-12	94	40	are	be	AUX
ma-12	94	41	sequences	sequence	NOUN
ma-12	94	42	in	in	ADP
ma-12	94	43	e	e	ADP
ma-12	94	44	such	such	ADJ
ma-12	94	45	that	that	SCONJ
ma-12	94	46	lim	lim	PROPN
ma-12	94	47	sup	sup	PROPN
ma-12	94	48	n→∞	n→∞	NUM
ma-12	94	49	‖xn‖	‖xn‖	ADJ
ma-12	94	50	≤	≤	ADJ
ma-12	94	51	r	r	NOUN
ma-12	94	52	,	,	PUNCT
ma-12	94	53	lim	lim	PROPN
ma-12	94	54	sup	sup	PROPN
ma-12	94	55	n→∞	n→∞	NUM
ma-12	94	56	‖yn‖	‖yn‖	PROPN
ma-12	94	57	≤	≤	PROPN
ma-12	94	58	r	r	NOUN
ma-12	95	1	and	and	CCONJ
ma-12	95	2	lim	lim	PROPN
ma-12	95	3	n→∞	n→∞	PROPN
ma-12	95	4	‖tnxn	‖tnxn	PROPN
ma-12	96	1	+	+	CCONJ
ma-12	96	2	(	(	PUNCT
ma-12	96	3	1−	1−	NUM
ma-12	96	4	tn)yn‖	tn)yn‖	NOUN
ma-12	96	5	=	=	SYM
ma-12	96	6	r	r	NOUN
ma-12	96	7	,	,	PUNCT
ma-12	96	8	(	(	PUNCT
ma-12	96	9	2.3	2.3	NUM
ma-12	96	10	)	)	PUNCT
ma-12	96	11	hold	hold	VERB
ma-12	96	12	for	for	ADP
ma-12	96	13	some	some	DET
ma-12	96	14	r	r	NOUN
ma-12	96	15	≥	≥	NOUN
ma-12	96	16	0	0	NUM
ma-12	96	17	.	.	PUNCT
ma-12	97	1	then	then	ADV
ma-12	97	2	limn→∞	limn→∞	PROPN
ma-12	97	3	‖xn	‖xn	PROPN
ma-12	97	4	−	−	PROPN
ma-12	97	5	yn‖	yn‖	NOUN
ma-12	97	6	=	=	SYM
ma-12	97	7	0	0	PROPN
ma-12	97	8	.	.	PUNCT
ma-12	98	1	lemma	lemma	PROPN
ma-12	98	2	2.3	2.3	NUM
ma-12	98	3	.	.	PUNCT
ma-12	99	1	(	(	PUNCT
ma-12	99	2	see	see	VERB
ma-12	99	3	[	[	X
ma-12	99	4	14	14	NUM
ma-12	99	5	]	]	PUNCT
ma-12	99	6	):	):	PUNCT
ma-12	99	7	let	let	VERB
ma-12	99	8	e	e	PRON
ma-12	99	9	be	be	AUX
ma-12	99	10	a	a	DET
ma-12	99	11	real	real	ADJ
ma-12	99	12	reflexive	reflexive	ADJ
ma-12	99	13	banach	banach	NOUN
ma-12	99	14	space	space	NOUN
ma-12	99	15	such	such	ADJ
ma-12	99	16	that	that	SCONJ
ma-12	99	17	its	its	PRON
ma-12	99	18	dual	dual	ADJ
ma-12	99	19	e	e	NOUN
ma-12	99	20	?	?	PROPN
ma-12	99	21	has	have	VERB
ma-12	99	22	the	the	DET
ma-12	99	23	kadec	kadec	NOUN
ma-12	99	24	-	-	PUNCT
ma-12	99	25	klec	klec	NOUN
ma-12	99	26	property	property	NOUN
ma-12	99	27	.	.	PUNCT
ma-12	100	1	let	let	VERB
ma-12	100	2	{	{	PUNCT
ma-12	100	3	xn	xn	VERB
ma-12	100	4	}	}	PUNCT
ma-12	100	5	be	be	AUX
ma-12	100	6	a	a	DET
ma-12	100	7	bounded	bounded	ADJ
ma-12	100	8	sequence	sequence	NOUN
ma-12	100	9	in	in	ADP
ma-12	100	10	e	e	PROPN
ma-12	100	11	and	and	CCONJ
ma-12	100	12	p	p	X
ma-12	100	13	,	,	PUNCT
ma-12	100	14	q	q	NOUN
ma-12	100	15	∈	∈	PROPN
ma-12	100	16	ωω(xn	ωω(xn	NUM
ma-12	100	17	)	)	PUNCT
ma-12	100	18	(	(	PUNCT
ma-12	100	19	where	where	SCONJ
ma-12	100	20	ωω(xn	ωω(xn	NUM
ma-12	100	21	)	)	PUNCT
ma-12	100	22	denotes	denote	VERB
ma-12	100	23	the	the	DET
ma-12	100	24	set	set	NOUN
ma-12	100	25	of	of	ADP
ma-12	100	26	all	all	DET
ma-12	100	27	weak	weak	ADJ
ma-12	100	28	subsequential	subsequential	ADJ
ma-12	100	29	limits	limit	NOUN
ma-12	100	30	of	of	ADP
ma-12	100	31	{	{	PUNCT
ma-12	100	32	xn	xn	NOUN
ma-12	100	33	}	}	PUNCT
ma-12	100	34	)	)	PUNCT
ma-12	100	35	.	.	PUNCT
ma-12	101	1	suppose	suppose	VERB
ma-12	101	2	limn→∞	limn→∞	PROPN
ma-12	101	3	‖txn	‖txn	PROPN
ma-12	101	4	+	+	X
ma-12	101	5	(	(	PUNCT
ma-12	101	6	1	1	NUM
ma-12	101	7	−	−	NOUN
ma-12	101	8	t)p	t)p	NOUN
ma-12	101	9	−	−	NOUN
ma-12	101	10	q‖	q‖	NOUN
ma-12	101	11	exists	exist	VERB
ma-12	101	12	for	for	ADP
ma-12	101	13	all	all	DET
ma-12	101	14	t	t	NOUN
ma-12	101	15	∈	∈	PROPN
ma-12	102	1	[	[	X
ma-12	102	2	0	0	NUM
ma-12	102	3	,	,	PUNCT
ma-12	102	4	1	1	NUM
ma-12	102	5	]	]	PUNCT
ma-12	102	6	.	.	PUNCT
ma-12	103	1	then	then	ADV
ma-12	103	2	,	,	PUNCT
ma-12	103	3	p	p	PROPN
ma-12	103	4	=	=	PROPN
ma-12	103	5	q.	q.	PROPN
ma-12	103	6	lemma	lemma	PROPN
ma-12	103	7	2.4	2.4	NUM
ma-12	103	8	.	.	PUNCT
ma-12	104	1	(	(	PUNCT
ma-12	104	2	see	see	VERB
ma-12	104	3	[	[	X
ma-12	104	4	14	14	NUM
ma-12	104	5	]	]	PUNCT
ma-12	104	6	):	):	PUNCT
ma-12	104	7	let	let	VERB
ma-12	104	8	k	k	PRON
ma-12	104	9	be	be	AUX
ma-12	104	10	a	a	DET
ma-12	104	11	nonempty	nonempty	ADJ
ma-12	104	12	convex	convex	NOUN
ma-12	104	13	subset	subset	NOUN
ma-12	104	14	of	of	ADP
ma-12	104	15	a	a	DET
ma-12	104	16	uniformly	uniformly	ADJ
ma-12	104	17	convex	convex	NOUN
ma-12	104	18	banach	banach	NOUN
ma-12	104	19	space	space	NOUN
ma-12	104	20	e.	e.	PROPN
ma-12	104	21	then	then	ADV
ma-12	104	22	,	,	PUNCT
ma-12	104	23	there	there	PRON
ma-12	104	24	exists	exist	VERB
ma-12	104	25	a	a	DET
ma-12	104	26	strictly	strictly	ADV
ma-12	104	27	incraesing	incraese	VERB
ma-12	104	28	continous	continous	ADJ
ma-12	104	29	convex	convex	NOUN
ma-12	104	30	function	function	NOUN
ma-12	104	31	φ	φ	NOUN
ma-12	104	32	:	:	PUNCT
ma-12	105	1	[	[	X
ma-12	105	2	0,∞)→	0,∞)→	NOUN
ma-12	105	3	[	[	X
ma-12	105	4	0,∞	0,∞	NOUN
ma-12	105	5	)	)	PUNCT
ma-12	105	6	with	with	ADP
ma-12	105	7	φ(0	φ(0	ADJ
ma-12	105	8	)	)	PUNCT
ma-12	105	9	=	=	SYM
ma-12	105	10	0	0	NUM
ma-12	105	11	such	such	ADJ
ma-12	105	12	that	that	PRON
ma-12	105	13	for	for	ADP
ma-12	105	14	each	each	DET
ma-12	105	15	lipshitizian	lipshitizian	PROPN
ma-12	105	16	mapping	mapping	NOUN
ma-12	105	17	t	t	NOUN
ma-12	105	18	:	:	PUNCT
ma-12	105	19	c	c	AUX
ma-12	105	20	−→	−→	NOUN
ma-12	105	21	c	c	PROPN
ma-12	105	22	with	with	ADP
ma-12	105	23	the	the	DET
ma-12	105	24	lipschiz	lipschiz	NOUN
ma-12	105	25	constant	constant	ADJ
ma-12	105	26	l	l	NOUN
ma-12	105	27	,	,	PUNCT
ma-12	105	28	‖tt	‖tt	PROPN
ma-12	105	29	x	x	SYM
ma-12	105	30	−	−	PROPN
ma-12	105	31	(	(	PUNCT
ma-12	105	32	1−	1−	NUM
ma-12	105	33	t)ty	t)ty	PROPN
ma-12	105	34	−	−	PROPN
ma-12	105	35	t	t	PROPN
ma-12	105	36	(	(	PUNCT
ma-12	105	37	tx	tx	PROPN
ma-12	105	38	−	−	PROPN
ma-12	105	39	(	(	PUNCT
ma-12	105	40	1−	1−	NUM
ma-12	105	41	t)y)‖	t)y)‖	PROPN
ma-12	105	42	≤	≤	ADV
ma-12	105	43	lφ−1(‖x	lφ−1(‖x	NUM
ma-12	105	44	−	−	PROPN
ma-12	105	45	y‖	y‖	NOUN
ma-12	105	46	−	−	NOUN
ma-12	105	47	1	1	NUM
ma-12	105	48	l	l	NOUN
ma-12	105	49	‖tx	‖tx	NUM
ma-12	105	50	−	−	PROPN
ma-12	105	51	ty‖	ty‖	PROPN
ma-12	105	52	)	)	PUNCT
ma-12	105	53	(	(	PUNCT
ma-12	105	54	2.4	2.4	NUM
ma-12	105	55	)	)	PUNCT
ma-12	105	56	for	for	ADP
ma-12	105	57	all	all	DET
ma-12	105	58	x	x	NOUN
ma-12	105	59	,	,	PUNCT
ma-12	105	60	y	y	PROPN
ma-12	105	61	∈	∈	PROPN
ma-12	105	62	k	k	PROPN
ma-12	105	63	and	and	CCONJ
ma-12	105	64	for	for	ADP
ma-12	105	65	all	all	DET
ma-12	105	66	t	t	NOUN
ma-12	105	67	∈	∈	PROPN
ma-12	106	1	[	[	X
ma-12	106	2	0.1	0.1	NUM
ma-12	106	3	]	]	PUNCT
ma-12	106	4	.	.	PUNCT
ma-12	107	1	lemma	lemma	PROPN
ma-12	107	2	2.5	2.5	NUM
ma-12	107	3	.	.	PUNCT
ma-12	108	1	(	(	PUNCT
ma-12	108	2	see	see	VERB
ma-12	108	3	[	[	X
ma-12	108	4	2	2	NUM
ma-12	108	5	]	]	PUNCT
ma-12	108	6	)	)	PUNCT
ma-12	108	7	let	let	VERB
ma-12	108	8	e	e	PRON
ma-12	108	9	be	be	AUX
ma-12	108	10	a	a	DET
ma-12	108	11	uniformly	uniformly	ADV
ma-12	108	12	convex	convex	NOUN
ma-12	108	13	banach	banach	NOUN
ma-12	108	14	space	space	NOUN
ma-12	108	15	,	,	PUNCT
ma-12	108	16	k	k	PROPN
ma-12	108	17	a	a	DET
ma-12	108	18	nonempty	nonempty	ADV
ma-12	108	19	bounded	bound	VERB
ma-12	108	20	close	close	ADJ
ma-12	108	21	convex	convex	NOUN
ma-12	108	22	subset	subset	NOUN
ma-12	108	23	of	of	ADP
ma-12	108	24	e.	e.	PROPN
ma-12	108	25	then	then	ADV
ma-12	108	26	,	,	PUNCT
ma-12	108	27	there	there	PRON
ma-12	108	28	exists	exist	VERB
ma-12	108	29	a	a	DET
ma-12	108	30	strictly	strictly	ADV
ma-12	108	31	increasing	increase	VERB
ma-12	108	32	continous	continous	ADJ
ma-12	108	33	convex	convex	NOUN
ma-12	108	34	function	function	NOUN
ma-12	108	35	φ	φ	NOUN
ma-12	108	36	:	:	PUNCT
ma-12	109	1	[	[	X
ma-12	109	2	0,∞	0,∞	X
ma-12	109	3	)	)	PUNCT
ma-12	109	4	−→	−→	NOUN
ma-12	109	5	[	[	X
ma-12	109	6	0,∞	0,∞	NOUN
ma-12	109	7	)	)	PUNCT
ma-12	109	8	with	with	ADP
ma-12	109	9	φ(0	φ(0	ADJ
ma-12	109	10	)	)	PUNCT
ma-12	109	11	=	=	SYM
ma-12	109	12	0	0	NUM
ma-12	109	13	such	such	ADJ
ma-12	109	14	that	that	PRON
ma-12	109	15	for	for	ADP
ma-12	109	16	any	any	DET
ma-12	109	17	lipschitizian	lipschitizian	ADJ
ma-12	109	18	mapping	mapping	NOUN
ma-12	109	19	t	t	NOUN
ma-12	109	20	:	:	PUNCT
ma-12	109	21	k	k	X
ma-12	109	22	−→	−→	NOUN
ma-12	109	23	e	e	NOUN
ma-12	109	24	with	with	ADP
ma-12	109	25	lipschitz	lipschitz	NOUN
ma-12	109	26	constant	constant	ADJ
ma-12	109	27	l	l	PROPN
ma-12	109	28	≥	≥	NUM
ma-12	109	29	1	1	NUM
ma-12	109	30	and	and	CCONJ
ma-12	109	31	elements	element	NOUN
ma-12	109	32	{	{	PUNCT
ma-12	109	33	xn}nj	xn}nj	SYM
ma-12	109	34	=	=	NOUN
ma-12	109	35	i	i	PROPN
ma-12	109	36	in	in	ADP
ma-12	109	37	k	k	PROPN
ma-12	109	38	and	and	CCONJ
ma-12	109	39	any	any	DET
ma-12	109	40	nonnegative	nonnegative	ADJ
ma-12	109	41	numbers	number	NOUN
ma-12	109	42	{	{	PUNCT
ma-12	109	43	tj}nj=1	tj}nj=1	PROPN
ma-12	109	44	with	with	ADP
ma-12	109	45	∑n	∑n	PROPN
ma-12	109	46	j=1	j=1	PROPN
ma-12	109	47	tj	tj	PROPN
ma-12	109	48	=	=	SYM
ma-12	109	49	1	1	NUM
ma-12	109	50	,	,	PUNCT
ma-12	109	51	the	the	DET
ma-12	109	52	following	follow	VERB
ma-12	109	53	inequality	inequality	NOUN
ma-12	109	54	holds	hold	VERB
ma-12	109	55	:	:	PUNCT
ma-12	109	56	‖t	‖t	NOUN
ma-12	109	57	(	(	PUNCT
ma-12	109	58	n∑	n∑	NOUN
ma-12	109	59	j=1	j=1	PROPN
ma-12	109	60	tjxj)−	tjxj)−	PROPN
ma-12	109	61	n∑	n∑	PROPN
ma-12	109	62	j=1	j=1	PROPN
ma-12	109	63	tjtxj‖	tjtxj‖	VERB
ma-12	109	64	≤	≤	NOUN
ma-12	109	65	lφ−1{max1≤j	lφ−1{max1≤j	NOUN
ma-12	109	66	,	,	PUNCT
ma-12	109	67	k≤n(‖xj	k≤n(‖xj	PROPN
ma-12	109	68	−	−	PROPN
ma-12	109	69	xk‖	xk‖	NUM
ma-12	110	1	−	−	PROPN
ma-12	110	2	l−1‖txj	l−1‖txj	PROPN
ma-12	110	3	−	−	PROPN
ma-12	110	4	txk‖	txk‖	NUM
ma-12	110	5	)	)	PUNCT
ma-12	111	1	}	}	PUNCT
ma-12	111	2	lemma	lemma	PROPN
ma-12	111	3	2.6	2.6	NUM
ma-12	111	4	.	.	PUNCT
ma-12	112	1	(	(	PUNCT
ma-12	112	2	see	see	VERB
ma-12	112	3	[	[	X
ma-12	112	4	21	21	NUM
ma-12	112	5	]	]	PUNCT
ma-12	112	6	)	)	PUNCT
ma-12	112	7	if	if	SCONJ
ma-12	112	8	the	the	DET
ma-12	112	9	sequence	sequence	NOUN
ma-12	112	10	{	{	PUNCT
ma-12	112	11	xn}∞n=1	xn}∞n=1	X
ma-12	112	12	converges	converge	VERB
ma-12	112	13	weakly	weakly	ADV
ma-12	112	14	to	to	ADP
ma-12	112	15	x	x	PRON
ma-12	112	16	,	,	PUNCT
ma-12	112	17	then	then	ADV
ma-12	112	18	there	there	PRON
ma-12	112	19	exists	exist	VERB
ma-12	112	20	a	a	DET
ma-12	112	21	sequence	sequence	NOUN
ma-12	112	22	of	of	ADP
ma-12	112	23	convex	convex	ADJ
ma-12	112	24	combination	combination	NOUN
ma-12	112	25	yj	yj	NOUN
ma-12	112	26	=	=	SYM
ma-12	112	27	∑n(j	∑n(j	PROPN
ma-12	112	28	)	)	PUNCT
ma-12	112	29	k=1	k=1	PUNCT
ma-12	113	1	λ	λ	INTJ
ma-12	113	2	(	(	PUNCT
ma-12	113	3	j	j	PROPN
ma-12	113	4	)	)	PUNCT
ma-12	114	1	k	k	PROPN
ma-12	114	2	xk+j	xk+j	PROPN
ma-12	114	3	,	,	PUNCT
ma-12	114	4	λ	λ	PROPN
ma-12	114	5	(	(	PUNCT
ma-12	114	6	j	j	PROPN
ma-12	114	7	)	)	PUNCT
ma-12	114	8	k	k	PROPN
ma-12	114	9	≥	≥	NOUN
ma-12	114	10	0	0	NUM
ma-12	114	11	and	and	CCONJ
ma-12	114	12	∑n(j	∑n(j	NUM
ma-12	114	13	)	)	PUNCT
ma-12	114	14	k=1	k=1	PUNCT
ma-12	115	1	λ	λ	INTJ
ma-12	115	2	(	(	PUNCT
ma-12	115	3	j	j	NOUN
ma-12	115	4	)	)	PUNCT
ma-12	115	5	=	=	SYM
ma-12	115	6	1	1	NUM
ma-12	115	7	,	,	PUNCT
ma-12	115	8	such	such	ADJ
ma-12	115	9	that	that	SCONJ
ma-12	115	10	‖yj	‖yj	PROPN
ma-12	115	11	−	−	PROPN
ma-12	115	12	x‖	x‖	PROPN
ma-12	115	13	→	→	SYM
ma-12	115	14	0	0	PROPN
ma-12	115	15	.	.	PUNCT
ma-12	116	1	as	as	ADP
ma-12	116	2	n	n	X
ma-12	116	3	→∞.	→∞.	PROPN
ma-12	116	4	3	3	X
ma-12	116	5	.	.	X
ma-12	116	6	main	main	ADJ
ma-12	116	7	results	result	NOUN
ma-12	116	8	lemma	lemma	PROPN
ma-12	116	9	3.1	3.1	NUM
ma-12	116	10	.	.	PUNCT
ma-12	117	1	(	(	PUNCT
ma-12	117	2	demiclosedness	demiclosedness	NOUN
ma-12	117	3	p	p	NOUN
ma-12	117	4	r	r	NOUN
ma-12	117	5	inciple	inciple	NOUN
ma-12	117	6	f	f	PROPN
ma-12	117	7	or	or	CCONJ
ma-12	117	8	nonself	nonself	PRON
ma-12	117	9	total	total	ADJ
ma-12	117	10	asymptotical	asymptotical	ADJ
ma-12	117	11	ly	ly	X
ma-12	117	12	nonexpansive	nonexpansive	ADJ
ma-12	117	13	maps	map	NOUN
ma-12	117	14	)	)	PUNCT
ma-12	117	15	let	let	VERB
ma-12	117	16	k	k	X
ma-12	117	17	be	be	AUX
ma-12	117	18	a	a	DET
ma-12	117	19	nonempty	nonempty	ADV
ma-12	117	20	closed	close	VERB
ma-12	117	21	convex	convex	NOUN
ma-12	117	22	and	and	CCONJ
ma-12	117	23	bounded	bound	VERB
ma-12	117	24	subset	subset	NOUN
ma-12	117	25	of	of	ADP
ma-12	117	26	a	a	DET
ma-12	117	27	uniformly	uniformly	ADJ
ma-12	117	28	convex	convex	NOUN
ma-12	117	29	banach	banach	NOUN
ma-12	117	30	space	space	NOUN
ma-12	117	31	e	e	NOUN
ma-12	117	32	and	and	CCONJ
ma-12	117	33	t	t	PROPN
ma-12	117	34	:	:	PUNCT
ma-12	118	1	k	k	X
ma-12	118	2	−→	−→	NOUN
ma-12	118	3	e	e	AUX
ma-12	118	4	be	be	VERB
ma-12	118	5	l	l	ADJ
ma-12	118	6	-	-	ADJ
ma-12	118	7	lipschitz	lipschitz	ADJ
ma-12	118	8	continuous	continuous	ADJ
ma-12	118	9	and	and	CCONJ
ma-12	118	10	total	total	ADJ
ma-12	118	11	asymptotically	asymptotically	ADV
ma-12	118	12	nonexpansive	nonexpansive	ADJ
ma-12	118	13	mapping	mapping	NOUN
ma-12	118	14	with	with	ADP
ma-12	118	15	the	the	DET
ma-12	118	16	function	function	NOUN
ma-12	118	17	φ	φ	NOUN
ma-12	118	18	:	:	PUNCT
ma-12	119	1	[	[	X
ma-12	119	2	0,∞	0,∞	X
ma-12	119	3	)	)	PUNCT
ma-12	119	4	−→	−→	NOUN
ma-12	119	5	[	[	X
ma-12	119	6	0,∞	0,∞	NUM
ma-12	119	7	)	)	PUNCT
ma-12	119	8	(	(	PUNCT
ma-12	119	9	such	such	ADJ
ma-12	119	10	that	that	DET
ma-12	119	11	φ(0	φ(0	ADJ
ma-12	119	12	)	)	PUNCT
ma-12	119	13	=	=	SYM
ma-12	119	14	0	0	X
ma-12	119	15	)	)	PUNCT
ma-12	119	16	and	and	CCONJ
ma-12	119	17	nonnegative	nonnegative	ADJ
ma-12	119	18	sequences	sequence	NOUN
ma-12	119	19	{	{	PUNCT
ma-12	119	20	k(1)n	k(1)n	X
ma-12	119	21	}	}	PUNCT
ma-12	119	22	,	,	PUNCT
ma-12	119	23	{	{	PUNCT
ma-12	119	24	k(2)n	k(2)n	X
ma-12	119	25	}	}	PUNCT
ma-12	119	26	such	such	ADJ
ma-12	119	27	that	that	SCONJ
ma-12	119	28	k(1)n	k(1)n	X
ma-12	119	29	,	,	PUNCT
ma-12	119	30	k	k	X
ma-12	119	31	(	(	PUNCT
ma-12	119	32	2	2	NUM
ma-12	119	33	)	)	PUNCT
ma-12	119	34	n	n	NOUN
ma-12	119	35	→	→	SYM
ma-12	119	36	0	0	NUM
ma-12	119	37	as	as	ADP
ma-12	119	38	n	n	X
ma-12	119	39	→∞.	→∞.	PROPN
ma-12	119	40	then	then	ADV
ma-12	119	41	,	,	PUNCT
ma-12	119	42	i	i	PRON
ma-12	119	43	−	−	PROPN
ma-12	119	44	t	t	PROPN
ma-12	119	45	is	be	AUX
ma-12	119	46	demiclosed	demiclose	VERB
ma-12	119	47	at	at	ADP
ma-12	119	48	0	0	NUM
ma-12	119	49	.	.	PUNCT
ma-12	120	1	proof	proof	NOUN
ma-12	120	2	.	.	PUNCT
ma-12	121	1	let	let	VERB
ma-12	121	2	{	{	PUNCT
ma-12	121	3	xn	xn	ADJ
ma-12	121	4	}	}	PUNCT
ma-12	121	5	converge	converge	VERB
ma-12	121	6	weakly	weakly	ADV
ma-12	121	7	to	to	ADP
ma-12	121	8	ω	ω	PROPN
ma-12	121	9	∈	∈	PROPN
ma-12	121	10	k	k	PROPN
ma-12	121	11	and	and	CCONJ
ma-12	121	12	{	{	PUNCT
ma-12	121	13	xn	xn	PROPN
ma-12	121	14	−	−	PROPN
ma-12	121	15	txn	txn	PROPN
ma-12	121	16	}	}	PUNCT
ma-12	121	17	converge	converge	VERB
ma-12	121	18	strongly	strongly	ADV
ma-12	121	19	to	to	ADP
ma-12	121	20	0	0	NUM
ma-12	121	21	.	.	PUNCT
ma-12	122	1	we	we	PRON
ma-12	122	2	prove	prove	VERB
ma-12	122	3	that	that	SCONJ
ma-12	122	4	(	(	PUNCT
ma-12	122	5	i	i	PRON
ma-12	122	6	−	−	PROPN
ma-12	122	7	t	t	NOUN
ma-12	122	8	)	)	PUNCT
ma-12	122	9	ω	ω	PROPN
ma-12	122	10	=	=	SYM
ma-12	122	11	0	0	X
ma-12	122	12	.	.	PUNCT
ma-12	123	1	clearly	clearly	ADV
ma-12	123	2	,	,	PUNCT
ma-12	123	3	{	{	PUNCT
ma-12	123	4	xn	xn	X
ma-12	123	5	}	}	PUNCT
ma-12	123	6	is	be	AUX
ma-12	123	7	bounded	bound	VERB
ma-12	123	8	.	.	PUNCT
ma-12	124	1	so	so	ADV
ma-12	124	2	,	,	PUNCT
ma-12	124	3	there	there	PRON
ma-12	124	4	exists	exist	VERB
ma-12	124	5	ρ	ρ	PROPN
ma-12	124	6	>	>	X
ma-12	124	7	0	0	NUM
ma-12	124	8	such	such	ADJ
ma-12	124	9	that	that	SCONJ
ma-12	124	10	{	{	PUNCT
ma-12	124	11	xn	xn	X
ma-12	124	12	}	}	PUNCT
ma-12	124	13	⊂	⊂	PROPN
ma-12	124	14	c	c	X
ma-12	124	15	=	=	SYM
ma-12	124	16	k	k	PROPN
ma-12	124	17	∩	∩	NOUN
ma-12	124	18	bρ(0),where	bρ(0),where	ADP
ma-12	124	19	bρ(0	bρ(0	PROPN
ma-12	124	20	)	)	PUNCT
ma-12	124	21	is	be	AUX
ma-12	124	22	a	a	DET
ma-12	124	23	closed	closed	ADJ
ma-12	124	24	ball	ball	NOUN
ma-12	124	25	in	in	ADP
ma-12	124	26	e	e	PROPN
ma-12	124	27	with	with	ADP
ma-12	124	28	centre	centre	NOUN
ma-12	124	29	0	0	PUNCT
ma-12	124	30	and	and	CCONJ
ma-12	124	31	radius	radius	PROPN
ma-12	124	32	ρ	ρ	PROPN
ma-12	124	33	.	.	PUNCT
ma-12	125	1	thus	thus	ADV
ma-12	125	2	,	,	PUNCT
ma-12	125	3	c	c	PROPN
ma-12	125	4	is	be	AUX
ma-12	125	5	nonempty	nonempty	ADJ
ma-12	125	6	,	,	PUNCT
ma-12	125	7	closed	closed	ADJ
ma-12	125	8	,	,	PUNCT
ma-12	125	9	eur	eur	PROPN
ma-12	125	10	.	.	PUNCT
ma-12	126	1	j.	j.	PROPN
ma-12	126	2	math	math	PROPN
ma-12	126	3	.	.	PUNCT
ma-12	127	1	anal	anal	ADJ
ma-12	127	2	.	.	PUNCT
ma-12	128	1	1	1	NUM
ma-12	128	2	(	(	PUNCT
ma-12	128	3	2021	2021	NUM
ma-12	128	4	)	)	PUNCT
ma-12	128	5	50bounded	50bounded	NUM
ma-12	128	6	and	and	CCONJ
ma-12	128	7	convex	convex	PROPN
ma-12	128	8	subset	subset	VERB
ma-12	128	9	in	in	ADP
ma-12	128	10	k.claim	k.claim	NOUN
ma-12	128	11	:	:	PUNCT
ma-12	128	12	t	t	PROPN
ma-12	128	13	(	(	PUNCT
ma-12	128	14	pt	pt	INTJ
ma-12	128	15	)	)	PUNCT
ma-12	128	16	n−1ω	n−1ω	NOUN
ma-12	128	17	→	→	SYM
ma-12	128	18	ω	ω	PROPN
ma-12	128	19	as	as	ADP
ma-12	128	20	n	n	PROPN
ma-12	128	21	→	→	SYM
ma-12	128	22	∞.	∞.	PROPN
ma-12	128	23	in	in	ADP
ma-12	128	24	fact	fact	NOUN
ma-12	128	25	,	,	PUNCT
ma-12	128	26	since	since	SCONJ
ma-12	128	27	{	{	PUNCT
ma-12	128	28	xn	xn	NOUN
ma-12	128	29	}	}	PUNCT
ma-12	128	30	converges	converge	VERB
ma-12	128	31	weakly	weakly	ADV
ma-12	128	32	to	to	ADP
ma-12	128	33	ω	ω	NUM
ma-12	128	34	,	,	PUNCT
ma-12	128	35	by	by	ADP
ma-12	128	36	lemma6(see	lemma6(see	PROPN
ma-12	129	1	[	[	X
ma-12	129	2	21	21	NUM
ma-12	129	3	]	]	PUNCT
ma-12	129	4	)	)	PUNCT
ma-12	129	5	,	,	PUNCT
ma-12	129	6	we	we	PRON
ma-12	129	7	have	have	VERB
ma-12	129	8	for	for	ADP
ma-12	129	9	all	all	DET
ma-12	129	10	n	n	SYM
ma-12	129	11	>	>	X
ma-12	129	12	1	1	NUM
ma-12	129	13	,	,	PUNCT
ma-12	129	14	there	there	PRON
ma-12	129	15	exists	exist	VERB
ma-12	129	16	a	a	DET
ma-12	129	17	convex	convex	NOUN
ma-12	129	18	combination	combination	NOUN
ma-12	129	19	yn	yn	X
ma-12	129	20	=	=	PUNCT
ma-12	129	21	m(n)∑	m(n)∑	PUNCT
ma-12	129	22	i=1	i=1	PROPN
ma-12	129	23	t	t	PROPN
ma-12	129	24	(	(	PUNCT
ma-12	129	25	n	n	CCONJ
ma-12	129	26	)	)	PUNCT
ma-12	129	27	i	i	PRON
ma-12	129	28	xi+n	xi+n	PROPN
ma-12	129	29	,	,	PUNCT
ma-12	129	30	t	t	PROPN
ma-12	129	31	(	(	PUNCT
ma-12	129	32	n	n	CCONJ
ma-12	129	33	)	)	PUNCT
ma-12	129	34	i	i	PRON
ma-12	129	35	≥	≥	VERB
ma-12	129	36	0	0	NUM
ma-12	129	37	and	and	CCONJ
ma-12	129	38	m(n)∑	m(n)∑	PUNCT
ma-12	129	39	i=1	i=1	PROPN
ma-12	129	40	t	t	PROPN
ma-12	129	41	(	(	PUNCT
ma-12	129	42	n	n	CCONJ
ma-12	129	43	)	)	PUNCT
ma-12	129	44	i	i	PRON
ma-12	129	45	=	=	NOUN
ma-12	130	1	1	1	NUM
ma-12	130	2	such	such	ADJ
ma-12	130	3	that	that	SCONJ
ma-12	130	4	‖yn	‖yn	PROPN
ma-12	130	5	−	−	NOUN
ma-12	130	6	ω‖	ω‖	NOUN
ma-12	130	7	→	→	SYM
ma-12	130	8	0	0	NUM
ma-12	130	9	as	as	ADP
ma-12	130	10	n	n	X
ma-12	130	11	→∞.	→∞.	X
ma-12	130	12	(	(	PUNCT
ma-12	130	13	3.1	3.1	NUM
ma-12	130	14	)	)	PUNCT
ma-12	130	15	also	also	ADV
ma-12	130	16	,	,	PUNCT
ma-12	130	17	since	since	SCONJ
ma-12	130	18	{	{	PUNCT
ma-12	130	19	xn−txn	xn−txn	PROPN
ma-12	130	20	}	}	PUNCT
ma-12	130	21	converges	converge	VERB
ma-12	130	22	to	to	ADP
ma-12	130	23	0	0	NUM
ma-12	130	24	,	,	PUNCT
ma-12	130	25	then	then	ADV
ma-12	130	26	for	for	ADP
ma-12	130	27	any	any	DET
ma-12	130	28	ε	ε	PROPN
ma-12	130	29	>	>	X
ma-12	130	30	0	0	PROPN
ma-12	130	31	and	and	CCONJ
ma-12	130	32	a	a	DET
ma-12	130	33	positive	positive	ADJ
ma-12	130	34	integer	integer	NOUN
ma-12	130	35	m	m	VERB
ma-12	130	36	≥	≥	NOUN
ma-12	130	37	1	1	NUM
ma-12	130	38	,	,	PUNCT
ma-12	130	39	there	there	PRON
ma-12	130	40	exists	exist	VERB
ma-12	130	41	n1	n1	PROPN
ma-12	130	42	=	=	SYM
ma-12	130	43	n(ε	n(ε	NOUN
ma-12	130	44	)	)	PUNCT
ma-12	130	45	>	>	X
ma-12	130	46	0	0	NUM
ma-12	130	47	such	such	ADJ
ma-12	130	48	that	that	DET
ma-12	130	49	‖(i	‖(i	NOUN
ma-12	130	50	−	−	PROPN
ma-12	130	51	t	t	NOUN
ma-12	130	52	)	)	PUNCT
ma-12	130	53	xn‖	xn‖	PROPN
ma-12	130	54	<	<	X
ma-12	130	55	ε	ε	PROPN
ma-12	130	56	1	1	NUM
ma-12	130	57	+	+	NOUN
ma-12	130	58	m	m	NOUN
ma-12	130	59	,	,	PUNCT
ma-12	130	60	∀n	∀n	NUM
ma-12	130	61	≥	≥	NOUN
ma-12	130	62	n1	n1	NOUN
ma-12	130	63	.	.	PUNCT
ma-12	131	1	(	(	PUNCT
ma-12	131	2	3.2	3.2	NUM
ma-12	131	3	)	)	PUNCT
ma-12	131	4	hence	hence	ADV
ma-12	131	5	,	,	PUNCT
ma-12	131	6	∀n	∀n	NUM
ma-12	131	7	≥	≥	NOUN
ma-12	131	8	n1	n1	NOUN
ma-12	131	9	,	,	PUNCT
ma-12	131	10	using	use	VERB
ma-12	131	11	definition	definition	NOUN
ma-12	131	12	1.4	1.4	NUM
ma-12	131	13	and	and	CCONJ
ma-12	131	14	the	the	DET
ma-12	131	15	fact	fact	NOUN
ma-12	131	16	that	that	SCONJ
ma-12	131	17	p	p	NOUN
ma-12	131	18	is	be	AUX
ma-12	131	19	nonexpansive	nonexpansive	ADJ
ma-12	131	20	,	,	PUNCT
ma-12	131	21	we	we	PRON
ma-12	131	22	have	have	VERB
ma-12	131	23	the	the	DET
ma-12	131	24	followingestimates	followingestimate	NOUN
ma-12	131	25	:	:	PUNCT
ma-12	131	26	for	for	ADP
ma-12	131	27	arbitrary	arbitrary	ADJ
ma-12	131	28	but	but	CCONJ
ma-12	131	29	fixed	fix	VERB
ma-12	131	30	j	j	PROPN
ma-12	131	31	≥	≥	NUM
ma-12	131	32	1	1	NUM
ma-12	131	33	,	,	PUNCT
ma-12	131	34	we	we	PRON
ma-12	131	35	have	have	VERB
ma-12	131	36	‖xn	‖xn	PROPN
ma-12	131	37	−	−	PROPN
ma-12	131	38	t	t	PROPN
ma-12	131	39	(	(	PUNCT
ma-12	131	40	pt	pt	PROPN
ma-12	131	41	)	)	PUNCT
ma-12	131	42	(	(	PUNCT
ma-12	131	43	j−1)xn‖	j−1)xn‖	NOUN
ma-12	131	44	≤	≤	NUM
ma-12	131	45	‖(i	‖(i	NOUN
ma-12	131	46	−	−	PROPN
ma-12	131	47	t	t	NOUN
ma-12	131	48	)	)	PUNCT
ma-12	131	49	xn‖+	xn‖+	PROPN
ma-12	132	1	‖(t	‖(t	PRON
ma-12	133	1	−	−	PROPN
ma-12	133	2	t	t	PROPN
ma-12	133	3	(	(	PUNCT
ma-12	133	4	pt	pt	PROPN
ma-12	133	5	)	)	PUNCT
ma-12	133	6	)	)	PUNCT
ma-12	133	7	xn‖	xn‖	PROPN
ma-12	134	1	+	+	PUNCT
ma-12	134	2	‖(t	‖(t	PRON
ma-12	134	3	(	(	PUNCT
ma-12	134	4	pt	pt	NOUN
ma-12	134	5	)	)	PUNCT
ma-12	134	6	−	−	PROPN
ma-12	134	7	t	t	PROPN
ma-12	134	8	(	(	PUNCT
ma-12	134	9	pt	pt	INTJ
ma-12	134	10	)	)	PUNCT
ma-12	134	11	2)xn‖	2)xn‖	NOUN
ma-12	135	1	+	+	ADV
ma-12	135	2	‖(t	‖(t	NUM
ma-12	135	3	(	(	PUNCT
ma-12	135	4	pt	pt	NOUN
ma-12	135	5	)	)	PUNCT
ma-12	135	6	2	2	NUM
ma-12	135	7	−	−	PROPN
ma-12	135	8	t	t	NOUN
ma-12	135	9	(	(	PUNCT
ma-12	135	10	pt	pt	INTJ
ma-12	135	11	)	)	PUNCT
ma-12	135	12	3)xn‖	3)xn‖	PROPN
ma-12	136	1	+	+	X
ma-12	136	2	·	·	PUNCT
ma-12	136	3	·	·	PUNCT
ma-12	136	4	·	·	PUNCT
ma-12	136	5	+	+	PUNCT
ma-12	136	6	‖(t	‖(t	NUM
ma-12	136	7	(	(	PUNCT
ma-12	136	8	pt	pt	NOUN
ma-12	136	9	)	)	PUNCT
ma-12	136	10	j−2	j−2	PROPN
ma-12	136	11	−	−	PROPN
ma-12	136	12	t	t	PROPN
ma-12	136	13	(	(	PUNCT
ma-12	136	14	pt	pt	INTJ
ma-12	136	15	)	)	PUNCT
ma-12	136	16	j−1))xn‖	j−1))xn‖	NOUN
ma-12	136	17	≤	≤	NUM
ma-12	136	18	‖(i	‖(i	NOUN
ma-12	136	19	−	−	PROPN
ma-12	136	20	t	t	NOUN
ma-12	136	21	)	)	PUNCT
ma-12	136	22	xn‖+	xn‖+	PROPN
ma-12	136	23	(	(	PUNCT
ma-12	136	24	‖(i	‖(i	NOUN
ma-12	136	25	−	−	PROPN
ma-12	136	26	t	t	NOUN
ma-12	136	27	)	)	PUNCT
ma-12	136	28	xn‖+	xn‖+	PROPN
ma-12	137	1	µ	µ	X
ma-12	137	2	(	(	PUNCT
ma-12	137	3	1	1	NUM
ma-12	137	4	)	)	PUNCT
ma-12	137	5	n	n	PRON
ma-12	137	6	φ(‖(i	φ(‖(i	NOUN
ma-12	137	7	−	−	PROPN
ma-12	137	8	t	t	NOUN
ma-12	137	9	)	)	PUNCT
ma-12	137	10	xn‖	xn‖	PROPN
ma-12	137	11	)	)	PUNCT
ma-12	138	1	+	+	SYM
ma-12	138	2	ξ	ξ	X
ma-12	138	3	(	(	PUNCT
ma-12	138	4	1	1	NUM
ma-12	138	5	)	)	PUNCT
ma-12	138	6	n	n	CCONJ
ma-12	138	7	)	)	PUNCT
ma-12	139	1	+	+	CCONJ
ma-12	139	2	(	(	PUNCT
ma-12	139	3	‖(i	‖(i	NOUN
ma-12	139	4	−	−	PROPN
ma-12	139	5	t	t	NOUN
ma-12	139	6	)	)	PUNCT
ma-12	139	7	xn‖+	xn‖+	PROPN
ma-12	140	1	µ	µ	X
ma-12	140	2	(	(	PUNCT
ma-12	140	3	2	2	NUM
ma-12	140	4	)	)	PUNCT
ma-12	140	5	n	n	PRON
ma-12	140	6	φ(‖(i	φ(‖(i	NOUN
ma-12	140	7	−	−	PROPN
ma-12	140	8	t	t	NOUN
ma-12	140	9	)	)	PUNCT
ma-12	140	10	xn‖	xn‖	PROPN
ma-12	140	11	)	)	PUNCT
ma-12	141	1	+	+	CCONJ
ma-12	142	1	ξ	ξ	X
ma-12	142	2	(	(	PUNCT
ma-12	142	3	2	2	NUM
ma-12	142	4	)	)	PUNCT
ma-12	142	5	n	n	CCONJ
ma-12	142	6	)	)	PUNCT
ma-12	143	1	+	+	ADJ
ma-12	143	2	(	(	PUNCT
ma-12	143	3	‖(i	‖(i	NOUN
ma-12	143	4	−	−	PROPN
ma-12	143	5	t	t	NOUN
ma-12	143	6	)	)	PUNCT
ma-12	143	7	xn‖+	xn‖+	PROPN
ma-12	144	1	µ	µ	X
ma-12	144	2	(	(	PUNCT
ma-12	144	3	3	3	NUM
ma-12	144	4	)	)	PUNCT
ma-12	144	5	n	n	PRON
ma-12	144	6	φ(‖(i	φ(‖(i	NOUN
ma-12	144	7	−	−	PROPN
ma-12	144	8	t	t	NOUN
ma-12	144	9	)	)	PUNCT
ma-12	144	10	xn‖	xn‖	PROPN
ma-12	144	11	)	)	PUNCT
ma-12	145	1	+	+	CCONJ
ma-12	146	1	ξ	ξ	X
ma-12	146	2	(	(	PUNCT
ma-12	146	3	3	3	NUM
ma-12	146	4	)	)	PUNCT
ma-12	146	5	n	n	CCONJ
ma-12	146	6	)	)	PUNCT
ma-12	146	7	+	+	CCONJ
ma-12	146	8	·	·	PUNCT
ma-12	146	9	·	·	PUNCT
ma-12	146	10	·	·	PUNCT
ma-12	146	11	+	+	CCONJ
ma-12	146	12	(	(	PUNCT
ma-12	146	13	‖(i	‖(i	NOUN
ma-12	146	14	−	−	PROPN
ma-12	146	15	t	t	NOUN
ma-12	146	16	)	)	PUNCT
ma-12	146	17	xn‖+	xn‖+	PROPN
ma-12	147	1	µ	µ	X
ma-12	147	2	(	(	PUNCT
ma-12	147	3	j−1	j−1	PROPN
ma-12	147	4	)	)	PUNCT
ma-12	147	5	n	n	NUM
ma-12	147	6	φ(‖(i	φ(‖(i	NOUN
ma-12	147	7	−	−	PROPN
ma-12	147	8	t	t	NOUN
ma-12	147	9	)	)	PUNCT
ma-12	147	10	xn‖	xn‖	PROPN
ma-12	147	11	)	)	PUNCT
ma-12	148	1	+	+	NUM
ma-12	149	1	ξ	ξ	X
ma-12	149	2	(	(	PUNCT
ma-12	149	3	j−1	j−1	PROPN
ma-12	149	4	)	)	PUNCT
ma-12	149	5	n	n	NOUN
ma-12	149	6	)	)	PUNCT
ma-12	149	7	=	=	PUNCT
ma-12	149	8	‖(i	‖(i	NOUN
ma-12	149	9	−	−	PROPN
ma-12	149	10	t	t	NOUN
ma-12	149	11	)	)	PUNCT
ma-12	149	12	xn‖+	xn‖+	PROPN
ma-12	150	1	m−1∑	m−1∑	PRON
ma-12	150	2	j=1	j=1	PROPN
ma-12	150	3	‖(i	‖(i	NOUN
ma-12	150	4	−	−	PROPN
ma-12	150	5	t	t	NOUN
ma-12	150	6	)	)	PUNCT
ma-12	150	7	xn‖+	xn‖+	PROPN
ma-12	150	8	m−1∑	m−1∑	PRON
ma-12	150	9	j=1	j=1	PROPN
ma-12	150	10	µ	µ	X
ma-12	150	11	(	(	PUNCT
ma-12	150	12	j	j	NOUN
ma-12	150	13	)	)	PUNCT
ma-12	150	14	n	n	PRON
ma-12	150	15	φ(‖(i	φ(‖(i	NOUN
ma-12	150	16	−	−	PROPN
ma-12	150	17	t	t	NOUN
ma-12	150	18	)	)	PUNCT
ma-12	150	19	xn‖	xn‖	PROPN
ma-12	150	20	)	)	PUNCT
ma-12	151	1	+	+	CCONJ
ma-12	152	1	m−1∑	m−1∑	NUM
ma-12	152	2	j=1	j=1	PROPN
ma-12	152	3	ξ	ξ	X
ma-12	152	4	(	(	PUNCT
ma-12	152	5	j	j	NOUN
ma-12	152	6	)	)	PUNCT
ma-12	152	7	n	n	PRON
ma-12	152	8	≤	≤	ADV
ma-12	152	9	m‖xn	m‖xn	PROPN
ma-12	152	10	−	−	PROPN
ma-12	152	11	txn‖+mµnφ(‖(i	txn‖+mµnφ(‖(i	NOUN
ma-12	152	12	−	−	PROPN
ma-12	152	13	t	t	NOUN
ma-12	152	14	)	)	PUNCT
ma-12	152	15	xn‖	xn‖	PROPN
ma-12	152	16	)	)	PUNCT
ma-12	153	1	+	+	NUM
ma-12	153	2	mξn	mξn	X
ma-12	153	3	,	,	PUNCT
ma-12	153	4	(	(	PUNCT
ma-12	153	5	3.3	3.3	NUM
ma-12	153	6	)	)	PUNCT
ma-12	153	7	where	where	SCONJ
ma-12	153	8	µn	µn	PROPN
ma-12	153	9	=	=	SYM
ma-12	153	10	max1≤j≤m−1{µ(j)n	max1≤j≤m−1{µ(j)n	X
ma-12	153	11	}	}	PUNCT
ma-12	153	12	and	and	CCONJ
ma-12	153	13	ξn	ξn	X
ma-12	153	14	=	=	PUNCT
ma-12	153	15	max1≤j≤m−1{ξ(j)n	max1≤j≤m−1{ξ(j)n	PROPN
ma-12	153	16	}	}	PUNCT
ma-12	153	17	.from	.from	ADP
ma-12	153	18	(	(	PUNCT
ma-12	153	19	3.2	3.2	NUM
ma-12	153	20	)	)	PUNCT
ma-12	153	21	and	and	CCONJ
ma-12	153	22	(	(	PUNCT
ma-12	153	23	3.3	3.3	NUM
ma-12	153	24	)	)	PUNCT
ma-12	153	25	,	,	PUNCT
ma-12	153	26	we	we	PRON
ma-12	153	27	get	get	VERB
ma-12	153	28	‖xn	‖xn	PROPN
ma-12	153	29	−	−	PROPN
ma-12	153	30	t	t	PROPN
ma-12	153	31	(	(	PUNCT
ma-12	153	32	pt	pt	PROPN
ma-12	153	33	)	)	PUNCT
ma-12	153	34	j−1xn‖	j−1xn‖	PROPN
ma-12	153	35	<	<	X
ma-12	153	36	ε	ε	PROPN
ma-12	153	37	.	.	PUNCT
ma-12	154	1	(	(	PUNCT
ma-12	154	2	3.4	3.4	NUM
ma-12	154	3	)	)	PUNCT
ma-12	154	4	now	now	ADV
ma-12	154	5	,	,	PUNCT
ma-12	154	6	since	since	SCONJ
ma-12	154	7	t	t	NOUN
ma-12	154	8	:	:	PUNCT
ma-12	154	9	k	k	X
ma-12	154	10	−→	−→	NOUN
ma-12	154	11	e	e	NOUN
ma-12	154	12	is	be	AUX
ma-12	154	13	l	l	NOUN
ma-12	154	14	-	-	NOUN
ma-12	154	15	lipschitizian	lipschitizian	NOUN
ma-12	154	16	and	and	CCONJ
ma-12	154	17	total	total	ADJ
ma-12	154	18	asymptotically	asymptotically	ADV
ma-12	154	19	nonexpansive	nonexpansive	ADJ
ma-12	154	20	,	,	PUNCT
ma-12	154	21	so	so	ADV
ma-12	154	22	is	be	AUX
ma-12	154	23	t	t	NOUN
ma-12	154	24	:	:	PUNCT
ma-12	154	25	c	c	PROPN
ma-12	154	26	−→	−→	PROPN
ma-12	154	27	e.	e.	PROPN
ma-12	154	28	therefore	therefore	ADV
ma-12	154	29	,	,	PUNCT
ma-12	154	30	∀j	∀j	PROPN
ma-12	154	31	≥	≥	NUM
ma-12	154	32	1	1	NUM
ma-12	154	33	,	,	PUNCT
ma-12	154	34	t	t	PROPN
ma-12	154	35	(	(	PUNCT
ma-12	154	36	pt	pt	INTJ
ma-12	154	37	)	)	PUNCT
ma-12	154	38	j−1	j−1	NOUN
ma-12	154	39	:	:	PUNCT
ma-12	155	1	c	c	AUX
ma-12	155	2	−→	−→	NOUN
ma-12	155	3	e	e	NOUN
ma-12	155	4	is	be	AUX
ma-12	155	5	lipschitizian	lipschitizian	ADJ
ma-12	155	6	mapping	mapping	NOUN
ma-12	155	7	with	with	ADP
ma-12	155	8	the	the	DET
ma-12	155	9	lipschitz	lipschitz	NOUN
ma-12	155	10	constant	constant	ADJ
ma-12	155	11	µj	µj	ADP
ma-12	155	12	≥	≥	NUM
ma-12	155	13	1	1	NUM
ma-12	155	14	.	.	PUNCT
ma-12	155	15	eur	eur	PROPN
ma-12	155	16	.	.	PUNCT
ma-12	156	1	j.	j.	PROPN
ma-12	156	2	math	math	PROPN
ma-12	156	3	.	.	PUNCT
ma-12	157	1	anal	anal	ADJ
ma-12	157	2	.	.	PUNCT
ma-12	158	1	1	1	NUM
ma-12	158	2	(	(	PUNCT
ma-12	158	3	2021	2021	NUM
ma-12	158	4	)	)	PUNCT
ma-12	158	5	51	51	NUM
ma-12	159	1	in	in	ADP
ma-12	159	2	addition	addition	NOUN
ma-12	159	3	,	,	PUNCT
ma-12	159	4	‖t	‖t	NOUN
ma-12	159	5	(	(	PUNCT
ma-12	159	6	pt	pt	NOUN
ma-12	159	7	)	)	PUNCT
ma-12	159	8	j−1yn	j−1yn	PROPN
ma-12	159	9	−	−	PROPN
ma-12	159	10	yn‖	yn‖	NOUN
ma-12	159	11	=	=	PROPN
ma-12	159	12	‖t	‖t	PROPN
ma-12	159	13	(	(	PUNCT
ma-12	159	14	pt	pt	NOUN
ma-12	159	15	)	)	PUNCT
ma-12	159	16	j−1yn	j−1yn	PROPN
ma-12	159	17	−	−	PROPN
ma-12	159	18	m(n)∑	m(n)∑	PUNCT
ma-12	159	19	i=1	i=1	PROPN
ma-12	159	20	t	t	PROPN
ma-12	159	21	(	(	PUNCT
ma-12	159	22	n	n	CCONJ
ma-12	159	23	)	)	PUNCT
ma-12	160	1	i	i	PRON
ma-12	160	2	t	t	PROPN
ma-12	160	3	(	(	PUNCT
ma-12	160	4	pt	pt	INTJ
ma-12	160	5	)	)	PUNCT
ma-12	160	6	j−1xi+n	j−1xi+n	PROPN
ma-12	161	1	+	+	NUM
ma-12	161	2	m(n)∑	m(n)∑	PUNCT
ma-12	161	3	i=1	i=1	PROPN
ma-12	161	4	t	t	PROPN
ma-12	161	5	(	(	PUNCT
ma-12	161	6	n	n	CCONJ
ma-12	161	7	)	)	PUNCT
ma-12	162	1	i	i	PRON
ma-12	162	2	t	t	PROPN
ma-12	162	3	(	(	PUNCT
ma-12	162	4	pt	pt	INTJ
