id	sid	tid	token	lemma	pos
ma-221	1	1	2024	2024	NUM
ma-221	1	2	ada	ada	PROPN
ma-221	1	3	academica	academica	PROPN
ma-221	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-221	1	5	.	.	PUNCT
ma-221	2	1	j.	j.	PROPN
ma-221	2	2	math	math	PROPN
ma-221	2	3	.	.	PUNCT
ma-221	3	1	anal	anal	ADJ
ma-221	3	2	.	.	PUNCT
ma-221	4	1	4	4	NUM
ma-221	4	2	(	(	PUNCT
ma-221	4	3	2024	2024	NUM
ma-221	4	4	)	)	PUNCT
ma-221	4	5	13doi	13doi	NOUN
ma-221	4	6	:	:	PUNCT
ma-221	4	7	10.28924	10.28924	NUM
ma-221	4	8	/	/	SYM
ma-221	4	9	ada	ada	PROPN
ma-221	4	10	/	/	SYM
ma-221	4	11	ma.4.13	ma.4.13	PROPN
ma-221	4	12	best	good	ADJ
ma-221	4	13	proximity	proximity	NOUN
ma-221	4	14	point	point	NOUN
ma-221	4	15	of	of	ADP
ma-221	4	16	generalized	generalized	ADJ
ma-221	4	17	θ	θ	PROPN
ma-221	4	18	−	−	X
ma-221	4	19	φ−proximal	φ−proximal	ADJ
ma-221	4	20	non	non	ADJ
ma-221	4	21	-	-	ADJ
ma-221	4	22	self	self	ADJ
ma-221	4	23	contractions	contraction	NOUN
ma-221	4	24	mohamed	mohamed	PROPN
ma-221	4	25	rossafi1,∗	rossafi1,∗	PROPN
ma-221	4	26	,	,	PUNCT
ma-221	4	27	abdelkarim	abdelkarim	PROPN
ma-221	4	28	kari2	kari2	X
ma-221	4	29	1faculty	1faculty	NUM
ma-221	4	30	of	of	ADP
ma-221	4	31	sciences	sciences	PROPN
ma-221	4	32	dhar	dhar	PROPN
ma-221	4	33	el	el	PROPN
ma-221	4	34	mahraz	mahraz	PROPN
ma-221	4	35	,	,	PUNCT
ma-221	4	36	university	university	NOUN
ma-221	4	37	sidi	sidi	NOUN
ma-221	4	38	mohamed	mohamed	PROPN
ma-221	4	39	ben	ben	PROPN
ma-221	4	40	abdellah	abdellah	PROPN
ma-221	4	41	,	,	PUNCT
ma-221	4	42	fes	fes	PROPN
ma-221	4	43	,	,	PUNCT
ma-221	4	44	morocco	morocco	PROPN
ma-221	4	45	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-221	5	1	2faculty	2faculty	NUM
ma-221	5	2	of	of	ADP
ma-221	5	3	sciences	sciences	PROPN
ma-221	5	4	ben	ben	PROPN
ma-221	5	5	m’sik	m’sik	PROPN
ma-221	5	6	,	,	PUNCT
ma-221	5	7	hassan	hassan	PROPN
ma-221	5	8	ii	ii	PROPN
ma-221	5	9	university	university	PROPN
ma-221	5	10	,	,	PUNCT
ma-221	5	11	casablanca	casablanca	PROPN
ma-221	5	12	,	,	PUNCT
ma-221	5	13	morocco	morocco	PROPN
ma-221	5	14	abdkrimkariprofes@gmail.com	abdkrimkariprofes@gmail.com	PUNCT
ma-221	6	1	∗correspondence	∗correspondence	NOUN
ma-221	6	2	:	:	PUNCT
ma-221	6	3	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-221	7	1	abstract	abstract	ADJ
ma-221	7	2	.	.	PUNCT
ma-221	8	1	in	in	ADP
ma-221	8	2	this	this	DET
ma-221	8	3	manuscript	manuscript	NOUN
ma-221	8	4	,	,	PUNCT
ma-221	8	5	motivated	motivated	ADJ
ma-221	8	6	and	and	CCONJ
ma-221	8	7	inspired	inspire	VERB
ma-221	8	8	by	by	ADP
ma-221	8	9	results	result	NOUN
ma-221	8	10	of	of	ADP
ma-221	8	11	best	good	ADJ
ma-221	8	12	proximity	proximity	NOUN
ma-221	8	13	point	point	NOUN
ma-221	8	14	of	of	ADP
ma-221	8	15	generalized	generalized	ADJ
ma-221	8	16	f	f	PROPN
ma-221	8	17	-proximal	-proximal	PROPN
ma-221	8	18	non	non	ADJ
ma-221	8	19	-	-	ADJ
ma-221	8	20	self	self	ADJ
ma-221	8	21	contractions	contraction	NOUN
ma-221	8	22	,	,	PUNCT
ma-221	8	23	we	we	PRON
ma-221	8	24	introduce	introduce	VERB
ma-221	8	25	the	the	DET
ma-221	8	26	concept	concept	NOUN
ma-221	8	27	of	of	ADP
ma-221	8	28	generalized	generalized	ADJ
ma-221	8	29	θ−φ−proximal	θ−φ−proximal	ADJ
ma-221	8	30	contractionand	contractionand	NOUN
ma-221	8	31	prove	prove	VERB
ma-221	8	32	new	new	ADJ
ma-221	8	33	best	good	ADJ
ma-221	8	34	proximity	proximity	NOUN
ma-221	8	35	results	result	NOUN
ma-221	8	36	for	for	ADP
ma-221	8	37	these	these	DET
ma-221	8	38	contractions	contraction	NOUN
ma-221	8	39	in	in	ADP
ma-221	8	40	the	the	DET
ma-221	8	41	setting	setting	NOUN
ma-221	8	42	of	of	ADP
ma-221	8	43	a	a	DET
ma-221	8	44	metric	metric	ADJ
ma-221	8	45	space	space	NOUN
ma-221	8	46	.	.	PUNCT
ma-221	9	1	our	our	PRON
ma-221	9	2	resultsgeneralize	resultsgeneralize	NOUN
ma-221	9	3	and	and	CCONJ
ma-221	9	4	extend	extend	VERB
ma-221	9	5	many	many	ADJ
ma-221	9	6	recent	recent	ADJ
ma-221	9	7	results	result	NOUN
ma-221	9	8	appearing	appear	VERB
ma-221	9	9	in	in	ADP
ma-221	9	10	the	the	DET
ma-221	9	11	literature	literature	NOUN
ma-221	9	12	.	.	PUNCT
ma-221	10	1	an	an	DET
ma-221	10	2	example	example	NOUN
ma-221	10	3	is	be	AUX
ma-221	10	4	being	be	AUX
ma-221	10	5	given	give	VERB
ma-221	10	6	todemonstrate	todemonstrate	ADP
ma-221	10	7	the	the	DET
ma-221	10	8	usefulness	usefulness	NOUN
ma-221	10	9	of	of	ADP
ma-221	10	10	our	our	PRON
ma-221	10	11	results	result	NOUN
ma-221	10	12	.	.	PUNCT
ma-221	11	1	1	1	X
ma-221	11	2	.	.	X
ma-221	11	3	introduction	introduction	NOUN
ma-221	11	4	it	it	PRON
ma-221	11	5	is	be	AUX
ma-221	11	6	well	well	ADV
ma-221	11	7	known	know	VERB
ma-221	11	8	that	that	SCONJ
ma-221	11	9	the	the	DET
ma-221	11	10	banach	banach	NOUN
ma-221	11	11	contraction	contraction	NOUN
ma-221	11	12	theorem	theorem	VERB
ma-221	11	13	is	be	AUX
ma-221	11	14	the	the	DET
ma-221	11	15	first	first	ADJ
ma-221	11	16	outstanding	outstanding	ADJ
ma-221	11	17	result	result	NOUN
ma-221	11	18	in	in	ADP
ma-221	11	19	thefield	thefield	NOUN
ma-221	11	20	of	of	ADP
ma-221	11	21	the	the	DET
ma-221	11	22	fixed	fix	VERB
ma-221	11	23	point	point	NOUN
ma-221	11	24	theory	theory	NOUN
ma-221	11	25	that	that	PRON
ma-221	11	26	ensure	ensure	VERB
ma-221	11	27	the	the	DET
ma-221	11	28	existence	existence	NOUN
ma-221	11	29	of	of	ADP
ma-221	11	30	unique	unique	ADJ
ma-221	11	31	fixed	fix	VERB
ma-221	11	32	point	point	NOUN
ma-221	11	33	in	in	ADP
ma-221	11	34	complete	complete	ADJ
ma-221	11	35	metricspaces	metricspace	NOUN
ma-221	11	36	.	.	PUNCT
ma-221	12	1	due	due	ADP
ma-221	12	2	to	to	ADP
ma-221	12	3	its	its	PRON
ma-221	12	4	importance	importance	NOUN
ma-221	12	5	,	,	PUNCT
ma-221	12	6	various	various	ADJ
ma-221	12	7	mathematics	mathematic	NOUN
ma-221	12	8	steadied	steady	VERB
ma-221	12	9	many	many	ADJ
ma-221	12	10	interesting	interesting	ADJ
ma-221	12	11	extensions	extension	NOUN
ma-221	12	12	andgeneralizations	andgeneralization	NOUN
ma-221	13	1	[	[	X
ma-221	13	2	7,8,12,14	7,8,12,14	NOUN
ma-221	13	3	]	]	PUNCT
ma-221	13	4	.	.	PUNCT
ma-221	14	1	one	one	NUM
ma-221	14	2	of	of	ADP
ma-221	14	3	the	the	DET
ma-221	14	4	famous	famous	ADJ
ma-221	14	5	generalizations	generalization	NOUN
ma-221	14	6	of	of	ADP
ma-221	14	7	the	the	DET
ma-221	14	8	banach	banach	NOUN
ma-221	14	9	contraction	contraction	NOUN
ma-221	14	10	principle	principle	NOUN
ma-221	14	11	[	[	X
ma-221	14	12	2]for	2]for	ADJ
ma-221	14	13	existence	existence	NOUN
ma-221	14	14	of	of	ADP
ma-221	14	15	fixed	fix	VERB
ma-221	14	16	point	point	NOUN
ma-221	14	17	for	for	ADP
ma-221	14	18	self	self	NOUN
ma-221	14	19	-	-	PUNCT
ma-221	14	20	mapping	mapping	NOUN
ma-221	14	21	on	on	ADP
ma-221	14	22	metric	metric	ADJ
ma-221	14	23	space	space	NOUN
ma-221	14	24	is	be	AUX
ma-221	14	25	the	the	DET
ma-221	14	26	theorem	theorem	NOUN
ma-221	14	27	by	by	ADP
ma-221	14	28	zheng	zheng	PROPN
ma-221	14	29	et	et	PROPN
ma-221	14	30	al	al	PROPN
ma-221	14	31	.	.	PUNCT
ma-221	15	1	[	[	X
ma-221	15	2	14	14	NUM
ma-221	15	3	]	]	X
ma-221	15	4	andthe	andthe	ADJ
ma-221	15	5	contraction	contraction	NOUN
ma-221	15	6	introduced	introduce	VERB
ma-221	15	7	by	by	ADP
ma-221	15	8	jleli	jleli	ADJ
ma-221	15	9	and	and	CCONJ
ma-221	15	10	samet	samet	VERB
ma-221	15	11	in	in	ADP
ma-221	15	12	[	[	X
ma-221	15	13	6].best	6].best	NUM
ma-221	15	14	proximity	proximity	NOUN
ma-221	15	15	point	point	NOUN
ma-221	15	16	theorem	theorem	VERB
ma-221	15	17	analyses	analyse	VERB
ma-221	15	18	the	the	DET
ma-221	15	19	condition	condition	NOUN
ma-221	15	20	under	under	ADP
ma-221	15	21	which	which	PRON
ma-221	15	22	the	the	DET
ma-221	15	23	optimisation	optimisation	NOUN
ma-221	15	24	problem	problem	NOUN
ma-221	15	25	,	,	PUNCT
ma-221	15	26	namely	namely	ADV
ma-221	15	27	infx∈a	infx∈a	ADJ
ma-221	15	28	d(x	d(x	PROPN
ma-221	15	29	,	,	PUNCT
ma-221	15	30	t	t	PROPN
ma-221	15	31	x	x	PROPN
ma-221	15	32	)	)	PUNCT
ma-221	15	33	,	,	PUNCT
ma-221	15	34	has	have	VERB
ma-221	15	35	a	a	DET
ma-221	15	36	solution	solution	NOUN
ma-221	15	37	.	.	PUNCT
ma-221	16	1	the	the	DET
ma-221	16	2	point	point	NOUN
ma-221	16	3	x	x	PUNCT
ma-221	16	4	is	be	AUX
ma-221	16	5	called	call	VERB
ma-221	16	6	the	the	DET
ma-221	16	7	best	good	ADJ
ma-221	16	8	proximity	proximity	NOUN
ma-221	16	9	of	of	ADP
ma-221	16	10	t	t	PROPN
ma-221	16	11	:	:	PUNCT
ma-221	16	12	a	a	DET
ma-221	16	13	→	→	SYM
ma-221	16	14	b	b	NOUN
ma-221	16	15	,	,	PUNCT
ma-221	16	16	if	if	SCONJ
ma-221	16	17	d(x	d(x	PROPN
ma-221	16	18	,	,	PUNCT
ma-221	16	19	t	t	NOUN
ma-221	16	20	x	x	X
ma-221	16	21	)	)	PUNCT
ma-221	16	22	=	=	SYM
ma-221	17	1	d(a	d(a	PROPN
ma-221	17	2	,	,	PUNCT
ma-221	17	3	b	b	NOUN
ma-221	17	4	)	)	PUNCT
ma-221	17	5	,	,	PUNCT
ma-221	17	6	where	where	SCONJ
ma-221	17	7	{	{	PUNCT
ma-221	17	8	d(a	d(a	PROPN
ma-221	17	9	,	,	PUNCT
ma-221	17	10	b	b	NOUN
ma-221	17	11	)	)	PUNCT
ma-221	17	12	=	=	SYM
ma-221	17	13	inf	inf	PROPN
ma-221	17	14	d(x	d(x	PROPN
ma-221	17	15	,	,	PUNCT
ma-221	17	16	y	y	PROPN
ma-221	17	17	)	)	PUNCT
ma-221	17	18	:	:	PUNCT
ma-221	18	1	x	x	X
ma-221	18	2	∈	∈	PROPN
ma-221	18	3	a	a	X
ma-221	18	4	,	,	PUNCT
ma-221	18	5	y	y	PROPN
ma-221	18	6	∈	∈	PROPN
ma-221	18	7	b	b	PROPN
ma-221	18	8	}	}	PUNCT
ma-221	18	9	.	.	PUNCT
ma-221	19	1	note	note	VERB
ma-221	19	2	that	that	SCONJ
ma-221	19	3	the	the	DET
ma-221	19	4	best	good	ADJ
ma-221	19	5	proximitypoint	proximitypoint	NOUN
ma-221	19	6	reduces	reduce	VERB
ma-221	19	7	to	to	ADP
ma-221	19	8	a	a	DET
ma-221	19	9	fixed	fix	VERB
ma-221	19	10	point	point	NOUN
ma-221	19	11	if	if	SCONJ
ma-221	19	12	t	t	PROPN
ma-221	19	13	is	be	AUX
ma-221	19	14	a	a	DET
ma-221	19	15	self	self	NOUN
ma-221	19	16	-	-	PUNCT
ma-221	19	17	mapping	mapping	NOUN
ma-221	19	18	.	.	PUNCT
ma-221	20	1	various	various	ADJ
ma-221	20	2	best	good	ADJ
ma-221	20	3	proximity	proximity	NOUN
ma-221	20	4	point	point	NOUN
ma-221	20	5	results	result	NOUN
ma-221	20	6	wereestablished	wereestablishe	VERB
ma-221	20	7	on	on	ADP
ma-221	20	8	such	such	ADJ
ma-221	20	9	spaces	space	NOUN
ma-221	20	10	[	[	X
ma-221	20	11	1	1	NUM
ma-221	20	12	,	,	PUNCT
ma-221	20	13	9	9	NUM
ma-221	20	14	,	,	PUNCT
ma-221	20	15	12].sankar	12].sankar	PROPN
ma-221	20	16	raj	raj	NOUN
ma-221	21	1	[	[	X
ma-221	21	2	10	10	NUM
ma-221	21	3	]	]	PUNCT
ma-221	21	4	and	and	CCONJ
ma-221	21	5	zhang	zhang	PROPN
ma-221	21	6	et	et	PROPN
ma-221	21	7	al	al	PROPN
ma-221	21	8	.	.	PUNCT
ma-221	22	1	[	[	X
ma-221	22	2	13	13	NUM
ma-221	22	3	]	]	PUNCT
ma-221	22	4	defined	define	VERB
ma-221	22	5	the	the	DET
ma-221	22	6	notion	notion	NOUN
ma-221	22	7	of	of	ADP
ma-221	22	8	p−property	p−property	NOUN
ma-221	22	9	and	and	CCONJ
ma-221	22	10	weak	weak	ADJ
ma-221	22	11	p−propertyrespectively	p−propertyrespectively	ADV
ma-221	22	12	.	.	PUNCT
ma-221	23	1	beg	beg	VERB
ma-221	23	2	et	et	PROPN
ma-221	23	3	al	al	PROPN
ma-221	23	4	.	.	PUNCT
ma-221	24	1	[	[	X
ma-221	24	2	4	4	X
ma-221	24	3	]	]	PUNCT
ma-221	24	4	defined	define	VERB
ma-221	24	5	the	the	DET
ma-221	24	6	concept	concept	NOUN
ma-221	24	7	of	of	ADP
ma-221	24	8	generalized	generalized	ADJ
ma-221	24	9	f	f	PROPN
ma-221	24	10	-proximal	-proximal	PROPN
ma-221	24	11	non	non	ADJ
ma-221	24	12	-	-	ADJ
ma-221	24	13	self	self	ADJ
ma-221	24	14	contractions	contraction	NOUN
ma-221	24	15	andobtained	andobtaine	VERB
ma-221	24	16	some	some	DET
ma-221	24	17	best	good	ADJ
ma-221	24	18	proximity	proximity	NOUN
ma-221	24	19	point	point	NOUN
ma-221	24	20	theorems	theorem	NOUN
ma-221	24	21	for	for	ADP
ma-221	24	22	self-mappings.in	self-mappings.in	NOUN
ma-221	24	23	this	this	DET
ma-221	24	24	paper	paper	NOUN
ma-221	24	25	,	,	PUNCT
ma-221	24	26	inspired	inspire	VERB
ma-221	24	27	by	by	ADP
ma-221	24	28	the	the	DET
ma-221	24	29	idea	idea	NOUN
ma-221	24	30	of	of	ADP
ma-221	24	31	generalized	generalized	ADJ
ma-221	24	32	f	f	PROPN
ma-221	24	33	-proximal	-proximal	PROPN
ma-221	24	34	non	non	ADJ
ma-221	24	35	-	-	ADJ
ma-221	24	36	self	self	ADJ
ma-221	24	37	contractions	contraction	NOUN
ma-221	24	38	,	,	PUNCT
ma-221	24	39	introducedby	introducedby	NOUN
ma-221	24	40	beg	beg	VERB
ma-221	24	41	et	et	PROPN
ma-221	24	42	al	al	PROPN
ma-221	24	43	.	.	PUNCT
ma-221	25	1	[	[	X
ma-221	25	2	4	4	X
ma-221	25	3	]	]	PUNCT
ma-221	25	4	in	in	ADP
ma-221	25	5	metric	metric	ADJ
ma-221	25	6	spaces	space	NOUN
ma-221	25	7	,	,	PUNCT
ma-221	25	8	we	we	PRON
ma-221	25	9	prove	prove	VERB
ma-221	25	10	a	a	DET
ma-221	25	11	new	new	ADJ
ma-221	25	12	existence	existence	NOUN
ma-221	25	13	of	of	ADP
ma-221	25	14	best	good	ADJ
ma-221	25	15	proximity	proximity	NOUN
ma-221	25	16	point	point	NOUN
ma-221	25	17	for	for	ADP
ma-221	25	18	generalized	generalized	ADJ
ma-221	25	19	received	receive	VERB
ma-221	25	20	:	:	PUNCT
ma-221	25	21	21	21	NUM
ma-221	25	22	jan	jan	PROPN
ma-221	25	23	2024	2024	NUM
ma-221	25	24	.	.	PUNCT
ma-221	26	1	key	key	ADJ
ma-221	26	2	words	word	NOUN
ma-221	26	3	and	and	CCONJ
ma-221	26	4	phrases	phrase	NOUN
ma-221	26	5	.	.	PUNCT
ma-221	27	1	p	p	PRON
ma-221	27	2	-property	-property	NOUN
ma-221	27	3	,	,	PUNCT
ma-221	27	4	best	good	ADJ
ma-221	27	5	proximity	proximity	NOUN
ma-221	27	6	point	point	NOUN
ma-221	27	7	,	,	PUNCT
ma-221	27	8	generalized	generalized	ADJ
ma-221	27	9	θ	θ	PROPN
ma-221	27	10	−	−	PROPN
ma-221	27	11	φ	φ	NUM
ma-221	27	12	-	-	ADJ
ma-221	27	13	proximal	proximal	ADJ
ma-221	27	14	contraction.1	contraction.1	PROPN
ma-221	27	15	https://adac.ee	https://adac.ee	PROPN
ma-221	27	16	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	27	17	https://orcid.org/0000-0002-5662-6921	https://orcid.org/0000-0002-5662-6921	PROPN
ma-221	27	18	https://orcid.org/0000-0003-0088-9404	https://orcid.org/0000-0003-0088-9404	PROPN
ma-221	27	19	eur	eur	PROPN
ma-221	27	20	.	.	PUNCT
ma-221	28	1	j.	j.	PROPN
ma-221	28	2	math	math	PROPN
ma-221	28	3	.	.	PUNCT
ma-221	29	1	anal	anal	PROPN
ma-221	29	2	.	.	PUNCT
ma-221	30	1	10.28924	10.28924	NUM
ma-221	30	2	/	/	SYM
ma-221	30	3	ada	ada	PROPN
ma-221	30	4	/	/	SYM
ma-221	30	5	ma.4.13	ma.4.13	PROPN
ma-221	31	1	2	2	NUM
ma-221	31	2	θ	θ	NOUN
ma-221	31	3	−	−	NOUN
ma-221	31	4	φ−proximal	φ−proximal	NUM
ma-221	31	5	contraction	contraction	NOUN
ma-221	31	6	defined	define	VERB
ma-221	31	7	on	on	ADP
ma-221	31	8	a	a	DET
ma-221	31	9	closed	closed	ADJ
ma-221	31	10	subset	subset	NOUN
ma-221	31	11	of	of	ADP
ma-221	31	12	a	a	DET
ma-221	31	13	complete	complete	ADJ
ma-221	31	14	metric	metric	ADJ
ma-221	31	15	space	space	NOUN
ma-221	31	16	.	.	PUNCT
ma-221	32	1	our	our	PRON
ma-221	32	2	theoremsextend	theoremsextend	NOUN
ma-221	32	3	,	,	PUNCT
ma-221	32	4	generalize	generalize	VERB
ma-221	32	5	and	and	CCONJ
ma-221	32	6	improve	improve	VERB
ma-221	32	7	many	many	ADJ
ma-221	32	8	existing	exist	VERB
ma-221	32	9	results	result	NOUN
ma-221	32	10	.	.	PUNCT
ma-221	33	1	2	2	X
ma-221	33	2	.	.	X
ma-221	33	3	preliminaries	preliminary	NOUN
ma-221	33	4	let	let	VERB
ma-221	33	5	(	(	PUNCT
ma-221	33	6	a	a	DET
ma-221	33	7	,	,	PUNCT
ma-221	33	8	b	b	NOUN
ma-221	33	9	)	)	PUNCT
ma-221	33	10	be	be	AUX
ma-221	33	11	a	a	DET
ma-221	33	12	pair	pair	NOUN
ma-221	33	13	of	of	ADP
ma-221	33	14	non	non	ADJ
ma-221	33	15	empty	empty	ADJ
ma-221	33	16	subsets	subset	NOUN
ma-221	33	17	of	of	ADP
ma-221	33	18	a	a	DET
ma-221	33	19	metric	metric	ADJ
ma-221	33	20	space	space	NOUN
ma-221	33	21	(	(	PUNCT
ma-221	33	22	x	x	X
ma-221	33	23	,	,	PUNCT
ma-221	33	24	d	d	NOUN
ma-221	33	25	)	)	PUNCT
ma-221	33	26	.	.	PUNCT
ma-221	34	1	we	we	PRON
ma-221	34	2	adopt	adopt	VERB
ma-221	34	3	the	the	DET
ma-221	34	4	followingnotations	followingnotation	NOUN
ma-221	34	5	:	:	PUNCT
ma-221	35	1	d(a	d(a	PROPN
ma-221	35	2	,	,	PUNCT
ma-221	35	3	b	b	NOUN
ma-221	35	4	)	)	PUNCT
ma-221	35	5	=	=	SYM
ma-221	35	6	{	{	PUNCT
ma-221	35	7	inf	inf	PROPN
ma-221	35	8	d	d	X
ma-221	35	9	(	(	PUNCT
ma-221	35	10	a	a	DET
ma-221	35	11	,	,	PUNCT
ma-221	35	12	b	b	NOUN
ma-221	35	13	)	)	PUNCT
ma-221	35	14	:	:	PUNCT
ma-221	35	15	a	a	DET
ma-221	35	16	∈	∈	PROPN
ma-221	35	17	a	a	PRON
ma-221	35	18	,	,	PUNCT
ma-221	35	19	b	b	PROPN
ma-221	35	20	∈	∈	PROPN
ma-221	35	21	b	b	NOUN
ma-221	35	22	}	}	PUNCT
ma-221	35	23	;	;	PUNCT
ma-221	35	24	a0	a0	PROPN
ma-221	35	25	=	=	SYM
ma-221	35	26	{	{	PUNCT
ma-221	35	27	a	a	PRON
ma-221	35	28	∈	∈	PROPN
ma-221	35	29	a	a	DET
ma-221	35	30	there	there	PRON
ma-221	35	31	exists	exist	VERB
ma-221	35	32	b	b	PROPN
ma-221	35	33	∈	∈	PROPN
ma-221	35	34	a	a	DET
ma-221	35	35	such	such	ADJ
ma-221	35	36	that	that	SCONJ
ma-221	35	37	d	d	NOUN
ma-221	35	38	(	(	PUNCT
ma-221	35	39	a	a	DET
ma-221	35	40	,	,	PUNCT
ma-221	35	41	b	b	NOUN
ma-221	35	42	)	)	PUNCT
ma-221	36	1	=	=	SYM
ma-221	36	2	d	d	PROPN
ma-221	36	3	(	(	PUNCT
ma-221	36	4	a	a	DET
ma-221	36	5	,	,	PUNCT
ma-221	36	6	b	b	NOUN
ma-221	36	7	)	)	PUNCT
ma-221	36	8	}	}	PUNCT
ma-221	36	9	;	;	PUNCT
ma-221	36	10	b0	b0	NOUN
ma-221	36	11	=	=	SYM
ma-221	36	12	{	{	PUNCT
ma-221	36	13	b	b	X
ma-221	36	14	∈	∈	ADP
ma-221	36	15	b	b	NOUN
ma-221	36	16	there	there	PRON
ma-221	36	17	exists	exist	VERB
ma-221	36	18	a	a	DET
ma-221	36	19	∈	∈	NOUN
ma-221	36	20	a	a	DET
ma-221	36	21	such	such	ADJ
ma-221	36	22	that	that	SCONJ
ma-221	36	23	d	d	NOUN
ma-221	36	24	(	(	PUNCT
ma-221	36	25	a	a	DET
ma-221	36	26	,	,	PUNCT
ma-221	36	27	b	b	NOUN
ma-221	36	28	)	)	PUNCT
ma-221	37	1	=	=	SYM
ma-221	37	2	d	d	PROPN
ma-221	37	3	(	(	PUNCT
ma-221	37	4	a	a	DET
ma-221	37	5	,	,	PUNCT
ma-221	37	6	b	b	NOUN
ma-221	37	7	)	)	PUNCT
ma-221	37	8	}	}	PUNCT
ma-221	37	9	.	.	PUNCT
ma-221	38	1	definition	definition	NOUN
ma-221	38	2	2.1	2.1	NUM
ma-221	38	3	.	.	PUNCT
ma-221	39	1	[	[	X
ma-221	39	2	5	5	X
ma-221	39	3	]	]	PUNCT
ma-221	39	4	let	let	VERB
ma-221	39	5	t	t	NOUN
ma-221	39	6	:	:	PUNCT
ma-221	39	7	a	a	PRON
ma-221	39	8	→	→	SYM
ma-221	39	9	b	b	X
ma-221	39	10	be	be	AUX
ma-221	39	11	a	a	DET
ma-221	39	12	mapping	mapping	NOUN
ma-221	39	13	.	.	PUNCT
ma-221	40	1	an	an	DET
ma-221	40	2	element	element	NOUN
ma-221	40	3	x∗	x∗	PROPN
ma-221	40	4	is	be	AUX
ma-221	40	5	said	say	VERB
ma-221	40	6	to	to	PART
ma-221	40	7	be	be	AUX
ma-221	40	8	a	a	DET
ma-221	40	9	best	good	ADJ
ma-221	40	10	proximitypoint	proximitypoint	NOUN
ma-221	40	11	of	of	ADP
ma-221	40	12	t	t	PROPN
ma-221	40	13	if	if	SCONJ
ma-221	40	14	d	d	PROPN
ma-221	40	15	(	(	PUNCT
ma-221	40	16	x∗	x∗	PROPN
ma-221	40	17	,	,	PUNCT
ma-221	40	18	t	t	PROPN
ma-221	40	19	x∗	x∗	X
ma-221	40	20	)	)	PUNCT
ma-221	41	1	=	=	SYM
ma-221	41	2	d	d	X
ma-221	41	3	(	(	PUNCT
ma-221	41	4	a	a	DET
ma-221	41	5	,	,	PUNCT
ma-221	41	6	b	b	NOUN
ma-221	41	7	)	)	PUNCT
ma-221	41	8	.	.	PUNCT
ma-221	42	1	definition	definition	NOUN
ma-221	42	2	2.2	2.2	NUM
ma-221	42	3	.	.	PUNCT
ma-221	43	1	[	[	X
ma-221	43	2	10	10	NUM
ma-221	43	3	]	]	X
ma-221	43	4	let	let	VERB
ma-221	43	5	(	(	PUNCT
ma-221	43	6	a	a	DET
ma-221	43	7	,	,	PUNCT
ma-221	43	8	b	b	NOUN
ma-221	43	9	)	)	PUNCT
ma-221	43	10	be	be	AUX
ma-221	43	11	a	a	DET
ma-221	43	12	pair	pair	NOUN
ma-221	43	13	of	of	ADP
ma-221	43	14	non	non	ADJ
ma-221	43	15	empty	empty	ADJ
ma-221	43	16	subsets	subset	NOUN
ma-221	43	17	of	of	ADP
ma-221	43	18	a	a	DET
ma-221	43	19	metric	metric	ADJ
ma-221	43	20	space	space	NOUN
ma-221	43	21	(	(	PUNCT
ma-221	43	22	x	x	X
ma-221	43	23	,	,	PUNCT