ma-12	162	5	)	)	PUNCT
ma-12	162	6	j−1xi+n	j−1xi+n	PROPN
ma-12	162	7	−	−	PROPN
ma-12	162	8	m(n)∑	m(n)∑	PUNCT
ma-12	162	9	i=1	i=1	PROPN
ma-12	162	10	t	t	PROPN
ma-12	162	11	(	(	PUNCT
ma-12	162	12	n	n	CCONJ
ma-12	162	13	)	)	PUNCT
ma-12	163	1	i	i	PRON
ma-12	163	2	xi+n‖	xi+n‖	VERB
ma-12	164	1	≤	≤	ADJ
ma-12	164	2	‖t	‖t	NOUN
ma-12	164	3	(	(	PUNCT
ma-12	164	4	pt	pt	NOUN
ma-12	164	5	)	)	PUNCT
ma-12	164	6	j−1yn	j−1yn	PROPN
ma-12	165	1	−	−	PROPN
ma-12	165	2	m(n)∑	m(n)∑	PUNCT
ma-12	165	3	i=1	i=1	PROPN
ma-12	165	4	t	t	PROPN
ma-12	165	5	(	(	PUNCT
ma-12	165	6	n	n	CCONJ
ma-12	165	7	)	)	PUNCT
ma-12	166	1	i	i	PRON
ma-12	166	2	t	t	PROPN
ma-12	166	3	(	(	PUNCT
ma-12	166	4	pt	pt	INTJ
ma-12	166	5	)	)	PUNCT
ma-12	166	6	j−1xi+n‖	j−1xi+n‖	NOUN
ma-12	166	7	+	+	CCONJ
ma-12	166	8	m(n)∑	m(n)∑	PUNCT
ma-12	166	9	i=1	i=1	PROPN
ma-12	166	10	t	t	PROPN
ma-12	166	11	(	(	PUNCT
ma-12	166	12	n	n	CCONJ
ma-12	166	13	)	)	PUNCT
ma-12	167	1	i	i	PRON
ma-12	167	2	‖t	‖t	NOUN
ma-12	167	3	(	(	PUNCT
ma-12	167	4	pt	pt	INTJ
ma-12	167	5	)	)	PUNCT
ma-12	167	6	j−1xi+n	j−1xi+n	PROPN
ma-12	167	7	−	−	PROPN
ma-12	167	8	xi+n‖.	xi+n‖.	NOUN
ma-12	167	9	(	(	PUNCT
ma-12	167	10	3.5	3.5	NUM
ma-12	167	11	)	)	PUNCT
ma-12	167	12	using	use	VERB
ma-12	167	13	(	(	PUNCT
ma-12	167	14	3.4	3.4	NUM
ma-12	167	15	)	)	PUNCT
ma-12	167	16	,	,	PUNCT
ma-12	167	17	we	we	PRON
ma-12	167	18	get	get	VERB
ma-12	167	19	m(n)∑	m(n)∑	PUNCT
ma-12	167	20	i=1	i=1	PROPN
ma-12	167	21	t	t	PROPN
ma-12	167	22	(	(	PUNCT
ma-12	167	23	n	n	CCONJ
ma-12	167	24	)	)	PUNCT
ma-12	168	1	i	i	PRON
ma-12	168	2	‖t	‖t	NOUN
ma-12	168	3	(	(	PUNCT
ma-12	168	4	pt	pt	INTJ
ma-12	168	5	)	)	PUNCT
ma-12	168	6	j−1xi+n	j−1xi+n	PROPN
ma-12	168	7	−	−	PROPN
ma-12	168	8	xi+n‖	xi+n‖	PUNCT
ma-12	169	1	<	<	X
ma-12	169	2	ε,∀n	ε,∀n	PROPN
ma-12	169	3	≥	≥	X
ma-12	169	4	n.	n.	NOUN
ma-12	169	5	(	(	PUNCT
ma-12	169	6	3.6	3.6	NUM
ma-12	169	7	)	)	PUNCT
ma-12	169	8	furthermore	furthermore	ADV
ma-12	169	9	,	,	PUNCT
ma-12	169	10	by	by	ADP
ma-12	169	11	lemma	lemma	PROPN
ma-12	169	12	2.5	2.5	NUM
ma-12	169	13	,	,	PUNCT
ma-12	169	14	there	there	PRON
ma-12	169	15	exists	exist	VERB
ma-12	169	16	a	a	DET
ma-12	169	17	strictly	strictly	ADV
ma-12	169	18	increasing	increase	VERB
ma-12	169	19	continous	continous	ADJ
ma-12	169	20	function	function	NOUN
ma-12	169	21	φ	φ	NOUN
ma-12	169	22	:	:	PUNCT
ma-12	170	1	[	[	X
ma-12	170	2	0,∞	0,∞	X
ma-12	170	3	)	)	PUNCT
ma-12	170	4	−→	−→	NOUN
ma-12	170	5	[	[	X
ma-12	170	6	0,∞	0,∞	NOUN
ma-12	170	7	)	)	PUNCT
ma-12	170	8	with	with	ADP
ma-12	170	9	φ(0	φ(0	ADJ
ma-12	170	10	)	)	PUNCT
ma-12	170	11	=	=	SYM
ma-12	170	12	0	0	NUM
ma-12	170	13	such	such	ADJ
ma-12	170	14	that	that	PRON
ma-12	170	15	for	for	ADP
ma-12	170	16	all	all	DET
ma-12	170	17	n	n	DET
ma-12	170	18	≥	≥	NOUN
ma-12	170	19	n	n	NOUN
ma-12	170	20	,	,	PUNCT
ma-12	170	21	we	we	PRON
ma-12	170	22	have	have	VERB
ma-12	170	23	‖t	‖t	NOUN
ma-12	170	24	(	(	PUNCT
ma-12	170	25	pt	pt	NOUN
ma-12	170	26	)	)	PUNCT
ma-12	170	27	j−1yn	j−1yn	PROPN
ma-12	171	1	−	−	PROPN
ma-12	171	2	m(n)∑	m(n)∑	PUNCT
ma-12	171	3	i=1	i=1	PROPN
ma-12	171	4	t	t	PROPN
ma-12	171	5	(	(	PUNCT
ma-12	171	6	n	n	CCONJ
ma-12	171	7	)	)	PUNCT
ma-12	172	1	i	i	PRON
ma-12	172	2	t	t	PROPN
ma-12	172	3	(	(	PUNCT
ma-12	172	4	pt	pt	INTJ
ma-12	172	5	)	)	PUNCT
ma-12	172	6	j−1xi+n‖	j−1xi+n‖	NOUN
ma-12	172	7	=	=	SYM
ma-12	172	8	‖t	‖t	PROPN
ma-12	172	9	(	(	PUNCT
ma-12	172	10	pt	pt	INTJ
ma-12	172	11	)	)	PUNCT
ma-12	172	12	j−1	j−1	PROPN
ma-12	172	13	(	(	PUNCT
ma-12	172	14	m(n)∑	m(n)∑	PUNCT
ma-12	172	15	i=1	i=1	PROPN
ma-12	172	16	t	t	PROPN
ma-12	172	17	(	(	PUNCT
ma-12	172	18	n	n	CCONJ
ma-12	172	19	)	)	PUNCT
ma-12	173	1	i	i	PRON
ma-12	173	2	xi+n)−	xi+n)−	X
ma-12	173	3	m(n)∑	m(n)∑	PUNCT
ma-12	173	4	i=1	i=1	PROPN
ma-12	173	5	t	t	PROPN
ma-12	173	6	(	(	PUNCT
ma-12	173	7	n	n	CCONJ
ma-12	173	8	)	)	PUNCT
ma-12	174	1	i	i	PRON
ma-12	174	2	t	t	PROPN
ma-12	174	3	(	(	PUNCT
ma-12	174	4	pt	pt	INTJ
ma-12	174	5	)	)	PUNCT
ma-12	174	6	j−1xi+n‖	j−1xi+n‖	PROPN
ma-12	174	7	≤	≤	PROPN
ma-12	174	8	µjφ	µjφ	NOUN
ma-12	174	9	−1{max1≤j	−1{max1≤j	NOUN
ma-12	174	10	,	,	PUNCT
ma-12	174	11	k≤n(‖xi+n	k≤n(‖xi+n	NOUN
ma-12	174	12	−	−	PROPN
ma-12	174	13	xi+k‖	xi+k‖	PROPN
ma-12	175	1	−µ−1j	−µ−1j	PROPN
ma-12	175	2	‖t	‖t	NOUN
ma-12	175	3	(	(	PUNCT
ma-12	175	4	pt	pt	INTJ
ma-12	175	5	)	)	PUNCT
ma-12	175	6	j−1xi+n	j−1xi+n	PROPN
ma-12	175	7	−	−	PROPN
ma-12	175	8	t	t	PROPN
ma-12	175	9	(	(	PUNCT
ma-12	175	10	pt	pt	INTJ
ma-12	175	11	)	)	PUNCT
ma-12	175	12	j−1xk+n‖	j−1xk+n‖	PROPN
ma-12	175	13	)	)	PUNCT
ma-12	175	14	}	}	PUNCT
ma-12	175	15	=	=	SYM
ma-12	175	16	µjφ	µjφ	PRON
ma-12	175	17	−1{max1≤j	−1{max1≤j	NOUN
ma-12	175	18	,	,	PUNCT
ma-12	175	19	k≤n(‖xi+n	k≤n(‖xi+n	PROPN
ma-12	175	20	−	−	PROPN
ma-12	175	21	t	t	PROPN
ma-12	175	22	(	(	PUNCT
ma-12	175	23	pt	pt	INTJ
ma-12	175	24	)	)	PUNCT
ma-12	175	25	j−1xi+n	j−1xi+n	PROPN
ma-12	176	1	+	+	PROPN
ma-12	176	2	t	t	PROPN
ma-12	176	3	(	(	PUNCT
ma-12	176	4	pt	pt	INTJ
ma-12	176	5	)	)	PUNCT
ma-12	176	6	j−1xi+n	j−1xi+n	PROPN
ma-12	176	7	−	−	PROPN
ma-12	176	8	t	t	PROPN
ma-12	176	9	(	(	PUNCT
ma-12	176	10	pt	pt	INTJ
ma-12	176	11	)	)	PUNCT
ma-12	176	12	j−1xk+n	j−1xk+n	PROPN
ma-12	176	13	+	+	PROPN
ma-12	176	14	t	t	PROPN
ma-12	176	15	(	(	PUNCT
ma-12	176	16	pt	pt	INTJ
ma-12	176	17	)	)	PUNCT
ma-12	176	18	j−1xk+n	j−1xk+n	PROPN
ma-12	176	19	−	−	PROPN
ma-12	176	20	xi+k‖	xi+k‖	PROPN
ma-12	177	1	−µ−1j	−µ−1j	PROPN
ma-12	177	2	‖t	‖t	NOUN
ma-12	177	3	(	(	PUNCT
ma-12	177	4	pt	pt	INTJ
ma-12	177	5	)	)	PUNCT
ma-12	177	6	j−1xi+n	j−1xi+n	PROPN
ma-12	177	7	−	−	PROPN
ma-12	177	8	t	t	PROPN
ma-12	177	9	(	(	PUNCT
ma-12	177	10	pt	pt	INTJ
ma-12	177	11	)	)	PUNCT
ma-12	177	12	j−1xk+n‖	j−1xk+n‖	PROPN
ma-12	177	13	)	)	PUNCT
ma-12	177	14	}	}	PUNCT
ma-12	177	15	≤	≤	PROPN
ma-12	177	16	µjφ	µjφ	NUM
ma-12	177	17	−1{max1≤j	−1{max1≤j	NOUN
ma-12	177	18	,	,	PUNCT
ma-12	177	19	k≤n(‖xi+n	k≤n(‖xi+n	PROPN
ma-12	177	20	−	−	PROPN
ma-12	177	21	t	t	PROPN
ma-12	177	22	(	(	PUNCT
ma-12	177	23	pt	pt	INTJ
ma-12	177	24	)	)	PUNCT
ma-12	177	25	j−1xi+n‖	j−1xi+n‖	PROPN
ma-12	177	26	+	+	NOUN
ma-12	177	27	‖t	‖t	NOUN
ma-12	177	28	(	(	PUNCT
ma-12	177	29	pt	pt	INTJ
ma-12	177	30	)	)	PUNCT
ma-12	177	31	j−1xi+n	j−1xi+n	PROPN
ma-12	177	32	−	−	PROPN
ma-12	177	33	t	t	PROPN
ma-12	177	34	(	(	PUNCT
ma-12	177	35	pt	pt	PROPN
ma-12	177	36	)	)	PUNCT
ma-12	177	37	j−1xk+n‖	j−1xk+n‖	PROPN
ma-12	178	1	+	+	PROPN
ma-12	178	2	‖t	‖t	PROPN
ma-12	178	3	(	(	PUNCT
ma-12	178	4	pt	pt	X
ma-12	178	5	)	)	PUNCT
ma-12	178	6	j−1xk+n	j−1xk+n	PROPN
ma-12	178	7	−	−	PROPN
ma-12	178	8	xi+k‖	xi+k‖	PROPN
ma-12	179	1	−µ−1j	−µ−1j	PROPN
ma-12	179	2	‖t	‖t	NOUN
ma-12	179	3	(	(	PUNCT
ma-12	179	4	pt	pt	INTJ
ma-12	179	5	)	)	PUNCT
ma-12	179	6	j−1xi+n	j−1xi+n	PROPN
ma-12	179	7	−	−	PROPN
ma-12	179	8	t	t	PROPN
ma-12	179	9	(	(	PUNCT
ma-12	179	10	pt	pt	INTJ
ma-12	179	11	)	)	PUNCT
ma-12	179	12	j−1xk+n‖	j−1xk+n‖	PROPN
ma-12	179	13	)	)	PUNCT
ma-12	179	14	}	}	PUNCT
ma-12	179	15	≤	≤	PROPN
ma-12	179	16	µjφ	µjφ	NUM
ma-12	179	17	−1{max1≤j	−1{max1≤j	NOUN
ma-12	179	18	,	,	PUNCT
ma-12	179	19	k≤n(ε+	k≤n(ε+	PROPN
ma-12	179	20	ε+	ε+	PUNCT
ma-12	179	21	(	(	PUNCT
ma-12	179	22	1−	1−	NUM
ma-12	179	23	µ−1j	µ−1j	NOUN
ma-12	179	24	)	)	PUNCT
ma-12	179	25	×‖t	×‖t	PROPN
ma-12	179	26	(	(	PUNCT
ma-12	179	27	pt	pt	NOUN
ma-12	179	28	)	)	PUNCT
ma-12	179	29	j−1xi+n	j−1xi+n	PROPN
ma-12	179	30	−	−	PROPN
ma-12	179	31	t	t	PROPN
ma-12	179	32	(	(	PUNCT
ma-12	179	33	pt	pt	INTJ
ma-12	179	34	)	)	PUNCT
ma-12	179	35	j−1xk+n‖	j−1xk+n‖	PROPN
ma-12	179	36	)	)	PUNCT
ma-12	179	37	}	}	PUNCT
ma-12	179	38	≤	≤	PROPN
ma-12	179	39	µjφ	µjφ	NUM
ma-12	179	40	−1{max1≤j	−1{max1≤j	NOUN
ma-12	179	41	,	,	PUNCT
ma-12	179	42	k≤n(ε+	k≤n(ε+	PROPN
ma-12	179	43	ε+	ε+	PUNCT
ma-12	179	44	(	(	PUNCT
ma-12	179	45	1−	1−	NUM
ma-12	179	46	µ−1j	µ−1j	NOUN
ma-12	179	47	)	)	PUNCT
ma-12	179	48	µj	µj	ADP
ma-12	179	49	×‖xi+n	×‖xi+n	PROPN
ma-12	179	50	−	−	PROPN
ma-12	179	51	xk+n‖	xk+n‖	NOUN
ma-12	179	52	}	}	PUNCT
ma-12	179	53	≤	≤	PROPN
ma-12	179	54	µjφ	µjφ	NUM
ma-12	179	55	−1{max1≤j	−1{max1≤j	NOUN
ma-12	179	56	,	,	PUNCT
ma-12	179	57	k≤n(ε+	k≤n(ε+	PROPN
ma-12	179	58	ε+	ε+	PUNCT
ma-12	179	59	(	(	PUNCT
ma-12	179	60	1−	1−	NUM
ma-12	179	61	µ−1j	µ−1j	NOUN
ma-12	179	62	)	)	PUNCT
ma-12	180	1	µj	µj	PROPN
ma-12	180	2	×(‖xi+n‖+	×(‖xi+n‖+	PROPN
ma-12	180	3	‖xk+n‖	‖xk+n‖	PROPN
ma-12	180	4	}	}	PUNCT
ma-12	180	5	.	.	PUNCT
ma-12	181	1	eur	eur	PROPN
ma-12	181	2	.	.	PUNCT
ma-12	182	1	j.	j.	PROPN
ma-12	182	2	math	math	PROPN
ma-12	182	3	.	.	PUNCT
ma-12	183	1	anal	anal	ADJ
ma-12	183	2	.	.	PUNCT
ma-12	184	1	1	1	NUM
ma-12	184	2	(	(	PUNCT
ma-12	184	3	2021	2021	NUM
ma-12	184	4	)	)	PUNCT
ma-12	185	1	52thus	52thus	NUM
ma-12	185	2	,	,	PUNCT
ma-12	185	3	‖t	‖t	NOUN
ma-12	185	4	(	(	PUNCT
ma-12	185	5	pt	pt	NOUN
ma-12	185	6	)	)	PUNCT
ma-12	185	7	j−1yn	j−1yn	PROPN
ma-12	186	1	−	−	PROPN
ma-12	186	2	m(n)∑	m(n)∑	PUNCT
ma-12	186	3	i=1	i=1	PROPN
ma-12	186	4	t	t	PROPN
ma-12	186	5	(	(	PUNCT
ma-12	186	6	n	n	CCONJ
ma-12	186	7	)	)	PUNCT
ma-12	186	8	i	i	PRON
ma-12	186	9	t	t	PROPN
ma-12	186	10	(	(	PUNCT
ma-12	186	11	pt	pt	INTJ
ma-12	186	12	)	)	PUNCT
ma-12	186	13	j−1xi+n‖	j−1xi+n‖	PROPN
ma-12	186	14	≤	≤	NUM
ma-12	186	15	µjφ−1(ε+	µjφ−1(ε+	PUNCT
ma-12	186	16	ε+	ε+	X
ma-12	186	17	2r(1−	2r(1−	PROPN
ma-12	186	18	µ−1j	µ−1j	NOUN
ma-12	186	19	)	)	PUNCT
ma-12	186	20	µj	µj	PROPN
ma-12	186	21	)	)	PUNCT
ma-12	186	22	,	,	PUNCT
ma-12	186	23	(	(	PUNCT
ma-12	186	24	3.7	3.7	NUM
ma-12	186	25	)	)	PUNCT
ma-12	186	26	since	since	SCONJ
ma-12	186	27	xi+n	xi+n	PROPN
ma-12	186	28	and	and	CCONJ
ma-12	186	29	xk+n	xk+n	PROPN
ma-12	186	30	are	be	AUX
ma-12	186	31	both	both	PRON
ma-12	186	32	in	in	ADP
ma-12	186	33	c.also	c.also	ADV
ma-12	186	34	,	,	PUNCT
ma-12	186	35	(	(	PUNCT
ma-12	186	36	3.5	3.5	NUM
ma-12	186	37	)	)	PUNCT
ma-12	186	38	,	,	PUNCT
ma-12	186	39	(	(	PUNCT
ma-12	186	40	3.6	3.6	NUM
ma-12	186	41	)	)	PUNCT
ma-12	186	42	and	and	CCONJ
ma-12	186	43	(	(	PUNCT
ma-12	186	44	3.7	3.7	NUM
ma-12	186	45	)	)	PUNCT
ma-12	186	46	imply	imply	VERB
ma-12	186	47	that	that	SCONJ
ma-12	186	48	‖t	‖t	NOUN
ma-12	186	49	(	(	PUNCT
ma-12	186	50	pt	pt	NOUN
ma-12	186	51	)	)	PUNCT
ma-12	186	52	j−1yn	j−1yn	PROPN
ma-12	187	1	−	−	PROPN
ma-12	187	2	yn‖	yn‖	NOUN
ma-12	187	3	≤	≤	ADV
ma-12	187	4	µjφ−1(ε+	µjφ−1(ε+	PUNCT
ma-12	187	5	ε+	ε+	X
ma-12	187	6	2r(1−	2r(1−	PROPN
ma-12	187	7	µ−1j	µ−1j	NOUN
ma-12	187	8	)	)	PUNCT
ma-12	187	9	µj	µj	PROPN
ma-12	187	10	)	)	PUNCT
ma-12	187	11	.	.	PUNCT
ma-12	188	1	(	(	PUNCT
ma-12	188	2	3.8	3.8	NUM
ma-12	188	3	)	)	PUNCT
ma-12	188	4	taking	take	VERB
ma-12	188	5	lim	lim	PROPN
ma-12	188	6	supn→∞	supn→∞	PROPN
ma-12	188	7	on	on	ADP
ma-12	188	8	both	both	DET
ma-12	188	9	sides	side	NOUN
ma-12	188	10	of	of	ADP
ma-12	188	11	(	(	PUNCT
ma-12	188	12	3.8	3.8	NUM
ma-12	188	13	)	)	PUNCT
ma-12	188	14	and	and	CCONJ
ma-12	188	15	noting	note	VERB
ma-12	188	16	that	that	SCONJ
ma-12	188	17	ε	ε	PROPN
ma-12	188	18	>	>	X
ma-12	188	19	0	0	PUNCT
ma-12	188	20	is	be	AUX
ma-12	188	21	arbitrary	arbitrary	ADJ
ma-12	188	22	,	,	PUNCT
ma-12	188	23	we	we	PRON
ma-12	188	24	have	have	VERB
ma-12	188	25	that	that	SCONJ
ma-12	188	26	lim	lim	PROPN
ma-12	188	27	sup	sup	VERB
ma-12	188	28	n→∞	n→∞	NUM
ma-12	188	29	‖t	‖t	NOUN
ma-12	188	30	(	(	PUNCT
ma-12	188	31	pt	pt	NOUN
ma-12	188	32	)	)	PUNCT
ma-12	188	33	j−1yn	j−1yn	PROPN
ma-12	189	1	−	−	PROPN
ma-12	189	2	yn‖	yn‖	NOUN
ma-12	189	3	≤	≤	PROPN
ma-12	189	4	µjφ−1(2r(1−	µjφ−1(2r(1−	PROPN
ma-12	189	5	µ−1j	µ−1j	PROPN
ma-12	189	6	)	)	PUNCT
ma-12	189	7	µj	µj	PROPN
ma-12	189	8	)	)	PUNCT
ma-12	189	9	.	.	PUNCT
ma-12	190	1	(	(	PUNCT
ma-12	190	2	3.9	3.9	NUM
ma-12	190	3	)	)	PUNCT
ma-12	190	4	on	on	ADP
ma-12	190	5	the	the	DET
ma-12	190	6	other	other	ADJ
ma-12	190	7	hand	hand	NOUN
ma-12	190	8	,	,	PUNCT
ma-12	190	9	for	for	ADP
ma-12	190	10	any	any	DET
ma-12	190	11	j	j	PROPN
ma-12	190	12	≥	≥	NUM
ma-12	190	13	1	1	NUM
ma-12	190	14	,	,	PUNCT
ma-12	190	15	it	it	PRON
ma-12	190	16	follows	follow	VERB
ma-12	190	17	from	from	ADP
ma-12	190	18	(	(	PUNCT
ma-12	190	19	3.1	3.1	NUM
ma-12	190	20	)	)	PUNCT
ma-12	190	21	that	that	PRON
ma-12	190	22	‖t	‖t	NOUN
ma-12	190	23	(	(	PUNCT
ma-12	190	24	pt	pt	PROPN
ma-12	190	25	)	)	PUNCT
ma-12	190	26	j−1ω	j−1ω	NOUN
ma-12	190	27	−	−	NOUN
ma-12	191	1	ω‖	ω‖	NOUN
ma-12	191	2	≤	≤	ADJ
ma-12	191	3	‖t	‖t	NOUN
ma-12	191	4	(	(	PUNCT
ma-12	191	5	pt	pt	PROPN
ma-12	191	6	)	)	PUNCT
ma-12	191	7	j−1ω	j−1ω	NOUN
ma-12	191	8	−	−	PROPN
ma-12	191	9	t	t	PROPN
ma-12	191	10	(	(	PUNCT
ma-12	191	11	pt	pt	INTJ
ma-12	191	12	)	)	PUNCT
ma-12	191	13	j−1yn‖+	j−1yn‖+	PROPN
ma-12	191	14	‖t	‖t	PROPN
ma-12	191	15	(	(	PUNCT
ma-12	191	16	pt	pt	NOUN
ma-12	191	17	)	)	PUNCT
ma-12	191	18	j−1yn	j−1yn	PROPN
ma-12	191	19	−	−	PROPN
ma-12	191	20	yn‖+	yn‖+	PROPN
ma-12	192	1	‖yn	‖yn	PROPN
ma-12	192	2	−	−	PROPN
ma-12	192	3	ω‖	ω‖	NOUN
ma-12	192	4	≤	≤	ADV
ma-12	192	5	µj‖yn	µj‖yn	PUNCT
ma-12	192	6	−	−	PROPN
ma-12	192	7	ω‖+	ω‖+	PROPN
ma-12	192	8	‖t	‖t	PROPN
ma-12	192	9	(	(	PUNCT
ma-12	192	10	pt	pt	NOUN
ma-12	192	11	)	)	PUNCT
ma-12	192	12	j−1yn	j−1yn	PROPN
ma-12	192	13	−	−	PROPN
ma-12	192	14	yn‖+	yn‖+	PROPN
ma-12	192	15	‖yn	‖yn	PROPN
ma-12	192	16	−	−	PROPN
ma-12	192	17	ω‖.	ω‖.	NUM
ma-12	192	18	(	(	PUNCT
ma-12	192	19	3.10	3.10	NUM
ma-12	192	20	)	)	PUNCT
ma-12	192	21	taking	take	VERB
ma-12	192	22	lim	lim	PROPN
ma-12	192	23	supn→∞	supn→∞	PROPN
ma-12	192	24	on	on	ADP
ma-12	192	25	both	both	DET
ma-12	192	26	sides	side	NOUN
ma-12	192	27	of	of	ADP
ma-12	192	28	the	the	DET
ma-12	192	29	above	above	ADJ
ma-12	192	30	inequality	inequality	NOUN
ma-12	192	31	and	and	CCONJ
ma-12	192	32	using	use	VERB
ma-12	192	33	(	(	PUNCT
ma-12	192	34	3.1	3.1	NUM
ma-12	192	35	)	)	PUNCT
ma-12	192	36	and	and	CCONJ
ma-12	192	37	(	(	PUNCT
ma-12	192	38	3.9	3.9	NUM
ma-12	192	39	)	)	PUNCT
ma-12	192	40	,	,	PUNCT
ma-12	192	41	we	we	PRON
ma-12	192	42	have	have	VERB
ma-12	192	43	‖t	‖t	NOUN
ma-12	192	44	(	(	PUNCT
ma-12	192	45	pt	pt	NOUN
ma-12	192	46	)	)	PUNCT
ma-12	192	47	j−1ω	j−1ω	NOUN
ma-12	192	48	−	−	NOUN
ma-12	192	49	ω‖	ω‖	NOUN
ma-12	192	50	≤	≤	NUM
ma-12	192	51	µjφ−1(2r(1−	µjφ−1(2r(1−	PROPN
ma-12	192	52	µ−1j	µ−1j	PROPN
ma-12	192	53	)	)	PUNCT
ma-12	192	54	µj	µj	PROPN
ma-12	192	55	)	)	PUNCT
ma-12	192	56	.	.	PUNCT
ma-12	193	1	again	again	ADV
ma-12	193	2	,	,	PUNCT
ma-12	193	3	taking	take	VERB
ma-12	193	4	lim	lim	PROPN
ma-12	193	5	supj→∞	supj→∞	PROPN
ma-12	193	6	on	on	ADP
ma-12	193	7	both	both	DET
ma-12	193	8	sides	side	NOUN
ma-12	193	9	of	of	ADP
ma-12	193	10	the	the	DET
ma-12	193	11	above	above	ADJ
ma-12	193	12	inequality	inequality	NOUN
ma-12	193	13	,	,	PUNCT
ma-12	193	14	we	we	PRON
ma-12	193	15	have	have	VERB
ma-12	193	16	lim	lim	PROPN
ma-12	193	17	sup	sup	PROPN
ma-12	193	18	j→∞	j→∞	NUM
ma-12	193	19	‖t	‖t	PROPN
ma-12	193	20	(	(	PUNCT
ma-12	193	21	pt	pt	NOUN
ma-12	193	22	)	)	PUNCT
ma-12	193	23	j−1ω	j−1ω	NOUN
ma-12	194	1	−	−	NOUN
ma-12	194	2	ω‖	ω‖	NOUN
ma-12	194	3	≤	≤	NUM
ma-12	194	4	φ−1(0	φ−1(0	NOUN
ma-12	194	5	)	)	PUNCT
ma-12	194	6	=	=	SYM
ma-12	194	7	0	0	NUM
ma-12	194	8	,	,	PUNCT
ma-12	194	9	which	which	PRON
ma-12	194	10	implies	imply	VERB
ma-12	194	11	that	that	SCONJ
ma-12	194	12	‖t	‖t	NOUN
ma-12	194	13	(	(	PUNCT
ma-12	194	14	pt	pt	INTJ
ma-12	194	15	)	)	PUNCT
ma-12	194	16	j−1ω	j−1ω	NOUN
ma-12	194	17	−	−	NOUN
ma-12	194	18	ω‖	ω‖	NOUN
ma-12	194	19	→	→	SYM
ma-12	194	20	0	0	PUNCT
ma-12	194	21	as	as	ADP
ma-12	194	22	j	j	PROPN
ma-12	194	23	→∞	→∞	PROPN
ma-12	194	24	,	,	PUNCT
ma-12	194	25	and	and	CCONJ
ma-12	194	26	hence	hence	ADV
ma-12	194	27	proving	prove	VERB
ma-12	194	28	our	our	PRON
ma-12	194	29	claim	claim	NOUN
ma-12	194	30	.	.	PUNCT
ma-12	195	1	by	by	ADP
ma-12	195	2	continuityof	continuityof	PROPN
ma-12	195	3	tp	tp	NOUN
ma-12	195	4	,	,	PUNCT
ma-12	195	5	we	we	PRON
ma-12	195	6	have	have	VERB
ma-12	195	7	that	that	DET
ma-12	195	8	lim	lim	PROPN
ma-12	195	9	j→∞	j→∞	PROPN
ma-12	195	10	tp	tp	PROPN
ma-12	195	11	(	(	PUNCT
ma-12	195	12	t	t	PROPN
ma-12	195	13	(	(	PUNCT
ma-12	195	14	pt	pt	INTJ
ma-12	195	15	)	)	PUNCT
ma-12	195	16	j−1ω	j−1ω	PROPN
ma-12	195	17	)	)	PUNCT
ma-12	195	18	=	=	SYM
ma-12	196	1	tpω	tpω	NOUN
ma-12	197	1	=	=	PUNCT
ma-12	197	2	tω	tω	PROPN
ma-12	197	3	=	=	SYM
ma-12	197	4	ω	ω	PROPN
ma-12	197	5	.	.	PUNCT
ma-12	198	1	this	this	PRON
ma-12	198	2	completes	complete	VERB
ma-12	198	3	the	the	DET
ma-12	198	4	proof	proof	NOUN
ma-12	198	5	.	.	PUNCT
ma-12	199	1	�	�	PROPN
ma-12	199	2	lemma	lemma	PROPN
ma-12	199	3	3.2	3.2	NUM
ma-12	199	4	.	.	PUNCT
ma-12	200	1	let	let	VERB
ma-12	200	2	e	e	PRON
ma-12	200	3	be	be	AUX
ma-12	200	4	a	a	DET
ma-12	200	5	uniformly	uniformly	ADV
ma-12	200	6	convex	convex	NOUN
ma-12	200	7	banach	banach	NOUN
ma-12	200	8	space	space	NOUN
ma-12	200	9	and	and	CCONJ
ma-12	200	10	k	k	PROPN
ma-12	200	11	a	a	DET
ma-12	200	12	nonempty	nonempty	ADV
ma-12	200	13	closed	close	VERB
ma-12	200	14	convex	convex	NOUN
ma-12	200	15	subset	subset	NOUN
ma-12	200	16	of	of	ADP
ma-12	200	17	e.	e.	PROPN
ma-12	200	18	let	let	VERB
ma-12	200	19	s1	s1	PROPN
ma-12	200	20	,	,	PUNCT
ma-12	200	21	s2	s2	PROPN
ma-12	200	22	,	,	PUNCT
ma-12	200	23	s3	s3	PROPN
ma-12	200	24	:	:	PUNCT
ma-12	200	25	k	k	PROPN
ma-12	200	26	−→	−→	NOUN
ma-12	200	27	k	k	PROPN
ma-12	200	28	be	be	AUX
ma-12	200	29	three	three	NUM
ma-12	200	30	total	total	ADJ
ma-12	200	31	asymptotically	asymptotically	ADV
ma-12	200	32	nonexpansive	nonexpansive	ADJ
ma-12	200	33	self	self	NOUN
ma-12	200	34	mapping	mapping	NOUN
ma-12	200	35	with	with	ADP
ma-12	200	36	sequences	sequence	NOUN
ma-12	200	37	{	{	PUNCT
ma-12	200	38	k(1)n	k(1)n	X
ma-12	200	39	}	}	PUNCT
ma-12	200	40	,	,	PUNCT
ma-12	200	41	{	{	PUNCT
ma-12	200	42	k(2)n	k(2)n	X
ma-12	200	43	}	}	PUNCT
ma-12	200	44	,	,	PUNCT
ma-12	200	45	{	{	PUNCT
ma-12	200	46	k(3)n	k(3)n	NOUN
ma-12	200	47	}	}	PUNCT
ma-12	200	48	∈	∈	PROPN
ma-12	201	1	[	[	X
ma-12	201	2	1,∞	1,∞	NUM
ma-12	201	3	)	)	PUNCT
ma-12	201	4	,	,	PUNCT
ma-12	201	5	{	{	PUNCT
ma-12	201	6	w	w	X
ma-12	201	7	(	(	PUNCT
ma-12	201	8	1)n	1)n	X
ma-12	201	9	}	}	PUNCT
ma-12	201	10	,	,	PUNCT
ma-12	201	11	{	{	PUNCT
ma-12	201	12	w	w	NOUN
ma-12	201	13	(	(	PUNCT
ma-12	201	14	2)}n	2)}n	NOUN
ma-12	201	15	,	,	PUNCT
ma-12	201	16	{	{	PUNCT
ma-12	201	17	w	w	NOUN
ma-12	201	18	(	(	PUNCT
ma-12	201	19	3)n	3)n	NUM
ma-12	201	20	}	}	PUNCT
ma-12	201	21	∈	∈	PROPN
ma-12	202	1	[	[	X
ma-12	202	2	1,∞	1,∞	NUM
ma-12	202	3	)	)	PUNCT
ma-12	202	4	and	and	CCONJ
ma-12	202	5	t1	t1	NOUN
ma-12	202	6	,	,	PUNCT
ma-12	202	7	t2	t2	NOUN
ma-12	202	8	,	,	PUNCT
ma-12	202	9	t3	t3	NOUN
ma-12	202	10	:	:	PUNCT
ma-12	202	11	k	k	X
ma-12	202	12	−→	−→	NOUN
ma-12	202	13	e	e	NOUN
ma-12	202	14	are	be	AUX
ma-12	202	15	three	three	NUM
ma-12	202	16	total	total	ADJ
ma-12	202	17	asymptotically	asymptotically	ADV
ma-12	202	18	nonexpansive	nonexpansive	ADJ
ma-12	202	19	nonself	nonself	PROPN
ma-12	202	20	mappings	mapping	NOUN
ma-12	202	21	with	with	ADP
ma-12	202	22	sequences	sequence	NOUN
ma-12	202	23	{	{	PUNCT
ma-12	202	24	µ(1)n	µ(1)n	X
ma-12	202	25	}	}	PUNCT
ma-12	202	26	,	,	PUNCT
ma-12	202	27	{	{	PUNCT
ma-12	202	28	µ(2)n	µ(2)n	NOUN
ma-12	202	29	}	}	PUNCT
ma-12	202	30	,	,	PUNCT
ma-12	202	31	{	{	PUNCT
ma-12	202	32	µ(3)n	µ(3)n	PROPN
ma-12	202	33	}	}	PUNCT
ma-12	202	34	∈	∈	PROPN
ma-12	203	1	[	[	X
ma-12	203	2	1,∞	1,∞	NUM
ma-12	203	3	)	)	PUNCT
ma-12	203	4	,	,	PUNCT
ma-12	203	5	{	{	PUNCT
ma-12	203	6	ν(1)n	ν(1)n	X
ma-12	203	7	}	}	PUNCT
ma-12	203	8	,	,	PUNCT
ma-12	203	9	{	{	PUNCT
ma-12	203	10	ν(2)n	ν(2)n	NOUN
ma-12	203	11	}	}	PUNCT
ma-12	203	12	,	,	PUNCT
ma-12	203	13	{	{	PUNCT
ma-12	203	14	ν(3)n	ν(3)n	NOUN
ma-12	203	15	}	}	PUNCT
ma-12	203	16	∈	∈	PROPN
ma-12	204	1	[	[	X
ma-12	204	2	1,∞	1,∞	NUM
ma-12	204	3	)	)	PUNCT
ma-12	204	4	.	.	PUNCT
ma-12	205	1	let	let	VERB
ma-12	205	2	{	{	PUNCT
ma-12	205	3	xn	xn	VERB
ma-12	205	4	}	}	PUNCT
ma-12	205	5	be	be	VERB
ma-12	205	6	the	the	DET
ma-12	205	7	sequence	sequence	NOUN
ma-12	205	8	defined	define	VERB
ma-12	205	9	by	by	ADP
ma-12	205	10	(	(	PUNCT
ma-12	205	11	1.7	1.7	NUM
ma-12	205	12	)	)	PUNCT
ma-12	205	13	,	,	PUNCT
ma-12	205	14	where	where	SCONJ
ma-12	205	15	{	{	PUNCT
ma-12	205	16	αn	αn	NOUN
ma-12	205	17	}	}	PUNCT
ma-12	205	18	and	and	CCONJ
ma-12	205	19	{	{	PUNCT
ma-12	205	20	βn	βn	VERB
ma-12	205	21	}	}	PUNCT
ma-12	205	22	are	be	AUX
ma-12	205	23	real	real	ADJ
ma-12	205	24	sequences	sequence	NOUN
ma-12	205	25	∈	∈	PROPN
ma-12	206	1	[	[	X
ma-12	206	2	0	0	NUM
ma-12	206	3	,	,	PUNCT
ma-12	206	4	1	1	NUM
ma-12	206	5	)	)	PUNCT
ma-12	206	6	.	.	PUNCT
ma-12	207	1	suppose	suppose	VERB
ma-12	207	2	f	f	X
ma-12	207	3	=	=	PRON
ma-12	207	4	(	(	PUNCT
ma-12	207	5	f	f	X
ma-12	207	6	(	(	PUNCT
ma-12	207	7	ti	ti	NOUN
ma-12	207	8	)	)	PUNCT
ma-12	207	9	∩	∩	ADJ
ma-12	207	10	f	f	X
ma-12	207	11	(	(	PUNCT
ma-12	207	12	si	si	NOUN
ma-12	207	13	)	)	PUNCT
ma-12	207	14	)	)	PUNCT
ma-12	208	1	6=	6=	ADP
ma-12	208	2	∅.	∅.	VERB
ma-12	208	3	if	if	SCONJ
ma-12	208	4	the	the	DET
ma-12	208	5	following	follow	VERB
ma-12	208	6	conditions	condition	NOUN
ma-12	208	7	hold	hold	VERB
ma-12	208	8	:	:	PUNCT
ma-12	208	9	i.	i.	NOUN
ma-12	208	10	∑∞	∑∞	PROPN
ma-12	208	11	n=1	n=1	PROPN
ma-12	208	12	k	k	PROPN
ma-12	208	13	(	(	PUNCT
ma-12	208	14	1	1	NUM
ma-12	208	15	)	)	PUNCT
ma-12	208	16	n	n	CCONJ
ma-12	208	17	<	<	X
ma-12	208	18	∞	∞	PROPN
ma-12	208	19	,	,	PUNCT
ma-12	208	20	∑∞	∑∞	NOUN
ma-12	208	21	n=1	n=1	PROPN
ma-12	208	22	k	k	PROPN
ma-12	208	23	(	(	PUNCT
ma-12	208	24	2	2	NUM
ma-12	208	25	)	)	PUNCT
ma-12	208	26	n	n	CCONJ
ma-12	208	27	<	<	X
ma-12	208	28	∞	∞	PROPN
ma-12	208	29	,	,	PUNCT
ma-12	208	30	∑∞	∑∞	NOUN
ma-12	208	31	n=1	n=1	PROPN
ma-12	208	32	k	k	PROPN
ma-12	208	33	(	(	PUNCT
ma-12	208	34	3	3	NUM
ma-12	208	35	)	)	PUNCT
ma-12	208	36	n	n	CCONJ
ma-12	208	37	<	<	X
ma-12	208	38	∞	∞	PROPN
ma-12	208	39	,	,	PUNCT
ma-12	208	40	∑∞	∑∞	NOUN
ma-12	208	41	n=1	n=1	PROPN
ma-12	208	42	µ	µ	X
ma-12	208	43	(	(	PUNCT
ma-12	208	44	1	1	NUM
ma-12	208	45	)	)	PUNCT
ma-12	208	46	n	n	CCONJ
ma-12	208	47	<	<	X
ma-12	208	48	∞	∞	PROPN
ma-12	208	49	,	,	PUNCT
ma-12	208	50	∑∞	∑∞	NOUN
ma-12	208	51	n=1	n=1	PROPN
ma-12	208	52	µ	µ	X
ma-12	208	53	(	(	PUNCT
ma-12	208	54	2	2	NUM
ma-12	208	55	)	)	PUNCT
ma-12	208	56	n	n	NOUN
ma-12	208	57	<	<	X
ma-12	208	58	∞,∑∞	∞,∑∞	ADJ
ma-12	208	59	n=1	n=1	PROPN
ma-12	208	60	µ	µ	X
ma-12	208	61	(	(	PUNCT
ma-12	208	62	3	3	NUM
ma-12	208	63	)	)	PUNCT
ma-12	208	64	n	n	CCONJ
ma-12	208	65	<	<	X
ma-12	208	66	∞	∞	PROPN
ma-12	208	67	,	,	PUNCT
ma-12	208	68	∑∞	∑∞	NOUN
ma-12	208	69	n=1	n=1	PROPN
ma-12	208	70	ν	ν	NOUN
ma-12	208	71	(	(	PUNCT
ma-12	208	72	1	1	NUM
ma-12	208	73	)	)	PUNCT
ma-12	208	74	n	n	CCONJ
ma-12	208	75	<	<	X
ma-12	208	76	∞	∞	PROPN
ma-12	208	77	,	,	PUNCT
ma-12	208	78	∑∞	∑∞	NOUN
ma-12	208	79	n=1	n=1	PROPN
ma-12	208	80	ν	ν	NOUN
ma-12	208	81	(	(	PUNCT
ma-12	208	82	2	2	NUM
ma-12	208	83	)	)	PUNCT
ma-12	208	84	n	n	CCONJ
ma-12	208	85	<	<	X
ma-12	208	86	∞	∞	PROPN
ma-12	208	87	,	,	PUNCT
ma-12	208	88	∑∞	∑∞	NOUN
ma-12	208	89	n=1	n=1	PROPN
ma-12	208	90	ν	ν	NOUN
ma-12	208	91	(	(	PUNCT
ma-12	208	92	3	3	NUM
ma-12	208	93	)	)	PUNCT
ma-12	208	94	n	n	CCONJ
ma-12	208	95	<	<	X
ma-12	208	96	∞,ii	∞,ii	NUM
ma-12	208	97	.	.	PUNCT
ma-12	209	1	there	there	PRON
ma-12	209	2	exists	exist	VERB
ma-12	209	3	a	a	DET
ma-12	209	4	constant	constant	ADJ
ma-12	209	5	m	m	NOUN
ma-12	209	6	>	>	X
ma-12	209	7	0	0	NUM
ma-12	209	8	such	such	ADJ
ma-12	209	9	thatψ(t	thatψ(t	PROPN
ma-12	209	10	)	)	PUNCT
ma-12	209	11	=	=	SYM
ma-12	209	12	φ(t	φ(t	PROPN
ma-12	209	13	)	)	PUNCT
ma-12	209	14	≤	≤	PROPN
ma-12	209	15	mt	mt	PROPN
ma-12	209	16	,	,	PUNCT
ma-12	209	17	t	t	PROPN
ma-12	209	18	≤	≤	NUM
ma-12	209	19	0	0	NUM
ma-12	209	20	.	.	PUNCT
ma-12	210	1	then	then	ADV
ma-12	210	2	,	,	PUNCT
ma-12	210	3	limn∞	limn∞	PROPN
ma-12	210	4	‖xn	‖xn	PROPN
ma-12	210	5	−	−	NOUN
ma-12	210	6	q‖	q‖	NOUN
ma-12	210	7	and	and	CCONJ
ma-12	210	8	limn∞	limn∞	VERB
ma-12	210	9	d(xn	d(xn	NUM
ma-12	211	1	−	−	PROPN
ma-12	211	2	f	f	NOUN
ma-12	211	3	)	)	PUNCT
ma-12	211	4	both	both	PRON
ma-12	211	5	exist	exist	VERB
ma-12	211	6	for	for	ADP
ma-12	211	7	all	all	PRON
ma-12	211	8	q	q	PROPN
ma-12	211	9	∈	∈	PROPN
ma-12	212	1	f	f	X
ma-12	212	2	.	.	PUNCT
ma-12	213	1	eur	eur	PROPN
ma-12	213	2	.	.	PUNCT
ma-12	214	1	j.	j.	PROPN
ma-12	214	2	math	math	PROPN
ma-12	214	3	.	.	PUNCT
ma-12	215	1	anal	anal	ADJ
ma-12	215	2	.	.	PUNCT
ma-12	216	1	1	1	NUM
ma-12	216	2	(	(	PUNCT
ma-12	216	3	2021	2021	NUM
ma-12	216	4	)	)	PUNCT
ma-12	216	5	53	53	NUM
ma-12	216	6	proof	proof	NOUN
ma-12	216	7	.	.	PUNCT
ma-12	217	1	set	set	VERB
ma-12	217	2	hn	hn	NOUN
ma-12	217	3	=	=	PUNCT
ma-12	217	4	max(k	max(k	PROPN
ma-12	217	5	(	(	PUNCT
ma-12	217	6	1	1	NUM
ma-12	217	7	)	)	PUNCT
ma-12	217	8	n	n	NOUN
ma-12	217	9	,	,	PUNCT
ma-12	217	10	k	k	X
ma-12	217	11	(	(	PUNCT
ma-12	217	12	2	2	NUM
ma-12	217	13	)	)	PUNCT
ma-12	217	14	n	n	NOUN
ma-12	217	15	,	,	PUNCT
ma-12	217	16	k	k	X
ma-12	217	17	(	(	PUNCT
ma-12	217	18	3	3	NUM
ma-12	217	19	)	)	PUNCT
ma-12	217	20	n	n	NOUN
ma-12	217	21	,	,	PUNCT
ma-12	217	22	µ	µ	X
ma-12	217	23	(	(	PUNCT
ma-12	217	24	1	1	NUM
ma-12	217	25	)	)	PUNCT
ma-12	217	26	n	n	NOUN
ma-12	217	27	,	,	PUNCT
ma-12	217	28	µ	µ	X
ma-12	217	29	(	(	PUNCT
ma-12	217	30	2	2	NUM
ma-12	217	31	)	)	PUNCT
ma-12	217	32	n	n	NOUN
ma-12	217	33	,	,	PUNCT
ma-12	217	34	µ	µ	X
ma-12	217	35	(	(	PUNCT
ma-12	217	36	3	3	NUM
ma-12	217	37	)	)	PUNCT
ma-12	217	38	n	n	CCONJ
ma-12	217	39	)	)	PUNCT
ma-12	217	40	,	,	PUNCT
ma-12	217	41	m	m	VERB
ma-12	217	42	=	=	SYM
ma-12	217	43	max(m1,m2,m3,m4,m5,m6	max(m1,m2,m3,m4,m5,m6	PROPN
ma-12	217	44	)	)	PUNCT
ma-12	217	45	and	and	CCONJ
ma-12	217	46	θn	θn	X
ma-12	217	47	=	=	PROPN
ma-12	217	48	max(ν	max(ν	PROPN
ma-12	217	49	(	(	PUNCT
ma-12	217	50	1	1	NUM
ma-12	217	51	)	)	PUNCT
ma-12	217	52	n	n	NOUN
ma-12	217	53	,	,	PUNCT
ma-12	217	54	ν	ν	X
ma-12	217	55	(	(	PUNCT
ma-12	217	56	2	2	NUM
ma-12	217	57	)	)	PUNCT
ma-12	217	58	n	n	NOUN
ma-12	217	59	,	,	PUNCT
ma-12	217	60	ν	ν	X
ma-12	217	61	(	(	PUNCT
ma-12	217	62	3	3	NUM
ma-12	217	63	)	)	PUNCT
ma-12	217	64	n	n	NOUN
ma-12	217	65	,	,	PUNCT
ma-12	217	66	ω	ω	PROPN
ma-12	217	67	(	(	PUNCT
ma-12	217	68	1	1	NUM
ma-12	217	69	)	)	PUNCT
ma-12	217	70	n	n	NOUN
ma-12	217	71	,	,	PUNCT
ma-12	217	72	ω	ω	PROPN
ma-12	217	73	(	(	PUNCT
ma-12	217	74	2	2	NUM
ma-12	217	75	)	)	PUNCT
ma-12	217	76	n	n	NOUN
ma-12	217	77	,	,	PUNCT
ma-12	217	78	ω	ω	PROPN
ma-12	217	79	(	(	PUNCT
ma-12	217	80	3	3	NUM
ma-12	217	81	)	)	PUNCT
ma-12	217	82	n	n	CCONJ
ma-12	217	83	)	)	PUNCT
ma-12	217	84	.	.	PUNCT
ma-12	218	1	then	then	ADV
ma-12	218	2	,	,	PUNCT
ma-12	218	3	∑∞	∑∞	NOUN
ma-12	218	4	n=1	n=1	PUNCT
ma-12	218	5	hn	hn	PROPN
ma-12	218	6	<	<	X
ma-12	218	7	∞	∞	PROPN
ma-12	218	8	and	and	CCONJ
ma-12	218	9	∑∞	∑∞	NOUN
ma-12	218	10	n=1	n=1	PUNCT
ma-12	218	11	θn	θn	PROPN
ma-12	218	12	<	<	X
ma-12	218	13	∞.	∞.	PROPN
ma-12	218	14	for	for	ADP
ma-12	218	15	any	any	DET
ma-12	218	16	q	q	NOUN
ma-12	218	17	∈	∈	PROPN
ma-12	218	18	f	f	PROPN
ma-12	218	19	,	,	PUNCT
ma-12	218	20	itfollows	itfollow	VERB
ma-12	218	21	from	from	ADP
ma-12	218	22	(	(	PUNCT
ma-12	218	23	3.1	3.1	NUM
ma-12	218	24	)	)	PUNCT
ma-12	218	25	that	that	PRON
ma-12	219	1	‖zn	‖zn	NUM
ma-12	219	2	−	−	NUM
ma-12	219	3	q‖	q‖	NOUN
ma-12	219	4	=	=	SYM
ma-12	219	5	|p	|p	X
ma-12	219	6	(	(	PUNCT
ma-12	219	7	(	(	PUNCT
ma-12	219	8	1−	1−	NUM
ma-12	219	9	βn)sn3xn	βn)sn3xn	X
ma-12	219	10	+	+	NUM
ma-12	219	11	βnt3(pt3	βnt3(pt3	X
ma-12	219	12	)	)	PUNCT
ma-12	219	13	n−1xn)−	n−1xn)−	PROPN
ma-12	219	14	p	p	X
ma-12	219	15	(	(	PUNCT
ma-12	219	16	q)‖	q)‖	NOUN
ma-12	219	17	≤	≤	ADV
ma-12	219	18	‖(1−	‖(1−	PROPN
ma-12	219	19	βn)sn3xn	βn)sn3xn	X
ma-12	220	1	+	+	CCONJ
ma-12	220	2	βnt3(pt	βnt3(pt	PROPN
ma-12	220	3	n−1	n−1	PROPN
ma-12	220	4	3	3	NUM
ma-12	220	5	xn	xn	NOUN
ma-12	220	6	−	−	NOUN
ma-12	220	7	q‖	q‖	NOUN
ma-12	220	8	=	=	SYM
ma-12	220	9	‖(1−	‖(1−	X
ma-12	220	10	βn)sn3xn	βn)sn3xn	X
ma-12	220	11	+	+	NUM
ma-12	220	12	βnq	βnq	NOUN
ma-12	221	1	−	−	NOUN
ma-12	221	2	q	q	NOUN
ma-12	221	3	−	−	NOUN
ma-12	221	4	βnq	βnq	NOUN
ma-12	222	1	+	+	CCONJ
ma-12	222	2	βnt3(pt3	βnt3(pt3	X
ma-12	222	3	)	)	PUNCT
ma-12	223	1	n−1xn‖	n−1xn‖	NUM
ma-12	223	2	=	=	SYM
ma-12	223	3	‖(1−	‖(1−	X
ma-12	223	4	βn)sn3xn	βn)sn3xn	X
ma-12	223	5	−	−	PROPN
ma-12	223	6	(	(	PUNCT
ma-12	223	7	1−	1−	NUM
ma-12	223	8	βn)q	βn)q	PUNCT
ma-12	224	1	+	+	NUM
ma-12	224	2	βn(t3(pt3	βn(t3(pt3	X
ma-12	224	3	)	)	PUNCT
ma-12	224	4	n−1xn	n−1xn	ADJ
ma-12	224	5	−	−	NOUN
ma-12	224	6	q)‖	q)‖	NOUN
ma-12	224	7	=	=	SYM
ma-12	224	8	‖(1−	‖(1−	PROPN
ma-12	224	9	βn)(sn3xn	βn)(sn3xn	NOUN
ma-12	224	10	−	−	NOUN
ma-12	224	11	q	q	NOUN
ma-12	224	12	)	)	PUNCT
ma-12	224	13	+	+	CCONJ
ma-12	224	14	βn(t3(pt3	βn(t3(pt3	X
ma-12	224	15	)	)	PUNCT
ma-12	224	16	n−1xn	n−1xn	ADJ
ma-12	224	17	−	−	PROPN
ma-12	224	18	q)‖	q)‖	NOUN
ma-12	224	19	(	(	PUNCT
ma-12	224	20	3.11	3.11	NUM
ma-12	224	21	)	)	PUNCT
ma-12	224	22	≤	≤	NOUN
ma-12	224	23	(	(	PUNCT
ma-12	224	24	1−	1−	NUM
ma-12	224	25	βn)‖sn3xn	βn)‖sn3xn	NOUN
ma-12	225	1	−	−	ADP
ma-12	225	2	q‖+	q‖+	PROPN
ma-12	226	1	βn‖t3(pt3)n−1xn	βn‖t3(pt3)n−1xn	NOUN
ma-12	227	1	−	−	NOUN
ma-12	227	2	q‖	q‖	NOUN
ma-12	227	3	≤	≤	X
ma-12	227	4	(	(	PUNCT
ma-12	227	5	1−	1−	NUM
ma-12	227	6	βn)[‖xn	βn)[‖xn	SYM
ma-12	228	1	−	−	PROPN
ma-12	228	2	q‖+	q‖+	PROPN
ma-12	228	3	k	k	PROPN
ma-12	228	4	(	(	PUNCT
ma-12	228	5	3	3	NUM
ma-12	228	6	)	)	PUNCT
ma-12	228	7	n	n	CCONJ
ma-12	228	8	ψ(‖xn	ψ(‖xn	PRON
ma-12	228	9	−	−	PROPN
ma-12	228	10	q‖	q‖	CCONJ
ma-12	228	11	)	)	PUNCT
ma-12	229	1	+	+	CCONJ
ma-12	229	2	ω	ω	NUM
ma-12	229	3	(	(	PUNCT
ma-12	229	4	3	3	NUM
ma-12	229	5	)	)	PUNCT
ma-12	229	6	n	n	NOUN
ma-12	229	7	]	]	PUNCT
ma-12	230	1	+	+	CCONJ
ma-12	230	2	βn[‖xn	βn[‖xn	PROPN
ma-12	230	3	−	−	NOUN
ma-12	230	4	q‖+	q‖+	PROPN
ma-12	230	5	µ	µ	X
ma-12	230	6	(	(	PUNCT
ma-12	230	7	3	3	NUM