ma-221	43	24	d	d	NOUN
ma-221	43	25	)	)	PUNCT
ma-221	43	26	such	such	ADJ
ma-221	43	27	that	that	SCONJ
ma-221	43	28	a0	a0	PROPN
ma-221	43	29	is	be	AUX
ma-221	43	30	non	non	X
ma-221	43	31	empty	empty	ADJ
ma-221	43	32	.	.	PUNCT
ma-221	44	1	then	then	ADV
ma-221	44	2	the	the	DET
ma-221	44	3	pair	pair	NOUN
ma-221	44	4	(	(	PUNCT
ma-221	44	5	a	a	DET
ma-221	44	6	,	,	PUNCT
ma-221	44	7	b	b	NOUN
ma-221	44	8	)	)	PUNCT
ma-221	44	9	is	be	AUX
ma-221	44	10	to	to	PART
ma-221	44	11	have	have	VERB
ma-221	44	12	p	p	NOUN
ma-221	44	13	-property	-property	NOUN
ma-221	44	14	if	if	SCONJ
ma-221	44	15	and	and	CCONJ
ma-221	44	16	only	only	ADV
ma-221	44	17	ifd	ifd	ADP
ma-221	44	18	(	(	PUNCT
ma-221	44	19	x1	x1	PROPN
ma-221	44	20	,	,	PUNCT
ma-221	44	21	y1	y1	NOUN
ma-221	44	22	)	)	PUNCT
ma-221	44	23	=	=	SYM
ma-221	45	1	d	d	PROPN
ma-221	45	2	(	(	PUNCT
ma-221	45	3	a	a	DET
ma-221	45	4	,	,	PUNCT
ma-221	45	5	b	b	NOUN
ma-221	45	6	)	)	PUNCT
ma-221	45	7	d	d	NOUN
ma-221	45	8	(	(	PUNCT
ma-221	45	9	x2	x2	PROPN
ma-221	45	10	,	,	PUNCT
ma-221	45	11	y2	y2	NOUN
ma-221	45	12	)	)	PUNCT
ma-221	46	1	=	=	SYM
ma-221	47	1	d	d	X
ma-221	47	2	(	(	PUNCT
ma-221	47	3	a	a	DET
ma-221	47	4	,	,	PUNCT
ma-221	47	5	b	b	NOUN
ma-221	47	6	)	)	PUNCT
ma-221	47	7	⇒	⇒	NOUN
ma-221	47	8	d(x1	d(x1	NOUN
ma-221	47	9	,	,	PUNCT
ma-221	47	10	x2	x2	NUM
ma-221	47	11	)	)	PUNCT
ma-221	47	12	=	=	SYM
ma-221	47	13	d(y1	d(y1	NOUN
ma-221	47	14	,	,	PUNCT
ma-221	47	15	y2	y2	PROPN
ma-221	47	16	)	)	PUNCT
ma-221	47	17	where	where	SCONJ
ma-221	47	18	x1	x1	X
ma-221	47	19	,	,	PUNCT
ma-221	47	20	x2	x2	PROPN
ma-221	47	21	∈	∈	PROPN
ma-221	47	22	a0	a0	NOUN
ma-221	47	23	and	and	CCONJ
ma-221	47	24	y1	y1	PROPN
ma-221	47	25	,	,	PUNCT
ma-221	47	26	y2	y2	PROPN
ma-221	47	27	∈	∈	PROPN
ma-221	47	28	b0	b0	NOUN
ma-221	47	29	.	.	PUNCT
ma-221	48	1	definition	definition	NOUN
ma-221	48	2	2.3	2.3	NUM
ma-221	48	3	.	.	PUNCT
ma-221	49	1	[	[	X
ma-221	49	2	3	3	X
ma-221	49	3	]	]	PUNCT
ma-221	49	4	a	a	DET
ma-221	49	5	set	set	NOUN
ma-221	49	6	b	b	NOUN
ma-221	49	7	is	be	AUX
ma-221	49	8	called	call	VERB
ma-221	49	9	approximately	approximately	ADV
ma-221	49	10	compact	compact	ADJ
ma-221	49	11	with	with	ADP
ma-221	49	12	respect	respect	NOUN
ma-221	49	13	to	to	ADP
ma-221	49	14	a	a	DET
ma-221	49	15	if	if	SCONJ
ma-221	49	16	every	every	DET
ma-221	49	17	sequence	sequence	NOUN
ma-221	49	18	{	{	PUNCT
ma-221	49	19	xn	xn	NOUN
ma-221	49	20	}	}	PUNCT
ma-221	49	21	of	of	ADP
ma-221	49	22	b	b	NOUN
ma-221	49	23	with	with	ADP
ma-221	49	24	d(y	d(y	NOUN
ma-221	49	25	,	,	PUNCT
ma-221	49	26	xn)→	xn)→	PROPN
ma-221	49	27	d(y	d(y	PROPN
ma-221	49	28	,	,	PUNCT
ma-221	49	29	b	b	X
ma-221	49	30	)	)	PUNCT
ma-221	49	31	for	for	ADP
ma-221	49	32	some	some	DET
ma-221	49	33	y	y	PROPN
ma-221	49	34	∈	∈	PROPN
ma-221	49	35	a	a	PRON
ma-221	49	36	has	have	AUX
ma-221	49	37	a	a	DET
ma-221	49	38	convergent	convergent	NOUN
ma-221	49	39	subsequence	subsequence	NOUN
ma-221	49	40	.	.	PUNCT
ma-221	50	1	definition	definition	NOUN
ma-221	50	2	2.4	2.4	NUM
ma-221	50	3	.	.	PUNCT
ma-221	51	1	[	[	X
ma-221	51	2	6	6	NUM
ma-221	51	3	]	]	PUNCT
ma-221	51	4	let	let	VERB
ma-221	51	5	θ	θ	PROPN
ma-221	51	6	be	be	AUX
ma-221	51	7	the	the	DET
ma-221	51	8	family	family	NOUN
ma-221	51	9	of	of	ADP
ma-221	51	10	all	all	DET
ma-221	51	11	functions	function	NOUN
ma-221	51	12	θ	θ	NOUN
ma-221	51	13	:	:	PUNCT
ma-221	51	14	]	]	PUNCT
ma-221	51	15	0,+∞	0,+∞	NUM
ma-221	51	16	[	[	PUNCT
ma-221	51	17	→	→	X
ma-221	51	18	]	]	X
ma-221	51	19	1,+∞	1,+∞	NUM
ma-221	51	20	[	[	PUNCT
ma-221	51	21	such	such	ADJ
ma-221	51	22	that	that	SCONJ
ma-221	51	23	(	(	PUNCT
ma-221	51	24	θ1	θ1	NOUN
ma-221	51	25	)	)	PUNCT
ma-221	51	26	θ	θ	PROPN
ma-221	51	27	is	be	AUX
ma-221	51	28	strictly	strictly	ADV
ma-221	51	29	increasing	increase	VERB
ma-221	51	30	;	;	PUNCT
ma-221	51	31	(	(	PUNCT
ma-221	51	32	θ2	θ2	PROPN
ma-221	51	33	)	)	PUNCT
ma-221	51	34	for	for	ADP
ma-221	51	35	each	each	DET
ma-221	51	36	sequence	sequence	NOUN
ma-221	51	37	xn	xn	PROPN
ma-221	51	38	∈	∈	PROPN
ma-221	51	39	]	]	PUNCT
ma-221	51	40	0,+∞	0,+∞	NUM
ma-221	52	1	[	[	X
ma-221	52	2	;	;	PUNCT
ma-221	52	3	lim	lim	PROPN
ma-221	52	4	n→0	n→0	PROPN
ma-221	52	5	xn	xn	PROPN
ma-221	53	1	=	=	SYM
ma-221	53	2	0	0	NUM
ma-221	53	3	,	,	PUNCT
ma-221	53	4	if	if	SCONJ
ma-221	53	5	and	and	CCONJ
ma-221	53	6	only	only	ADV
ma-221	54	1	if	if	SCONJ
ma-221	54	2	lim	lim	PROPN
ma-221	54	3	n→∞	n→∞	NUM
ma-221	54	4	θ	θ	PROPN
ma-221	54	5	(	(	PUNCT
ma-221	54	6	xn	xn	PROPN
ma-221	54	7	)	)	PUNCT
ma-221	54	8	=	=	SYM
ma-221	54	9	1	1	NUM
ma-221	54	10	;	;	PUNCT
ma-221	54	11	(	(	PUNCT
ma-221	54	12	θ3	θ3	NOUN
ma-221	54	13	)	)	PUNCT
ma-221	54	14	θ	θ	PROPN
ma-221	54	15	is	be	AUX
ma-221	54	16	continuous	continuous	ADJ
ma-221	54	17	.	.	PUNCT
ma-221	55	1	definition	definition	NOUN
ma-221	55	2	2.5	2.5	NUM
ma-221	55	3	.	.	PUNCT
ma-221	56	1	[	[	X
ma-221	56	2	14	14	NUM
ma-221	56	3	]	]	PUNCT
ma-221	56	4	let	let	VERB
ma-221	56	5	φ	φ	PROPN
ma-221	56	6	be	be	AUX
ma-221	56	7	the	the	DET
ma-221	56	8	family	family	NOUN
ma-221	56	9	of	of	ADP
ma-221	56	10	all	all	DET
ma-221	56	11	functions	function	NOUN
ma-221	56	12	φ	φ	NOUN
ma-221	56	13	:	:	PUNCT
ma-221	57	1	[	[	X
ma-221	57	2	1,+∞	1,+∞	X
ma-221	57	3	[	[	PUNCT
ma-221	57	4	→	→	X
ma-221	57	5	[	[	X
ma-221	57	6	1,+∞	1,+∞	NUM
ma-221	57	7	[	[	X
ma-221	57	8	,	,	PUNCT
ma-221	57	9	such	such	ADJ
ma-221	57	10	that	that	SCONJ
ma-221	57	11	(	(	PUNCT
ma-221	57	12	φ1	φ1	PROPN
ma-221	57	13	)	)	PUNCT
ma-221	57	14	φ	φ	PROPN
ma-221	57	15	is	be	AUX
ma-221	57	16	increasing	increase	VERB
ma-221	57	17	;	;	PUNCT
ma-221	57	18	(	(	PUNCT
ma-221	57	19	φ2	φ2	PROPN
ma-221	57	20	)	)	PUNCT
ma-221	57	21	for	for	ADP
ma-221	57	22	each	each	DET
ma-221	57	23	t	t	NOUN
ma-221	57	24	∈	∈	PROPN
ma-221	57	25	]	]	PUNCT
ma-221	57	26	1,+∞	1,+∞	NUM
ma-221	57	27	[	[	X
ma-221	57	28	,	,	PUNCT
ma-221	57	29	l	l	NOUN
ma-221	57	30	imn→∞φn(t	imn→∞φn(t	NOUN
ma-221	57	31	)	)	PUNCT
ma-221	57	32	=	=	SYM
ma-221	57	33	1	1	NUM
ma-221	57	34	;	;	PUNCT
ma-221	57	35	(	(	PUNCT
ma-221	57	36	φ3	φ3	NOUN
ma-221	57	37	)	)	PUNCT
ma-221	57	38	φ	φ	PROPN
ma-221	57	39	is	be	AUX
ma-221	57	40	continuous	continuous	ADJ
ma-221	57	41	.	.	PUNCT
ma-221	58	1	lemma	lemma	PROPN
ma-221	58	2	2.6	2.6	NUM
ma-221	58	3	.	.	PUNCT
ma-221	59	1	[	[	X
ma-221	59	2	14	14	NUM
ma-221	59	3	]	]	X
ma-221	59	4	if	if	SCONJ
ma-221	59	5	φ	φ	PROPN
ma-221	59	6	∈	∈	PROPN
ma-221	59	7	φ	φ	X
ma-221	59	8	then	then	ADV
ma-221	59	9	φ(1)=1	φ(1)=1	NUM
ma-221	59	10	,	,	PUNCT
ma-221	59	11	and	and	CCONJ
ma-221	59	12	φ(t	φ(t	PROPN
ma-221	59	13	)	)	PUNCT
ma-221	59	14	<	<	X
ma-221	59	15	t	t	PROPN
ma-221	59	16	.	.	PUNCT
ma-221	60	1	definition	definition	NOUN
ma-221	60	2	2.7	2.7	NUM
ma-221	60	3	.	.	PUNCT
ma-221	61	1	[	[	X
ma-221	61	2	14	14	NUM
ma-221	61	3	]	]	PUNCT
ma-221	61	4	.	.	PUNCT
ma-221	62	1	let	let	VERB
ma-221	62	2	(	(	PUNCT
ma-221	62	3	x	x	X
ma-221	62	4	,	,	PUNCT
ma-221	62	5	d	d	NOUN
ma-221	62	6	)	)	PUNCT
ma-221	62	7	be	be	AUX
ma-221	62	8	a	a	DET
ma-221	62	9	metric	metric	ADJ
ma-221	62	10	space	space	NOUN
ma-221	62	11	and	and	CCONJ
ma-221	62	12	t	t	NOUN
ma-221	62	13	:	:	PUNCT
ma-221	62	14	x	x	X
ma-221	62	15	→	→	PUNCT
ma-221	62	16	x	x	PUNCT
ma-221	62	17	be	be	AUX
ma-221	62	18	a	a	DET
ma-221	62	19	mapping	mapping	NOUN
ma-221	62	20	.	.	PUNCT
ma-221	63	1	t	t	PROPN
ma-221	63	2	is	be	AUX
ma-221	63	3	said	say	VERB
ma-221	63	4	to	to	PART
ma-221	63	5	be	be	AUX
ma-221	63	6	a	a	DET
ma-221	63	7	θ	θ	NOUN
ma-221	63	8	−	−	NOUN
ma-221	63	9	φ−contraction	φ−contraction	NOUN
ma-221	63	10	if	if	SCONJ
ma-221	63	11	there	there	PRON
ma-221	63	12	exist	exist	VERB
ma-221	63	13	θ	θ	PROPN
ma-221	63	14	∈	∈	PROPN
ma-221	63	15	θ	θ	PROPN
ma-221	63	16	and	and	CCONJ
ma-221	63	17	φ	φ	PROPN
ma-221	63	18	∈	∈	PROPN
ma-221	63	19	φ	φ	NOUN
ma-221	63	20	such	such	ADJ
ma-221	63	21	that	that	PRON
ma-221	63	22	for	for	ADP
ma-221	63	23	any	any	DET
ma-221	63	24	x	x	NOUN
ma-221	63	25	,	,	PUNCT
ma-221	63	26	y	y	PROPN
ma-221	63	27	∈	∈	PROPN
ma-221	63	28	x	x	X
ma-221	63	29	,	,	PUNCT
ma-221	63	30	d	d	X
ma-221	63	31	(	(	PUNCT
ma-221	63	32	tx	tx	PROPN
ma-221	63	33	,	,	PUNCT
ma-221	63	34	t	t	PROPN
ma-221	63	35	y	y	PROPN
ma-221	63	36	)	)	PUNCT
ma-221	63	37	>	>	X
ma-221	63	38	0⇒	0⇒	NUM
ma-221	64	1	θ	θ	X
ma-221	65	1	[	[	X
ma-221	65	2	d	d	X
ma-221	65	3	(	(	PUNCT
ma-221	65	4	tx	tx	PROPN
ma-221	65	5	,	,	PUNCT
ma-221	65	6	t	t	PROPN
ma-221	65	7	y	y	PROPN
ma-221	65	8	)	)	PUNCT
ma-221	65	9	]	]	PUNCT
ma-221	65	10	≤	≤	NUM
ma-221	65	11	φ	φ	PROPN
ma-221	65	12	[	[	X
ma-221	65	13	θ	θ	X
ma-221	65	14	(	(	PUNCT
ma-221	65	15	d	d	X
ma-221	65	16	(	(	PUNCT
ma-221	65	17	x	x	NOUN
ma-221	65	18	,	,	PUNCT
ma-221	65	19	y	y	PROPN
ma-221	65	20	)	)	PUNCT
ma-221	65	21	)	)	PUNCT
ma-221	65	22	]	]	PUNCT
ma-221	65	23	,	,	PUNCT
ma-221	65	24	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	65	25	eur	eur	PROPN
ma-221	65	26	.	.	PUNCT
ma-221	66	1	j.	j.	PROPN
ma-221	66	2	math	math	PROPN
ma-221	66	3	.	.	PUNCT
ma-221	67	1	anal	anal	PROPN
ma-221	67	2	.	.	PUNCT
ma-221	68	1	10.28924	10.28924	NUM
ma-221	68	2	/	/	SYM
ma-221	68	3	ada	ada	PROPN
ma-221	68	4	/	/	SYM
ma-221	68	5	ma.4.13	ma.4.13	PROPN
ma-221	68	6	33	33	NUM
ma-221	68	7	.	.	PUNCT
ma-221	69	1	main	main	ADJ
ma-221	69	2	result	result	NOUN
ma-221	69	3	in	in	ADP
ma-221	69	4	this	this	DET
ma-221	69	5	section	section	NOUN
ma-221	69	6	,	,	PUNCT
ma-221	69	7	inspired	inspire	VERB
ma-221	69	8	by	by	ADP
ma-221	69	9	the	the	DET
ma-221	69	10	notion	notion	NOUN
ma-221	69	11	of	of	ADP
ma-221	69	12	f	f	PROPN
ma-221	69	13	-proximal	-proximal	PROPN
ma-221	69	14	contraction	contraction	NOUN
ma-221	69	15	of	of	ADP
ma-221	69	16	the	the	DET
ma-221	69	17	first	first	ADJ
ma-221	69	18	kind	kind	NOUN
ma-221	69	19	and	and	CCONJ
ma-221	69	20	second	second	ADJ
ma-221	69	21	kind	kind	NOUN
ma-221	69	22	,	,	PUNCT
ma-221	69	23	we	we	PRON
ma-221	69	24	introduce	introduce	VERB
ma-221	69	25	new	new	ADJ
ma-221	69	26	generalized	generalize	VERB
ma-221	69	27	θ−	θ−	PROPN
ma-221	69	28	φ	φ	VERB
ma-221	69	29	-	-	ADJ
ma-221	69	30	proximal	proximal	ADJ
ma-221	69	31	first	first	ADJ
ma-221	69	32	kind	kind	NOUN
ma-221	69	33	and	and	CCONJ
ma-221	69	34	second	second	ADJ
ma-221	69	35	kind	kind	NOUN
ma-221	69	36	on	on	ADP
ma-221	69	37	complete	complete	ADJ
ma-221	69	38	metric	metric	ADJ
ma-221	69	39	space	space	NOUN
ma-221	69	40	.	.	PUNCT
ma-221	70	1	definition	definition	NOUN
ma-221	70	2	3.1	3.1	NUM
ma-221	70	3	.	.	PUNCT
ma-221	71	1	the	the	DET
ma-221	71	2	mapping	mapping	NOUN
ma-221	71	3	t	t	NOUN
ma-221	71	4	:	:	PUNCT
ma-221	71	5	a→	a→	PROPN
ma-221	71	6	b	b	NOUN
ma-221	71	7	is	be	AUX
ma-221	71	8	said	say	VERB
ma-221	71	9	to	to	PART
ma-221	71	10	be	be	AUX
ma-221	71	11	a	a	DET
ma-221	71	12	generalized	generalized	ADJ
ma-221	71	13	θ	θ	NOUN
ma-221	71	14	−	−	PROPN
ma-221	71	15	φ	φ	NUM
ma-221	71	16	-	-	ADJ
ma-221	71	17	proximal	proximal	ADJ
ma-221	71	18	contraction	contraction	NOUN
ma-221	71	19	offirst	offirst	ADV
ma-221	71	20	kind	kind	ADV
ma-221	71	21	if	if	SCONJ
ma-221	71	22	there	there	PRON
ma-221	71	23	exist	exist	VERB
ma-221	71	24	θ	θ	PROPN
ma-221	71	25	∈	∈	PROPN
ma-221	71	26	θ	θ	PROPN
ma-221	71	27	,	,	PUNCT
ma-221	71	28	φ	φ	PROPN
ma-221	71	29	∈	∈	PROPN
ma-221	71	30	φ	φ	PROPN
ma-221	71	31	and	and	CCONJ
ma-221	71	32	a	a	DET
ma-221	71	33	,	,	PUNCT
ma-221	71	34	b	b	NOUN
ma-221	71	35	,	,	PUNCT
ma-221	71	36	c	c	NOUN
ma-221	71	37	,	,	PUNCT
ma-221	71	38	h	h	PROPN
ma-221	71	39	≥	≥	NOUN
ma-221	71	40	0	0	NUM
ma-221	71	41	with	with	ADP
ma-221	71	42	a	a	DET
ma-221	71	43	+	+	NOUN
ma-221	71	44	b	b	NOUN
ma-221	71	45	+	+	CCONJ
ma-221	71	46	+2ch	+2ch	NUM
ma-221	71	47	,	,	PUNCT
ma-221	71	48	c	c	NOUN
ma-221	71	49	6=	6=	NUM
ma-221	71	50	1	1	NUM
ma-221	71	51	such	such	ADJ
ma-221	71	52	thatd	thatd	PROPN
ma-221	71	53	(	(	PUNCT
ma-221	71	54	u1	u1	PROPN
ma-221	71	55	,	,	PUNCT
ma-221	71	56	t	t	NOUN
ma-221	71	57	v1	v1	NOUN
ma-221	71	58	)	)	PUNCT
ma-221	72	1	=	=	SYM
ma-221	72	2	d	d	PROPN
ma-221	72	3	(	(	PUNCT
ma-221	72	4	a	a	DET
ma-221	72	5	,	,	PUNCT
ma-221	72	6	b	b	NOUN
ma-221	72	7	)	)	PUNCT
ma-221	72	8	d	d	NOUN
ma-221	72	9	(	(	PUNCT
ma-221	72	10	u2	u2	PROPN
ma-221	72	11	,	,	PUNCT
ma-221	72	12	t	t	NOUN
ma-221	72	13	v2	v2	PROPN
ma-221	72	14	)	)	PUNCT
ma-221	72	15	=	=	SYM
ma-221	73	1	d	d	X
ma-221	73	2	(	(	PUNCT
ma-221	73	3	a	a	DET
ma-221	73	4	,	,	PUNCT
ma-221	73	5	b	b	NOUN
ma-221	73	6	)	)	PUNCT
ma-221	73	7	⇒	⇒	NOUN
ma-221	73	8	θ(d(u1	θ(d(u1	NOUN
ma-221	73	9	,	,	PUNCT
ma-221	73	10	u2	u2	NOUN
ma-221	73	11	)	)	PUNCT
ma-221	73	12	)	)	PUNCT
ma-221	73	13	≤	≤	NUM
ma-221	74	1	φ	φ	PROPN
ma-221	74	2	[	[	X
ma-221	74	3	θ	θ	X
ma-221	74	4	[	[	X
ma-221	74	5	ad	ad	X
ma-221	74	6	(	(	PUNCT
ma-221	74	7	v1	v1	NOUN
ma-221	74	8	,	,	PUNCT
ma-221	74	9	v2	v2	PROPN
ma-221	74	10	)	)	PUNCT
ma-221	75	1	+	+	CCONJ
ma-221	75	2	bd	bd	PROPN
ma-221	75	3	(	(	PUNCT
ma-221	75	4	u1	u1	NOUN
ma-221	75	5	,	,	PUNCT
ma-221	75	6	v1	v1	NOUN
ma-221	75	7	)	)	PUNCT
ma-221	75	8	+	+	CCONJ
ma-221	75	9	cd	cd	PROPN
ma-221	75	10	(	(	PUNCT
ma-221	75	11	u2	u2	PROPN
ma-221	75	12	,	,	PUNCT
ma-221	75	13	v2	v2	PROPN
ma-221	75	14	)	)	PUNCT
ma-221	76	1	+	+	NUM
ma-221	76	2	h	h	NOUN
ma-221	76	3	(	(	PUNCT
ma-221	76	4	d	d	X
ma-221	76	5	(	(	PUNCT
ma-221	76	6	v1	v1	NOUN
ma-221	76	7	,	,	PUNCT
ma-221	76	8	u2	u2	NOUN
ma-221	76	9	)	)	PUNCT
ma-221	76	10	+	+	PROPN
ma-221	77	1	d	d	X
ma-221	77	2	(	(	PUNCT
ma-221	77	3	v2	v2	PROPN
ma-221	77	4	,	,	PUNCT
ma-221	77	5	u1))]]for	u1))]]for	PROPN
ma-221	77	6	all	all	DET
ma-221	77	7	u1	u1	NOUN
ma-221	77	8	,	,	PUNCT
ma-221	77	9	u2	u2	NOUN
ma-221	77	10	,	,	PUNCT
ma-221	77	11	v1	v1	NOUN
ma-221	77	12	,	,	PUNCT
ma-221	77	13	v2	v2	PROPN
ma-221	77	14	∈	∈	PROPN
ma-221	77	15	a	a	PRON
ma-221	77	16	and	and	CCONJ
ma-221	77	17	u1	u1	NOUN
ma-221	77	18	6=	6=	SYM
ma-221	77	19	v1	v1	NOUN
ma-221	77	20	.	.	PUNCT
ma-221	78	1	definition	definition	NOUN
ma-221	78	2	3.2	3.2	NUM
ma-221	78	3	.	.	PUNCT
ma-221	79	1	the	the	DET
ma-221	79	2	mapping	mapping	NOUN
ma-221	79	3	t	t	NOUN
ma-221	79	4	:	:	PUNCT
ma-221	79	5	a→	a→	PROPN
ma-221	79	6	b	b	NOUN
ma-221	79	7	is	be	AUX
ma-221	79	8	said	say	VERB
ma-221	79	9	to	to	PART
ma-221	79	10	be	be	AUX
ma-221	79	11	a	a	DET
ma-221	79	12	generalized	generalized	ADJ
ma-221	79	13	θ	θ	NOUN
ma-221	79	14	−	−	PROPN
ma-221	79	15	φ	φ	NUM
ma-221	79	16	-	-	ADJ
ma-221	79	17	proximal	proximal	ADJ
ma-221	79	18	contraction	contraction	NOUN
ma-221	79	19	ofsecond	ofsecond	NOUN
ma-221	79	20	kind	kind	INTJ
ma-221	79	21	if	if	SCONJ
ma-221	79	22	there	there	PRON
ma-221	79	23	exist	exist	VERB
ma-221	79	24	θ	θ	PROPN
ma-221	79	25	∈	∈	PROPN
ma-221	79	26	θ	θ	PROPN
ma-221	79	27	,	,	PUNCT
ma-221	79	28	φ	φ	PROPN
ma-221	79	29	∈	∈	PROPN
ma-221	79	30	φ	φ	PROPN
ma-221	79	31	and	and	CCONJ
ma-221	79	32	a	a	DET
ma-221	79	33	,	,	PUNCT
ma-221	79	34	b	b	NOUN
ma-221	79	35	,	,	PUNCT
ma-221	79	36	c	c	NOUN
ma-221	79	37	,	,	PUNCT
ma-221	79	38	h	h	PROPN
ma-221	79	39	≥	≥	NOUN
ma-221	79	40	0	0	NUM
ma-221	79	41	with	with	ADP
ma-221	79	42	a	a	DET
ma-221	79	43	+	+	NOUN
ma-221	79	44	b	b	NOUN
ma-221	79	45	+	+	CCONJ
ma-221	79	46	+2ch	+2ch	NUM
ma-221	79	47	,	,	PUNCT
ma-221	79	48	c	c	NOUN
ma-221	79	49	6=	6=	NUM
ma-221	79	50	1	1	NUM
ma-221	79	51	such	such	ADJ
ma-221	79	52	thatd	thatd	PROPN
ma-221	79	53	(	(	PUNCT
ma-221	79	54	u1	u1	PROPN
ma-221	79	55	,	,	PUNCT
ma-221	79	56	t	t	NOUN
ma-221	79	57	v1	v1	NOUN
ma-221	79	58	)	)	PUNCT
ma-221	80	1	=	=	SYM
ma-221	80	2	d	d	PROPN
ma-221	80	3	(	(	PUNCT
ma-221	80	4	a	a	DET
ma-221	80	5	,	,	PUNCT
ma-221	80	6	b	b	NOUN
ma-221	80	7	)	)	PUNCT
ma-221	80	8	d	d	NOUN
ma-221	80	9	(	(	PUNCT
ma-221	80	10	u2	u2	PROPN
ma-221	80	11	,	,	PUNCT
ma-221	80	12	t	t	NOUN
ma-221	80	13	v2	v2	PROPN
ma-221	80	14	)	)	PUNCT
ma-221	80	15	=	=	SYM
ma-221	81	1	d	d	X
ma-221	81	2	(	(	PUNCT
ma-221	81	3	a	a	DET
ma-221	81	4	,	,	PUNCT
ma-221	81	5	b	b	NOUN
ma-221	81	6	)	)	PUNCT
ma-221	81	7	⇒	⇒	PROPN
ma-221	81	8	θ(d(tu1	θ(d(tu1	PROPN
ma-221	81	9	,	,	PUNCT
ma-221	81	10	t	t	PROPN
ma-221	81	11	u2	u2	PROPN
ma-221	81	12	)	)	PUNCT
ma-221	81	13	)	)	PUNCT
ma-221	81	14	≤	≤	NUM
ma-221	82	1	φ	φ	PROPN
ma-221	83	1	[	[	X
ma-221	83	2	θ	θ	X
ma-221	84	1	[	[	X
ma-221	84	2	ad	ad	X
ma-221	84	3	(	(	PUNCT
ma-221	84	4	tv1	tv1	PROPN
ma-221	84	5	,	,	PUNCT
ma-221	84	6	t	t	PROPN
ma-221	84	7	v2	v2	PROPN
ma-221	84	8	)	)	PUNCT
ma-221	84	9	+	+	CCONJ
ma-221	84	10	bd	bd	PROPN
ma-221	84	11	(	(	PUNCT
ma-221	84	12	tu1	tu1	PROPN
ma-221	84	13	,	,	PUNCT
ma-221	84	14	t	t	NOUN
ma-221	84	15	v1	v1	NOUN
ma-221	84	16	)	)	PUNCT
ma-221	84	17	+	+	CCONJ
ma-221	84	18	cd	cd	PROPN
ma-221	84	19	(	(	PUNCT
ma-221	84	20	tu2	tu2	PROPN
ma-221	84	21	,	,	PUNCT
ma-221	84	22	t	t	PROPN
ma-221	84	23	v2	v2	PROPN
ma-221	84	24	)	)	PUNCT
ma-221	85	1	+	+	NUM
ma-221	85	2	h	h	NOUN
ma-221	85	3	(	(	PUNCT
ma-221	85	4	d	d	X
ma-221	85	5	(	(	PUNCT
ma-221	85	6	tv1	tv1	PROPN
ma-221	85	7	,	,	PUNCT
ma-221	85	8	t	t	PROPN
ma-221	85	9	u2	u2	PROPN
ma-221	85	10	)	)	PUNCT
ma-221	85	11	+	+	PROPN
ma-221	86	1	d	d	PROPN
ma-221	86	2	(	(	PUNCT
ma-221	86	3	tv2	tv2	NOUN
ma-221	86	4	,	,	PUNCT
ma-221	86	5	t	t	PROPN
ma-221	86	6	u1))]]for	u1))]]for	PROPN
ma-221	86	7	all	all	DET
ma-221	86	8	u1	u1	NOUN
ma-221	86	9	,	,	PUNCT
ma-221	86	10	u2	u2	NOUN
ma-221	86	11	,	,	PUNCT
ma-221	86	12	v1	v1	NOUN
ma-221	86	13	,	,	PUNCT
ma-221	86	14	v2	v2	PROPN
ma-221	86	15	∈	∈	PROPN
ma-221	86	16	a	a	PRON
ma-221	86	17	and	and	CCONJ
ma-221	86	18	tu1	tu1	PROPN
ma-221	86	19	6=	6=	NUM
ma-221	86	20	tv1	tv1	PROPN
ma-221	86	21	.	.	PUNCT
ma-221	87	1	theorem	theorem	VERB
ma-221	87	2	3.3	3.3	NUM
ma-221	87	3	.	.	PUNCT
ma-221	88	1	let	let	VERB
ma-221	88	2	(	(	PUNCT
ma-221	88	3	x	x	NOUN
ma-221	88	4	,	,	PUNCT
ma-221	88	5	d	d	NOUN
ma-221	88	6	)	)	PUNCT
ma-221	88	7	be	be	AUX
ma-221	88	8	a	a	DET
ma-221	88	9	complete	complete	ADJ
ma-221	88	10	metric	metric	ADJ
ma-221	88	11	space	space	NOUN
ma-221	88	12	and	and	CCONJ
ma-221	88	13	(	(	PUNCT
ma-221	88	14	a	a	DET
ma-221	88	15	,	,	PUNCT
ma-221	88	16	b	b	NOUN
ma-221	88	17	)	)	PUNCT
ma-221	88	18	be	be	AUX
ma-221	88	19	a	a	DET
ma-221	88	20	pair	pair	NOUN
ma-221	88	21	of	of	ADP
ma-221	88	22	non	non	ADJ
ma-221	88	23	-	-	ADJ
ma-221	88	24	void	void	ADJ
ma-221	88	25	closed	closed	ADJ
ma-221	88	26	subsets	subset	NOUN
ma-221	88	27	of	of	ADP
ma-221	88	28	(	(	PUNCT
ma-221	88	29	x	x	X
ma-221	88	30	,	,	PUNCT
ma-221	88	31	d	d	NOUN
ma-221	88	32	)	)	PUNCT
ma-221	88	33	.	.	PUNCT
ma-221	89	1	if	if	SCONJ
ma-221	89	2	b	b	NOUN
ma-221	89	3	is	be	AUX
ma-221	89	4	approximately	approximately	ADV
ma-221	89	5	compact	compact	ADJ
ma-221	89	6	with	with	ADP
ma-221	89	7	respect	respect	NOUN
ma-221	89	8	to	to	ADP
ma-221	89	9	a	a	PRON
ma-221	89	10	and	and	CCONJ
ma-221	89	11	t	t	NOUN
ma-221	89	12	:	:	PUNCT
ma-221	89	13	a	a	DET
ma-221	89	14	→	→	SYM
ma-221	89	15	b	b	NOUN
ma-221	89	16	satisfy	satisfy	NOUN
ma-221	89	17	the	the	DET
ma-221	89	18	following	follow	VERB
ma-221	89	19	conditions	condition	NOUN
ma-221	89	20	:(	:(	PUNCT
ma-221	90	1	i	i	PRON
ma-221	90	2	)	)	PUNCT
ma-221	90	3	t	t	PROPN
ma-221	90	4	(	(	PUNCT
ma-221	90	5	a0	a0	PROPN