ma-12	230	8	)	)	PUNCT
ma-12	230	9	n	n	PROPN
ma-12	230	10	φ(‖xn	φ(‖xn	PROPN
ma-12	230	11	−	−	PROPN
ma-12	230	12	q‖	q‖	CCONJ
ma-12	230	13	)	)	PUNCT
ma-12	231	1	+	+	NUM
ma-12	231	2	ν	ν	X
ma-12	231	3	(	(	PUNCT
ma-12	231	4	3	3	NUM
ma-12	231	5	)	)	PUNCT
ma-12	231	6	n	n	NOUN
ma-12	231	7	]	]	PUNCT
ma-12	231	8	=	=	PUNCT
ma-12	231	9	(	(	PUNCT
ma-12	231	10	1−	1−	NUM
ma-12	231	11	βn)‖xn	βn)‖xn	SYM
ma-12	232	1	−	−	PUNCT
ma-12	232	2	q‖+	q‖+	PROPN
ma-12	232	3	(	(	PUNCT
ma-12	232	4	1−	1−	NUM
ma-12	232	5	βn)hnψ(‖xn	βn)hnψ(‖xn	PROPN
ma-12	232	6	−	−	PROPN
ma-12	232	7	q‖	q‖	CCONJ
ma-12	232	8	)	)	PUNCT
ma-12	233	1	+	+	CCONJ
ma-12	233	2	(	(	PUNCT
ma-12	233	3	1−	1−	NUM
ma-12	233	4	βn)θn	βn)θn	PUNCT
ma-12	233	5	+	+	CCONJ
ma-12	233	6	βn‖xn	βn‖xn	NUM
ma-12	233	7	−	−	PROPN
ma-12	233	8	q‖	q‖	PROPN
ma-12	233	9	+	+	PROPN
ma-12	233	10	βnhnφ(‖xn	βnhnφ(‖xn	PUNCT
ma-12	233	11	−	−	NOUN
ma-12	233	12	q‖	q‖	NOUN
ma-12	233	13	)	)	PUNCT
ma-12	234	1	+	+	NUM
ma-12	234	2	βnθn	βnθn	NOUN
ma-12	234	3	≤	≤	NOUN
ma-12	234	4	(	(	PUNCT
ma-12	234	5	1−	1−	NUM
ma-12	234	6	βn)(1	βn)(1	SYM
ma-12	234	7	+	+	NUM
ma-12	234	8	hnm5)‖xn	hnm5)‖xn	NOUN
ma-12	234	9	−	−	PROPN
ma-12	234	10	q‖+	q‖+	PROPN
ma-12	234	11	βn(1	βn(1	NOUN
ma-12	234	12	+	+	CCONJ
ma-12	234	13	hnm6)‖xn	hnm6)‖xn	NOUN
ma-12	234	14	−	−	ADP
ma-12	234	15	q‖+	q‖+	PROPN
ma-12	234	16	θn	θn	ADP
ma-12	234	17	≤	≤	NOUN
ma-12	234	18	(	(	PUNCT
ma-12	234	19	1−	1−	NUM
ma-12	234	20	βn)(1	βn)(1	PUNCT
ma-12	235	1	+	+	CCONJ
ma-12	235	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	235	3	−	−	PROPN
ma-12	235	4	q‖+	q‖+	ADJ
ma-12	235	5	βn(1	βn(1	PROPN
ma-12	235	6	+	+	CCONJ
ma-12	236	1	hnm)‖xn	hnm)‖xn	PROPN
ma-12	236	2	−	−	PROPN
ma-12	236	3	q‖+	q‖+	PROPN
ma-12	236	4	θn	θn	ADP
ma-12	236	5	≤	≤	NOUN
ma-12	236	6	(	(	PUNCT
ma-12	236	7	1	1	NUM
ma-12	236	8	+	+	CCONJ
ma-12	236	9	hnm)‖xn	hnm)‖xn	PROPN
ma-12	236	10	−	−	PROPN
ma-12	236	11	q‖+	q‖+	PROPN
ma-12	236	12	θn	θn	PROPN
ma-12	236	13	.	.	PROPN
ma-12	236	14	(	(	PUNCT
ma-12	236	15	3.12	3.12	NUM
ma-12	236	16	)	)	PUNCT
ma-12	236	17	also	also	ADV
ma-12	236	18	,	,	PUNCT
ma-12	236	19	form	form	NOUN
ma-12	236	20	(	(	PUNCT
ma-12	236	21	1.7	1.7	NUM
ma-12	236	22	)	)	PUNCT
ma-12	236	23	,	,	PUNCT
ma-12	236	24	we	we	PRON
ma-12	236	25	get	get	VERB
ma-12	236	26	‖yn	‖yn	PUNCT
ma-12	236	27	−	−	PROPN
ma-12	236	28	q‖	q‖	NOUN
ma-12	236	29	=	=	SYM
ma-12	236	30	|p	|p	X
ma-12	236	31	(	(	PUNCT
ma-12	236	32	(	(	PUNCT
ma-12	236	33	1−	1−	NUM
ma-12	236	34	βn)sn2xn	βn)sn2xn	X
ma-12	236	35	+	+	CCONJ
ma-12	236	36	βnt2(pt2	βnt2(pt2	NOUN
ma-12	236	37	)	)	PUNCT
ma-12	236	38	n−1zn)−	n−1zn)−	PROPN
ma-12	236	39	p	p	X
ma-12	236	40	(	(	PUNCT
ma-12	236	41	q)‖	q)‖	NOUN
ma-12	236	42	≤	≤	NUM
ma-12	236	43	‖(1−	‖(1−	PROPN
ma-12	236	44	βn)sn2xn	βn)sn2xn	X
ma-12	236	45	+	+	CCONJ
ma-12	236	46	βnt2(pt2	βnt2(pt2	NOUN
ma-12	236	47	)	)	PUNCT
ma-12	236	48	n−1xn	n−1xn	NOUN
ma-12	236	49	−	−	PROPN
ma-12	237	1	q‖	q‖	NOUN
ma-12	237	2	=	=	SYM
ma-12	237	3	‖(1−	‖(1−	X
ma-12	237	4	βn)sn2xn	βn)sn2xn	X
ma-12	237	5	+	+	PUNCT
ma-12	237	6	βnq	βnq	NOUN
ma-12	238	1	−	−	NOUN
ma-12	238	2	q	q	NOUN
ma-12	238	3	−	−	NOUN
ma-12	238	4	βnq	βnq	NOUN
ma-12	239	1	+	+	CCONJ
ma-12	239	2	βnt2(pt2	βnt2(pt2	NOUN
ma-12	239	3	)	)	PUNCT
ma-12	239	4	n−1zn‖	n−1zn‖	ADV
ma-12	239	5	=	=	SYM
ma-12	239	6	‖(1−	‖(1−	X
ma-12	239	7	βn)sn2xn	βn)sn2xn	X
ma-12	239	8	−	−	PROPN
ma-12	240	1	(	(	PUNCT
ma-12	240	2	1−	1−	NUM
ma-12	240	3	βn)q	βn)q	PUNCT
ma-12	240	4	+	+	NUM
ma-12	240	5	βn(t2(pt2	βn(t2(pt2	NOUN
ma-12	240	6	)	)	PUNCT
ma-12	240	7	n−1zn	n−1zn	NOUN
ma-12	240	8	−	−	PROPN
ma-12	241	1	q)‖	q)‖	NOUN
ma-12	242	1	(	(	PUNCT
ma-12	242	2	3.13	3.13	NUM
ma-12	242	3	)	)	PUNCT
ma-12	242	4	=	=	SYM
ma-12	243	1	‖(1−	‖(1−	X
ma-12	243	2	βn)(sn2xn	βn)(sn2xn	NOUN
ma-12	243	3	−	−	NOUN
ma-12	243	4	q	q	NOUN
ma-12	243	5	)	)	PUNCT
ma-12	243	6	+	+	CCONJ
ma-12	243	7	βn(t2(pt2	βn(t2(pt2	NOUN
ma-12	243	8	)	)	PUNCT
ma-12	243	9	n−1zn	n−1zn	NOUN
ma-12	243	10	−	−	PROPN
ma-12	243	11	q)‖	q)‖	NOUN
ma-12	243	12	≤	≤	NUM
ma-12	243	13	(	(	PUNCT
ma-12	243	14	1−	1−	NUM
ma-12	243	15	βn)‖sn2xn	βn)‖sn2xn	PUNCT
ma-12	243	16	−	−	ADP
ma-12	243	17	q‖+	q‖+	ADJ
ma-12	243	18	βn‖t2(pt2)n−1zn	βn‖t2(pt2)n−1zn	NOUN
ma-12	243	19	−	−	NOUN
ma-12	243	20	q‖	q‖	NOUN
ma-12	243	21	≤	≤	X
ma-12	243	22	(	(	PUNCT
ma-12	243	23	1−	1−	NUM
ma-12	243	24	βn)[‖xn	βn)[‖xn	SYM
ma-12	243	25	−	−	PROPN
ma-12	243	26	q‖+	q‖+	PROPN
ma-12	243	27	k	k	X
ma-12	243	28	(	(	PUNCT
ma-12	243	29	2	2	NUM
ma-12	243	30	)	)	PUNCT
ma-12	243	31	n	n	CCONJ
ma-12	243	32	ψ(‖xn	ψ(‖xn	PRON
ma-12	243	33	−	−	PROPN
ma-12	243	34	q‖	q‖	CCONJ
ma-12	243	35	)	)	PUNCT
ma-12	244	1	+	+	CCONJ
ma-12	244	2	ω	ω	NUM
ma-12	244	3	(	(	PUNCT
ma-12	244	4	2	2	NUM
ma-12	244	5	)	)	PUNCT
ma-12	244	6	n	n	NOUN
ma-12	244	7	]	]	PUNCT
ma-12	245	1	+	+	CCONJ
ma-12	245	2	βn[‖zn	βn[‖zn	INTJ
ma-12	245	3	−	−	X
ma-12	245	4	q‖+	q‖+	PROPN
ma-12	245	5	µ	µ	X
ma-12	245	6	(	(	PUNCT
ma-12	245	7	2	2	NUM
ma-12	245	8	)	)	PUNCT
ma-12	245	9	n	n	PROPN
ma-12	245	10	φ(‖xn	φ(‖xn	PROPN
ma-12	245	11	−	−	PROPN
ma-12	245	12	q‖	q‖	CCONJ
ma-12	245	13	)	)	PUNCT
ma-12	246	1	+	+	NUM
ma-12	246	2	ν	ν	NOUN
ma-12	246	3	(	(	PUNCT
ma-12	246	4	2	2	NUM
ma-12	246	5	)	)	PUNCT
ma-12	246	6	n	n	NOUN
ma-12	246	7	]	]	PUNCT
ma-12	246	8	=	=	PUNCT
ma-12	246	9	(	(	PUNCT
ma-12	246	10	1−	1−	NUM
ma-12	246	11	βn)‖xn	βn)‖xn	SYM
ma-12	247	1	−	−	PUNCT
ma-12	247	2	q‖+	q‖+	PROPN
ma-12	247	3	(	(	PUNCT
ma-12	247	4	1−	1−	NUM
ma-12	247	5	βn)hnψ(‖xn	βn)hnψ(‖xn	PROPN
ma-12	247	6	−	−	PROPN
ma-12	247	7	q‖	q‖	CCONJ
ma-12	247	8	)	)	PUNCT
ma-12	248	1	+	+	CCONJ
ma-12	248	2	(	(	PUNCT
ma-12	248	3	1−	1−	NUM
ma-12	248	4	βn)θn	βn)θn	PUNCT
ma-12	248	5	+	+	CCONJ
ma-12	248	6	βn‖xn	βn‖xn	NUM
ma-12	248	7	−	−	PROPN
ma-12	248	8	q‖	q‖	NOUN
ma-12	248	9	+	+	PROPN
ma-12	248	10	βnhnφ(‖zn	βnhnφ(‖zn	PROPN
ma-12	248	11	−	−	NOUN
ma-12	248	12	q‖	q‖	NOUN
ma-12	248	13	)	)	PUNCT
ma-12	249	1	+	+	NUM
ma-12	249	2	βnθn	βnθn	NOUN
ma-12	249	3	≤	≤	NOUN
ma-12	249	4	(	(	PUNCT
ma-12	249	5	1−	1−	NUM
ma-12	249	6	βn)(1	βn)(1	PUNCT
ma-12	249	7	+	+	NUM
ma-12	249	8	hnm3)‖xn	hnm3)‖xn	PROPN
ma-12	249	9	−	−	VERB
ma-12	249	10	q‖+	q‖+	PROPN
ma-12	249	11	βn(1	βn(1	X
ma-12	249	12	+	+	CCONJ
ma-12	249	13	hnm4)‖zn	hnm4)‖zn	PROPN
ma-12	249	14	−	−	PROPN
ma-12	249	15	q‖+	q‖+	PROPN
ma-12	249	16	θn	θn	ADP
ma-12	249	17	≤	≤	NOUN
ma-12	249	18	(	(	PUNCT
ma-12	249	19	1−	1−	NUM
ma-12	249	20	βn)(1	βn)(1	PUNCT
ma-12	250	1	+	+	CCONJ
ma-12	250	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	250	3	−	−	PROPN
ma-12	250	4	q‖+	q‖+	ADJ
ma-12	250	5	βn(1	βn(1	X
ma-12	250	6	+	+	CCONJ
ma-12	251	1	hnm)‖zn	hnm)‖zn	ADP
ma-12	251	2	−	−	PROPN
ma-12	251	3	q‖+	q‖+	PROPN
ma-12	251	4	θn	θn	PROPN
ma-12	251	5	.	.	PROPN
ma-12	251	6	(	(	PUNCT
ma-12	251	7	3.14	3.14	NUM
ma-12	251	8	)	)	PUNCT
ma-12	251	9	eur	eur	PROPN
ma-12	251	10	.	.	PUNCT
ma-12	252	1	j.	j.	PROPN
ma-12	252	2	math	math	PROPN
ma-12	252	3	.	.	PUNCT
ma-12	253	1	anal	anal	ADJ
ma-12	253	2	.	.	PUNCT
ma-12	254	1	1	1	NUM
ma-12	254	2	(	(	PUNCT
ma-12	254	3	2021	2021	NUM
ma-12	254	4	)	)	PUNCT
ma-12	254	5	54putting	54putting	NOUN
ma-12	254	6	(	(	PUNCT
ma-12	254	7	3.12	3.12	NUM
ma-12	254	8	)	)	PUNCT
ma-12	254	9	into	into	ADP
ma-12	254	10	(	(	PUNCT
ma-12	254	11	3.14	3.14	NUM
ma-12	254	12	)	)	PUNCT
ma-12	254	13	,	,	PUNCT
ma-12	254	14	we	we	PRON
ma-12	254	15	have	have	VERB
ma-12	254	16	‖yn	‖yn	NUM
ma-12	254	17	−	−	NUM
ma-12	254	18	q‖	q‖	NOUN
ma-12	254	19	≤	≤	X
ma-12	254	20	(	(	PUNCT
ma-12	254	21	1−	1−	NUM
ma-12	254	22	βn)(1	βn)(1	PUNCT
ma-12	255	1	+	+	CCONJ
ma-12	255	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	255	3	−	−	PROPN
ma-12	255	4	q‖+	q‖+	ADJ
ma-12	255	5	βn(1	βn(1	X
ma-12	255	6	+	+	CCONJ
ma-12	255	7	hnm)[(1	hnm)[(1	PROPN
ma-12	255	8	+	+	CCONJ
ma-12	255	9	hnm)‖xn	hnm)‖xn	PROPN
ma-12	255	10	−	−	PROPN
ma-12	255	11	q‖+	q‖+	PROPN
ma-12	255	12	θn	θn	PROPN
ma-12	255	13	]	]	X
ma-12	255	14	+	+	NUM
ma-12	255	15	θn	θn	NOUN
ma-12	255	16	=	=	SYM
ma-12	255	17	(	(	PUNCT
ma-12	255	18	1	1	NUM
ma-12	255	19	+	+	ADP
ma-12	255	20	hnm)[(1−	hnm)[(1−	PROPN
ma-12	255	21	βn)‖xn	βn)‖xn	PRON
ma-12	255	22	−	−	PUNCT
ma-12	255	23	q‖+	q‖+	PROPN
ma-12	255	24	βn((1	βn((1	PUNCT
ma-12	256	1	+	+	CCONJ
ma-12	256	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	256	3	−	−	PROPN
ma-12	256	4	q‖+	q‖+	PROPN
ma-12	256	5	θn	θn	NOUN
ma-12	256	6	)	)	PUNCT
ma-12	256	7	]	]	PUNCT
ma-12	257	1	+	+	CCONJ
ma-12	257	2	θn	θn	ADJ
ma-12	257	3	=	=	SYM
ma-12	257	4	(	(	PUNCT
ma-12	257	5	1	1	NUM
ma-12	257	6	+	+	CCONJ
ma-12	257	7	hnm)[(1−	hnm)[(1−	NOUN
ma-12	257	8	βn	βn	ADJ
ma-12	257	9	+	+	CCONJ
ma-12	257	10	βn	βn	X
ma-12	257	11	+	+	CCONJ
ma-12	257	12	βnhnm))‖xn	βnhnm))‖xn	PROPN
ma-12	257	13	−	−	PROPN
ma-12	257	14	q‖+	q‖+	PROPN
ma-12	257	15	θn	θn	NOUN
ma-12	257	16	)	)	PUNCT
ma-12	257	17	]	]	PUNCT
ma-12	258	1	+	+	CCONJ
ma-12	258	2	θn	θn	ADJ
ma-12	258	3	≤	≤	NOUN
ma-12	258	4	(	(	PUNCT
ma-12	258	5	1	1	NUM
ma-12	258	6	+	+	NUM
ma-12	258	7	hnm)[1	hnm)[1	PROPN
ma-12	258	8	+	+	CCONJ
ma-12	258	9	hnm))‖xn	hnm))‖xn	PROPN
ma-12	258	10	−	−	PROPN
ma-12	258	11	q‖+	q‖+	PROPN
ma-12	258	12	θn	θn	NOUN
ma-12	258	13	)	)	PUNCT
ma-12	258	14	]	]	PUNCT
ma-12	259	1	+	+	CCONJ
ma-12	259	2	θn	θn	ADJ
ma-12	259	3	=	=	SYM
ma-12	259	4	(	(	PUNCT
ma-12	259	5	1	1	NUM
ma-12	259	6	+	+	NUM
ma-12	259	7	hnm)2‖xn	hnm)2‖xn	PROPN
ma-12	259	8	−	−	PUNCT
ma-12	259	9	q‖+	q‖+	ADJ
ma-12	259	10	(	(	PUNCT
ma-12	259	11	2	2	NUM
ma-12	259	12	+	+	CCONJ
ma-12	259	13	hnm)θn	hnm)θn	ADJ
ma-12	259	14	.	.	PUNCT
ma-12	260	1	(	(	PUNCT
ma-12	260	2	3.15	3.15	NUM
ma-12	260	3	)	)	PUNCT
ma-12	260	4	again	again	ADV
ma-12	260	5	,	,	PUNCT
ma-12	260	6	using	use	VERB
ma-12	260	7	(	(	PUNCT
ma-12	260	8	1.7	1.7	NUM
ma-12	260	9	)	)	PUNCT
ma-12	260	10	,	,	PUNCT
ma-12	260	11	we	we	PRON
ma-12	260	12	have	have	VERB
ma-12	260	13	‖xn+1	‖xn+1	NUM
ma-12	260	14	−	−	PROPN
ma-12	260	15	q‖	q‖	PROPN
ma-12	260	16	=	=	SYM
ma-12	260	17	|p	|p	X
ma-12	260	18	(	(	PUNCT
ma-12	260	19	(	(	PUNCT
ma-12	260	20	1−	1−	NUM
ma-12	260	21	αn)sn1xn	αn)sn1xn	NOUN
ma-12	260	22	+	+	CCONJ
ma-12	260	23	αnt1(pt1	αnt1(pt1	ADJ
ma-12	260	24	)	)	PUNCT
ma-12	260	25	n−1yn)−	n−1yn)−	NOUN
ma-12	260	26	p	p	NOUN
ma-12	260	27	(	(	PUNCT
ma-12	260	28	q)‖	q)‖	NOUN
ma-12	260	29	≤	≤	NUM
ma-12	260	30	‖(1−	‖(1−	PROPN
ma-12	260	31	αn)sn1xn	αn)sn1xn	NOUN
ma-12	260	32	+	+	CCONJ
ma-12	260	33	αnt1(pt1	αnt1(pt1	ADJ
ma-12	260	34	)	)	PUNCT
ma-12	260	35	n−1yn	n−1yn	NOUN
ma-12	260	36	−	−	PROPN
ma-12	261	1	q‖	q‖	NOUN
ma-12	261	2	=	=	SYM
ma-12	261	3	‖(1−	‖(1−	PROPN
ma-12	261	4	αn)sn1xn	αn)sn1xn	NOUN
ma-12	261	5	+	+	CCONJ
ma-12	261	6	αnq	αnq	PROPN
ma-12	261	7	−	−	PROPN
ma-12	261	8	q	q	PROPN
ma-12	261	9	−	−	PROPN
ma-12	261	10	αnq	αnq	PROPN
ma-12	261	11	+	+	CCONJ
ma-12	261	12	αnt1(pt1	αnt1(pt1	X
ma-12	261	13	)	)	PUNCT
ma-12	261	14	n−1yn‖	n−1yn‖	PROPN
ma-12	261	15	=	=	SYM
ma-12	261	16	‖(1−	‖(1−	PROPN
ma-12	261	17	αn)sn1xn	αn)sn1xn	NOUN
ma-12	261	18	−	−	PROPN
ma-12	261	19	(	(	PUNCT
ma-12	261	20	1−	1−	NUM
ma-12	261	21	αn)q	αn)q	NUM
ma-12	261	22	+	+	CCONJ
ma-12	261	23	αn(t1(pt1	αn(t1(pt1	NOUN
ma-12	261	24	)	)	PUNCT
ma-12	261	25	n−1yn	n−1yn	NOUN
ma-12	261	26	−	−	PROPN
ma-12	261	27	q)‖	q)‖	NOUN
ma-12	261	28	=	=	SYM
ma-12	261	29	‖(1−	‖(1−	PROPN
ma-12	261	30	αn)(sn1xn	αn)(sn1xn	NUM
ma-12	261	31	−	−	PROPN
ma-12	261	32	q	q	NOUN
ma-12	261	33	)	)	PUNCT
ma-12	261	34	+	+	CCONJ
ma-12	261	35	αn(t1(pt1	αn(t1(pt1	VERB
ma-12	261	36	)	)	PUNCT
ma-12	261	37	n−1yn	n−1yn	NOUN
ma-12	261	38	−	−	PROPN
ma-12	261	39	q)‖	q)‖	NOUN
ma-12	261	40	(	(	PUNCT
ma-12	261	41	3.16	3.16	NUM
ma-12	261	42	)	)	PUNCT
ma-12	261	43	≤	≤	NOUN
ma-12	261	44	(	(	PUNCT
ma-12	261	45	1−	1−	NUM
ma-12	261	46	αn)‖sn1xn	αn)‖sn1xn	NOUN
ma-12	261	47	−	−	NOUN
ma-12	262	1	q‖+	q‖+	ADJ
ma-12	262	2	αn‖t1(pt1)n−1yn	αn‖t1(pt1)n−1yn	ADP
ma-12	262	3	−	−	PROPN
ma-12	262	4	q‖	q‖	NOUN
ma-12	262	5	≤	≤	X
ma-12	262	6	(	(	PUNCT
ma-12	262	7	1−	1−	NUM
ma-12	262	8	αn)[‖xn	αn)[‖xn	SYM
ma-12	263	1	−	−	NOUN
ma-12	263	2	q‖+	q‖+	PROPN
ma-12	263	3	k	k	X
ma-12	263	4	(	(	PUNCT
ma-12	263	5	1	1	NUM
ma-12	263	6	)	)	PUNCT
ma-12	263	7	n	n	CCONJ
ma-12	263	8	ψ(‖xn	ψ(‖xn	PRON
ma-12	263	9	−	−	PROPN
ma-12	263	10	q‖	q‖	CCONJ
ma-12	263	11	)	)	PUNCT
ma-12	264	1	+	+	CCONJ
ma-12	264	2	ω	ω	NUM
ma-12	264	3	(	(	PUNCT
ma-12	264	4	1	1	NUM
ma-12	264	5	)	)	PUNCT
ma-12	264	6	n	n	NOUN
ma-12	264	7	]	]	PUNCT
ma-12	264	8	+	+	NUM
ma-12	264	9	αn[‖yn	αn[‖yn	NOUN
ma-12	264	10	−	−	NOUN
ma-12	264	11	q‖	q‖	NOUN
ma-12	264	12	+	+	PROPN
ma-12	264	13	µ	µ	X
ma-12	264	14	(	(	PUNCT
ma-12	264	15	1	1	NUM
ma-12	264	16	)	)	PUNCT
ma-12	264	17	n	n	PRON
ma-12	264	18	φ(‖yn	φ(‖yn	NOUN
ma-12	264	19	−	−	PROPN
ma-12	264	20	q‖	q‖	NOUN
ma-12	264	21	)	)	PUNCT
ma-12	265	1	+	+	CCONJ
ma-12	265	2	ν	ν	X
ma-12	265	3	(	(	PUNCT
ma-12	265	4	1	1	NUM
ma-12	265	5	)	)	PUNCT
ma-12	265	6	n	n	CCONJ
ma-12	265	7	]	]	PUNCT
ma-12	265	8	≤	≤	X
ma-12	265	9	(	(	PUNCT
ma-12	265	10	1−	1−	NUM
ma-12	265	11	αn)‖xn	αn)‖xn	NUM
ma-12	265	12	−	−	NOUN
ma-12	266	1	q‖+	q‖+	PROPN
ma-12	266	2	(	(	PUNCT
ma-12	266	3	1−	1−	NUM
ma-12	266	4	αn)hnψ(‖xn	αn)hnψ(‖xn	PROPN
ma-12	266	5	−	−	PROPN
ma-12	266	6	q‖	q‖	NOUN
ma-12	266	7	)	)	PUNCT
ma-12	267	1	+	+	CCONJ
ma-12	267	2	(	(	PUNCT
ma-12	267	3	1−	1−	NUM
ma-12	267	4	αn)θn	αn)θn	SYM
ma-12	267	5	+	+	CCONJ
ma-12	267	6	αn‖yn	αn‖yn	ADP
ma-12	267	7	−	−	PROPN
ma-12	267	8	q‖	q‖	PROPN
ma-12	267	9	+	+	NOUN
ma-12	267	10	αnhnφ(‖yn	αnhnφ(‖yn	NOUN
ma-12	267	11	−	−	NOUN
ma-12	267	12	q‖	q‖	NOUN
ma-12	267	13	)	)	PUNCT
ma-12	268	1	+	+	NUM
ma-12	268	2	αnθn	αnθn	NOUN
ma-12	268	3	≤	≤	NOUN
ma-12	268	4	(	(	PUNCT
ma-12	268	5	1−	1−	NUM
ma-12	268	6	αn)(1	αn)(1	NOUN
ma-12	268	7	+	+	CCONJ
ma-12	268	8	hnm1)‖xn	hnm1)‖xn	PROPN
ma-12	268	9	−	−	NOUN
ma-12	268	10	q‖+	q‖+	PROPN
ma-12	268	11	αn(1	αn(1	PROPN
ma-12	268	12	+	+	NOUN
ma-12	268	13	hnm2)‖yn	hnm2)‖yn	NOUN
ma-12	268	14	−	−	ADP
ma-12	268	15	q‖+	q‖+	PROPN
ma-12	268	16	θn	θn	ADP
ma-12	268	17	≤	≤	NOUN
ma-12	268	18	(	(	PUNCT
ma-12	268	19	1−	1−	NUM
ma-12	268	20	αn)(1	αn)(1	NOUN
ma-12	268	21	+	+	CCONJ
ma-12	268	22	hnm)‖xn	hnm)‖xn	PROPN
ma-12	268	23	−	−	PROPN
ma-12	268	24	q‖+	q‖+	VERB
ma-12	268	25	αn(1	αn(1	PROPN
ma-12	268	26	+	+	NUM
ma-12	268	27	hnm)‖yn	hnm)‖yn	ADJ
ma-12	268	28	−	−	PROPN
ma-12	268	29	q‖+	q‖+	PROPN
ma-12	268	30	θn	θn	PROPN
ma-12	268	31	.	.	PROPN
ma-12	269	1	(	(	PUNCT
ma-12	269	2	3.17	3.17	NUM
ma-12	269	3	)	)	PUNCT
ma-12	269	4	putting	put	VERB
ma-12	269	5	(	(	PUNCT
ma-12	269	6	3.15	3.15	NUM
ma-12	269	7	)	)	PUNCT
ma-12	269	8	into	into	ADP
ma-12	269	9	(	(	PUNCT
ma-12	269	10	3.17	3.17	NUM
ma-12	269	11	)	)	PUNCT
ma-12	269	12	,	,	PUNCT
ma-12	269	13	we	we	PRON
ma-12	269	14	obtain	obtain	VERB
ma-12	269	15	‖xn+1	‖xn+1	PUNCT
ma-12	269	16	−	−	PROPN
ma-12	269	17	q‖	q‖	NOUN
ma-12	269	18	≤	≤	X
ma-12	269	19	(	(	PUNCT
ma-12	269	20	1−	1−	NUM
ma-12	269	21	αn)(1	αn)(1	NOUN
ma-12	270	1	+	+	CCONJ
ma-12	270	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	270	3	−	−	PROPN
ma-12	270	4	q‖+	q‖+	VERB
ma-12	270	5	αn(1	αn(1	PROPN
ma-12	270	6	+	+	NOUN
ma-12	270	7	hnm)[(1	hnm)[(1	PROPN
ma-12	270	8	+	+	CCONJ
ma-12	270	9	hnm)2‖xn	hnm)2‖xn	PROPN
ma-12	270	10	−	−	PROPN
ma-12	270	11	q‖	q‖	PROPN
ma-12	270	12	+	+	PROPN
ma-12	270	13	(	(	PUNCT
ma-12	270	14	2	2	NUM
ma-12	270	15	+	+	CCONJ
ma-12	270	16	hnm)θn	hnm)θn	ADJ
ma-12	270	17	]	]	X
ma-12	271	1	+	+	CCONJ
ma-12	271	2	θn	θn	X
ma-12	271	3	]	]	X
ma-12	271	4	=	=	SYM
ma-12	271	5	(	(	PUNCT
ma-12	271	6	1	1	NUM
ma-12	271	7	+	+	NUM
ma-12	271	8	hnm)‖xn	hnm)‖xn	PROPN
ma-12	271	9	−	−	PROPN
ma-12	271	10	q‖	q‖	NOUN
ma-12	271	11	−	−	ADP
ma-12	271	12	αn(1	αn(1	PROPN
ma-12	271	13	+	+	NUM
ma-12	271	14	hnm)‖xn	hnm)‖xn	PROPN
ma-12	271	15	−	−	PROPN
ma-12	271	16	q‖+	q‖+	VERB
ma-12	271	17	αn(1	αn(1	PROPN
ma-12	271	18	+	+	NUM
ma-12	271	19	hnm)3‖xn	hnm)3‖xn	NOUN
ma-12	271	20	−	−	PUNCT
ma-12	271	21	q‖	q‖	NOUN
ma-12	272	1	+	+	PROPN
ma-12	272	2	αn(1	αn(1	X
ma-12	272	3	+	+	NUM
ma-12	272	4	hnm)(2	hnm)(2	NOUN
ma-12	272	5	+	+	CCONJ
ma-12	272	6	hnm)θn	hnm)θn	ADJ
ma-12	272	7	+	+	CCONJ
ma-12	272	8	θn	θn	ADJ
ma-12	272	9	≤	≤	NOUN
ma-12	272	10	[	[	X
ma-12	272	11	1	1	NUM
ma-12	272	12	+	+	CCONJ
ma-12	272	13	(	(	PUNCT
ma-12	272	14	3	3	NUM
ma-12	272	15	+	+	NUM
ma-12	272	16	3hnm	3hnm	NUM
ma-12	272	17	+	+	CCONJ
ma-12	272	18	h2	h2	PROPN
ma-12	272	19	nm	nm	ADJ
ma-12	272	20	2)hnm]‖xn	2)hnm]‖xn	NUM
ma-12	272	21	−	−	NOUN
ma-12	272	22	q‖+	q‖+	PROPN
ma-12	272	23	[	[	X
ma-12	272	24	1	1	NUM
ma-12	272	25	+	+	CCONJ
ma-12	272	26	(	(	PUNCT
ma-12	272	27	1	1	NUM
ma-12	272	28	+	+	NUM
ma-12	272	29	hnm)(2	hnm)(2	NOUN
ma-12	272	30	+	+	CCONJ
ma-12	272	31	hnm]θn	hnm]θn	NOUN
ma-12	272	32	=	=	SYM
ma-12	272	33	(	(	PUNCT
ma-12	272	34	1	1	NUM
ma-12	272	35	+	+	NUM
ma-12	272	36	δn)‖xn	δn)‖xn	NOUN
ma-12	272	37	−	−	VERB
ma-12	272	38	q‖+	q‖+	PROPN
ma-12	272	39	ρn	ρn	INTJ
ma-12	272	40	.	.	PUNCT
ma-12	272	41	(	(	PUNCT
ma-12	272	42	3.18	3.18	NUM
ma-12	272	43	)	)	PUNCT
ma-12	272	44	where	where	SCONJ
ma-12	272	45	δn	δn	NOUN
ma-12	272	46	=	=	SYM
ma-12	272	47	1+(3	1+(3	NUM
ma-12	272	48	+	+	PROPN
ma-12	272	49	3hnm+h2	3hnm+h2	PROPN
ma-12	272	50	nm	nm	ADJ
ma-12	272	51	2)hnm	2)hnm	NUM
ma-12	272	52	and	and	CCONJ
ma-12	272	53	ρn	ρn	NOUN
ma-12	272	54	=	=	PUNCT
ma-12	273	1	[	[	X
ma-12	273	2	1+(1+hnm)(2+hnm]θn	1+(1+hnm)(2+hnm]θn	NUM
ma-12	273	3	.	.	PUNCT
ma-12	274	1	since	since	SCONJ
ma-12	274	2	∑∞	∑∞	NOUN
ma-12	274	3	n=1	n=1	PROPN
ma-12	274	4	δn	δn	ADP
ma-12	274	5	<	<	X
ma-12	274	6	∞and	∞and	X
ma-12	274	7	∑∞	∑∞	NOUN
ma-12	274	8	n=1	n=1	PUNCT
ma-12	274	9	ρn	ρn	PROPN
ma-12	274	10	<	<	X
ma-12	274	11	∞	∞	PROPN
ma-12	274	12	,	,	PUNCT
ma-12	274	13	it	it	PRON
ma-12	274	14	follows	follow	VERB
ma-12	274	15	from	from	ADP
ma-12	274	16	lemma	lemma	PROPN
ma-12	274	17	2.1	2.1	NUM
ma-12	274	18	that	that	PRON
ma-12	274	19	limn→∞	limn→∞	PROPN
ma-12	274	20	‖xn	‖xn	PROPN
ma-12	274	21	−	−	PROPN
ma-12	274	22	q‖	q‖	NOUN
ma-12	274	23	exists	exist	VERB
ma-12	274	24	.	.	PUNCT
ma-12	275	1	eur	eur	PROPN
ma-12	275	2	.	.	PUNCT
ma-12	276	1	j.	j.	PROPN
ma-12	276	2	math	math	PROPN
ma-12	276	3	.	.	PUNCT
ma-12	277	1	anal	anal	ADJ
ma-12	277	2	.	.	PUNCT
ma-12	278	1	1	1	NUM
ma-12	278	2	(	(	PUNCT
ma-12	278	3	2021	2021	NUM
ma-12	278	4	)	)	PUNCT
ma-12	278	5	55now	55now	NOUN
ma-12	278	6	taking	take	VERB
ma-12	278	7	the	the	DET
ma-12	278	8	infimum	infimum	NOUN
ma-12	278	9	over	over	ADP
ma-12	278	10	all	all	DET
ma-12	278	11	q	q	PROPN
ma-12	278	12	∈	∈	PROPN
ma-12	278	13	f	f	X
ma-12	278	14	in	in	ADP
ma-12	278	15	(	(	PUNCT
ma-12	278	16	3.18	3.18	NUM
ma-12	278	17	)	)	PUNCT
ma-12	278	18	,	,	PUNCT
ma-12	278	19	we	we	PRON
ma-12	278	20	get	get	VERB
ma-12	278	21	d(xn+1	d(xn+1	PROPN
ma-12	278	22	,	,	PUNCT
ma-12	278	23	f	f	PROPN
ma-12	278	24	)	)	PUNCT
ma-12	278	25	≤	≤	NOUN
ma-12	279	1	(	(	PUNCT
ma-12	279	2	1	1	NUM
ma-12	279	3	+	+	SYM
ma-12	279	4	δn)d(xn	δn)d(xn	PROPN
ma-12	279	5	,	,	PUNCT
ma-12	279	6	f	f	PROPN
ma-12	279	7	)	)	PUNCT
ma-12	280	1	+	+	CCONJ
ma-12	280	2	ρn,∀n	ρn,∀n	NUM
ma-12	280	3	∈	∈	PROPN
ma-12	280	4	n.	n.	NOUN
ma-12	280	5	(	(	PUNCT
ma-12	280	6	3.19	3.19	NUM
ma-12	280	7	)	)	PUNCT
ma-12	280	8	again	again	ADV
ma-12	280	9	,	,	PUNCT
ma-12	280	10	since	since	SCONJ
ma-12	280	11	∑∞	∑∞	NOUN
ma-12	280	12	n=1	n=1	PUNCT
ma-12	280	13	δn	δn	ADP
ma-12	280	14	<	<	NOUN
ma-12	280	15	∞	∞	NUM
ma-12	280	16	and	and	CCONJ
ma-12	280	17	∑∞	∑∞	NOUN
ma-12	280	18	n=1	n=1	PROPN
ma-12	280	19	ρn	ρn	PROPN
ma-12	280	20	<	<	X
ma-12	280	21	∞	∞	PROPN
ma-12	280	22	,	,	PUNCT
ma-12	280	23	it	it	PRON
ma-12	280	24	follows	follow	VERB
ma-12	280	25	from	from	ADP
ma-12	280	26	lemma	lemma	PROPN
ma-12	280	27	2.1	2.1	NUM
ma-12	280	28	and	and	CCONJ
ma-12	280	29	(	(	PUNCT
ma-12	280	30	3.19	3.19	NUM
ma-12	280	31	)	)	PUNCT
ma-12	280	32	that	that	SCONJ
ma-12	280	33	limn→∞	limn→∞	PROPN
ma-12	280	34	d(xn	d(xn	X
ma-12	280	35	,	,	PUNCT
ma-12	280	36	f	f	NOUN
ma-12	280	37	)	)	PUNCT
ma-12	280	38	exists	exist	VERB
ma-12	280	39	.	.	PUNCT
ma-12	281	1	this	this	PRON
ma-12	281	2	completes	complete	VERB
ma-12	281	3	the	the	DET
ma-12	281	4	proof	proof	NOUN
ma-12	281	5	.	.	PUNCT
ma-12	282	1	�	�	PROPN
ma-12	282	2	lemma	lemma	PROPN
ma-12	282	3	3.3	3.3	NUM
ma-12	282	4	.	.	PUNCT
ma-12	283	1	let	let	VERB
ma-12	283	2	e	e	PRON
ma-12	283	3	be	be	AUX
ma-12	283	4	a	a	DET
ma-12	283	5	uniformly	uniformly	ADV
ma-12	283	6	convex	convex	NOUN
ma-12	283	7	banach	banach	NOUN
ma-12	283	8	space	space	NOUN
ma-12	283	9	and	and	CCONJ
ma-12	283	10	k	k	PROPN
ma-12	283	11	a	a	DET
ma-12	283	12	nonempty	nonempty	ADV
ma-12	283	13	closed	close	VERB
ma-12	283	14	convex	convex	NOUN
ma-12	283	15	subset	subset	NOUN
ma-12	283	16	of	of	ADP
ma-12	283	17	e.	e.	PROPN
ma-12	283	18	let	let	VERB
ma-12	283	19	s1	s1	PROPN
ma-12	283	20	,	,	PUNCT
ma-12	283	21	s2	s2	PROPN
ma-12	283	22	,	,	PUNCT
ma-12	283	23	s3	s3	PROPN
ma-12	283	24	:	:	PUNCT
ma-12	283	25	k	k	PROPN
ma-12	283	26	−→	−→	NOUN
ma-12	283	27	k	k	PROPN
ma-12	283	28	be	be	AUX
ma-12	283	29	three	three	NUM
ma-12	283	30	total	total	ADJ
ma-12	283	31	asymptotically	asymptotically	ADV
ma-12	283	32	nonexpansive	nonexpansive	ADJ
ma-12	283	33	self	self	NOUN
ma-12	283	34	mapping	mapping	NOUN
ma-12	283	35	with	with	ADP
ma-12	283	36	sequences	sequence	NOUN
ma-12	283	37	{	{	PUNCT
ma-12	283	38	k(1)n	k(1)n	X
ma-12	283	39	}	}	PUNCT
ma-12	283	40	,	,	PUNCT
ma-12	283	41	{	{	PUNCT
ma-12	283	42	k(2)n	k(2)n	X
ma-12	283	43	}	}	PUNCT
ma-12	283	44	,	,	PUNCT
ma-12	283	45	{	{	PUNCT
ma-12	283	46	k(3)n	k(3)n	NOUN
ma-12	283	47	}	}	PUNCT
ma-12	283	48	∈	∈	PROPN
ma-12	284	1	[	[	X
ma-12	284	2	1,∞	1,∞	NUM
ma-12	284	3	)	)	PUNCT
ma-12	284	4	,	,	PUNCT
ma-12	284	5	{	{	PUNCT
ma-12	284	6	w	w	X
ma-12	284	7	(	(	PUNCT
ma-12	284	8	1)n	1)n	X
ma-12	284	9	}	}	PUNCT
ma-12	284	10	,	,	PUNCT
ma-12	284	11	{	{	PUNCT
ma-12	284	12	w	w	NOUN
ma-12	284	13	(	(	PUNCT
ma-12	284	14	2)}n	2)}n	NOUN
ma-12	284	15	,	,	PUNCT
ma-12	284	16	{	{	PUNCT
ma-12	284	17	w	w	NOUN
ma-12	284	18	(	(	PUNCT
ma-12	284	19	3)n	3)n	NUM
ma-12	284	20	}	}	PUNCT
ma-12	284	21	∈	∈	PROPN
ma-12	285	1	[	[	X
ma-12	285	2	1,∞	1,∞	NUM
ma-12	285	3	)	)	PUNCT
ma-12	285	4	and	and	CCONJ
ma-12	285	5	t1	t1	NOUN
ma-12	285	6	,	,	PUNCT
ma-12	285	7	t2	t2	NOUN
ma-12	285	8	,	,	PUNCT
ma-12	285	9	t3	t3	NOUN
ma-12	285	10	:	:	PUNCT
ma-12	285	11	k	k	X
ma-12	285	12	−→	−→	NOUN
ma-12	285	13	e	e	NOUN
ma-12	285	14	are	be	AUX
ma-12	285	15	three	three	NUM
ma-12	285	16	total	total	ADJ
ma-12	285	17	asymptotically	asymptotically	ADV
ma-12	285	18	nonexpansive	nonexpansive	ADJ
ma-12	285	19	nonself	nonself	PROPN
ma-12	285	20	mappings	mapping	NOUN
ma-12	285	21	with	with	ADP
ma-12	285	22	sequences	sequence	NOUN
ma-12	285	23	{	{	PUNCT
ma-12	285	24	µ(1)n	µ(1)n	X
ma-12	285	25	}	}	PUNCT
ma-12	285	26	,	,	PUNCT
ma-12	285	27	{	{	PUNCT
ma-12	285	28	µ(2)n	µ(2)n	NOUN
ma-12	285	29	}	}	PUNCT
ma-12	285	30	,	,	PUNCT
ma-12	285	31	{	{	PUNCT
ma-12	285	32	µ(3)n	µ(3)n	PROPN
ma-12	285	33	}	}	PUNCT
ma-12	285	34	∈	∈	PROPN
ma-12	286	1	[	[	X
ma-12	286	2	1,∞	1,∞	NUM
ma-12	286	3	)	)	PUNCT
ma-12	286	4	,	,	PUNCT
ma-12	286	5	{	{	PUNCT
ma-12	286	6	ν(1)n	ν(1)n	X
ma-12	286	7	}	}	PUNCT
ma-12	286	8	,	,	PUNCT
ma-12	286	9	{	{	PUNCT
ma-12	286	10	ν(2)n	ν(2)n	NOUN
ma-12	286	11	}	}	PUNCT
ma-12	286	12	,	,	PUNCT
ma-12	286	13	{	{	PUNCT
ma-12	286	14	ν(3)n	ν(3)n	NOUN
ma-12	286	15	}	}	PUNCT
ma-12	286	16	∈	∈	PROPN
ma-12	287	1	[	[	X
ma-12	287	2	1,∞	1,∞	NUM
ma-12	287	3	)	)	PUNCT
ma-12	287	4	.	.	PUNCT
ma-12	288	1	let	let	VERB
ma-12	288	2	{	{	PUNCT
ma-12	288	3	xn	xn	VERB
ma-12	288	4	}	}	PUNCT
ma-12	288	5	be	be	VERB
ma-12	288	6	the	the	DET
ma-12	288	7	sequence	sequence	NOUN
ma-12	288	8	defined	define	VERB
ma-12	288	9	by	by	ADP
ma-12	288	10	(	(	PUNCT
ma-12	288	11	1.7	1.7	NUM
ma-12	288	12	)	)	PUNCT
ma-12	288	13	,	,	PUNCT
ma-12	288	14	where	where	SCONJ
ma-12	288	15	{	{	PUNCT
ma-12	288	16	αn	αn	NOUN
ma-12	288	17	}	}	PUNCT
ma-12	288	18	and	and	CCONJ
ma-12	288	19	{	{	PUNCT
ma-12	288	20	βn	βn	VERB
ma-12	288	21	}	}	PUNCT
ma-12	288	22	are	be	AUX
ma-12	288	23	real	real	ADJ
ma-12	288	24	sequences	sequence	NOUN
ma-12	288	25	∈	∈	PROPN
ma-12	289	1	[	[	X
ma-12	289	2	0	0	NUM
ma-12	289	3	,	,	PUNCT
ma-12	289	4	1	1	NUM
ma-12	289	5	)	)	PUNCT
ma-12	289	6	.	.	PUNCT
ma-12	290	1	suppose	suppose	VERB
ma-12	291	1	f	f	X
ma-12	291	2	=	=	PRON
ma-12	291	3	(	(	PUNCT
ma-12	291	4	f	f	X
ma-12	291	5	(	(	PUNCT
ma-12	291	6	ti)∩f	ti)∩f	X
ma-12	291	7	(	(	PUNCT
ma-12	291	8	si	si	NOUN
ma-12	291	9	)	)	PUNCT
ma-12	291	10	)	)	PUNCT
ma-12	292	1	6=	6=	ADP
ma-12	292	2	∅.	∅.	VERB
ma-12	292	3	if	if	SCONJ
ma-12	292	4	the	the	DET
ma-12	292	5	following	follow	VERB
ma-12	292	6	conditions	condition	NOUN
ma-12	292	7	hold	hold	VERB
ma-12	292	8	:	:	PUNCT
ma-12	292	9	i.	i.	NOUN
ma-12	292	10	∑∞	∑∞	PROPN
ma-12	292	11	n=1	n=1	PROPN
ma-12	292	12	k	k	PROPN
ma-12	292	13	(	(	PUNCT
ma-12	292	14	1	1	NUM
ma-12	292	15	)	)	PUNCT
ma-12	292	16	n	n	CCONJ
ma-12	292	17	<	<	X
ma-12	292	18	∞	∞	PROPN
ma-12	292	19	,	,	PUNCT
ma-12	292	20	∑∞	∑∞	NOUN
ma-12	292	21	n=1	n=1	PROPN
ma-12	292	22	k	k	PROPN
ma-12	292	23	(	(	PUNCT
ma-12	292	24	2	2	NUM
ma-12	292	25	)	)	PUNCT
ma-12	292	26	n	n	CCONJ
ma-12	292	27	<	<	X
ma-12	292	28	∞	∞	PROPN
ma-12	292	29	,	,	PUNCT
ma-12	292	30	∑∞	∑∞	NOUN
ma-12	292	31	n=1	n=1	PROPN
ma-12	292	32	k	k	PROPN
ma-12	292	33	(	(	PUNCT
ma-12	292	34	3	3	NUM
ma-12	292	35	)	)	PUNCT
ma-12	292	36	n	n	CCONJ
ma-12	292	37	<	<	X
ma-12	292	38	∞	∞	PROPN
ma-12	292	39	,	,	PUNCT
ma-12	292	40	∑∞	∑∞	NOUN
ma-12	292	41	n=1	n=1	PROPN
ma-12	292	42	µ	µ	X
ma-12	292	43	(	(	PUNCT
ma-12	292	44	1	1	NUM
ma-12	292	45	)	)	PUNCT
ma-12	292	46	n	n	CCONJ
ma-12	292	47	<	<	X
ma-12	292	48	∞	∞	PROPN
ma-12	292	49	,	,	PUNCT
ma-12	292	50	∑∞	∑∞	NOUN
ma-12	292	51	n=1	n=1	PROPN
ma-12	292	52	µ	µ	X
ma-12	292	53	(	(	PUNCT
ma-12	292	54	2	2	NUM
ma-12	292	55	)	)	PUNCT
ma-12	292	56	n	n	NOUN
ma-12	292	57	<	<	X
ma-12	292	58	∞,∑∞	∞,∑∞	ADJ
ma-12	292	59	n=1	n=1	PROPN
ma-12	292	60	µ	µ	X
ma-12	292	61	(	(	PUNCT
ma-12	292	62	3	3	NUM
ma-12	292	63	)	)	PUNCT
ma-12	292	64	n	n	CCONJ
ma-12	292	65	<	<	X
ma-12	292	66	∞	∞	PROPN
ma-12	292	67	,	,	PUNCT
ma-12	292	68	∑∞	∑∞	NOUN
ma-12	292	69	n=1	n=1	PROPN
ma-12	292	70	ν	ν	NOUN
ma-12	292	71	(	(	PUNCT
ma-12	292	72	1	1	NUM
ma-12	292	73	)	)	PUNCT
ma-12	292	74	n	n	CCONJ
ma-12	292	75	<	<	X
ma-12	292	76	∞	∞	PROPN
ma-12	292	77	,	,	PUNCT
ma-12	292	78	∑∞	∑∞	NOUN
ma-12	292	79	n=1	n=1	PROPN
ma-12	292	80	ν	ν	NOUN
ma-12	292	81	(	(	PUNCT
ma-12	292	82	2	2	NUM
ma-12	292	83	)	)	PUNCT
ma-12	292	84	n	n	CCONJ
ma-12	292	85	<	<	X
ma-12	292	86	∞	∞	PROPN
ma-12	292	87	,	,	PUNCT
ma-12	292	88	∑∞	∑∞	NOUN
ma-12	292	89	n=1	n=1	PROPN
ma-12	292	90	ν	ν	NOUN
ma-12	292	91	(	(	PUNCT
ma-12	292	92	3	3	NUM
ma-12	292	93	)	)	PUNCT
ma-12	292	94	n	n	CCONJ
ma-12	292	95	<	<	X
ma-12	292	96	∞,ii	∞,ii	NUM
ma-12	292	97	.	.	PUNCT
ma-12	292	98	‖x−t1(pt1)n−1y‖	‖x−t1(pt1)n−1y‖	NOUN
ma-12	292	99	≤	≤	NUM
ma-12	292	100	‖sn1x−t1(pt1)n−1y‖	‖sn1x−t1(pt1)n−1y‖	NOUN
ma-12	292	101	,	,	PUNCT
ma-12	292	102	‖x−t2(pt2)n−1y‖	‖x−t2(pt2)n−1y‖	NUM
ma-12	292	103	≤	≤	NOUN
ma-12	292	104	‖sn2x−t2(pt2)n−1y‖	‖sn2x−t2(pt2)n−1y‖	PROPN
ma-12	292	105	,	,	PUNCT
ma-12	292	106	‖x	‖x	NOUN
ma-12	292	107	−	−	PROPN
ma-12	292	108	t3(pt3)n−1y‖	t3(pt3)n−1y‖	PROPN
ma-12	292	109	≤	≤	NOUN
ma-12	293	1	‖sn3x	‖sn3x	ADP
ma-12	293	2	−	−	PROPN
ma-12	293	3	t3(pt3)n−1y‖iii	t3(pt3)n−1y‖iii	NUM
ma-12	293	4	.	.	PUNCT
ma-12	294	1	there	there	PRON
ma-12	294	2	exists	exist	VERB
ma-12	294	3	a	a	DET
ma-12	294	4	constant	constant	ADJ
ma-12	294	5	m1,m2	m1,m2	PROPN
ma-12	294	6	>	>	PUNCT
ma-12	294	7	0	0	NUM
ma-12	294	8	such	such	ADJ
ma-12	294	9	that	that	SCONJ
ma-12	294	10	ψ(t	ψ(t	PROPN
ma-12	294	11	)	)	PUNCT
ma-12	294	12	≤	≤	NOUN
ma-12	294	13	m1	m1	PROPN
ma-12	294	14	t	t	PROPN
ma-12	294	15	,	,	PUNCT
ma-12	294	16	φ(t	φ(t	PROPN
ma-12	294	17	)	)	PUNCT
ma-12	294	18	≤	≤	NUM
ma-12	294	19	m2	m2	PROPN
ma-12	294	20	t	t	PROPN
ma-12	294	21	,	,	PUNCT
ma-12	294	22	t	t	PROPN
ma-12	294	23	≥	≥	PROPN
ma-12	294	24	0	0	NUM
ma-12	294	25	.	.	PUNCT
ma-12	295	1	then	then	ADV
ma-12	295	2	,	,	PUNCT
ma-12	295	3	limn∞	limn∞	PROPN
ma-12	295	4	‖xn	‖xn	PROPN
ma-12	295	5	−	−	NOUN
ma-12	295	6	sixn‖	sixn‖	PUNCT
ma-12	295	7	=	=	SYM
ma-12	295	8	0	0	NUM
ma-12	296	1	and	and	CCONJ
ma-12	296	2	limn∞	limn∞	ADJ
ma-12	296	3	‖xn	‖xn	PROPN
ma-12	296	4	−	−	PROPN
ma-12	296	5	tixn‖	tixn‖	PROPN
ma-12	296	6	=	=	SYM
ma-12	296	7	0	0	NUM
ma-12	296	8	,	,	PUNCT
ma-12	296	9	for	for	ADP
ma-12	296	10	i	i	PROPN
ma-12	296	11	=	=	SYM
ma-12	296	12	1	1	NUM
ma-12	296	13	,	,	PUNCT
ma-12	296	14	2	2	NUM
ma-12	296	15	,	,	PUNCT
ma-12	296	16	,	,	PUNCT
ma-12	296	17	3	3	X
ma-12	296	18	.	.	PUNCT
ma-12	296	19	proof	proof	NOUN
ma-12	296	20	.	.	PUNCT
ma-12	297	1	set	set	VERB
ma-12	297	2	hn	hn	NOUN
ma-12	297	3	=	=	PUNCT
ma-12	297	4	max(k	max(k	PROPN
ma-12	297	5	(	(	PUNCT
ma-12	297	6	1	1	NUM
ma-12	297	7	)	)	PUNCT
ma-12	297	8	n	n	NOUN
ma-12	297	9	,	,	PUNCT
ma-12	297	10	k	k	X
ma-12	297	11	(	(	PUNCT
ma-12	297	12	2	2	NUM
ma-12	297	13	)	)	PUNCT
ma-12	297	14	n	n	NOUN
ma-12	297	15	,	,	PUNCT
ma-12	297	16	k	k	X
ma-12	297	17	(	(	PUNCT
ma-12	297	18	3	3	NUM
ma-12	297	19	)	)	PUNCT
ma-12	297	20	n	n	NOUN
ma-12	297	21	,	,	PUNCT
ma-12	297	22	µ	µ	X
ma-12	297	23	(	(	PUNCT
ma-12	297	24	1	1	NUM
ma-12	297	25	)	)	PUNCT
ma-12	297	26	n	n	NOUN
ma-12	297	27	,	,	PUNCT
ma-12	297	28	µ	µ	X
ma-12	297	29	(	(	PUNCT
ma-12	297	30	2	2	NUM
ma-12	297	31	)	)	PUNCT
ma-12	297	32	n	n	NOUN
ma-12	297	33	,	,	PUNCT
ma-12	297	34	µ	µ	X
ma-12	297	35	(	(	PUNCT
ma-12	297	36	3	3	NUM
ma-12	297	37	)	)	PUNCT
ma-12	297	38	n	n	CCONJ
ma-12	297	39	)	)	PUNCT
ma-12	297	40	,	,	PUNCT
ma-12	297	41	m	m	VERB
ma-12	297	42	=	=	SYM
ma-12	297	43	max(m1,m2,m3,m4,m5,m6	max(m1,m2,m3,m4,m5,m6	PROPN
ma-12	297	44	)	)	PUNCT
ma-12	297	45	and	and	CCONJ
ma-12	297	46	θn	θn	X
ma-12	297	47	=	=	PROPN
ma-12	297	48	max(ν	max(ν	PROPN
ma-12	297	49	(	(	PUNCT
ma-12	297	50	1	1	NUM
ma-12	297	51	)	)	PUNCT
ma-12	297	52	n	n	NOUN
ma-12	297	53	,	,	PUNCT
ma-12	297	54	ν	ν	X
ma-12	297	55	(	(	PUNCT
ma-12	297	56	2	2	NUM
ma-12	297	57	)	)	PUNCT
ma-12	297	58	n	n	NOUN
ma-12	297	59	,	,	PUNCT
ma-12	297	60	ν	ν	X
ma-12	297	61	(	(	PUNCT
ma-12	297	62	3	3	NUM
ma-12	297	63	)	)	PUNCT
ma-12	297	64	n	n	NOUN
ma-12	297	65	,	,	PUNCT
ma-12	297	66	ω	ω	PROPN
ma-12	297	67	(	(	PUNCT
ma-12	297	68	1	1	NUM
ma-12	297	69	)	)	PUNCT
ma-12	297	70	n	n	NOUN
ma-12	297	71	,	,	PUNCT
ma-12	297	72	ω	ω	PROPN
ma-12	297	73	(	(	PUNCT
ma-12	297	74	2	2	NUM
ma-12	297	75	)	)	PUNCT
ma-12	297	76	n	n	NOUN
ma-12	297	77	,	,	PUNCT
ma-12	297	78	ω	ω	PROPN
ma-12	297	79	(	(	PUNCT
ma-12	297	80	3	3	NUM
ma-12	297	81	)	)	PUNCT
ma-12	297	82	n	n	CCONJ
ma-12	297	83	)	)	PUNCT
ma-12	297	84	.	.	PUNCT
ma-12	298	1	then	then	ADV
ma-12	298	2	,	,	PUNCT
ma-12	298	3	∑∞	∑∞	NOUN
ma-12	298	4	n=1	n=1	PROPN
ma-12	298	5	hn	hn	PROPN
ma-12	298	6	<	<	X
ma-12	298	7	∞	∞	NUM
ma-12	298	8	and	and	CCONJ
ma-12	298	9	∑∞	∑∞	NOUN
ma-12	298	10	n=1	n=1	PUNCT
ma-12	298	11	θn	θn	PROPN
ma-12	298	12	<	<	X
ma-12	298	13	∞.	∞.	PROPN
ma-12	298	14	for	for	ADP
ma-12	298	15	any	any	DET
ma-12	298	16	given	give	VERB
ma-12	298	17	q	q	PROPN
ma-12	298	18	∈	∈	PROPN
ma-12	298	19	f	f	PROPN
ma-12	298	20	,	,	PUNCT
ma-12	298	21	limn∞	limn∞	PROPN
ma-12	298	22	‖xn	‖xn	PROPN
ma-12	298	23	−	−	PROPN
ma-12	298	24	q‖	q‖	NOUN
ma-12	298	25	exists	exist	VERB
ma-12	298	26	by	by	ADP
ma-12	298	27	lemma	lemma	PROPN
ma-12	298	28	3.2	3.2	NUM
ma-12	298	29	.	.	PUNCT
ma-12	299	1	now	now	ADV
ma-12	299	2	,	,	PUNCT
ma-12	299	3	assume	assume	VERB
ma-12	299	4	that	that	SCONJ
ma-12	299	5	limn∞	limn∞	PROPN
ma-12	299	6	‖xn	‖xn	PROPN
ma-12	299	7	−	−	PROPN
ma-12	299	8	q‖	q‖	PROPN
ma-12	299	9	=	=	SYM
ma-12	299	10	c.	c.	PROPN
ma-12	299	11	it	it	PRON
ma-12	299	12	follows	follow	VERB
ma-12	299	13	from	from	ADP
ma-12	299	14	(	(	PUNCT
ma-12	299	15	3.15),(3.16	3.15),(3.16	NUM
ma-12	299	16	)	)	PUNCT
ma-12	299	17	and	and	CCONJ
ma-12	299	18	the	the	DET
ma-12	299	19	fact	fact	NOUN
ma-12	299	20	that	that	SCONJ
ma-12	299	21	∑∞	∑∞	NOUN
ma-12	299	22	n=1	n=1	PUNCT
ma-12	299	23	hn	hn	PROPN
ma-12	299	24	<	<	X
ma-12	299	25	∞	∞	NUM
ma-12	299	26	and	and	CCONJ
ma-12	299	27	∑∞	∑∞	NOUN
ma-12	299	28	n=1	n=1	PUNCT
ma-12	299	29	θn	θn	PROPN
ma-12	299	30	<	<	X
ma-12	299	31	∞	∞	PROPN
ma-12	299	32	that	that	PRON
ma-12	299	33	lim	lim	PROPN
ma-12	299	34	‖(1−	‖(1−	PROPN
ma-12	299	35	αn)(sn1xn	αn)(sn1xn	PROPN
ma-12	299	36	−	−	PROPN
ma-12	299	37	q	q	NOUN
ma-12	299	38	)	)	PUNCT
ma-12	299	39	+	+	CCONJ
ma-12	299	40	αnt1(pt1	αnt1(pt1	X
ma-12	299	41	)	)	PUNCT
ma-12	299	42	n−1yn	n−1yn	NOUN
ma-12	299	43	−	−	PROPN
ma-12	299	44	q)‖	q)‖	NOUN
ma-12	299	45	=	=	PROPN
ma-12	299	46	c.	c.	NOUN
ma-12	299	47	(	(	PUNCT
ma-12	299	48	3.20	3.20	NUM
ma-12	299	49	)	)	PUNCT
ma-12	299	50	also	also	ADV
ma-12	299	51	,	,	PUNCT
ma-12	299	52	we	we	PRON
ma-12	299	53	have	have	VERB
ma-12	299	54	‖sn1xn	‖sn1xn	NOUN
ma-12	299	55	−	−	PROPN
ma-12	300	1	q‖	q‖	NOUN
ma-12	300	2	≤	≤	SCONJ
ma-12	300	3	‖xn	‖xn	PUNCT
ma-12	300	4	−	−	PUNCT
ma-12	300	5	q‖+	q‖+	PROPN
ma-12	300	6	k	k	X
ma-12	300	7	(	(	PUNCT
ma-12	300	8	1	1	NUM
ma-12	300	9	)	)	PUNCT
ma-12	300	10	n	n	CCONJ
ma-12	300	11	ψ(‖xn	ψ(‖xn	PRON
ma-12	300	12	−	−	PROPN
ma-12	301	1	q‖	q‖	CCONJ
ma-12	301	2	)	)	PUNCT
ma-12	302	1	+	+	CCONJ
ma-12	303	1	ω	ω	NUM
ma-12	303	2	(	(	PUNCT
ma-12	303	3	1	1	NUM
ma-12	303	4	)	)	PUNCT
ma-12	303	5	n	n	PRON
ma-12	303	6	≤	≤	NOUN
ma-12	303	7	‖xn	‖xn	PUNCT
ma-12	303	8	−	−	PUNCT
ma-12	303	9	q‖+	q‖+	PROPN
ma-12	303	10	k	k	X
ma-12	303	11	(	(	PUNCT
ma-12	303	12	1	1	X
ma-12	303	13	)	)	PUNCT
ma-12	303	14	n	n	PRON
ma-12	303	15	m‖xn	m‖xn	PROPN
ma-12	303	16	−	−	PROPN
ma-12	303	17	q‖	q‖	NOUN
ma-12	303	18	)	)	PUNCT
ma-12	304	1	+	+	CCONJ
ma-12	305	1	ω	ω	NUM
ma-12	305	2	(	(	PUNCT
ma-12	305	3	1	1	NUM
ma-12	305	4	)	)	PUNCT
ma-12	305	5	n	n	CCONJ
ma-12	305	6	≤	≤	NUM
ma-12	305	7	(	(	PUNCT
ma-12	305	8	1	1	NUM
ma-12	305	9	+	+	NUM
ma-12	305	10	k	k	X
ma-12	305	11	(	(	PUNCT
ma-12	305	12	1	1	NUM
ma-12	305	13	)	)	PUNCT
ma-12	305	14	n	n	CCONJ
ma-12	305	15	m)‖xn	m)‖xn	NUM
ma-12	305	16	−	−	ADP
ma-12	305	17	q‖+	q‖+	PROPN
ma-12	305	18	ω	ω	PROPN
ma-12	305	19	(	(	PUNCT
ma-12	305	20	1	1	NUM
ma-12	305	21	)	)	PUNCT
ma-12	305	22	n	n	CCONJ
ma-12	305	23	≤	≤	NUM
ma-12	305	24	(	(	PUNCT
ma-12	305	25	1	1	NUM
ma-12	305	26	+	+	CCONJ
ma-12	305	27	hnm)‖xn	hnm)‖xn	PROPN
ma-12	305	28	−	−	PROPN
ma-12	305	29	q‖+	q‖+	PROPN
ma-12	305	30	θn	θn	PROPN
ma-12	305	31	⇒	⇒	PROPN
ma-12	305	32	lim	lim	PROPN
ma-12	305	33	sup	sup	PROPN
ma-12	305	34	‖sn1xn	‖sn1xn	PROPN
ma-12	305	35	−	−	PROPN
ma-12	305	36	q‖	q‖	PROPN
ma-12	305	37	≤	≤	PROPN
ma-12	305	38	lim	lim	PROPN
ma-12	305	39	sup[(1	sup[(1	PROPN
ma-12	306	1	+	+	CCONJ
ma-12	306	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	306	3	−	−	PROPN
ma-12	306	4	q‖+	q‖+	PROPN
ma-12	306	5	θn	θn	PROPN
ma-12	306	6	]	]	X
ma-12	306	7	=	=	PUNCT
ma-12	306	8	c.	c.	NOUN
ma-12	306	9	(	(	PUNCT
ma-12	306	10	3.21	3.21	NUM
ma-12	306	11	)	)	PUNCT
ma-12	306	12	eur	eur	PROPN
ma-12	306	13	.	.	PUNCT
ma-12	307	1	j.	j.	PROPN
ma-12	307	2	math	math	PROPN
ma-12	307	3	.	.	PUNCT
ma-12	308	1	anal	anal	ADJ
ma-12	308	2	.	.	PUNCT
ma-12	309	1	1	1	NUM
ma-12	309	2	(	(	PUNCT
ma-12	309	3	2021	2021	NUM
ma-12	309	4	)	)	PUNCT
ma-12	309	5	56furthermore	56furthermore	NUM
ma-12	309	6	,	,	PUNCT
ma-12	309	7	‖t1(pt1)yn	‖t1(pt1)yn	PROPN
ma-12	309	8	−	−	DET
ma-12	309	9	q‖	q‖	NOUN
ma-12	309	10	≤	≤	SCONJ
ma-12	309	11	‖yn	‖yn	NUM
ma-12	309	12	−	−	ADP
ma-12	309	13	q‖+	q‖+	PROPN
ma-12	309	14	µ	µ	X
ma-12	309	15	(	(	PUNCT
ma-12	309	16	1	1	NUM
ma-12	309	17	)	)	PUNCT
ma-12	309	18	n	n	PRON
ma-12	309	19	φ(‖yn	φ(‖yn	NOUN
ma-12	309	20	−	−	PROPN
ma-12	309	21	q‖	q‖	NOUN
ma-12	309	22	)	)	PUNCT
ma-12	310	1	+	+	CCONJ
ma-12	310	2	ν	ν	X
ma-12	310	3	(	(	PUNCT
ma-12	310	4	1	1	NUM
ma-12	310	5	)	)	PUNCT
ma-12	310	6	n	n	PRON
ma-12	310	7	≤	≤	NOUN
ma-12	310	8	‖yn	‖yn	PUNCT
ma-12	310	9	−	−	ADP
ma-12	310	10	q‖+	q‖+	PROPN
ma-12	310	11	µ	µ	X
ma-12	310	12	(	(	PUNCT
ma-12	310	13	1	1	NUM
ma-12	310	14	)	)	PUNCT
ma-12	310	15	n	n	CCONJ
ma-12	310	16	m‖yn	m‖yn	PROPN
ma-12	310	17	−	−	NOUN
ma-12	310	18	q‖	q‖	NOUN
ma-12	310	19	)	)	PUNCT
ma-12	311	1	+	+	CCONJ
ma-12	311	2	ν	ν	X
ma-12	311	3	(	(	PUNCT
ma-12	311	4	1	1	NUM
ma-12	311	5	)	)	PUNCT
ma-12	311	6	n	n	CCONJ
ma-12	311	7	≤	≤	NUM
ma-12	311	8	(	(	PUNCT
ma-12	311	9	1	1	NUM
ma-12	311	10	+	+	SYM
ma-12	311	11	µ	µ	X
ma-12	311	12	(	(	PUNCT
ma-12	311	13	1	1	NUM
ma-12	311	14	)	)	PUNCT
ma-12	311	15	n	n	PRON
ma-12	311	16	m)‖yn	m)‖yn	NOUN
ma-12	311	17	−	−	NOUN
ma-12	311	18	q‖+	q‖+	ADJ
ma-12	311	19	ν	ν	NOUN
ma-12	311	20	(	(	PUNCT
ma-12	311	21	1	1	NUM
ma-12	311	22	)	)	PUNCT
ma-12	311	23	n	n	CCONJ
ma-12	311	24	≤	≤	NUM
ma-12	311	25	(	(	PUNCT
ma-12	311	26	1	1	NUM
ma-12	311	27	+	+	NUM
ma-12	311	28	hnm)‖yn	hnm)‖yn	ADJ
ma-12	311	29	−	−	NOUN
ma-12	311	30	q‖+	q‖+	PROPN
ma-12	311	31	θn	θn	ADP
ma-12	311	32	taking	take	VERB
ma-12	311	33	limsup	limsup	NOUN
ma-12	311	34	on	on	ADP
ma-12	311	35	both	both	DET
ma-12	311	36	sides	side	NOUN
ma-12	311	37	of	of	ADP
ma-12	311	38	(	(	PUNCT
ma-12	311	39	3.15	3.15	NUM
ma-12	311	40	)	)	PUNCT
ma-12	311	41	,	,	PUNCT
ma-12	311	42	we	we	PRON
ma-12	311	43	obtain	obtain	VERB
ma-12	311	44	lim	lim	PROPN
ma-12	311	45	sup	sup	PROPN
ma-12	311	46	‖yn−q‖	‖yn−q‖	PROPN
ma-12	311	47	≤	≤	PROPN
ma-12	311	48	c	c	NOUN
ma-12	312	1	and	and	CCONJ
ma-12	312	2	so	so	ADV
ma-12	312	3	lim	lim	PROPN
ma-12	312	4	sup	sup	PROPN
ma-12	312	5	‖t1(pt1)yn−q‖	‖t1(pt1)yn−q‖	PROPN
ma-12	312	6	≤	≤	PROPN
ma-12	312	7	lim	lim	NOUN
ma-12	312	8	sup[(1	sup[(1	PROPN
ma-12	313	1	+	+	PROPN
ma-12	313	2	hnm)‖yn−q‖+	hnm)‖yn−q‖+	PROPN
ma-12	313	3	θn	θn	NOUN
ma-12	313	4	]	]	PUNCT
ma-12	313	5	≤	≤	PROPN
ma-12	313	6	c	c	NOUN
ma-12	313	7	.	.	PUNCT
ma-12	314	1	thus	thus	ADV
ma-12	314	2	,	,	PUNCT
ma-12	314	3	lim	lim	PROPN
ma-12	314	4	sup	sup	NOUN
ma-12	314	5	‖t1(pt1)yn	‖t1(pt1)yn	PROPN
ma-12	314	6	−	−	PROPN
ma-12	314	7	q‖	q‖	PROPN
ma-12	314	8	≤	≤	PROPN
ma-12	314	9	lim	lim	PROPN
ma-12	314	10	sup[(1	sup[(1	PROPN
ma-12	315	1	+	+	CCONJ
ma-12	315	2	hnm)‖yn	hnm)‖yn	X
ma-12	315	3	−	−	PROPN
ma-12	315	4	q‖+	q‖+	PROPN
ma-12	315	5	θn	θn	PROPN
ma-12	315	6	]	]	X
ma-12	315	7	=	=	PUNCT
ma-12	315	8	c.	c.	NOUN
ma-12	315	9	(	(	PUNCT
ma-12	315	10	3.22	3.22	NUM
ma-12	315	11	)	)	PUNCT
ma-12	315	12	using	use	VERB
ma-12	315	13	lemma	lemma	PROPN
ma-12	315	14	2.2	2.2	NUM
ma-12	315	15	,	,	PUNCT
ma-12	315	16	we	we	PRON
ma-12	315	17	get	get	VERB
ma-12	315	18	lim	lim	PROPN
ma-12	315	19	n→∞	n→∞	NUM
ma-12	315	20	‖sn1xn	‖sn1xn	NOUN
ma-12	315	21	−	−	PROPN
ma-12	315	22	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	315	23	=	=	SYM
ma-12	315	24	0	0	NUM
ma-12	315	25	.	.	PUNCT
ma-12	316	1	(	(	PUNCT
ma-12	316	2	3.23)by	3.23)by	NUM
ma-12	316	3	condition	condition	NOUN
ma-12	316	4	(	(	PUNCT
ma-12	316	5	ii	ii	NOUN
ma-12	316	6	)	)	PUNCT
ma-12	316	7	,	,	PUNCT
ma-12	316	8	it	it	PRON
ma-12	316	9	follows	follow	VERB
ma-12	316	10	that	that	SCONJ
ma-12	316	11	‖xn	‖xn	PROPN
ma-12	316	12	−	−	PROPN
ma-12	316	13	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	316	14	≤	≤	PROPN
ma-12	316	15	‖sn1xn	‖sn1xn	NOUN
ma-12	316	16	−	−	PROPN
ma-12	316	17	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	316	18	,	,	PUNCT
ma-12	316	19	and	and	CCONJ
ma-12	316	20	so	so	ADV
ma-12	316	21	from	from	ADP
ma-12	316	22	(	(	PUNCT
ma-12	316	23	3.23	3.23	NUM
ma-12	316	24	)	)	PUNCT
ma-12	316	25	,	,	PUNCT
ma-12	316	26	we	we	PRON
ma-12	316	27	have	have	VERB
ma-12	316	28	lim	lim	PROPN
ma-12	316	29	n→∞	n→∞	X
ma-12	317	1	‖xn	‖xn	PROPN
ma-12	317	2	−	−	PROPN
ma-12	317	3	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	317	4	=	=	SYM
ma-12	317	5	0	0	NUM
ma-12	317	6	.	.	PUNCT
ma-12	318	1	(	(	PUNCT
ma-12	318	2	3.24)also	3.24)also	NUM
ma-12	318	3	,	,	PUNCT
ma-12	318	4	we	we	PRON
ma-12	318	5	have	have	AUX
ma-12	318	6	‖sn2xn	‖sn2xn	VERB
ma-12	318	7	−	−	PROPN
ma-12	318	8	q‖	q‖	NOUN
ma-12	318	9	≤	≤	ADV
ma-12	318	10	‖xn	‖xn	PUNCT
ma-12	318	11	−	−	PUNCT
ma-12	318	12	q‖+	q‖+	PROPN
ma-12	318	13	k	k	X
ma-12	318	14	(	(	PUNCT
ma-12	318	15	2	2	NUM
ma-12	318	16	)	)	PUNCT
ma-12	318	17	n	n	CCONJ
ma-12	318	18	ψ(‖xn	ψ(‖xn	PRON
ma-12	318	19	−	−	PROPN
ma-12	318	20	q‖	q‖	CCONJ
ma-12	318	21	)	)	PUNCT
ma-12	319	1	+	+	CCONJ
ma-12	320	1	ω	ω	NUM
ma-12	320	2	(	(	PUNCT
ma-12	320	3	2	2	NUM
ma-12	320	4	)	)	PUNCT
ma-12	320	5	n	n	PRON
ma-12	320	6	≤	≤	NOUN
ma-12	320	7	‖xn	‖xn	PUNCT
ma-12	320	8	−	−	PUNCT
ma-12	320	9	q‖+	q‖+	PROPN
ma-12	320	10	k	k	X
ma-12	320	11	(	(	PUNCT
ma-12	320	12	2	2	NUM
ma-12	320	13	)	)	PUNCT
ma-12	320	14	n	n	PRON
ma-12	320	15	m‖xn	m‖xn	PROPN
ma-12	320	16	−	−	PROPN
ma-12	320	17	q‖	q‖	NOUN
ma-12	320	18	)	)	PUNCT
ma-12	321	1	+	+	CCONJ
ma-12	322	1	ω	ω	NUM
ma-12	322	2	(	(	PUNCT
ma-12	322	3	2	2	NUM
ma-12	322	4	)	)	PUNCT
ma-12	322	5	n	n	PRON
ma-12	322	6	≤	≤	NUM
ma-12	322	7	(	(	PUNCT
ma-12	322	8	1	1	NUM
ma-12	322	9	+	+	NUM
ma-12	322	10	k	k	X
ma-12	322	11	(	(	PUNCT
ma-12	322	12	2	2	NUM
ma-12	322	13	)	)	PUNCT
ma-12	322	14	n	n	CCONJ
ma-12	322	15	m)‖xn	m)‖xn	NUM
ma-12	322	16	−	−	ADP
ma-12	322	17	q‖+	q‖+	PROPN
ma-12	322	18	ω	ω	PROPN
ma-12	322	19	(	(	PUNCT
ma-12	322	20	2	2	NUM
ma-12	322	21	)	)	PUNCT
ma-12	322	22	n	n	PRON
ma-12	322	23	≤	≤	NUM
ma-12	322	24	(	(	PUNCT
ma-12	322	25	1	1	NUM
ma-12	322	26	+	+	CCONJ
ma-12	322	27	hnm)‖xn	hnm)‖xn	PROPN
ma-12	322	28	−	−	PROPN
ma-12	322	29	q‖+	q‖+	PROPN
ma-12	322	30	θn	θn	PROPN
ma-12	322	31	⇒	⇒	PROPN
ma-12	322	32	lim	lim	PROPN
ma-12	322	33	sup	sup	PROPN
ma-12	322	34	‖sn2xn	‖sn2xn	ADV
ma-12	322	35	−	−	PROPN
ma-12	322	36	q‖	q‖	PROPN
ma-12	322	37	≤	≤	PROPN
ma-12	322	38	lim	lim	PROPN
ma-12	322	39	sup[(1	sup[(1	PROPN
ma-12	323	1	+	+	CCONJ
ma-12	323	2	hnm)‖xn	hnm)‖xn	PROPN
ma-12	323	3	−	−	PROPN
ma-12	323	4	q‖+	q‖+	PROPN
ma-12	323	5	θn	θn	PROPN
ma-12	323	6	]	]	X
ma-12	323	7	=	=	PUNCT
ma-12	323	8	c.	c.	NOUN
ma-12	323	9	(	(	PUNCT
ma-12	323	10	3.25	3.25	NUM
ma-12	323	11	)	)	PUNCT
ma-12	323	12	furthermore	furthermore	ADV
ma-12	323	13	,	,	PUNCT
ma-12	323	14	‖t2(pt2)zn	‖t2(pt2)zn	PROPN
ma-12	323	15	−	−	PROPN
ma-12	323	16	q‖	q‖	NOUN
ma-12	323	17	≤	≤	SCONJ
ma-12	324	1	‖zn	‖zn	NUM
ma-12	324	2	−	−	NOUN
ma-12	324	3	q‖+	q‖+	VERB
ma-12	324	4	µ(2)n	µ(2)n	PROPN
ma-12	324	5	φ(‖zn	φ(‖zn	ADV
ma-12	324	6	−	−	PROPN
ma-12	324	7	q‖	q‖	NOUN
ma-12	324	8	)	)	PUNCT
ma-12	325	1	+	+	NUM
ma-12	325	2	ν(2)n	ν(2)n	X
ma-12	325	3	≤	≤	NUM
ma-12	325	4	‖zn	‖zn	NUM
ma-12	325	5	−	−	NOUN
ma-12	325	6	q‖+	q‖+	ADJ
ma-12	325	7	µ(2)n	µ(2)n	PROPN
ma-12	326	1	m‖zn	m‖zn	ADP
ma-12	326	2	−	−	PROPN
ma-12	326	3	q‖	q‖	NOUN
ma-12	326	4	)	)	PUNCT
ma-12	327	1	+	+	NUM
ma-12	327	2	ν(2)n	ν(2)n	X
ma-12	327	3	≤	≤	NOUN
ma-12	327	4	(	(	PUNCT
ma-12	327	5	1	1	NUM
ma-12	327	6	+	+	CCONJ
ma-12	327	7	µ(2)n	µ(2)n	PROPN
ma-12	327	8	m)‖zn	m)‖zn	NUM
ma-12	327	9	−	−	PROPN
ma-12	327	10	q‖+	q‖+	PROPN
ma-12	327	11	ν(2)n	ν(2)n	NOUN
ma-12	327	12	≤	≤	NOUN
ma-12	327	13	(	(	PUNCT
ma-12	327	14	1	1	NUM
ma-12	327	15	+	+	CCONJ
ma-12	327	16	hnm)‖zn	hnm)‖zn	ADP
ma-12	327	17	−	−	NOUN
ma-12	327	18	q‖+	q‖+	PROPN
ma-12	327	19	θn	θn	PROPN
ma-12	327	20	eur	eur	PROPN
ma-12	327	21	.	.	PUNCT
ma-12	328	1	j.	j.	PROPN
ma-12	328	2	math	math	PROPN
ma-12	328	3	.	.	PUNCT
ma-12	329	1	anal	anal	ADJ
ma-12	329	2	.	.	PUNCT
ma-12	330	1	1	1	NUM
ma-12	330	2	(	(	PUNCT
ma-12	330	3	2021	2021	NUM
ma-12	330	4	)	)	PUNCT
ma-12	330	5	57	57	NUM
ma-12	331	1	taking	take	VERB
ma-12	331	2	lim	lim	PROPN
ma-12	331	3	sup	sup	NOUN
ma-12	331	4	on	on	ADP
ma-12	331	5	both	both	DET
ma-12	331	6	sides	side	NOUN
ma-12	331	7	of	of	ADP
ma-12	331	8	(	(	PUNCT
ma-12	331	9	3.12	3.12	NUM
ma-12	331	10	)	)	PUNCT
ma-12	331	11	,	,	PUNCT
ma-12	331	12	we	we	PRON
ma-12	331	13	obtain	obtain	VERB
ma-12	331	14	lim	lim	NOUN
ma-12	331	15	supn→∞	supn→∞	PROPN
ma-12	332	1	‖zn	‖zn	NUM
ma-12	332	2	−	−	PROPN
ma-12	332	3	q‖	q‖	NOUN
ma-12	332	4	≤	≤	PROPN
ma-12	332	5	c	c	PROPN
ma-12	333	1	and	and	CCONJ
ma-12	333	2	so	so	ADV
ma-12	333	3	lim	lim	PROPN
ma-12	333	4	sup	sup	PROPN
ma-12	333	5	‖t2(pt1)zn	‖t2(pt1)zn	NOUN
ma-12	333	6	−	−	PROPN
ma-12	334	1	q‖	q‖	PROPN
ma-12	334	2	≤	≤	PROPN
ma-12	334	3	lim	lim	PROPN
ma-12	334	4	sup[(1	sup[(1	PROPN
ma-12	335	1	+	+	CCONJ
ma-12	335	2	hnm)‖zn	hnm)‖zn	ADV
ma-12	335	3	−	−	PROPN
ma-12	335	4	q‖+	q‖+	PROPN
ma-12	335	5	θn	θn	PROPN
ma-12	335	6	]	]	PUNCT
ma-12	335	7	≤	≤	ADJ
ma-12	335	8	c.	c.	NOUN
ma-12	335	9	(	(	PUNCT
ma-12	335	10	3.26	3.26	NUM
ma-12	335	11	)	)	PUNCT
ma-12	335	12	(	(	PUNCT
ma-12	335	13	3.13	3.13	NUM
ma-12	335	14	)	)	PUNCT
ma-12	335	15	,	,	PUNCT
ma-12	335	16	(	(	PUNCT
ma-12	335	17	3.25	3.25	NUM
ma-12	335	18	)	)	PUNCT
ma-12	335	19	,	,	PUNCT
ma-12	335	20	(	(	PUNCT
ma-12	335	21	3.26	3.26	NUM
ma-12	335	22	)	)	PUNCT
ma-12	335	23	and	and	CCONJ
ma-12	335	24	lemma	lemma	PROPN
ma-12	335	25	2.2	2.2	NUM
ma-12	335	26	imply	imply	NOUN
ma-12	335	27	lim	lim	PROPN
ma-12	335	28	n→∞	n→∞	NUM
ma-12	335	29	‖sn2xn	‖sn2xn	ADJ
ma-12	335	30	−	−	PROPN
ma-12	336	1	t2(pt2)n−1zn‖	t2(pt2)n−1zn‖	PROPN
ma-12	336	2	=	=	NOUN
ma-12	336	3	0	0	X
ma-12	336	4	.	.	PUNCT
ma-12	337	1	(	(	PUNCT
ma-12	337	2	3.27	3.27	NUM
ma-12	337	3	)	)	PUNCT
ma-12	337	4	(	(	PUNCT
ma-12	337	5	3.27	3.27	NUM
ma-12	337	6	)	)	PUNCT
ma-12	337	7	and	and	CCONJ
ma-12	337	8	condition	condition	NOUN
ma-12	337	9	(	(	PUNCT
ma-12	337	10	ii	ii	NOUN
ma-12	337	11	)	)	PUNCT
ma-12	337	12	yields	yield	NOUN
ma-12	337	13	lim	lim	PROPN
ma-12	337	14	n→∞	n→∞	X
ma-12	338	1	‖xn	‖xn	PROPN
ma-12	338	2	−	−	X
ma-12	338	3	t2(pt2)n−1zn‖	t2(pt2)n−1zn‖	PROPN
ma-12	338	4	=	=	NOUN
ma-12	338	5	0	0	X
ma-12	338	6	.	.	PUNCT
ma-12	338	7	(	(	PUNCT
ma-12	338	8	3.28	3.28	NUM
ma-12	338	9	)	)	PUNCT
ma-12	338	10	from	from	ADP
ma-12	338	11	(	(	PUNCT
ma-12	338	12	3.11	3.11	NUM
ma-12	338	13	)	)	PUNCT
ma-12	338	14	,	,	PUNCT
ma-12	338	15	using	use	VERB
ma-12	338	16	the	the	DET
ma-12	338	17	same	same	ADJ
ma-12	338	18	argument	argument	NOUN
ma-12	338	19	as	as	SCONJ
ma-12	338	20	was	be	AUX
ma-12	338	21	used	use	VERB
ma-12	338	22	in	in	ADP
ma-12	338	23	obtaining	obtain	VERB
ma-12	338	24	(	(	PUNCT
ma-12	338	25	3.27	3.27	NUM
ma-12	338	26	)	)	PUNCT
ma-12	338	27	above	above	ADV
ma-12	338	28	,	,	PUNCT
ma-12	338	29	we	we	PRON
ma-12	338	30	get	get	VERB
ma-12	338	31	lim	lim	PROPN
ma-12	338	32	n→∞	n→∞	NUM
ma-12	338	33	‖sn3xn	‖sn3xn	NOUN
ma-12	339	1	−	−	PROPN
ma-12	339	2	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	339	3	=	=	SYM
ma-12	339	4	0	0	PROPN
ma-12	339	5	.	.	PUNCT
ma-12	340	1	(	(	PUNCT
ma-12	340	2	3.29)now	3.29)now	NUM
ma-12	340	3	,	,	PUNCT
ma-12	340	4	we	we	PRON
ma-12	340	5	prove	prove	VERB
ma-12	340	6	that	that	SCONJ
ma-12	340	7	lim	lim	PROPN
ma-12	340	8	n→∞	n→∞	PRON
ma-12	340	9	‖xn	‖xn	PROPN
ma-12	340	10	−	−	PROPN
ma-12	340	11	t1(pt1)n−1xn‖	t1(pt1)n−1xn‖	PROPN
ma-12	340	12	=	=	SYM
ma-12	340	13	lim	lim	PROPN
ma-12	340	14	n→∞	n→∞	X
ma-12	341	1	‖xn	‖xn	PROPN
ma-12	341	2	−	−	PROPN
ma-12	341	3	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	341	4	lim	lim	PROPN
ma-12	341	5	n→∞	n→∞	X
ma-12	342	1	‖xn	‖xn	PROPN
ma-12	342	2	−	−	PUNCT
ma-12	342	3	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	342	4	=	=	SYM
ma-12	342	5	0.indeed	0.indeed	NUM
ma-12	342	6	,	,	PUNCT
ma-12	342	7	since	since	SCONJ
ma-12	342	8	‖xn	‖xn	PROPN
ma-12	342	9	−	−	PROPN
ma-12	342	10	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	342	11	≤	≤	ADJ
ma-12	342	12	‖sn3xn	‖sn3xn	NOUN
ma-12	342	13	−	−	PROPN
ma-12	342	14	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	342	15	,	,	PUNCT
ma-12	342	16	(	(	PUNCT
ma-12	342	17	by	by	ADP
ma-12	342	18	condition	condition	NOUN
ma-12	342	19	(	(	PUNCT
ma-12	342	20	ii	ii	NOUN
ma-12	342	21	)	)	PUNCT
ma-12	342	22	)	)	PUNCT
ma-12	342	23	,	,	PUNCT
ma-12	342	24	it	it	PRON
ma-12	342	25	follows	follow	VERB
ma-12	342	26	from(3.29	from(3.29	NOUN
ma-12	342	27	)	)	PUNCT
ma-12	342	28	that	that	PRON
ma-12	342	29	lim	lim	PROPN
ma-12	342	30	n→∞	n→∞	PRON
ma-12	342	31	‖xn	‖xn	PROPN
ma-12	342	32	−	−	PROPN
ma-12	342	33	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	342	34	=	=	SYM
ma-12	342	35	0	0	PROPN
ma-12	342	36	.	.	PUNCT
ma-12	343	1	(	(	PUNCT
ma-12	343	2	3.30)since	3.30)since	NUM
ma-12	343	3	,	,	PUNCT
ma-12	343	4	p	p	X
ma-12	343	5	(	(	PUNCT
ma-12	343	6	snxn	snxn	NOUN
ma-12	343	7	)	)	PUNCT
ma-12	343	8	=	=	VERB
ma-12	344	1	snxn	snxn	NOUN
ma-12	344	2	and	and	CCONJ
ma-12	344	3	p	p	NOUN
ma-12	344	4	:	:	PUNCT
ma-12	344	5	e	e	X
ma-12	344	6	−→	−→	NOUN
ma-12	344	7	k	k	PROPN
ma-12	344	8	is	be	AUX
ma-12	344	9	a	a	DET
ma-12	344	10	nonexpansive	nonexpansive	ADJ
ma-12	344	11	retraction	retraction	NOUN
ma-12	344	12	of	of	ADP
ma-12	344	13	e	e	PROPN
ma-12	344	14	onto	onto	ADP
ma-12	344	15	k	k	PROPN
ma-12	344	16	,	,	PUNCT
ma-12	344	17	we	we	PRON
ma-12	344	18	get	get	VERB
ma-12	344	19	‖zn	‖zn	NUM
ma-12	344	20	−	−	NOUN
ma-12	344	21	sn3xn‖	sn3xn‖	NOUN
ma-12	345	1	=	=	SYM
ma-12	345	2	‖p	‖p	PROPN
ma-12	345	3	(	(	PUNCT
ma-12	345	4	(	(	PUNCT
ma-12	345	5	1−	1−	NUM
ma-12	345	6	γn)sn3xn	γn)sn3xn	X
ma-12	345	7	+	+	CCONJ
ma-12	345	8	γnt3(pt3	γnt3(pt3	X
ma-12	345	9	)	)	PUNCT
ma-12	345	10	n−1xn)−	n−1xn)−	NOUN
ma-12	345	11	sn3xn‖	sn3xn‖	VERB
ma-12	345	12	≤	≤	NUM
ma-12	345	13	‖(1−	‖(1−	PROPN
ma-12	345	14	γn)sn3xn	γn)sn3xn	ADJ
ma-12	345	15	+	+	ADJ
ma-12	345	16	γnt3(pt3	γnt3(pt3	X
ma-12	345	17	)	)	PUNCT
ma-12	345	18	n−1xn	n−1xn	NOUN
ma-12	345	19	−	−	NOUN
ma-12	345	20	sn3xn‖	sn3xn‖	PROPN
ma-12	345	21	=	=	PUNCT
ma-12	346	1	‖	‖	PROPN
ma-12	346	2	−	−	PROPN
ma-12	346	3	γn(sn3xn	γn(sn3xn	PROPN
ma-12	346	4	−	−	PROPN
ma-12	346	5	γnt3(pt3)n−1xn)‖	γnt3(pt3)n−1xn)‖	X
ma-12	346	6	=	=	SYM
ma-12	346	7	γn‖(sn3xn	γn‖(sn3xn	NOUN
ma-12	346	8	−	−	PROPN
ma-12	347	1	γnt3(pt3)n−1xn)‖	γnt3(pt3)n−1xn)‖	PROPN
ma-12	347	2	,	,	PUNCT
ma-12	347	3	which	which	PRON
ma-12	347	4	by	by	ADP
ma-12	347	5	(	(	PUNCT
ma-12	347	6	3.29	3.29	NUM
ma-12	347	7	)	)	PUNCT
ma-12	347	8	gives	give	VERB
ma-12	347	9	lim	lim	PROPN
ma-12	347	10	n→∞	n→∞	X
ma-12	348	1	‖zn	‖zn	NUM
ma-12	348	2	−	−	NOUN
ma-12	348	3	sn3xn‖	sn3xn‖	NOUN
ma-12	349	1	=	=	SYM
ma-12	350	1	0	0	X
ma-12	350	2	.	.	PUNCT
ma-12	351	1	(	(	PUNCT
ma-12	351	2	3.31)observe	3.31)observe	NUM
ma-12	351	3	that	that	PRON
ma-12	352	1	‖zn	‖zn	NUM
ma-12	352	2	−	−	NOUN
ma-12	352	3	xn‖	xn‖	PROPN
ma-12	353	1	=	=	PUNCT
ma-12	354	1	‖zn	‖zn	NUM
ma-12	354	2	−	−	NOUN
ma-12	354	3	sn3xn	sn3xn	AUX
ma-12	354	4	+	+	CCONJ
ma-12	354	5	sn3xn	sn3xn	ADJ
ma-12	354	6	−	−	ADV
ma-12	354	7	t3(pt3)n−1xn	t3(pt3)n−1xn	PROPN
ma-12	354	8	+	+	CCONJ
ma-12	354	9	t3(pt3	t3(pt3	PROPN
ma-12	354	10	)	)	PUNCT
ma-12	354	11	n−1xn	n−1xn	NOUN
ma-12	354	12	−	−	NOUN
ma-12	354	13	xn‖	xn‖	PROPN
ma-12	354	14	≤	≤	PROPN
ma-12	355	1	‖zn	‖zn	NUM
ma-12	355	2	−	−	NOUN
ma-12	355	3	sn3xn‖+	sn3xn‖+	PROPN
ma-12	355	4	‖sn3xn	‖sn3xn	NOUN
ma-12	355	5	−	−	PROPN
ma-12	356	1	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	356	2	+	+	PROPN
ma-12	356	3	‖t3(pt3)n−1xn	‖t3(pt3)n−1xn	PROPN
ma-12	356	4	−	−	PROPN
ma-12	356	5	xn‖.	xn‖.	PROPN
ma-12	356	6	(	(	PUNCT
ma-12	356	7	3.32	3.32	NUM
ma-12	356	8	)	)	PUNCT
ma-12	356	9	thus	thus	ADV
ma-12	356	10	,	,	PUNCT
ma-12	356	11	it	it	PRON
ma-12	356	12	follows	follow	VERB
ma-12	356	13	from	from	ADP
ma-12	356	14	(	(	PUNCT
ma-12	356	15	3.29	3.29	NUM
ma-12	356	16	)	)	PUNCT
ma-12	356	17	,	,	PUNCT
ma-12	356	18	(	(	PUNCT
ma-12	356	19	3.30),(3.31	3.30),(3.31	X
ma-12	356	20	)	)	PUNCT
ma-12	356	21	and	and	CCONJ
ma-12	356	22	(	(	PUNCT
ma-12	356	23	3.32	3.32	NUM
ma-12	356	24	)	)	PUNCT
ma-12	356	25	that	that	PRON
ma-12	356	26	lim	lim	PROPN
ma-12	356	27	n→∞	n→∞	PRON
ma-12	357	1	‖zn	‖zn	NUM
ma-12	357	2	−	−	NOUN
ma-12	357	3	xn‖	xn‖	PROPN
ma-12	358	1	=	=	SYM
ma-12	358	2	0	0	PROPN
ma-12	358	3	.	.	PUNCT
ma-12	359	1	(	(	PUNCT
ma-12	359	2	3.33	3.33	NUM
ma-12	359	3	)	)	PUNCT
ma-12	359	4	eur	eur	PROPN
ma-12	359	5	.	.	PUNCT
ma-12	360	1	j.	j.	PROPN
ma-12	360	2	math	math	PROPN
ma-12	360	3	.	.	PUNCT
ma-12	361	1	anal	anal	ADJ
ma-12	361	2	.	.	PUNCT
ma-12	362	1	1	1	NUM
ma-12	362	2	(	(	PUNCT
ma-12	362	3	2021	2021	NUM
ma-12	362	4	)	)	PUNCT
ma-12	363	1	58again	58again	NUM
ma-12	363	2	,	,	PUNCT
ma-12	363	3	observe	observe	VERB
ma-12	363	4	that	that	SCONJ
ma-12	363	5	‖sn2xn	‖sn2xn	ADJ
ma-12	363	6	−	−	PROPN
ma-12	363	7	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	363	8	≤	≤	PUNCT
ma-12	363	9	‖sn2xn	‖sn2xn	ADJ
ma-12	363	10	−	−	PROPN
ma-12	363	11	t2(pt2)n−1zn‖+	t2(pt2)n−1zn‖+	NOUN
ma-12	363	12	‖t2(pt2)n−1zn	‖t2(pt2)n−1zn	X
ma-12	363	13	−	−	PROPN
ma-12	363	14	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	363	15	≤	≤	PUNCT
ma-12	363	16	‖sn2xn	‖sn2xn	ADJ
ma-12	363	17	−	−	PROPN
ma-12	363	18	t2(pt2)n−1zn‖+	t2(pt2)n−1zn‖+	ADP
ma-12	363	19	(	(	PUNCT
ma-12	364	1	‖zn	‖zn	NUM
ma-12	364	2	−	−	PROPN
ma-12	364	3	xn‖+	xn‖+	PROPN
ma-12	364	4	k	k	PROPN
ma-12	364	5	(	(	PUNCT
ma-12	364	6	2	2	NUM
ma-12	364	7	)	)	PUNCT
ma-12	364	8	n	n	CCONJ
ma-12	364	9	φ(‖zn	φ(‖zn	ADV
ma-12	364	10	−	−	PROPN
ma-12	364	11	xn‖	xn‖	PROPN
ma-12	364	12	)	)	PUNCT
ma-12	365	1	+	+	NUM
ma-12	365	2	ν	ν	NOUN
ma-12	365	3	(	(	PUNCT
ma-12	365	4	2	2	NUM
ma-12	365	5	)	)	PUNCT
ma-12	365	6	n	n	PRON
ma-12	365	7	≤	≤	NOUN
ma-12	365	8	‖sn2xn	‖sn2xn	ADJ
ma-12	365	9	−	−	PROPN
ma-12	365	10	t2(pt2)n−1zn‖+	t2(pt2)n−1zn‖+	PROPN
ma-12	365	11	‖zn	‖zn	NUM
ma-12	365	12	−	−	PROPN
ma-12	365	13	xn‖+mhn(‖zn	xn‖+mhn(‖zn	SYM
ma-12	366	1	−	−	PROPN
ma-12	366	2	xn‖	xn‖	PROPN
ma-12	366	3	)	)	PUNCT
ma-12	367	1	+	+	NUM
ma-12	367	2	θn	θn	ADJ
ma-12	367	3	=	=	PROPN
ma-12	367	4	‖sn2xn	‖sn2xn	PROPN
ma-12	367	5	−	−	PROPN
ma-12	367	6	t2(pt2)n−1zn‖+	t2(pt2)n−1zn‖+	X
ma-12	367	7	(	(	PUNCT
ma-12	367	8	1	1	NUM
ma-12	367	9	+	+	NOUN
ma-12	367	10	mhn)‖zn	mhn)‖zn	NOUN
ma-12	367	11	−	−	PROPN
ma-12	367	12	xn‖+	xn‖+	PROPN
ma-12	368	1	θn	θn	PROPN
ma-12	368	2	.	.	PROPN
ma-12	368	3	(	(	PUNCT
ma-12	368	4	3.34	3.34	NUM
ma-12	368	5	)	)	PUNCT
ma-12	368	6	from	from	ADP
ma-12	368	7	(	(	PUNCT
ma-12	368	8	3.27),(3.33	3.27),(3.33	NUM
ma-12	368	9	)	)	PUNCT
ma-12	368	10	,	,	PUNCT
ma-12	368	11	(	(	PUNCT
ma-12	368	12	3.34	3.34	NUM
ma-12	368	13	)	)	PUNCT
ma-12	368	14	and	and	CCONJ
ma-12	368	15	the	the	DET
ma-12	368	16	fact	fact	NOUN
ma-12	368	17	that	that	SCONJ
ma-12	368	18	∑∞	∑∞	NOUN
ma-12	368	19	n=1	n=1	PUNCT
ma-12	368	20	θn	θn	PROPN
ma-12	368	21	<	<	PROPN
ma-12	368	22	∞	∞	PROPN
ma-12	368	23	,	,	PUNCT
ma-12	368	24	we	we	PRON
ma-12	368	25	get	get	VERB
ma-12	368	26	lim	lim	PROPN
ma-12	368	27	n→∞	n→∞	NUM
ma-12	368	28	‖sn2xn	‖sn2xn	ADJ
ma-12	368	29	−	−	PROPN
ma-12	368	30	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	368	31	=	=	SYM
ma-12	368	32	0	0	PROPN
ma-12	368	33	.	.	PUNCT
ma-12	369	1	(	(	PUNCT
ma-12	369	2	3.35	3.35	NUM
ma-12	369	3	)	)	PUNCT
ma-12	369	4	since	since	SCONJ
ma-12	369	5	‖xn−t2(pt2)n−1xn‖	‖xn−t2(pt2)n−1xn‖	NOUN
ma-12	369	6	≤	≤	PROPN
ma-12	369	7	‖sn2xn−t2(pt2)n−1xn‖	‖sn2xn−t2(pt2)n−1xn‖	PROPN
ma-12	369	8	(	(	PUNCT
ma-12	369	9	by	by	ADP
ma-12	369	10	condition	condition	NOUN
ma-12	369	11	(	(	PUNCT
ma-12	369	12	ii	ii	NOUN
ma-12	369	13	)	)	PUNCT
ma-12	369	14	,	,	PUNCT
ma-12	369	15	it	it	PRON
ma-12	369	16	follows	follow	VERB
ma-12	369	17	from	from	ADP
ma-12	369	18	(	(	PUNCT
ma-12	369	19	3.35	3.35	NUM
ma-12	369	20	)	)	PUNCT
ma-12	370	1	that	that	PRON
ma-12	370	2	lim	lim	PROPN
ma-12	370	3	n→∞	n→∞	PRON
ma-12	370	4	‖xn	‖xn	PROPN
ma-12	370	5	−	−	PROPN
ma-12	370	6	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	370	7	=	=	SYM
ma-12	370	8	0	0	PROPN
ma-12	370	9	.	.	PUNCT
ma-12	371	1	(	(	PUNCT
ma-12	371	2	3.36	3.36	NUM
ma-12	371	3	)	)	PUNCT
ma-12	371	4	also	also	ADV
ma-12	371	5	,	,	PUNCT
ma-12	371	6	since	since	SCONJ
ma-12	371	7	p	p	X
ma-12	371	8	(	(	PUNCT
ma-12	371	9	snxn	snxn	NOUN
ma-12	371	10	)	)	PUNCT
ma-12	371	11	=	=	VERB
ma-12	372	1	snxn	snxn	NOUN
ma-12	372	2	and	and	CCONJ
ma-12	372	3	p	p	NOUN
ma-12	372	4	:	:	PUNCT
ma-12	372	5	e	e	X
ma-12	372	6	−→	−→	NOUN
ma-12	372	7	k	k	PROPN
ma-12	372	8	is	be	AUX
ma-12	372	9	a	a	DET
ma-12	372	10	nonexpansive	nonexpansive	ADJ
ma-12	372	11	retraction	retraction	NOUN
ma-12	372	12	of	of	ADP
ma-12	372	13	e	e	PROPN
ma-12	372	14	onto	onto	ADP
ma-12	372	15	k	k	PROPN
ma-12	372	16	,	,	PUNCT
ma-12	372	17	we	we	PRON
ma-12	372	18	get	get	VERB
ma-12	372	19	‖yn	‖yn	PUNCT
ma-12	372	20	−	−	NOUN
ma-12	372	21	sn2xn‖	sn2xn‖	PROPN
ma-12	372	22	=	=	SYM
ma-12	372	23	‖p	‖p	PROPN
ma-12	372	24	(	(	PUNCT
ma-12	372	25	(	(	PUNCT
ma-12	372	26	1−	1−	NUM
ma-12	372	27	βn)sn2xn	βn)sn2xn	X
ma-12	372	28	+	+	CCONJ
ma-12	372	29	βnt2(pt2	βnt2(pt2	NOUN
ma-12	372	30	)	)	PUNCT
ma-12	372	31	n−1zn)−	n−1zn)−	NOUN
ma-12	372	32	sn2xn‖	sn2xn‖	PROPN
ma-12	372	33	≤	≤	NUM
ma-12	372	34	‖(1−	‖(1−	X
ma-12	372	35	βn)sn2xn	βn)sn2xn	X
ma-12	372	36	+	+	CCONJ
ma-12	372	37	βnt2(pt2	βnt2(pt2	NOUN
ma-12	372	38	)	)	PUNCT
ma-12	372	39	n−1zn	n−1zn	NOUN
ma-12	373	1	−	−	PROPN
ma-12	373	2	sn2xn‖	sn2xn‖	PROPN
ma-12	373	3	=	=	SYM
ma-12	373	4	‖	‖	PROPN
ma-12	373	5	−	−	PROPN
ma-12	373	6	βn(sn2xn	βn(sn2xn	PUNCT
ma-12	373	7	−	−	PROPN
ma-12	373	8	βnt2(pt2)n−1zn)‖	βnt2(pt2)n−1zn)‖	NOUN
ma-12	373	9	=	=	SYM
ma-12	373	10	βn‖(sn2xn	βn‖(sn2xn	NOUN
ma-12	373	11	−	−	PROPN
ma-12	374	1	βnt2(pt2)n−1zn)‖	βnt2(pt2)n−1zn)‖	NOUN
ma-12	374	2	,	,	PUNCT
ma-12	374	3	which	which	PRON
ma-12	374	4	by	by	ADP
ma-12	374	5	(	(	PUNCT
ma-12	374	6	3.27	3.27	NUM
ma-12	374	7	)	)	PUNCT
ma-12	374	8	gives	give	VERB
ma-12	374	9	lim	lim	PROPN
ma-12	374	10	n→∞	n→∞	X
ma-12	375	1	‖yn	‖yn	PUNCT
ma-12	375	2	−	−	NOUN
ma-12	375	3	sn2xn‖	sn2xn‖	PROPN
ma-12	376	1	=	=	SYM
ma-12	376	2	0	0	X
ma-12	376	3	.	.	PUNCT
ma-12	377	1	(	(	PUNCT
ma-12	377	2	3.37)moreover	3.37)moreover	NUM
ma-12	377	3	,	,	PUNCT
ma-12	377	4	since	since	SCONJ
ma-12	377	5	‖yn	‖yn	PROPN
ma-12	377	6	−	−	NOUN
ma-12	377	7	xn‖	xn‖	PROPN
ma-12	377	8	=	=	PUNCT
ma-12	378	1	‖yn	‖yn	PROPN
ma-12	378	2	−	−	NOUN
ma-12	378	3	sn2xn	sn2xn	NOUN
ma-12	378	4	+	+	CCONJ
ma-12	378	5	sn2xn	sn2xn	NUM
ma-12	378	6	−	−	NOUN
ma-12	378	7	t2(pt2)n−1zn	t2(pt2)n−1zn	PROPN
ma-12	378	8	+	+	NUM
ma-12	378	9	t2(pt2	t2(pt2	NOUN
ma-12	378	10	)	)	PUNCT
ma-12	378	11	n−1xn	n−1xn	NOUN
ma-12	378	12	−	−	PROPN
ma-12	378	13	zn‖	zn‖	PROPN
ma-12	378	14	≤	≤	PROPN
ma-12	379	1	‖yn	‖yn	PUNCT
ma-12	379	2	−	−	PROPN
ma-12	379	3	sn2xn‖+	sn2xn‖+	PROPN
ma-12	379	4	‖sn2xn	‖sn2xn	PROPN
ma-12	379	5	−	−	PROPN
ma-12	379	6	t2(pt2)n−1zn‖+	t2(pt2)n−1zn‖+	ADP
ma-12	379	7	‖t2(pt2)n−1zn	‖t2(pt2)n−1zn	PROPN
ma-12	379	8	−	−	NOUN
ma-12	379	9	xn‖	xn‖	PROPN
ma-12	379	10	,	,	PUNCT
ma-12	379	11	it	it	PRON
ma-12	379	12	follows	follow	VERB
ma-12	379	13	from	from	ADP
ma-12	379	14	(	(	PUNCT
ma-12	379	15	3.27	3.27	NUM
ma-12	379	16	)	)	PUNCT
ma-12	379	17	,	,	PUNCT
ma-12	379	18	(	(	PUNCT
ma-12	379	19	3.28	3.28	NUM
ma-12	379	20	)	)	PUNCT
ma-12	379	21	and	and	CCONJ
ma-12	379	22	(	(	PUNCT
ma-12	379	23	3.37	3.37	NUM
ma-12	379	24	)	)	PUNCT
ma-12	379	25	that	that	PRON
ma-12	379	26	lim	lim	PROPN
ma-12	379	27	n→∞	n→∞	X
ma-12	380	1	‖yn	‖yn	PROPN
ma-12	380	2	−	−	NOUN
ma-12	380	3	xn‖	xn‖	PROPN
ma-12	380	4	=	=	SYM
ma-12	380	5	0	0	PROPN
ma-12	380	6	.	.	PUNCT
ma-12	381	1	(	(	PUNCT
ma-12	381	2	3.38	3.38	NUM
ma-12	381	3	)	)	PUNCT
ma-12	381	4	observe	observe	VERB
ma-12	381	5	that	that	DET
ma-12	381	6	‖sn1xn	‖sn1xn	NOUN
ma-12	382	1	−	−	PROPN
ma-12	383	1	t1(pt1)n−1xn‖	t1(pt1)n−1xn‖	PROPN
ma-12	383	2	≤	≤	PROPN
ma-12	383	3	‖sn1xn	‖sn1xn	NOUN
ma-12	383	4	−	−	PROPN
ma-12	383	5	t1(pt1)n−1yn‖+	t1(pt1)n−1yn‖+	PRON
ma-12	383	6	‖t1(pt1)n−1yn	‖t1(pt1)n−1yn	ADP
ma-12	383	7	−	−	PROPN
ma-12	383	8	t1(pt1)n−1xn‖	t1(pt1)n−1xn‖	PROPN
ma-12	383	9	≤	≤	NUM
ma-12	383	10	‖sn1xn	‖sn1xn	NOUN
ma-12	383	11	−	−	ADP
ma-12	383	12	t1(pt1)n−1yn‖+	t1(pt1)n−1yn‖+	PUNCT