ma-221	90	6	)	)	PUNCT
ma-221	90	7	∈	∈	PROPN
ma-221	90	8	b0	b0	NOUN
ma-221	90	9	and	and	CCONJ
ma-221	90	10	the	the	DET
ma-221	90	11	pair	pair	NOUN
ma-221	90	12	(	(	PUNCT
ma-221	90	13	a	a	DET
ma-221	90	14	,	,	PUNCT
ma-221	90	15	b	b	NOUN
ma-221	90	16	)	)	PUNCT
ma-221	90	17	satisfies	satisfy	VERB
ma-221	90	18	the	the	DET
ma-221	90	19	weak	weak	ADJ
ma-221	90	20	p	p	NOUN
ma-221	90	21	-property;(ii	-property;(ii	PROPN
ma-221	90	22	)	)	PUNCT
ma-221	90	23	t	t	PROPN
ma-221	90	24	is	be	AUX
ma-221	90	25	a	a	DET
ma-221	90	26	generalized	generalized	ADJ
ma-221	90	27	θ	θ	NOUN
ma-221	90	28	−	−	PROPN
ma-221	90	29	φ	φ	NUM
ma-221	90	30	-	-	ADJ
ma-221	90	31	proximal	proximal	ADJ
ma-221	90	32	contraction	contraction	NOUN
ma-221	90	33	of	of	ADP
ma-221	90	34	first	first	ADJ
ma-221	90	35	kind	kind	NOUN
ma-221	90	36	.	.	PUNCT
ma-221	91	1	then	then	ADV
ma-221	91	2	there	there	PRON
ma-221	91	3	exists	exist	VERB
ma-221	91	4	a	a	DET
ma-221	91	5	unique	unique	ADJ
ma-221	91	6	u	u	NOUN
ma-221	91	7	∈	∈	PROPN
ma-221	91	8	a	a	DET
ma-221	91	9	such	such	ADJ
ma-221	91	10	that	that	SCONJ
ma-221	91	11	d(u	d(u	PROPN
ma-221	91	12	,	,	PUNCT
ma-221	91	13	tu	tu	PROPN
ma-221	91	14	)	)	PUNCT
ma-221	91	15	=	=	SYM
ma-221	92	1	d(a	d(a	PROPN
ma-221	92	2	,	,	PUNCT
ma-221	92	3	b	b	NOUN
ma-221	92	4	)	)	PUNCT
ma-221	92	5	.	.	PUNCT
ma-221	93	1	in	in	ADP
ma-221	93	2	addition	addition	NOUN
ma-221	93	3	,	,	PUNCT
ma-221	93	4	for	for	ADP
ma-221	93	5	any	any	DET
ma-221	93	6	fixed	fix	VERB
ma-221	93	7	element	element	NOUN
ma-221	93	8	u0	u0	PROPN
ma-221	93	9	∈	∈	PROPN
ma-221	93	10	a0	a0	PROPN
ma-221	93	11	,	,	PUNCT
ma-221	93	12	sequence	sequence	NOUN
ma-221	93	13	{	{	PUNCT
ma-221	93	14	un	un	PROPN
ma-221	93	15	}	}	PUNCT
ma-221	93	16	defined	define	VERB
ma-221	93	17	by	by	ADP
ma-221	93	18	d(un+1	d(un+1	PROPN
ma-221	93	19	,	,	PUNCT
ma-221	93	20	t	t	PROPN
ma-221	93	21	un	un	PROPN
ma-221	93	22	)	)	PUNCT
ma-221	93	23	=	=	SYM
ma-221	93	24	d(a	d(a	PROPN
ma-221	93	25	,	,	PUNCT
ma-221	93	26	b	b	NOUN
ma-221	93	27	)	)	PUNCT
ma-221	93	28	,	,	PUNCT
ma-221	93	29	converges	converge	VERB
ma-221	93	30	to	to	ADP
ma-221	93	31	the	the	DET
ma-221	93	32	proximity	proximity	NOUN
ma-221	93	33	point	point	NOUN
ma-221	93	34	.	.	PUNCT
ma-221	94	1	proof	proof	NOUN
ma-221	94	2	.	.	PUNCT
ma-221	95	1	choose	choose	VERB
ma-221	95	2	an	an	DET
ma-221	95	3	element	element	NOUN
ma-221	95	4	u0	u0	PROPN
ma-221	95	5	∈	∈	PROPN
ma-221	95	6	a0	a0	PROPN
ma-221	95	7	.	.	PUNCT
ma-221	96	1	as	as	ADP
ma-221	96	2	,	,	PUNCT
ma-221	96	3	t	t	PROPN
ma-221	96	4	(	(	PUNCT
ma-221	96	5	a0	a0	PROPN
ma-221	96	6	)	)	PUNCT
ma-221	96	7	∈	∈	PROPN
ma-221	96	8	b0	b0	NOUN
ma-221	96	9	,	,	PUNCT
ma-221	96	10	therefore	therefore	ADV
ma-221	96	11	there	there	PRON
ma-221	96	12	is	be	VERB
ma-221	96	13	an	an	DET
ma-221	96	14	element	element	ADJ
ma-221	96	15	u1	u1	NOUN
ma-221	96	16	∈	∈	PROPN
ma-221	96	17	a0satisfying	a0satisfying	NOUN
ma-221	96	18	d(u1	d(u1	NOUN
ma-221	96	19	,	,	PUNCT
ma-221	96	20	t	t	NOUN
ma-221	96	21	u0	u0	PROPN
ma-221	96	22	)	)	PUNCT
ma-221	97	1	=	=	SYM
ma-221	97	2	d(a	d(a	PROPN
ma-221	97	3	,	,	PUNCT
ma-221	97	4	b).since	b).since	PROPN
ma-221	97	5	t	t	PROPN
ma-221	97	6	(	(	PUNCT
ma-221	97	7	a0	a0	PROPN
ma-221	97	8	)	)	PUNCT
ma-221	97	9	∈	∈	PROPN
ma-221	97	10	b0	b0	NOUN
ma-221	97	11	,	,	PUNCT
ma-221	97	12	there	there	PRON
ma-221	97	13	exists	exist	VERB
ma-221	97	14	u2	u2	PROPN
ma-221	97	15	∈	∈	PROPN
ma-221	97	16	a0	a0	NOUN
ma-221	97	17	such	such	ADJ
ma-221	97	18	that	that	DET
ma-221	97	19	d(u2	d(u2	NOUN
ma-221	97	20	,	,	PUNCT
ma-221	97	21	t	t	NOUN
ma-221	97	22	u1	u1	NOUN
ma-221	97	23	)	)	PUNCT
ma-221	98	1	=	=	SYM
ma-221	99	1	d(a	d(a	PROPN
ma-221	99	2	,	,	PUNCT
ma-221	99	3	b	b	NOUN
ma-221	99	4	)	)	PUNCT
ma-221	99	5	.	.	PUNCT
ma-221	100	1	again	again	ADV
ma-221	100	2	,	,	PUNCT
ma-221	100	3	since	since	SCONJ
ma-221	100	4	t	t	PROPN
ma-221	100	5	(	(	PUNCT
ma-221	100	6	a0	a0	PROPN
ma-221	100	7	)	)	PUNCT
ma-221	100	8	∈	∈	PROPN
ma-221	100	9	b0	b0	NOUN
ma-221	100	10	,	,	PUNCT
ma-221	100	11	there	there	PRON
ma-221	100	12	exists	exist	VERB
ma-221	100	13	u3	u3	PROPN
ma-221	100	14	∈	∈	PROPN
ma-221	100	15	a0	a0	NOUN
ma-221	100	16	such	such	ADJ
ma-221	100	17	that	that	DET
ma-221	100	18	d(u3	d(u3	NOUN
ma-221	100	19	,	,	PUNCT
ma-221	100	20	t	t	PROPN
ma-221	100	21	u2	u2	PROPN
ma-221	100	22	)	)	PUNCT
ma-221	100	23	=	=	SYM
ma-221	101	1	d(a	d(a	PROPN
ma-221	101	2	,	,	PUNCT
ma-221	101	3	b	b	NOUN
ma-221	101	4	)	)	PUNCT
ma-221	101	5	.	.	PUNCT
ma-221	102	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	102	2	eur	eur	PROPN
ma-221	102	3	.	.	PUNCT
ma-221	103	1	j.	j.	PROPN
ma-221	103	2	math	math	PROPN
ma-221	103	3	.	.	PUNCT
ma-221	104	1	anal	anal	PROPN
ma-221	104	2	.	.	PUNCT
ma-221	105	1	10.28924	10.28924	NUM
ma-221	105	2	/	/	SYM
ma-221	105	3	ada	ada	PROPN
ma-221	105	4	/	/	SYM
ma-221	105	5	ma.4.13	ma.4.13	PROPN
ma-221	105	6	4continuing	4continue	VERB
ma-221	105	7	this	this	DET
ma-221	105	8	process	process	NOUN
ma-221	105	9	,	,	PUNCT
ma-221	105	10	by	by	ADP
ma-221	105	11	induction	induction	NOUN
ma-221	105	12	,	,	PUNCT
ma-221	105	13	we	we	PRON
ma-221	105	14	construct	construct	VERB
ma-221	105	15	a	a	DET
ma-221	105	16	sequence	sequence	NOUN
ma-221	105	17	xn	xn	PROPN
ma-221	105	18	∈	∈	PROPN
ma-221	105	19	a0	a0	PROPN
ma-221	105	20	such	such	ADJ
ma-221	105	21	that	that	SCONJ
ma-221	105	22	d	d	X
ma-221	105	23	(	(	PUNCT
ma-221	105	24	un+1	un+1	PROPN
ma-221	105	25	,	,	PUNCT
ma-221	105	26	t	t	PROPN
ma-221	105	27	un	un	PROPN
ma-221	105	28	)	)	PUNCT
ma-221	105	29	=	=	SYM
ma-221	105	30	d(a	d(a	PROPN
ma-221	105	31	,	,	PUNCT
ma-221	105	32	b),∀n	b),∀n	PROPN
ma-221	105	33	∈	∈	PROPN
ma-221	105	34	n.	n.	NOUN
ma-221	105	35	since	since	SCONJ
ma-221	105	36	(	(	PUNCT
ma-221	105	37	a	a	DET
ma-221	105	38	,	,	PUNCT
ma-221	105	39	b	b	NOUN
ma-221	105	40	)	)	PUNCT
ma-221	105	41	satisfies	satisfy	VERB
ma-221	105	42	the	the	DET
ma-221	105	43	p	p	PROPN
ma-221	105	44	property	property	NOUN
ma-221	105	45	,	,	PUNCT
ma-221	105	46	we	we	PRON
ma-221	105	47	conclude	conclude	VERB
ma-221	105	48	that	that	SCONJ
ma-221	105	49	d(un	d(un	PROPN
ma-221	105	50	,	,	PUNCT
ma-221	105	51	un+1	un+1	NOUN
ma-221	105	52	)	)	PUNCT
ma-221	105	53	=	=	SYM
ma-221	105	54	d(tun	d(tun	PROPN
ma-221	105	55	,	,	PUNCT
ma-221	105	56	t	t	PROPN
ma-221	105	57	un+1),∀n	un+1),∀n	NOUN
ma-221	105	58	∈	∈	PROPN
ma-221	105	59	n.	n.	NOUN
ma-221	105	60	(	(	PUNCT
ma-221	105	61	3.1	3.1	NUM
ma-221	105	62	)	)	PUNCT
ma-221	105	63	if	if	SCONJ
ma-221	105	64	un0	un0	NOUN
ma-221	105	65	=	=	SYM
ma-221	105	66	un0	un0	NOUN
ma-221	105	67	+	+	NOUN
ma-221	105	68	1	1	NUM
ma-221	105	69	for	for	ADP
ma-221	105	70	some	some	DET
ma-221	105	71	n0	n0	PROPN
ma-221	105	72	∈	∈	PROPN
ma-221	105	73	n	n	CCONJ
ma-221	105	74	,	,	PUNCT
ma-221	105	75	from	from	ADP
ma-221	105	76	(	(	PUNCT
ma-221	105	77	3	3	X
ma-221	105	78	)	)	PUNCT
ma-221	105	79	one	one	NOUN
ma-221	105	80	obtains	obtain	VERB
ma-221	105	81	d	d	NOUN
ma-221	105	82	(	(	PUNCT
ma-221	105	83	un0	un0	NOUN
ma-221	105	84	,	,	PUNCT
ma-221	105	85	t	t	PROPN
ma-221	105	86	un0	un0	NOUN
ma-221	105	87	)	)	PUNCT
ma-221	106	1	=	=	SYM
ma-221	106	2	d	d	NOUN
ma-221	106	3	(	(	PUNCT
ma-221	106	4	un0	un0	NOUN
ma-221	106	5	+	+	NOUN
ma-221	106	6	1	1	NUM
ma-221	106	7	,	,	PUNCT
ma-221	106	8	t	t	PROPN
ma-221	106	9	un0	un0	NOUN
ma-221	106	10	)	)	PUNCT
ma-221	106	11	=	=	SYM
ma-221	107	1	d(a	d(a	PROPN
ma-221	107	2	,	,	PUNCT
ma-221	107	3	b	b	NOUN
ma-221	107	4	)	)	PUNCT
ma-221	107	5	(	(	PUNCT
ma-221	107	6	3.2	3.2	NUM
ma-221	107	7	)	)	PUNCT
ma-221	107	8	that	that	PRON
ma-221	107	9	is	be	AUX
ma-221	107	10	,	,	PUNCT
ma-221	107	11	un0	un0	NOUN
ma-221	107	12	∈	∈	NOUN
ma-221	107	13	bpp	bpp	NOUN
ma-221	107	14	.	.	PUNCT
ma-221	108	1	thus	thus	ADV
ma-221	108	2	,	,	PUNCT
ma-221	108	3	we	we	PRON
ma-221	108	4	suppose	suppose	VERB
ma-221	108	5	that	that	SCONJ
ma-221	108	6	d(un	d(un	PROPN
ma-221	108	7	,	,	PUNCT
ma-221	108	8	xn+1	xn+1	NUM
ma-221	108	9	)	)	PUNCT
ma-221	108	10	>	>	X
ma-221	108	11	0	0	PUNCT
ma-221	108	12	for	for	SCONJ
ma-221	108	13	all	all	DET
ma-221	108	14	n	n	PRON
ma-221	108	15	∈	∈	NOUN
ma-221	108	16	n.we	n.we	NOUN
ma-221	108	17	shall	shall	AUX
ma-221	108	18	prove	prove	VERB
ma-221	108	19	that	that	SCONJ
ma-221	108	20	the	the	DET
ma-221	108	21	sequence	sequence	NOUN
ma-221	108	22	un	un	PROPN
ma-221	108	23	is	be	AUX
ma-221	108	24	a	a	DET
ma-221	108	25	cauchy	cauchy	ADJ
ma-221	108	26	sequence	sequence	NOUN
ma-221	108	27	.	.	PUNCT
ma-221	109	1	let	let	VERB
ma-221	109	2	us	we	PRON
ma-221	109	3	first	first	ADV
ma-221	109	4	prove	prove	VERB
ma-221	109	5	that	that	SCONJ
ma-221	109	6	lim	lim	PROPN
ma-221	109	7	n→∞	n→∞	X
ma-221	109	8	d	d	NOUN
ma-221	109	9	(	(	PUNCT
ma-221	109	10	un	un	PROPN
ma-221	109	11	,	,	PUNCT
ma-221	109	12	un+1	un+1	NOUN
ma-221	109	13	)	)	PUNCT
ma-221	109	14	=	=	SYM
ma-221	110	1	0	0	X
ma-221	110	2	.	.	PUNCT
ma-221	111	1	as	as	SCONJ
ma-221	111	2	t	t	PROPN
ma-221	111	3	is	be	AUX
ma-221	111	4	generalized	generalize	VERB
ma-221	111	5	(	(	PUNCT
ma-221	111	6	θ	θ	NOUN
ma-221	111	7	,	,	PUNCT
ma-221	111	8	φ)-proximal	φ)-proximal	ADJ
ma-221	111	9	contraction	contraction	NOUN
ma-221	111	10	of	of	ADP
ma-221	111	11	the	the	DET
ma-221	111	12	first	first	ADJ
ma-221	111	13	kind	kind	NOUN
ma-221	111	14	,	,	PUNCT
ma-221	111	15	we	we	PRON
ma-221	111	16	have	have	VERB
ma-221	111	17	that	that	PRON
ma-221	111	18	θ	θ	PROPN
ma-221	111	19	(	(	PUNCT
ma-221	111	20	d	d	X
ma-221	111	21	(	(	PUNCT
ma-221	111	22	un	un	PROPN
ma-221	111	23	,	,	PUNCT
ma-221	111	24	un+1	un+1	NOUN
ma-221	111	25	)	)	PUNCT
ma-221	111	26	)	)	PUNCT
ma-221	112	1	≤	≤	NUM
ma-221	112	2	φ	φ	PROPN
ma-221	113	1	[	[	X
ma-221	113	2	θ	θ	X
ma-221	114	1	[	[	X
ma-221	114	2	ad	ad	X
ma-221	114	3	(	(	PUNCT
ma-221	114	4	un−1	un−1	PROPN
ma-221	114	5	,	,	PUNCT
ma-221	114	6	un	un	ADJ
ma-221	114	7	)	)	PUNCT
ma-221	114	8	+	+	CCONJ
ma-221	114	9	bd	bd	PROPN
ma-221	114	10	(	(	PUNCT
ma-221	114	11	un−1	un−1	PROPN
ma-221	114	12	,	,	PUNCT
ma-221	114	13	un	un	ADJ
ma-221	114	14	)	)	PUNCT
ma-221	114	15	+	+	CCONJ
ma-221	114	16	cd	cd	PROPN
ma-221	114	17	(	(	PUNCT
ma-221	114	18	un	un	PROPN
ma-221	114	19	,	,	PUNCT
ma-221	114	20	un+1	un+1	NOUN
ma-221	114	21	)	)	PUNCT
ma-221	114	22	+	+	NUM
ma-221	114	23	h	h	NOUN
ma-221	114	24	(	(	PUNCT
ma-221	114	25	d	d	X
ma-221	114	26	(	(	PUNCT
ma-221	114	27	un−1	un−1	PROPN
ma-221	114	28	,	,	PUNCT
ma-221	114	29	un+1	un+1	NOUN
ma-221	114	30	)	)	PUNCT
ma-221	115	1	+	+	CCONJ
ma-221	115	2	d	d	PROPN
ma-221	115	3	(	(	PUNCT
ma-221	115	4	un	un	PROPN
ma-221	115	5	,	,	PUNCT
ma-221	115	6	un	un	NOUN
ma-221	115	7	)	)	PUNCT
ma-221	115	8	)	)	PUNCT
ma-221	116	1	]	]	PUNCT
ma-221	116	2	]	]	X
ma-221	116	3	=	=	PUNCT
ma-221	116	4	φ	φ	PROPN
ma-221	117	1	[	[	X
ma-221	117	2	θ	θ	X
ma-221	118	1	[	[	X
ma-221	118	2	ad	ad	X
ma-221	118	3	(	(	PUNCT
ma-221	118	4	un−1	un−1	PROPN
ma-221	118	5	,	,	PUNCT
ma-221	118	6	un	un	ADJ
ma-221	118	7	)	)	PUNCT
ma-221	118	8	+	+	CCONJ
ma-221	118	9	bd	bd	PROPN
ma-221	118	10	(	(	PUNCT
ma-221	118	11	un−1	un−1	PROPN
ma-221	118	12	,	,	PUNCT
ma-221	118	13	un	un	ADJ
ma-221	118	14	)	)	PUNCT
ma-221	118	15	+	+	CCONJ
ma-221	118	16	cd	cd	PROPN
ma-221	118	17	(	(	PUNCT
ma-221	118	18	un	un	PROPN
ma-221	118	19	,	,	PUNCT
ma-221	118	20	un+1	un+1	NOUN
ma-221	118	21	)	)	PUNCT
ma-221	118	22	+	+	NUM
ma-221	118	23	h	h	NOUN
ma-221	118	24	(	(	PUNCT
ma-221	118	25	d	d	X
ma-221	118	26	(	(	PUNCT
ma-221	118	27	un−1	un−1	PROPN
ma-221	118	28	,	,	PUNCT
ma-221	118	29	un+1	un+1	NOUN
ma-221	118	30	)	)	PUNCT
ma-221	118	31	)	)	PUNCT
ma-221	118	32	]	]	PUNCT
ma-221	118	33	]	]	X
ma-221	118	34	≤	≤	NUM
ma-221	118	35	φ	φ	PROPN
ma-221	119	1	[	[	X
ma-221	119	2	θ	θ	X
ma-221	120	1	[	[	X
ma-221	120	2	ad	ad	X
ma-221	120	3	(	(	PUNCT
ma-221	120	4	un−1	un−1	PROPN
ma-221	120	5	,	,	PUNCT
ma-221	120	6	un	un	ADJ
ma-221	120	7	)	)	PUNCT
ma-221	120	8	+	+	CCONJ
ma-221	120	9	bd	bd	PROPN
ma-221	120	10	(	(	PUNCT
ma-221	120	11	xn−1	xn−1	PROPN
ma-221	120	12	,	,	PUNCT
ma-221	120	13	xn	xn	PUNCT
ma-221	120	14	)	)	PUNCT
ma-221	120	15	+	+	CCONJ
ma-221	120	16	cd	cd	PROPN
ma-221	120	17	(	(	PUNCT
ma-221	120	18	un	un	PROPN
ma-221	120	19	,	,	PUNCT
ma-221	120	20	un+1	un+1	NOUN
ma-221	120	21	)	)	PUNCT
ma-221	121	1	+	+	NUM
ma-221	121	2	h	h	NOUN
ma-221	121	3	(	(	PUNCT
ma-221	121	4	d	d	X
ma-221	121	5	(	(	PUNCT
ma-221	121	6	un−1	un−1	PROPN
ma-221	121	7	,	,	PUNCT
ma-221	121	8	un	un	ADJ
ma-221	121	9	)	)	PUNCT
ma-221	122	1	+	+	CCONJ
ma-221	122	2	d	d	PROPN
ma-221	122	3	(	(	PUNCT
ma-221	122	4	un	un	PROPN
ma-221	122	5	,	,	PUNCT
ma-221	122	6	un+1	un+1	NOUN
ma-221	122	7	)	)	PUNCT
ma-221	122	8	)	)	PUNCT
ma-221	123	1	]	]	PUNCT
ma-221	123	2	]	]	X
ma-221	123	3	=	=	PUNCT
ma-221	123	4	φ	φ	PROPN
ma-221	124	1	[	[	X
ma-221	124	2	θ	θ	X
ma-221	124	3	[	[	X
ma-221	124	4	(	(	PUNCT
ma-221	124	5	a	a	DET
ma-221	124	6	+	+	NOUN
ma-221	124	7	b	b	NOUN
ma-221	124	8	+	+	NUM
ma-221	124	9	h)d	h)d	NOUN
ma-221	124	10	(	(	PUNCT
ma-221	124	11	un−1	un−1	PROPN
ma-221	124	12	,	,	PUNCT
ma-221	124	13	un	un	ADJ
ma-221	124	14	)	)	PUNCT
ma-221	124	15	+	+	CCONJ
ma-221	124	16	(	(	PUNCT
ma-221	124	17	c	c	NOUN
ma-221	124	18	+	+	NOUN
ma-221	124	19	h)d	h)d	NOUN
ma-221	124	20	(	(	PUNCT
ma-221	124	21	un	un	PROPN
ma-221	124	22	,	,	PUNCT
ma-221	124	23	un+1	un+1	NOUN
ma-221	124	24	)	)	PUNCT
ma-221	124	25	]	]	X
ma-221	124	26	]	]	X
ma-221	124	27	since	since	SCONJ
ma-221	124	28	θ	θ	PROPN
ma-221	124	29	is	be	AUX
ma-221	124	30	strictly	strictly	ADV
ma-221	124	31	increasing	increase	VERB
ma-221	124	32	and	and	CCONJ
ma-221	124	33	by	by	ADP
ma-221	124	34	lemma	lemma	PROPN
ma-221	124	35	2.6	2.6	NUM
ma-221	124	36	,	,	PUNCT
ma-221	124	37	we	we	PRON
ma-221	124	38	deduce	deduce	VERB
ma-221	124	39	d	d	X
ma-221	124	40	(	(	PUNCT
ma-221	124	41	xn	xn	PROPN
ma-221	124	42	,	,	PUNCT
ma-221	124	43	xn+1	xn+1	NUM
ma-221	124	44	)	)	PUNCT
ma-221	124	45	<	<	X
ma-221	124	46	(	(	PUNCT
ma-221	124	47	a	a	DET
ma-221	124	48	+	+	NOUN
ma-221	124	49	b	b	NOUN
ma-221	124	50	+	+	NUM
ma-221	124	51	h)d	h)d	NOUN
ma-221	124	52	(	(	PUNCT
ma-221	124	53	xn−1	xn−1	PROPN
ma-221	124	54	,	,	PUNCT
ma-221	124	55	xn	xn	PUNCT
ma-221	124	56	)	)	PUNCT
ma-221	125	1	+	+	CCONJ
ma-221	125	2	(	(	PUNCT
ma-221	125	3	c	c	NOUN
ma-221	125	4	+	+	NOUN
ma-221	125	5	h)d	h)d	NOUN
ma-221	125	6	(	(	PUNCT
ma-221	125	7	xn	xn	X
ma-221	125	8	,	,	PUNCT
ma-221	125	9	xn+1	xn+1	NUM
ma-221	125	10	)	)	PUNCT
ma-221	125	11	.	.	PUNCT
ma-221	126	1	thus	thus	ADV
ma-221	126	2	d	d	X
ma-221	126	3	(	(	PUNCT
ma-221	126	4	un	un	PROPN
ma-221	126	5	,	,	PUNCT
ma-221	126	6	un+1	un+1	NOUN
ma-221	126	7	)	)	PUNCT
ma-221	126	8	<	<	X
ma-221	126	9	a	a	DET
ma-221	126	10	+	+	NUM
ma-221	126	11	b	b	NOUN
ma-221	126	12	+	+	NUM
ma-221	126	13	h	h	NOUN
ma-221	126	14	1−	1−	NUM
ma-221	126	15	c	c	NOUN
ma-221	126	16	−	−	PROPN
ma-221	127	1	h	h	NOUN
ma-221	127	2	(	(	PUNCT
ma-221	127	3	d	d	X
ma-221	127	4	(	(	PUNCT
ma-221	127	5	un−1	un−1	PROPN
ma-221	127	6	,	,	PUNCT
ma-221	127	7	xn	xn	PROPN
ma-221	127	8	)	)	PUNCT
ma-221	127	9	)	)	PUNCT
ma-221	127	10	.	.	PUNCT
ma-221	128	1	if	if	SCONJ
ma-221	128	2	b	b	PROPN
ma-221	128	3	+	+	SYM
ma-221	128	4	b	b	NOUN
ma-221	128	5	+	+	CCONJ
ma-221	128	6	c	c	NOUN
ma-221	128	7	+	+	CCONJ
ma-221	128	8	2h	2h	NUM
ma-221	128	9	=	=	SYM
ma-221	128	10	1	1	NUM
ma-221	128	11	,	,	PUNCT
ma-221	128	12	we	we	PRON
ma-221	128	13	have	have	VERB
ma-221	128	14	0	0	NUM
ma-221	128	15	<	<	X
ma-221	128	16	1−	1−	NUM
ma-221	128	17	c	c	NOUN
ma-221	128	18	−	−	PROPN
ma-221	129	1	h	h	NOUN
ma-221	130	1	and	and	CCONJ
ma-221	130	2	so	so	ADV
ma-221	130	3	d	d	X
ma-221	130	4	(	(	PUNCT
ma-221	130	5	un	un	PROPN
ma-221	130	6	,	,	PUNCT
ma-221	130	7	un+1	un+1	NOUN
ma-221	130	8	)	)	PUNCT
ma-221	130	9	≤	≤	NOUN
ma-221	131	1	a	a	DET
ma-221	131	2	+	+	NUM
ma-221	131	3	b	b	NOUN
ma-221	131	4	+	+	NUM
ma-221	131	5	h	h	NOUN
ma-221	131	6	1−	1−	NUM
ma-221	131	7	c	c	NOUN
ma-221	131	8	−	−	PROPN
ma-221	131	9	h	h	NOUN
ma-221	131	10	(	(	PUNCT
ma-221	131	11	d	d	X
ma-221	131	12	(	(	PUNCT
ma-221	131	13	un−1	un−1	PROPN
ma-221	131	14	,	,	PUNCT
ma-221	131	15	un	un	NOUN
ma-221	131	16	)	)	PUNCT
ma-221	131	17	)	)	PUNCT
ma-221	132	1	=	=	PUNCT
ma-221	132	2	d	d	X
ma-221	132	3	(	(	PUNCT
ma-221	132	4	un−1	un−1	PROPN
ma-221	132	5	,	,	PUNCT
ma-221	132	6	un	un	ADJ
ma-221	132	7	)	)	PUNCT
ma-221	132	8	,	,	PUNCT
ma-221	132	9	∀n	∀n	NUM
ma-221	132	10	∈	∈	PROPN
ma-221	132	11	n	n	CCONJ
ma-221	132	12	;	;	PUNCT
ma-221	132	13	consequently	consequently	ADV
ma-221	132	14	,	,	PUNCT
ma-221	132	15	θ	θ	PROPN
ma-221	132	16	(	(	PUNCT
ma-221	132	17	d	d	X
ma-221	132	18	(	(	PUNCT
ma-221	132	19	un	un	PROPN
ma-221	132	20	,	,	PUNCT
ma-221	132	21	un+1	un+1	NOUN
ma-221	132	22	)	)	PUNCT
ma-221	132	23	)	)	PUNCT
ma-221	133	1	≤	≤	NUM
ma-221	133	2	φ	φ	PROPN
ma-221	134	1	[	[	X
ma-221	134	2	θ	θ	X
ma-221	134	3	(	(	PUNCT
ma-221	134	4	d	d	X
ma-221	134	5	(	(	PUNCT
ma-221	134	6	un−1	un−1	PROPN
ma-221	134	7	,	,	PUNCT
ma-221	134	8	un	un	NOUN
ma-221	134	9	)	)	PUNCT
ma-221	134	10	)	)	PUNCT
ma-221	134	11	]	]	PUNCT
ma-221	135	1	if	if	SCONJ
ma-221	135	2	b	b	PROPN
ma-221	135	3	+	+	SYM
ma-221	135	4	b	b	NOUN
ma-221	135	5	+	+	CCONJ
ma-221	135	6	c	c	NOUN
ma-221	135	7	+	+	CCONJ
ma-221	135	8	2h	2h	NUM
ma-221	135	9	<	<	X
ma-221	135	10	1	1	NUM
ma-221	135	11	,	,	PUNCT
ma-221	135	12	we	we	PRON
ma-221	135	13	have	have	VERB
ma-221	135	14	0	0	NUM
ma-221	135	15	<	<	X
ma-221	136	1	1−	1−	NUM
ma-221	136	2	c	c	NOUN
ma-221	136	3	−	−	PROPN
ma-221	137	1	h	h	NOUN
ma-221	138	1	and	and	CCONJ
ma-221	138	2	so	so	ADV
ma-221	138	3	d	d	X
ma-221	138	4	(	(	PUNCT
ma-221	138	5	un	un	PROPN
ma-221	138	6	,	,	PUNCT
ma-221	138	7	un+1	un+1	NOUN
ma-221	138	8	)	)	PUNCT
ma-221	138	9	<	<	X
ma-221	139	1	d	d	X
ma-221	139	2	(	(	PUNCT
ma-221	139	3	un−1	un−1	PROPN
ma-221	139	4	,	,	PUNCT
ma-221	139	5	un	un	ADJ
ma-221	139	6	)	)	PUNCT
ma-221	139	7	,	,	PUNCT
ma-221	139	8	∀n	∀n	NUM
ma-221	139	9	∈	∈	PROPN
ma-221	139	10	n	n	CCONJ
ma-221	139	11	;	;	PUNCT
ma-221	139	12	consequently	consequently	ADV
ma-221	139	13	,	,	PUNCT
ma-221	139	14	θ	θ	PROPN
ma-221	139	15	(	(	PUNCT
ma-221	139	16	d	d	X
ma-221	139	17	(	(	PUNCT
ma-221	139	18	un	un	PROPN
ma-221	139	19	,	,	PUNCT
ma-221	139	20	un+1	un+1	NOUN
ma-221	139	21	)	)	PUNCT
ma-221	139	22	)	)	PUNCT
ma-221	139	23	≤	≤	NUM
ma-221	139	24	φ	φ	PROPN
ma-221	140	1	[	[	X
ma-221	140	2	θ	θ	X
ma-221	140	3	(	(	PUNCT
ma-221	140	4	d	d	X
ma-221	140	5	(	(	PUNCT
ma-221	140	6	un−1	un−1	PROPN
ma-221	140	7	,	,	PUNCT
ma-221	140	8	un	un	NOUN
ma-221	140	9	)	)	PUNCT
ma-221	140	10	)	)	PUNCT
ma-221	140	11	]	]	PUNCT
ma-221	141	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	141	2	eur	eur	PROPN
ma-221	141	3	.	.	PUNCT
ma-221	142	1	j.	j.	PROPN
ma-221	142	2	math	math	PROPN
ma-221	142	3	.	.	PUNCT
ma-221	143	1	anal	anal	PROPN
ma-221	143	2	.	.	PUNCT
ma-221	144	1	10.28924	10.28924	NUM
ma-221	144	2	/	/	SYM
ma-221	144	3	ada	ada	PROPN
ma-221	144	4	/	/	SYM
ma-221	144	5	ma.4.13	ma.4.13	PROPN
ma-221	144	6	5it	5it	NOUN
ma-221	144	7	implies	imply	VERB
ma-221	144	8	θ	θ	PROPN