ma-12	383	13	(	(	PUNCT
ma-12	383	14	‖yn	‖yn	PROPN
ma-12	383	15	−	−	PROPN
ma-12	383	16	xn‖+	xn‖+	PROPN
ma-12	384	1	k	k	PROPN
ma-12	384	2	(	(	PUNCT
ma-12	384	3	1	1	X
ma-12	384	4	)	)	PUNCT
ma-12	384	5	n	n	CCONJ
ma-12	384	6	ψ(‖yn	ψ(‖yn	ADJ
ma-12	384	7	−	−	NOUN
ma-12	384	8	xn‖	xn‖	PROPN
ma-12	384	9	)	)	PUNCT
ma-12	385	1	+	+	NUM
ma-12	385	2	ν	ν	X
ma-12	385	3	(	(	PUNCT
ma-12	385	4	1	1	NUM
ma-12	385	5	)	)	PUNCT
ma-12	385	6	n	n	PRON
ma-12	385	7	≤	≤	NOUN
ma-12	385	8	‖sn2xn	‖sn2xn	ADJ
ma-12	385	9	−	−	PROPN
ma-12	385	10	t2(pt2)n−1zn‖+	t2(pt2)n−1zn‖+	X
ma-12	385	11	‖yn	‖yn	PROPN
ma-12	385	12	−	−	PROPN
ma-12	386	1	xn‖+mhn(‖zn	xn‖+mhn(‖zn	NOUN
ma-12	387	1	−	−	PROPN
ma-12	387	2	xn‖	xn‖	PROPN
ma-12	387	3	)	)	PUNCT
ma-12	388	1	+	+	NUM
ma-12	388	2	θn	θn	ADJ
ma-12	388	3	=	=	PROPN
ma-12	388	4	‖sn1xn	‖sn1xn	PROPN
ma-12	388	5	−	−	PROPN
ma-12	388	6	t1(pt1)n−1yn‖+	t1(pt1)n−1yn‖+	NOUN
ma-12	388	7	(	(	PUNCT
ma-12	388	8	1	1	NUM
ma-12	388	9	+	+	NOUN
ma-12	388	10	mhn)‖yn	mhn)‖yn	NOUN
ma-12	388	11	−	−	PROPN
ma-12	388	12	xn‖+	xn‖+	PROPN
ma-12	388	13	θn	θn	PROPN
ma-12	388	14	.	.	PROPN
ma-12	388	15	(	(	PUNCT
ma-12	388	16	3.39	3.39	NUM
ma-12	388	17	)	)	PUNCT
ma-12	388	18	from	from	ADP
ma-12	388	19	(	(	PUNCT
ma-12	388	20	3.23	3.23	NUM
ma-12	388	21	)	)	PUNCT
ma-12	388	22	,	,	PUNCT
ma-12	388	23	(	(	PUNCT
ma-12	388	24	3.38	3.38	NUM
ma-12	388	25	)	)	PUNCT
ma-12	388	26	,	,	PUNCT
ma-12	388	27	(	(	PUNCT
ma-12	388	28	3.39	3.39	NUM
ma-12	388	29	)	)	PUNCT
ma-12	388	30	and	and	CCONJ
ma-12	388	31	the	the	DET
ma-12	388	32	fact	fact	NOUN
ma-12	388	33	that	that	SCONJ
ma-12	388	34	∑∞	∑∞	NOUN
ma-12	388	35	n=1	n=1	PUNCT
ma-12	388	36	θn	θn	PROPN
ma-12	388	37	<	<	PROPN
ma-12	388	38	∞	∞	PROPN
ma-12	388	39	lim	lim	PROPN
ma-12	388	40	n→∞	n→∞	NUM
ma-12	388	41	‖sn1xn	‖sn1xn	NOUN
ma-12	389	1	−	−	PROPN
ma-12	390	1	t1(pt1)n−1xn‖	t1(pt1)n−1xn‖	NOUN
ma-12	390	2	=	=	NOUN
ma-12	390	3	0	0	NUM
ma-12	390	4	.	.	PUNCT
ma-12	391	1	(	(	PUNCT
ma-12	391	2	3.40	3.40	NUM
ma-12	391	3	)	)	PUNCT
ma-12	391	4	eur	eur	PROPN
ma-12	391	5	.	.	PUNCT
ma-12	392	1	j.	j.	PROPN
ma-12	392	2	math	math	PROPN
ma-12	392	3	.	.	PUNCT
ma-12	393	1	anal	anal	ADJ
ma-12	393	2	.	.	PUNCT
ma-12	394	1	1	1	NUM
ma-12	394	2	(	(	PUNCT
ma-12	394	3	2021	2021	NUM
ma-12	394	4	)	)	PUNCT
ma-12	394	5	59now	59now	NOUN
ma-12	394	6	,	,	PUNCT
ma-12	394	7	since	since	SCONJ
ma-12	394	8	‖xn−t1(pt1)n−1xn‖	‖xn−t1(pt1)n−1xn‖	PRON
ma-12	394	9	≤	≤	NOUN
ma-12	394	10	‖sn1xn−t1(pt1)n−1xn‖	‖sn1xn−t1(pt1)n−1xn‖	NOUN
ma-12	394	11	(	(	PUNCT
ma-12	394	12	by	by	ADP
ma-12	394	13	condition	condition	NOUN
ma-12	394	14	(	(	PUNCT
ma-12	394	15	ii	ii	NOUN
ma-12	394	16	)	)	PUNCT
ma-12	394	17	,	,	PUNCT
ma-12	394	18	it	it	PRON
ma-12	394	19	follows	follow	VERB
ma-12	394	20	from	from	ADP
ma-12	394	21	(	(	PUNCT
ma-12	394	22	3.40)that	3.40)that	NUM
ma-12	394	23	lim	lim	NOUN
ma-12	394	24	n→∞	n→∞	X
ma-12	395	1	‖xn	‖xn	PROPN
ma-12	395	2	−	−	PROPN
ma-12	395	3	t1(pt1)n−1xn‖	t1(pt1)n−1xn‖	PROPN
ma-12	395	4	=	=	NOUN
ma-12	395	5	0	0	NUM
ma-12	395	6	.	.	PUNCT
ma-12	395	7	(	(	PUNCT
ma-12	395	8	3.41	3.41	NUM
ma-12	395	9	)	)	PUNCT
ma-12	395	10	from	from	ADP
ma-12	395	11	‖xn+1	‖xn+1	NOUN
ma-12	395	12	−	−	PROPN
ma-12	395	13	sn1xn‖	sn1xn‖	PROPN
ma-12	395	14	=	=	PUNCT
ma-12	395	15	‖p	‖p	PROPN
ma-12	396	1	[	[	X
ma-12	396	2	(	(	PUNCT
ma-12	396	3	1−	1−	NUM
ma-12	396	4	αn)sn1xn	αn)sn1xn	NOUN
ma-12	396	5	+	+	CCONJ
ma-12	396	6	αnt1(pt1	αnt1(pt1	ADJ
ma-12	396	7	)	)	PUNCT
ma-12	396	8	n−1yn]−	n−1yn]−	NOUN
ma-12	396	9	sn1xn‖	sn1xn‖	PROPN
ma-12	396	10	≤	≤	PROPN
ma-12	396	11	‖(1−	‖(1−	PROPN
ma-12	396	12	αn)sn1xn	αn)sn1xn	NOUN
ma-12	396	13	+	+	CCONJ
ma-12	396	14	αnt1(pt1	αnt1(pt1	ADJ
ma-12	396	15	)	)	PUNCT
ma-12	396	16	n−1yn	n−1yn	NOUN
ma-12	396	17	−	−	PROPN
ma-12	397	1	sn1xn‖	sn1xn‖	PROPN
ma-12	397	2	=	=	PUNCT
ma-12	397	3	‖	‖	PROPN
ma-12	397	4	−	−	PROPN
ma-12	398	1	αn(sn1xn	αn(sn1xn	NOUN
ma-12	398	2	−	−	NOUN
ma-12	398	3	t1(pt1)n−1yn])‖	t1(pt1)n−1yn])‖	PROPN
ma-12	398	4	=	=	SYM
ma-12	398	5	αn‖sn1xn	αn‖sn1xn	ADJ
ma-12	398	6	−	−	NOUN
ma-12	398	7	t1(pt1)n−1yn]‖	t1(pt1)n−1yn]‖	NUM
ma-12	398	8	and	and	CCONJ
ma-12	398	9	(	(	PUNCT
ma-12	398	10	3.23	3.23	NUM
ma-12	398	11	)	)	PUNCT
ma-12	398	12	,	,	PUNCT
ma-12	398	13	we	we	PRON
ma-12	398	14	obtain	obtain	VERB
ma-12	398	15	lim	lim	PROPN
ma-12	398	16	n→∞	n→∞	X
ma-12	398	17	‖xn+1	‖xn+1	NUM
ma-12	398	18	−	−	PROPN
ma-12	399	1	sn1xn‖	sn1xn‖	PROPN
ma-12	399	2	=	=	NOUN
ma-12	399	3	0	0	PROPN
ma-12	399	4	.	.	PUNCT
ma-12	399	5	(	(	PUNCT
ma-12	399	6	3.42	3.42	NUM
ma-12	399	7	)	)	PUNCT
ma-12	399	8	from	from	ADP
ma-12	399	9	‖xn+1	‖xn+1	NOUN
ma-12	399	10	−	−	PROPN
ma-12	399	11	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	399	12	≤	≤	NOUN
ma-12	399	13	‖xn+1	‖xn+1	NUM
ma-12	399	14	−	−	PROPN
ma-12	399	15	sn1xn‖+	sn1xn‖+	PROPN
ma-12	399	16	‖sn1xn	‖sn1xn	NOUN
ma-12	399	17	−	−	PROPN
ma-12	399	18	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	399	19	,	,	PUNCT
ma-12	399	20	(	(	PUNCT
ma-12	399	21	3.23	3.23	NUM
ma-12	399	22	)	)	PUNCT
ma-12	399	23	and	and	CCONJ
ma-12	399	24	(	(	PUNCT
ma-12	399	25	3.42	3.42	NUM
ma-12	399	26	)	)	PUNCT
ma-12	399	27	,	,	PUNCT
ma-12	399	28	we	we	PRON
ma-12	399	29	get	get	VERB
ma-12	399	30	lim	lim	PROPN
ma-12	399	31	n→∞	n→∞	X
ma-12	399	32	‖xn+1	‖xn+1	NUM
ma-12	400	1	−	−	PROPN
ma-12	401	1	t1(pt1)n−1yn‖	t1(pt1)n−1yn‖	PROPN
ma-12	401	2	=	=	SYM
ma-12	401	3	0	0	NUM
ma-12	401	4	.	.	PUNCT
ma-12	402	1	(	(	PUNCT
ma-12	402	2	3.43	3.43	NUM
ma-12	402	3	)	)	PUNCT
ma-12	402	4	also	also	ADV
ma-12	402	5	,	,	PUNCT
ma-12	402	6	from	from	ADP
ma-12	402	7	(	(	PUNCT
ma-12	402	8	3.23	3.23	NUM
ma-12	402	9	)	)	PUNCT
ma-12	402	10	,	,	PUNCT
ma-12	402	11	(	(	PUNCT
ma-12	402	12	3.24	3.24	NUM
ma-12	402	13	)	)	PUNCT
ma-12	402	14	and	and	CCONJ
ma-12	402	15	the	the	DET
ma-12	402	16	inequality	inequality	NOUN
ma-12	402	17	‖sn1xn	‖sn1xn	NOUN
ma-12	402	18	−	−	PROPN
ma-12	402	19	xn‖	xn‖	PROPN
ma-12	402	20	≤	≤	PROPN
ma-12	402	21	‖sn1xn	‖sn1xn	NOUN
ma-12	402	22	−	−	PROPN
ma-12	402	23	t1(pt1)n−1yn‖+	t1(pt1)n−1yn‖+	PRON
ma-12	402	24	‖t1(pt1)n−1yn	‖t1(pt1)n−1yn	ADP
ma-12	402	25	−	−	PROPN
ma-12	402	26	xn‖	xn‖	PROPN
ma-12	402	27	,	,	PUNCT
ma-12	402	28	we	we	PRON
ma-12	402	29	have	have	VERB
ma-12	402	30	lim	lim	PROPN
ma-12	402	31	n→∞	n→∞	NUM
ma-12	402	32	‖sn1xn	‖sn1xn	PROPN
ma-12	402	33	−	−	PROPN
ma-12	402	34	xn‖	xn‖	PROPN
ma-12	403	1	=	=	SYM
ma-12	403	2	0	0	PROPN
ma-12	403	3	.	.	PUNCT
ma-12	404	1	(	(	PUNCT
ma-12	404	2	3.44	3.44	NUM
ma-12	404	3	)	)	PUNCT
ma-12	404	4	again	again	ADV
ma-12	404	5	,	,	PUNCT
ma-12	404	6	from	from	ADP
ma-12	404	7	(	(	PUNCT
ma-12	404	8	3.41	3.41	NUM
ma-12	404	9	)	)	PUNCT
ma-12	404	10	,	,	PUNCT
ma-12	404	11	(	(	PUNCT
ma-12	404	12	3.44	3.44	NUM
ma-12	404	13	)	)	PUNCT
ma-12	404	14	and	and	CCONJ
ma-12	404	15	the	the	DET
ma-12	404	16	inequality	inequality	NOUN
ma-12	404	17	‖sn1xn	‖sn1xn	NOUN
ma-12	405	1	−	−	PROPN
ma-12	406	1	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	406	2	≤	≤	PROPN
ma-12	406	3	‖sn1xn	‖sn1xn	NOUN
ma-12	406	4	−	−	PROPN
ma-12	406	5	xn‖+	xn‖+	PUNCT
ma-12	407	1	‖xn	‖xn	PROPN
ma-12	407	2	−	−	PROPN
ma-12	407	3	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	407	4	,	,	PUNCT
ma-12	407	5	we	we	PRON
ma-12	407	6	have	have	VERB
ma-12	407	7	lim	lim	PROPN
ma-12	407	8	n→∞	n→∞	NUM
ma-12	407	9	‖sn1xn	‖sn1xn	NOUN
ma-12	407	10	−	−	PROPN
ma-12	407	11	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	NOUN
ma-12	407	12	=	=	SYM
ma-12	407	13	0	0	PROPN
ma-12	407	14	.	.	PUNCT
ma-12	408	1	(	(	PUNCT
ma-12	408	2	3.45	3.45	NUM
ma-12	408	3	)	)	PUNCT
ma-12	408	4	eur	eur	PROPN
ma-12	408	5	.	.	PUNCT
ma-12	409	1	j.	j.	PROPN
ma-12	409	2	math	math	PROPN
ma-12	409	3	.	.	PUNCT
ma-12	410	1	anal	anal	ADJ
ma-12	410	2	.	.	PUNCT
ma-12	411	1	1	1	NUM
ma-12	411	2	(	(	PUNCT
ma-12	411	3	2021	2021	NUM
ma-12	411	4	)	)	PUNCT
ma-12	412	1	60since	60since	NUM
ma-12	412	2	‖xn+1	‖xn+1	NUM
ma-12	413	1	−	−	PROPN
ma-12	413	2	t2(pt2)n−1yn‖	t2(pt2)n−1yn‖	PROPN
ma-12	413	3	≤	≤	NOUN
ma-12	413	4	‖xn+1	‖xn+1	PUNCT
ma-12	413	5	−	−	PROPN
ma-12	413	6	sn1xn‖+	sn1xn‖+	PROPN
ma-12	413	7	‖sn1xn	‖sn1xn	NOUN
ma-12	413	8	−	−	NOUN
ma-12	414	1	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	414	2	+	+	PROPN
ma-12	414	3	‖t2(pt2)n−1xn	‖t2(pt2)n−1xn	PROPN
ma-12	414	4	−	−	PROPN
ma-12	414	5	t2(pt2)n−1yn‖	t2(pt2)n−1yn‖	PROPN
ma-12	414	6	≤	≤	NOUN
ma-12	414	7	‖xn+1	‖xn+1	PUNCT
ma-12	414	8	−	−	PROPN
ma-12	414	9	sn1xn‖+	sn1xn‖+	PROPN
ma-12	414	10	‖sn1xn	‖sn1xn	NOUN
ma-12	414	11	−	−	PROPN
ma-12	414	12	t2(pt2)n−1xn‖+	t2(pt2)n−1xn‖+	ADP
ma-12	414	13	(	(	PUNCT
ma-12	414	14	‖xn	‖xn	PROPN
ma-12	414	15	−	−	NOUN
ma-12	414	16	yn‖	yn‖	PROPN
ma-12	415	1	+	+	PROPN
ma-12	415	2	k	k	PROPN
ma-12	415	3	(	(	PUNCT
ma-12	415	4	2	2	NUM
ma-12	415	5	)	)	PUNCT
ma-12	415	6	n	n	PROPN
ma-12	415	7	φ(‖xn	φ(‖xn	PROPN
ma-12	415	8	−	−	PROPN
ma-12	415	9	yn‖	yn‖	NOUN
ma-12	415	10	)	)	PUNCT
ma-12	415	11	+	+	NUM
ma-12	415	12	ν	ν	NOUN
ma-12	415	13	(	(	PUNCT
ma-12	415	14	2	2	NUM
ma-12	415	15	)	)	PUNCT
ma-12	415	16	n	n	CCONJ
ma-12	415	17	)	)	PUNCT
ma-12	415	18	≤	≤	NUM
ma-12	415	19	‖xn+1	‖xn+1	PUNCT
ma-12	415	20	−	−	PROPN
ma-12	415	21	sn1xn‖+	sn1xn‖+	PROPN
ma-12	415	22	‖sn1xn	‖sn1xn	NOUN
ma-12	415	23	−	−	NOUN
ma-12	415	24	t2(pt2)n−1xn‖+	t2(pt2)n−1xn‖+	ADP
ma-12	415	25	‖xn	‖xn	NUM
ma-12	415	26	−	−	PROPN
ma-12	415	27	yn‖	yn‖	PROPN
ma-12	416	1	+	+	PROPN
ma-12	416	2	mhn‖xn	mhn‖xn	PROPN
ma-12	416	3	−	−	PROPN
ma-12	416	4	yn‖	yn‖	PROPN
ma-12	416	5	)	)	PUNCT
ma-12	417	1	+	+	NUM
ma-12	417	2	θn	θn	NOUN
ma-12	417	3	=	=	SYM
ma-12	417	4	‖xn+1	‖xn+1	PUNCT
ma-12	417	5	−	−	PROPN
ma-12	417	6	sn1xn‖+	sn1xn‖+	PROPN
ma-12	417	7	‖sn1xn	‖sn1xn	NOUN
ma-12	417	8	−	−	NOUN
ma-12	418	1	t2(pt2)n−1xn‖	t2(pt2)n−1xn‖	PROPN
ma-12	418	2	+	+	PROPN
ma-12	418	3	(	(	PUNCT
ma-12	418	4	1	1	NUM
ma-12	418	5	+	+	NOUN
ma-12	418	6	mhn)‖xn	mhn)‖xn	PROPN
ma-12	418	7	−	−	PROPN
ma-12	418	8	yn‖	yn‖	PROPN
ma-12	418	9	)	)	PUNCT
ma-12	418	10	+	+	CCONJ
ma-12	419	1	θn	θn	ADP
ma-12	419	2	,	,	PUNCT
ma-12	419	3	it	it	PRON
ma-12	419	4	follows	follow	VERB
ma-12	419	5	from	from	ADP
ma-12	419	6	(	(	PUNCT
ma-12	419	7	3.38	3.38	NUM
ma-12	419	8	)	)	PUNCT
ma-12	419	9	,	,	PUNCT
ma-12	419	10	(	(	PUNCT
ma-12	419	11	3.42	3.42	NUM
ma-12	419	12	)	)	PUNCT
ma-12	419	13	,	,	PUNCT
ma-12	419	14	(	(	PUNCT
ma-12	419	15	3.45	3.45	NUM
ma-12	419	16	)	)	PUNCT
ma-12	419	17	and	and	CCONJ
ma-12	419	18	the	the	DET
ma-12	419	19	fact	fact	NOUN
ma-12	419	20	that	that	SCONJ
ma-12	419	21	∑∞	∑∞	NOUN
ma-12	419	22	n=1	n=1	PUNCT
ma-12	419	23	θn	θn	PROPN
ma-12	419	24	<	<	X
ma-12	419	25	∞	∞	PROPN
ma-12	419	26	that	that	PRON
ma-12	419	27	lim	lim	PROPN
ma-12	419	28	n→∞	n→∞	PRON
ma-12	419	29	‖xn+1	‖xn+1	NUM
ma-12	419	30	−	−	PROPN
ma-12	419	31	t2(pt2)n−1yn‖	t2(pt2)n−1yn‖	PROPN
ma-12	419	32	=	=	SYM
ma-12	419	33	0	0	PROPN
ma-12	419	34	.	.	PUNCT
ma-12	420	1	(	(	PUNCT
ma-12	420	2	3.46	3.46	NUM
ma-12	420	3	)	)	PUNCT
ma-12	420	4	now	now	ADV
ma-12	420	5	,	,	PUNCT
ma-12	420	6	from	from	ADP
ma-12	420	7	(	(	PUNCT
ma-12	420	8	3.30	3.30	NUM
ma-12	420	9	)	)	PUNCT
ma-12	420	10	,	,	PUNCT
ma-12	420	11	(	(	PUNCT
ma-12	420	12	3.41	3.41	NUM
ma-12	420	13	)	)	PUNCT
ma-12	420	14	and	and	CCONJ
ma-12	420	15	the	the	DET
ma-12	420	16	inequality	inequality	NOUN
ma-12	420	17	‖sn1xn	‖sn1xn	NOUN
ma-12	421	1	−	−	PROPN
ma-12	421	2	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	421	3	≤	≤	PROPN
ma-12	421	4	‖sn1xn	‖sn1xn	NOUN
ma-12	421	5	−	−	PROPN
ma-12	421	6	xn‖+	xn‖+	PROPN
ma-12	422	1	‖xn	‖xn	PROPN
ma-12	422	2	−	−	PROPN
ma-12	422	3	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	422	4	,	,	PUNCT
ma-12	422	5	we	we	PRON
ma-12	422	6	obtain	obtain	VERB
ma-12	422	7	lim	lim	PROPN
ma-12	422	8	n→∞	n→∞	NUM
ma-12	422	9	‖sn1xn	‖sn1xn	NOUN
ma-12	423	1	−	−	PROPN
ma-12	423	2	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	423	3	=	=	SYM
ma-12	423	4	0	0	PROPN
ma-12	423	5	.	.	PUNCT
ma-12	424	1	(	(	PUNCT
ma-12	424	2	3.47	3.47	NUM
ma-12	424	3	)	)	PUNCT
ma-12	424	4	since	since	SCONJ
ma-12	424	5	‖xn+1	‖xn+1	NUM
ma-12	424	6	−	−	PROPN
ma-12	424	7	t3(pt3)n−1yn‖	t3(pt3)n−1yn‖	PROPN
ma-12	424	8	≤	≤	PROPN
ma-12	424	9	‖xn+1	‖xn+1	PUNCT
ma-12	424	10	−	−	PROPN
ma-12	424	11	sn1xn‖+	sn1xn‖+	PROPN
ma-12	424	12	‖sn1xn	‖sn1xn	NOUN
ma-12	424	13	−	−	ADP
ma-12	425	1	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	425	2	+	+	PROPN
ma-12	425	3	‖t3(pt3)n−1xn	‖t3(pt3)n−1xn	PROPN
ma-12	425	4	−	−	PROPN
ma-12	425	5	t3(pt3)n−1yn‖	t3(pt3)n−1yn‖	PROPN
ma-12	425	6	≤	≤	PROPN
ma-12	425	7	‖xn+1	‖xn+1	PUNCT
ma-12	425	8	−	−	PROPN
ma-12	425	9	sn1xn‖+	sn1xn‖+	PROPN
ma-12	425	10	‖sn1xn	‖sn1xn	NOUN
ma-12	425	11	−	−	PROPN
ma-12	425	12	t3(pt3)n−1xn‖+	t3(pt3)n−1xn‖+	PROPN
ma-12	425	13	(	(	PUNCT
ma-12	425	14	‖xn	‖xn	PROPN
ma-12	425	15	−	−	NOUN
ma-12	425	16	yn‖	yn‖	PROPN
ma-12	426	1	+	+	PROPN
ma-12	426	2	k	k	PROPN
ma-12	426	3	(	(	PUNCT
ma-12	426	4	3	3	NUM
ma-12	426	5	)	)	PUNCT
ma-12	426	6	n	n	PROPN
ma-12	426	7	φ(‖xn	φ(‖xn	PROPN
ma-12	426	8	−	−	PROPN
ma-12	426	9	yn‖	yn‖	NOUN
ma-12	426	10	)	)	PUNCT
ma-12	426	11	+	+	NUM
ma-12	426	12	ν	ν	NOUN
ma-12	426	13	(	(	PUNCT
ma-12	426	14	3	3	NUM
ma-12	426	15	)	)	PUNCT
ma-12	426	16	n	n	CCONJ
ma-12	426	17	)	)	PUNCT
ma-12	426	18	≤	≤	NUM
ma-12	426	19	‖xn+1	‖xn+1	PUNCT
ma-12	426	20	−	−	PROPN
ma-12	426	21	sn1xn‖+	sn1xn‖+	PROPN
ma-12	426	22	‖sn1xn	‖sn1xn	NOUN
ma-12	426	23	−	−	NOUN
ma-12	426	24	t3(pt3)n−1xn‖+	t3(pt3)n−1xn‖+	PUNCT
ma-12	426	25	‖xn	‖xn	NUM
ma-12	426	26	−	−	NOUN
ma-12	426	27	yn‖	yn‖	PROPN
ma-12	427	1	+	+	PROPN
ma-12	427	2	mhn‖xn	mhn‖xn	PROPN
ma-12	427	3	−	−	PROPN
ma-12	427	4	yn‖	yn‖	PROPN
ma-12	427	5	)	)	PUNCT
ma-12	428	1	+	+	NUM
ma-12	428	2	θn	θn	NOUN
ma-12	428	3	=	=	SYM
ma-12	428	4	‖xn+1	‖xn+1	PUNCT
ma-12	428	5	−	−	PROPN
ma-12	428	6	sn1xn‖+	sn1xn‖+	PROPN
ma-12	428	7	‖sn1xn	‖sn1xn	NOUN
ma-12	428	8	−	−	ADP
ma-12	429	1	t3(pt3)n−1xn‖	t3(pt3)n−1xn‖	PROPN
ma-12	429	2	+	+	PROPN
ma-12	429	3	(	(	PUNCT
ma-12	429	4	1	1	NUM
ma-12	429	5	+	+	NOUN
ma-12	429	6	mhn)‖xn	mhn)‖xn	PROPN
ma-12	429	7	−	−	PROPN
ma-12	429	8	yn‖	yn‖	PROPN
ma-12	429	9	)	)	PUNCT
ma-12	430	1	+	+	CCONJ
ma-12	430	2	θn	θn	ADP
ma-12	430	3	it	it	PRON
ma-12	430	4	follows	follow	VERB
ma-12	430	5	from	from	ADP
ma-12	430	6	(	(	PUNCT
ma-12	430	7	3.38	3.38	NUM
ma-12	430	8	)	)	PUNCT
ma-12	430	9	,	,	PUNCT
ma-12	430	10	(	(	PUNCT
ma-12	430	11	3.42	3.42	NUM
ma-12	430	12	)	)	PUNCT
ma-12	430	13	,	,	PUNCT
ma-12	430	14	(	(	PUNCT
ma-12	430	15	3.47	3.47	NUM
ma-12	430	16	)	)	PUNCT
ma-12	430	17	and	and	CCONJ
ma-12	430	18	the	the	DET
ma-12	430	19	fact	fact	NOUN
ma-12	430	20	that	that	SCONJ
ma-12	430	21	∑∞	∑∞	NOUN
ma-12	430	22	n=1	n=1	PUNCT
ma-12	430	23	θn	θn	PROPN
ma-12	430	24	<	<	X
ma-12	430	25	∞	∞	PROPN
ma-12	430	26	that	that	PRON
ma-12	430	27	lim	lim	PROPN
ma-12	430	28	n→∞	n→∞	PRON
ma-12	430	29	‖xn+1	‖xn+1	NUM
ma-12	430	30	−	−	PROPN
ma-12	430	31	t3(pt3)n−1yn‖	t3(pt3)n−1yn‖	PROPN
ma-12	430	32	=	=	SYM
ma-12	430	33	0	0	PROPN
ma-12	430	34	.	.	PUNCT
ma-12	431	1	(	(	PUNCT
ma-12	431	2	3.48	3.48	NUM
ma-12	431	3	)	)	PUNCT
ma-12	431	4	eur	eur	PROPN
ma-12	431	5	.	.	PUNCT
ma-12	432	1	j.	j.	PROPN
ma-12	432	2	math	math	PROPN
ma-12	432	3	.	.	PUNCT
ma-12	433	1	anal	anal	ADJ
ma-12	433	2	.	.	PUNCT
ma-12	434	1	1	1	NUM
ma-12	434	2	(	(	PUNCT
ma-12	434	3	2021	2021	NUM
ma-12	434	4	)	)	PUNCT
ma-12	434	5	61again	61again	NUM
ma-12	434	6	,	,	PUNCT
ma-12	434	7	since	since	SCONJ
ma-12	434	8	(	(	PUNCT
ma-12	434	9	pt	pt	X
ma-12	434	10	i)(pt	i)(pt	PROPN
ma-12	434	11	i)n−2yn−1	i)n−2yn−1	PROPN
ma-12	434	12	,	,	PUNCT
ma-12	435	1	xn	xn	PROPN
ma-12	435	2	∈	∈	PROPN
ma-12	435	3	k	k	PROPN
ma-12	435	4	for	for	ADP
ma-12	435	5	i	i	PRON
ma-12	435	6	=	=	NOUN
ma-12	435	7	1	1	NUM
ma-12	435	8	,	,	PUNCT
ma-12	435	9	2	2	NUM
ma-12	435	10	,	,	PUNCT
ma-12	435	11	3	3	NUM
ma-12	435	12	and	and	CCONJ
ma-12	435	13	t1	t1	NOUN
ma-12	435	14	,	,	PUNCT
ma-12	435	15	t2	t2	NOUN
ma-12	435	16	,	,	PUNCT
ma-12	435	17	t3	t3	PROPN
ma-12	435	18	are	be	AUX
ma-12	435	19	three	three	NUM
ma-12	435	20	total	total	ADJ
ma-12	435	21	asymptoti	asymptoti	NOUN
ma-12	435	22	-	-	PUNCT
ma-12	435	23	cally	cally	ADV
ma-12	435	24	nonexpansive	nonexpansive	ADJ
ma-12	435	25	nonself	nonself	PROPN
ma-12	435	26	mappings	mapping	NOUN
ma-12	435	27	,	,	PUNCT
ma-12	435	28	we	we	PRON
ma-12	435	29	have	have	VERB
ma-12	435	30	‖ti(pti)n−1yn−1	‖ti(pti)n−1yn−1	PROPN
ma-12	435	31	−	−	PROPN
ma-12	435	32	tixn‖	tixn‖	PROPN
ma-12	435	33	=	=	SYM
ma-12	435	34	‖ti(pti)(pti	‖ti(pti)(pti	PROPN
ma-12	435	35	)	)	PUNCT
ma-12	435	36	n−2yn−1	n−2yn−1	ADV
ma-12	435	37	−	−	PROPN
ma-12	435	38	ti(pxn)‖	ti(pxn)‖	NOUN
ma-12	435	39	≤	≤	NUM
ma-12	435	40	‖(pti)(pti	‖(pti)(pti	NOUN
ma-12	435	41	)	)	PUNCT
ma-12	435	42	n−2yn−1	n−2yn−1	ADV
ma-12	435	43	−	−	PROPN
ma-12	435	44	p	p	NOUN
ma-12	435	45	(	(	PUNCT
ma-12	435	46	xn)‖	xn)‖	PROPN
ma-12	435	47	+	+	PROPN
ma-12	435	48	k	k	PROPN
ma-12	435	49	(	(	PUNCT
ma-12	435	50	i	i	NOUN
ma-12	435	51	)	)	PUNCT
ma-12	435	52	n	n	PRON
ma-12	435	53	φ(‖(pti)(pti	φ(‖(pti)(pti	PROPN
ma-12	435	54	)	)	PUNCT
ma-12	435	55	n−2yn−1	n−2yn−1	ADV
ma-12	435	56	−	−	PROPN
ma-12	435	57	p	p	NOUN
ma-12	435	58	(	(	PUNCT
ma-12	435	59	xn)‖	xn)‖	PROPN
ma-12	435	60	)	)	PUNCT
ma-12	436	1	+	+	CCONJ
ma-12	436	2	ν	ν	X
ma-12	436	3	(	(	PUNCT
ma-12	436	4	i	i	NOUN
ma-12	436	5	)	)	PUNCT
ma-12	436	6	n	n	CCONJ
ma-12	436	7	≤	≤	NUM
ma-12	436	8	‖(pti)(pti	‖(pti)(pti	PROPN
ma-12	436	9	)	)	PUNCT
ma-12	436	10	n−2yn−1	n−2yn−1	ADV
ma-12	436	11	−	−	PROPN
ma-12	436	12	p	p	NOUN
ma-12	436	13	(	(	PUNCT
ma-12	436	14	xn)‖	xn)‖	PROPN
ma-12	436	15	+	+	PROPN
ma-12	436	16	mhn‖(pt	mhn‖(pt	PROPN
ma-12	436	17	i)(pt	i)(pt	X
ma-12	436	18	i)n−2yn−1	i)n−2yn−1	PROPN
ma-12	437	1	−	−	PROPN
ma-12	437	2	p	p	X
ma-12	437	3	(	(	PUNCT
ma-12	437	4	xn)‖+	xn)‖+	PROPN
ma-12	437	5	θn	θn	NOUN
ma-12	437	6	=	=	SYM
ma-12	437	7	(	(	PUNCT
ma-12	437	8	1	1	NUM
ma-12	437	9	+	+	NOUN
ma-12	437	10	mhn)‖(pti)(pti	mhn)‖(pti)(pti	PROPN
ma-12	437	11	)	)	PUNCT
ma-12	437	12	n−2yn−1	n−2yn−1	ADV
ma-12	437	13	−	−	PROPN
ma-12	437	14	p	p	X
ma-12	437	15	(	(	PUNCT
ma-12	437	16	xn)‖+	xn)‖+	PROPN
ma-12	437	17	θn	θn	NOUN
ma-12	437	18	=	=	SYM
ma-12	437	19	(	(	PUNCT
ma-12	437	20	1	1	NUM
ma-12	437	21	+	+	NOUN
ma-12	437	22	mhn)‖ti(pti)n−2yn−1	mhn)‖ti(pti)n−2yn−1	PROPN
ma-12	437	23	−	−	PROPN
ma-12	437	24	xn‖+	xn‖+	PROPN
ma-12	437	25	θn	θn	PROPN
ma-12	437	26	.	.	PROPN
ma-12	437	27	(	(	PUNCT
ma-12	437	28	3.49	3.49	NUM
ma-12	437	29	)	)	PUNCT
ma-12	437	30	for	for	ADP
ma-12	437	31	i	i	PROPN
ma-12	437	32	=	=	SYM
ma-12	437	33	1.2.3	1.2.3	NUM
ma-12	437	34	,	,	PUNCT
ma-12	437	35	,	,	PUNCT
ma-12	437	36	it	it	PRON
ma-12	437	37	follows	follow	VERB
ma-12	437	38	from	from	ADP
ma-12	437	39	(	(	PUNCT
ma-12	437	40	3.43	3.43	NUM
ma-12	437	41	)	)	PUNCT
ma-12	437	42	,	,	PUNCT
ma-12	437	43	(	(	PUNCT
ma-12	437	44	3.46	3.46	NUM
ma-12	437	45	)	)	PUNCT
ma-12	437	46	and	and	CCONJ
ma-12	437	47	(	(	PUNCT
ma-12	437	48	3.48	3.48	NUM
ma-12	437	49	)	)	PUNCT
ma-12	437	50	that	that	PRON
ma-12	437	51	lim	lim	PROPN
ma-12	437	52	n→∞	n→∞	NUM
ma-12	437	53	‖t	‖t	PROPN
ma-12	437	54	i(pt	i(pt	PROPN
ma-12	437	55	i)n−1yn−1	i)n−1yn−1	ADJ
ma-12	437	56	−	−	PROPN
ma-12	437	57	t	t	NOUN
ma-12	437	58	ixn‖	ixn‖	PUNCT
ma-12	437	59	=	=	NOUN
ma-12	437	60	0	0	X
ma-12	437	61	.	.	PUNCT
ma-12	438	1	(	(	PUNCT
ma-12	438	2	3.50	3.50	NUM
ma-12	438	3	)	)	PUNCT
ma-12	438	4	observe	observe	VERB
ma-12	438	5	that	that	PRON
ma-12	438	6	‖xn+1	‖xn+1	VERB
ma-12	438	7	−	−	NUM
ma-12	438	8	yn‖	yn‖	NOUN
ma-12	438	9	≤	≤	PROPN
ma-12	438	10	‖xn+1	‖xn+1	PUNCT
ma-12	438	11	−	−	PROPN
ma-12	438	12	t1(pt1)n−1yn‖+	t1(pt1)n−1yn‖+	PUNCT
ma-12	438	13	‖t1(pt1)n−1yn	‖t1(pt1)n−1yn	ADP
ma-12	438	14	−	−	PROPN
ma-12	438	15	xn‖+	xn‖+	PUNCT
ma-12	439	1	‖xn	‖xn	PROPN
ma-12	439	2	−	−	NUM
ma-12	439	3	yn‖	yn‖	PROPN
ma-12	439	4	,	,	PUNCT
ma-12	439	5	so	so	SCONJ
ma-12	439	6	that	that	SCONJ
ma-12	439	7	,	,	PUNCT
ma-12	439	8	by	by	ADP
ma-12	439	9	(	(	PUNCT
ma-12	439	10	3.24	3.24	NUM
ma-12	439	11	)	)	PUNCT
ma-12	439	12	,	,	PUNCT
ma-12	439	13	(	(	PUNCT
ma-12	439	14	3.38	3.38	NUM
ma-12	439	15	)	)	PUNCT
ma-12	439	16	and	and	CCONJ
ma-12	439	17	(	(	PUNCT
ma-12	439	18	3.43	3.43	NUM
ma-12	439	19	)	)	PUNCT
ma-12	439	20	,	,	PUNCT
ma-12	439	21	we	we	PRON
ma-12	439	22	get	get	VERB
ma-12	439	23	lim	lim	PROPN
ma-12	439	24	n→∞	n→∞	X
ma-12	439	25	‖xn+1	‖xn+1	NUM
ma-12	439	26	−	−	NOUN
ma-12	439	27	yn‖	yn‖	NOUN
ma-12	439	28	=	=	NOUN
ma-12	439	29	0	0	PROPN
ma-12	439	30	.	.	PUNCT
ma-12	439	31	(	(	PUNCT
ma-12	439	32	3.51	3.51	NUM
ma-12	439	33	)	)	PUNCT
ma-12	439	34	next	next	ADV
ma-12	439	35	,	,	PUNCT
ma-12	439	36	observe	observe	VERB
ma-12	439	37	,	,	PUNCT
ma-12	439	38	for	for	ADP
ma-12	439	39	i	i	PROPN
ma-12	439	40	=	=	SYM
ma-12	439	41	1	1	NUM
ma-12	439	42	,	,	PUNCT
ma-12	439	43	2	2	NUM
ma-12	439	44	,	,	PUNCT
ma-12	439	45	3	3	NUM
ma-12	439	46	,	,	PUNCT
ma-12	439	47	that	that	SCONJ
ma-12	439	48	‖xn	‖xn	PROPN
ma-12	439	49	−	−	SYM
ma-12	439	50	tixn‖	tixn‖	PROPN
ma-12	439	51	≤	≤	PUNCT
ma-12	440	1	‖xn	‖xn	PUNCT
ma-12	440	2	−	−	NUM
ma-12	440	3	ti(pti)n−1xn‖+	ti(pti)n−1xn‖+	NUM
ma-12	440	4	‖ti(pti)n−1xn	‖ti(pti)n−1xn	NOUN
ma-12	440	5	−	−	PROPN
ma-12	441	1	ti(pti)n−1yn−1‖	ti(pti)n−1yn−1‖	NUM
ma-12	442	1	+	+	NOUN
ma-12	442	2	‖ti(pti)n−1yn−1	‖ti(pti)n−1yn−1	PROPN
ma-12	442	3	−	−	NUM
ma-12	442	4	tixn‖	tixn‖	PROPN
ma-12	442	5	≤	≤	PUNCT
ma-12	442	6	‖xn	‖xn	NUM
ma-12	442	7	−	−	X
ma-12	442	8	ti(pti)n−1xn‖+	ti(pti)n−1xn‖+	NUM
ma-12	443	1	[	[	X
ma-12	443	2	‖xn	‖xn	NUM
ma-12	443	3	−	−	NOUN
ma-12	443	4	yn−1‖+	yn−1‖+	NOUN
ma-12	443	5	k	k	PROPN
ma-12	443	6	(	(	PUNCT
ma-12	443	7	i	i	NOUN
ma-12	443	8	)	)	PUNCT
ma-12	443	9	n	n	CCONJ
ma-12	443	10	φ(‖xn	φ(‖xn	PROPN
ma-12	443	11	−	−	PROPN
ma-12	443	12	yn−1‖	yn−1‖	PROPN
ma-12	443	13	)	)	PUNCT
ma-12	444	1	+	+	ADJ
ma-12	444	2	ν	ν	X
ma-12	444	3	(	(	PUNCT
ma-12	444	4	i	i	NOUN
ma-12	444	5	)	)	PUNCT
ma-12	444	6	n	n	CCONJ
ma-12	444	7	]	]	PUNCT
ma-12	445	1	+	+	CCONJ
ma-12	445	2	‖ti(pti)n−1yn−1	‖ti(pti)n−1yn−1	PROPN
ma-12	445	3	−	−	NUM
ma-12	445	4	tixn‖	tixn‖	PROPN
ma-12	445	5	≤	≤	PUNCT
ma-12	445	6	‖xn	‖xn	NUM
ma-12	445	7	−	−	X
ma-12	445	8	ti(pti)n−1xn‖+	ti(pti)n−1xn‖+	NUM
ma-12	445	9	‖xn	‖xn	PROPN
ma-12	445	10	−	−	NOUN
ma-12	445	11	yn−1‖+	yn−1‖+	NUM
ma-12	446	1	k	k	PROPN
ma-12	446	2	(	(	PUNCT
ma-12	446	3	i	i	NOUN
ma-12	446	4	)	)	PUNCT
ma-12	446	5	n	n	PROPN
ma-12	446	6	m‖xn	m‖xn	PROPN
ma-12	446	7	−	−	PROPN
ma-12	446	8	yn−1‖	yn−1‖	PROPN
ma-12	447	1	+	+	NOUN
ma-12	447	2	ν	ν	X
ma-12	447	3	(	(	PUNCT
ma-12	447	4	i	i	NOUN
ma-12	447	5	)	)	PUNCT
ma-12	447	6	n	n	PROPN
ma-12	447	7	+	+	CCONJ
ma-12	447	8	‖ti(pt1)n−1yn−1	‖ti(pt1)n−1yn−1	ADJ
ma-12	447	9	−	−	NOUN
ma-12	447	10	tixn‖	tixn‖	PROPN
ma-12	448	1	=	=	PUNCT
ma-12	448	2	‖xn	‖xn	PROPN
ma-12	449	1	−	−	NOUN
ma-12	449	2	ti(pti)n−1xn‖+	ti(pti)n−1xn‖+	NUM
ma-12	449	3	(	(	PUNCT
ma-12	449	4	1	1	NUM
ma-12	449	5	+	+	NUM
ma-12	449	6	k	k	PROPN
ma-12	449	7	(	(	PUNCT
ma-12	449	8	i	i	NOUN
ma-12	449	9	)	)	PUNCT
ma-12	449	10	n	n	CCONJ
ma-12	449	11	m)‖xn	m)‖xn	NUM
ma-12	449	12	−	−	ADP
ma-12	449	13	yn−1‖+	yn−1‖+	NUM
ma-12	449	14	ν	ν	NOUN
ma-12	449	15	(	(	PUNCT
ma-12	449	16	i	i	NOUN
ma-12	449	17	)	)	PUNCT
ma-12	449	18	n	n	CCONJ
ma-12	449	19	]	]	PUNCT
ma-12	450	1	+	+	ADJ
ma-12	450	2	‖ti(pti)n−1yn−1	‖ti(pti)n−1yn−1	PROPN
ma-12	450	3	−	−	NUM
ma-12	450	4	tixn‖	tixn‖	PROPN
ma-12	450	5	≤	≤	PUNCT
ma-12	450	6	‖xn	‖xn	PUNCT
ma-12	450	7	−	−	PUNCT
ma-12	450	8	ti(pti)n−1xn‖+max	ti(pti)n−1xn‖+max	PUNCT
ma-12	451	1	[	[	X
ma-12	451	2	supn≥1(1	supn≥1(1	X
ma-12	451	3	+	+	X
ma-12	451	4	k	k	PROPN
ma-12	451	5	(	(	PUNCT
ma-12	451	6	i	i	NOUN
ma-12	451	7	)	)	PUNCT
ma-12	451	8	n	n	PROPN
ma-12	451	9	m)]‖xn	m)]‖xn	PROPN
ma-12	451	10	−	−	PROPN
ma-12	451	11	yn−1‖	yn−1‖	PROPN
ma-12	452	1	+	+	SYM
ma-12	452	2	max	max	PROPN
ma-12	452	3	[	[	X
ma-12	452	4	supn≥1]ν	supn≥1]ν	PROPN
ma-12	452	5	(	(	PUNCT
ma-12	452	6	i	i	NOUN
ma-12	452	7	)	)	PUNCT
ma-12	452	8	n	n	CCONJ
ma-12	452	9	]	]	PUNCT
ma-12	453	1	+	+	CCONJ
ma-12	453	2	‖ti(pti)n−1yn−1	‖ti(pti)n−1yn−1	PROPN
ma-12	453	3	−	−	ADP
ma-12	453	4	tixn‖	tixn‖	PROPN
ma-12	453	5	thus	thus	ADV
ma-12	453	6	,	,	PUNCT
ma-12	453	7	it	it	PRON
ma-12	453	8	follows	follow	VERB
ma-12	453	9	from	from	ADP
ma-12	453	10	(	(	PUNCT
ma-12	453	11	3.30	3.30	NUM
ma-12	453	12	)	)	PUNCT
ma-12	453	13	,	,	PUNCT
ma-12	453	14	(	(	PUNCT
ma-12	453	15	3.36	3.36	NUM
ma-12	453	16	)	)	PUNCT
ma-12	453	17	,	,	PUNCT
ma-12	453	18	(	(	PUNCT
ma-12	453	19	3.41	3.41	NUM
ma-12	453	20	)	)	PUNCT
ma-12	453	21	,	,	PUNCT
ma-12	453	22	(	(	PUNCT
ma-12	453	23	3.50	3.50	NUM
ma-12	453	24	)	)	PUNCT
ma-12	453	25	and	and	CCONJ
ma-12	453	26	(	(	PUNCT
ma-12	453	27	3.51	3.51	NUM
ma-12	453	28	)	)	PUNCT
ma-12	453	29	that	that	PRON
ma-12	453	30	limn→∞	limn→∞	VERB
ma-12	453	31	‖xn	‖xn	PROPN
ma-12	453	32	−	−	PROPN
ma-12	453	33	tixn‖	tixn‖	PROPN
ma-12	453	34	=	=	SYM
ma-12	453	35	0	0	NUM
ma-12	453	36	,	,	PUNCT
ma-12	453	37	for	for	ADP
ma-12	453	38	i	i	PROPN
ma-12	453	39	=	=	NOUN
ma-12	453	40	1	1	NUM
ma-12	453	41	,	,	PUNCT
ma-12	453	42	,	,	PUNCT
ma-12	453	43	2	2	NUM
ma-12	453	44	,	,	PUNCT
ma-12	453	45	3	3	NUM
ma-12	453	46	.	.	X
ma-12	453	47	eur	eur	PROPN
ma-12	453	48	.	.	PUNCT
ma-12	454	1	j.	j.	PROPN
ma-12	454	2	math	math	PROPN
ma-12	454	3	.	.	PUNCT
ma-12	455	1	anal	anal	ADJ
ma-12	455	2	.	.	PUNCT
ma-12	456	1	1	1	NUM
ma-12	456	2	(	(	PUNCT
ma-12	456	3	2021	2021	NUM
ma-12	456	4	)	)	PUNCT
ma-12	456	5	62finally	62finally	ADV
ma-12	456	6	,	,	PUNCT
ma-12	456	7	we	we	PRON
ma-12	456	8	prove	prove	VERB
ma-12	456	9	that	that	SCONJ
ma-12	456	10	limn→∞	limn→∞	PROPN
ma-12	456	11	‖xn	‖xn	PROPN
ma-12	456	12	−	−	PROPN
ma-12	456	13	sni	sni	PROPN
ma-12	456	14	xn‖	xn‖	PROPN
ma-12	457	1	=	=	SYM
ma-12	457	2	0	0	NUM
ma-12	457	3	,	,	PUNCT
ma-12	457	4	for	for	ADP
ma-12	457	5	i	i	PROPN
ma-12	457	6	=	=	NOUN
ma-12	457	7	1	1	NUM
ma-12	457	8	,	,	PUNCT
ma-12	457	9	,	,	PUNCT
ma-12	457	10	2	2	NUM
ma-12	457	11	,	,	PUNCT
ma-12	457	12	3.infact	3.infact	NUM
ma-12	457	13	,	,	PUNCT
ma-12	457	14	by	by	ADP
ma-12	457	15	condition	condition	NOUN
ma-12	457	16	(	(	PUNCT
ma-12	457	17	ii	ii	NOUN
ma-12	457	18	)	)	PUNCT
ma-12	457	19	,	,	PUNCT
ma-12	457	20	we	we	PRON
ma-12	457	21	have	have	VERB
ma-12	457	22	for	for	ADP
ma-12	457	23	i	i	PRON
ma-12	457	24	=	=	SYM
ma-12	457	25	1	1	NUM
ma-12	457	26	,	,	PUNCT
ma-12	457	27	2	2	NUM
ma-12	457	28	,	,	PUNCT
ma-12	457	29	3	3	NUM
ma-12	457	30	,	,	PUNCT
ma-12	457	31	that	that	SCONJ
ma-12	457	32	‖xn	‖xn	PROPN
ma-12	457	33	−	−	PROPN
ma-12	457	34	sni	sni	PROPN
ma-12	457	35	xn	xn	PROPN
ma-12	457	36	≤	≤	PROPN
ma-12	458	1	‖xn	‖xn	NUM
ma-12	458	2	−	−	PROPN
ma-12	458	3	ti(pti)n−1xn‖+	ti(pti)n−1xn‖+	NUM
ma-12	458	4	‖sni	‖sni	PROPN
ma-12	458	5	xn	xn	PROPN
ma-12	458	6	−	−	PROPN
ma-12	458	7	ti(pti)n−1xn‖	ti(pti)n−1xn‖	NOUN
ma-12	458	8	thus	thus	ADV
ma-12	458	9	,	,	PUNCT
ma-12	458	10	it	it	PRON
ma-12	458	11	follows	follow	VERB
ma-12	458	12	from	from	ADP
ma-12	458	13	(	(	PUNCT
ma-12	458	14	3.29	3.29	NUM
ma-12	458	15	)	)	PUNCT
ma-12	458	16	,	,	PUNCT
ma-12	458	17	(	(	PUNCT
ma-12	458	18	3.30	3.30	NUM
ma-12	458	19	)	)	PUNCT
ma-12	458	20	,	,	PUNCT
ma-12	458	21	(	(	PUNCT
ma-12	458	22	3.36	3.36	NUM
ma-12	458	23	)	)	PUNCT
ma-12	458	24	,	,	PUNCT
ma-12	458	25	(	(	PUNCT
ma-12	458	26	3.40	3.40	NUM
ma-12	458	27	)	)	PUNCT
ma-12	458	28	,	,	PUNCT
ma-12	458	29	(	(	PUNCT
ma-12	458	30	3.41	3.41	NUM
ma-12	458	31	)	)	PUNCT
ma-12	458	32	and	and	CCONJ
ma-12	458	33	(	(	PUNCT
ma-12	458	34	3.45	3.45	NUM
ma-12	458	35	)	)	PUNCT
ma-12	459	1	that	that	PRON
ma-12	459	2	lim	lim	PROPN
ma-12	459	3	n→∞	n→∞	X
ma-12	460	1	‖xn	‖xn	PROPN
ma-12	460	2	−	−	PROPN
ma-12	460	3	sni	sni	PROPN
ma-12	460	4	xn‖	xn‖	PROPN
ma-12	460	5	=	=	SYM
ma-12	460	6	0	0	NUM
ma-12	460	7	,	,	PUNCT
ma-12	460	8	f	f	PROPN
ma-12	460	9	or	or	CCONJ
ma-12	460	10	i	i	PRON
ma-12	460	11	=	=	NOUN
ma-12	460	12	1	1	NUM
ma-12	460	13	,	,	PUNCT
ma-12	460	14	2	2	NUM
ma-12	460	15	,	,	PUNCT
ma-12	460	16	3	3	NUM
ma-12	460	17	.	.	PUNCT
ma-12	460	18	(	(	PUNCT
ma-12	460	19	3.52	3.52	NUM
ma-12	460	20	)	)	PUNCT
ma-12	460	21	this	this	PRON
ma-12	460	22	completes	complete	VERB
ma-12	460	23	the	the	DET
ma-12	460	24	proof	proof	NOUN
ma-12	460	25	of	of	ADP
ma-12	460	26	lemma	lemma	PROPN
ma-12	460	27	3.3	3.3	NUM
ma-12	460	28	.	.	PUNCT
ma-12	461	1	�	�	PROPN
ma-12	461	2	lemma	lemma	PROPN
ma-12	461	3	3.4	3.4	NUM
ma-12	461	4	.	.	PUNCT
ma-12	462	1	under	under	ADP
ma-12	462	2	the	the	DET
ma-12	462	3	assumption	assumption	NOUN
ma-12	462	4	of	of	ADP
ma-12	462	5	lemma	lemma	PROPN
ma-12	462	6	3.2	3.2	NUM
ma-12	462	7	,	,	PUNCT
ma-12	462	8	for	for	ADP
ma-12	462	9	all	all	DET
ma-12	462	10	p1	p1	NOUN
ma-12	462	11	,	,	PUNCT
ma-12	462	12	p2	p2	PROPN
ma-12	462	13	∈	∈	PROPN
ma-12	462	14	∩3i1(f	∩3i1(f	VERB
ma-12	462	15	(	(	PUNCT
ma-12	462	16	si	si	NOUN
ma-12	462	17	)	)	PUNCT
ma-12	462	18	∩	∩	ADJ
ma-12	462	19	f	f	X
ma-12	462	20	(	(	PUNCT
ma-12	462	21	ti	ti	NOUN
ma-12	462	22	)	)	PUNCT
ma-12	462	23	)	)	PUNCT
ma-12	462	24	,	,	PUNCT
ma-12	462	25	the	the	DET
ma-12	462	26	limit	limit	NOUN
ma-12	462	27	limn→∞	limn→∞	VERB
ma-12	462	28	‖xn	‖xn	PUNCT
ma-12	462	29	+	+	ADJ
ma-12	462	30	(	(	PUNCT
ma-12	462	31	1−	1−	NUM
ma-12	462	32	t)p1	t)p1	PROPN
ma-12	462	33	−	−	PROPN
ma-12	462	34	p2‖	p2‖	PROPN
ma-12	462	35	exists	exist	VERB
ma-12	462	36	for	for	ADP
ma-12	462	37	all	all	DET
ma-12	462	38	t	t	NOUN
ma-12	462	39	∈	∈	PROPN
ma-12	463	1	[	[	X
ma-12	463	2	0	0	NUM
ma-12	463	3	,	,	PUNCT
ma-12	463	4	1	1	NUM
ma-12	463	5	]	]	PUNCT
ma-12	463	6	,	,	PUNCT
ma-12	463	7	where	where	SCONJ
ma-12	463	8	{	{	PUNCT
ma-12	463	9	xn	xn	X
ma-12	463	10	}	}	PUNCT
ma-12	463	11	is	be	AUX
ma-12	463	12	the	the	DET
ma-12	463	13	sequence	sequence	NOUN
ma-12	463	14	defined	define	VERB