ma-221	144	9	(	(	PUNCT
ma-221	144	10	d	d	X
ma-221	144	11	(	(	PUNCT
ma-221	144	12	un	un	PROPN
ma-221	144	13	,	,	PUNCT
ma-221	144	14	un+1	un+1	NOUN
ma-221	144	15	)	)	PUNCT
ma-221	144	16	)	)	PUNCT
ma-221	145	1	≤	≤	NUM
ma-221	145	2	φ	φ	PROPN
ma-221	145	3	[	[	X
ma-221	145	4	θ	θ	X
ma-221	145	5	(	(	PUNCT
ma-221	145	6	d(xn−1	d(xn−1	PROPN
ma-221	145	7	,	,	PUNCT
ma-221	145	8	un	un	PROPN
ma-221	145	9	)	)	PUNCT
ma-221	145	10	]	]	PUNCT
ma-221	145	11	≤	≤	NUM
ma-221	145	12	φ2	φ2	PROPN
ma-221	146	1	[	[	X
ma-221	146	2	θ	θ	X
ma-221	146	3	(	(	PUNCT
ma-221	146	4	d(un−2	d(un−2	PROPN
ma-221	146	5	,	,	PUNCT
ma-221	146	6	un−1	un−1	ADJ
ma-221	146	7	)	)	PUNCT
ma-221	146	8	]	]	PUNCT
ma-221	146	9	≤	≤	NUM
ma-221	146	10	...	...	PUNCT
ma-221	146	11	≤	≤	NUM
ma-221	146	12	φn	φn	ADP
ma-221	147	1	[	[	X
ma-221	147	2	θ	θ	X
ma-221	147	3	(	(	PUNCT
ma-221	147	4	d(u0	d(u0	NOUN
ma-221	147	5	,	,	PUNCT
ma-221	147	6	u1	u1	NOUN
ma-221	147	7	)	)	PUNCT
ma-221	147	8	]	]	PUNCT
ma-221	147	9	.	.	PUNCT
ma-221	148	1	taking	take	VERB
ma-221	148	2	the	the	DET
ma-221	148	3	limit	limit	NOUN
ma-221	148	4	as	as	ADP
ma-221	148	5	n	n	PROPN
ma-221	148	6	→∞	→∞	NUM
ma-221	148	7	,	,	PUNCT
ma-221	148	8	we	we	PRON
ma-221	148	9	have	have	VERB
ma-221	148	10	1	1	NUM
ma-221	148	11	≤	≤	NUM
ma-221	148	12	θ(d	θ(d	NOUN
ma-221	148	13	(	(	PUNCT
ma-221	148	14	un	un	PROPN
ma-221	148	15	,	,	PUNCT
ma-221	148	16	un+1	un+1	NOUN
ma-221	148	17	)	)	PUNCT
ma-221	148	18	)	)	PUNCT
ma-221	148	19	≤	≤	PROPN
ma-221	148	20	lim	lim	PROPN
ma-221	148	21	n→∞	n→∞	X
ma-221	149	1	φn	φn	ADP
ma-221	150	1	[	[	X
ma-221	150	2	θ(d	θ(d	NOUN
ma-221	150	3	(	(	PUNCT
ma-221	150	4	u0	u0	ADJ
ma-221	150	5	,	,	PUNCT
ma-221	150	6	u1	u1	NOUN
ma-221	150	7	)	)	PUNCT
ma-221	150	8	)	)	PUNCT
ma-221	150	9	]	]	PUNCT
ma-221	151	1	=	=	PUNCT
ma-221	151	2	1	1	X
ma-221	151	3	.	.	PUNCT
ma-221	151	4	since	since	SCONJ
ma-221	151	5	θ	θ	PROPN
ma-221	151	6	∈	∈	PROPN
ma-221	151	7	θ	θ	PROPN
ma-221	151	8	,	,	PUNCT
ma-221	151	9	we	we	PRON
ma-221	151	10	obtain	obtain	VERB
ma-221	151	11	lim	lim	PROPN
ma-221	151	12	n→∞	n→∞	PROPN
ma-221	152	1	d	d	NOUN
ma-221	152	2	(	(	PUNCT
ma-221	152	3	un	un	PROPN
ma-221	152	4	,	,	PUNCT
ma-221	152	5	un+1	un+1	NOUN
ma-221	152	6	)	)	PUNCT
ma-221	152	7	=	=	SYM
ma-221	153	1	0	0	X
ma-221	153	2	.	.	PUNCT
ma-221	154	1	(	(	PUNCT
ma-221	154	2	3.3	3.3	NUM
ma-221	154	3	)	)	PUNCT
ma-221	154	4	next	next	ADV
ma-221	154	5	,	,	PUNCT
ma-221	154	6	we	we	PRON
ma-221	154	7	shall	shall	AUX
ma-221	154	8	prove	prove	VERB
ma-221	154	9	that	that	SCONJ
ma-221	154	10	{	{	PUNCT
ma-221	154	11	un}n∈n	un}n∈n	NOUN
ma-221	154	12	is	be	AUX
ma-221	154	13	a	a	DET
ma-221	154	14	cauchy	cauchy	ADJ
ma-221	154	15	sequence	sequence	NOUN
ma-221	154	16	,	,	PUNCT
ma-221	154	17	i.e	i.e	PROPN
ma-221	154	18	,	,	PUNCT
ma-221	154	19	limn→∞	limn→∞	PROPN
ma-221	154	20	d	d	X
ma-221	154	21	(	(	PUNCT
ma-221	154	22	un	un	PROPN
ma-221	154	23	,	,	PUNCT
ma-221	154	24	um	um	INTJ
ma-221	154	25	)	)	PUNCT
ma-221	154	26	=	=	SYM
ma-221	154	27	0	0	NUM
ma-221	154	28	,	,	PUNCT
ma-221	154	29	for	for	ADP
ma-221	154	30	all	all	DET
ma-221	154	31	n	n	PRON
ma-221	154	32	∈	∈	NOUN
ma-221	154	33	n.suppose	n.suppose	NOUN
ma-221	154	34	to	to	ADP
ma-221	154	35	the	the	DET
ma-221	154	36	contrary	contrary	NOUN
ma-221	154	37	that	that	PRON
ma-221	154	38	exists	exist	VERB
ma-221	154	39	ε	ε	PROPN
ma-221	154	40	>	>	PUNCT
ma-221	154	41	0	0	PUNCT
ma-221	155	1	and	and	CCONJ
ma-221	155	2	sequences	sequence	NOUN
ma-221	155	3	n(k	n(k	PROPN
ma-221	155	4	)	)	PUNCT
ma-221	155	5	and	and	CCONJ
ma-221	155	6	m(k	m(k	PROPN
ma-221	155	7	)	)	PUNCT
ma-221	155	8	of	of	ADP
ma-221	155	9	natural	natural	ADJ
ma-221	155	10	numbers	number	NOUN
ma-221	155	11	suchthat	suchthat	VERB
ma-221	155	12	m(k	m(k	PROPN
ma-221	155	13	)	)	PUNCT
ma-221	155	14	>	>	X
ma-221	156	1	n(k	n(k	PROPN
ma-221	156	2	)	)	PUNCT
ma-221	156	3	>	>	X
ma-221	157	1	k	k	X
ma-221	157	2	,	,	PUNCT
ma-221	157	3	d	d	X
ma-221	157	4	(	(	PUNCT
ma-221	157	5	xm(k	xm(k	X
ma-221	157	6	)	)	PUNCT
ma-221	157	7	,	,	PUNCT
ma-221	157	8	xn(k	xn(k	NUM
ma-221	157	9	)	)	PUNCT
ma-221	157	10	)	)	PUNCT
ma-221	157	11	≥	≥	X
ma-221	157	12	ε	ε	PROPN
ma-221	157	13	,	,	PUNCT
ma-221	157	14	d	d	X
ma-221	157	15	(	(	PUNCT
ma-221	157	16	um(k)−1	um(k)−1	X
ma-221	157	17	,	,	PUNCT
ma-221	157	18	un(k	un(k	NOUN
ma-221	157	19	)	)	PUNCT
ma-221	157	20	)	)	PUNCT
ma-221	158	1	<	<	X
ma-221	158	2	ε	ε	PROPN
ma-221	158	3	.	.	PUNCT
ma-221	159	1	(	(	PUNCT
ma-221	159	2	3.4	3.4	NUM
ma-221	159	3	)	)	PUNCT
ma-221	159	4	using	use	VERB
ma-221	159	5	the	the	DET
ma-221	159	6	triangular	triangular	NOUN
ma-221	159	7	inequality	inequality	NOUN
ma-221	159	8	,	,	PUNCT
ma-221	159	9	we	we	PRON
ma-221	159	10	find	find	VERB
ma-221	159	11	that	that	SCONJ
ma-221	159	12	,	,	PUNCT
ma-221	159	13	ε	ε	PROPN
ma-221	159	14	≤	≤	PROPN
ma-221	159	15	d	d	PROPN
ma-221	159	16	(	(	PUNCT
ma-221	159	17	um(k	um(k	PROPN
ma-221	159	18	)	)	PUNCT
ma-221	159	19	,	,	PUNCT
ma-221	159	20	un(k	un(k	NOUN
ma-221	159	21	)	)	PUNCT
ma-221	159	22	)	)	PUNCT
ma-221	160	1	≤	≤	NUM
ma-221	160	2	d	d	X
ma-221	160	3	(	(	PUNCT
ma-221	160	4	um(k	um(k	PROPN
ma-221	160	5	)	)	PUNCT
ma-221	160	6	,	,	PUNCT
ma-221	160	7	un(k)−1	un(k)−1	NOUN
ma-221	160	8	)	)	PUNCT
ma-221	161	1	+	+	CCONJ
ma-221	161	2	d	d	X
ma-221	161	3	(	(	PUNCT
ma-221	161	4	xn(k)−1	xn(k)−1	NOUN
ma-221	161	5	,	,	PUNCT
ma-221	161	6	xn(k	xn(k	NUM
ma-221	161	7	)	)	PUNCT
ma-221	161	8	)	)	PUNCT
ma-221	161	9	(	(	PUNCT
ma-221	161	10	3.5	3.5	NUM
ma-221	161	11	)	)	PUNCT
ma-221	161	12	<	<	X
ma-221	161	13	ε+	ε+	X
ma-221	161	14	d	d	X
ma-221	161	15	(	(	PUNCT
ma-221	161	16	un(k)−1	un(k)−1	NOUN
ma-221	161	17	,	,	PUNCT
ma-221	161	18	un(k	un(k	NOUN
ma-221	161	19	)	)	PUNCT
ma-221	161	20	)	)	PUNCT
ma-221	161	21	.	.	PUNCT
ma-221	162	1	(	(	PUNCT
ma-221	162	2	3.6	3.6	NUM
ma-221	162	3	)	)	PUNCT
ma-221	162	4	then	then	ADV
ma-221	162	5	,	,	PUNCT
ma-221	162	6	by	by	ADP
ma-221	162	7	3.4	3.4	NUM
ma-221	162	8	and	and	CCONJ
ma-221	162	9	3.22	3.22	NUM
ma-221	162	10	,	,	PUNCT
ma-221	162	11	it	it	PRON
ma-221	162	12	follows	follow	VERB
ma-221	162	13	that	that	SCONJ
ma-221	162	14	lim	lim	PROPN
ma-221	162	15	k→∞	k→∞	PROPN
ma-221	162	16	d	d	PROPN
ma-221	162	17	(	(	PUNCT
ma-221	162	18	um(k	um(k	PROPN
ma-221	162	19	)	)	PUNCT
ma-221	162	20	,	,	PUNCT
ma-221	162	21	un(k	un(k	NOUN
ma-221	162	22	)	)	PUNCT
ma-221	162	23	)	)	PUNCT
ma-221	163	1	=	=	PUNCT
ma-221	163	2	ε	ε	PROPN
ma-221	163	3	.	.	PUNCT
ma-221	163	4	(	(	PUNCT
ma-221	163	5	3.7	3.7	NUM
ma-221	163	6	)	)	PUNCT
ma-221	163	7	using	use	VERB
ma-221	163	8	the	the	DET
ma-221	163	9	triangular	triangular	NOUN
ma-221	163	10	inequality	inequality	NOUN
ma-221	163	11	,	,	PUNCT
ma-221	163	12	we	we	PRON
ma-221	163	13	find	find	VERB
ma-221	163	14	that	that	SCONJ
ma-221	163	15	,	,	PUNCT
ma-221	163	16	ε	ε	PROPN
ma-221	163	17	≤	≤	PROPN
ma-221	163	18	d	d	PROPN
ma-221	163	19	(	(	PUNCT
ma-221	163	20	um(k	um(k	PROPN
ma-221	163	21	)	)	PUNCT
ma-221	163	22	,	,	PUNCT
ma-221	163	23	un(k	un(k	NOUN
ma-221	163	24	)	)	PUNCT
ma-221	163	25	)	)	PUNCT
ma-221	164	1	≤	≤	NUM
ma-221	164	2	d	d	X
ma-221	164	3	(	(	PUNCT
ma-221	164	4	um(k	um(k	PROPN
ma-221	164	5	)	)	PUNCT
ma-221	164	6	,	,	PUNCT
ma-221	164	7	un(k)+1	un(k)+1	NOUN
ma-221	164	8	)	)	PUNCT
ma-221	165	1	+	+	CCONJ
ma-221	166	1	d	d	X
ma-221	166	2	(	(	PUNCT
ma-221	166	3	xn(k)+1	xn(k)+1	PROPN
ma-221	166	4	,	,	PUNCT
ma-221	166	5	un(k	un(k	NOUN
ma-221	166	6	)	)	PUNCT
ma-221	166	7	)	)	PUNCT
ma-221	166	8	(	(	PUNCT
ma-221	166	9	3.8	3.8	NUM
ma-221	166	10	)	)	PUNCT
ma-221	166	11	and	and	CCONJ
ma-221	166	12	ε	ε	PROPN
ma-221	166	13	≤	≤	PROPN
ma-221	166	14	d	d	PROPN
ma-221	166	15	(	(	PUNCT
ma-221	166	16	um(k	um(k	PROPN
ma-221	166	17	)	)	PUNCT
ma-221	166	18	,	,	PUNCT
ma-221	166	19	un(k)+1	un(k)+1	NOUN
ma-221	166	20	)	)	PUNCT
ma-221	166	21	≤	≤	NUM
ma-221	167	1	d	d	X
ma-221	167	2	(	(	PUNCT
ma-221	167	3	um(k	um(k	PROPN
ma-221	167	4	)	)	PUNCT
ma-221	167	5	,	,	PUNCT
ma-221	167	6	un(k	un(k	NOUN
ma-221	167	7	)	)	PUNCT
ma-221	167	8	)	)	PUNCT
ma-221	168	1	+	+	CCONJ
ma-221	168	2	d	d	X
ma-221	168	3	(	(	PUNCT
ma-221	168	4	un(k	un(k	NOUN
ma-221	168	5	)	)	PUNCT
ma-221	168	6	,	,	PUNCT
ma-221	168	7	un(k)+1	un(k)+1	NUM
ma-221	168	8	)	)	PUNCT
ma-221	168	9	(	(	PUNCT
ma-221	168	10	3.9	3.9	NUM
ma-221	168	11	)	)	PUNCT
ma-221	168	12	then	then	ADV
ma-221	168	13	,	,	PUNCT
ma-221	168	14	by	by	ADP
ma-221	168	15	(	(	PUNCT
ma-221	168	16	3.25	3.25	NUM
ma-221	168	17	)	)	PUNCT
ma-221	168	18	and	and	CCONJ
ma-221	168	19	(	(	PUNCT
ma-221	168	20	3.9	3.9	NUM
ma-221	168	21	)	)	PUNCT
ma-221	168	22	,	,	PUNCT
ma-221	168	23	it	it	PRON
ma-221	168	24	follows	follow	VERB
ma-221	168	25	that	that	SCONJ
ma-221	168	26	lim	lim	PROPN
ma-221	168	27	k→∞	k→∞	PROPN
ma-221	168	28	d	d	PROPN
ma-221	168	29	(	(	PUNCT
ma-221	168	30	um(k	um(k	PROPN
ma-221	168	31	)	)	PUNCT
ma-221	168	32	,	,	PUNCT
ma-221	168	33	un(k)+1	un(k)+1	NOUN
ma-221	168	34	)	)	PUNCT
ma-221	169	1	=	=	SYM
ma-221	169	2	ε	ε	PROPN
ma-221	169	3	.	.	PUNCT
ma-221	170	1	(	(	PUNCT
ma-221	170	2	3.10	3.10	NUM
ma-221	170	3	)	)	PUNCT
ma-221	170	4	similarly	similarly	ADV
ma-221	170	5	method	method	NOUN
ma-221	170	6	,	,	PUNCT
ma-221	170	7	we	we	PRON
ma-221	170	8	conclude	conclude	VERB
ma-221	170	9	that	that	SCONJ
ma-221	170	10	lim	lim	PROPN
ma-221	170	11	k→∞	k→∞	PROPN
ma-221	170	12	d	d	PROPN
ma-221	170	13	(	(	PUNCT
ma-221	170	14	um(k)+1	um(k)+1	PROPN
ma-221	170	15	,	,	PUNCT
ma-221	170	16	un(k	un(k	NOUN
ma-221	170	17	)	)	PUNCT
ma-221	170	18	)	)	PUNCT
ma-221	171	1	=	=	PUNCT
ma-221	171	2	ε	ε	AUX
ma-221	171	3	.	.	PUNCT
ma-221	171	4	(	(	PUNCT
ma-221	171	5	3.11	3.11	NUM
ma-221	171	6	)	)	PUNCT
ma-221	171	7	using	use	VERB
ma-221	171	8	again	again	ADV
ma-221	171	9	the	the	DET
ma-221	171	10	triangular	triangular	NOUN
ma-221	171	11	inequality	inequality	NOUN
ma-221	171	12	,	,	PUNCT
ma-221	171	13	d	d	X
ma-221	171	14	(	(	PUNCT
ma-221	171	15	um(k)+1	um(k)+1	PROPN
ma-221	171	16	,	,	PUNCT
ma-221	171	17	un(k)+1	un(k)+1	NOUN
ma-221	171	18	)	)	PUNCT
ma-221	171	19	≤	≤	NUM
ma-221	171	20	d	d	X
ma-221	171	21	(	(	PUNCT
ma-221	171	22	xm(k)+1	xm(k)+1	PROPN
ma-221	171	23	,	,	PUNCT
ma-221	171	24	xm(k	xm(k	PUNCT
ma-221	171	25	)	)	PUNCT
ma-221	171	26	)	)	PUNCT
ma-221	172	1	+	+	CCONJ
ma-221	172	2	d	d	X
ma-221	172	3	(	(	PUNCT
ma-221	172	4	um(k	um(k	ADJ
ma-221	172	5	)	)	PUNCT
ma-221	172	6	,	,	PUNCT
ma-221	172	7	un(k	un(k	NOUN
ma-221	172	8	)	)	PUNCT
ma-221	172	9	)	)	PUNCT
ma-221	173	1	+	+	CCONJ
ma-221	173	2	d	d	X
ma-221	173	3	(	(	PUNCT
ma-221	173	4	un(k	un(k	NOUN
ma-221	173	5	)	)	PUNCT
ma-221	173	6	,	,	PUNCT
ma-221	173	7	un(k)+1	un(k)+1	NUM
ma-221	173	8	)	)	PUNCT
ma-221	173	9	.	.	PUNCT
ma-221	174	1	(	(	PUNCT
ma-221	174	2	3.12	3.12	NUM
ma-221	174	3	)	)	PUNCT
ma-221	174	4	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	174	5	eur	eur	PROPN
ma-221	174	6	.	.	PUNCT
ma-221	175	1	j.	j.	PROPN
ma-221	175	2	math	math	PROPN
ma-221	175	3	.	.	PUNCT
ma-221	176	1	anal	anal	PROPN
ma-221	176	2	.	.	PUNCT
ma-221	177	1	10.28924	10.28924	NUM
ma-221	177	2	/	/	SYM
ma-221	177	3	ada	ada	PROPN
ma-221	177	4	/	/	SYM
ma-221	177	5	ma.4.13	ma.4.13	PROPN
ma-221	177	6	6on	6on	NOUN
ma-221	177	7	the	the	DET
ma-221	177	8	other	other	ADJ
ma-221	177	9	hand	hand	NOUN
ma-221	177	10	,	,	PUNCT
ma-221	177	11	using	use	VERB
ma-221	177	12	triangular	triangular	NOUN
ma-221	177	13	inequality	inequality	NOUN
ma-221	177	14	,	,	PUNCT
ma-221	177	15	we	we	PRON
ma-221	177	16	have	have	VERB
ma-221	177	17	d	d	X
ma-221	177	18	(	(	PUNCT
ma-221	177	19	um(k	um(k	PROPN
ma-221	177	20	)	)	PUNCT
ma-221	177	21	,	,	PUNCT
ma-221	177	22	un(k	un(k	NOUN
ma-221	177	23	)	)	PUNCT
ma-221	177	24	)	)	PUNCT
ma-221	178	1	≤	≤	NUM
ma-221	178	2	d	d	X
ma-221	178	3	(	(	PUNCT
ma-221	178	4	um(k	um(k	PROPN
ma-221	178	5	)	)	PUNCT
ma-221	178	6	,	,	PUNCT
ma-221	178	7	um(k)+1	um(k)+1	PROPN
ma-221	178	8	)	)	PUNCT
ma-221	179	1	+	+	CCONJ
ma-221	179	2	d	d	X
ma-221	179	3	(	(	PUNCT
ma-221	179	4	um(k)+1	um(k)+1	PROPN
ma-221	179	5	,	,	PUNCT
ma-221	179	6	un(k)+1	un(k)+1	NUM
ma-221	179	7	)	)	PUNCT
ma-221	180	1	+	+	CCONJ
ma-221	180	2	d	d	X
ma-221	180	3	(	(	PUNCT
ma-221	180	4	un(k)+1	un(k)+1	NOUN
ma-221	180	5	,	,	PUNCT
ma-221	180	6	un(k	un(k	NOUN
ma-221	180	7	)	)	PUNCT
ma-221	180	8	)	)	PUNCT
ma-221	180	9	.	.	PUNCT
ma-221	181	1	(	(	PUNCT
ma-221	181	2	3.13	3.13	NUM
ma-221	181	3	)	)	PUNCT
ma-221	181	4	letting	let	VERB
ma-221	181	5	k	k	PROPN
ma-221	181	6	→∞	→∞	PROPN
ma-221	181	7	in	in	ADP
ma-221	181	8	inequality	inequality	NOUN
ma-221	181	9	(	(	PUNCT
ma-221	181	10	3.12	3.12	NUM
ma-221	181	11	)	)	PUNCT
ma-221	181	12	and	and	CCONJ
ma-221	181	13	(	(	PUNCT
ma-221	181	14	3.13	3.13	NUM
ma-221	181	15	)	)	PUNCT
ma-221	181	16	,	,	PUNCT
ma-221	181	17	we	we	PRON
ma-221	181	18	obtain	obtain	VERB
ma-221	181	19	lim	lim	PROPN
ma-221	181	20	k→∞	k→∞	PROPN
ma-221	182	1	d	d	PROPN
ma-221	182	2	(	(	PUNCT
ma-221	182	3	um(k)+1	um(k)+1	PROPN
ma-221	182	4	,	,	PUNCT
ma-221	182	5	un(k)+1	un(k)+1	NOUN
ma-221	182	6	)	)	PUNCT
ma-221	182	7	=	=	SYM
ma-221	182	8	ε	ε	PROPN
ma-221	182	9	.	.	PUNCT
ma-221	182	10	(	(	PUNCT
ma-221	182	11	3.14	3.14	NUM
ma-221	182	12	)	)	PUNCT
ma-221	182	13	substituting	substitute	VERB
ma-221	182	14	u1	u1	NOUN
ma-221	182	15	=	=	SYM
ma-221	182	16	xm(k)+1	xm(k)+1	PROPN
ma-221	182	17	,	,	PUNCT
ma-221	182	18	u2	u2	PROPN
ma-221	182	19	=	=	PUNCT
ma-221	182	20	xn(k)+1	xn(k)+1	PROPN
ma-221	182	21	,	,	PUNCT
ma-221	182	22	v1	v1	NOUN
ma-221	182	23	=	=	SYM
ma-221	182	24	um(k	um(k	X
ma-221	182	25	)	)	PUNCT
ma-221	182	26	and	and	CCONJ
ma-221	182	27	v1	v1	NOUN
ma-221	182	28	=	=	SYM
ma-221	182	29	un(k	un(k	NOUN
ma-221	182	30	)	)	PUNCT
ma-221	182	31	in	in	ADP
ma-221	182	32	assumption	assumption	NOUN
ma-221	182	33	of	of	ADP
ma-221	182	34	the	the	DET
ma-221	182	35	theorem	theorem	NOUN
ma-221	182	36	,	,	PUNCT
ma-221	182	37	weget	weget	NOUN
ma-221	182	38	θ	θ	PROPN
ma-221	182	39	(	(	PUNCT
ma-221	182	40	d	d	X
ma-221	182	41	(	(	PUNCT
ma-221	182	42	um(k)+1	um(k)+1	PROPN
ma-221	182	43	,	,	PUNCT
ma-221	182	44	un(k)+1	un(k)+1	NUM
ma-221	182	45	)	)	PUNCT
ma-221	182	46	)	)	PUNCT
ma-221	182	47	≤	≤	NUM
ma-221	183	1	φ	φ	NUM
ma-221	183	2			NUM
ma-221	183	3	θ	θ	PROPN
ma-221	183	4			PROPN
ma-221	183	5	ad	ad	NOUN
ma-221	183	6	(	(	PUNCT
ma-221	183	7	um(k	um(k	PROPN
ma-221	183	8	)	)	PUNCT
ma-221	183	9	,	,	PUNCT
ma-221	183	10	un(k	un(k	NOUN
ma-221	183	11	)	)	PUNCT
ma-221	183	12	)	)	PUNCT
ma-221	184	1	+	+	CCONJ
ma-221	184	2	bd	bd	PROPN
ma-221	184	3	(	(	PUNCT
ma-221	184	4	um(k)1	um(k)1	PROPN
ma-221	184	5	,	,	PUNCT
ma-221	184	6	un(k	un(k	NOUN
ma-221	184	7	)	)	PUNCT
ma-221	184	8	)	)	PUNCT
ma-221	185	1	+	+	CCONJ
ma-221	186	1	cd	cd	PROPN
ma-221	186	2	(	(	PUNCT
ma-221	186	3	un(k)+1	un(k)+1	NOUN
ma-221	186	4	,	,	PUNCT
ma-221	186	5	un(k	un(k	NOUN
ma-221	186	6	)	)	PUNCT
ma-221	186	7	)	)	PUNCT
ma-221	187	1	+	+	CCONJ
ma-221	187	2	h(d	h(d	PRON
ma-221	187	3	(	(	PUNCT
ma-221	187	4	um(k	um(k	PROPN
ma-221	187	5	)	)	PUNCT
ma-221	187	6	,	,	PUNCT
ma-221	187	7	un(k)+1	un(k)+1	NOUN
ma-221	187	8	)	)	PUNCT
ma-221	188	1	+	+	CCONJ
ma-221	188	2	d	d	X
ma-221	188	3	(	(	PUNCT
ma-221	188	4	un(k	un(k	NOUN
ma-221	188	5	)	)	PUNCT
ma-221	188	6	,	,	PUNCT
ma-221	188	7	um(k)+1	um(k)+1	PROPN
ma-221	188	8	)	)	PUNCT
ma-221	188	9	)	)	PUNCT
ma-221	189	1			NOUN
ma-221	189	2			NOUN
ma-221	189	3	(	(	PUNCT
ma-221	189	4	3.15	3.15	NUM
ma-221	189	5	)	)	PUNCT
ma-221	189	6	letting	let	VERB
ma-221	189	7	letting	let	VERB
ma-221	189	8	k	k	PROPN
ma-221	189	9	→∞	→∞	PROPN
ma-221	189	10	in	in	ADP
ma-221	189	11	(	(	PUNCT
ma-221	189	12	3.15	3.15	NUM
ma-221	189	13	)	)	PUNCT
ma-221	189	14	,	,	PUNCT
ma-221	189	15	and	and	CCONJ
ma-221	189	16	using	use	VERB
ma-221	189	17	(	(	PUNCT
ma-221	189	18	θ1	θ1	NOUN
ma-221	189	19	)	)	PUNCT
ma-221	189	20	,	,	PUNCT
ma-221	189	21	(	(	PUNCT
ma-221	189	22	θ3	θ3	NOUN
ma-221	189	23	)	)	PUNCT
ma-221	189	24	,	,	PUNCT
ma-221	189	25	(	(	PUNCT
ma-221	189	26	φ3	φ3	PROPN
ma-221	189	27	)	)	PUNCT
ma-221	189	28	and	and	CCONJ
ma-221	189	29	lemma	lemma	PROPN
ma-221	189	30	(	(	PUNCT
ma-221	189	31	2.6	2.6	NUM
ma-221	189	32	)	)	PUNCT
ma-221	189	33	we	we	PRON
ma-221	189	34	obtain	obtain	VERB
ma-221	189	35	θ	θ	PROPN
ma-221	189	36	(	(	PUNCT
ma-221	189	37	ε	ε	PROPN
ma-221	189	38	)	)	PUNCT
ma-221	189	39	≤	≤	NOUN
ma-221	190	1	φ	φ	PROPN
ma-221	191	1	[	[	X
ma-221	191	2	θ	θ	X
ma-221	191	3	(	(	PUNCT
ma-221	191	4	aε+	aε+	PROPN
ma-221	191	5	bε+	bε+	NOUN
ma-221	191	6	cε+	cε+	PROPN
ma-221	191	7	2hε	2hε	NUM
ma-221	191	8	)	)	PUNCT
ma-221	191	9	]	]	PUNCT
ma-221	191	10	.	.	PUNCT
ma-221	192	1	we	we	PRON
ma-221	192	2	derive	derive	VERB
ma-221	192	3	ε	ε	PROPN
ma-221	192	4	<	<	X
ma-221	192	5	ε	ε	PROPN
ma-221	192	6	.	.	PROPN
ma-221	192	7	which	which	PRON
ma-221	192	8	is	be	AUX
ma-221	192	9	a	a	DET
ma-221	192	10	contradiction	contradiction	NOUN
ma-221	192	11	.	.	PUNCT
ma-221	193	1	thus	thus	ADV
ma-221	193	2	limn	limn	ADJ
ma-221	193	3	,	,	PUNCT
ma-221	193	4	m→∞	m→∞	NOUN
ma-221	193	5	d	d	NOUN
ma-221	193	6	(	(	PUNCT
ma-221	193	7	un	un	PROPN
ma-221	193	8	,	,	PUNCT
ma-221	193	9	um	um	INTJ
ma-221	193	10	)	)	PUNCT
ma-221	193	11	=	=	SYM
ma-221	193	12	0	0	NUM
ma-221	193	13	,	,	PUNCT
ma-221	193	14	which	which	PRON
ma-221	193	15	shows	show	VERB
ma-221	193	16	that	that	SCONJ
ma-221	193	17	{	{	PUNCT
ma-221	193	18	xn	xn	X
ma-221	193	19	}	}	PUNCT
ma-221	193	20	is	be	AUX
ma-221	193	21	a	a	DET
ma-221	193	22	cauchysequence	cauchysequence	NOUN
ma-221	193	23	.	.	PUNCT
ma-221	194	1	then	then	ADV
ma-221	194	2	there	there	PRON
ma-221	194	3	exists	exist	VERB
ma-221	194	4	z	z	PROPN
ma-221	194	5	∈	∈	PROPN
ma-221	194	6	a	a	DET
ma-221	194	7	such	such	ADJ
ma-221	194	8	that	that	SCONJ
ma-221	194	9	lim	lim	PROPN
ma-221	194	10	n→∞	n→∞	PROPN
ma-221	195	1	d	d	NOUN
ma-221	195	2	(	(	PUNCT
ma-221	195	3	un	un	PROPN
ma-221	195	4	,	,	PUNCT
ma-221	195	5	u	u	NOUN
ma-221	195	6	)	)	PUNCT
ma-221	195	7	=	=	SYM
ma-221	196	1	0	0	X
ma-221	196	2	.	.	PUNCT
ma-221	197	1	also	also	ADV
ma-221	197	2	,	,	PUNCT
ma-221	197	3	d	d	X
ma-221	197	4	(	(	PUNCT
ma-221	197	5	u	u	NOUN
ma-221	197	6	,	,	PUNCT
ma-221	197	7	b	b	NOUN
ma-221	197	8	)	)	PUNCT
ma-221	197	9	≤	≤	NUM
ma-221	197	10	d	d	PROPN
ma-221	197	11	(	(	PUNCT
ma-221	197	12	u	u	NOUN
ma-221	197	13	,	,	PUNCT
ma-221	197	14	tun	tun	NOUN
ma-221	197	15	)	)	PUNCT
ma-221	197	16	≤	≤	NOUN
ma-221	197	17	d	d	X
ma-221	197	18	(	(	PUNCT
ma-221	197	19	u	u	NOUN
ma-221	197	20	,	,	PUNCT
ma-221	197	21	xn+1	xn+1	NUM
ma-221	197	22	)	)	PUNCT
ma-221	198	1	+	+	CCONJ
ma-221	198	2	d	d	X
ma-221	198	3	(	(	PUNCT
ma-221	198	4	un+1	un+1	PROPN
ma-221	198	5	,	,	PUNCT
ma-221	198	6	t	t	PROPN
ma-221	198	7	un	un	PROPN
ma-221	198	8	)	)	PUNCT
ma-221	198	9	=	=	SYM
ma-221	199	1	d	d	PROPN
ma-221	199	2	(	(	PUNCT
ma-221	199	3	u	u	NOUN
ma-221	199	4	,	,	PUNCT
ma-221	199	5	un+1	un+1	NOUN
ma-221	199	6	)	)	PUNCT
ma-221	200	1	+	+	CCONJ
ma-221	200	2	d	d	X
ma-221	200	3	(	(	PUNCT
ma-221	200	4	a	a	DET
ma-221	200	5	,	,	PUNCT
ma-221	200	6	b	b	NOUN
ma-221	200	7	)	)	PUNCT
ma-221	200	8	≤	≤	NUM
ma-221	201	1	d	d	PROPN