ma-12	463	15	by	by	ADP
ma-12	463	16	(	(	PUNCT
ma-12	463	17	1.7	1.7	NUM
ma-12	463	18	)	)	PUNCT
ma-12	463	19	.	.	PUNCT
ma-12	464	1	proof	proof	NOUN
ma-12	464	2	.	.	PUNCT
ma-12	465	1	by	by	ADP
ma-12	465	2	lemma	lemma	PROPN
ma-12	465	3	3.2	3.2	NUM
ma-12	465	4	,	,	PUNCT
ma-12	465	5	limn→∞	limn→∞	PROPN
ma-12	465	6	‖xn	‖xn	PROPN
ma-12	465	7	−	−	PROPN
ma-12	465	8	q‖	q‖	NOUN
ma-12	465	9	exists	exist	VERB
ma-12	465	10	for	for	ADP
ma-12	465	11	all	all	DET
ma-12	465	12	q	q	PROPN
ma-12	465	13	∈	∈	PROPN
ma-12	465	14	f	f	PROPN
ma-12	465	15	and	and	CCONJ
ma-12	465	16	therefor	therefor	ADP
ma-12	465	17	{	{	PUNCT
ma-12	465	18	xn	xn	PROPN
ma-12	465	19	}	}	PUNCT
ma-12	465	20	is	be	AUX
ma-12	465	21	bounded	bound	VERB
ma-12	465	22	.	.	PUNCT
ma-12	466	1	let	let	VERB
ma-12	466	2	an(t	an(t	NOUN
ma-12	466	3	)	)	PUNCT
ma-12	467	1	=	=	PUNCT
ma-12	468	1	‖xn	‖xn	PUNCT
ma-12	468	2	+	+	CCONJ
ma-12	468	3	(	(	PUNCT
ma-12	468	4	1	1	NUM
ma-12	468	5	−	−	PROPN
ma-12	468	6	t)p1	t)p1	PROPN
ma-12	468	7	−	−	PROPN
ma-12	468	8	p2‖	p2‖	NOUN
ma-12	468	9	exists	exist	VERB
ma-12	468	10	for	for	ADP
ma-12	468	11	all	all	DET
ma-12	468	12	t	t	NOUN
ma-12	468	13	∈	∈	PROPN
ma-12	469	1	[	[	X
ma-12	469	2	0	0	NUM
ma-12	469	3	,	,	PUNCT
ma-12	469	4	1	1	NUM
ma-12	469	5	]	]	PUNCT
ma-12	469	6	.	.	PUNCT
ma-12	470	1	then	then	ADV
ma-12	470	2	,	,	PUNCT
ma-12	470	3	limn→∞	limn→∞	PROPN
ma-12	470	4	a(0	a(0	PROPN
ma-12	470	5	)	)	PUNCT
ma-12	471	1	=	=	SYM
ma-12	471	2	‖p1	‖p1	PROPN
ma-12	471	3	−	−	PROPN
ma-12	471	4	p2‖	p2‖	PROPN
ma-12	471	5	and	and	CCONJ
ma-12	471	6	limn→∞	limn→∞	PROPN
ma-12	471	7	a(1	a(1	NOUN
ma-12	471	8	)	)	PUNCT
ma-12	471	9	=	=	SYM
ma-12	471	10	‖xn−	‖xn−	PROPN
ma-12	471	11	p2‖	p2‖	PROPN
ma-12	471	12	exist	exist	VERB
ma-12	471	13	by	by	ADP
ma-12	471	14	lemma	lemma	PROPN
ma-12	471	15	3.2	3.2	NUM
ma-12	471	16	.	.	PUNCT
ma-12	472	1	it	it	PRON
ma-12	472	2	remains	remain	VERB
ma-12	472	3	therefor	therefor	ADJ
ma-12	472	4	to	to	PART
ma-12	472	5	prove	prove	VERB
ma-12	472	6	lemma	lemma	PROPN
ma-12	472	7	3.4	3.4	NUM
ma-12	472	8	for	for	ADP
ma-12	472	9	t	t	PROPN
ma-12	472	10	∈	∈	PROPN
ma-12	472	11	(	(	PUNCT
ma-12	472	12	0	0	NUM
ma-12	472	13	,	,	PUNCT
ma-12	472	14	1).for	1).for	NOUN
ma-12	472	15	all	all	PRON
ma-12	472	16	x	x	SYM
ma-12	472	17	∈	∈	PROPN
ma-12	472	18	k	k	NOUN
ma-12	472	19	,	,	PUNCT
ma-12	472	20	we	we	PRON
ma-12	472	21	define	define	VERB
ma-12	472	22	the	the	DET
ma-12	472	23	mapping	mapping	NOUN
ma-12	472	24			PROPN
ma-12	472	25	rn(x	rn(x	NUM
ma-12	472	26	)	)	PUNCT
ma-12	472	27	=	=	PUNCT
ma-12	473	1	p	p	X
ma-12	473	2	[	[	X
ma-12	473	3	(	(	PUNCT
ma-12	473	4	1−	1−	NUM
ma-12	473	5	γn)sn3	γn)sn3	NOUN
ma-12	473	6	+	+	CCONJ
ma-12	473	7	γt3(pt3	γt3(pt3	NOUN
ma-12	473	8	)	)	PUNCT
ma-12	473	9	n−1xn	n−1xn	NOUN
ma-12	473	10	]	]	PUNCT
ma-12	473	11	;	;	PUNCT
ma-12	473	12	wn(x	wn(x	X
ma-12	473	13	)	)	PUNCT
ma-12	473	14	=	=	SYM
ma-12	474	1	p	p	X
ma-12	474	2	[	[	X
ma-12	474	3	(	(	PUNCT
ma-12	474	4	1−	1−	NUM
ma-12	474	5	βn)sn2	βn)sn2	NOUN
ma-12	474	6	+	+	CCONJ
ma-12	474	7	βt2(pt2	βt2(pt2	NOUN
ma-12	474	8	)	)	PUNCT
ma-12	474	9	n−1xn	n−1xn	NOUN
ma-12	474	10	]	]	PUNCT
ma-12	474	11	;	;	PUNCT
ma-12	474	12	vn(x	vn(x	X
ma-12	474	13	)	)	PUNCT
ma-12	475	1	=	=	PUNCT
ma-12	476	1	p	p	X
ma-12	476	2	[	[	X
ma-12	476	3	(	(	PUNCT
ma-12	476	4	1−	1−	NUM
ma-12	476	5	αn)sn1	αn)sn1	NOUN
ma-12	476	6	+	+	CCONJ
ma-12	476	7	αt1(pt1	αt1(pt1	ADJ
ma-12	476	8	)	)	PUNCT
ma-12	476	9	n−1xn	n−1xn	NOUN
ma-12	476	10	]	]	PUNCT
ma-12	476	11	,	,	PUNCT
ma-12	476	12	n	n	X
ma-12	476	13	≥	≥	NOUN
ma-12	476	14	1	1	NUM
ma-12	476	15	.	.	PUNCT
ma-12	476	16	(	(	PUNCT
ma-12	476	17	3.53	3.53	NUM
ma-12	476	18	)	)	PUNCT
ma-12	476	19	then	then	ADV
ma-12	476	20	,	,	PUNCT
ma-12	476	21	it	it	PRON
ma-12	476	22	follows	follow	VERB
ma-12	476	23	that	that	PRON
ma-12	476	24	xn+1	xn+1	ADV
ma-12	476	25	=	=	SYM
ma-12	476	26	vnxn	vnxn	NOUN
ma-12	476	27	,	,	PUNCT
ma-12	476	28	vnp	vnp	NOUN
ma-12	476	29	=	=	SYM
ma-12	476	30	p,∀p	p,∀p	PROPN
ma-12	476	31	∈	∈	PROPN
ma-12	476	32	f	f	PROPN
ma-12	476	33	.	.	PUNCT
ma-12	477	1	now	now	ADV
ma-12	477	2	,	,	PUNCT
ma-12	477	3	from	from	ADP
ma-12	477	4	(	(	PUNCT
ma-12	477	5	3.12	3.12	NUM
ma-12	477	6	)	)	PUNCT
ma-12	477	7	,	,	PUNCT
ma-12	477	8	(	(	PUNCT
ma-12	477	9	3.15	3.15	NUM
ma-12	477	10	)	)	PUNCT
ma-12	477	11	and	and	CCONJ
ma-12	477	12	(	(	PUNCT
ma-12	477	13	3.18	3.18	NUM
ma-12	477	14	)	)	PUNCT
ma-12	477	15	of	of	ADP
ma-12	477	16	lemma3.2	lemma3.2	PROPN
ma-12	477	17	,	,	PUNCT
ma-12	477	18	we	we	PRON
ma-12	477	19	see	see	VERB
ma-12	477	20	that	that	SCONJ
ma-12	477	21			PROPN
ma-12	477	22	‖rn(x	‖rn(x	PROPN
ma-12	477	23	)	)	PUNCT
ma-12	477	24	−	−	PROPN
ma-12	477	25	rn(y)‖	rn(y)‖	PROPN
ma-12	477	26	≤	≤	PROPN
ma-12	477	27	(	(	PUNCT
ma-12	477	28	1	1	NUM
ma-12	478	1	+	+	NUM
ma-12	478	2	hn)m‖x	hn)m‖x	PROPN
ma-12	478	3	−	−	PROPN
ma-12	478	4	y‖+	y‖+	PROPN
ma-12	478	5	θn	θn	PROPN
ma-12	478	6	;	;	PUNCT
ma-12	478	7	‖wn(x	‖wn(x	NUM
ma-12	478	8	)	)	PUNCT
ma-12	479	1	−wn(y)‖	−wn(y)‖	ADP
ma-12	479	2	≤	≤	NUM
ma-12	479	3	(	(	PUNCT
ma-12	479	4	1	1	NUM
ma-12	479	5	+	+	CCONJ
ma-12	479	6	rn)m‖x	rn)m‖x	VERB
ma-12	479	7	−	−	PROPN
ma-12	479	8	y‖+	y‖+	PROPN
ma-12	479	9	δnθn	δnθn	NOUN
ma-12	479	10	;	;	PUNCT
ma-12	479	11	‖vn(x	‖vn(x	X
ma-12	479	12	)	)	PUNCT
ma-12	480	1	−	−	PROPN
ma-12	481	1	vn(y)‖	vn(y)‖	PROPN
ma-12	481	2	≤	≤	NOUN
ma-12	481	3	(	(	PUNCT
ma-12	481	4	1	1	NUM
ma-12	481	5	+	+	CCONJ
ma-12	481	6	en)m‖x	en)m‖x	VERB
ma-12	481	7	−	−	PROPN
ma-12	481	8	y‖+	y‖+	PROPN
ma-12	481	9	θn	θn	NOUN
ma-12	481	10	=	=	PROPN
ma-12	481	11	fn‖x	fn‖x	PROPN
ma-12	481	12	−	−	PROPN
ma-12	481	13	y‖+	y‖+	PROPN
ma-12	481	14	gn	gn	PROPN
ma-12	481	15	,	,	PUNCT
ma-12	481	16	(	(	PUNCT
ma-12	481	17	3.54	3.54	NUM
ma-12	481	18	)	)	PUNCT
ma-12	481	19	where	where	SCONJ
ma-12	481	20	rn	rn	PROPN
ma-12	481	21	=	=	PROPN
ma-12	481	22	2hn+h2	2hn+h2	NUM
ma-12	481	23	nm	nm	PRON
ma-12	481	24	2	2	NUM
ma-12	481	25	,	,	PUNCT
ma-12	481	26	δn	δn	ADJ
ma-12	481	27	=	=	SYM
ma-12	481	28	2+hnm	2+hnm	NOUN
ma-12	481	29	,	,	PUNCT
ma-12	481	30	en	en	PROPN
ma-12	481	31	=	=	SYM
ma-12	481	32	3hnm+3h2	3hnm+3h2	NUM
ma-12	481	33	nm	nm	NOUN
ma-12	481	34	2+h3	2+h3	NUM
ma-12	481	35	nm	nm	NOUN
ma-12	481	36	3	3	NUM
ma-12	481	37	and	and	CCONJ
ma-12	481	38	gn	gn	PROPN
ma-12	481	39	=	=	PUNCT
ma-12	481	40	(	(	PUNCT
ma-12	481	41	1+hnm)(2+hnm)θnwith	1+hnm)(2+hnm)θnwith	NUM
ma-12	481	42	∑∞	∑∞	NOUN
ma-12	481	43	n=1	n=1	PUNCT
ma-12	481	44	en	en	ADP
ma-12	481	45	<	<	PROPN
ma-12	481	46	∞	∞	PROPN
ma-12	481	47	,	,	PUNCT
ma-12	481	48	∑∞	∑∞	NOUN
ma-12	481	49	n=1	n=1	PUNCT
ma-12	481	50	gn	gn	PROPN
ma-12	482	1	<	<	X
ma-12	482	2	∞	∞	PROPN
ma-12	482	3	and	and	CCONJ
ma-12	482	4	fn	fn	NOUN
ma-12	482	5	=	=	SYM
ma-12	482	6	1	1	NUM
ma-12	482	7	+	+	CCONJ
ma-12	482	8	en	en	X
ma-12	482	9	.	.	PUNCT
ma-12	482	10	since	since	SCONJ
ma-12	482	11	∑∞	∑∞	NOUN
ma-12	482	12	n=1	n=1	PROPN
ma-12	482	13	en	en	ADP
ma-12	482	14	<	<	PROPN
ma-12	482	15	∞	∞	PROPN
ma-12	482	16	,	,	PUNCT
ma-12	482	17	it	it	PRON
ma-12	482	18	follows	follow	VERB
ma-12	482	19	that	that	SCONJ
ma-12	482	20	fn	fn	PROPN
ma-12	482	21	→	→	SYM
ma-12	482	22	1	1	NUM
ma-12	482	23	as	as	SCONJ
ma-12	482	24	n	n	NUM
ma-12	482	25	→∞.	→∞.	PUNCT
ma-12	482	26	set	set	VERB
ma-12	482	27	sn	sn	PROPN
ma-12	482	28	,	,	PUNCT
ma-12	482	29	m	m	VERB
ma-12	482	30	=	=	SYM
ma-12	482	31	vn+m−1vn+m−2	vn+m−1vn+m−2	NUM
ma-12	482	32	·	·	PUNCT
ma-12	482	33	·	·	PUNCT
ma-12	482	34	·	·	PUNCT
ma-12	483	1	vn	vn	INTJ
ma-12	483	2	,	,	PUNCT
ma-12	483	3	m	m	PROPN
ma-12	483	4	∈	∈	PROPN
ma-12	483	5	n	n	CCONJ
ma-12	483	6	;	;	PUNCT
ma-12	483	7	bn	bn	X
ma-12	483	8	,	,	PUNCT
ma-12	483	9	m	m	VERB
ma-12	483	10	=	=	SYM
ma-12	483	11	‖sn	‖sn	PROPN
ma-12	483	12	,	,	PUNCT
ma-12	483	13	m(txn	m(txn	NOUN
ma-12	483	14	+	+	CCONJ
ma-12	483	15	(	(	PUNCT
ma-12	483	16	1−	1−	NUM
ma-12	483	17	t)p1)−	t)p1)−	NOUN
ma-12	483	18	sn.m(txm	sn.m(txm	PROPN
ma-12	483	19	+	+	CCONJ
ma-12	483	20	(	(	PUNCT
ma-12	483	21	1−	1−	NUM
ma-12	483	22	t)p2‖.	t)p2‖.	NOUN
ma-12	483	23	(	(	PUNCT
ma-12	483	24	3.55	3.55	NUM
ma-12	483	25	)	)	PUNCT
ma-12	483	26	eur	eur	PROPN
ma-12	483	27	.	.	PUNCT
ma-12	484	1	j.	j.	PROPN
ma-12	484	2	math	math	PROPN
ma-12	484	3	.	.	PUNCT
ma-12	485	1	anal	anal	ADJ
ma-12	485	2	.	.	PUNCT
ma-12	486	1	1	1	NUM
ma-12	486	2	(	(	PUNCT
ma-12	486	3	2021	2021	NUM
ma-12	486	4	)	)	PUNCT
ma-12	487	1	63from	63from	NUM
ma-12	487	2	(	(	PUNCT
ma-12	487	3	3.54	3.54	NUM
ma-12	487	4	)	)	PUNCT
ma-12	487	5	and	and	CCONJ
ma-12	488	1	(	(	PUNCT
ma-12	488	2	3.55	3.55	NUM
ma-12	488	3	)	)	PUNCT
ma-12	488	4	,	,	PUNCT
ma-12	488	5	we	we	PRON
ma-12	488	6	have	have	VERB
ma-12	488	7	‖sn	‖sn	PROPN
ma-12	488	8	,	,	PUNCT
ma-12	488	9	m(x)−	m(x)−	PROPN
ma-12	488	10	sn	sn	PROPN
ma-12	488	11	,	,	PUNCT
ma-12	488	12	m(y)‖	m(y)‖	PROPN
ma-12	488	13	=	=	PUNCT
ma-12	489	1	‖vn+m−1vn+m−2	‖vn+m−1vn+m−2	NUM
ma-12	489	2	·	·	PUNCT
ma-12	489	3	·	·	PUNCT
ma-12	489	4	·	·	PUNCT
ma-12	490	1	vn(x)−	vn(x)−	NOUN
ma-12	490	2	vn+m−1vn+m−2	vn+m−1vn+m−2	PROPN
ma-12	490	3	·	·	PUNCT
ma-12	490	4	·	·	PUNCT
ma-12	490	5	·	·	PUNCT
ma-12	491	1	vn(y)‖	vn(y)‖	PROPN
ma-12	491	2	≤	≤	ADJ
ma-12	491	3	fn+m−1‖vn+m−2vn+m−3	fn+m−1‖vn+m−2vn+m−3	PROPN
ma-12	491	4	·	·	PUNCT
ma-12	491	5	·	·	PUNCT
ma-12	491	6	·	·	PUNCT
ma-12	492	1	vn(x)−	vn(x)−	NOUN
ma-12	492	2	vn+m−2vn+m−3	vn+m−2vn+m−3	NUM
ma-12	492	3	·	·	PUNCT
ma-12	492	4	·	·	PUNCT
ma-12	492	5	·	·	PUNCT
ma-12	493	1	vn(y)‖	vn(y)‖	PUNCT
ma-12	493	2	+	+	ADJ
ma-12	493	3	gn+m−1	gn+m−1	NOUN
ma-12	493	4	≤	≤	NUM
ma-12	493	5	(	(	PUNCT
ma-12	493	6	fn+m−1)(fn+m−2)‖vn+m−3vn+m−4	fn+m−1)(fn+m−2)‖vn+m−3vn+m−4	PROPN
ma-12	493	7	·	·	PUNCT
ma-12	493	8	·	·	PUNCT
ma-12	493	9	·	·	PUNCT
ma-12	493	10	vn(x	vn(x	X
ma-12	493	11	)	)	PUNCT
ma-12	493	12	−vn+m−3vn+m−4	−vn+m−3vn+m−4	NOUN
ma-12	493	13	·	·	PUNCT
ma-12	493	14	·	·	PUNCT
ma-12	493	15	·	·	PUNCT
ma-12	493	16	vn(y)‖+	vn(y)‖+	PROPN
ma-12	493	17	gn+m−1	gn+m−1	NOUN
ma-12	493	18	+	+	CCONJ
ma-12	493	19	gn+m−2	gn+m−2	NOUN
ma-12	493	20	...	...	PUNCT
ma-12	493	21	≤	≤	NUM
ma-12	493	22	(	(	PUNCT
ma-12	493	23	n+m−1∏	n+m−1∏	NOUN
ma-12	493	24	i	i	PROPN
ma-12	493	25	=	=	PROPN
ma-12	493	26	n	n	PRON
ma-12	493	27	fi)‖x	fi)‖x	NOUN
ma-12	493	28	−	−	PUNCT
ma-12	493	29	y‖+	y‖+	PROPN
ma-12	493	30	n+m−1∑	n+m−1∑	PROPN
ma-12	493	31	i	i	PROPN
ma-12	493	32	=	=	VERB
ma-12	493	33	n	n	NOUN
ma-12	493	34	gi	gi	NOUN
ma-12	493	35	=	=	SYM
ma-12	493	36	bn‖x	bn‖x	PROPN
ma-12	493	37	−	−	NOUN
ma-12	494	1	y‖+	y‖+	PROPN
ma-12	494	2	n+m−1∑	n+m−1∑	PROPN
ma-12	494	3	i	i	PROPN
ma-12	494	4	=	=	VERB
ma-12	494	5	n	n	NOUN
ma-12	494	6	gi	gi	VERB
ma-12	494	7	,	,	PUNCT
ma-12	494	8	(	(	PUNCT
ma-12	494	9	3.56	3.56	NUM
ma-12	494	10	)	)	PUNCT
ma-12	494	11	for	for	ADP
ma-12	494	12	all	all	DET
ma-12	494	13	x	x	NOUN
ma-12	494	14	,	,	PUNCT
ma-12	494	15	y	y	PROPN
ma-12	494	16	∈	∈	PROPN
ma-12	494	17	k	k	NOUN
ma-12	494	18	,	,	PUNCT
ma-12	494	19	where	where	SCONJ
ma-12	494	20	bn	bn	NOUN
ma-12	494	21	=	=	PUNCT
ma-12	494	22	∏n+m−1	∏n+m−1	PROPN
ma-12	494	23	i	i	PROPN
ma-12	494	24	=	=	PROPN
ma-12	494	25	n	n	PROPN
ma-12	494	26	fi	fi	NOUN
ma-12	494	27	,	,	PUNCT
ma-12	494	28	sn	sn	PROPN
ma-12	494	29	,	,	PUNCT
ma-12	494	30	mxn	mxn	PROPN
ma-12	494	31	=	=	SYM
ma-12	494	32	xn	xn	PROPN
ma-12	494	33	and	and	CCONJ
ma-12	494	34	sn	sn	PROPN
ma-12	494	35	,	,	PUNCT
ma-12	494	36	mp	mp	PROPN
ma-12	494	37	=	=	PUNCT
ma-12	494	38	p	p	PROPN
ma-12	494	39	for	for	ADP
ma-12	494	40	all	all	DET
ma-12	494	41	p	p	NOUN
ma-12	494	42	∈	∈	PROPN
ma-12	494	43	f	f	X
ma-12	494	44	.	.	PUNCT
ma-12	495	1	thus	thus	ADV
ma-12	495	2	,	,	PUNCT
ma-12	495	3	an+m(t	an+m(t	PROPN
ma-12	495	4	)	)	PUNCT
ma-12	495	5	=	=	SYM
ma-12	496	1	‖txn	‖txn	PROPN
ma-12	496	2	+	+	CCONJ
ma-12	496	3	(	(	PUNCT
ma-12	496	4	1−	1−	NUM
ma-12	496	5	t)p1	t)p1	PROPN
ma-12	496	6	−	−	PROPN
ma-12	496	7	p2‖	p2‖	NOUN
ma-12	496	8	=	=	SYM
ma-12	496	9	‖sn	‖sn	PROPN
ma-12	496	10	,	,	PUNCT
ma-12	496	11	m(txn	m(txn	NOUN
ma-12	496	12	+	+	CCONJ
ma-12	496	13	(	(	PUNCT
ma-12	496	14	1−	1−	NUM
ma-12	496	15	t)p1	t)p1	PROPN
ma-12	496	16	−	−	PROPN
ma-12	496	17	p2‖	p2‖	PROPN
ma-12	496	18	≤	≤	PROPN
ma-12	496	19	bn	bn	PROPN
ma-12	496	20	,	,	PUNCT
ma-12	496	21	m	m	VERB
ma-12	496	22	+	+	ADJ
ma-12	496	23	‖sn	‖sn	ADJ
ma-12	496	24	,	,	PUNCT
ma-12	496	25	m(txn	m(txn	NOUN
ma-12	496	26	+	+	CCONJ
ma-12	496	27	(	(	PUNCT
ma-12	496	28	1−	1−	NUM
ma-12	496	29	t)p1	t)p1	PROPN
ma-12	496	30	−	−	PROPN
ma-12	496	31	p2‖.	p2‖.	NOUN
ma-12	496	32	(	(	PUNCT
ma-12	496	33	3.57	3.57	NUM
ma-12	496	34	)	)	PUNCT
ma-12	496	35	by	by	ADP
ma-12	496	36	using	use	VERB
ma-12	496	37	theorem	theorem	ADJ
ma-12	496	38	2.3	2.3	NUM
ma-12	496	39	in	in	ADP
ma-12	496	40	[	[	X
ma-12	496	41	5	5	NUM
ma-12	496	42	]	]	PUNCT
ma-12	496	43	,	,	PUNCT
ma-12	496	44	we	we	PRON
ma-12	496	45	have	have	AUX
ma-12	496	46	bn	bn	NUM
ma-12	496	47	,	,	PUNCT
ma-12	496	48	m	m	VERB
ma-12	496	49	≤	≤	ADJ
ma-12	496	50	ψ−1(‖(xn	ψ−1(‖(xn	NUM
ma-12	496	51	−	−	NOUN
ma-12	497	1	u‖	u‖	NOUN
ma-12	497	2	−	−	PROPN
ma-12	497	3	‖xn+1	‖xn+1	PUNCT
ma-12	497	4	−	−	PROPN
ma-12	497	5	sn	sn	PROPN
ma-12	497	6	,	,	PUNCT
ma-12	497	7	mu‖	mu‖	NOUN
ma-12	497	8	)	)	PUNCT
ma-12	497	9	=	=	NOUN
ma-12	497	10	ψ−1(‖(xn	ψ−1(‖(xn	NUM
ma-12	497	11	−	−	NOUN
ma-12	497	12	u‖	u‖	NOUN
ma-12	497	13	−	−	PROPN
ma-12	497	14	‖xn+1	‖xn+1	PUNCT
ma-12	497	15	−	−	PROPN
ma-12	497	16	u	u	NOUN
ma-12	497	17	+	+	NOUN
ma-12	497	18	u	u	NOUN
ma-12	497	19	−	−	PROPN
ma-12	497	20	sn	sn	PROPN
ma-12	497	21	,	,	PUNCT
ma-12	497	22	mu‖	mu‖	NOUN
ma-12	497	23	)	)	PUNCT
ma-12	497	24	≤	≤	NOUN
ma-12	497	25	ψ−1(‖(xn	ψ−1(‖(xn	NUM
ma-12	497	26	−	−	NOUN
ma-12	497	27	u‖	u‖	NOUN
ma-12	497	28	−	−	PROPN
ma-12	497	29	(	(	PUNCT
ma-12	497	30	‖xn+1	‖xn+1	PROPN
ma-12	497	31	−	−	PROPN
ma-12	497	32	u‖+	u‖+	PROPN
ma-12	497	33	‖sn	‖sn	PROPN
ma-12	497	34	,	,	PUNCT
ma-12	497	35	mu	mu	NOUN
ma-12	497	36	−	−	PROPN
ma-12	497	37	u‖	u‖	NOUN
ma-12	497	38	)	)	PUNCT
ma-12	497	39	)	)	PUNCT
ma-12	497	40	,	,	PUNCT
ma-12	497	41	(	(	PUNCT
ma-12	497	42	3.58	3.58	NUM
ma-12	497	43	)	)	PUNCT
ma-12	497	44	so	so	SCONJ
ma-12	497	45	that	that	SCONJ
ma-12	497	46	the	the	DET
ma-12	497	47	sequence	sequence	NOUN
ma-12	497	48	{	{	PUNCT
ma-12	497	49	bn.m	bn.m	NOUN
ma-12	497	50	}	}	PUNCT
ma-12	497	51	converges	converge	VERB
ma-12	497	52	uniformly	uniformly	ADV
ma-12	497	53	to	to	ADP
ma-12	497	54	0	0	NUM
ma-12	497	55	,	,	PUNCT
ma-12	497	56	i.e	i.e	X
ma-12	497	57	,	,	PUNCT
ma-12	497	58	bn	bn	NOUN
ma-12	497	59	,	,	PUNCT
ma-12	497	60	m	m	PROPN
ma-12	497	61	→	→	SYM
ma-12	497	62	0	0	NUM
ma-12	497	63	as	as	ADP
ma-12	497	64	n	n	X
ma-12	497	65	→∞.	→∞.	X
ma-12	497	66	since	since	SCONJ
ma-12	497	67	limn→bn	limn→bn	PROPN
ma-12	497	68	=	=	PROPN
ma-12	497	69	1and	1and	NUM
ma-12	497	70	limn→∞	limn→∞	PROPN
ma-12	497	71	bn	bn	NOUN
ma-12	497	72	,	,	PUNCT
ma-12	497	73	m	m	PROPN
ma-12	497	74	=	=	ADJ
ma-12	497	75	0	0	NUM
ma-12	497	76	,	,	PUNCT
ma-12	497	77	it	it	PRON
ma-12	497	78	follows	follow	VERB
ma-12	497	79	from	from	ADP
ma-12	497	80	(	(	PUNCT
ma-12	497	81	3.57	3.57	NUM
ma-12	497	82	)	)	PUNCT
ma-12	497	83	that	that	PRON
ma-12	497	84	lim	lim	PROPN
ma-12	497	85	supn→∞	supn→∞	PROPN
ma-12	497	86	an(t	an(t	NUM
ma-12	497	87	)	)	PUNCT
ma-12	497	88	≤	≤	VERB
ma-12	498	1	lim	lim	PROPN
ma-12	498	2	infb→∞	infb→∞	NOUN
ma-12	498	3	bn.m	bn.m	NOUN
ma-12	498	4	≤	≤	NUM
ma-12	498	5	lim	lim	PROPN
ma-12	498	6	infn→∞	infn→∞	PROPN
ma-12	499	1	an(t).this	an(t).this	PROPN
ma-12	499	2	shows	show	VERB
ma-12	499	3	that	that	SCONJ
ma-12	499	4	limn→∞	limn→∞	PROPN
ma-12	499	5	an(t	an(t	X
ma-12	499	6	)	)	PUNCT
ma-12	499	7	exists	exist	VERB
ma-12	499	8	,	,	PUNCT
ma-12	499	9	i.e	i.e	PRON
ma-12	499	10	,	,	PUNCT
ma-12	499	11	limn→∞	limn→∞	X
ma-12	499	12	‖txn	‖txn	NOUN
ma-12	499	13	+	+	X
ma-12	499	14	(	(	PUNCT
ma-12	499	15	1	1	NUM
ma-12	499	16	−	−	PROPN
ma-12	499	17	t)p1	t)p1	PROPN
ma-12	499	18	−	−	PROPN
ma-12	499	19	p2‖	p2‖	NOUN
ma-12	499	20	exists	exist	VERB
ma-12	499	21	for	for	ADP
ma-12	499	22	all	all	DET
ma-12	499	23	t	t	NOUN
ma-12	499	24	∈	∈	PROPN
ma-12	500	1	[	[	X
ma-12	500	2	0	0	NUM
ma-12	500	3	,	,	PUNCT
ma-12	500	4	1].this	1].this	NUM
ma-12	500	5	completes	complete	VERB
ma-12	500	6	the	the	DET
ma-12	500	7	proof	proof	NOUN
ma-12	500	8	lemma	lemma	PROPN
ma-12	500	9	3.4	3.4	NUM
ma-12	500	10	.	.	PUNCT
ma-12	500	11	�	�	PROPN
ma-12	500	12	lemma	lemma	PROPN
ma-12	500	13	3.5	3.5	NUM
ma-12	500	14	.	.	PUNCT
ma-12	501	1	under	under	ADP
ma-12	501	2	the	the	DET
ma-12	501	3	assumption	assumption	NOUN
ma-12	501	4	of	of	ADP
ma-12	501	5	lemma	lemma	PROPN
ma-12	501	6	3.2	3.2	NUM
ma-12	501	7	,	,	PUNCT
ma-12	501	8	if	if	SCONJ
ma-12	501	9	e	e	NOUN
ma-12	501	10	has	have	VERB
ma-12	501	11	frechet	frechet	VERB
ma-12	501	12	differentiable	differentiable	ADJ
ma-12	501	13	norm	norm	NOUN
ma-12	501	14	,	,	PUNCT
ma-12	501	15	then	then	ADV
ma-12	501	16	for	for	ADP
ma-12	501	17	all	all	DET
ma-12	501	18	p1	p1	NOUN
ma-12	501	19	,	,	PUNCT
ma-12	501	20	jp2	jp2	PROPN
ma-12	501	21	∈	∈	PROPN
ma-12	502	1	f	f	PROPN
ma-12	502	2	=	=	PRON
ma-12	502	3	∩3i=1(f	∩3i=1(f	X
ma-12	502	4	(	(	PUNCT
ma-12	502	5	ti	ti	NOUN
ma-12	502	6	)	)	PUNCT
ma-12	502	7	∩	∩	ADJ
ma-12	502	8	f	f	X
ma-12	502	9	(	(	PUNCT
ma-12	502	10	si	si	NOUN
ma-12	502	11	)	)	PUNCT
ma-12	502	12	)	)	PUNCT
ma-12	502	13	,	,	PUNCT
ma-12	502	14	the	the	DET
ma-12	502	15	limn→∞(〈xn	limn→∞(〈xn	NOUN
ma-12	502	16	,	,	PUNCT
ma-12	502	17	j(p1	j(p1	ADJ
ma-12	502	18	−	−	PROPN
ma-12	502	19	p2	p2	NOUN
ma-12	502	20	)	)	PUNCT
ma-12	502	21	〉	〉	NOUN
ma-12	502	22	exists	exist	VERB
ma-12	502	23	,	,	PUNCT
ma-12	502	24	where	where	SCONJ
ma-12	502	25	{	{	PUNCT
ma-12	502	26	xn	xn	X
ma-12	502	27	}	}	PUNCT
ma-12	502	28	is	be	AUX
ma-12	502	29	the	the	DET
ma-12	502	30	sequence	sequence	NOUN
ma-12	502	31	defined	define	VERB
ma-12	502	32	by	by	ADP
ma-12	502	33	(	(	PUNCT
ma-12	502	34	1.7	1.7	NUM
ma-12	502	35	)	)	PUNCT
ma-12	502	36	.	.	PUNCT
ma-12	503	1	if	if	SCONJ
ma-12	503	2	ωω(xn	ωω(xn	NUM
ma-12	503	3	)	)	PUNCT
ma-12	503	4	denotes	denote	VERB
ma-12	503	5	the	the	DET
ma-12	503	6	set	set	NOUN
ma-12	503	7	of	of	ADP
ma-12	503	8	all	all	DET
ma-12	503	9	weak	weak	ADJ
ma-12	503	10	subsequential	subsequential	ADJ
ma-12	503	11	limits	limit	NOUN
ma-12	503	12	of	of	ADP
ma-12	503	13	{	{	PUNCT
ma-12	503	14	xn	xn	NUM
ma-12	503	15	}	}	PUNCT
ma-12	503	16	,	,	PUNCT
ma-12	503	17	then	then	ADV
ma-12	503	18	〈	〈	PROPN
ma-12	503	19	q1	q1	PROPN
ma-12	503	20	−	−	PROPN
ma-12	503	21	q2	q2	PROPN
ma-12	503	22	,	,	PUNCT
ma-12	503	23	j(p1	j(p1	ADJ
ma-12	503	24	−	−	PROPN
ma-12	503	25	p2	p2	PROPN
ma-12	503	26	〉	〉	NOUN
ma-12	503	27	=	=	NOUN
ma-12	503	28	0	0	NUM
ma-12	503	29	for	for	ADP
ma-12	503	30	all	all	DET
ma-12	503	31	p1	p1	NOUN
ma-12	503	32	,	,	PUNCT
ma-12	503	33	p2	p2	PROPN
ma-12	503	34	∈	∈	PROPN
ma-12	503	35	f	f	PROPN
ma-12	503	36	and	and	CCONJ
ma-12	503	37	for	for	ADP
ma-12	503	38	all	all	DET
ma-12	503	39	q1	q1	NOUN
ma-12	503	40	,	,	PUNCT
ma-12	503	41	q2	q2	PROPN
ma-12	503	42	∈	∈	PROPN
ma-12	503	43	ωω(xn	ωω(xn	PROPN
ma-12	503	44	)	)	PUNCT
ma-12	503	45	.	.	PUNCT
ma-12	504	1	proof	proof	NOUN
ma-12	504	2	.	.	PUNCT
ma-12	505	1	suppose	suppose	VERB
ma-12	505	2	that	that	SCONJ
ma-12	505	3	x	x	SYM
ma-12	505	4	=	=	PUNCT
ma-12	505	5	p1	p1	PROPN
ma-12	505	6	−	−	PROPN
ma-12	505	7	p2	p2	PROPN
ma-12	505	8	with	with	ADP
ma-12	505	9	p1	p1	PROPN
ma-12	505	10	6=	6=	SYM
ma-12	505	11	p2	p2	PROPN
ma-12	505	12	and	and	CCONJ
ma-12	505	13	h	h	NOUN
ma-12	505	14	=	=	SYM
ma-12	505	15	t(xn	t(xn	NUM
ma-12	505	16	−	−	PROPN
ma-12	505	17	p1	p1	PROPN
ma-12	505	18	)	)	PUNCT
ma-12	505	19	in(2.1	in(2.1	NUM
ma-12	505	20	)	)	PUNCT
ma-12	505	21	.	.	PUNCT
ma-12	506	1	then	then	ADV
ma-12	506	2	,	,	PUNCT
ma-12	506	3	we	we	PRON
ma-12	506	4	have	have	VERB
ma-12	506	5	t(〈xn	t(〈xn	PROPN
ma-12	506	6	,	,	PUNCT
ma-12	506	7	j(p1	j(p1	ADJ
ma-12	506	8	−	−	PROPN
ma-12	506	9	p2)〉+	p2)〉+	NOUN
ma-12	507	1	1	1	NUM
ma-12	507	2	2	2	NUM
ma-12	507	3	‖p1	‖p1	PRON
ma-12	507	4	−	−	PROPN
ma-12	507	5	p2‖2	p2‖2	NOUN
ma-12	507	6	≤	≤	NUM
ma-12	507	7	1	1	NUM
ma-12	507	8	2	2	NUM
ma-12	507	9	‖txn	‖txn	NOUN
ma-12	507	10	+	+	CCONJ
ma-12	507	11	(	(	PUNCT
ma-12	507	12	1−	1−	NUM
ma-12	507	13	t)p1	t)p1	PROPN
ma-12	507	14	−	−	PROPN
ma-12	507	15	p2‖2	p2‖2	NOUN
ma-12	507	16	≤	≤	PUNCT
ma-12	507	17	t(〈xn	t(〈xn	PROPN
ma-12	507	18	,	,	PUNCT
ma-12	507	19	j(p1	j(p1	ADJ
ma-12	507	20	−	−	PROPN
ma-12	507	21	p2)〉+	p2)〉+	NOUN
ma-12	507	22	1	1	NUM
ma-12	507	23	2	2	NUM
ma-12	507	24	‖p1	‖p1	ADP
ma-12	507	25	−	−	PROPN
ma-12	507	26	p2‖2	p2‖2	NOUN
ma-12	507	27	+	+	NUM
ma-12	507	28	b(t‖xn	b(t‖xn	PROPN
ma-12	507	29	−	−	PROPN
ma-12	507	30	p1‖	p1‖	PROPN
ma-12	507	31	)	)	PUNCT
ma-12	507	32	eur	eur	NOUN
ma-12	507	33	.	.	PUNCT
ma-12	508	1	j.	j.	PROPN
ma-12	508	2	math	math	PROPN
ma-12	508	3	.	.	PUNCT
ma-12	509	1	anal	anal	ADJ
ma-12	509	2	.	.	PUNCT
ma-12	510	1	1	1	NUM
ma-12	510	2	(	(	PUNCT
ma-12	510	3	2021	2021	NUM
ma-12	510	4	)	)	PUNCT
ma-12	510	5	64since	64since	NOUN
ma-12	511	1	supn≥1	supn≥1	NOUN
ma-12	511	2	‖xn	‖xn	NUM
ma-12	511	3	−	−	PROPN
ma-12	511	4	p‖	p‖	NOUN
ma-12	511	5	≤	≤	X
ma-12	511	6	q	q	NOUN
ma-12	511	7	for	for	ADP
ma-12	511	8	some	some	PRON
ma-12	511	9	q	q	NOUN
ma-12	511	10	>	>	X
ma-12	511	11	0	0	NUM
ma-12	511	12	,	,	PUNCT
ma-12	511	13	we	we	PRON
ma-12	511	14	have	have	VERB
ma-12	511	15	t	t	PROPN
ma-12	511	16	lim	lim	PROPN
ma-12	511	17	n→∞	n→∞	NUM
ma-12	511	18	sup(〈xn	sup(〈xn	PROPN
ma-12	511	19	,	,	PUNCT
ma-12	511	20	j(p1	j(p1	ADJ
ma-12	511	21	−	−	PROPN
ma-12	511	22	p2)〉+	p2)〉+	NOUN
ma-12	511	23	1	1	NUM
ma-12	511	24	2	2	NUM
ma-12	511	25	‖p1	‖p1	PRON
ma-12	511	26	−	−	PROPN
ma-12	511	27	p2‖2	p2‖2	NOUN
ma-12	511	28	≤	≤	NUM
ma-12	511	29	1	1	NUM
ma-12	511	30	2	2	NUM
ma-12	511	31	lim	lim	NOUN
ma-12	511	32	n→∞	n→∞	PRON
ma-12	511	33	sup‖txn	sup‖txn	PROPN
ma-12	511	34	+	+	CCONJ
ma-12	511	35	(	(	PUNCT
ma-12	511	36	1−	1−	NUM
ma-12	511	37	t)p1	t)p1	PROPN
ma-12	511	38	−	−	PROPN
ma-12	511	39	p2‖2	p2‖2	NOUN
ma-12	511	40	≤	≤	NUM
ma-12	512	1	t	t	PROPN
ma-12	512	2	lim	lim	PROPN
ma-12	512	3	n→∞	n→∞	PROPN
ma-12	512	4	inf	inf	PROPN
ma-12	512	5	(	(	PUNCT
ma-12	512	6	〈	〈	PROPN
ma-12	512	7	xn	xn	PROPN
ma-12	512	8	,	,	PUNCT
ma-12	512	9	j(p1	j(p1	ADJ
ma-12	512	10	−	−	PROPN
ma-12	512	11	p2)〉+	p2)〉+	NOUN
ma-12	512	12	1	1	NUM
ma-12	512	13	2	2	NUM
ma-12	512	14	‖p1	‖p1	ADP
ma-12	512	15	−	−	PROPN
ma-12	512	16	p2‖2	p2‖2	NOUN
ma-12	512	17	+	+	NOUN
ma-12	512	18	b(tq	b(tq	NOUN
ma-12	512	19	)	)	PUNCT
ma-12	512	20	that	that	PRON
ma-12	512	21	is	be	AUX
ma-12	512	22	,	,	PUNCT
ma-12	512	23	t	t	PROPN
ma-12	512	24	limn→∞	limn→∞	PROPN
ma-12	512	25	sup(〈xn	sup(〈xn	PROPN
ma-12	512	26	,	,	PUNCT
ma-12	512	27	j(p1	j(p1	ADJ
ma-12	512	28	−	−	PROPN
ma-12	512	29	p2	p2	NOUN
ma-12	512	30	)	)	PUNCT
ma-12	512	31	〉	〉	PROPN
ma-12	512	32	≤	≤	PROPN
ma-12	512	33	t	t	PROPN
ma-12	512	34	lim	lim	PROPN
ma-12	512	35	infn→∞(〈xn	infn→∞(〈xn	PROPN
ma-12	512	36	,	,	PUNCT
ma-12	512	37	j(p1	j(p1	ADJ
ma-12	512	38	−	−	PROPN
ma-12	512	39	p2	p2	PROPN
ma-12	512	40	)	)	PUNCT
ma-12	512	41	〉	〉	PROPN
ma-12	512	42	+	+	SYM
ma-12	512	43	b(tq	b(tq	PROPN
ma-12	512	44	)	)	PUNCT
ma-12	512	45	.	.	PUNCT
ma-12	513	1	if	if	SCONJ
ma-12	513	2	t	t	PROPN
ma-12	513	3	→	→	SYM
ma-12	513	4	0	0	NUM
ma-12	513	5	,	,	PUNCT
ma-12	513	6	then	then	ADV
ma-12	513	7	limn→∞〈xn	limn→∞〈xn	PROPN
ma-12	513	8	−	−	PROPN
ma-12	513	9	p1	p1	PROPN
ma-12	513	10	,	,	PUNCT
ma-12	513	11	j(p1	j(p1	ADJ
ma-12	513	12	−	−	PROPN
ma-12	513	13	p2	p2	NOUN
ma-12	513	14	)	)	PUNCT
ma-12	513	15	〉	〉	NOUN
ma-12	513	16	exists	exist	VERB
ma-12	513	17	for	for	ADP
ma-12	513	18	all	all	DET
ma-12	513	19	p1	p1	NOUN
ma-12	513	20	,	,	PUNCT
ma-12	513	21	p2	p2	PROPN
ma-12	513	22	∈	∈	PROPN
ma-12	513	23	f	f	PROPN
ma-12	513	24	and	and	CCONJ
ma-12	513	25	for	for	ADP
ma-12	513	26	all	all	DET
ma-12	513	27	q2	q2	NOUN
ma-12	513	28	,	,	PUNCT
ma-12	513	29	q2	q2	PROPN
ma-12	513	30	∈	∈	PROPN
ma-12	513	31	ωω(xn	ωω(xn	PROPN
ma-12	513	32	)	)	PUNCT
ma-12	513	33	;	;	PUNCT
ma-12	513	34	in	in	ADP
ma-12	513	35	particular	particular	ADJ
ma-12	513	36	,	,	PUNCT
ma-12	513	37	(	(	PUNCT
ma-12	513	38	〈	〈	PROPN
ma-12	513	39	q1	q1	PROPN
ma-12	513	40	−	−	PROPN
ma-12	513	41	q2	q2	PROPN
ma-12	513	42	,	,	PUNCT
ma-12	513	43	j(p1	j(p1	ADJ
ma-12	513	44	−	−	PROPN
ma-12	513	45	p2	p2	NOUN
ma-12	513	46	)	)	PUNCT
ma-12	513	47	〉	〉	NOUN
ma-12	513	48	=	=	NOUN
ma-12	513	49	0	0	NUM
ma-12	513	50	for	for	ADP
ma-12	513	51	all	all	DET
ma-12	513	52	q2	q2	NOUN
ma-12	513	53	,	,	PUNCT
ma-12	513	54	q2	q2	PROPN
ma-12	513	55	∈	∈	PROPN
ma-12	513	56	ωω(xn	ωω(xn	PROPN
ma-12	513	57	)	)	PUNCT
ma-12	513	58	.	.	PUNCT
ma-12	514	1	this	this	PRON
ma-12	514	2	completes	complete	VERB
ma-12	514	3	the	the	DET
ma-12	514	4	proof	proof	NOUN
ma-12	514	5	lemma	lemma	PROPN
ma-12	514	6	3.5	3.5	NUM
ma-12	514	7	.	.	PUNCT
ma-12	515	1	�	�	PROPN
ma-12	515	2	theorem	theorem	VERB
ma-12	515	3	3.6	3.6	NUM
ma-12	515	4	.	.	PUNCT
ma-12	516	1	under	under	ADP
ma-12	516	2	the	the	DET
ma-12	516	3	assumption	assumption	NOUN
ma-12	516	4	of	of	ADP
ma-12	516	5	lemma	lemma	PROPN
ma-12	516	6	3.2	3.2	NUM
ma-12	516	7	,	,	PUNCT
ma-12	516	8	if	if	SCONJ
ma-12	516	9	e	e	NOUN
ma-12	516	10	has	have	VERB
ma-12	516	11	frechet	frechet	VERB
ma-12	516	12	differentiable	differentiable	ADJ
ma-12	516	13	norm	norm	NOUN
ma-12	516	14	,	,	PUNCT
ma-12	516	15	then	then	ADV
ma-12	516	16	the	the	DET
ma-12	516	17	sequence	sequence	NOUN
ma-12	516	18	{	{	PUNCT
ma-12	516	19	xn	xn	NOUN
ma-12	516	20	}	}	PUNCT
ma-12	516	21	defined	define	VERB
ma-12	516	22	by	by	ADP
ma-12	516	23	(	(	PUNCT
ma-12	516	24	1.7	1.7	NUM
ma-12	516	25	)	)	PUNCT
ma-12	516	26	converges	converge	VERB
ma-12	516	27	weakly	weakly	ADJ
ma-12	516	28	to	to	ADP
ma-12	516	29	a	a	DET
ma-12	516	30	common	common	ADJ
ma-12	516	31	fixed	fix	VERB
ma-12	516	32	point	point	NOUN
ma-12	516	33	in	in	ADP
ma-12	516	34	f	f	PROPN
ma-12	516	35	=	=	X
ma-12	516	36	∩3i=1f	∩3i=1f	X
ma-12	516	37	(	(	PUNCT
ma-12	516	38	ti)∩f	ti)∩f	X
ma-12	516	39	(	(	PUNCT
ma-12	516	40	si	si	NOUN
ma-12	516	41	)	)	PUNCT
ma-12	516	42	.	.	PUNCT
ma-12	517	1	proof	proof	NOUN
ma-12	517	2	.	.	PUNCT
ma-12	518	1	by	by	ADP
ma-12	518	2	lemma	lemma	PROPN
ma-12	518	3	3.5	3.5	NUM
ma-12	518	4	,	,	PUNCT
ma-12	518	5	(	(	PUNCT
ma-12	518	6	〈	〈	NOUN
ma-12	518	7	q1−	q1−	PRON
ma-12	518	8	q2	q2	NOUN
ma-12	518	9	,	,	PUNCT
ma-12	518	10	j(p1−	j(p1−	PROPN
ma-12	518	11	p2	p2	NOUN
ma-12	518	12	)	)	PUNCT
ma-12	518	13	〉	〉	NOUN
ma-12	518	14	=	=	NOUN
ma-12	518	15	0	0	NUM
ma-12	518	16	for	for	ADP
ma-12	518	17	all	all	DET
ma-12	518	18	q2	q2	NOUN
ma-12	518	19	,	,	PUNCT
ma-12	518	20	q2	q2	PROPN
ma-12	518	21	∈	∈	PROPN
ma-12	518	22	ωω(xn	ωω(xn	PROPN
ma-12	518	23	)	)	PUNCT
ma-12	518	24	.	.	PUNCT
ma-12	519	1	therefore	therefore	ADV
ma-12	519	2	,	,	PUNCT
ma-12	519	3	‖q?−	‖q?−	PROPN
ma-12	519	4	p	p	NOUN
ma-12	519	5	?	?	PUNCT
ma-12	519	6	‖2	‖2	NOUN
ma-12	520	1	=	=	PUNCT
ma-12	520	2	〈	〈	NOUN
ma-12	520	3	q	q	NOUN
ma-12	520	4	?	?	PUNCT
ma-12	521	1	−	−	PROPN
ma-12	522	1	p	p	X
ma-12	522	2	?	?	PUNCT
ma-12	522	3	,	,	PUNCT
ma-12	522	4	j(q	j(q	PROPN
ma-12	522	5	?	?	PUNCT
ma-12	523	1	−	−	PROPN
ma-12	524	1	p	p	X
ma-12	524	2	?	?	PUNCT
ma-12	524	3	)	)	PUNCT
ma-12	524	4	〉	〉	NOUN
ma-12	524	5	=	=	SYM
ma-12	524	6	0	0	X
ma-12	524	7	.	.	PUNCT
ma-12	525	1	this	this	PRON
ma-12	525	2	implies	imply	VERB
ma-12	525	3	that	that	SCONJ
ma-12	525	4	p	p	X
ma-12	525	5	?	?	PUNCT
ma-12	526	1	=	=	PUNCT
ma-12	526	2	q	q	NOUN
ma-12	526	3	?	?	PUNCT
ma-12	526	4	.	.	PUNCT
ma-12	527	1	consequently	consequently	ADV
ma-12	527	2	,	,	PUNCT
ma-12	527	3	{	{	PUNCT
ma-12	527	4	xn	xn	X
ma-12	527	5	}	}	PUNCT
ma-12	527	6	converges	converge	NOUN
ma-12	527	7	to	to	ADP
ma-12	527	8	a	a	DET
ma-12	527	9	commonfixed	commonfixed	NOUN
ma-12	527	10	point	point	NOUN
ma-12	527	11	of	of	ADP
ma-12	527	12	f	f	PROPN
ma-12	527	13	=	=	X
ma-12	527	14	∩3i=1f	∩3i=1f	X
ma-12	527	15	(	(	PUNCT
ma-12	527	16	ti	ti	NOUN
ma-12	527	17	)	)	PUNCT
ma-12	527	18	∩	∩	ADJ
ma-12	527	19	f	f	X
ma-12	527	20	(	(	PUNCT
ma-12	527	21	si	si	X
ma-12	527	22	)	)	PUNCT
ma-12	527	23	.	.	PUNCT
ma-12	528	1	this	this	PRON
ma-12	528	2	completes	complete	VERB
ma-12	528	3	the	the	DET
ma-12	528	4	proof	proof	NOUN
ma-12	528	5	theorem	theorem	VERB
ma-12	528	6	3.6	3.6	NUM
ma-12	528	7	.	.	PUNCT
ma-12	528	8	�	�	PROPN
ma-12	528	9	theorem	theorem	VERB
ma-12	528	10	3.7	3.7	NUM
ma-12	528	11	.	.	PUNCT
ma-12	529	1	under	under	ADP
ma-12	529	2	the	the	DET
ma-12	529	3	assumption	assumption	NOUN
ma-12	529	4	of	of	ADP
ma-12	529	5	lemma	lemma	PROPN
ma-12	529	6	3.2	3.2	NUM
ma-12	529	7	,	,	PUNCT
ma-12	529	8	if	if	SCONJ
ma-12	529	9	the	the	DET
ma-12	529	10	dual	dual	ADJ
ma-12	529	11	space	space	NOUN
ma-12	529	12	e	e	NOUN
ma-12	529	13	?	?	PUNCT
ma-12	529	14	of	of	ADP
ma-12	529	15	e	e	PROPN
ma-12	529	16	has	have	VERB
ma-12	529	17	the	the	DET
ma-12	529	18	kadec	kadec	PROPN
ma-12	529	19	klec	klec	PROPN
ma-12	529	20	(	(	PUNCT
ma-12	529	21	kk	kk	NOUN
ma-12	529	22	)	)	PUNCT
ma-12	529	23	property	property	NOUN
ma-12	529	24	and	and	CCONJ
ma-12	529	25	the	the	DET
ma-12	529	26	mappings	mapping	NOUN
ma-12	530	1	i	i	PRON
ma-12	530	2	−	−	VERB
ma-12	531	1	si	si	INTJ
ma-12	532	1	and	and	CCONJ
ma-12	532	2	i	i	PRON
ma-12	532	3	−	−	PROPN
ma-12	532	4	ti	ti	NOUN
ma-12	533	1	for	for	ADP
ma-12	533	2	i	i	PRON
ma-12	533	3	=	=	NOUN
ma-12	533	4	1	1	NUM
ma-12	533	5	,	,	PUNCT
ma-12	533	6	2	2	NUM
ma-12	533	7	,	,	PUNCT
ma-12	533	8	3	3	NUM
ma-12	533	9	,	,	PUNCT
ma-12	533	10	where	where	SCONJ
ma-12	533	11	i	i	PRON
ma-12	533	12	denotes	denote	VERB
ma-12	533	13	the	the	DET
ma-12	533	14	identity	identity	NOUN
ma-12	533	15	mapping	mapping	NOUN
ma-12	533	16	,	,	PUNCT
ma-12	533	17	are	be	AUX
ma-12	533	18	demiclosed	demiclose	VERB
ma-12	533	19	at	at	ADP
ma-12	533	20	zero	zero	NUM
ma-12	533	21	,	,	PUNCT
ma-12	533	22	then	then	ADV
ma-12	533	23	the	the	DET
ma-12	533	24	sequence	sequence	NOUN
ma-12	533	25	{	{	PUNCT
ma-12	533	26	xn	xn	NOUN
ma-12	533	27	}	}	PUNCT
ma-12	533	28	defined	define	VERB
ma-12	533	29	by	by	ADP
ma-12	533	30	(	(	PUNCT
ma-12	533	31	1.7	1.7	NUM
ma-12	533	32	)	)	PUNCT
ma-12	533	33	converges	converge	VERB
ma-12	533	34	weakly	weakly	ADJ
ma-12	533	35	to	to	ADP
ma-12	533	36	a	a	DET
ma-12	533	37	common	common	ADJ
ma-12	533	38	fixed	fix	VERB
ma-12	533	39	point	point	NOUN
ma-12	533	40	in	in	ADP
ma-12	533	41	f	f	PROPN
ma-12	533	42	=	=	PRON
ma-12	533	43	∩3i=1(f	∩3i=1(f	X
ma-12	533	44	(	(	PUNCT
ma-12	533	45	ti	ti	NOUN
ma-12	533	46	)	)	PUNCT
ma-12	533	47	∩	∩	ADJ
ma-12	533	48	f	f	X
ma-12	533	49	(	(	PUNCT
ma-12	533	50	si	si	NOUN
ma-12	533	51	)	)	PUNCT
ma-12	533	52	)	)	PUNCT
ma-12	533	53	.	.	PUNCT
ma-12	534	1	proof	proof	NOUN
ma-12	534	2	.	.	PUNCT
ma-12	535	1	by	by	ADP
ma-12	535	2	lemma	lemma	PROPN
ma-12	535	3	3.2	3.2	NUM
ma-12	535	4	{	{	PUNCT
ma-12	535	5	xn	xn	PRON
ma-12	535	6	}	}	PUNCT
ma-12	535	7	is	be	AUX
ma-12	535	8	bounded	bound	VERB
ma-12	535	9	and	and	CCONJ
ma-12	535	10	since	since	SCONJ
ma-12	535	11	e	e	NOUN
ma-12	535	12	is	be	AUX
ma-12	535	13	reflexive	reflexive	ADJ
ma-12	535	14	,	,	PUNCT
ma-12	535	15	there	there	PRON
ma-12	535	16	exists	exist	VERB
ma-12	535	17	a	a	DET
ma-12	535	18	subsequence	subsequence	NOUN
ma-12	535	19	{	{	PUNCT
ma-12	535	20	xnk	xnk	PROPN
ma-12	535	21	}	}	PUNCT
ma-12	535	22	of	of	ADP
ma-12	535	23	{	{	PUNCT
ma-12	535	24	xn	xn	NOUN
ma-12	535	25	}	}	PUNCT
ma-12	535	26	which	which	PRON
ma-12	535	27	converges	converge	VERB
ma-12	535	28	weakly	weakly	ADV
ma-12	535	29	to	to	ADP
ma-12	535	30	some	some	DET
ma-12	535	31	q	q	NOUN
ma-12	535	32	?	?	PUNCT
ma-12	536	1	∈	∈	PROPN
ma-12	536	2	k.	k.	PROPN
ma-12	536	3	by	by	ADP
ma-12	536	4	lemma	lemma	PROPN
ma-12	536	5	3.3	3.3	NUM
ma-12	536	6	,	,	PUNCT
ma-12	536	7	we	we	PRON
ma-12	536	8	have	have	VERB
ma-12	536	9	limn→∞	limn→∞	PRON
ma-12	536	10	‖xnk	‖xnk	PROPN
ma-12	536	11	−	−	PROPN
ma-12	536	12	sixnk‖	sixnk‖	PROPN
ma-12	537	1	=	=	SYM
ma-12	537	2	0and	0and	PROPN
ma-12	537	3	limn→∞	limn→∞	PROPN
ma-12	537	4	‖xnk	‖xnk	PROPN
ma-12	537	5	−	−	NOUN
ma-12	537	6	tixnk‖	tixnk‖	PROPN
ma-12	537	7	=	=	SYM
ma-12	537	8	0	0	PROPN
ma-12	537	9	for	for	ADP
ma-12	537	10	i	i	PRON
ma-12	537	11	=	=	NOUN
ma-12	537	12	1	1	NUM
ma-12	537	13	,	,	PUNCT
ma-12	537	14	2	2	NUM
ma-12	537	15	,	,	PUNCT
ma-12	537	16	3	3	NUM
ma-12	537	17	.	.	PUNCT
ma-12	537	18	since	since	SCONJ
ma-12	537	19	by	by	ADP
ma-12	537	20	hypothesis	hypothesis	NOUN
ma-12	537	21	,	,	PUNCT
ma-12	537	22	the	the	DET
ma-12	537	23	mappings	mapping	NOUN
ma-12	538	1	i	i	PRON
ma-12	538	2	−	−	VERB
ma-12	539	1	si	si	INTJ
ma-12	540	1	and	and	CCONJ
ma-12	540	2	i	i	PRON
ma-12	540	3	−	−	PROPN
ma-12	541	1	tifor	tifor	VERB
ma-12	541	2	i	i	PRON
ma-12	541	3	=	=	NOUN
ma-12	541	4	1	1	NUM
ma-12	541	5	,	,	PUNCT
ma-12	541	6	2	2	NUM
ma-12	541	7	,	,	PUNCT
ma-12	541	8	3	3	NUM
ma-12	541	9	,	,	PUNCT
ma-12	541	10	where	where	SCONJ
ma-12	541	11	i	i	PRON
ma-12	541	12	denotes	denote	VERB
ma-12	541	13	the	the	DET
ma-12	541	14	identity	identity	NOUN
ma-12	541	15	mapping	mapping	NOUN
ma-12	541	16	,	,	PUNCT
ma-12	541	17	are	be	AUX
ma-12	541	18	demiclosed	demiclose	VERB
ma-12	541	19	at	at	ADP
ma-12	541	20	zero	zero	NUM
ma-12	541	21	,	,	PUNCT
ma-12	541	22	siq	siq	PROPN
ma-12	541	23	?	?	PUNCT
ma-12	542	1	=	=	PUNCT
ma-12	543	1	q	q	X
ma-12	543	2	?	?	PUNCT
ma-12	544	1	and	and	CCONJ
ma-12	544	2	tiq	tiq	PROPN
ma-12	544	3	?	?	PUNCT
ma-12	545	1	=	=	PUNCT
ma-12	546	1	q	q	X
ma-12	546	2	?	?	PUNCT
ma-12	547	1	for	for	ADP
ma-12	547	2	i	i	PRON
ma-12	547	3	=	=	SYM
ma-12	547	4	1	1	NUM
ma-12	547	5	,	,	PUNCT
ma-12	547	6	2	2	NUM
ma-12	547	7	,	,	PUNCT
ma-12	547	8	3	3	NUM
ma-12	547	9	.	.	NUM
ma-12	547	10	;	;	PUNCT
ma-12	548	1	which	which	PRON
ma-12	548	2	means	mean	VERB
ma-12	548	3	q	q	X
ma-12	548	4	?	?	PUNCT
ma-12	549	1	∈	∈	PROPN
ma-12	549	2	f	f	X
ma-12	550	1	=	=	PRON
ma-12	550	2	∩3i=1(f	∩3i=1(f	X
ma-12	550	3	(	(	PUNCT
ma-12	550	4	ti	ti	NOUN
ma-12	550	5	)	)	PUNCT
ma-12	550	6	∩	∩	ADJ
ma-12	550	7	f	f	X
ma-12	550	8	(	(	PUNCT
ma-12	550	9	si	si	NOUN
ma-12	550	10	)	)	PUNCT
ma-12	550	11	)	)	PUNCT
ma-12	550	12	.	.	PUNCT
ma-12	551	1	now	now	ADV
ma-12	551	2	,	,	PUNCT
ma-12	551	3	we	we	PRON
ma-12	551	4	show	show	VERB
ma-12	551	5	that	that	SCONJ
ma-12	551	6	{	{	PUNCT
ma-12	551	7	xn}converges	xn}converge	NOUN
ma-12	551	8	weakly	weakly	ADJ
ma-12	551	9	to	to	ADP
ma-12	551	10	q	q	NOUN
ma-12	551	11	?	?	PUNCT
ma-12	551	12	.	.	PUNCT
ma-12	552	1	suppose	suppose	VERB
ma-12	552	2	{	{	PUNCT
ma-12	552	3	xnj	xnj	PROPN
ma-12	552	4	}	}	PUNCT
ma-12	552	5	is	be	AUX
ma-12	552	6	another	another	DET
ma-12	552	7	subsequence	subsequence	NOUN
ma-12	552	8	of	of	ADP
ma-12	552	9	{	{	PUNCT
ma-12	552	10	xn	xn	NOUN
ma-12	552	11	}	}	PUNCT
ma-12	552	12	which	which	PRON
ma-12	552	13	converges	converge	VERB
ma-12	552	14	weakly	weakly	ADV
ma-12	552	15	to	to	ADP
ma-12	552	16	p	p	PRON
ma-12	552	17	?	?	PUNCT
ma-12	553	1	∈	∈	PROPN
ma-12	553	2	k.	k.	PROPN
ma-12	554	1	by	by	ADP
ma-12	554	2	the	the	DET
ma-12	554	3	same	same	ADJ
ma-12	554	4	method	method	NOUN
ma-12	554	5	as	as	ADP
ma-12	554	6	above	above	ADV
ma-12	554	7	,	,	PUNCT
ma-12	554	8	we	we	PRON
ma-12	554	9	have	have	VERB
ma-12	554	10	p	p	X
ma-12	554	11	?	?	PUNCT
ma-12	555	1	∈	∈	PROPN
ma-12	555	2	f	f	PROPN
ma-12	555	3	and	and	CCONJ
ma-12	555	4	q	q	NOUN
ma-12	555	5	?	?	PUNCT
ma-12	555	6	∈	∈	PROPN
ma-12	555	7	ωω(xn	ωω(xn	NUM
ma-12	555	8	)	)	PUNCT
ma-12	555	9	.	.	PUNCT
ma-12	556	1	by	by	ADP
ma-12	556	2	lemma	lemma	PROPN
ma-12	556	3	3.4	3.4	NUM
ma-12	556	4	,	,	PUNCT
ma-12	556	5	the	the	DET
ma-12	556	6	limit	limit	NOUN
ma-12	556	7	limn→∞	limn→∞	X
ma-12	556	8	‖txn	‖txn	NOUN
ma-12	557	1	+	+	X
ma-12	557	2	(	(	PUNCT
ma-12	557	3	1	1	NUM
ma-12	557	4	−	−	NUM
ma-12	557	5	t)q	t)q	PUNCT
ma-12	557	6	?	?	PUNCT
ma-12	558	1	−	−	NOUN
ma-12	558	2	p?‖	p?‖	PROPN
ma-12	558	3	exists	exist	VERB
ma-12	558	4	for	for	ADP
ma-12	558	5	all	all	DET
ma-12	558	6	t	t	NOUN
ma-12	558	7	∈	∈	PROPN
ma-12	559	1	[	[	X
ma-12	559	2	0	0	NUM
ma-12	559	3	,	,	PUNCT
ma-12	559	4	1	1	NUM
ma-12	559	5	]	]	PUNCT
ma-12	559	6	and	and	CCONJ
ma-12	559	7	so	so	ADV
ma-12	559	8	q	q	ADJ
ma-12	559	9	?	?	PUNCT
ma-12	560	1	=	=	PUNCT
ma-12	561	1	p	p	X
ma-12	561	2	?	?	PUNCT
ma-12	561	3	.	.	PUNCT
ma-12	562	1	thus	thus	ADV
ma-12	562	2	,	,	PUNCT
ma-12	562	3	the	the	DET
ma-12	562	4	sequence	sequence	NOUN
ma-12	562	5	{	{	PUNCT
ma-12	562	6	xn}converges	xn}converge	NOUN
ma-12	562	7	weakly	weakly	ADJ
ma-12	562	8	to	to	ADP
ma-12	562	9	q	q	PUNCT
ma-12	562	10	?	?	PUNCT
ma-12	563	1	∈	∈	PROPN
ma-12	564	1	f	f	NOUN
ma-12	564	2	.	.	PUNCT
ma-12	565	1	this	this	PRON
ma-12	565	2	completes	complete	VERB
ma-12	565	3	the	the	DET
ma-12	565	4	proof	proof	NOUN
ma-12	565	5	.	.	PUNCT
ma-12	566	1	�	�	PROPN
ma-12	566	2	theorem	theorem	VERB
ma-12	566	3	3.8	3.8	NUM
ma-12	566	4	.	.	PUNCT
ma-12	567	1	under	under	ADP
ma-12	567	2	the	the	DET
ma-12	567	3	assumption	assumption	NOUN
ma-12	567	4	of	of	ADP
ma-12	567	5	lemma	lemma	PROPN
ma-12	567	6	3.2	3.2	NUM
ma-12	567	7	,	,	PUNCT
ma-12	567	8	if	if	SCONJ
ma-12	567	9	e	e	NOUN
ma-12	567	10	satisfies	satisfy	VERB
ma-12	567	11	opial	opial	NOUN
ma-12	567	12	’s	’s	PART
ma-12	567	13	condition	condition	NOUN
ma-12	567	14	and	and	CCONJ
ma-12	567	15	the	the	DET
ma-12	567	16	mappings	mapping	NOUN
ma-12	567	17	i−si	i−si	NOUN
ma-12	567	18	and	and	CCONJ
ma-12	567	19	i−ti	i−ti	NOUN
ma-12	567	20	for	for	ADP
ma-12	567	21	i	i	PRON
ma-12	567	22	=	=	NOUN
ma-12	567	23	1	1	NUM
ma-12	567	24	,	,	PUNCT
ma-12	567	25	2	2	NUM
ma-12	567	26	,	,	PUNCT
ma-12	567	27	3	3	NUM
ma-12	567	28	,	,	PUNCT
ma-12	567	29	where	where	SCONJ
ma-12	567	30	i	i	PRON
ma-12	567	31	denotes	denote	VERB
ma-12	567	32	the	the	DET
ma-12	567	33	identity	identity	NOUN
ma-12	567	34	mapping	mapping	NOUN
ma-12	567	35	,	,	PUNCT
ma-12	567	36	are	be	AUX
ma-12	567	37	demiclosed	demiclose	VERB
ma-12	567	38	at	at	ADP
ma-12	567	39	zero	zero	NUM
ma-12	567	40	,	,	PUNCT
ma-12	567	41	then	then	ADV
ma-12	567	42	the	the	DET
ma-12	567	43	sequence	sequence	NOUN
ma-12	567	44	{	{	PUNCT
ma-12	567	45	xn	xn	NOUN
ma-12	567	46	}	}	PUNCT
ma-12	567	47	defined	define	VERB
ma-12	567	48	by	by	ADP
ma-12	567	49	(	(	PUNCT
ma-12	567	50	1.7	1.7	NUM
ma-12	567	51	)	)	PUNCT
ma-12	567	52	converges	converge	VERB
ma-12	567	53	weakly	weakly	ADJ
ma-12	567	54	to	to	ADP
ma-12	567	55	a	a	DET
ma-12	567	56	common	common	ADJ
ma-12	567	57	fixed	fix	VERB
ma-12	567	58	point	point	NOUN
ma-12	567	59	in	in	ADP
ma-12	567	60	f	f	PROPN
ma-12	568	1	=	=	PRON
ma-12	568	2	∩3i=1(f	∩3i=1(f	X
ma-12	569	1	(	(	PUNCT
ma-12	569	2	ti)∩	ti)∩	NOUN
ma-12	569	3	f	f	X
ma-12	569	4	(	(	PUNCT
ma-12	569	5	si	si	NOUN
ma-12	569	6	)	)	PUNCT
ma-12	569	7	)	)	PUNCT
ma-12	569	8	.	.	PUNCT
ma-12	570	1	proof	proof	NOUN
ma-12	570	2	.	.	PUNCT
ma-12	571	1	let	let	VERB
ma-12	571	2	q	q	X
ma-12	571	3	?	?	PUNCT
ma-12	572	1	∈	∈	PROPN
ma-12	573	1	f	f	X
ma-12	573	2	.	.	PUNCT
ma-12	574	1	from	from	ADP
ma-12	574	2	lemma	lemma	PROPN
ma-12	574	3	3.2	3.2	NUM
ma-12	574	4	,	,	PUNCT
ma-12	574	5	the	the	DET
ma-12	574	6	squence	squence	NOUN
ma-12	574	7	{	{	PUNCT
ma-12	574	8	‖xn	‖xn	PROPN
ma-12	574	9	−	−	PROPN
ma-12	574	10	p	p	NOUN
ma-12	574	11	?	?	PUNCT
ma-12	575	1	‖	‖	NUM
ma-12	575	2	}	}	PUNCT
ma-12	575	3	is	be	AUX
ma-12	575	4	convergent	convergent	ADJ
ma-12	575	5	and	and	CCONJ
ma-12	575	6	hence	hence	ADV
ma-12	575	7	bounded.since	bounded.since	NOUN
ma-12	575	8	,	,	PUNCT
ma-12	575	9	e	e	NOUN
ma-12	575	10	is	be	AUX
ma-12	575	11	uniformly	uniformly	ADV
ma-12	575	12	convex	convex	ADJ
ma-12	575	13	,	,	PUNCT
ma-12	575	14	every	every	DET
ma-12	575	15	bounded	bounded	ADJ
ma-12	575	16	subset	subset	NOUN
ma-12	575	17	of	of	ADP
ma-12	575	18	e	e	PROPN
ma-12	575	19	is	be	AUX
ma-12	575	20	weakly	weakly	ADV
ma-12	575	21	compact	compact	ADJ
ma-12	575	22	.	.	PUNCT
ma-12	576	1	thus	thus	ADV
ma-12	576	2	,	,	PUNCT
ma-12	576	3	the	the	DET
ma-12	576	4	existsa	existsa	NOUN
ma-12	576	5	subsequence	subsequence	PROPN
ma-12	576	6	{	{	PUNCT
ma-12	576	7	xnk	xnk	PROPN
ma-12	576	8	}	}	PUNCT
ma-12	576	9	of	of	ADP
ma-12	576	10	{	{	PUNCT
ma-12	576	11	xn	xn	NOUN
ma-12	576	12	}	}	PUNCT
ma-12	576	13	which	which	PRON
ma-12	576	14	converges	converge	VERB
ma-12	576	15	weakly	weakly	ADV
ma-12	576	16	to	to	ADP
ma-12	576	17	some	some	DET
ma-12	576	18	q	q	NOUN
ma-12	576	19	?	?	PUNCT
ma-12	577	1	∈	∈	PROPN
ma-12	577	2	k.	k.	PROPN
ma-12	577	3	by	by	ADP
ma-12	577	4	lemma	lemma	PROPN
ma-12	577	5	3.3	3.3	NUM
ma-12	577	6	,	,	PUNCT
ma-12	577	7	we	we	PRON
ma-12	577	8	have	have	VERB
ma-12	577	9	eur	eur	NOUN
ma-12	577	10	.	.	PUNCT
ma-12	578	1	j.	j.	PROPN
ma-12	578	2	math	math	PROPN
ma-12	578	3	.	.	PUNCT
ma-12	579	1	anal	anal	ADJ
ma-12	579	2	.	.	PUNCT
ma-12	580	1	1	1	NUM
ma-12	580	2	(	(	PUNCT
ma-12	580	3	2021	2021	NUM
ma-12	580	4	)	)	PUNCT
ma-12	581	1	65	65	NUM
ma-12	581	2	limn→∞	limn→∞	PROPN
ma-12	581	3	‖xnk	‖xnk	PROPN
ma-12	581	4	−	−	PROPN
ma-12	581	5	sixnk‖	sixnk‖	PROPN
ma-12	581	6	=	=	SYM
ma-12	581	7	0	0	NUM
ma-12	581	8	and	and	CCONJ
ma-12	581	9	limn→∞	limn→∞	PROPN
ma-12	581	10	‖xnk	‖xnk	PROPN
ma-12	581	11	−	−	NOUN
ma-12	581	12	tixnk‖	tixnk‖	PROPN
ma-12	581	13	=	=	SYM
ma-12	581	14	0	0	PROPN
ma-12	581	15	for	for	ADP
ma-12	581	16	i	i	PRON
ma-12	581	17	=	=	NOUN
ma-12	581	18	1	1	NUM
ma-12	581	19	,	,	PUNCT
ma-12	581	20	2	2	NUM
ma-12	581	21	,	,	PUNCT
ma-12	581	22	3	3	NUM
ma-12	581	23	.	.	PUNCT
ma-12	581	24	since	since	SCONJ
ma-12	581	25	by	by	ADP
ma-12	581	26	hypothesis	hypothesis	NOUN
ma-12	581	27	,	,	PUNCT
ma-12	581	28	themappings	themapping	NOUN
ma-12	581	29	i	i	PRON
ma-12	581	30	−	−	VERB
ma-12	581	31	si	si	INTJ
ma-12	581	32	and	and	CCONJ
ma-12	581	33	i	i	PRON
ma-12	581	34	−	−	PROPN
ma-12	581	35	ti	ti	NOUN
ma-12	581	36	for	for	ADP
ma-12	581	37	i	i	PRON
ma-12	581	38	=	=	NOUN
ma-12	581	39	1	1	NUM
ma-12	581	40	,	,	PUNCT
ma-12	581	41	2	2	NUM
ma-12	581	42	,	,	PUNCT
ma-12	581	43	3	3	NUM
ma-12	581	44	,	,	PUNCT
ma-12	581	45	where	where	SCONJ
ma-12	581	46	i	i	PRON
ma-12	581	47	denotes	denote	VERB
ma-12	581	48	the	the	DET
ma-12	581	49	identity	identity	NOUN
ma-12	581	50	mapping	mapping	NOUN
ma-12	581	51	,	,	PUNCT
ma-12	581	52	are	be	AUX
ma-12	581	53	demiclosedat	demiclosedat	PROPN
ma-12	581	54	zero	zero	NUM
ma-12	581	55	,	,	PUNCT
ma-12	581	56	siq	siq	PROPN
ma-12	581	57	?	?	PUNCT
ma-12	582	1	=	=	PUNCT
ma-12	583	1	q	q	X
ma-12	583	2	?	?	PUNCT
ma-12	583	3	and	and	CCONJ
ma-12	583	4	tiq	tiq	VERB
ma-12	583	5	?	?	PUNCT
ma-12	584	1	=	=	PUNCT
ma-12	585	1	q	q	X
ma-12	585	2	?	?	PUNCT
ma-12	586	1	for	for	ADP
ma-12	586	2	i	i	PRON
ma-12	586	3	=	=	SYM
ma-12	586	4	1	1	NUM
ma-12	586	5	,	,	PUNCT
ma-12	586	6	2	2	NUM
ma-12	586	7	,	,	PUNCT
ma-12	586	8	3	3	NUM
ma-12	586	9	.	.	NUM
ma-12	586	10	;	;	PUNCT
ma-12	587	1	which	which	PRON
ma-12	587	2	means	mean	VERB
ma-12	587	3	q	q	X
ma-12	587	4	?	?	PUNCT
ma-12	588	1	∈	∈	PROPN
ma-12	588	2	f	f	X
ma-12	589	1	=	=	PRON
ma-12	589	2	∩3i=1(f	∩3i=1(f	X
ma-12	589	3	(	(	PUNCT
ma-12	589	4	ti	ti	NOUN
ma-12	589	5	)	)	PUNCT
ma-12	589	6	∩	∩	ADJ
ma-12	589	7	f	f	PROPN
ma-12	589	8	(	(	PUNCT
ma-12	589	9	si)).finally	si)).finally	ADV
ma-12	589	10	,	,	PUNCT
ma-12	589	11	we	we	PRON
ma-12	589	12	show	show	VERB
ma-12	589	13	that	that	SCONJ
ma-12	589	14	{	{	PUNCT
ma-12	589	15	xn	xn	X
ma-12	589	16	}	}	PUNCT
ma-12	589	17	converges	converge	VERB
ma-12	589	18	weakly	weakly	ADV
ma-12	589	19	to	to	ADP
ma-12	589	20	q	q	NOUN
ma-12	589	21	?	?	PUNCT
ma-12	589	22	.	.	PUNCT
ma-12	590	1	suppose	suppose	VERB
ma-12	590	2	on	on	ADP
ma-12	590	3	the	the	DET
ma-12	590	4	contrary	contrary	NOUN
ma-12	590	5	that	that	SCONJ
ma-12	590	6	{	{	PUNCT
ma-12	590	7	xnj	xnj	PROPN
ma-12	590	8	}	}	PUNCT
ma-12	590	9	is	be	AUX
ma-12	590	10	anothersubsequence	anothersubsequence	NOUN
ma-12	590	11	of	of	ADP
ma-12	590	12	{	{	PUNCT
ma-12	590	13	xn	xn	PROPN
ma-12	590	14	}	}	PUNCT
ma-12	590	15	which	which	PRON
ma-12	590	16	converges	converge	VERB
ma-12	590	17	weakly	weakly	ADV
ma-12	590	18	to	to	ADP
ma-12	590	19	p	p	PRON
ma-12	590	20	?	?	PUNCT
ma-12	591	1	∈	∈	PROPN
ma-12	591	2	k	k	PROPN
ma-12	591	3	and	and	CCONJ
ma-12	591	4	q	q	NOUN
ma-12	591	5	?	?	PUNCT
ma-12	592	1	6=	6=	NOUN
ma-12	593	1	p	p	X
ma-12	593	2	?	?	PUNCT
ma-12	593	3	by	by	ADP
ma-12	593	4	lemma	lemma	PROPN
ma-12	593	5	3.2	3.2	NUM
ma-12	593	6	,	,	PUNCT
ma-12	593	7	limn→∞	limn→∞	PROPN
ma-12	593	8	‖xn−	‖xn−	NUM
ma-12	593	9	q?‖	q?‖	ADJ
ma-12	593	10	and	and	CCONJ
ma-12	593	11	limn→∞	limn→∞	PROPN
ma-12	593	12	‖xn	‖xn	PROPN
ma-12	593	13	−	−	PROPN
ma-12	593	14	p?‖	p?‖	NOUN
ma-12	593	15	exist	exist	VERB
ma-12	593	16	.	.	PUNCT
ma-12	594	1	by	by	ADP
ma-12	594	2	virtue	virtue	NOUN
ma-12	594	3	of	of	ADP
ma-12	594	4	opial	opial	NOUN
ma-12	594	5	’s	’s	PART
ma-12	594	6	condition	condition	NOUN
ma-12	594	7	on	on	ADP
ma-12	594	8	e	e	NOUN
ma-12	594	9	,	,	PUNCT
ma-12	594	10	we	we	PRON
ma-12	594	11	obtain	obtain	VERB
ma-12	594	12	lim	lim	PROPN
ma-12	594	13	n→∞	n→∞	X
ma-12	595	1	‖xn	‖xn	PROPN
ma-12	595	2	−	−	PROPN
ma-12	595	3	q?‖	q?‖	PROPN
ma-12	595	4	=	=	SYM
ma-12	595	5	lim	lim	PROPN
ma-12	595	6	n→∞	n→∞	NUM
ma-12	595	7	‖xnk	‖xnk	PROPN
ma-12	595	8	−	−	PROPN
ma-12	595	9	q	q	NOUN
ma-12	595	10	?	?	PUNCT
ma-12	596	1	‖	‖	PROPN
ma-12	596	2	<	<	X
ma-12	596	3	lim	lim	PROPN
ma-12	596	4	n→∞	n→∞	NUM
ma-12	596	5	‖xnk	‖xnk	PROPN
ma-12	596	6	−	−	PROPN
ma-12	596	7	p	p	NOUN
ma-12	596	8	?	?	PUNCT
ma-12	596	9	‖	‖	PROPN
ma-12	597	1	=	=	SYM
ma-12	597	2	lim	lim	PROPN
ma-12	597	3	n→∞	n→∞	X
ma-12	598	1	‖xn	‖xn	PROPN
ma-12	598	2	−	−	PROPN
ma-12	598	3	p?‖	p?‖	NOUN
ma-12	598	4	=	=	SYM
ma-12	598	5	lim	lim	PROPN
ma-12	598	6	n→∞	n→∞	X
ma-12	599	1	‖xnj	‖xnj	X
ma-12	599	2	−	−	PROPN
ma-12	600	1	p	p	X
ma-12	600	2	?	?	PUNCT
ma-12	600	3	‖	‖	PROPN
ma-12	600	4	<	<	X
ma-12	600	5	lim	lim	PROPN
ma-12	600	6	n→∞	n→∞	X
ma-12	600	7	‖xnj	‖xnj	X
ma-12	600	8	−	−	PROPN
ma-12	600	9	q	q	NOUN
ma-12	600	10	?	?	PUNCT
ma-12	600	11	‖	‖	PROPN
ma-12	600	12	=	=	SYM
ma-12	600	13	lim	lim	PROPN
ma-12	600	14	n→∞	n→∞	X
ma-12	601	1	‖xn	‖xn	PROPN
ma-12	601	2	−	−	PROPN
ma-12	601	3	q?‖	q?‖	PROPN
ma-12	601	4	,	,	PUNCT
ma-12	601	5	(	(	PUNCT
ma-12	601	6	3.59	3.59	NUM
ma-12	601	7	)	)	PUNCT
ma-12	601	8	which	which	PRON
ma-12	601	9	is	be	AUX
ma-12	601	10	a	a	DET
ma-12	601	11	contradiction	contradiction	NOUN
ma-12	601	12	,	,	PUNCT
ma-12	601	13	so	so	ADV
ma-12	601	14	q	q	ADJ
ma-12	601	15	?	?	PUNCT
ma-12	602	1	=	=	SYM
ma-12	603	1	p	p	X
ma-12	603	2	?	?	PUNCT
ma-12	603	3	therefore	therefore	ADV
ma-12	603	4	,	,	PUNCT
ma-12	603	5	the	the	DET
ma-12	603	6	sequence	sequence	NOUN
ma-12	603	7	{	{	PUNCT
ma-12	603	8	xn	xn	NOUN
ma-12	603	9	}	}	PUNCT
ma-12	603	10	defined	define	VERB
ma-12	603	11	by	by	ADP
ma-12	603	12	(	(	PUNCT
ma-12	603	13	1.7	1.7	NUM
ma-12	603	14	)	)	PUNCT
ma-12	603	15	converges	converge	VERB
ma-12	603	16	weaklyto	weaklyto	VERB
ma-12	603	17	q	q	ADJ
ma-12	603	18	?	?	PUNCT
ma-12	604	1	∈	∈	PROPN
ma-12	604	2	f	f	NOUN
ma-12	604	3	.	.	PUNCT
ma-12	605	1	this	this	PRON
ma-12	605	2	completes	complete	VERB
ma-12	605	3	the	the	DET
ma-12	605	4	proof	proof	NOUN
ma-12	605	5	.	.	PUNCT
ma-12	606	1	�	�	PROPN
ma-12	606	2	corollary	corollary	ADJ
ma-12	606	3	3.9	3.9	NUM
ma-12	606	4	.	.	PUNCT
ma-12	607	1	let	let	VERB
ma-12	607	2	e	e	PRON
ma-12	607	3	be	be	AUX
ma-12	607	4	a	a	DET
ma-12	607	5	uniformly	uniformly	ADV
ma-12	607	6	convex	convex	NOUN
ma-12	607	7	banach	banach	NOUN
ma-12	607	8	space	space	NOUN
ma-12	607	9	and	and	CCONJ
ma-12	607	10	k	k	PROPN
ma-12	607	11	a	a	DET
ma-12	607	12	nonempty	nonempty	ADV
ma-12	607	13	closed	close	VERB
ma-12	607	14	convex	convex	NOUN
ma-12	607	15	subset	subset	NOUN
ma-12	607	16	of	of	ADP
ma-12	607	17	e.	e.	PROPN
ma-12	607	18	let	let	VERB
ma-12	607	19	s1	s1	PROPN
ma-12	607	20	,	,	PUNCT
ma-12	607	21	s2	s2	PROPN
ma-12	607	22	,	,	PUNCT
ma-12	607	23	s3	s3	PROPN
ma-12	607	24	:	:	PUNCT
ma-12	607	25	k	k	PROPN
ma-12	607	26	−→	−→	NOUN
ma-12	607	27	k	k	PROPN
ma-12	607	28	be	be	AUX
ma-12	607	29	three	three	NUM
ma-12	607	30	generalize	generalize	VERB
ma-12	607	31	asymptotically	asymptotically	ADV
ma-12	607	32	nonexpansive	nonexpansive	ADJ
ma-12	607	33	self	self	NOUN
ma-12	607	34	mapping	mapping	NOUN
ma-12	607	35	with	with	ADP
ma-12	607	36	sequences	sequence	NOUN
ma-12	607	37	{	{	PUNCT
ma-12	607	38	k(1)n	k(1)n	X
ma-12	607	39	}	}	PUNCT
ma-12	607	40	,	,	PUNCT
ma-12	607	41	{	{	PUNCT
ma-12	607	42	k(2)n	k(2)n	X
ma-12	607	43	}	}	PUNCT
ma-12	607	44	,	,	PUNCT
ma-12	607	45	{	{	PUNCT
ma-12	607	46	k(3)n	k(3)n	NOUN
ma-12	607	47	}	}	PUNCT
ma-12	607	48	∈	∈	PROPN
ma-12	608	1	[	[	X
ma-12	608	2	1,∞	1,∞	NUM
ma-12	608	3	)	)	PUNCT
ma-12	608	4	,	,	PUNCT
ma-12	608	5	{	{	PUNCT
ma-12	608	6	w	w	X
ma-12	608	7	(	(	PUNCT
ma-12	608	8	1)n	1)n	X
ma-12	608	9	}	}	PUNCT
ma-12	608	10	,	,	PUNCT
ma-12	608	11	{	{	PUNCT
ma-12	608	12	w	w	NOUN
ma-12	608	13	(	(	PUNCT
ma-12	608	14	2)}n	2)}n	NOUN
ma-12	608	15	,	,	PUNCT
ma-12	608	16	{	{	PUNCT
ma-12	608	17	w	w	NOUN
ma-12	608	18	(	(	PUNCT
ma-12	608	19	3)n	3)n	NUM
ma-12	608	20	}	}	PUNCT
ma-12	608	21	∈	∈	PROPN
ma-12	609	1	[	[	X
ma-12	609	2	1,∞	1,∞	NUM
ma-12	609	3	)	)	PUNCT
ma-12	609	4	and	and	CCONJ
ma-12	609	5	t1	t1	NOUN
ma-12	609	6	,	,	PUNCT
ma-12	609	7	t2	t2	NOUN
ma-12	609	8	,	,	PUNCT
ma-12	609	9	t3	t3	NOUN
ma-12	609	10	:	:	PUNCT
ma-12	609	11	k	k	X
ma-12	609	12	−→	−→	NOUN
ma-12	609	13	e	e	NOUN
ma-12	609	14	are	be	AUX
ma-12	609	15	three	three	NUM
ma-12	609	16	generalize	generalize	VERB
ma-12	609	17	asymptotically	asymptotically	ADV
ma-12	609	18	nonexpansive	nonexpansive	ADJ
ma-12	609	19	nonself	nonself	PROPN
ma-12	609	20	mappings	mapping	NOUN
ma-12	609	21	with	with	ADP
ma-12	609	22	sequences	sequence	NOUN
ma-12	609	23	{	{	PUNCT
ma-12	609	24	µ(1)n	µ(1)n	X
ma-12	609	25	}	}	PUNCT
ma-12	609	26	,	,	PUNCT
ma-12	609	27	{	{	PUNCT
ma-12	609	28	µ(2)n	µ(2)n	NOUN
ma-12	609	29	}	}	PUNCT
ma-12	609	30	,	,	PUNCT
ma-12	609	31	{	{	PUNCT
ma-12	609	32	µ(3)n	µ(3)n	PROPN
ma-12	609	33	}	}	PUNCT
ma-12	609	34	∈	∈	PROPN
ma-12	610	1	[	[	X
ma-12	610	2	1,∞	1,∞	NUM
ma-12	610	3	)	)	PUNCT
ma-12	610	4	,	,	PUNCT
ma-12	610	5	{	{	PUNCT
ma-12	610	6	ν(1)n	ν(1)n	X
ma-12	610	7	}	}	PUNCT
ma-12	610	8	,	,	PUNCT
ma-12	610	9	{	{	PUNCT
ma-12	610	10	ν(2)n	ν(2)n	NOUN
ma-12	610	11	}	}	PUNCT
ma-12	610	12	,	,	PUNCT
ma-12	610	13	{	{	PUNCT
ma-12	610	14	ν(3)n	ν(3)n	NOUN
ma-12	610	15	}	}	PUNCT
ma-12	610	16	∈	∈	PROPN
ma-12	611	1	[	[	X
ma-12	611	2	1,∞	1,∞	NUM
ma-12	611	3	)	)	PUNCT
ma-12	611	4	.	.	PUNCT
ma-12	612	1	let	let	VERB
ma-12	612	2	{	{	PUNCT
ma-12	612	3	xn	xn	VERB
ma-12	612	4	}	}	PUNCT
ma-12	612	5	be	be	VERB
ma-12	612	6	the	the	DET
ma-12	612	7	sequence	sequence	NOUN
ma-12	612	8	defined	define	VERB
ma-12	612	9	by	by	ADP
ma-12	612	10	(	(	PUNCT
ma-12	612	11	1.7	1.7	NUM
ma-12	612	12	)	)	PUNCT
ma-12	612	13	,	,	PUNCT
ma-12	612	14	where	where	SCONJ
ma-12	612	15	{	{	PUNCT
ma-12	612	16	αn	αn	NOUN
ma-12	612	17	}	}	PUNCT
ma-12	612	18	and	and	CCONJ
ma-12	612	19	{	{	PUNCT
ma-12	612	20	βn	βn	VERB
ma-12	612	21	}	}	PUNCT
ma-12	612	22	are	be	AUX
ma-12	612	23	real	real	ADJ
ma-12	612	24	sequences	sequence	NOUN
ma-12	612	25	∈	∈	PROPN
ma-12	613	1	[	[	X
ma-12	613	2	0	0	NUM
ma-12	613	3	,	,	PUNCT
ma-12	613	4	1	1	NUM
ma-12	613	5	)	)	PUNCT
ma-12	613	6	..	..	PUNCT
ma-12	613	7	suppose	suppose	VERB
ma-12	613	8	f	f	PROPN
ma-12	613	9	=	=	PRON
ma-12	613	10	∩3i=1(f	∩3i=1(f	X
ma-12	613	11	(	(	PUNCT
ma-12	613	12	ti	ti	NOUN
ma-12	613	13	)	)	PUNCT
ma-12	613	14	∩	∩	ADJ
ma-12	613	15	f	f	X
ma-12	613	16	(	(	PUNCT
ma-12	613	17	si	si	NOUN
ma-12	613	18	)	)	PUNCT
ma-12	613	19	)	)	PUNCT
ma-12	614	1	6=	6=	ADP
ma-12	614	2	0	0	X
ma-12	614	3	.	.	PUNCT
ma-12	615	1	if	if	SCONJ
ma-12	615	2	the	the	DET
ma-12	615	3	following	follow	VERB
ma-12	615	4	conditions	condition	NOUN
ma-12	615	5	hold	hold	VERB
ma-12	615	6	:	:	PUNCT
ma-12	615	7	i.	i.	NOUN
ma-12	615	8	∑∞	∑∞	PROPN
ma-12	615	9	n=1	n=1	PROPN
ma-12	615	10	k	k	PROPN
ma-12	615	11	(	(	PUNCT
ma-12	615	12	1	1	NUM
ma-12	615	13	)	)	PUNCT
ma-12	615	14	n	n	CCONJ
ma-12	615	15	<	<	X
ma-12	615	16	∞	∞	PROPN
ma-12	615	17	,	,	PUNCT
ma-12	615	18	∑∞	∑∞	NOUN
ma-12	615	19	n=1	n=1	PROPN
ma-12	615	20	k	k	PROPN
ma-12	615	21	(	(	PUNCT
ma-12	615	22	2	2	NUM
ma-12	615	23	)	)	PUNCT
ma-12	615	24	n	n	CCONJ
ma-12	615	25	<	<	X
ma-12	615	26	∞	∞	PROPN
ma-12	615	27	,	,	PUNCT
ma-12	615	28	∑∞	∑∞	NOUN
ma-12	615	29	n=1	n=1	PROPN
ma-12	615	30	k	k	PROPN
ma-12	615	31	(	(	PUNCT
ma-12	615	32	3	3	NUM
ma-12	615	33	)	)	PUNCT
ma-12	615	34	n	n	CCONJ
ma-12	615	35	<	<	X
ma-12	615	36	∞	∞	PROPN
ma-12	615	37	,	,	PUNCT
ma-12	615	38	∑∞	∑∞	NOUN
ma-12	615	39	n=1	n=1	PROPN
ma-12	615	40	µ	µ	X
ma-12	615	41	(	(	PUNCT
ma-12	615	42	1	1	NUM
ma-12	615	43	)	)	PUNCT
ma-12	615	44	n	n	CCONJ
ma-12	615	45	<	<	X
ma-12	615	46	∞	∞	PROPN
ma-12	615	47	,	,	PUNCT
ma-12	615	48	∑∞	∑∞	NOUN
ma-12	615	49	n=1	n=1	PROPN
ma-12	615	50	µ	µ	X
ma-12	615	51	(	(	PUNCT
ma-12	615	52	2	2	NUM
ma-12	615	53	)	)	PUNCT
ma-12	615	54	n	n	CCONJ
ma-12	615	55	<	<	X
ma-12	615	56	∞	∞	PROPN
ma-12	615	57	,	,	PUNCT
ma-12	615	58	∑∞	∑∞	NOUN
ma-12	615	59	n=1	n=1	PROPN
ma-12	615	60	µ	µ	X
ma-12	615	61	(	(	PUNCT
ma-12	615	62	3	3	NUM
ma-12	615	63	)	)	PUNCT
ma-12	615	64	n	n	CCONJ
ma-12	615	65	<	<	X
ma-12	615	66	∞	∞	PROPN
ma-12	615	67	,	,	PUNCT
ma-12	615	68	∑∞	∑∞	NOUN
ma-12	615	69	n=1	n=1	PROPN
ma-12	615	70	ν	ν	NOUN
ma-12	615	71	(	(	PUNCT
ma-12	615	72	1	1	NUM
ma-12	615	73	)	)	PUNCT
ma-12	615	74	n	n	CCONJ
ma-12	615	75	<	<	X
ma-12	615	76	∞	∞	PROPN
ma-12	615	77	,	,	PUNCT
ma-12	615	78	∑∞	∑∞	NOUN
ma-12	615	79	n=1	n=1	PROPN
ma-12	615	80	ν	ν	NOUN
ma-12	615	81	(	(	PUNCT
ma-12	615	82	2	2	NUM
ma-12	615	83	)	)	PUNCT
ma-12	615	84	n	n	CCONJ
ma-12	615	85	<	<	X
ma-12	615	86	∞	∞	PROPN
ma-12	615	87	,	,	PUNCT
ma-12	615	88	∑∞	∑∞	NOUN
ma-12	615	89	n=1	n=1	PROPN
ma-12	615	90	ν	ν	NOUN
ma-12	615	91	(	(	PUNCT
ma-12	615	92	3	3	NUM
ma-12	615	93	)	)	PUNCT
ma-12	615	94	n	n	CCONJ
ma-12	615	95	<	<	X
ma-12	615	96	∞,ii	∞,ii	NUM
ma-12	615	97	.	.	PUNCT
ma-12	616	1	there	there	PRON
ma-12	616	2	exists	exist	VERB
ma-12	616	3	a	a	DET
ma-12	616	4	constant	constant	ADJ
ma-12	616	5	m	m	NOUN
ma-12	616	6	>	>	X
ma-12	616	7	0	0	NUM
ma-12	616	8	such	such	ADJ
ma-12	616	9	that	that	SCONJ
ma-12	616	10	ψ(t	ψ(t	PROPN
ma-12	616	11	)	)	PUNCT
ma-12	616	12	=	=	SYM
ma-12	616	13	φ(t	φ(t	PROPN
ma-12	616	14	)	)	PUNCT
ma-12	616	15	≤	≤	PROPN
ma-12	616	16	mt	mt	PROPN
ma-12	616	17	,	,	PUNCT
ma-12	616	18	t	t	PROPN
ma-12	616	19	≤	≤	NUM
ma-12	616	20	0	0	NUM
ma-12	616	21	.	.	PUNCT
ma-12	617	1	then	then	ADV
ma-12	617	2	,	,	PUNCT
ma-12	617	3	limn∞	limn∞	PROPN
ma-12	617	4	‖xn	‖xn	PROPN
ma-12	617	5	−	−	NOUN
ma-12	617	6	q‖	q‖	NOUN
ma-12	617	7	and	and	CCONJ
ma-12	617	8	limn∞	limn∞	VERB
ma-12	617	9	d(xn	d(xn	NUM
ma-12	617	10	−	−	PROPN
ma-12	617	11	f	f	NOUN
ma-12	617	12	)	)	PUNCT
ma-12	617	13	both	both	PRON
ma-12	617	14	exist	exist	VERB
ma-12	617	15	for	for	ADP
ma-12	617	16	all	all	DET
ma-12	617	17	q	q	PROPN
ma-12	617	18	∈	∈	PROPN
ma-12	617	19	f	f	X
ma-12	617	20	.	.	PUNCT
ma-12	618	1	corollary	corollary	ADJ
ma-12	618	2	3.10	3.10	NUM
ma-12	618	3	.	.	PUNCT
ma-12	619	1	let	let	VERB
ma-12	619	2	e	e	PRON
ma-12	619	3	be	be	AUX
ma-12	619	4	a	a	DET
ma-12	619	5	uniformly	uniformly	ADV
ma-12	619	6	convex	convex	NOUN
ma-12	619	7	banach	banach	NOUN
ma-12	619	8	space	space	NOUN
ma-12	619	9	and	and	CCONJ
ma-12	619	10	k	k	PROPN
ma-12	619	11	a	a	DET
ma-12	619	12	nonempty	nonempty	ADV
ma-12	619	13	closed	close	VERB
ma-12	619	14	convex	convex	NOUN
ma-12	619	15	subset	subset	NOUN
ma-12	619	16	of	of	ADP
ma-12	619	17	e.	e.	PROPN
ma-12	619	18	let	let	VERB
ma-12	619	19	s1	s1	PROPN
ma-12	619	20	,	,	PUNCT
ma-12	619	21	s2	s2	PROPN
ma-12	619	22	,	,	PUNCT
ma-12	619	23	s3	s3	PROPN
ma-12	619	24	:	:	PUNCT
ma-12	619	25	k	k	PROPN
ma-12	619	26	−→	−→	NOUN
ma-12	619	27	k	k	PROPN
ma-12	619	28	be	be	AUX
ma-12	619	29	three	three	NUM
ma-12	619	30	generalize	generalize	VERB
ma-12	619	31	asymptotically	asymptotically	ADV
ma-12	619	32	nonexpansive	nonexpansive	ADJ
ma-12	619	33	self	self	NOUN
ma-12	619	34	mapping	mapping	NOUN
ma-12	619	35	with	with	ADP
ma-12	619	36	sequences	sequence	NOUN
ma-12	619	37	{	{	PUNCT
ma-12	619	38	k(1)n	k(1)n	X
ma-12	619	39	}	}	PUNCT
ma-12	619	40	,	,	PUNCT
ma-12	619	41	{	{	PUNCT
ma-12	619	42	k(2)n	k(2)n	X
ma-12	619	43	}	}	PUNCT
ma-12	619	44	,	,	PUNCT
ma-12	619	45	{	{	PUNCT
ma-12	619	46	k(3)n	k(3)n	NOUN
ma-12	619	47	}	}	PUNCT
ma-12	619	48	∈	∈	PROPN
ma-12	620	1	[	[	X
ma-12	620	2	1,∞	1,∞	NUM
ma-12	620	3	)	)	PUNCT
ma-12	620	4	,	,	PUNCT
ma-12	620	5	{	{	PUNCT
ma-12	620	6	w	w	X
ma-12	620	7	(	(	PUNCT
ma-12	620	8	1)n	1)n	X
ma-12	620	9	}	}	PUNCT
ma-12	620	10	,	,	PUNCT
ma-12	620	11	{	{	PUNCT
ma-12	620	12	w	w	NOUN
ma-12	620	13	(	(	PUNCT
ma-12	620	14	2)}n	2)}n	NOUN
ma-12	620	15	,	,	PUNCT
ma-12	620	16	{	{	PUNCT
ma-12	620	17	w	w	NOUN
ma-12	620	18	(	(	PUNCT
ma-12	620	19	3)n	3)n	NUM
ma-12	620	20	}	}	PUNCT
ma-12	620	21	∈	∈	PROPN
ma-12	621	1	[	[	X
ma-12	621	2	1,∞	1,∞	NUM
ma-12	621	3	)	)	PUNCT
ma-12	621	4	and	and	CCONJ
ma-12	621	5	t1	t1	NOUN
ma-12	621	6	,	,	PUNCT
ma-12	621	7	t2	t2	NOUN
ma-12	621	8	,	,	PUNCT
ma-12	621	9	t3	t3	NOUN
ma-12	621	10	:	:	PUNCT
ma-12	621	11	k	k	X
ma-12	621	12	−→	−→	NOUN
ma-12	621	13	e	e	NOUN
ma-12	621	14	are	be	AUX
ma-12	621	15	three	three	NUM
ma-12	621	16	generalize	generalize	VERB
ma-12	621	17	asymptotically	asymptotically	ADV
ma-12	621	18	nonexpansive	nonexpansive	ADJ
ma-12	621	19	nonself	nonself	PROPN
ma-12	621	20	mappings	mapping	NOUN
ma-12	621	21	with	with	ADP
ma-12	621	22	sequences	sequence	NOUN
ma-12	621	23	{	{	PUNCT
ma-12	621	24	µ(1)n	µ(1)n	X
ma-12	621	25	}	}	PUNCT
ma-12	621	26	,	,	PUNCT
ma-12	621	27	{	{	PUNCT
ma-12	621	28	µ(2)n	µ(2)n	NOUN
ma-12	621	29	}	}	PUNCT
ma-12	621	30	,	,	PUNCT
ma-12	621	31	{	{	PUNCT
ma-12	621	32	µ(3)n	µ(3)n	PROPN
ma-12	621	33	}	}	PUNCT
ma-12	621	34	∈	∈	PROPN
ma-12	622	1	[	[	X
ma-12	622	2	1,∞	1,∞	NUM
ma-12	622	3	)	)	PUNCT
ma-12	622	4	,	,	PUNCT
ma-12	622	5	{	{	PUNCT
ma-12	622	6	ν(1)n	ν(1)n	X
ma-12	622	7	}	}	PUNCT
ma-12	622	8	,	,	PUNCT
ma-12	622	9	{	{	PUNCT
ma-12	622	10	ν(2)n	ν(2)n	NOUN
ma-12	622	11	}	}	PUNCT
ma-12	622	12	,	,	PUNCT
ma-12	622	13	{	{	PUNCT
ma-12	622	14	ν(3)n	ν(3)n	NOUN
ma-12	622	15	}	}	PUNCT
ma-12	622	16	∈	∈	PROPN
ma-12	623	1	[	[	X
ma-12	623	2	1,∞	1,∞	NUM
ma-12	623	3	)	)	PUNCT
ma-12	623	4	.	.	PUNCT
ma-12	624	1	let	let	VERB
ma-12	624	2	{	{	PUNCT
ma-12	624	3	xn	xn	VERB
ma-12	624	4	}	}	PUNCT
ma-12	624	5	be	be	VERB
ma-12	624	6	the	the	DET
ma-12	624	7	sequence	sequence	NOUN
ma-12	624	8	defined	define	VERB
ma-12	624	9	by	by	ADP
ma-12	624	10	(	(	PUNCT
ma-12	624	11	1.7	1.7	NUM
ma-12	624	12	)	)	PUNCT
ma-12	624	13	,	,	PUNCT
ma-12	624	14	where	where	SCONJ
ma-12	624	15	{	{	PUNCT
ma-12	624	16	αn	αn	NOUN
ma-12	624	17	}	}	PUNCT
ma-12	624	18	and	and	CCONJ
ma-12	624	19	{	{	PUNCT
ma-12	624	20	βn	βn	VERB
ma-12	624	21	}	}	PUNCT
ma-12	624	22	are	be	AUX
ma-12	624	23	real	real	ADJ
ma-12	624	24	sequences	sequence	NOUN
ma-12	624	25	∈	∈	PROPN
ma-12	625	1	[	[	X
ma-12	625	2	0	0	NUM
ma-12	625	3	,	,	PUNCT
ma-12	625	4	1	1	NUM
ma-12	625	5	)	)	PUNCT
ma-12	625	6	.	.	PUNCT
ma-12	626	1	suppose	suppose	VERB
ma-12	626	2	f	f	PROPN
ma-12	626	3	=	=	PRON
ma-12	626	4	∩3i=1(f	∩3i=1(f	X
ma-12	626	5	(	(	PUNCT
ma-12	626	6	ti	ti	NOUN
ma-12	626	7	)	)	PUNCT
ma-12	626	8	∩	∩	ADJ
ma-12	626	9	f	f	X
ma-12	626	10	(	(	PUNCT
ma-12	626	11	si	si	NOUN
ma-12	626	12	)	)	PUNCT
ma-12	626	13	)	)	PUNCT
ma-12	627	1	6=	6=	ADP
ma-12	627	2	0	0	X
ma-12	627	3	.	.	PUNCT
ma-12	628	1	if	if	SCONJ