ma-221	201	2	(	(	PUNCT
ma-221	201	3	u	u	NOUN
ma-221	201	4	,	,	PUNCT
ma-221	201	5	un+1	un+1	NOUN
ma-221	201	6	)	)	PUNCT
ma-221	202	1	+	+	CCONJ
ma-221	202	2	d	d	X
ma-221	202	3	(	(	PUNCT
ma-221	202	4	u	u	NOUN
ma-221	202	5	,	,	PUNCT
ma-221	202	6	b	b	NOUN
ma-221	202	7	)	)	PUNCT
ma-221	202	8	.	.	PUNCT
ma-221	203	1	therefore	therefore	ADV
ma-221	203	2	,	,	PUNCT
ma-221	203	3	d	d	X
ma-221	203	4	(	(	PUNCT
ma-221	203	5	u	u	NOUN
ma-221	203	6	,	,	PUNCT
ma-221	203	7	tun)→	tun)→	NOUN
ma-221	203	8	d	d	X
ma-221	203	9	(	(	PUNCT
ma-221	203	10	u	u	NOUN
ma-221	203	11	,	,	PUNCT
ma-221	203	12	b	b	NOUN
ma-221	203	13	)	)	PUNCT
ma-221	203	14	.	.	PUNCT
ma-221	204	1	in	in	ADP
ma-221	204	2	spite	spite	NOUN
ma-221	204	3	of	of	ADP
ma-221	204	4	the	the	DET
ma-221	204	5	fact	fact	NOUN
ma-221	204	6	that	that	SCONJ
ma-221	204	7	b	b	NOUN
ma-221	204	8	is	be	AUX
ma-221	204	9	approximately	approximately	ADV
ma-221	204	10	compact	compact	ADJ
ma-221	204	11	with	with	ADP
ma-221	204	12	respectto	respectto	ADJ
ma-221	204	13	a	a	PRON
ma-221	204	14	,	,	PUNCT
ma-221	204	15	the	the	DET
ma-221	204	16	sequence	sequence	NOUN
ma-221	204	17	{	{	PUNCT
ma-221	204	18	tun	tun	NOUN
ma-221	204	19	}	}	PUNCT
ma-221	204	20	has	have	VERB
ma-221	204	21	a	a	DET
ma-221	204	22	subsequence	subsequence	NOUN
ma-221	204	23	{	{	PUNCT
ma-221	204	24	tunk	tunk	NOUN
ma-221	204	25	}	}	PUNCT
ma-221	204	26	converging	converge	VERB
ma-221	204	27	to	to	ADP
ma-221	204	28	some	some	DET
ma-221	204	29	element	element	NOUN
ma-221	204	30	v	v	ADP
ma-221	204	31	∈	∈	PROPN
ma-221	204	32	b.	b.	PROPN
ma-221	205	1	so	so	ADV
ma-221	205	2	itturns	itturn	VERB
ma-221	205	3	out	out	ADP
ma-221	205	4	that	that	SCONJ
ma-221	205	5	d(u	d(u	PROPN
ma-221	205	6	,	,	PUNCT
ma-221	205	7	v	v	NOUN
ma-221	205	8	)	)	PUNCT
ma-221	205	9	=	=	VERB
ma-221	205	10	lim	lim	PROPN
ma-221	205	11	n→∞	n→∞	X
ma-221	206	1	d	d	NOUN
ma-221	206	2	(	(	PUNCT
ma-221	206	3	unk+1	unk+1	PROPN
ma-221	206	4	,	,	PUNCT
ma-221	206	5	t	t	PROPN
ma-221	206	6	unk	unk	NOUN
ma-221	206	7	)	)	PUNCT
ma-221	206	8	=	=	SYM
ma-221	207	1	d(a	d(a	PROPN
ma-221	207	2	,	,	PUNCT
ma-221	207	3	b	b	NOUN
ma-221	207	4	)	)	PUNCT
ma-221	207	5	.	.	PUNCT
ma-221	208	1	(	(	PUNCT
ma-221	208	2	3.16	3.16	NUM
ma-221	208	3	)	)	PUNCT
ma-221	208	4	thus	thus	ADV
ma-221	208	5	u	u	PRON
ma-221	208	6	must	must	AUX
ma-221	208	7	be	be	AUX
ma-221	208	8	an	an	DET
ma-221	208	9	element	element	NOUN
ma-221	208	10	of	of	ADP
ma-221	208	11	a0	a0	PROPN
ma-221	208	12	.	.	PUNCT
ma-221	209	1	again	again	ADV
ma-221	209	2	,	,	PUNCT
ma-221	209	3	since	since	SCONJ
ma-221	209	4	t	t	PROPN
ma-221	209	5	(	(	PUNCT
ma-221	209	6	a0	a0	PROPN
ma-221	209	7	)	)	PUNCT
ma-221	209	8	∈	∈	PROPN
ma-221	209	9	b0	b0	NOUN
ma-221	209	10	,	,	PUNCT
ma-221	209	11	there	there	PRON
ma-221	209	12	exists	exist	VERB
ma-221	209	13	t	t	PROPN
ma-221	209	14	∈	∈	PROPN
ma-221	209	15	a0	a0	PROPN
ma-221	209	16	such	such	ADJ
ma-221	209	17	that	that	SCONJ
ma-221	209	18	d(t	d(t	PROPN
ma-221	209	19	,	,	PUNCT
ma-221	209	20	tu	tu	PROPN
ma-221	209	21	)	)	PUNCT
ma-221	209	22	=	=	SYM
ma-221	210	1	d(a	d(a	PROPN
ma-221	210	2	,	,	PUNCT
ma-221	210	3	b	b	NOUN
ma-221	210	4	)	)	PUNCT
ma-221	210	5	(	(	PUNCT
ma-221	210	6	3.17	3.17	NUM
ma-221	210	7	)	)	PUNCT
ma-221	210	8	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	210	9	eur	eur	PROPN
ma-221	210	10	.	.	PUNCT
ma-221	211	1	j.	j.	PROPN
ma-221	211	2	math	math	PROPN
ma-221	211	3	.	.	PUNCT
ma-221	212	1	anal	anal	PROPN
ma-221	212	2	.	.	PUNCT
ma-221	213	1	10.28924	10.28924	NUM
ma-221	213	2	/	/	SYM
ma-221	213	3	ada	ada	PROPN
ma-221	213	4	/	/	SYM
ma-221	213	5	ma.4.13	ma.4.13	PROPN
ma-221	213	6	7for	7for	NUM
ma-221	213	7	some	some	DET
ma-221	213	8	element	element	NOUN
ma-221	213	9	t	t	NOUN
ma-221	213	10	in	in	ADP
ma-221	213	11	a.	a.	NOUN
ma-221	213	12	using	use	VERB
ma-221	213	13	the	the	DET
ma-221	213	14	weak	weak	ADJ
ma-221	213	15	p	p	NOUN
ma-221	213	16	-	-	PUNCT
ma-221	213	17	property	property	NOUN
ma-221	213	18	and	and	CCONJ
ma-221	213	19	(	(	PUNCT
ma-221	213	20	3.33	3.33	NUM
ma-221	213	21	)	)	PUNCT
ma-221	213	22	we	we	PRON
ma-221	213	23	have	have	VERB
ma-221	213	24	d(unk+1	d(unk+1	NOUN
ma-221	213	25	,	,	PUNCT
ma-221	213	26	t	t	PROPN
ma-221	213	27	)	)	PUNCT
ma-221	213	28	=	=	VERB
ma-221	214	1	d(punk	d(punk	VERB
ma-221	214	2	,	,	PUNCT
ma-221	214	3	p	p	PROPN
ma-221	214	4	u),∀nk	u),∀nk	PROPN
ma-221	214	5	∈	∈	PROPN
ma-221	214	6	n.	n.	NOUN
ma-221	214	7	if	if	SCONJ
ma-221	214	8	for	for	ADP
ma-221	214	9	some	some	DET
ma-221	214	10	n0	n0	PROPN
ma-221	214	11	,	,	PUNCT
ma-221	214	12	d(t	d(t	PROPN
ma-221	214	13	,	,	PUNCT
ma-221	214	14	un0	un0	NOUN
ma-221	214	15	+	+	NOUN
ma-221	214	16	1	1	NUM
ma-221	214	17	)	)	PUNCT
ma-221	214	18	=	=	SYM
ma-221	214	19	0	0	NUM
ma-221	214	20	,	,	PUNCT
ma-221	214	21	consequently	consequently	ADV
ma-221	214	22	d(pun0	d(pun0	NOUN
ma-221	214	23	,	,	PUNCT
ma-221	214	24	t	t	PROPN
ma-221	214	25	u	u	NOUN
ma-221	214	26	)	)	PUNCT
ma-221	214	27	=	=	SYM
ma-221	215	1	0	0	X
ma-221	215	2	.	.	PUNCT
ma-221	216	1	so	so	ADV
ma-221	216	2	pun0	pun0	NOUN
ma-221	216	3	=	=	SYM
ma-221	216	4	tu	tu	PROPN
ma-221	216	5	,	,	PUNCT
ma-221	216	6	hence	hence	ADV
ma-221	216	7	d(a	d(a	PROPN
ma-221	216	8	,	,	PUNCT
ma-221	216	9	b	b	NOUN
ma-221	216	10	)	)	PUNCT
ma-221	216	11	=	=	SYM
ma-221	216	12	d(u	d(u	PROPN
ma-221	216	13	,	,	PUNCT
ma-221	216	14	tu	tu	PROPN
ma-221	216	15	)	)	PUNCT
ma-221	216	16	.	.	PUNCT
ma-221	217	1	thus	thus	ADV
ma-221	217	2	the	the	DET
ma-221	217	3	conclusion	conclusion	NOUN
ma-221	217	4	is	be	AUX
ma-221	217	5	immediate	immediate	ADJ
ma-221	217	6	.	.	PUNCT
ma-221	218	1	so	so	ADV
ma-221	218	2	let	let	VERB
ma-221	218	3	for	for	ADP
ma-221	218	4	any	any	DET
ma-221	218	5	n	n	PRON
ma-221	218	6	≥	≥	NOUN
ma-221	218	7	0	0	NUM
ma-221	218	8	,	,	PUNCT
ma-221	218	9	d(t	d(t	PROPN
ma-221	218	10	,	,	PUNCT
ma-221	218	11	un+1	un+1	NOUN
ma-221	218	12	)	)	PUNCT
ma-221	218	13	>	>	X
ma-221	219	1	0	0	X
ma-221	219	2	.	.	PUNCT
ma-221	220	1	since	since	SCONJ
ma-221	220	2	t	t	PROPN
ma-221	220	3	is	be	AUX
ma-221	220	4	ageneralized	ageneralize	VERB
ma-221	220	5	(	(	PUNCT
ma-221	220	6	θ	θ	NOUN
ma-221	220	7	,	,	PUNCT
ma-221	220	8	φ)-proximal	φ)-proximal	ADJ
ma-221	220	9	contraction	contraction	NOUN
ma-221	220	10	of	of	ADP
ma-221	220	11	the	the	DET
ma-221	220	12	first	first	ADJ
ma-221	220	13	kind	kind	NOUN
ma-221	220	14	,	,	PUNCT
ma-221	220	15	it	it	PRON
ma-221	220	16	follows	follow	VERB
ma-221	220	17	from	from	ADP
ma-221	220	18	this	this	PRON
ma-221	220	19	that	that	SCONJ
ma-221	220	20	θ(d(t	θ(d(t	NOUN
ma-221	220	21	,	,	PUNCT
ma-221	220	22	un+1	un+1	NOUN
ma-221	220	23	)	)	PUNCT
ma-221	220	24	)	)	PUNCT
ma-221	221	1	≤	≤	NUM
ma-221	221	2	φ	φ	PROPN
ma-221	221	3	[	[	X
ma-221	221	4	θ	θ	X
ma-221	221	5	[	[	X
ma-221	221	6	ad	ad	X
ma-221	221	7	(	(	PUNCT
ma-221	221	8	u	u	NOUN
ma-221	221	9	,	,	PUNCT
ma-221	221	10	un	un	PROPN
ma-221	221	11	)	)	PUNCT
ma-221	222	1	+	+	CCONJ
ma-221	222	2	bd	bd	PROPN
ma-221	222	3	(	(	PUNCT
ma-221	222	4	t	t	PROPN
ma-221	222	5	,	,	PUNCT
ma-221	222	6	u	u	NOUN
ma-221	222	7	)	)	PUNCT
ma-221	222	8	+	+	CCONJ
ma-221	222	9	cd	cd	PROPN
ma-221	222	10	(	(	PUNCT
ma-221	222	11	un	un	PROPN
ma-221	222	12	,	,	PUNCT
ma-221	222	13	un+1	un+1	NOUN
ma-221	222	14	)	)	PUNCT
ma-221	222	15	+	+	NUM
ma-221	223	1	h	h	NOUN
ma-221	223	2	(	(	PUNCT
ma-221	223	3	d	d	X
ma-221	223	4	(	(	PUNCT
ma-221	223	5	u	u	NOUN
ma-221	223	6	,	,	PUNCT
ma-221	223	7	un+1	un+1	NOUN
ma-221	223	8	)	)	PUNCT
ma-221	224	1	+	+	CCONJ
ma-221	224	2	d	d	PROPN
ma-221	224	3	(	(	PUNCT
ma-221	224	4	un	un	PROPN
ma-221	224	5	,	,	PUNCT
ma-221	224	6	t	t	PROPN
ma-221	224	7	)	)	PUNCT
ma-221	224	8	)	)	PUNCT
ma-221	225	1	]	]	PUNCT
ma-221	225	2	]	]	X
ma-221	225	3	(	(	PUNCT
ma-221	225	4	3.18	3.18	NUM
ma-221	225	5	)	)	PUNCT
ma-221	225	6	since	since	SCONJ
ma-221	225	7	θ	θ	PROPN
ma-221	225	8	and	and	CCONJ
ma-221	225	9	φ	φ	PROPN
ma-221	225	10	are	be	AUX
ma-221	225	11	two	two	NUM
ma-221	225	12	continuous	continuous	ADJ
ma-221	225	13	functions	function	NOUN
ma-221	225	14	,	,	PUNCT
ma-221	225	15	by	by	ADP
ma-221	225	16	letting	let	VERB
ma-221	225	17	n	n	PRON
ma-221	225	18	→∞	→∞	PROPN
ma-221	225	19	in	in	ADP
ma-221	225	20	inequality	inequality	NOUN
ma-221	225	21	(	(	PUNCT
ma-221	225	22	3.18	3.18	NUM
ma-221	225	23	)	)	PUNCT
ma-221	225	24	,	,	PUNCT
ma-221	225	25	we	we	PRON
ma-221	225	26	obtain	obtain	VERB
ma-221	225	27	θ(d(t	θ(d(t	NOUN
ma-221	225	28	,	,	PUNCT
ma-221	225	29	u	u	NOUN
ma-221	225	30	)	)	PUNCT
ma-221	225	31	)	)	PUNCT
ma-221	226	1	≤	≤	NUM
ma-221	226	2	φ	φ	PROPN
ma-221	226	3	[	[	X
ma-221	226	4	θ	θ	X
ma-221	226	5	[	[	X
ma-221	226	6	(	(	PUNCT
ma-221	226	7	b	b	NOUN
ma-221	226	8	+	+	NUM
ma-221	226	9	h	h	NOUN
ma-221	226	10	)	)	PUNCT
ma-221	226	11	(	(	PUNCT
ma-221	226	12	d	d	X
ma-221	226	13	(	(	PUNCT
ma-221	226	14	u	u	NOUN
ma-221	226	15	,	,	PUNCT
ma-221	226	16	t	t	PROPN
ma-221	226	17	)	)	PUNCT
ma-221	226	18	)	)	PUNCT
ma-221	227	1	]	]	PUNCT
ma-221	227	2	]	]	X
ma-221	227	3	≤	≤	NUM
ma-221	227	4	φ	φ	PROPN
ma-221	227	5	[	[	X
ma-221	227	6	θ	θ	X
ma-221	227	7	[	[	X
ma-221	227	8	(	(	PUNCT
ma-221	227	9	d	d	X
ma-221	227	10	(	(	PUNCT
ma-221	227	11	u	u	NOUN
ma-221	227	12	,	,	PUNCT
ma-221	227	13	t	t	PROPN
ma-221	227	14	)	)	PUNCT
ma-221	227	15	)	)	PUNCT
ma-221	228	1	]	]	PUNCT
ma-221	228	2	]	]	X
ma-221	228	3	<	<	X
ma-221	228	4	θ(d(t	θ(d(t	NOUN
ma-221	228	5	,	,	PUNCT
ma-221	228	6	u	u	NOUN
ma-221	228	7	)	)	PUNCT
ma-221	228	8	)	)	PUNCT
ma-221	228	9	.	.	PUNCT
ma-221	229	1	it	it	PRON
ma-221	229	2	is	be	AUX
ma-221	229	3	a	a	DET
ma-221	229	4	contradiction	contradiction	NOUN
ma-221	229	5	.	.	PUNCT
ma-221	230	1	therefore	therefore	ADV
ma-221	230	2	,	,	PUNCT
ma-221	230	3	u	u	PROPN
ma-221	230	4	=	=	PROPN
ma-221	230	5	t	t	PROPN
ma-221	230	6	,	,	PUNCT
ma-221	230	7	that	that	SCONJ
ma-221	230	8	d(u	d(u	PROPN
ma-221	230	9	,	,	PUNCT
ma-221	230	10	tu	tu	PROPN
ma-221	230	11	)	)	PUNCT
ma-221	230	12	=	=	SYM
ma-221	230	13	d(t	d(t	PROPN
ma-221	230	14	,	,	PUNCT
ma-221	230	15	tu	tu	PROPN
ma-221	230	16	)	)	PUNCT
ma-221	230	17	=	=	SYM
ma-221	231	1	d(a	d(a	PROPN
ma-221	231	2	,	,	PUNCT
ma-221	231	3	b	b	NOUN
ma-221	231	4	)	)	PUNCT
ma-221	231	5	.	.	PUNCT
ma-221	232	1	uniqueness	uniqueness	NOUN
ma-221	232	2	:	:	PUNCT
ma-221	232	3	suppose	suppose	VERB
ma-221	232	4	that	that	SCONJ
ma-221	232	5	there	there	PRON
ma-221	232	6	is	be	VERB
ma-221	232	7	another	another	DET
ma-221	232	8	best	good	ADJ
ma-221	232	9	proximity	proximity	NOUN
ma-221	232	10	point	point	NOUN
ma-221	232	11	z	z	NOUN
ma-221	232	12	of	of	ADP
ma-221	232	13	the	the	DET
ma-221	232	14	mapping	mapping	NOUN
ma-221	232	15	t	t	NOUN
ma-221	232	16	such	such	ADJ
ma-221	232	17	that	that	SCONJ
ma-221	232	18	d(z	d(z	PROPN
ma-221	232	19	,	,	PUNCT
ma-221	232	20	t	t	PROPN
ma-221	232	21	z	z	PROPN
ma-221	232	22	)	)	PUNCT
ma-221	232	23	=	=	SYM
ma-221	233	1	d(a	d(a	PROPN
ma-221	233	2	,	,	PUNCT
ma-221	233	3	b	b	NOUN
ma-221	233	4	)	)	PUNCT
ma-221	233	5	.	.	PUNCT
ma-221	234	1	since	since	SCONJ
ma-221	234	2	t	t	PROPN
ma-221	234	3	is	be	AUX
ma-221	234	4	a	a	DET
ma-221	234	5	generalized	generalized	ADJ
ma-221	234	6	(	(	PUNCT
ma-221	234	7	θ	θ	NOUN
ma-221	234	8	,	,	PUNCT
ma-221	234	9	φ)-proximal	φ)-proximal	ADJ
ma-221	234	10	contraction	contraction	NOUN
ma-221	234	11	of	of	ADP
ma-221	234	12	the	the	DET
ma-221	234	13	first	first	ADJ
ma-221	234	14	kind	kind	NOUN
ma-221	234	15	,	,	PUNCT
ma-221	234	16	it	it	PRON
ma-221	234	17	follows	follow	VERB
ma-221	234	18	from	from	ADP
ma-221	234	19	this	this	PRON
ma-221	234	20	that	that	SCONJ
ma-221	234	21	θ(d(z	θ(d(z	PROPN
ma-221	234	22	,	,	PUNCT
ma-221	234	23	u	u	NOUN
ma-221	234	24	)	)	PUNCT
ma-221	234	25	)	)	PUNCT
ma-221	234	26	≤	≤	NUM
ma-221	235	1	φ	φ	PROPN
ma-221	236	1	[	[	X
ma-221	236	2	θ	θ	X
ma-221	237	1	[	[	X
ma-221	237	2	ad	ad	X
ma-221	237	3	(	(	PUNCT
ma-221	237	4	z	z	NOUN
ma-221	237	5	,	,	PUNCT
ma-221	237	6	u	u	NOUN
ma-221	237	7	)	)	PUNCT
ma-221	237	8	+	+	CCONJ
ma-221	237	9	bd	bd	PROPN
ma-221	237	10	(	(	PUNCT
ma-221	237	11	z	z	NOUN
ma-221	237	12	,	,	PUNCT
ma-221	237	13	z	z	NOUN
ma-221	237	14	)	)	PUNCT
ma-221	237	15	+	+	CCONJ
ma-221	237	16	cd	cd	PROPN
ma-221	237	17	(	(	PUNCT
ma-221	237	18	u	u	NOUN
ma-221	237	19	,	,	PUNCT
ma-221	237	20	u	u	NOUN
ma-221	237	21	)	)	PUNCT
ma-221	237	22	+	+	NUM
ma-221	237	23	h	h	NOUN
ma-221	237	24	(	(	PUNCT
ma-221	237	25	d	d	X
ma-221	237	26	(	(	PUNCT
ma-221	237	27	z	z	NOUN
ma-221	237	28	,	,	PUNCT
ma-221	237	29	u	u	NOUN
ma-221	237	30	)	)	PUNCT
ma-221	238	1	+	+	CCONJ
ma-221	238	2	d	d	X
ma-221	238	3	(	(	PUNCT
ma-221	238	4	z	z	NOUN
ma-221	238	5	,	,	PUNCT
ma-221	238	6	u	u	NOUN
ma-221	238	7	)	)	PUNCT
ma-221	238	8	)	)	PUNCT
ma-221	239	1	]	]	PUNCT
ma-221	239	2	]	]	X
ma-221	239	3	=	=	PUNCT
ma-221	239	4	φ	φ	PROPN
ma-221	240	1	[	[	X
ma-221	240	2	θ	θ	X
ma-221	240	3	[	[	X
ma-221	240	4	(	(	PUNCT
ma-221	240	5	a	a	DET
ma-221	240	6	+	+	NUM
ma-221	240	7	2h)d	2h)d	NUM
ma-221	240	8	(	(	PUNCT
ma-221	240	9	z	z	NOUN
ma-221	240	10	,	,	PUNCT
ma-221	240	11	u	u	NOUN
ma-221	240	12	)	)	PUNCT
ma-221	240	13	]	]	PUNCT
ma-221	240	14	]	]	PUNCT
ma-221	240	15	,	,	PUNCT
ma-221	240	16	which	which	PRON
ma-221	240	17	is	be	AUX
ma-221	240	18	a	a	DET
ma-221	240	19	contradiction	contradiction	NOUN
ma-221	240	20	.	.	PUNCT
ma-221	241	1	thus	thus	ADV
ma-221	241	2	,	,	PUNCT
ma-221	241	3	z	z	NOUN
ma-221	241	4	and	and	CCONJ
ma-221	241	5	u	u	NOUN
ma-221	241	6	must	must	AUX
ma-221	241	7	be	be	AUX
ma-221	241	8	identical	identical	ADJ
ma-221	241	9	.	.	PUNCT
ma-221	242	1	hence	hence	ADV
ma-221	242	2	,	,	PUNCT
ma-221	242	3	t	t	PROPN
ma-221	242	4	has	have	VERB
ma-221	242	5	a	a	DET
ma-221	242	6	unique	unique	ADJ
ma-221	242	7	best	good	ADJ
ma-221	242	8	proximitypoint	proximitypoint	NOUN
ma-221	242	9	.	.	PUNCT
ma-221	243	1	�	�	PROPN
ma-221	243	2	next	next	ADV
ma-221	243	3	,	,	PUNCT
ma-221	243	4	we	we	PRON
ma-221	243	5	state	state	VERB
ma-221	243	6	and	and	CCONJ
ma-221	243	7	prove	prove	VERB
ma-221	243	8	the	the	DET
ma-221	243	9	best	good	ADJ
ma-221	243	10	proximity	proximity	NOUN
ma-221	243	11	point	point	NOUN
ma-221	243	12	theorem	theorem	NOUN
ma-221	243	13	for	for	ADP
ma-221	243	14	non	non	ADJ
ma-221	243	15	-	-	ADJ
ma-221	243	16	self	self	ADJ
ma-221	243	17	generalized	generalized	ADJ
ma-221	243	18	(	(	PUNCT
ma-221	243	19	θ	θ	NOUN
ma-221	243	20	,	,	PUNCT
ma-221	243	21	φ)-proximalcontraction	φ)-proximalcontraction	NOUN
ma-221	243	22	of	of	ADP
ma-221	243	23	the	the	DET
ma-221	243	24	second	second	ADJ
ma-221	243	25	kind	kind	NOUN
ma-221	243	26	.	.	PUNCT
ma-221	244	1	theorem	theorem	VERB
ma-221	244	2	3.4	3.4	NUM
ma-221	244	3	.	.	PUNCT
ma-221	245	1	let	let	AUX
ma-221	245	2	(	(	PUNCT
ma-221	245	3	x	x	NOUN
ma-221	245	4	,	,	PUNCT
ma-221	245	5	d	d	NOUN
ma-221	245	6	)	)	PUNCT
ma-221	245	7	be	be	AUX
ma-221	245	8	a	a	DET
ma-221	245	9	complete	complete	ADJ
ma-221	245	10	metric	metric	ADJ
ma-221	245	11	space	space	NOUN
ma-221	245	12	and	and	CCONJ
ma-221	245	13	(	(	PUNCT
ma-221	245	14	a	a	DET
ma-221	245	15	,	,	PUNCT
ma-221	245	16	b	b	NOUN
ma-221	245	17	)	)	PUNCT
ma-221	245	18	be	be	AUX
ma-221	245	19	a	a	DET
ma-221	245	20	pair	pair	NOUN
ma-221	245	21	of	of	ADP
ma-221	245	22	non	non	ADJ
ma-221	245	23	-	-	ADJ
ma-221	245	24	void	void	ADJ
ma-221	245	25	closed	closed	ADJ
ma-221	245	26	subsets	subset	NOUN
ma-221	245	27	of	of	ADP
ma-221	245	28	(	(	PUNCT
ma-221	245	29	x	x	X
ma-221	245	30	,	,	PUNCT
ma-221	245	31	d	d	NOUN
ma-221	245	32	)	)	PUNCT
ma-221	245	33	.	.	PUNCT
ma-221	246	1	if	if	SCONJ
ma-221	246	2	a	a	PRON
ma-221	246	3	is	be	AUX
ma-221	246	4	approximately	approximately	ADV
ma-221	246	5	compact	compact	ADJ
ma-221	246	6	with	with	ADP
ma-221	246	7	respect	respect	NOUN
ma-221	246	8	to	to	ADP
ma-221	246	9	b	b	NOUN
ma-221	246	10	and	and	CCONJ
ma-221	246	11	t	t	PROPN
ma-221	246	12	:	:	PUNCT
ma-221	246	13	a	a	DET
ma-221	246	14	→	→	SYM
ma-221	246	15	b	b	NOUN
ma-221	246	16	satisfy	satisfy	NOUN
ma-221	246	17	the	the	DET
ma-221	246	18	following	follow	VERB
ma-221	246	19	conditions	condition	NOUN
ma-221	246	20	:(	:(	PUNCT
ma-221	247	1	i	i	PRON
ma-221	247	2	)	)	PUNCT
ma-221	247	3	t	t	PROPN
ma-221	247	4	(	(	PUNCT
ma-221	247	5	a0	a0	PROPN
ma-221	247	6	)	)	PUNCT
ma-221	247	7	∈	∈	PROPN
ma-221	247	8	b0	b0	NOUN
ma-221	247	9	and	and	CCONJ
ma-221	247	10	the	the	DET
ma-221	247	11	pair	pair	NOUN
ma-221	247	12	(	(	PUNCT
ma-221	247	13	a	a	DET
ma-221	247	14	,	,	PUNCT
ma-221	247	15	b	b	NOUN
ma-221	247	16	)	)	PUNCT
ma-221	247	17	satisfies	satisfy	VERB
ma-221	247	18	the	the	DET
ma-221	247	19	weak	weak	ADJ
ma-221	247	20	p	p	NOUN
ma-221	247	21	-property;(ii	-property;(ii	PROPN
ma-221	247	22	)	)	PUNCT
ma-221	247	23	t	t	PROPN
ma-221	247	24	is	be	AUX
ma-221	247	25	continuous	continuous	ADJ
ma-221	247	26	generalized	generalized	ADJ
ma-221	247	27	(	(	PUNCT
ma-221	247	28	θ	θ	NOUN
ma-221	247	29	,	,	PUNCT
ma-221	247	30	φ)-proximal	φ)-proximal	ADJ
ma-221	247	31	contraction	contraction	NOUN
ma-221	247	32	of	of	ADP
ma-221	247	33	second	second	ADJ
ma-221	247	34	kind	kind	NOUN
ma-221	247	35	.	.	PUNCT
ma-221	248	1	then	then	ADV
ma-221	248	2	there	there	PRON
ma-221	248	3	exists	exist	VERB
ma-221	248	4	a	a	DET
ma-221	248	5	unique	unique	ADJ
ma-221	248	6	u	u	NOUN
ma-221	248	7	∈	∈	PROPN
ma-221	248	8	a	a	DET
ma-221	248	9	such	such	ADJ
ma-221	248	10	that	that	SCONJ
ma-221	248	11	d(u	d(u	PROPN
ma-221	248	12	,	,	PUNCT
ma-221	248	13	tu	tu	PROPN
ma-221	248	14	)	)	PUNCT
ma-221	248	15	=	=	SYM
ma-221	249	1	d(a	d(a	PROPN
ma-221	249	2	,	,	PUNCT
ma-221	249	3	b	b	NOUN
ma-221	249	4	)	)	PUNCT
ma-221	249	5	and	and	CCONJ
ma-221	249	6	un	un	PROPN
ma-221	249	7	→	→	SYM
ma-221	249	8	u	u	PROPN
ma-221	249	9	,	,	PUNCT
ma-221	249	10	where	where	SCONJ
ma-221	249	11	u0	u0	ADJ
ma-221	249	12	is	be	AUX
ma-221	249	13	any	any	DET
ma-221	249	14	fixed	fixed	ADJ
ma-221	249	15	point	point	NOUN
ma-221	249	16	in	in	ADP
ma-221	249	17	a0	a0	PROPN
ma-221	249	18	and	and	CCONJ
ma-221	249	19	d(un+1	d(un+1	PROPN
ma-221	249	20	,	,	PUNCT
ma-221	249	21	t	t	PROPN
ma-221	249	22	un	un	PROPN
ma-221	249	23	)	)	PUNCT
ma-221	249	24	=	=	SYM
ma-221	250	1	d(a	d(a	PROPN
ma-221	250	2	,	,	PUNCT
ma-221	250	3	b	b	NOUN
ma-221	250	4	)	)	PUNCT
ma-221	250	5	for	for	ADP
ma-221	250	6	n	n	PRON
ma-221	250	7	≥	≥	NOUN
ma-221	250	8	0	0	NUM
ma-221	250	9	.	.	PUNCT
ma-221	251	1	further	far	ADV
ma-221	251	2	,	,	PUNCT
ma-221	251	3	if	if	SCONJ
ma-221	251	4	z	z	NOUN
ma-221	251	5	is	be	AUX
ma-221	251	6	another	another	DET
ma-221	251	7	best	good	ADJ
ma-221	251	8	proximity	proximity	NOUN
ma-221	251	9	point	point	NOUN
ma-221	251	10	of	of	ADP
ma-221	251	11	t	t	PROPN
ma-221	251	12	,	,	PUNCT
ma-221	251	13	then	then	ADV
ma-221	251	14	tu	tu	PROPN
ma-221	251	15	=	=	PUNCT
ma-221	251	16	tz	tz	PROPN
ma-221	251	17	.	.	PUNCT
ma-221	252	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	252	2	eur	eur	PROPN
ma-221	252	3	.	.	PUNCT
ma-221	253	1	j.	j.	PROPN
ma-221	253	2	math	math	PROPN
ma-221	253	3	.	.	PUNCT
ma-221	254	1	anal	anal	PROPN
ma-221	254	2	.	.	PUNCT
ma-221	255	1	10.28924	10.28924	NUM
ma-221	255	2	/	/	SYM
ma-221	255	3	ada	ada	PROPN
ma-221	255	4	/	/	SYM
ma-221	255	5	ma.4.13	ma.4.13	PROPN
ma-221	255	6	8	8	NUM
ma-221	255	7	proof	proof	NOUN
ma-221	255	8	.	.	PUNCT
ma-221	256	1	similar	similar	ADJ
ma-221	256	2	to	to	ADP
ma-221	256	3	theorem	theorem	VERB
ma-221	256	4	3.3	3.3	NUM
ma-221	256	5	,	,	PUNCT
ma-221	256	6	we	we	PRON
ma-221	256	7	can	can	AUX
ma-221	256	8	find	find	VERB
ma-221	256	9	a	a	DET
ma-221	256	10	sequence	sequence	NOUN
ma-221	256	11	{	{	PUNCT
ma-221	256	12	un	un	PROPN
ma-221	256	13	}	}	PUNCT
ma-221	256	14	in	in	ADP
ma-221	256	15	a0	a0	NOUN
ma-221	256	16	such	such	ADJ
ma-221	256	17	that	that	PRON
ma-221	256	18	d(un+1	d(un+1	PROPN
ma-221	256	19	,	,	PUNCT
ma-221	256	20	t	t	PROPN
ma-221	256	21	un	un	PROPN
ma-221	256	22	)	)	PUNCT
ma-221	256	23	=	=	SYM
ma-221	257	1	d(a	d(a	PROPN
ma-221	257	2	,	,	PUNCT
ma-221	257	3	b	b	NOUN
ma-221	257	4	)	)	PUNCT
ma-221	257	5	.	.	PUNCT
ma-221	258	1	(	(	PUNCT
ma-221	258	2	3.19	3.19	NUM
ma-221	258	3	)	)	PUNCT
ma-221	258	4	for	for	ADP
ma-221	258	5	all	all	DET
ma-221	258	6	non	non	ADJ
ma-221	258	7	-	-	ADJ
ma-221	258	8	negative	negative	ADJ
ma-221	258	9	integral	integral	ADJ
ma-221	258	10	values	value	NOUN
ma-221	258	11	of	of	ADP
ma-221	258	12	n.	n.	NOUN
ma-221	258	13	from	from	ADP
ma-221	258	14	the	the	DET
ma-221	258	15	p	p	NOUN
ma-221	258	16	-	-	PUNCT
ma-221	258	17	property	property	NOUN
ma-221	258	18	and	and	CCONJ
ma-221	258	19	(	(	PUNCT
ma-221	258	20	3.19	3.19	NUM
ma-221	258	21	)	)	PUNCT
ma-221	258	22	we	we	PRON
ma-221	258	23	get	get	VERB
ma-221	258	24	d(un	d(un	PROPN
ma-221	258	25	,	,	PUNCT
ma-221	258	26	un+1	un+1	NOUN
ma-221	258	27	)	)	PUNCT
ma-221	258	28	=	=	SYM
ma-221	258	29	d(tun−1	d(tun−1	PROPN
ma-221	258	30	,	,	PUNCT
ma-221	258	31	t	t	X
ma-221	258	32	un),∀n	un),∀n	X
ma-221	258	33	∈	∈	PROPN
ma-221	258	34	n.	n.	NOUN
ma-221	258	35	if	if	SCONJ
ma-221	258	36	for	for	ADP
ma-221	258	37	some	some	DET
ma-221	258	38	n0	n0	ADJ
ma-221	258	39	,	,	PUNCT
ma-221	258	40	d(un0	d(un0	PROPN
ma-221	258	41	+	+	ADJ
ma-221	258	42	1	1	NUM
ma-221	258	43	,	,	PUNCT
ma-221	258	44	un0	un0	NOUN
ma-221	258	45	+	+	NOUN
ma-221	258	46	2	2	NUM
ma-221	258	47	)	)	PUNCT
ma-221	258	48	=	=	SYM
ma-221	258	49	0	0	NUM
ma-221	258	50	,	,	PUNCT
ma-221	258	51	consequently	consequently	ADV
ma-221	258	52	d(tun0	d(tun0	NOUN
ma-221	258	53	,	,	PUNCT
ma-221	258	54	t	t	PROPN
ma-221	258	55	un0	un0	NOUN
ma-221	258	56	+	+	PROPN
ma-221	258	57	1	1	NUM
ma-221	258	58	)	)	PUNCT
ma-221	258	59	=	=	SYM
ma-221	259	1	0	0	X
ma-221	259	2	.	.	PUNCT
ma-221	260	1	so	so	ADV
ma-221	260	2	tun0	tun0	VERB
ma-221	260	3	=	=	PUNCT
ma-221	260	4	tun0	tun0	PROPN
ma-221	260	5	+	+	PROPN
ma-221	260	6	1,hence	1,hence	PROPN
ma-221	260	7	d(a	d(a	PROPN
ma-221	260	8	,	,	PUNCT
ma-221	260	9	b	b	NOUN
ma-221	260	10	)	)	PUNCT
ma-221	260	11	=	=	SYM
ma-221	261	1	d(tun0	d(tun0	NOUN
ma-221	261	2	,	,	PUNCT
ma-221	261	3	tn0	tn0	NOUN
ma-221	261	4	+	+	NOUN
ma-221	261	5	1	1	NUM
ma-221	261	6	)	)	PUNCT
ma-221	261	7	.	.	PUNCT
ma-221	262	1	thus	thus	ADV
ma-221	262	2	the	the	DET
ma-221	262	3	conclusion	conclusion	NOUN
ma-221	262	4	is	be	AUX
ma-221	262	5	immediate	immediate	ADJ
ma-221	262	6	.	.	PUNCT
ma-221	263	1	so	so	ADV
ma-221	263	2	let	let	VERB
ma-221	263	3	for	for	ADP
ma-221	263	4	any	any	DET
ma-221	263	5	n	n	PRON
ma-221	263	6	≥	≥	NOUN
ma-221	263	7	0	0	NUM
ma-221	263	8	,	,	PUNCT
ma-221	263	9	d(tun	d(tun	PROPN
ma-221	263	10	,	,	PUNCT
ma-221	263	11	t	t	PROPN
ma-221	263	12	un+1	un+1	PROPN
ma-221	263	13	)	)	PUNCT
ma-221	263	14	>	>	X
ma-221	264	1	0	0	X
ma-221	264	2	.	.	PUNCT
ma-221	265	1	we	we	PRON
ma-221	265	2	shall	shall	AUX
ma-221	265	3	prove	prove	VERB
ma-221	265	4	that	that	SCONJ
ma-221	265	5	the	the	DET
ma-221	265	6	sequence	sequence	NOUN
ma-221	265	7	un	un	PROPN
ma-221	265	8	is	be	AUX
ma-221	265	9	a	a	DET
ma-221	265	10	cauchy	cauchy	ADJ
ma-221	265	11	sequence	sequence	NOUN
ma-221	265	12	.	.	PUNCT
ma-221	266	1	let	let	VERB
ma-221	266	2	us	we	PRON
ma-221	266	3	firstprove	firstprove	VERB
ma-221	266	4	that	that	SCONJ
ma-221	266	5	lim	lim	PROPN
ma-221	266	6	n→∞	n→∞	X
ma-221	267	1	d	d	NOUN
ma-221	267	2	(	(	PUNCT
ma-221	267	3	un	un	PROPN
ma-221	267	4	,	,	PUNCT
ma-221	267	5	un+1	un+1	NOUN
ma-221	267	6	)	)	PUNCT
ma-221	267	7	=	=	SYM
ma-221	268	1	0	0	X
ma-221	268	2	.	.	PUNCT
ma-221	269	1	as	as	SCONJ
ma-221	269	2	t	t	PROPN
ma-221	269	3	is	be	AUX
ma-221	269	4	generalized	generalize	VERB
ma-221	269	5	(	(	PUNCT
ma-221	269	6	θ	θ	NOUN
ma-221	269	7	,	,	PUNCT
ma-221	269	8	φ)-proximal	φ)-proximal	ADJ
ma-221	269	9	contraction	contraction	NOUN
ma-221	269	10	of	of	ADP
ma-221	269	11	the	the	DET
ma-221	269	12	second	second	ADJ
ma-221	269	13	kind	kind	NOUN
ma-221	269	14	,	,	PUNCT
ma-221	269	15	we	we	PRON
ma-221	269	16	have	have	VERB
ma-221	269	17	that	that	PRON
ma-221	269	18	θ	θ	PROPN
ma-221	269	19	(	(	PUNCT
ma-221	269	20	d	d	X
ma-221	269	21	(	(	PUNCT
ma-221	269	22	tun	tun	PROPN
ma-221	269	23	,	,	PUNCT
ma-221	269	24	t	t	PROPN
ma-221	269	25	un+1	un+1	NUM
ma-221	269	26	)	)	PUNCT
ma-221	269	27	)	)	PUNCT
ma-221	270	1	≤	≤	NUM
ma-221	270	2	φ	φ	PROPN
ma-221	271	1	[	[	X
ma-221	271	2	θ	θ	X
ma-221	272	1	[	[	X
ma-221	272	2	ad	ad	X
ma-221	272	3	(	(	PUNCT
ma-221	272	4	tun−1	tun−1	PROPN
ma-221	272	5	,	,	PUNCT
ma-221	272	6	t	t	PROPN
ma-221	272	7	un	un	PROPN
ma-221	272	8	)	)	PUNCT
ma-221	272	9	+	+	CCONJ
ma-221	272	10	bd	bd	PROPN
ma-221	272	11	(	(	PUNCT
ma-221	272	12	tun−1	tun−1	PROPN
ma-221	272	13	,	,	PUNCT
ma-221	272	14	t	t	PROPN
ma-221	272	15	un	un	PROPN
ma-221	272	16	)	)	PUNCT
ma-221	272	17	+	+	CCONJ
ma-221	272	18	cd	cd	PROPN
ma-221	272	19	(	(	PUNCT
ma-221	272	20	tun	tun	PROPN
ma-221	272	21	,	,	PUNCT
ma-221	272	22	t	t	PROPN
ma-221	272	23	un+1	un+1	PROPN
ma-221	272	24	)	)	PUNCT
ma-221	273	1	+	+	NUM
ma-221	273	2	h	h	NOUN
ma-221	273	3	(	(	PUNCT
ma-221	273	4	d	d	X
ma-221	273	5	(	(	PUNCT
ma-221	273	6	tun−1	tun−1	PROPN
ma-221	273	7	,	,	PUNCT
ma-221	273	8	t	t	PROPN
ma-221	273	9	un+1	un+1	PROPN
ma-221	273	10	)	)	PUNCT
ma-221	274	1	+	+	CCONJ
ma-221	274	2	d	d	X
ma-221	274	3	(	(	PUNCT
ma-221	274	4	tun	tun	PROPN
ma-221	274	5	,	,	PUNCT
ma-221	274	6	t	t	PROPN
ma-221	274	7	un	un	PROPN
ma-221	274	8	)	)	PUNCT
ma-221	274	9	)	)	PUNCT
ma-221	275	1	]	]	PUNCT
ma-221	275	2	]	]	X
ma-221	275	3	=	=	PUNCT
ma-221	275	4	φ	φ	PROPN
ma-221	276	1	[	[	X
ma-221	276	2	θ	θ	X
ma-221	277	1	[	[	X
ma-221	277	2	ad	ad	X
ma-221	277	3	(	(	PUNCT
ma-221	277	4	tun−1	tun−1	PROPN
ma-221	277	5	,	,	PUNCT
ma-221	277	6	t	t	PROPN
ma-221	277	7	un	un	PROPN
ma-221	277	8	)	)	PUNCT
ma-221	277	9	+	+	CCONJ
ma-221	277	10	bd	bd	PROPN
ma-221	277	11	(	(	PUNCT
ma-221	277	12	tun−1	tun−1	PROPN
ma-221	277	13	,	,	PUNCT
ma-221	277	14	t	t	PROPN
ma-221	277	15	un	un	PROPN
ma-221	277	16	)	)	PUNCT
ma-221	277	17	+	+	CCONJ
ma-221	277	18	cd	cd	PROPN
ma-221	277	19	(	(	PUNCT
ma-221	277	20	tun	tun	PROPN
ma-221	277	21	,	,	PUNCT
ma-221	277	22	t	t	PROPN
ma-221	277	23	un+1	un+1	PROPN
ma-221	277	24	)	)	PUNCT
ma-221	278	1	+	+	NUM
ma-221	278	2	h	h	NOUN
ma-221	278	3	(	(	PUNCT
ma-221	278	4	d	d	X
ma-221	278	5	(	(	PUNCT
ma-221	278	6	tun−1	tun−1	PROPN
ma-221	278	7	,	,	PUNCT
ma-221	278	8	t	t	PROPN
ma-221	278	9	un+1	un+1	NUM
ma-221	278	10	)	)	PUNCT
ma-221	278	11	)	)	PUNCT
ma-221	278	12	]	]	PUNCT
ma-221	278	13	]	]	X
ma-221	278	14	≤	≤	NUM
ma-221	278	15	φ	φ	PROPN
ma-221	279	1	[	[	X
ma-221	279	2	θ	θ	X
ma-221	280	1	[	[	X
ma-221	280	2	ad	ad	X
ma-221	280	3	(	(	PUNCT
ma-221	280	4	tun−1	tun−1	PROPN
ma-221	280	5	,	,	PUNCT
ma-221	280	6	t	t	PROPN
ma-221	280	7	un	un	PROPN
ma-221	280	8	)	)	PUNCT
ma-221	280	9	+	+	CCONJ
ma-221	280	10	bd	bd	PROPN
ma-221	280	11	(	(	PUNCT
ma-221	280	12	tun−1	tun−1	PROPN
ma-221	280	13	,	,	PUNCT
ma-221	280	14	t	t	PROPN
ma-221	280	15	un	un	PROPN
ma-221	280	16	)	)	PUNCT
ma-221	280	17	+	+	CCONJ
ma-221	280	18	cd	cd	PROPN
ma-221	280	19	(	(	PUNCT
ma-221	280	20	tun	tun	PROPN
ma-221	280	21	,	,	PUNCT
ma-221	280	22	t	t	PROPN
ma-221	280	23	un+1	un+1	PROPN
ma-221	280	24	)	)	PUNCT
ma-221	281	1	+	+	NUM
ma-221	281	2	h	h	NOUN
ma-221	281	3	(	(	PUNCT
ma-221	281	4	d	d	X
ma-221	281	5	(	(	PUNCT
ma-221	281	6	tun−1	tun−1	PROPN
ma-221	281	7	,	,	PUNCT
ma-221	281	8	t	t	PROPN
ma-221	281	9	un	un	PROPN
ma-221	281	10	)	)	PUNCT
ma-221	281	11	+	+	CCONJ
ma-221	281	12	d	d	PROPN
ma-221	281	13	(	(	PUNCT
ma-221	281	14	tun	tun	PROPN
ma-221	281	15	,	,	PUNCT
ma-221	281	16	t	t	PROPN
ma-221	281	17	un+1	un+1	NUM
ma-221	281	18	)	)	PUNCT
ma-221	281	19	)	)	PUNCT
ma-221	281	20	]	]	PUNCT
ma-221	281	21	]	]	X
ma-221	282	1	=	=	PUNCT
ma-221	282	2	φ	φ	PROPN
ma-221	283	1	[	[	X
ma-221	283	2	θ	θ	X
ma-221	283	3	[	[	X
ma-221	283	4	(	(	PUNCT
ma-221	283	5	a	a	DET
ma-221	283	6	+	+	NOUN
ma-221	283	7	b	b	NOUN
ma-221	283	8	+	+	NUM
ma-221	283	9	h)d	h)d	NOUN
ma-221	283	10	(	(	PUNCT
ma-221	283	11	tun−1	tun−1	PROPN
ma-221	283	12	,	,	PUNCT
ma-221	283	13	t	t	PROPN
ma-221	283	14	un	un	PROPN
ma-221	283	15	)	)	PUNCT
ma-221	284	1	+	+	CCONJ
ma-221	284	2	(	(	PUNCT
ma-221	284	3	c	c	NOUN
ma-221	284	4	+	+	NOUN
ma-221	284	5	h)d	h)d	NOUN
ma-221	284	6	(	(	PUNCT
ma-221	284	7	tun	tun	NOUN
ma-221	284	8	,	,	PUNCT
ma-221	284	9	t	t	PROPN
ma-221	284	10	un+1	un+1	PROPN
ma-221	284	11	)	)	PUNCT
ma-221	284	12	]	]	X
ma-221	284	13	]	]	X
ma-221	284	14	since	since	SCONJ
ma-221	284	15	θ	θ	PROPN
ma-221	284	16	is	be	AUX
ma-221	284	17	strictly	strictly	ADV
ma-221	284	18	increasing	increase	VERB
ma-221	284	19	and	and	CCONJ
ma-221	284	20	by	by	ADP
ma-221	284	21	lemma	lemma	PROPN
ma-221	284	22	2.6	2.6	NUM
ma-221	284	23	,	,	PUNCT
ma-221	284	24	we	we	PRON
ma-221	284	25	deduce	deduce	VERB
ma-221	284	26	d	d	X
ma-221	284	27	(	(	PUNCT
ma-221	284	28	tun	tun	PROPN
ma-221	284	29	,	,	PUNCT
ma-221	284	30	t	t	PROPN
ma-221	284	31	un+1	un+1	PROPN
ma-221	284	32	)	)	PUNCT
ma-221	284	33	<	<	X
ma-221	284	34	(	(	PUNCT
ma-221	284	35	a	a	DET
ma-221	284	36	+	+	NOUN
ma-221	284	37	b	b	NOUN
ma-221	284	38	+	+	NUM
ma-221	284	39	h)d	h)d	NOUN
ma-221	284	40	(	(	PUNCT
ma-221	284	41	tun−1	tun−1	PROPN
ma-221	284	42	,	,	PUNCT
ma-221	284	43	t	t	PROPN
ma-221	284	44	un	un	PROPN
ma-221	284	45	)	)	PUNCT
ma-221	285	1	+	+	CCONJ
ma-221	285	2	(	(	PUNCT
ma-221	285	3	c	c	NOUN
ma-221	285	4	+	+	NOUN
ma-221	285	5	h)d	h)d	NOUN
ma-221	285	6	(	(	PUNCT
ma-221	285	7	tun	tun	NOUN
ma-221	285	8	,	,	PUNCT
ma-221	285	9	t	t	PROPN
ma-221	285	10	un+1	un+1	NUM
ma-221	285	11	)	)	PUNCT
ma-221	285	12	.	.	PUNCT
ma-221	286	1	thus	thus	ADV
ma-221	286	2	d	d	X
ma-221	286	3	(	(	PUNCT
ma-221	286	4	tun	tun	PROPN
ma-221	286	5	,	,	PUNCT
ma-221	286	6	t	t	PROPN
ma-221	286	7	un+1	un+1	PROPN
ma-221	286	8	)	)	PUNCT
ma-221	286	9	<	<	X
ma-221	286	10	a	a	DET
ma-221	286	11	+	+	NUM
ma-221	286	12	b	b	NOUN
ma-221	286	13	+	+	NUM
ma-221	286	14	h	h	NOUN
ma-221	286	15	1−	1−	NUM
ma-221	286	16	c	c	NOUN
ma-221	286	17	−	−	PROPN
ma-221	287	1	h	h	NOUN
ma-221	287	2	(	(	PUNCT
ma-221	287	3	d	d	X
ma-221	287	4	(	(	PUNCT
ma-221	287	5	tun−1	tun−1	PROPN
ma-221	287	6	,	,	PUNCT
ma-221	287	7	t	t	PROPN
ma-221	287	8	un	un	PROPN
ma-221	287	9	)	)	PUNCT
ma-221	287	10	)	)	PUNCT
ma-221	287	11	.	.	PUNCT
ma-221	288	1	if	if	SCONJ
ma-221	288	2	b	b	PROPN
ma-221	288	3	+	+	SYM
ma-221	288	4	b	b	NOUN
ma-221	288	5	+	+	CCONJ
ma-221	288	6	c	c	NOUN
ma-221	288	7	+	+	CCONJ
ma-221	288	8	2h	2h	NUM
ma-221	288	9	=	=	SYM
ma-221	288	10	1	1	NUM
ma-221	288	11	,	,	PUNCT
ma-221	288	12	we	we	PRON
ma-221	288	13	have	have	VERB
ma-221	288	14	0	0	NUM
ma-221	288	15	<	<	X
ma-221	288	16	1−	1−	NUM
ma-221	288	17	c	c	NOUN
ma-221	288	18	−	−	PROPN
ma-221	289	1	h	h	NOUN
ma-221	290	1	and	and	CCONJ
ma-221	290	2	so	so	ADV
ma-221	290	3	d	d	X
ma-221	290	4	(	(	PUNCT
ma-221	290	5	tun	tun	PROPN
ma-221	290	6	,	,	PUNCT
ma-221	290	7	t	t	PROPN
ma-221	290	8	un+1	un+1	NOUN
ma-221	290	9	)	)	PUNCT
ma-221	290	10	≤	≤	NOUN
ma-221	291	1	a	a	DET
ma-221	291	2	+	+	NUM
ma-221	291	3	b	b	NOUN
ma-221	291	4	+	+	NUM
ma-221	291	5	h	h	NOUN
ma-221	291	6	1−	1−	NUM
ma-221	291	7	c	c	NOUN
ma-221	291	8	−	−	PROPN
ma-221	291	9	h	h	NOUN
ma-221	291	10	(	(	PUNCT
ma-221	291	11	d	d	X
ma-221	291	12	(	(	PUNCT
ma-221	291	13	tun−1	tun−1	PROPN
ma-221	291	14	,	,	PUNCT
ma-221	291	15	t	t	PROPN
ma-221	291	16	un	un	PROPN
ma-221	291	17	)	)	PUNCT
ma-221	291	18	)	)	PUNCT
ma-221	292	1	=	=	PUNCT
ma-221	292	2	d	d	X
ma-221	292	3	(	(	PUNCT
ma-221	292	4	tun−1	tun−1	PROPN
ma-221	292	5	,	,	PUNCT
ma-221	292	6	t	t	PROPN
ma-221	292	7	un	un	PROPN
ma-221	292	8	)	)	PUNCT
ma-221	292	9	,	,	PUNCT
ma-221	292	10	∀n	∀n	NUM
ma-221	292	11	∈	∈	PROPN
ma-221	292	12	n	n	CCONJ
ma-221	292	13	;	;	PUNCT
ma-221	292	14	consequently	consequently	ADV
ma-221	292	15	,	,	PUNCT
ma-221	292	16	θ	θ	PROPN
ma-221	292	17	(	(	PUNCT
ma-221	292	18	d	d	X
ma-221	292	19	(	(	PUNCT
ma-221	292	20	tun	tun	PROPN
ma-221	292	21	,	,	PUNCT
ma-221	292	22	t	t	PROPN
ma-221	292	23	un+1	un+1	NUM
ma-221	292	24	)	)	PUNCT
ma-221	292	25	)	)	PUNCT
ma-221	293	1	≤	≤	NUM
ma-221	293	2	φ	φ	PROPN
ma-221	294	1	[	[	X
ma-221	294	2	θ	θ	X
ma-221	294	3	(	(	PUNCT
ma-221	294	4	d	d	X
ma-221	294	5	(	(	PUNCT
ma-221	294	6	tun−1	tun−1	PROPN
ma-221	294	7	,	,	PUNCT
ma-221	294	8	t	t	PROPN
ma-221	294	9	un	un	PROPN
ma-221	294	10	)	)	PUNCT
ma-221	294	11	)	)	PUNCT
ma-221	294	12	]	]	PUNCT
ma-221	295	1	if	if	SCONJ
ma-221	295	2	b	b	PROPN
ma-221	295	3	+	+	SYM
ma-221	295	4	b	b	NOUN
ma-221	295	5	+	+	CCONJ
ma-221	295	6	c	c	NOUN
ma-221	295	7	+	+	CCONJ
ma-221	295	8	2h	2h	NUM
ma-221	295	9	<	<	X
ma-221	295	10	1	1	NUM
ma-221	295	11	,	,	PUNCT
ma-221	295	12	we	we	PRON
ma-221	295	13	have	have	VERB
ma-221	295	14	0	0	NUM
ma-221	295	15	<	<	X
ma-221	296	1	1−	1−	NUM
ma-221	296	2	c	c	NOUN
ma-221	296	3	−	−	PROPN
ma-221	297	1	h	h	NOUN
ma-221	298	1	and	and	CCONJ
ma-221	298	2	so	so	ADV
ma-221	298	3	d	d	X
ma-221	298	4	(	(	PUNCT
ma-221	298	5	tun	tun	PROPN
ma-221	298	6	,	,	PUNCT
ma-221	298	7	t	t	PROPN
ma-221	298	8	un+1	un+1	PROPN
ma-221	298	9	)	)	PUNCT
ma-221	298	10	<	<	X
ma-221	299	1	d	d	X
ma-221	299	2	(	(	PUNCT
ma-221	299	3	tun−1	tun−1	PROPN
ma-221	299	4	,	,	PUNCT
ma-221	299	5	t	t	PROPN
ma-221	299	6	un	un	PROPN
ma-221	299	7	)	)	PUNCT
ma-221	299	8	,	,	PUNCT
ma-221	299	9	∀n	∀n	NUM
ma-221	299	10	∈	∈	PROPN
ma-221	299	11	n	n	CCONJ
ma-221	299	12	;	;	PUNCT
ma-221	299	13	consequently	consequently	ADV
ma-221	299	14	,	,	PUNCT
ma-221	299	15	θ	θ	PROPN
ma-221	299	16	(	(	PUNCT
ma-221	299	17	d	d	X
ma-221	299	18	(	(	PUNCT
ma-221	299	19	tun	tun	PROPN
ma-221	299	20	,	,	PUNCT
ma-221	299	21	t	t	PROPN
ma-221	299	22	un+1	un+1	NUM
ma-221	299	23	)	)	PUNCT
ma-221	299	24	)	)	PUNCT
ma-221	299	25	≤	≤	NUM
ma-221	299	26	φ	φ	PROPN
ma-221	300	1	[	[	X
ma-221	300	2	θ	θ	X
ma-221	300	3	(	(	PUNCT
ma-221	300	4	d	d	X
ma-221	300	5	(	(	PUNCT
ma-221	300	6	tun−1	tun−1	PROPN
ma-221	300	7	,	,	PUNCT
ma-221	300	8	t	t	PROPN
ma-221	300	9	un	un	PROPN
ma-221	300	10	)	)	PUNCT
ma-221	300	11	)	)	PUNCT
ma-221	300	12	]	]	PUNCT
ma-221	301	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	301	2	eur	eur	PROPN
ma-221	301	3	.	.	PUNCT
ma-221	302	1	j.	j.	PROPN
ma-221	302	2	math	math	PROPN
ma-221	302	3	.	.	PUNCT
ma-221	303	1	anal	anal	PROPN
ma-221	303	2	.	.	PUNCT
ma-221	304	1	10.28924	10.28924	NUM
ma-221	304	2	/	/	SYM
ma-221	304	3	ada	ada	PROPN
ma-221	304	4	/	/	SYM
ma-221	304	5	ma.4.13	ma.4.13	PROPN
ma-221	304	6	9it	9it	NOUN
ma-221	304	7	implies	imply	VERB
ma-221	304	8	θ	θ	PROPN
ma-221	304	9	(	(	PUNCT
ma-221	304	10	d	d	X
ma-221	304	11	(	(	PUNCT
ma-221	304	12	tun	tun	PROPN
ma-221	304	13	,	,	PUNCT
ma-221	304	14	t	t	PROPN
ma-221	304	15	un+1	un+1	NUM
ma-221	304	16	)	)	PUNCT
ma-221	304	17	)	)	PUNCT
ma-221	304	18	≤	≤	NUM
ma-221	305	1	φ	φ	PROPN
ma-221	305	2	[	[	X
ma-221	305	3	θ	θ	X
ma-221	305	4	(	(	PUNCT
ma-221	305	5	d(tun−1	d(tun−1	PROPN
ma-221	305	6	,	,	PUNCT
ma-221	305	7	t	t	PROPN
ma-221	305	8	un	un	PROPN
ma-221	305	9	)	)	PUNCT
ma-221	305	10	]	]	PUNCT
ma-221	306	1	≤	≤	NUM
ma-221	306	2	φ2	φ2	PROPN
ma-221	306	3	[	[	X
ma-221	306	4	θ	θ	PROPN
ma-221	306	5	(	(	PUNCT
ma-221	306	6	d(tun−2	d(tun−2	PROPN
ma-221	306	7	,	,	PUNCT
ma-221	306	8	t	t	PROPN
ma-221	306	9	un−1	un−1	PROPN
ma-221	306	10	)	)	PUNCT
ma-221	306	11	]	]	PUNCT
ma-221	306	12	≤	≤	NUM
ma-221	306	13	...	...	PUNCT
ma-221	306	14	≤	≤	NUM
ma-221	307	1	φn	φn	ADP
ma-221	308	1	[	[	X
ma-221	308	2	θ	θ	X
ma-221	308	3	(	(	PUNCT
ma-221	308	4	d(tu0	d(tu0	PROPN
ma-221	308	5	,	,	PUNCT
ma-221	308	6	t	t	NOUN
ma-221	308	7	u1	u1	NOUN
ma-221	308	8	)	)	PUNCT
ma-221	308	9	]	]	PUNCT
ma-221	308	10	.	.	PUNCT
ma-221	309	1	taking	take	VERB
ma-221	309	2	the	the	DET
ma-221	309	3	limit	limit	NOUN
ma-221	309	4	as	as	ADP
ma-221	309	5	n	n	PROPN
ma-221	309	6	→∞	→∞	NUM
ma-221	309	7	,	,	PUNCT
ma-221	309	8	we	we	PRON
ma-221	309	9	have	have	VERB
ma-221	309	10	1	1	NUM
ma-221	309	11	≤	≤	NUM
ma-221	309	12	θ(d	θ(d	NOUN
ma-221	309	13	(	(	PUNCT
ma-221	309	14	tun	tun	PROPN
ma-221	309	15	,	,	PUNCT
ma-221	309	16	t	t	PROPN
ma-221	309	17	un+1	un+1	NUM
ma-221	309	18	)	)	PUNCT
ma-221	309	19	)	)	PUNCT
ma-221	309	20	≤	≤	PROPN
ma-221	309	21	lim	lim	PROPN
ma-221	309	22	n→∞	n→∞	X
ma-221	310	1	φn	φn	ADP
ma-221	311	1	[	[	X
ma-221	311	2	θ(d	θ(d	NOUN
ma-221	311	3	(	(	PUNCT
ma-221	311	4	tu0	tu0	PROPN
ma-221	311	5	,	,	PUNCT
ma-221	311	6	t	t	NOUN
ma-221	311	7	u1	u1	NOUN
ma-221	311	8	)	)	PUNCT
ma-221	311	9	)	)	PUNCT
ma-221	311	10	]	]	PUNCT
ma-221	312	1	=	=	PUNCT
ma-221	312	2	1	1	X
ma-221	312	3	.	.	PUNCT
ma-221	312	4	since	since	SCONJ
ma-221	312	5	θ	θ	PROPN
ma-221	312	6	∈	∈	PROPN
ma-221	312	7	θ	θ	PROPN
ma-221	312	8	,	,	PUNCT
ma-221	312	9	we	we	PRON
ma-221	312	10	obtain	obtain	VERB
ma-221	312	11	lim	lim	PROPN
ma-221	312	12	n→∞	n→∞	PROPN
ma-221	313	1	d	d	NOUN
ma-221	313	2	(	(	PUNCT
ma-221	313	3	tun	tun	PROPN
ma-221	313	4	,	,	PUNCT
ma-221	313	5	t	t	PROPN
ma-221	313	6	un+1	un+1	PROPN
ma-221	313	7	)	)	PUNCT
ma-221	313	8	=	=	SYM
ma-221	313	9	0	0	X
ma-221	313	10	.	.	PUNCT
ma-221	313	11	(	(	PUNCT
ma-221	313	12	3.20	3.20	NUM
ma-221	313	13	)	)	PUNCT
ma-221	313	14	next	next	ADV
ma-221	313	15	,	,	PUNCT
ma-221	313	16	we	we	PRON
ma-221	313	17	shall	shall	AUX
ma-221	313	18	prove	prove	VERB
ma-221	313	19	that	that	SCONJ
ma-221	313	20	{	{	PUNCT
ma-221	313	21	tun}n∈n	tun}n∈n	PRON
ma-221	313	22	is	be	AUX
ma-221	313	23	a	a	DET
ma-221	313	24	cauchy	cauchy	ADJ
ma-221	313	25	sequence	sequence	NOUN
ma-221	313	26	,	,	PUNCT
ma-221	313	27	i.e	i.e	PROPN
ma-221	313	28	,	,	PUNCT
ma-221	313	29	limn→∞	limn→∞	PROPN
ma-221	313	30	d	d	X
ma-221	313	31	(	(	PUNCT
ma-221	313	32	tun	tun	NOUN
ma-221	313	33	,	,	PUNCT
ma-221	313	34	tum	tum	NOUN
ma-221	313	35	)	)	PUNCT
ma-221	313	36	=	=	SYM
ma-221	314	1	0	0	NUM
ma-221	314	2	,	,	PUNCT
ma-221	314	3	for	for	ADP
ma-221	314	4	all	all	PRON
ma-221	314	5	n	n	DET
ma-221	314	6	∈	∈	PROPN
ma-221	314	7	n.	n.	NOUN
ma-221	314	8	suppose	suppose	VERB
ma-221	314	9	to	to	ADP
ma-221	314	10	the	the	DET
ma-221	314	11	contrary	contrary	NOUN
ma-221	314	12	that	that	PRON
ma-221	314	13	exists	exist	VERB
ma-221	314	14	ε	ε	PROPN
ma-221	314	15	>	>	PUNCT
ma-221	314	16	0	0	PUNCT
ma-221	314	17	and	and	CCONJ
ma-221	314	18	sequences	sequence	NOUN
ma-221	314	19	tn(k	tn(k	NOUN