ma-12	628	2	the	the	DET
ma-12	628	3	following	follow	VERB
ma-12	628	4	conditions	condition	NOUN
ma-12	628	5	hold	hold	VERB
ma-12	628	6	:	:	PUNCT
ma-12	628	7	eur	eur	PROPN
ma-12	628	8	.	.	PUNCT
ma-12	629	1	j.	j.	PROPN
ma-12	629	2	math	math	PROPN
ma-12	629	3	.	.	PUNCT
ma-12	630	1	anal	anal	ADJ
ma-12	630	2	.	.	PUNCT
ma-12	631	1	1	1	NUM
ma-12	631	2	(	(	PUNCT
ma-12	631	3	2021	2021	NUM
ma-12	631	4	)	)	PUNCT
ma-12	631	5	66	66	NUM
ma-12	631	6	i.	i.	NOUN
ma-12	631	7	∑∞	∑∞	NOUN
ma-12	631	8	n=1	n=1	PROPN
ma-12	631	9	k	k	PROPN
ma-12	631	10	(	(	PUNCT
ma-12	631	11	1	1	NUM
ma-12	631	12	)	)	PUNCT
ma-12	631	13	n	n	CCONJ
ma-12	631	14	<	<	X
ma-12	631	15	∞	∞	PROPN
ma-12	631	16	,	,	PUNCT
ma-12	631	17	∑∞	∑∞	NOUN
ma-12	631	18	n=1	n=1	PROPN
ma-12	631	19	k	k	PROPN
ma-12	631	20	(	(	PUNCT
ma-12	631	21	2	2	NUM
ma-12	631	22	)	)	PUNCT
ma-12	631	23	n	n	CCONJ
ma-12	631	24	<	<	X
ma-12	631	25	∞	∞	PROPN
ma-12	631	26	,	,	PUNCT
ma-12	631	27	∑∞	∑∞	NOUN
ma-12	631	28	n=1	n=1	PROPN
ma-12	631	29	k	k	PROPN
ma-12	631	30	(	(	PUNCT
ma-12	631	31	3	3	NUM
ma-12	631	32	)	)	PUNCT
ma-12	631	33	n	n	CCONJ
ma-12	631	34	<	<	X
ma-12	631	35	∞	∞	PROPN
ma-12	631	36	,	,	PUNCT
ma-12	631	37	∑∞	∑∞	NOUN
ma-12	631	38	n=1	n=1	PROPN
ma-12	631	39	µ	µ	X
ma-12	631	40	(	(	PUNCT
ma-12	631	41	1	1	NUM
ma-12	631	42	)	)	PUNCT
ma-12	631	43	n	n	CCONJ
ma-12	631	44	<	<	X
ma-12	631	45	∞	∞	PROPN
ma-12	631	46	,	,	PUNCT
ma-12	631	47	∑∞	∑∞	NOUN
ma-12	631	48	n=1	n=1	PROPN
ma-12	631	49	µ	µ	X
ma-12	631	50	(	(	PUNCT
ma-12	631	51	2	2	NUM
ma-12	631	52	)	)	PUNCT
ma-12	631	53	n	n	CCONJ
ma-12	631	54	<	<	X
ma-12	631	55	∞	∞	PROPN
ma-12	631	56	,	,	PUNCT
ma-12	631	57	∑∞	∑∞	NOUN
ma-12	631	58	n=1	n=1	PROPN
ma-12	631	59	µ	µ	X
ma-12	631	60	(	(	PUNCT
ma-12	631	61	3	3	NUM
ma-12	631	62	)	)	PUNCT
ma-12	631	63	n	n	CCONJ
ma-12	631	64	<	<	X
ma-12	631	65	∞	∞	PROPN
ma-12	631	66	,	,	PUNCT
ma-12	631	67	∑∞	∑∞	NOUN
ma-12	631	68	n=1	n=1	PROPN
ma-12	631	69	ν	ν	NOUN
ma-12	631	70	(	(	PUNCT
ma-12	631	71	1	1	NUM
ma-12	631	72	)	)	PUNCT
ma-12	631	73	n	n	CCONJ
ma-12	631	74	<	<	X
ma-12	631	75	∞	∞	PROPN
ma-12	631	76	,	,	PUNCT
ma-12	631	77	∑∞	∑∞	NOUN
ma-12	631	78	n=1	n=1	PROPN
ma-12	631	79	ν	ν	NOUN
ma-12	631	80	(	(	PUNCT
ma-12	631	81	2	2	NUM
ma-12	631	82	)	)	PUNCT
ma-12	631	83	n	n	CCONJ
ma-12	631	84	<	<	X
ma-12	631	85	∞	∞	PROPN
ma-12	631	86	,	,	PUNCT
ma-12	631	87	∑∞	∑∞	NOUN
ma-12	631	88	n=1	n=1	PROPN
ma-12	631	89	ν	ν	NOUN
ma-12	631	90	(	(	PUNCT
ma-12	631	91	3	3	NUM
ma-12	631	92	)	)	PUNCT
ma-12	631	93	n	n	CCONJ
ma-12	631	94	<	<	X
ma-12	631	95	∞,ii	∞,ii	NUM
ma-12	631	96	.	.	PUNCT
ma-12	631	97	‖x−t1(pt1)n−1y‖	‖x−t1(pt1)n−1y‖	NOUN
ma-12	632	1	≤	≤	NUM
ma-12	632	2	‖sn1x−t1(pt1)n−1y‖	‖sn1x−t1(pt1)n−1y‖	NOUN
ma-12	632	3	,	,	PUNCT
ma-12	632	4	‖x−t2(pt2)n−1y‖	‖x−t2(pt2)n−1y‖	NUM
ma-12	632	5	≤	≤	NOUN
ma-12	632	6	‖sn2x−t2(pt2)n−1y‖	‖sn2x−t2(pt2)n−1y‖	PROPN
ma-12	632	7	,	,	PUNCT
ma-12	632	8	‖x	‖x	NOUN
ma-12	632	9	−	−	PROPN
ma-12	633	1	t3(pt3)n−1y‖	t3(pt3)n−1y‖	PROPN
ma-12	633	2	≤	≤	NOUN
ma-12	633	3	‖sn3x	‖sn3x	ADP
ma-12	633	4	−	−	PROPN
ma-12	633	5	t3(pt3)n−1y‖iii	t3(pt3)n−1y‖iii	NUM
ma-12	633	6	.	.	PUNCT
ma-12	634	1	there	there	PRON
ma-12	634	2	exists	exist	VERB
ma-12	634	3	a	a	DET
ma-12	634	4	constant	constant	ADJ
ma-12	634	5	m1,m2	m1,m2	PROPN
ma-12	634	6	>	>	PUNCT
ma-12	634	7	0	0	NUM
ma-12	634	8	such	such	ADJ
ma-12	634	9	that	that	SCONJ
ma-12	634	10	ψ(t	ψ(t	PROPN
ma-12	634	11	)	)	PUNCT
ma-12	634	12	≤	≤	NOUN
ma-12	634	13	m1	m1	PROPN
ma-12	634	14	t	t	PROPN
ma-12	634	15	,	,	PUNCT
ma-12	634	16	φ(t	φ(t	PROPN
ma-12	634	17	)	)	PUNCT
ma-12	634	18	≤	≤	NUM
ma-12	634	19	m2	m2	PROPN
ma-12	634	20	t	t	PROPN
ma-12	634	21	,	,	PUNCT
ma-12	634	22	t	t	PROPN
ma-12	634	23	≥	≥	PROPN
ma-12	634	24	0	0	NUM
ma-12	634	25	.	.	PUNCT
ma-12	635	1	then	then	ADV
ma-12	635	2	,	,	PUNCT
ma-12	635	3	limn∞	limn∞	PROPN
ma-12	635	4	‖xn	‖xn	PROPN
ma-12	635	5	−	−	NOUN
ma-12	635	6	sixn‖	sixn‖	PUNCT
ma-12	635	7	=	=	SYM
ma-12	635	8	0	0	NUM
ma-12	636	1	and	and	CCONJ
ma-12	636	2	limn∞	limn∞	ADJ
ma-12	636	3	‖xn	‖xn	PROPN
ma-12	636	4	−	−	PROPN
ma-12	636	5	tixn‖	tixn‖	PROPN
ma-12	636	6	=	=	SYM
ma-12	636	7	0	0	NUM
ma-12	636	8	,	,	PUNCT
ma-12	636	9	for	for	ADP
ma-12	636	10	i	i	PROPN
ma-12	636	11	=	=	SYM
ma-12	636	12	1	1	NUM
ma-12	636	13	,	,	PUNCT
ma-12	636	14	2	2	NUM
ma-12	636	15	,	,	PUNCT
ma-12	636	16	,	,	PUNCT
ma-12	636	17	3	3	X
ma-12	636	18	.	.	X
ma-12	636	19	abbreviations	abbreviation	NOUN
ma-12	636	20	usednot	usednot	ADV
ma-12	636	21	applicable	applicable	ADJ
ma-12	636	22	declaration	declaration	NOUN
ma-12	636	23	:	:	PUNCT
ma-12	636	24	availability	availability	NOUN
ma-12	636	25	of	of	ADP
ma-12	636	26	data	datum	NOUN
ma-12	636	27	and	and	CCONJ
ma-12	636	28	materialnot	materialnot	ADV
ma-12	636	29	applicable	applicable	ADJ
ma-12	636	30	competing	compete	VERB
ma-12	636	31	lnterestthe	lnterestthe	ADJ
ma-12	636	32	authors	author	NOUN
ma-12	636	33	declare	declare	VERB
ma-12	636	34	that	that	SCONJ
ma-12	636	35	there	there	PRON
ma-12	636	36	is	be	VERB
ma-12	636	37	no	no	DET
ma-12	636	38	conflict	conflict	NOUN
ma-12	636	39	of	of	ADP
ma-12	636	40	interest	interest	NOUN
ma-12	636	41	.	.	PUNCT
ma-12	637	1	fundingno	fundingno	ADP
ma-12	637	2	specific	specific	ADJ
ma-12	637	3	funding	funding	NOUN
ma-12	637	4	received	receive	VERB
ma-12	637	5	for	for	ADP
ma-12	637	6	this	this	DET
ma-12	637	7	work	work	NOUN
ma-12	637	8	authors	author	NOUN
ma-12	637	9	contributionika	contributionika	VERB
ma-12	637	10	and	and	CCONJ
ma-12	637	11	ncu	ncu	PROPN
ma-12	637	12	wrote	write	VERB
ma-12	637	13	the	the	DET
ma-12	637	14	paper	paper	NOUN
ma-12	637	15	while	while	SCONJ
ma-12	637	16	dii	dii	NOUN
ma-12	637	17	suggested	suggest	VERB
ma-12	637	18	the	the	DET
ma-12	637	19	idea	idea	NOUN
ma-12	637	20	and	and	CCONJ
ma-12	637	21	did	do	VERB
ma-12	637	22	the	the	DET
ma-12	637	23	analysis	analysis	NOUN
ma-12	637	24	.	.	PUNCT
ma-12	638	1	the	the	DET
ma-12	638	2	three	three	NUM
ma-12	638	3	authorsread	authorsread	NOUN
ma-12	638	4	and	and	CCONJ
ma-12	638	5	approved	approve	VERB
ma-12	638	6	the	the	DET
ma-12	638	7	final	final	ADJ
ma-12	638	8	manuscript	manuscript	NOUN
ma-12	638	9	.	.	PUNCT
ma-12	639	1	acknowledgementthe	acknowledgementthe	PROPN
ma-12	639	2	authors	author	NOUN
ma-12	639	3	thank	thank	VERB
ma-12	639	4	the	the	DET
ma-12	639	5	anonymous	anonymous	ADJ
ma-12	639	6	reviewers	reviewer	NOUN
ma-12	639	7	for	for	ADP
ma-12	639	8	their	their	PRON
ma-12	639	9	careful	careful	ADJ
ma-12	639	10	reading	reading	NOUN
ma-12	639	11	of	of	ADP
ma-12	639	12	this	this	DET
ma-12	639	13	paper	paper	NOUN
ma-12	639	14	and	and	CCONJ
ma-12	639	15	approvedthe	approvedthe	DET
ma-12	639	16	final	final	ADJ
ma-12	639	17	manuscript	manuscript	NOUN
ma-12	639	18	.	.	PUNCT
ma-12	640	1	eur	eur	PROPN
ma-12	640	2	.	.	PUNCT
ma-12	641	1	j.	j.	PROPN
ma-12	641	2	math	math	PROPN
ma-12	641	3	.	.	PUNCT
ma-12	642	1	anal	anal	ADJ
ma-12	642	2	.	.	PUNCT
ma-12	643	1	1	1	NUM
ma-12	643	2	(	(	PUNCT
ma-12	643	3	2021	2021	NUM
ma-12	643	4	)	)	PUNCT
ma-12	643	5	67references	67reference	NOUN
ma-12	644	1	[	[	X
ma-12	644	2	1	1	X
ma-12	644	3	]	]	PUNCT
ma-12	644	4	ya.i	ya.i	PROPN
ma-12	644	5	.	.	PUNCT
ma-12	644	6	alber	alber	PROPN
ma-12	644	7	,	,	PUNCT
ma-12	644	8	c.e	c.e	PROPN
ma-12	644	9	.	.	PROPN
ma-12	644	10	chidume	chidume	PROPN
ma-12	644	11	,	,	PUNCT
ma-12	644	12	h.	h.	PROPN
ma-12	644	13	zegeye	zegeye	PROPN
ma-12	644	14	,	,	PUNCT
ma-12	644	15	approximating	approximate	VERB
ma-12	644	16	fixed	fix	VERB
ma-12	644	17	points	point	NOUN
ma-12	644	18	of	of	ADP
ma-12	644	19	total	total	ADJ
ma-12	644	20	asymptotically	asymptotically	ADV
ma-12	644	21	nonexpansive	nonexpansive	ADJ
ma-12	644	22	mappings	mapping	NOUN
ma-12	644	23	,	,	PUNCT
ma-12	644	24	fixed	fix	VERB
ma-12	644	25	point	point	NOUN
ma-12	644	26	theory	theory	NOUN
ma-12	644	27	appl	appl	NOUN
ma-12	644	28	.	.	PUNCT
ma-12	645	1	2006	2006	NUM
ma-12	645	2	(	(	PUNCT
ma-12	645	3	2006	2006	NUM
ma-12	645	4	)	)	PUNCT
ma-12	645	5	10673	10673	NUM
ma-12	645	6	.	.	PUNCT
ma-12	646	1	https://doi.org/10.1155/fpta/2006/10673.[2	https://doi.org/10.1155/fpta/2006/10673.[2	PROPN
ma-12	646	2	]	]	X
ma-12	646	3	c.e	c.e	PROPN
ma-12	646	4	.	.	PROPN
ma-12	646	5	chidume	chidume	PROPN
ma-12	646	6	,	,	PUNCT
ma-12	646	7	e.u	e.u	PROPN
ma-12	646	8	.	.	PROPN
ma-12	646	9	ofoedu	ofoedu	PROPN
ma-12	646	10	,	,	PUNCT
ma-12	646	11	h.	h.	PROPN
ma-12	646	12	zegeye	zegeye	PROPN
ma-12	646	13	,	,	PUNCT
ma-12	646	14	strong	strong	ADJ
ma-12	646	15	and	and	CCONJ
ma-12	646	16	weak	weak	ADJ
ma-12	646	17	convergence	convergence	NOUN
ma-12	646	18	theorems	theorem	NOUN
ma-12	646	19	for	for	ADP
ma-12	646	20	asymptotically	asymptotically	ADJ
ma-12	646	21	nonexpansivemappings	nonexpansivemapping	NOUN
ma-12	646	22	,	,	PUNCT
ma-12	646	23	j.	j.	PROPN
ma-12	646	24	math	math	PROPN
ma-12	646	25	.	.	PUNCT
ma-12	647	1	anal	anal	PROPN
ma-12	647	2	.	.	PUNCT
ma-12	648	1	appl	appl	PROPN
ma-12	648	2	.	.	PROPN
ma-12	649	1	280	280	NUM
ma-12	649	2	(	(	PUNCT
ma-12	649	3	2003	2003	NUM
ma-12	649	4	)	)	PUNCT
ma-12	650	1	364?374	364?374	ADV
ma-12	650	2	.	.	PUNCT
ma-12	651	1	https://doi.org/10.1016/s0022-247x(03)00061-1.[3	https://doi.org/10.1016/s0022-247x(03)00061-1.[3	PROPN
ma-12	651	2	]	]	X
ma-12	651	3	c.e	c.e	PROPN
ma-12	651	4	.	.	PROPN
ma-12	651	5	chidume	chidume	PROPN
ma-12	651	6	,	,	PUNCT
ma-12	651	7	n.	n.	PROPN
ma-12	651	8	shahzad	shahzad	PROPN
ma-12	651	9	,	,	PUNCT
ma-12	651	10	h.	h.	PROPN
ma-12	651	11	zegeye	zegeye	PROPN
ma-12	651	12	,	,	PUNCT
ma-12	651	13	convergence	convergence	NOUN
ma-12	651	14	theorems	theorem	NOUN
ma-12	651	15	for	for	ADP
ma-12	651	16	mappings	mapping	NOUN
ma-12	651	17	which	which	PRON
ma-12	651	18	are	be	AUX
ma-12	651	19	asymptotically	asymptotically	ADV
ma-12	651	20	nonexpan	nonexpan	NOUN
ma-12	651	21	-	-	PUNCT
ma-12	651	22	sive	sive	ADJ
ma-12	651	23	in	in	ADP
ma-12	651	24	the	the	DET
ma-12	651	25	intermediate	intermediate	ADJ
ma-12	651	26	sense	sense	NOUN
ma-12	651	27	,	,	PUNCT
ma-12	651	28	numer	numer	PROPN
ma-12	651	29	.	.	PUNCT
ma-12	652	1	funct	funct	PROPN
ma-12	652	2	.	.	PUNCT
ma-12	653	1	anal	anal	PROPN
ma-12	653	2	.	.	PUNCT
ma-12	654	1	optim	optim	PROPN
ma-12	654	2	.	.	PUNCT
ma-12	655	1	25	25	NUM
ma-12	655	2	(	(	PUNCT
ma-12	655	3	2005	2005	NUM
ma-12	655	4	)	)	PUNCT
ma-12	655	5	239?257	239?257	NOUN
ma-12	655	6	.	.	PUNCT
ma-12	656	1	https://doi.org/10.1081/	https://doi.org/10.1081/	PROPN
ma-12	656	2	nfa-120039611.[4	nfa-120039611.[4	NOUN
ma-12	656	3	]	]	X
ma-12	656	4	c.e	c.e	PROPN
ma-12	656	5	.	.	PROPN
ma-12	656	6	chidume	chidume	PROPN
ma-12	656	7	,	,	PUNCT
ma-12	656	8	e.u	e.u	PROPN
ma-12	656	9	.	.	PROPN
ma-12	656	10	ofoedu	ofoedu	PROPN
ma-12	656	11	,	,	PUNCT
ma-12	656	12	approximation	approximation	NOUN
ma-12	656	13	of	of	ADP
ma-12	656	14	common	common	ADJ
ma-12	656	15	fixed	fix	VERB
ma-12	656	16	points	point	NOUN
ma-12	656	17	for	for	ADP
ma-12	656	18	finite	finite	ADJ
ma-12	656	19	families	family	NOUN
ma-12	656	20	of	of	ADP
ma-12	656	21	total	total	ADJ
ma-12	656	22	asymptotically	asymptotically	ADV
ma-12	656	23	nonex	nonex	ADJ
ma-12	656	24	-	-	PUNCT
ma-12	656	25	pansive	pansive	ADJ
ma-12	656	26	mappings	mapping	NOUN
ma-12	656	27	,	,	PUNCT
ma-12	656	28	j.	j.	PROPN
ma-12	656	29	math	math	PROPN
ma-12	656	30	.	.	PUNCT
ma-12	657	1	anal	anal	PROPN
ma-12	657	2	.	.	PUNCT
ma-12	657	3	appl	appl	PROPN
ma-12	657	4	.	.	PROPN
ma-12	658	1	333	333	NUM
ma-12	658	2	(	(	PUNCT
ma-12	658	3	2007	2007	NUM
ma-12	658	4	)	)	PUNCT
ma-12	658	5	128?141	128?141	NUM
ma-12	658	6	.	.	PUNCT
ma-12	659	1	https://doi.org/10.1016/j.jmaa.2006.09.023.[5	https://doi.org/10.1016/j.jmaa.2006.09.023.[5	PROPN
ma-12	659	2	]	]	PUNCT
ma-12	659	3	j.g	j.g	PROPN
ma-12	659	4	.	.	PROPN
ma-12	659	5	falset	falset	PROPN
ma-12	659	6	,	,	PUNCT
ma-12	659	7	w.	w.	PROPN
ma-12	659	8	awa	awa	PROPN
ma-12	659	9	kaczor	kaczor	PROPN
ma-12	659	10	,	,	PUNCT
ma-12	659	11	t.	t.	PROPN
ma-12	659	12	kuczumow	kuczumow	PROPN
ma-12	659	13	,	,	PUNCT
ma-12	659	14	s.	s.	PROPN
ma-12	659	15	reich	reich	PROPN
ma-12	659	16	,	,	PUNCT
ma-12	659	17	weak	weak	ADJ
ma-12	659	18	convergence	convergence	NOUN
ma-12	659	19	theorems	theorem	NOUN
ma-12	659	20	for	for	ADP
ma-12	659	21	asymptotically	asymptotically	ADV
ma-12	659	22	nonexpansivemappings	nonexpansivemapping	NOUN
ma-12	659	23	and	and	CCONJ
ma-12	659	24	semigroups	semigroup	NOUN
ma-12	659	25	,	,	PUNCT
ma-12	659	26	nonlinear	nonlinear	ADJ
ma-12	659	27	anal	anal	NOUN
ma-12	659	28	.	.	PUNCT
ma-12	659	29	:	:	PUNCT
ma-12	660	1	theory	theory	NOUN
ma-12	660	2	meth	meth	NOUN
ma-12	660	3	.	.	PUNCT
ma-12	661	1	appl	appl	PROPN
ma-12	661	2	.	.	PUNCT
ma-12	662	1	43	43	NUM
ma-12	662	2	(	(	PUNCT
ma-12	662	3	2001	2001	NUM
ma-12	662	4	)	)	PUNCT
ma-12	663	1	377?401	377?401	NUM
ma-12	663	2	.	.	PUNCT
ma-12	664	1	https://doi.org/10.1016/	https://doi.org/10.1016/	PROPN
ma-12	664	2	s0362	s0362	PROPN
ma-12	664	3	-	-	PUNCT
ma-12	664	4	546x(99)00200	546x(99)00200	PROPN
ma-12	664	5	-	-	PUNCT
ma-12	664	6	x.[6	x.[6	PROPN
ma-12	664	7	]	]	PUNCT
ma-12	664	8	k.	k.	PROPN
ma-12	664	9	goebel	goebel	PROPN
ma-12	664	10	,	,	PUNCT
ma-12	664	11	w.a	w.a	PROPN
ma-12	664	12	.	.	PROPN
ma-12	664	13	kirk	kirk	PROPN
ma-12	664	14	,	,	PUNCT
ma-12	664	15	a	a	DET
ma-12	664	16	fixed	fix	VERB
ma-12	664	17	point	point	NOUN
ma-12	664	18	theorem	theorem	NOUN
ma-12	664	19	for	for	ADP
ma-12	664	20	asymptotically	asymptotically	ADV
ma-12	664	21	nonexpansive	nonexpansive	ADJ
ma-12	664	22	mappings	mapping	NOUN
ma-12	664	23	,	,	PUNCT
ma-12	664	24	proc	proc	NOUN
ma-12	664	25	.	.	PUNCT
ma-12	665	1	amer	amer	PROPN
ma-12	665	2	.	.	PUNCT
ma-12	665	3	math	math	PROPN
ma-12	665	4	.	.	PUNCT
ma-12	666	1	soc	soc	PROPN
ma-12	666	2	.	.	PUNCT
ma-12	667	1	35(1972	35(1972	NUM
ma-12	667	2	)	)	PUNCT
ma-12	668	1	171?171	171?171	NUM
ma-12	668	2	.	.	PUNCT
ma-12	669	1	https://doi.org/10.1090/s0002-9939-1972-0298500-3.[7	https://doi.org/10.1090/s0002-9939-1972-0298500-3.[7	PROPN
ma-12	669	2	]	]	X
ma-12	669	3	w.	w.	PROPN
ma-12	669	4	guo	guo	PROPN
ma-12	669	5	,	,	PUNCT
ma-12	669	6	w.	w.	PROPN
ma-12	669	7	guo	guo	PROPN
ma-12	669	8	,	,	PUNCT
ma-12	669	9	weak	weak	ADJ
ma-12	669	10	convergence	convergence	NOUN
ma-12	669	11	theorems	theorem	NOUN
ma-12	669	12	for	for	ADP
ma-12	669	13	asymptotically	asymptotically	ADV
ma-12	669	14	nonexpansive	nonexpansive	ADJ
ma-12	669	15	nonself	nonself	PROPN
ma-12	669	16	-	-	PUNCT
ma-12	669	17	mappings	mapping	NOUN
ma-12	669	18	,	,	PUNCT
ma-12	669	19	appl	appl	PROPN
ma-12	669	20	.	.	PROPN
ma-12	669	21	math	math	NOUN
ma-12	669	22	.	.	PUNCT
ma-12	670	1	lett.24	lett.24	PROPN
ma-12	670	2	(	(	PUNCT
ma-12	670	3	2011	2011	NUM
ma-12	670	4	)	)	PUNCT
ma-12	670	5	2181?2185	2181?2185	NUM
ma-12	670	6	.	.	PUNCT
ma-12	670	7	https://doi.org/10.1016/j.aml.2011.06.022.[8	https://doi.org/10.1016/j.aml.2011.06.022.[8	X
ma-12	670	8	]	]	PUNCT
ma-12	670	9	w.	w.	PROPN
ma-12	670	10	guo	guo	PROPN
ma-12	670	11	,	,	PUNCT
ma-12	670	12	y.j	y.j	PROPN
ma-12	670	13	.	.	PUNCT
ma-12	670	14	cho	cho	PROPN
ma-12	670	15	,	,	PUNCT
ma-12	670	16	w.	w.	PROPN
ma-12	670	17	guo	guo	PROPN
ma-12	670	18	,	,	PUNCT
ma-12	670	19	convergence	convergence	NOUN
ma-12	670	20	theorems	theorem	NOUN
ma-12	670	21	for	for	ADP
ma-12	670	22	mixed	mixed	ADJ
ma-12	670	23	type	type	NOUN
ma-12	670	24	asymptotically	asymptotically	ADV
ma-12	670	25	nonexpansive	nonexpansive	ADJ
ma-12	670	26	mappings	mapping	NOUN
ma-12	670	27	,	,	PUNCT
ma-12	670	28	fixedpoint	fixedpoint	NOUN
ma-12	670	29	theory	theory	NOUN
ma-12	670	30	appl	appl	NOUN
ma-12	670	31	.	.	PUNCT
ma-12	670	32	2012	2012	NUM
ma-12	670	33	(	(	PUNCT
ma-12	670	34	2012	2012	NUM
ma-12	670	35	)	)	PUNCT
ma-12	670	36	224	224	NUM
ma-12	670	37	.	.	PUNCT
ma-12	671	1	https://doi.org/10.1186/1687-1812-2012-224.[9	https://doi.org/10.1186/1687-1812-2012-224.[9	PROPN
ma-12	671	2	]	]	PUNCT
ma-12	671	3	s.h	s.h	PROPN
ma-12	671	4	.	.	PROPN
ma-12	671	5	khan	khan	PROPN
ma-12	671	6	,	,	PUNCT
ma-12	671	7	w.	w.	PROPN
ma-12	671	8	takahashi	takahashi	PROPN
ma-12	671	9	,	,	PUNCT
ma-12	671	10	approximating	approximate	VERB
ma-12	671	11	common	common	ADJ
ma-12	671	12	fixed	fix	VERB
ma-12	671	13	points	point	NOUN
ma-12	671	14	of	of	ADP
ma-12	671	15	two	two	NUM
ma-12	671	16	asymptotically	asymptotically	ADV
ma-12	671	17	nonexpansive	nonexpansive	ADJ
ma-12	671	18	mappings	mapping	NOUN
ma-12	671	19	,	,	PUNCT
ma-12	671	20	sci.math	sci.math	PROPN
ma-12	671	21	.	.	PUNCT
ma-12	672	1	japon	japon	PROPN
ma-12	672	2	.	.	PUNCT
ma-12	673	1	53	53	NUM
ma-12	673	2	(	(	PUNCT
ma-12	673	3	2001	2001	NUM
ma-12	673	4	)	)	PUNCT
ma-12	673	5	143	143	NUM
ma-12	673	6	-	-	SYM
ma-12	673	7	148.[10	148.[10	NUM
ma-12	673	8	]	]	PUNCT
ma-12	673	9	z.	z.	PROPN
ma-12	673	10	opial	opial	PROPN
ma-12	673	11	,	,	PUNCT
ma-12	673	12	weak	weak	ADJ
ma-12	673	13	convergence	convergence	NOUN
ma-12	673	14	of	of	ADP
ma-12	673	15	the	the	DET
ma-12	673	16	sequence	sequence	NOUN
ma-12	673	17	of	of	ADP
ma-12	673	18	successive	successive	ADJ
ma-12	673	19	approximation	approximation	NOUN
ma-12	673	20	for	for	ADP
ma-12	673	21	nonexpansive	nonexpansive	ADJ
ma-12	673	22	mappings	mapping	NOUN
ma-12	673	23	,	,	PUNCT
ma-12	673	24	bull	bull	NOUN
ma-12	673	25	.	.	PUNCT
ma-12	674	1	amer.math	amer.math	PROPN
ma-12	674	2	.	.	PUNCT
ma-12	675	1	soc	soc	PROPN
ma-12	675	2	.	.	PUNCT
ma-12	676	1	73	73	NUM
ma-12	676	2	(	(	PUNCT
ma-12	676	3	1967	1967	NUM
ma-12	676	4	)	)	PUNCT
ma-12	676	5	591	591	NUM
ma-12	676	6	-	-	SYM
ma-12	676	7	597.[11	597.[11	NUM
ma-12	676	8	]	]	X
ma-12	676	9	m.o	m.o	PROPN
ma-12	676	10	.	.	PROPN
ma-12	676	11	osilike	osilike	PROPN
ma-12	676	12	,	,	PUNCT
ma-12	676	13	s.c	s.c	PROPN
ma-12	676	14	.	.	PROPN
ma-12	676	15	aniagbosor	aniagbosor	PROPN
ma-12	676	16	,	,	PUNCT
ma-12	676	17	weak	weak	ADJ
ma-12	676	18	and	and	CCONJ
ma-12	676	19	strong	strong	ADJ
ma-12	676	20	convergence	convergence	NOUN
ma-12	676	21	theorems	theorem	NOUN
ma-12	676	22	for	for	ADP
ma-12	676	23	fixed	fix	VERB
ma-12	676	24	points	point	NOUN
ma-12	676	25	of	of	ADP
ma-12	676	26	asymptotically	asymptotically	ADV
ma-12	676	27	nonex	nonex	ADJ
ma-12	676	28	-	-	PUNCT
ma-12	676	29	pensive	pensive	ADJ
ma-12	676	30	mappings	mapping	NOUN
ma-12	676	31	,	,	PUNCT
ma-12	676	32	math	math	NOUN
ma-12	676	33	.	.	PUNCT
ma-12	677	1	computer	computer	NOUN
ma-12	677	2	model	model	NOUN
ma-12	677	3	.	.	PUNCT
ma-12	678	1	32	32	NUM
ma-12	678	2	(	(	PUNCT
ma-12	678	3	2000	2000	NUM
ma-12	678	4	)	)	PUNCT
ma-12	678	5	1181?1191	1181?1191	NUM
ma-12	678	6	.	.	PUNCT
ma-12	679	1	https://doi.org/10.1016/s0895-7177(00	https://doi.org/10.1016/s0895-7177(00	PROPN
ma-12	679	2	)	)	PUNCT
ma-12	679	3	00199	00199	NUM
ma-12	679	4	-	-	PUNCT
ma-12	679	5	0.[12	0.[12	PROPN
ma-12	679	6	]	]	X
ma-12	679	7	b.	b.	PROPN
ma-12	679	8	e.	e.	PROPN
ma-12	679	9	rhoades	rhoades	PROPN
ma-12	679	10	,	,	PUNCT
ma-12	679	11	fixed	fix	VERB
ma-12	679	12	point	point	NOUN
ma-12	679	13	iteration	iteration	NOUN
ma-12	679	14	for	for	ADP
ma-12	679	15	certain	certain	ADJ
ma-12	679	16	nonlinear	nonlinear	ADJ
ma-12	679	17	mappings	mapping	NOUN
ma-12	679	18	,	,	PUNCT
ma-12	679	19	j.	j.	PROPN
ma-12	679	20	math	math	PROPN
ma-12	679	21	.	.	PUNCT
ma-12	680	1	anal	anal	PROPN
ma-12	680	2	.	.	PUNCT
ma-12	680	3	appl	appl	PROPN
ma-12	680	4	.	.	PUNCT
ma-12	680	5	,	,	PUNCT
ma-12	680	6	183(1994	183(1994	NUM
ma-12	680	7	)	)	PUNCT
ma-12	680	8	,	,	PUNCT
ma-12	680	9	118	118	NUM
ma-12	680	10	-	-	SYM
ma-12	680	11	120.[13	120.[13	PROPN
ma-12	680	12	]	]	X
ma-12	680	13	j.	j.	PROPN
ma-12	680	14	schu	schu	PROPN
ma-12	680	15	,	,	PUNCT
ma-12	680	16	weak	weak	ADJ
ma-12	680	17	and	and	CCONJ
ma-12	680	18	strong	strong	ADJ
ma-12	680	19	convergence	convergence	NOUN
ma-12	680	20	theorems	theorem	NOUN
ma-12	680	21	for	for	ADP
ma-12	680	22	fixed	fix	VERB
ma-12	680	23	point	point	NOUN
ma-12	680	24	of	of	ADP
ma-12	680	25	asymptotically	asymptotically	ADV
ma-12	680	26	nonexpansive	nonexpansive	ADJ
ma-12	680	27	mappings	mapping	NOUN
ma-12	680	28	,	,	PUNCT
ma-12	680	29	bull.austral	bull.austral	PROPN
ma-12	680	30	.	.	PUNCT
ma-12	680	31	math	math	NOUN
ma-12	680	32	.	.	PUNCT
ma-12	681	1	soc	soc	PROPN
ma-12	681	2	.	.	PUNCT
ma-12	682	1	43	43	NUM
ma-12	682	2	(	(	PUNCT
ma-12	682	3	1991	1991	NUM
ma-12	682	4	)	)	PUNCT
ma-12	682	5	,	,	PUNCT
ma-12	683	1	153	153	NUM
ma-12	683	2	-	-	SYM
ma-12	683	3	159.[14	159.[14	PROPN
ma-12	683	4	]	]	PUNCT
ma-12	683	5	k.	k.	NOUN
ma-12	683	6	sithikul	sithikul	PROPN
ma-12	683	7	,	,	PUNCT
ma-12	683	8	s.	s.	PROPN
ma-12	683	9	saejung	saejung	PROPN
ma-12	683	10	,	,	PUNCT
ma-12	683	11	convergence	convergence	NOUN
ma-12	683	12	theorems	theorem	NOUN
ma-12	683	13	for	for	ADP
ma-12	683	14	a	a	DET
ma-12	683	15	finite	finite	ADJ
ma-12	683	16	family	family	NOUN
ma-12	683	17	of	of	ADP
ma-12	683	18	nonexpansive	nonexpansive	ADJ
ma-12	683	19	and	and	CCONJ
ma-12	683	20	asymptotically	asymptotically	ADV
ma-12	683	21	nonexpansivemappings	nonexpansivemapping	NOUN
ma-12	683	22	,	,	PUNCT
ma-12	683	23	acta	acta	PROPN
ma-12	683	24	univ	univ	PROPN
ma-12	683	25	.	.	PUNCT
ma-12	684	1	palack	palack	NOUN
ma-12	684	2	.	.	PUNCT
ma-12	685	1	olomuc	olomuc	PROPN
ma-12	685	2	.	.	PUNCT
ma-12	686	1	math	math	NOUN
ma-12	686	2	.	.	PUNCT
ma-12	687	1	48	48	NUM
ma-12	687	2	(	(	PUNCT
ma-12	687	3	2009	2009	NUM
ma-12	687	4	)	)	PUNCT
ma-12	687	5	,	,	PUNCT
ma-12	687	6	139	139	NUM
ma-12	687	7	-	-	SYM
ma-12	687	8	152.[15	152.[15	NUM
ma-12	687	9	]	]	PUNCT
ma-12	687	10	w.	w.	PROPN
ma-12	687	11	takahashi	takahashi	PROPN
ma-12	687	12	,	,	PUNCT
ma-12	687	13	g.e	g.e	PROPN
ma-12	687	14	.	.	PUNCT
ma-12	687	15	kim	kim	PROPN
ma-12	687	16	,	,	PUNCT
ma-12	687	17	approximating	approximate	VERB
ma-12	687	18	fixed	fix	VERB
ma-12	687	19	points	point	NOUN
ma-12	687	20	of	of	ADP
ma-12	687	21	nonexpansive	nonexpansive	ADJ
ma-12	687	22	mappings	mapping	NOUN
ma-12	687	23	in	in	ADP
ma-12	687	24	banach	banach	NOUN
ma-12	687	25	spaces	space	NOUN
ma-12	687	26	,	,	PUNCT
ma-12	687	27	math	math	NOUN
ma-12	687	28	.	.	PUNCT
ma-12	688	1	japon	japon	PROPN
ma-12	688	2	.	.	PUNCT
ma-12	689	1	48(1998	48(1998	NUM
ma-12	689	2	)	)	PUNCT
ma-12	690	1	,	,	PUNCT
ma-12	690	2	1	1	NUM
ma-12	690	3	-	-	SYM
ma-12	690	4	9.[16	9.[16	NUM
ma-12	690	5	]	]	X
ma-12	690	6	k.k	k.k	PROPN
ma-12	690	7	.	.	PROPN
ma-12	690	8	tan	tan	PROPN
ma-12	690	9	,	,	PUNCT
ma-12	690	10	h.k	h.k	PROPN
ma-12	690	11	.	.	PROPN
ma-12	690	12	xu	xu	PROPN
ma-12	690	13	,	,	PUNCT
ma-12	690	14	approximating	approximate	VERB
ma-12	690	15	fixed	fix	VERB
ma-12	690	16	point	point	NOUN
ma-12	690	17	of	of	ADP
ma-12	690	18	nonexpansive	nonexpansive	ADJ
ma-12	690	19	mappings	mapping	NOUN
ma-12	690	20	by	by	ADP
ma-12	690	21	the	the	DET
ma-12	690	22	lshikawa	lshikawa	PROPN
ma-12	690	23	iteration	iteration	NOUN
ma-12	690	24	process	process	NOUN
ma-12	690	25	,	,	PUNCT
ma-12	690	26	j.	j.	PROPN
ma-12	690	27	math.anal	math.anal	PROPN
ma-12	690	28	.	.	PUNCT
ma-12	690	29	appl	appl	PROPN
ma-12	690	30	.	.	PROPN
ma-12	690	31	178	178	NUM
ma-12	690	32	(	(	PUNCT
ma-12	690	33	1993	1993	NUM
ma-12	690	34	)	)	PUNCT
ma-12	690	35	,	,	PUNCT
ma-12	690	36	301	301	NUM
ma-12	690	37	-	-	SYM
ma-12	690	38	308.[17	308.[17	NUM
ma-12	690	39	]	]	PUNCT
ma-12	690	40	l.	l.	PROPN
ma-12	690	41	wang	wang	PROPN
ma-12	690	42	,	,	PUNCT
ma-12	690	43	strong	strong	ADJ
ma-12	690	44	and	and	CCONJ
ma-12	690	45	weak	weak	ADJ
ma-12	690	46	convergence	convergence	NOUN
ma-12	690	47	theorems	theorem	NOUN
ma-12	690	48	for	for	ADP
ma-12	690	49	common	common	ADJ
ma-12	690	50	fixed	fix	VERB
ma-12	690	51	points	point	NOUN
ma-12	690	52	of	of	ADP
ma-12	690	53	nonself	nonself	PRON
ma-12	690	54	asymptotically	asymptotically	ADV
ma-12	690	55	nonexpansivemappings	nonexpansivemapping	NOUN
ma-12	690	56	,	,	PUNCT
ma-12	690	57	j.	j.	PROPN
ma-12	690	58	math	math	PROPN
ma-12	690	59	.	.	PUNCT
ma-12	691	1	anal	anal	PROPN
ma-12	691	2	.	.	PUNCT
ma-12	691	3	appl	appl	PROPN
ma-12	691	4	.	.	PROPN
ma-12	692	1	323	323	NUM
ma-12	692	2	(	(	PUNCT
ma-12	692	3	2006	2006	NUM
ma-12	692	4	)	)	PUNCT
ma-12	693	1	550?557	550?557	PROPN
ma-12	693	2	.	.	PUNCT
ma-12	693	3	https://doi.org/10.1016/j.jmaa.2005.10.062.[18	https://doi.org/10.1016/j.jmaa.2005.10.062.[18	PROPN
ma-12	693	4	]	]	X
ma-12	693	5	e.	e.	PROPN
ma-12	693	6	yolacan	yolacan	PROPN
ma-12	693	7	,	,	PUNCT
ma-12	693	8	h.	h.	PROPN
ma-12	693	9	kiziltune	kiziltune	PROPN
ma-12	693	10	,	,	PUNCT
ma-12	693	11	on	on	ADP
ma-12	693	12	convergence	convergence	NOUN
ma-12	693	13	theorems	theorem	NOUN
ma-12	693	14	for	for	ADP
ma-12	693	15	total	total	ADJ
ma-12	693	16	asymptotically	asymptotically	ADV
ma-12	693	17	nonexpansive	nonexpansive	ADJ
ma-12	693	18	nonself	nonself	PROPN
ma-12	693	19	mappings	mapping	VERB
ma-12	693	20	inbanach	inbanach	ADJ
ma-12	693	21	space	space	NOUN
ma-12	693	22	,	,	PUNCT
ma-12	693	23	j.	j.	PROPN
ma-12	693	24	nonlinear	nonlinear	PROPN
ma-12	693	25	sci	sci	PROPN
ma-12	693	26	.	.	PUNCT
ma-12	693	27	appl	appl	PROPN
ma-12	693	28	.	.	PROPN
ma-12	693	29	5	5	NUM
ma-12	693	30	(	(	PUNCT
ma-12	693	31	2012	2012	NUM
ma-12	693	32	)	)	PUNCT
ma-12	693	33	,	,	PUNCT
ma-12	693	34	389?402.[19	389?402.[19	NUM
ma-12	693	35	]	]	X
ma-12	693	36	d.i	d.i	PROPN
ma-12	693	37	.	.	PROPN
ma-12	693	38	igbokwe	igbokwe	PROPN
ma-12	693	39	,	,	PUNCT
ma-12	693	40	s.j	s.j	PROPN
ma-12	693	41	.	.	PROPN
ma-12	693	42	uko	uko	PROPN
ma-12	693	43	,	,	PUNCT
ma-12	693	44	weak	weak	ADJ
ma-12	693	45	and	and	CCONJ
ma-12	693	46	strong	strong	ADJ
ma-12	693	47	convergence	convergence	NOUN
ma-12	693	48	theorems	theorem	NOUN
ma-12	693	49	for	for	ADP
ma-12	693	50	approximating	approximate	VERB
ma-12	693	51	fixed	fix	VERB
ma-12	693	52	points	point	NOUN
ma-12	693	53	of	of	ADP
ma-12	693	54	nonexpansivemappings	nonexpansivemapping	NOUN
ma-12	693	55	using	use	VERB
ma-12	693	56	composite	composite	ADJ
ma-12	693	57	hybrid	hybrid	ADJ
ma-12	693	58	iteration	iteration	NOUN
ma-12	693	59	method	method	NOUN
ma-12	693	60	,	,	PUNCT
ma-12	693	61	j.	j.	PROPN
ma-12	693	62	nig	nig	PROPN
ma-12	693	63	.	.	PUNCT
ma-12	694	1	math	math	PROPN
ma-12	694	2	.	.	PUNCT
ma-12	695	1	soc	soc	PROPN
ma-12	695	2	.	.	PUNCT
ma-12	696	1	33	33	NUM
ma-12	696	2	(	(	PUNCT
ma-12	696	3	2014	2014	NUM
ma-12	696	4	)	)	PUNCT
ma-12	696	5	,	,	PUNCT
ma-12	696	6	129	129	NUM
ma-12	696	7	-	-	SYM
ma-12	696	8	144.[20	144.[20	NUM
ma-12	696	9	]	]	X
ma-12	696	10	d.i	d.i	PROPN
ma-12	696	11	.	.	PROPN
ma-12	696	12	igbokwe	igbokwe	PROPN
ma-12	696	13	,	,	PUNCT
ma-12	696	14	s.j	s.j	PROPN
ma-12	696	15	.	.	PROPN
ma-12	696	16	uko	uko	PROPN
ma-12	696	17	,	,	PUNCT
ma-12	696	18	weak	weak	ADJ
ma-12	696	19	and	and	CCONJ
ma-12	696	20	strong	strong	ADJ
ma-12	696	21	convergence	convergence	NOUN
ma-12	696	22	of	of	ADP
ma-12	696	23	hybrid	hybrid	ADJ
ma-12	696	24	iteration	iteration	NOUN
ma-12	696	25	methods	method	NOUN
ma-12	696	26	for	for	ADP
ma-12	696	27	fixed	fix	VERB
ma-12	696	28	points	point	NOUN
ma-12	696	29	of	of	ADP
ma-12	696	30	asymptoticallynonexpansive	asymptoticallynonexpansive	ADJ
ma-12	696	31	mappings	mapping	NOUN
ma-12	696	32	,	,	PUNCT
ma-12	696	33	adv	adv	PROPN
ma-12	696	34	.	.	PUNCT
ma-12	696	35	fixed	fix	VERB
ma-12	696	36	point	point	NOUN
ma-12	696	37	theory	theory	NOUN
ma-12	696	38	.	.	PUNCT
ma-12	697	1	5	5	NUM
ma-12	697	2	(	(	PUNCT
ma-12	697	3	2015	2015	NUM
ma-12	697	4	)	)	PUNCT
ma-12	697	5	,	,	PUNCT
ma-12	697	6	120	120	NUM
ma-12	697	7	-	-	SYM
ma-12	697	8	134.[21	134.[21	NUM
ma-12	697	9	]	]	PUNCT
ma-12	697	10	p.	p.	NOUN
ma-12	697	11	wojtaszczyk	wojtaszczyk	NOUN
ma-12	697	12	,	,	PUNCT
ma-12	697	13	banach	banach	NOUN
ma-12	697	14	space	space	NOUN
ma-12	697	15	for	for	ADP
ma-12	697	16	analyst	analyst	NOUN
ma-12	697	17	,	,	PUNCT
ma-12	697	18	cambridge	cambridge	PROPN
ma-12	697	19	university	university	PROPN
ma-12	697	20	press	press	NOUN
ma-12	697	21	,	,	PUNCT
ma-12	697	22	1991	1991	NUM
ma-12	697	23	.	.	PUNCT
ma-12	698	1	https://doi.org/10.1155/fpta/2006/10673	https://doi.org/10.1155/fpta/2006/10673	PROPN
ma-12	698	2	https://doi.org/10.1016/s0022-247x(03)00061-1	https://doi.org/10.1016/s0022-247x(03)00061-1	PROPN
ma-12	698	3	https://doi.org/10.1081/nfa-120039611	https://doi.org/10.1081/nfa-120039611	NOUN
ma-12	698	4	https://doi.org/10.1081/nfa-120039611	https://doi.org/10.1081/nfa-120039611	NOUN
ma-12	698	5	https://doi.org/10.1016/j.jmaa.2006.09.023	https://doi.org/10.1016/j.jmaa.2006.09.023	PROPN
ma-12	698	6	https://doi.org/10.1016/s0362-546x(99)00200-x	https://doi.org/10.1016/s0362-546x(99)00200-x	NOUN
ma-12	698	7	https://doi.org/10.1016/s0362-546x(99)00200-x	https://doi.org/10.1016/s0362-546x(99)00200-x	NOUN
ma-12	698	8	https://doi.org/10.1090/s0002-9939-1972-0298500-3	https://doi.org/10.1090/s0002-9939-1972-0298500-3	PROPN
ma-12	698	9	https://doi.org/10.1016/j.aml.2011.06.022	https://doi.org/10.1016/j.aml.2011.06.022	NOUN
ma-12	698	10	https://doi.org/10.1186/1687-1812-2012-224	https://doi.org/10.1186/1687-1812-2012-224	NOUN
ma-12	698	11	https://doi.org/10.1016/s0895-7177(00)00199-0	https://doi.org/10.1016/s0895-7177(00)00199-0	PROPN
ma-12	698	12	https://doi.org/10.1016/s0895-7177(00)00199-0	https://doi.org/10.1016/s0895-7177(00)00199-0	VERB
ma-12	698	13	https://doi.org/10.1016/j.jmaa.2005.10.062	https://doi.org/10.1016/j.jmaa.2005.10.062	NUM
ma-12	698	14	1	1	NUM
ma-12	698	15	.	.	PUNCT
ma-12	699	1	introduction	introduction	NOUN
ma-12	699	2	2	2	NUM
ma-12	699	3	.	.	PUNCT
ma-12	699	4	preliminary	preliminary	ADJ
ma-12	699	5	3	3	NUM
ma-12	699	6	.	.	PUNCT
ma-12	699	7	main	main	ADJ
ma-12	699	8	results	result	NOUN
ma-12	699	9	references	reference	NOUN