ma-221	314	20	)	)	PUNCT
ma-221	314	21	and	and	CCONJ
ma-221	314	22	tm(k	tm(k	NUM
ma-221	314	23	)	)	PUNCT
ma-221	314	24	of	of	ADP
ma-221	314	25	naturalnumbers	naturalnumber	NOUN
ma-221	314	26	such	such	ADJ
ma-221	314	27	that	that	PRON
ma-221	314	28	tm(k	tm(k	NUM
ma-221	314	29	)	)	PUNCT
ma-221	314	30	>	>	X
ma-221	314	31	tn(k	tn(k	PROPN
ma-221	314	32	)	)	PUNCT
ma-221	314	33	>	>	X
ma-221	315	1	k	k	X
ma-221	315	2	,	,	PUNCT
ma-221	315	3	d	d	X
ma-221	315	4	(	(	PUNCT
ma-221	315	5	tum(k	tum(k	PROPN
ma-221	315	6	)	)	PUNCT
ma-221	315	7	,	,	PUNCT
ma-221	315	8	t	t	PROPN
ma-221	315	9	un(k	un(k	NUM
ma-221	315	10	)	)	PUNCT
ma-221	315	11	)	)	PUNCT
ma-221	315	12	≥	≥	X
ma-221	315	13	ε	ε	PROPN
ma-221	315	14	,	,	PUNCT
ma-221	315	15	d	d	X
ma-221	315	16	(	(	PUNCT
ma-221	315	17	tum(k)−1	tum(k)−1	NOUN
ma-221	315	18	,	,	PUNCT
ma-221	315	19	t	t	PROPN
ma-221	315	20	un(k	un(k	NUM
ma-221	315	21	)	)	PUNCT
ma-221	315	22	)	)	PUNCT
ma-221	316	1	<	<	X
ma-221	316	2	ε	ε	PROPN
ma-221	316	3	.	.	PUNCT
ma-221	317	1	(	(	PUNCT
ma-221	317	2	3.21	3.21	NUM
ma-221	317	3	)	)	PUNCT
ma-221	317	4	using	use	VERB
ma-221	317	5	the	the	DET
ma-221	317	6	triangular	triangular	NOUN
ma-221	317	7	inequality	inequality	NOUN
ma-221	317	8	,	,	PUNCT
ma-221	317	9	we	we	PRON
ma-221	317	10	find	find	VERB
ma-221	317	11	that	that	SCONJ
ma-221	317	12	,	,	PUNCT
ma-221	318	1	ε	ε	PROPN
ma-221	318	2	≤	≤	PROPN
ma-221	318	3	d	d	PROPN
ma-221	318	4	(	(	PUNCT
ma-221	318	5	tum(k	tum(k	PROPN
ma-221	318	6	)	)	PUNCT
ma-221	318	7	,	,	PUNCT
ma-221	318	8	t	t	PROPN
ma-221	318	9	un(k	un(k	NOUN
ma-221	318	10	)	)	PUNCT
ma-221	318	11	)	)	PUNCT
ma-221	319	1	≤	≤	NUM
ma-221	319	2	d	d	X
ma-221	319	3	(	(	PUNCT
ma-221	319	4	tum(k	tum(k	PROPN
ma-221	319	5	)	)	PUNCT
ma-221	319	6	,	,	PUNCT
ma-221	319	7	t	t	NOUN
ma-221	319	8	xn(k)−1	xn(k)−1	NUM
ma-221	319	9	)	)	PUNCT
ma-221	320	1	+	+	CCONJ
ma-221	320	2	d	d	X
ma-221	320	3	(	(	PUNCT
ma-221	320	4	tun(k)−1	tun(k)−1	NOUN
ma-221	320	5	,	,	PUNCT
ma-221	320	6	t	t	PROPN
ma-221	320	7	un(k	un(k	NUM
ma-221	320	8	)	)	PUNCT
ma-221	320	9	)	)	PUNCT
ma-221	321	1	(	(	PUNCT
ma-221	321	2	3.22	3.22	NUM
ma-221	321	3	)	)	PUNCT
ma-221	321	4	<	<	X
ma-221	321	5	ε+	ε+	X
ma-221	321	6	d	d	X
ma-221	321	7	(	(	PUNCT
ma-221	321	8	tun(k)−1	tun(k)−1	PROPN
ma-221	321	9	,	,	PUNCT
ma-221	321	10	t	t	PROPN
ma-221	321	11	un(k	un(k	NUM
ma-221	321	12	)	)	PUNCT
ma-221	321	13	)	)	PUNCT
ma-221	321	14	.	.	PUNCT
ma-221	322	1	(	(	PUNCT
ma-221	322	2	3.23	3.23	NUM
ma-221	322	3	)	)	PUNCT
ma-221	322	4	then	then	ADV
ma-221	322	5	,	,	PUNCT
ma-221	322	6	by	by	ADP
ma-221	322	7	3.4	3.4	NUM
ma-221	322	8	and	and	CCONJ
ma-221	322	9	3.22	3.22	NUM
ma-221	322	10	,	,	PUNCT
ma-221	322	11	it	it	PRON
ma-221	322	12	follows	follow	VERB
ma-221	322	13	that	that	SCONJ
ma-221	322	14	lim	lim	PROPN
ma-221	322	15	k→∞	k→∞	PROPN
ma-221	322	16	d	d	PROPN
ma-221	322	17	(	(	PUNCT
ma-221	322	18	tum(k	tum(k	PROPN
ma-221	322	19	)	)	PUNCT
ma-221	322	20	,	,	PUNCT
ma-221	322	21	t	t	PROPN
ma-221	322	22	un(k	un(k	NUM
ma-221	322	23	)	)	PUNCT
ma-221	322	24	)	)	PUNCT
ma-221	323	1	=	=	PUNCT
ma-221	323	2	ε	ε	PROPN
ma-221	323	3	.	.	PUNCT
ma-221	324	1	(	(	PUNCT
ma-221	324	2	3.24	3.24	NUM
ma-221	324	3	)	)	PUNCT
ma-221	324	4	using	use	VERB
ma-221	324	5	the	the	DET
ma-221	324	6	triangular	triangular	NOUN
ma-221	324	7	inequality	inequality	NOUN
ma-221	324	8	,	,	PUNCT
ma-221	324	9	we	we	PRON
ma-221	324	10	find	find	VERB
ma-221	324	11	that	that	SCONJ
ma-221	324	12	,	,	PUNCT
ma-221	325	1	ε	ε	PROPN
ma-221	325	2	≤	≤	PROPN
ma-221	325	3	d	d	PROPN
ma-221	325	4	(	(	PUNCT
ma-221	325	5	tum(k	tum(k	PROPN
ma-221	325	6	)	)	PUNCT
ma-221	325	7	,	,	PUNCT
ma-221	325	8	t	t	PROPN
ma-221	325	9	un(k	un(k	NOUN
ma-221	325	10	)	)	PUNCT
ma-221	325	11	)	)	PUNCT
ma-221	326	1	≤	≤	NUM
ma-221	326	2	d	d	X
ma-221	326	3	(	(	PUNCT
ma-221	326	4	tum(k	tum(k	PROPN
ma-221	326	5	)	)	PUNCT
ma-221	326	6	,	,	PUNCT
ma-221	326	7	t	t	PROPN
ma-221	326	8	un(k)+1	un(k)+1	PROPN
ma-221	326	9	)	)	PUNCT
ma-221	327	1	+	+	CCONJ
ma-221	327	2	d	d	X
ma-221	327	3	(	(	PUNCT
ma-221	327	4	tun(k)+1	tun(k)+1	PROPN
ma-221	327	5	,	,	PUNCT
ma-221	327	6	t	t	NOUN
ma-221	327	7	un(k	un(k	NUM
ma-221	327	8	)	)	PUNCT
ma-221	327	9	)	)	PUNCT
ma-221	328	1	(	(	PUNCT
ma-221	328	2	3.25	3.25	NUM
ma-221	328	3	)	)	PUNCT
ma-221	328	4	and	and	CCONJ
ma-221	328	5	ε	ε	PROPN
ma-221	329	1	≤	≤	PROPN
ma-221	329	2	d	d	PROPN
ma-221	329	3	(	(	PUNCT
ma-221	329	4	tum(k	tum(k	PROPN
ma-221	329	5	)	)	PUNCT
ma-221	329	6	,	,	PUNCT
ma-221	329	7	t	t	PROPN
ma-221	329	8	un(k)+1	un(k)+1	CCONJ
ma-221	329	9	)	)	PUNCT
ma-221	329	10	≤	≤	NUM
ma-221	330	1	d	d	X
ma-221	330	2	(	(	PUNCT
ma-221	330	3	tum(k	tum(k	PROPN
ma-221	330	4	)	)	PUNCT
ma-221	330	5	,	,	PUNCT
ma-221	330	6	t	t	PROPN
ma-221	330	7	un(k	un(k	NUM
ma-221	330	8	)	)	PUNCT
ma-221	330	9	)	)	PUNCT
ma-221	331	1	+	+	CCONJ
ma-221	331	2	d	d	X
ma-221	331	3	(	(	PUNCT
ma-221	331	4	tun(k	tun(k	NOUN
ma-221	331	5	)	)	PUNCT
ma-221	331	6	,	,	PUNCT
ma-221	331	7	t	t	PROPN
ma-221	331	8	un(k)+1	un(k)+1	PROPN
ma-221	331	9	)	)	PUNCT
ma-221	331	10	(	(	PUNCT
ma-221	331	11	3.26	3.26	NUM
ma-221	331	12	)	)	PUNCT
ma-221	331	13	then	then	ADV
ma-221	331	14	,	,	PUNCT
ma-221	331	15	by	by	ADP
ma-221	331	16	(	(	PUNCT
ma-221	331	17	3.25	3.25	NUM
ma-221	331	18	)	)	PUNCT
ma-221	331	19	and	and	CCONJ
ma-221	331	20	(	(	PUNCT
ma-221	331	21	3.9	3.9	NUM
ma-221	331	22	)	)	PUNCT
ma-221	331	23	,	,	PUNCT
ma-221	331	24	it	it	PRON
ma-221	331	25	follows	follow	VERB
ma-221	331	26	that	that	SCONJ
ma-221	331	27	lim	lim	PROPN
ma-221	331	28	k→∞	k→∞	PROPN
ma-221	331	29	d	d	PROPN
ma-221	331	30	(	(	PUNCT
ma-221	331	31	tum(k	tum(k	PROPN
ma-221	331	32	)	)	PUNCT
ma-221	331	33	,	,	PUNCT
ma-221	331	34	t	t	PROPN
ma-221	331	35	un(k)+1	un(k)+1	PROPN
ma-221	331	36	)	)	PUNCT
ma-221	332	1	=	=	PUNCT
ma-221	332	2	ε	ε	PROPN
ma-221	332	3	.	.	PUNCT
ma-221	333	1	(	(	PUNCT
ma-221	333	2	3.27	3.27	NUM
ma-221	333	3	)	)	PUNCT
ma-221	333	4	similarly	similarly	ADV
ma-221	333	5	method	method	NOUN
ma-221	333	6	,	,	PUNCT
ma-221	333	7	we	we	PRON
ma-221	333	8	conclude	conclude	VERB
ma-221	333	9	that	that	SCONJ
ma-221	333	10	lim	lim	PROPN
ma-221	333	11	k→∞	k→∞	PROPN
ma-221	333	12	d	d	PROPN
ma-221	333	13	(	(	PUNCT
ma-221	333	14	tum(k)+1	tum(k)+1	NOUN
ma-221	333	15	,	,	PUNCT
ma-221	333	16	t	t	PROPN
ma-221	333	17	un(k	un(k	NOUN
ma-221	333	18	)	)	PUNCT
ma-221	333	19	)	)	PUNCT
ma-221	334	1	=	=	PUNCT
ma-221	334	2	ε	ε	AUX
ma-221	334	3	.	.	PUNCT
ma-221	334	4	(	(	PUNCT
ma-221	334	5	3.28	3.28	NUM
ma-221	334	6	)	)	PUNCT
ma-221	334	7	using	use	VERB
ma-221	334	8	again	again	ADV
ma-221	334	9	the	the	DET
ma-221	334	10	triangular	triangular	NOUN
ma-221	334	11	inequality	inequality	NOUN
ma-221	334	12	,	,	PUNCT
ma-221	334	13	d	d	X
ma-221	334	14	(	(	PUNCT
ma-221	334	15	tum(k)+1	tum(k)+1	NOUN
ma-221	334	16	,	,	PUNCT
ma-221	334	17	t	t	PROPN
ma-221	334	18	un(k)+1	un(k)+1	CCONJ
ma-221	334	19	)	)	PUNCT
ma-221	334	20	≤	≤	NUM
ma-221	335	1	d	d	X
ma-221	335	2	(	(	PUNCT
ma-221	335	3	um(k)+1	um(k)+1	PROPN
ma-221	335	4	,	,	PUNCT
ma-221	335	5	t	t	PROPN
ma-221	335	6	um(k	um(k	PROPN
ma-221	335	7	)	)	PUNCT
ma-221	335	8	)	)	PUNCT
ma-221	336	1	+	+	CCONJ
ma-221	336	2	d	d	X
ma-221	336	3	(	(	PUNCT
ma-221	336	4	tum(k	tum(k	PROPN
ma-221	336	5	)	)	PUNCT
ma-221	336	6	,	,	PUNCT
ma-221	336	7	t	t	PROPN
ma-221	336	8	un(k	un(k	NUM
ma-221	336	9	)	)	PUNCT
ma-221	336	10	)	)	PUNCT
ma-221	337	1	+	+	CCONJ
ma-221	337	2	d	d	X
ma-221	337	3	(	(	PUNCT
ma-221	337	4	tun(k	tun(k	NOUN
ma-221	337	5	)	)	PUNCT
ma-221	337	6	,	,	PUNCT
ma-221	337	7	t	t	PROPN
ma-221	337	8	un(k)+1	un(k)+1	NUM
ma-221	337	9	)	)	PUNCT
ma-221	337	10	.	.	PUNCT
ma-221	338	1	(	(	PUNCT
ma-221	338	2	3.29	3.29	NUM
ma-221	338	3	)	)	PUNCT
ma-221	338	4	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	338	5	eur	eur	PROPN
ma-221	338	6	.	.	PUNCT
ma-221	339	1	j.	j.	PROPN
ma-221	339	2	math	math	PROPN
ma-221	339	3	.	.	PUNCT
ma-221	340	1	anal	anal	PROPN
ma-221	340	2	.	.	PUNCT
ma-221	341	1	10.28924	10.28924	NUM
ma-221	341	2	/	/	SYM
ma-221	341	3	ada	ada	PROPN
ma-221	341	4	/	/	SYM
ma-221	341	5	ma.4.13	ma.4.13	PROPN
ma-221	341	6	10on	10on	NOUN
ma-221	341	7	the	the	DET
ma-221	341	8	other	other	ADJ
ma-221	341	9	hand	hand	NOUN
ma-221	341	10	,	,	PUNCT
ma-221	341	11	using	use	VERB
ma-221	341	12	triangular	triangular	NOUN
ma-221	341	13	inequality	inequality	NOUN
ma-221	341	14	,	,	PUNCT
ma-221	341	15	we	we	PRON
ma-221	341	16	have	have	VERB
ma-221	341	17	d	d	X
ma-221	341	18	(	(	PUNCT
ma-221	341	19	tum(k	tum(k	PROPN
ma-221	341	20	)	)	PUNCT
ma-221	341	21	,	,	PUNCT
ma-221	341	22	t	t	PROPN
ma-221	341	23	un(k	un(k	NOUN
ma-221	341	24	)	)	PUNCT
ma-221	341	25	)	)	PUNCT
ma-221	342	1	≤	≤	NUM
ma-221	342	2	d	d	X
ma-221	342	3	(	(	PUNCT
ma-221	342	4	tum(k	tum(k	PROPN
ma-221	342	5	)	)	PUNCT
ma-221	342	6	,	,	PUNCT
ma-221	342	7	t	t	X
ma-221	342	8	um(k)+1	um(k)+1	PROPN
ma-221	342	9	)	)	PUNCT
ma-221	343	1	+	+	CCONJ
ma-221	343	2	d	d	NOUN
ma-221	343	3	(	(	PUNCT
ma-221	343	4	tum(k)+1	tum(k)+1	NOUN
ma-221	343	5	,	,	PUNCT
ma-221	343	6	t	t	PROPN
ma-221	343	7	un(k)+1	un(k)+1	PROPN
ma-221	343	8	)	)	PUNCT
ma-221	344	1	+	+	CCONJ
ma-221	344	2	d	d	X
ma-221	344	3	(	(	PUNCT
ma-221	344	4	tun(k)+1	tun(k)+1	NOUN
ma-221	344	5	,	,	PUNCT
ma-221	344	6	t	t	PROPN
ma-221	344	7	un(k	un(k	NUM
ma-221	344	8	)	)	PUNCT
ma-221	344	9	)	)	PUNCT
ma-221	344	10	.	.	PUNCT
ma-221	345	1	(	(	PUNCT
ma-221	345	2	3.30	3.30	X
ma-221	345	3	)	)	PUNCT
ma-221	345	4	letting	let	VERB
ma-221	345	5	k	k	PROPN
ma-221	345	6	→∞	→∞	PROPN
ma-221	345	7	in	in	ADP
ma-221	345	8	inequality	inequality	NOUN
ma-221	345	9	(	(	PUNCT
ma-221	345	10	3.29	3.29	NUM
ma-221	345	11	)	)	PUNCT
ma-221	345	12	and	and	CCONJ
ma-221	345	13	(	(	PUNCT
ma-221	345	14	3.30	3.30	NUM
ma-221	345	15	)	)	PUNCT
ma-221	345	16	,	,	PUNCT
ma-221	345	17	we	we	PRON
ma-221	345	18	obtain	obtain	VERB
ma-221	345	19	lim	lim	PROPN
ma-221	345	20	k→∞	k→∞	PROPN
ma-221	345	21	d	d	PROPN
ma-221	345	22	(	(	PUNCT
ma-221	345	23	tum(k)+1	tum(k)+1	NOUN
ma-221	345	24	,	,	PUNCT
ma-221	345	25	t	t	PROPN
ma-221	345	26	un(k)+1	un(k)+1	PROPN
ma-221	345	27	)	)	PUNCT
ma-221	346	1	=	=	PUNCT
ma-221	346	2	ε	ε	PROPN
ma-221	346	3	.	.	PUNCT
ma-221	346	4	(	(	PUNCT
ma-221	346	5	3.31	3.31	NUM
ma-221	346	6	)	)	PUNCT
ma-221	346	7	substituting	substitute	VERB
ma-221	346	8	u1	u1	NOUN
ma-221	346	9	=	=	PUNCT
ma-221	346	10	tum(k)+1	tum(k)+1	NOUN
ma-221	346	11	,	,	PUNCT
ma-221	346	12	u2	u2	PROPN
ma-221	346	13	=	=	PUNCT
ma-221	346	14	tun(k)+1	tun(k)+1	NOUN
ma-221	346	15	,	,	PUNCT
ma-221	346	16	v1	v1	NOUN
ma-221	346	17	=	=	SYM
ma-221	346	18	tum(k	tum(k	PROPN
ma-221	346	19	)	)	PUNCT
ma-221	346	20	and	and	CCONJ
ma-221	346	21	v1	v1	NOUN
ma-221	346	22	=	=	SYM
ma-221	346	23	tun(k	tun(k	PROPN
ma-221	346	24	)	)	PUNCT
ma-221	346	25	in	in	ADP
ma-221	346	26	assumption	assumption	NOUN
ma-221	346	27	of	of	ADP
ma-221	346	28	thetheorem	thetheorem	PROPN
ma-221	346	29	,	,	PUNCT
ma-221	346	30	we	we	PRON
ma-221	346	31	get	get	VERB
ma-221	346	32	θ	θ	NOUN
ma-221	346	33	(	(	PUNCT
ma-221	346	34	d	d	X
ma-221	346	35	(	(	PUNCT
ma-221	346	36	tum(k)+1	tum(k)+1	NOUN
ma-221	346	37	,	,	PUNCT
ma-221	346	38	t	t	PROPN
ma-221	346	39	un(k)+1	un(k)+1	PROPN
ma-221	346	40	)	)	PUNCT
ma-221	346	41	)	)	PUNCT
ma-221	347	1	≤	≤	NUM
ma-221	348	1	φ	φ	NUM
ma-221	348	2			NUM
ma-221	348	3	θ	θ	PROPN
ma-221	348	4			NUM
ma-221	348	5	ad	ad	NOUN
ma-221	348	6	(	(	PUNCT
ma-221	348	7	tum(k	tum(k	NOUN
ma-221	348	8	)	)	PUNCT
ma-221	348	9	,	,	PUNCT
ma-221	348	10	t	t	PROPN
ma-221	348	11	un(k	un(k	NUM
ma-221	348	12	)	)	PUNCT
ma-221	348	13	)	)	PUNCT
ma-221	349	1	+	+	CCONJ
ma-221	349	2	bd	bd	PROPN
ma-221	349	3	(	(	PUNCT
ma-221	349	4	tum(k)1	tum(k)1	NOUN
ma-221	349	5	,	,	PUNCT
ma-221	349	6	t	t	NOUN
ma-221	349	7	un(k	un(k	NUM
ma-221	349	8	)	)	PUNCT
ma-221	349	9	)	)	PUNCT
ma-221	350	1	+	+	CCONJ
ma-221	350	2	cd	cd	PROPN
ma-221	350	3	(	(	PUNCT
ma-221	350	4	tun(k)+1	tun(k)+1	PROPN
ma-221	350	5	,	,	PUNCT
ma-221	350	6	t	t	NOUN
ma-221	350	7	un(k	un(k	NUM
ma-221	350	8	)	)	PUNCT
ma-221	350	9	)	)	PUNCT
ma-221	351	1	+	+	CCONJ
ma-221	351	2	h(d	h(d	PRON
ma-221	351	3	(	(	PUNCT
ma-221	351	4	tum(k	tum(k	PROPN
ma-221	351	5	)	)	PUNCT
ma-221	351	6	,	,	PUNCT
ma-221	351	7	t	t	PROPN
ma-221	351	8	un(k)+1	un(k)+1	PROPN
ma-221	351	9	)	)	PUNCT
ma-221	352	1	+	+	CCONJ
ma-221	352	2	d	d	X
ma-221	352	3	(	(	PUNCT
ma-221	352	4	tun(k	tun(k	NOUN
ma-221	352	5	)	)	PUNCT
ma-221	352	6	,	,	PUNCT
ma-221	352	7	t	t	PROPN
ma-221	352	8	um(k)+1	um(k)+1	PROPN
ma-221	352	9	)	)	PUNCT
ma-221	352	10	)	)	PUNCT
ma-221	352	11			VERB
ma-221	352	12	(3.32)letting	(3.32)lette	VERB
ma-221	352	13	letting	let	VERB
ma-221	352	14	k	k	PROPN
ma-221	352	15	→∞	→∞	PROPN
ma-221	352	16	in	in	ADP
ma-221	352	17	(	(	PUNCT
ma-221	352	18	3.32	3.32	NUM
ma-221	352	19	)	)	PUNCT
ma-221	352	20	,	,	PUNCT
ma-221	352	21	and	and	CCONJ
ma-221	352	22	using	use	VERB
ma-221	352	23	(	(	PUNCT
ma-221	352	24	θ1	θ1	NOUN
ma-221	352	25	)	)	PUNCT
ma-221	352	26	,	,	PUNCT
ma-221	352	27	(	(	PUNCT
ma-221	352	28	θ3	θ3	NOUN
ma-221	352	29	)	)	PUNCT
ma-221	352	30	,	,	PUNCT
ma-221	352	31	(	(	PUNCT
ma-221	352	32	φ3	φ3	PROPN
ma-221	352	33	)	)	PUNCT
ma-221	352	34	and	and	CCONJ
ma-221	352	35	lemma	lemma	PROPN
ma-221	352	36	(	(	PUNCT
ma-221	352	37	2.6	2.6	NUM
ma-221	352	38	)	)	PUNCT
ma-221	352	39	we	we	PRON
ma-221	352	40	obtain	obtain	VERB
ma-221	352	41	θ	θ	PROPN
ma-221	352	42	(	(	PUNCT
ma-221	352	43	ε	ε	PROPN
ma-221	352	44	)	)	PUNCT
ma-221	352	45	≤	≤	NOUN
ma-221	352	46	φ	φ	PROPN
ma-221	353	1	[	[	X
ma-221	353	2	θ	θ	X
ma-221	353	3	(	(	PUNCT
ma-221	353	4	aε+	aε+	PROPN
ma-221	353	5	bε+	bε+	NOUN
ma-221	353	6	cε+	cε+	PROPN
ma-221	353	7	2hε	2hε	NUM
ma-221	353	8	)	)	PUNCT
ma-221	353	9	]	]	PUNCT
ma-221	353	10	.	.	PUNCT
ma-221	354	1	we	we	PRON
ma-221	354	2	derive	derive	VERB
ma-221	354	3	ε	ε	PROPN
ma-221	354	4	<	<	X
ma-221	354	5	ε	ε	PROPN
ma-221	354	6	.	.	PROPN
ma-221	354	7	which	which	PRON
ma-221	354	8	is	be	AUX
ma-221	354	9	a	a	DET
ma-221	354	10	contradiction	contradiction	NOUN
ma-221	354	11	.	.	PUNCT
ma-221	355	1	thus	thus	ADV
ma-221	355	2	limn	limn	ADJ
ma-221	355	3	,	,	PUNCT
ma-221	355	4	m→∞	m→∞	NOUN
ma-221	355	5	d	d	NOUN
ma-221	355	6	(	(	PUNCT
ma-221	355	7	tun	tun	PROPN
ma-221	355	8	,	,	PUNCT
ma-221	355	9	t	t	PROPN
ma-221	355	10	um	um	INTJ
ma-221	355	11	)	)	PUNCT
ma-221	355	12	=	=	SYM
ma-221	355	13	0	0	NUM
ma-221	355	14	,	,	PUNCT
ma-221	355	15	which	which	PRON
ma-221	355	16	shows	show	VERB
ma-221	355	17	that	that	SCONJ
ma-221	355	18	{	{	PUNCT
ma-221	355	19	tun	tun	NOUN
ma-221	355	20	}	}	PUNCT
ma-221	355	21	is	be	AUX
ma-221	355	22	a	a	DET
ma-221	355	23	cauchysequence	cauchysequence	NOUN
ma-221	355	24	.	.	PUNCT
ma-221	356	1	then	then	ADV
ma-221	356	2	there	there	PRON
ma-221	356	3	exists	exist	VERB
ma-221	356	4	v	v	ADP
ma-221	356	5	∈	∈	PROPN
ma-221	356	6	b	b	NOUN
ma-221	356	7	such	such	ADJ
ma-221	356	8	that	that	SCONJ
ma-221	356	9	lim	lim	PROPN
ma-221	356	10	n→∞	n→∞	PROPN
ma-221	357	1	d	d	NOUN
ma-221	357	2	(	(	PUNCT
ma-221	357	3	tun	tun	NOUN
ma-221	357	4	,	,	PUNCT
ma-221	357	5	v	v	NOUN
ma-221	357	6	)	)	PUNCT
ma-221	357	7	=	=	SYM
ma-221	357	8	0	0	X
ma-221	357	9	.	.	PUNCT
ma-221	358	1	also	also	ADV
ma-221	358	2	,	,	PUNCT
ma-221	358	3	d	d	X
ma-221	358	4	(	(	PUNCT
ma-221	358	5	v	v	NOUN
ma-221	358	6	,	,	PUNCT
ma-221	358	7	a	a	PRON
ma-221	358	8	)	)	PUNCT
ma-221	358	9	≤	≤	NUM
ma-221	358	10	d	d	NOUN
ma-221	358	11	(	(	PUNCT
ma-221	358	12	v	v	NOUN
ma-221	358	13	,	,	PUNCT
ma-221	358	14	tun	tun	NOUN
ma-221	358	15	)	)	PUNCT
ma-221	358	16	≤	≤	NOUN
ma-221	359	1	d	d	PROPN
ma-221	359	2	(	(	PUNCT
ma-221	359	3	v	v	NOUN
ma-221	359	4	,	,	PUNCT
ma-221	359	5	un+1	un+1	NOUN
ma-221	359	6	)	)	PUNCT
ma-221	360	1	+	+	CCONJ
ma-221	360	2	d	d	X
ma-221	360	3	(	(	PUNCT
ma-221	360	4	un+1	un+1	PROPN
ma-221	360	5	,	,	PUNCT
ma-221	360	6	t	t	PROPN
ma-221	360	7	un	un	PROPN
ma-221	360	8	)	)	PUNCT
ma-221	360	9	=	=	SYM
ma-221	361	1	d	d	PROPN
ma-221	361	2	(	(	PUNCT
ma-221	361	3	v	v	NOUN
ma-221	361	4	,	,	PUNCT
ma-221	361	5	un+1	un+1	NOUN
ma-221	361	6	)	)	PUNCT
ma-221	362	1	+	+	CCONJ
ma-221	362	2	d	d	X
ma-221	362	3	(	(	PUNCT
ma-221	362	4	a	a	DET
ma-221	362	5	,	,	PUNCT
ma-221	362	6	b	b	NOUN
ma-221	362	7	)	)	PUNCT
ma-221	362	8	≤	≤	NOUN
ma-221	363	1	d	d	NOUN
ma-221	363	2	(	(	PUNCT
ma-221	363	3	v	v	NOUN
ma-221	363	4	,	,	PUNCT
ma-221	363	5	un+1	un+1	NOUN
ma-221	363	6	)	)	PUNCT
ma-221	363	7	+	+	CCONJ
ma-221	364	1	d	d	X
ma-221	364	2	(	(	PUNCT
ma-221	364	3	v	v	NOUN
ma-221	364	4	,	,	PUNCT
ma-221	364	5	a	a	PRON
ma-221	364	6	)	)	PUNCT
ma-221	364	7	.	.	PUNCT
ma-221	365	1	therefore	therefore	ADV
ma-221	365	2	,	,	PUNCT
ma-221	365	3	d	d	X
ma-221	365	4	(	(	PUNCT
ma-221	365	5	v	v	NOUN
ma-221	365	6	,	,	PUNCT
ma-221	365	7	tun	tun	NOUN
ma-221	365	8	)	)	PUNCT
ma-221	366	1	→	→	SYM
ma-221	366	2	d	d	X
ma-221	366	3	(	(	PUNCT
ma-221	366	4	v	v	NOUN
ma-221	366	5	,	,	PUNCT
ma-221	366	6	a	a	PRON
ma-221	366	7	)	)	PUNCT
ma-221	366	8	.	.	PUNCT
ma-221	367	1	since	since	SCONJ
ma-221	367	2	a	a	PRON
ma-221	367	3	is	be	AUX
ma-221	367	4	approximately	approximately	ADV
ma-221	367	5	compact	compact	ADJ
ma-221	367	6	with	with	ADP
ma-221	367	7	respect	respect	NOUN
ma-221	367	8	to	to	ADP
ma-221	367	9	b	b	NOUN
ma-221	367	10	,	,	PUNCT
ma-221	367	11	thesequence	thesequence	NOUN
ma-221	367	12	{	{	PUNCT
ma-221	367	13	un	un	PROPN
ma-221	367	14	}	}	PUNCT
ma-221	367	15	has	have	VERB
ma-221	367	16	a	a	DET
ma-221	367	17	subsequence	subsequence	NOUN
ma-221	367	18	{	{	PUNCT
ma-221	367	19	unk	unk	NOUN
ma-221	367	20	}	}	PUNCT
ma-221	367	21	converging	converge	VERB
ma-221	367	22	to	to	ADP
ma-221	367	23	some	some	DET
ma-221	367	24	element	element	NOUN
ma-221	367	25	u	u	PROPN
ma-221	367	26	∈	∈	PROPN
ma-221	367	27	a.	a.	NOUN
ma-221	368	1	so	so	ADV
ma-221	368	2	it	it	PRON
ma-221	368	3	turns	turn	VERB
ma-221	368	4	out	out	ADP
ma-221	368	5	that	that	SCONJ
ma-221	368	6	d(u	d(u	PROPN
ma-221	368	7	,	,	PUNCT
ma-221	368	8	v	v	NOUN
ma-221	368	9	)	)	PUNCT
ma-221	368	10	=	=	VERB
ma-221	369	1	lim	lim	PROPN
ma-221	369	2	n→∞	n→∞	X
ma-221	370	1	d	d	NOUN
ma-221	370	2	(	(	PUNCT
ma-221	370	3	unk+1	unk+1	PROPN
ma-221	370	4	,	,	PUNCT
ma-221	370	5	t	t	PROPN
ma-221	370	6	unk	unk	NOUN
ma-221	370	7	)	)	PUNCT
ma-221	370	8	=	=	SYM
ma-221	371	1	d(a	d(a	PROPN
ma-221	371	2	,	,	PUNCT
ma-221	371	3	b	b	NOUN
ma-221	371	4	)	)	PUNCT
ma-221	371	5	.	.	PUNCT
ma-221	372	1	(	(	PUNCT
ma-221	372	2	3.33	3.33	NUM
ma-221	372	3	)	)	PUNCT
ma-221	372	4	because	because	SCONJ
ma-221	372	5	t	t	PROPN
ma-221	372	6	is	be	AUX
ma-221	372	7	a	a	DET
ma-221	372	8	continuous	continuous	ADJ
ma-221	372	9	mapping	mapping	NOUN
ma-221	372	10	,	,	PUNCT
ma-221	372	11	d(u	d(u	PROPN
ma-221	372	12	,	,	PUNCT
ma-221	372	13	tu	tu	PROPN
ma-221	372	14	)	)	PUNCT
ma-221	372	15	=	=	PROPN
ma-221	372	16	lim	lim	PROPN
ma-221	372	17	n→∞	n→∞	X
ma-221	372	18	d(un+1	d(un+1	PROPN
ma-221	372	19	,	,	PUNCT
ma-221	372	20	t	t	PROPN
ma-221	372	21	un	un	PROPN
ma-221	372	22	)	)	PUNCT
ma-221	372	23	=	=	SYM
ma-221	373	1	d(a	d(a	PROPN
ma-221	373	2	,	,	PUNCT
ma-221	373	3	b	b	NOUN
ma-221	373	4	)	)	PUNCT
ma-221	373	5	.	.	PUNCT
ma-221	374	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	374	2	eur	eur	PROPN
ma-221	374	3	.	.	PUNCT
ma-221	375	1	j.	j.	PROPN
ma-221	375	2	math	math	PROPN
ma-221	375	3	.	.	PUNCT
ma-221	376	1	anal	anal	PROPN
ma-221	376	2	.	.	PUNCT
ma-221	377	1	10.28924	10.28924	NUM
ma-221	377	2	/	/	SYM
ma-221	377	3	ada	ada	PROPN
ma-221	377	4	/	/	SYM
ma-221	377	5	ma.4.13	ma.4.13	PROPN
ma-221	377	6	11uniqueness	11uniqueness	NUM
ma-221	377	7	:	:	PUNCT
ma-221	377	8	suppose	suppose	VERB
ma-221	377	9	that	that	SCONJ
ma-221	377	10	there	there	PRON
ma-221	377	11	is	be	VERB
ma-221	377	12	another	another	DET
ma-221	377	13	best	good	ADJ
ma-221	377	14	proximity	proximity	NOUN
ma-221	377	15	point	point	NOUN
ma-221	377	16	z	z	NOUN
ma-221	377	17	of	of	ADP
ma-221	377	18	the	the	DET
ma-221	377	19	mapping	mapping	NOUN
ma-221	377	20	t	t	NOUN
ma-221	377	21	such	such	ADJ
ma-221	377	22	that	that	SCONJ
ma-221	377	23	d(z	d(z	PROPN
ma-221	377	24	,	,	PUNCT
ma-221	377	25	t	t	PROPN
ma-221	377	26	z	z	PROPN
ma-221	377	27	)	)	PUNCT
ma-221	377	28	=	=	SYM
ma-221	378	1	d(a	d(a	PROPN
ma-221	378	2	,	,	PUNCT
ma-221	378	3	b	b	NOUN
ma-221	378	4	)	)	PUNCT
ma-221	378	5	.	.	PUNCT
ma-221	379	1	since	since	SCONJ
ma-221	379	2	t	t	PROPN
ma-221	379	3	is	be	AUX
ma-221	379	4	a	a	DET
ma-221	379	5	generalized	generalized	ADJ
ma-221	379	6	(	(	PUNCT
ma-221	379	7	θ	θ	NOUN
ma-221	379	8	,	,	PUNCT
ma-221	379	9	φ)-proximal	φ)-proximal	ADJ
ma-221	379	10	contraction	contraction	NOUN
ma-221	379	11	of	of	ADP
ma-221	379	12	the	the	DET
ma-221	379	13	first	first	ADJ
ma-221	379	14	second	second	NOUN
ma-221	379	15	,	,	PUNCT
ma-221	379	16	it	it	PRON
ma-221	379	17	follows	follow	VERB
ma-221	379	18	from	from	ADP
ma-221	379	19	this	this	PRON
ma-221	379	20	that	that	SCONJ
ma-221	379	21	θ(d(tz	θ(d(tz	PROPN
ma-221	379	22	,	,	PUNCT
ma-221	379	23	tu	tu	PROPN
ma-221	379	24	)	)	PUNCT
ma-221	379	25	)	)	PUNCT
ma-221	379	26	≤	≤	NUM
ma-221	380	1	φ	φ	PROPN
ma-221	381	1	[	[	X
ma-221	381	2	θ	θ	X
ma-221	382	1	[	[	X
ma-221	382	2	ad	ad	X
ma-221	382	3	(	(	PUNCT
ma-221	382	4	tz	tz	PROPN
ma-221	382	5	,	,	PUNCT
ma-221	382	6	tu	tu	PROPN
ma-221	382	7	)	)	PUNCT
ma-221	383	1	+	+	CCONJ
ma-221	383	2	bd	bd	PROPN
ma-221	383	3	(	(	PUNCT
ma-221	383	4	tz	tz	PROPN
ma-221	383	5	,	,	PUNCT
ma-221	383	6	t	t	PROPN
ma-221	383	7	z	z	PROPN
ma-221	383	8	)	)	PUNCT
ma-221	383	9	+	+	CCONJ
ma-221	384	1	cd	cd	PROPN
ma-221	384	2	(	(	PUNCT
ma-221	384	3	tu	tu	PROPN
ma-221	384	4	,	,	PUNCT
ma-221	384	5	tu	tu	PROPN
ma-221	384	6	)	)	PUNCT
ma-221	385	1	+	+	NUM
ma-221	385	2	h	h	NOUN
ma-221	385	3	(	(	PUNCT
ma-221	385	4	d	d	X
ma-221	385	5	(	(	PUNCT
ma-221	385	6	tz	tz	PROPN
ma-221	385	7	,	,	PUNCT
ma-221	385	8	tu	tu	PROPN
ma-221	385	9	)	)	PUNCT
ma-221	386	1	+	+	CCONJ
ma-221	386	2	d	d	X
ma-221	386	3	(	(	PUNCT
ma-221	386	4	tz	tz	PROPN
ma-221	386	5	,	,	PUNCT
ma-221	386	6	tu	tu	PROPN
ma-221	386	7	)	)	PUNCT
ma-221	386	8	)	)	PUNCT
ma-221	387	1	]	]	PUNCT
ma-221	387	2	]	]	X
ma-221	387	3	=	=	PUNCT
ma-221	387	4	φ	φ	PROPN
ma-221	388	1	[	[	X
ma-221	388	2	θ	θ	X
ma-221	388	3	[	[	X
ma-221	388	4	(	(	PUNCT
ma-221	388	5	a	a	DET
ma-221	388	6	+	+	NUM
ma-221	388	7	2h)d	2h)d	NUM
ma-221	388	8	(	(	PUNCT
ma-221	388	9	tz	tz	PROPN
ma-221	388	10	,	,	PUNCT
ma-221	388	11	tu	tu	PROPN
ma-221	388	12	)	)	PUNCT
ma-221	388	13	]	]	X
ma-221	388	14	]	]	PUNCT
ma-221	388	15	,	,	PUNCT
ma-221	388	16	which	which	PRON
ma-221	388	17	is	be	AUX
ma-221	388	18	a	a	DET
ma-221	388	19	contradiction	contradiction	NOUN
ma-221	388	20	.	.	PUNCT
ma-221	389	1	thus	thus	ADV
ma-221	389	2	,	,	PUNCT
ma-221	389	3	z	z	NOUN
ma-221	389	4	and	and	CCONJ
ma-221	389	5	u	u	NOUN
ma-221	389	6	must	must	AUX
ma-221	389	7	be	be	AUX
ma-221	389	8	identical	identical	ADJ
ma-221	389	9	.	.	PUNCT
ma-221	390	1	hence	hence	ADV
ma-221	390	2	,	,	PUNCT
ma-221	390	3	t	t	PROPN
ma-221	390	4	has	have	VERB
ma-221	390	5	a	a	DET
ma-221	390	6	unique	unique	ADJ
ma-221	390	7	best	good	ADJ
ma-221	390	8	proximitypoint	proximitypoint	NOUN
ma-221	390	9	.	.	PUNCT
ma-221	391	1	�	�	PROPN
ma-221	391	2	theorem	theorem	VERB
ma-221	391	3	3.5	3.5	NUM
ma-221	391	4	.	.	PUNCT
ma-221	392	1	let	let	VERB
ma-221	392	2	(	(	PUNCT
ma-221	392	3	x	x	NOUN
ma-221	392	4	,	,	PUNCT
ma-221	392	5	d	d	NOUN
ma-221	392	6	)	)	PUNCT
ma-221	392	7	be	be	AUX
ma-221	392	8	a	a	DET
ma-221	392	9	complete	complete	ADJ
ma-221	392	10	metric	metric	ADJ
ma-221	392	11	space	space	NOUN
ma-221	392	12	and	and	CCONJ
ma-221	392	13	(	(	PUNCT
ma-221	392	14	a	a	DET
ma-221	392	15	,	,	PUNCT
ma-221	392	16	b	b	NOUN
ma-221	392	17	)	)	PUNCT
ma-221	392	18	be	be	AUX
ma-221	392	19	a	a	DET
ma-221	392	20	pair	pair	NOUN
ma-221	392	21	of	of	ADP
ma-221	392	22	non	non	ADJ
ma-221	392	23	-	-	ADJ
ma-221	392	24	void	void	ADJ
ma-221	392	25	closed	closed	ADJ
ma-221	392	26	subsets	subset	NOUN
ma-221	392	27	of	of	ADP
ma-221	392	28	(	(	PUNCT
ma-221	392	29	x	x	X
ma-221	392	30	,	,	PUNCT
ma-221	392	31	d	d	NOUN
ma-221	392	32	)	)	PUNCT
ma-221	392	33	.	.	PUNCT
ma-221	393	1	let	let	VERB
ma-221	393	2	t	t	NOUN
ma-221	393	3	:	:	PUNCT
ma-221	393	4	a→	a→	PROPN
ma-221	393	5	b	b	X
ma-221	393	6	satisfy	satisfy	VERB
ma-221	393	7	the	the	DET
ma-221	393	8	following	follow	VERB
ma-221	393	9	conditions	condition	NOUN
ma-221	393	10	:(	:(	PUNCT
ma-221	394	1	i	i	PRON
ma-221	394	2	)	)	PUNCT
ma-221	394	3	t	t	PROPN
ma-221	394	4	(	(	PUNCT
ma-221	394	5	a0	a0	PROPN
ma-221	394	6	)	)	PUNCT
ma-221	394	7	∈	∈	PROPN
ma-221	394	8	b0	b0	NOUN
ma-221	394	9	and	and	CCONJ
ma-221	394	10	the	the	DET
ma-221	394	11	pair	pair	NOUN
ma-221	394	12	(	(	PUNCT
ma-221	394	13	a	a	DET
ma-221	394	14	,	,	PUNCT
ma-221	394	15	b	b	NOUN
ma-221	394	16	)	)	PUNCT
ma-221	394	17	satisfies	satisfy	VERB
ma-221	394	18	the	the	DET
ma-221	394	19	weak	weak	ADJ
ma-221	394	20	p	p	NOUN
ma-221	394	21	-property;(ii	-property;(ii	PROPN
ma-221	394	22	)	)	PUNCT
ma-221	394	23	t	t	PROPN
ma-221	394	24	is	be	AUX
ma-221	394	25	a	a	DET
ma-221	394	26	generalized	generalized	ADJ
ma-221	394	27	(	(	PUNCT
ma-221	394	28	θ	θ	NOUN
ma-221	394	29	,	,	PUNCT
ma-221	394	30	φ)-proximal	φ)-proximal	ADJ
ma-221	394	31	contraction	contraction	NOUN
ma-221	394	32	of	of	ADP
ma-221	394	33	the	the	DET
ma-221	394	34	first	first	ADJ
ma-221	394	35	kind	kind	NOUN
ma-221	394	36	as	as	ADV
ma-221	394	37	well	well	ADV
ma-221	394	38	as	as	ADP
ma-221	394	39	a	a	DET
ma-221	394	40	generalized	generalized	ADJ
ma-221	394	41	(	(	PUNCT
ma-221	394	42	θ	θ	NOUN
ma-221	394	43	,	,	PUNCT
ma-221	394	44	φ)-proximal	φ)-proximal	ADJ
ma-221	394	45	contraction	contraction	NOUN
ma-221	394	46	of	of	ADP
ma-221	394	47	the	the	DET
ma-221	394	48	second	second	ADJ
ma-221	394	49	kind	kind	NOUN
ma-221	394	50	.	.	PUNCT
ma-221	395	1	then	then	ADV
ma-221	395	2	there	there	PRON
ma-221	395	3	exists	exist	VERB
ma-221	395	4	a	a	DET
ma-221	395	5	unique	unique	ADJ
ma-221	395	6	u	u	NOUN
ma-221	395	7	∈	∈	PROPN
ma-221	395	8	a	a	DET
ma-221	395	9	such	such	ADJ
ma-221	395	10	that	that	SCONJ
ma-221	395	11	d(u	d(u	PROPN
ma-221	395	12	,	,	PUNCT
ma-221	395	13	tu	tu	PROPN
ma-221	395	14	)	)	PUNCT
ma-221	395	15	=	=	SYM
ma-221	396	1	d(a	d(a	PROPN
ma-221	396	2	,	,	PUNCT
ma-221	396	3	b	b	NOUN
ma-221	396	4	)	)	PUNCT
ma-221	396	5	and	and	CCONJ
ma-221	396	6	un	un	PROPN
ma-221	396	7	→	→	SYM
ma-221	396	8	u	u	PROPN
ma-221	396	9	,	,	PUNCT
ma-221	396	10	where	where	SCONJ
ma-221	396	11	u0	u0	ADJ
ma-221	396	12	is	be	AUX
ma-221	396	13	any	any	DET
ma-221	396	14	fixed	fixed	ADJ
ma-221	396	15	point	point	NOUN
ma-221	396	16	in	in	ADP
ma-221	396	17	a0	a0	PROPN
ma-221	396	18	and	and	CCONJ
ma-221	396	19	d(un+1	d(un+1	PROPN
ma-221	396	20	,	,	PUNCT
ma-221	396	21	t	t	PROPN
ma-221	396	22	un	un	PROPN
ma-221	396	23	)	)	PUNCT
ma-221	396	24	=	=	SYM
ma-221	397	1	d(a	d(a	PROPN
ma-221	397	2	,	,	PUNCT
ma-221	397	3	b	b	NOUN
ma-221	397	4	)	)	PUNCT
ma-221	397	5	for	for	ADP
ma-221	397	6	n	n	PRON
ma-221	397	7	≥	≥	NOUN
ma-221	397	8	0	0	NUM
ma-221	397	9	.	.	PUNCT
ma-221	398	1	proof	proof	NOUN
ma-221	398	2	.	.	PUNCT
ma-221	399	1	similar	similar	ADJ
ma-221	399	2	to	to	ADP
ma-221	399	3	theorem	theorem	VERB
ma-221	399	4	3.3	3.3	NUM
ma-221	399	5	,	,	PUNCT
ma-221	399	6	we	we	PRON
ma-221	399	7	find	find	VERB
ma-221	399	8	a	a	DET
ma-221	399	9	sequence	sequence	NOUN
ma-221	399	10	{	{	PUNCT
ma-221	399	11	un	un	PROPN
ma-221	399	12	}	}	PUNCT
ma-221	399	13	in	in	ADP
ma-221	399	14	a0	a0	NOUN
ma-221	400	1	such	such	ADJ
ma-221	400	2	that	that	PRON
ma-221	400	3	d(un+1	d(un+1	PROPN
ma-221	400	4	,	,	PUNCT
ma-221	400	5	t	t	PROPN
ma-221	400	6	un	un	PROPN
ma-221	400	7	)	)	PUNCT
ma-221	400	8	=	=	SYM
ma-221	400	9	d(a	d(a	PROPN
ma-221	400	10	,	,	PUNCT
ma-221	400	11	b	b	NOUN
ma-221	400	12	)	)	PUNCT
ma-221	400	13	for	for	ADP
ma-221	400	14	all	all	DET
ma-221	400	15	non	non	ADJ
ma-221	400	16	-	-	ADJ
ma-221	400	17	negative	negative	ADJ
ma-221	400	18	integral	integral	ADJ
ma-221	400	19	values	value	NOUN
ma-221	400	20	of	of	ADP
ma-221	400	21	n.	n.	NOUN
ma-221	400	22	similar	similar	ADJ
ma-221	400	23	to	to	ADP
ma-221	400	24	theorem	theorem	VERB
ma-221	400	25	3.3	3.3	NUM
ma-221	400	26	,	,	PUNCT
ma-221	400	27	we	we	PRON
ma-221	400	28	can	can	AUX
ma-221	400	29	show	show	VERB
ma-221	400	30	that	that	DET
ma-221	400	31	sequence	sequence	NOUN
ma-221	400	32	{	{	PUNCT
ma-221	400	33	un}is	un}is	ADP
ma-221	400	34	a	a	DET
ma-221	400	35	cauchy	cauchy	ADJ
ma-221	400	36	sequence	sequence	NOUN
ma-221	400	37	.	.	PUNCT
ma-221	401	1	thus	thus	ADV
ma-221	401	2	converges	converge	VERB
ma-221	401	3	to	to	ADP
ma-221	401	4	some	some	DET
ma-221	401	5	element	element	NOUN
ma-221	401	6	u	u	NOUN
ma-221	401	7	in	in	ADP
ma-221	401	8	a.	a.	NOUN
ma-221	401	9	as	as	ADP
ma-221	401	10	in	in	ADP
ma-221	401	11	theorem	theorem	NOUN
ma-221	401	12	3.4	3.4	NUM
ma-221	401	13	,	,	PUNCT
ma-221	401	14	it	it	PRON
ma-221	401	15	can	can	AUX
ma-221	401	16	be	be	AUX
ma-221	401	17	shownthat	shownthat	NOUN
ma-221	401	18	the	the	DET
ma-221	401	19	sequence	sequence	NOUN
ma-221	401	20	{	{	PUNCT
ma-221	401	21	tun	tun	NOUN
ma-221	401	22	}	}	PUNCT
ma-221	401	23	is	be	AUX
ma-221	401	24	a	a	DET
ma-221	401	25	cauchy	cauchy	ADJ
ma-221	401	26	sequence	sequence	NOUN
ma-221	401	27	and	and	CCONJ
ma-221	401	28	converges	converge	VERB
ma-221	401	29	to	to	ADP
ma-221	401	30	some	some	DET
ma-221	401	31	element	element	NOUN
ma-221	401	32	v	v	NOUN
ma-221	401	33	in	in	ADP
ma-221	401	34	b.	b.	PROPN
ma-221	401	35	therefore	therefore	ADV
ma-221	401	36	,	,	PUNCT
ma-221	401	37	d(u	d(u	PROPN
ma-221	401	38	,	,	PUNCT
ma-221	401	39	v	v	NOUN
ma-221	401	40	)	)	PUNCT
ma-221	401	41	=	=	VERB
ma-221	402	1	lim	lim	PROPN
ma-221	402	2	n→∞	n→∞	X
ma-221	402	3	d(un+1	d(un+1	PROPN
ma-221	402	4	,	,	PUNCT
ma-221	402	5	t	t	PROPN
ma-221	402	6	un	un	PROPN
ma-221	402	7	)	)	PUNCT
ma-221	402	8	=	=	SYM
ma-221	403	1	d(a	d(a	PROPN
ma-221	403	2	,	,	PUNCT
ma-221	403	3	b	b	NOUN
ma-221	403	4	)	)	PUNCT
ma-221	403	5	.	.	PUNCT
ma-221	404	1	(	(	PUNCT
ma-221	404	2	3.34	3.34	NUM
ma-221	404	3	)	)	PUNCT
ma-221	404	4	eventually	eventually	ADV
ma-221	404	5	,	,	PUNCT
ma-221	404	6	u	u	NOUN
ma-221	404	7	becomes	become	VERB
ma-221	404	8	an	an	DET
ma-221	404	9	element	element	NOUN
ma-221	404	10	of	of	ADP
ma-221	404	11	a0	a0	PROPN
ma-221	404	12	.	.	PUNCT
ma-221	405	1	in	in	ADP
ma-221	405	2	light	light	NOUN
ma-221	405	3	of	of	ADP
ma-221	405	4	the	the	DET
ma-221	405	5	fact	fact	NOUN
ma-221	405	6	that	that	SCONJ
ma-221	405	7	t	t	PROPN
ma-221	405	8	(	(	PUNCT
ma-221	405	9	a0	a0	PROPN
ma-221	405	10	)	)	PUNCT
ma-221	405	11	∈	∈	PROPN
ma-221	405	12	b0	b0	NOUN
ma-221	405	13	,	,	PUNCT
ma-221	405	14	d(t	d(t	PROPN
ma-221	405	15	,	,	PUNCT
ma-221	405	16	tu	tu	PROPN
ma-221	405	17	)	)	PUNCT
ma-221	405	18	=	=	SYM
ma-221	406	1	d(a	d(a	PROPN
ma-221	406	2	,	,	PUNCT
ma-221	406	3	b	b	NOUN
ma-221	406	4	)	)	PUNCT
ma-221	406	5	for	for	ADP
ma-221	406	6	some	some	DET
ma-221	406	7	element	element	NOUN
ma-221	406	8	t	t	PROPN
ma-221	406	9	in	in	ADP
ma-221	406	10	a.	a.	NOUN
ma-221	406	11	from	from	ADP
ma-221	406	12	the	the	DET
ma-221	406	13	p	p	NOUN
ma-221	406	14	-	-	PUNCT
ma-221	406	15	property	property	NOUN
ma-221	406	16	framework	framework	NOUN
ma-221	406	17	and	and	CCONJ
ma-221	406	18	(	(	PUNCT
ma-221	406	19	3.34	3.34	NUM
ma-221	406	20	,	,	PUNCT
ma-221	406	21	)	)	PUNCT
ma-221	407	1	we	we	PRON
ma-221	407	2	get	get	VERB
ma-221	407	3	d(un+1	d(un+1	PROPN
ma-221	407	4	,	,	PUNCT
ma-221	407	5	t	t	NOUN
ma-221	407	6	)	)	PUNCT
ma-221	407	7	=	=	SYM
ma-221	407	8	d(tun	d(tun	PROPN
ma-221	407	9	,	,	PUNCT
ma-221	407	10	t	t	PROPN
ma-221	407	11	u),∀n	u),∀n	NOUN
ma-221	407	12	∈	∈	PROPN
ma-221	407	13	n.	n.	NOUN
ma-221	407	14	if	if	SCONJ
ma-221	407	15	for	for	ADP
ma-221	407	16	some	some	DET
ma-221	407	17	n0	n0	PROPN
ma-221	407	18	,	,	PUNCT
ma-221	407	19	d(t	d(t	PROPN
ma-221	407	20	,	,	PUNCT
ma-221	407	21	un0	un0	NOUN
ma-221	407	22	+	+	NOUN
ma-221	407	23	1	1	NUM
ma-221	407	24	)	)	PUNCT
ma-221	407	25	=	=	SYM
ma-221	407	26	0	0	NUM
ma-221	407	27	,	,	PUNCT
ma-221	407	28	consequently	consequently	ADV
ma-221	407	29	d(tun0	d(tun0	NOUN
ma-221	407	30	,	,	PUNCT
ma-221	407	31	t	t	PROPN
ma-221	407	32	u	u	NOUN
ma-221	407	33	)	)	PUNCT
ma-221	407	34	=	=	SYM
ma-221	407	35	0	0	NUM
ma-221	407	36	.	.	PUNCT
ma-221	408	1	so	so	ADV
ma-221	408	2	tun0	tun0	VERB
ma-221	408	3	=	=	SYM
ma-221	408	4	tu	tu	PROPN
ma-221	408	5	,	,	PUNCT
ma-221	408	6	hence	hence	ADV
ma-221	408	7	d(a	d(a	PROPN
ma-221	408	8	,	,	PUNCT
ma-221	408	9	b	b	NOUN
ma-221	408	10	)	)	PUNCT
ma-221	408	11	=	=	SYM
ma-221	408	12	d(u	d(u	PROPN
ma-221	408	13	,	,	PUNCT
ma-221	408	14	tu	tu	PROPN
ma-221	408	15	)	)	PUNCT
ma-221	408	16	.	.	PUNCT
ma-221	409	1	thus	thus	ADV
ma-221	409	2	the	the	DET
ma-221	409	3	conclusion	conclusion	NOUN
ma-221	409	4	is	be	AUX
ma-221	409	5	immediate	immediate	ADJ
ma-221	409	6	.	.	PUNCT
ma-221	410	1	so	so	ADV
ma-221	410	2	let	let	VERB
ma-221	410	3	for	for	ADP
ma-221	410	4	any	any	DET
ma-221	410	5	n	n	PRON
ma-221	410	6	≥	≥	NOUN
ma-221	410	7	0	0	NUM
ma-221	410	8	,	,	PUNCT
ma-221	410	9	d(t	d(t	PROPN
ma-221	410	10	,	,	PUNCT
ma-221	410	11	un+1	un+1	NOUN
ma-221	410	12	)	)	PUNCT
ma-221	410	13	>	>	X
ma-221	411	1	0	0	X
ma-221	411	2	.	.	PUNCT
ma-221	412	1	since	since	SCONJ
ma-221	412	2	t	t	PROPN
ma-221	412	3	is	be	AUX
ma-221	412	4	ageneralized	ageneralize	VERB
ma-221	412	5	(	(	PUNCT
ma-221	412	6	θ	θ	NOUN
ma-221	412	7	,	,	PUNCT
ma-221	412	8	φ)-proximal	φ)-proximal	ADJ
ma-221	412	9	contraction	contraction	NOUN
ma-221	412	10	of	of	ADP
ma-221	412	11	the	the	DET
ma-221	412	12	first	first	ADJ
ma-221	412	13	kind	kind	NOUN
ma-221	412	14	,	,	PUNCT
ma-221	412	15	it	it	PRON
ma-221	412	16	can	can	AUX
ma-221	412	17	be	be	AUX
ma-221	412	18	seen	see	VERB
ma-221	412	19	that	that	SCONJ
ma-221	412	20	θ(d(t	θ(d(t	NOUN
ma-221	412	21	,	,	PUNCT
ma-221	412	22	un+1	un+1	NOUN
ma-221	412	23	)	)	PUNCT
ma-221	412	24	)	)	PUNCT
ma-221	412	25	≤	≤	NUM
ma-221	413	1	φ	φ	PROPN
ma-221	414	1	[	[	X
ma-221	414	2	θ	θ	X
ma-221	414	3	(	(	PUNCT
ma-221	414	4	ad(u	ad(u	NOUN
ma-221	414	5	,	,	PUNCT
ma-221	414	6	un	un	PROPN
ma-221	414	7	)	)	PUNCT
ma-221	414	8	+	+	NUM
ma-221	414	9	bd(t	bd(t	NOUN
ma-221	414	10	,	,	PUNCT
ma-221	414	11	u	u	NOUN
ma-221	414	12	)	)	PUNCT
ma-221	414	13	+	+	CCONJ
ma-221	414	14	cd(un	cd(un	PROPN
ma-221	414	15	,	,	PUNCT
ma-221	414	16	un+1	un+1	NOUN
ma-221	414	17	)	)	PUNCT
ma-221	414	18	+	+	CCONJ
ma-221	414	19	h[d(u	h[d(u	X
ma-221	414	20	,	,	PUNCT
ma-221	414	21	un+1	un+1	NOUN
ma-221	414	22	)	)	PUNCT
ma-221	415	1	+	+	CCONJ
ma-221	415	2	d(un	d(un	PROPN
ma-221	415	3	,	,	PUNCT
ma-221	415	4	t	t	PROPN
ma-221	415	5	)	)	PUNCT
ma-221	415	6	)	)	PUNCT
ma-221	415	7	]	]	PUNCT
ma-221	415	8	.	.	PUNCT
ma-221	416	1	(	(	PUNCT
ma-221	416	2	3.35	3.35	NUM
ma-221	416	3	)	)	PUNCT
ma-221	416	4	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	416	5	eur	eur	PROPN
ma-221	416	6	.	.	PUNCT
ma-221	417	1	j.	j.	PROPN
ma-221	417	2	math	math	PROPN
ma-221	417	3	.	.	PUNCT
ma-221	418	1	anal	anal	PROPN
ma-221	418	2	.	.	PUNCT
ma-221	419	1	10.28924	10.28924	NUM
ma-221	419	2	/	/	SYM
ma-221	419	3	ada	ada	PROPN
ma-221	419	4	/	/	SYM
ma-221	419	5	ma.4.13	ma.4.13	PROPN
ma-221	419	6	12since	12since	NUM
ma-221	419	7	θ	θ	PROPN
ma-221	419	8	and	and	CCONJ
ma-221	419	9	φ	φ	PROPN
ma-221	419	10	are	be	AUX
ma-221	419	11	two	two	NUM
ma-221	419	12	continuous	continuous	ADJ
ma-221	419	13	functions	function	NOUN
ma-221	419	14	,	,	PUNCT
ma-221	419	15	by	by	ADP
ma-221	419	16	letting	let	VERB
ma-221	419	17	n	n	PRON
ma-221	419	18	→	→	SYM
ma-221	419	19	∞	∞	NUM
ma-221	419	20	in	in	ADP
ma-221	419	21	inequality	inequality	NOUN
ma-221	419	22	(	(	PUNCT
ma-221	419	23	3.35	3.35	NUM
ma-221	419	24	)	)	PUNCT
ma-221	419	25	,	,	PUNCT
ma-221	419	26	we	we	PRON
ma-221	419	27	obtain	obtain	VERB
ma-221	419	28	,	,	PUNCT
ma-221	419	29	d(u	d(u	PROPN
ma-221	419	30	,	,	PUNCT
ma-221	419	31	tu	tu	PROPN
ma-221	419	32	)	)	PUNCT
ma-221	419	33	=	=	SYM
ma-221	420	1	d(t	d(t	PROPN
ma-221	420	2	,	,	PUNCT
ma-221	420	3	tu	tu	PROPN
ma-221	420	4	)	)	PUNCT
ma-221	420	5	=	=	SYM
ma-221	421	1	d(a	d(a	PROPN
ma-221	421	2	,	,	PUNCT
ma-221	421	3	b	b	NOUN
ma-221	421	4	)	)	PUNCT
ma-221	421	5	.	.	PUNCT
ma-221	422	1	also	also	ADV
ma-221	422	2	,	,	PUNCT
ma-221	422	3	as	as	ADP
ma-221	422	4	in	in	ADP
ma-221	422	5	the	the	DET
ma-221	422	6	theorem	theorem	ADJ
ma-221	422	7	3.3	3.3	NUM
ma-221	422	8	,	,	PUNCT
ma-221	422	9	the	the	DET
ma-221	422	10	uniqueness	uniqueness	NOUN
ma-221	422	11	of	of	ADP
ma-221	422	12	the	the	DET
ma-221	422	13	best	good	ADJ
ma-221	422	14	proximitypoint	proximitypoint	NOUN
ma-221	422	15	of	of	ADP
ma-221	422	16	mapping	mapping	NOUN
ma-221	422	17	t	t	PROPN
ma-221	422	18	follows	follow	VERB
ma-221	422	19	.	.	PUNCT
ma-221	423	1	�	�	PROPN
ma-221	423	2	example	example	NOUN
ma-221	423	3	3.6	3.6	NUM
ma-221	423	4	.	.	PUNCT
ma-221	424	1	let	let	VERB
ma-221	424	2	x	x	PUNCT
ma-221	424	3	=	=	PRON
ma-221	424	4	{	{	PUNCT
ma-221	424	5	λn	λn	NOUN
ma-221	424	6	:	:	PUNCT
ma-221	424	7	n	n	CCONJ
ma-221	424	8	∈	∈	PROPN
ma-221	424	9	n	n	CCONJ
ma-221	424	10	}	}	PUNCT
ma-221	424	11	with	with	ADP
ma-221	424	12	the	the	DET
ma-221	424	13	metric	metric	ADJ
ma-221	424	14	d(x	d(x	PROPN
ma-221	424	15	,	,	PUNCT
ma-221	424	16	y	y	NOUN
ma-221	424	17	)	)	PUNCT
ma-221	424	18	=	=	NOUN
ma-221	424	19	|x	|x	NOUN
ma-221	424	20	−	−	NOUN
ma-221	424	21	y	y	PROPN
ma-221	424	22	|	|	ADV
ma-221	424	23	for	for	ADP
ma-221	424	24	all	all	DET
ma-221	424	25	x	x	NOUN
ma-221	424	26	,	,	PUNCT
ma-221	424	27	y	y	PROPN
ma-221	424	28	∈	∈	PROPN
ma-221	424	29	x	x	X
ma-221	424	30	,	,	PUNCT
ma-221	424	31	where	where	SCONJ
ma-221	424	32	thesequence	thesequence	NOUN
ma-221	424	33	gn	gn	PROPN
ma-221	424	34	,	,	PUNCT
ma-221	424	35	defined	define	VERB
ma-221	424	36	by	by	ADP
ma-221	424	37	λ1	λ1	PROPN
ma-221	424	38	=	=	SYM
ma-221	424	39	1	1	NUM
ma-221	424	40	λ2	λ2	NOUN
ma-221	424	41	=	=	SYM
ma-221	424	42	1	1	NUM
ma-221	424	43	+	+	NUM
ma-221	424	44	2	2	NUM
ma-221	424	45	λ3	λ3	NOUN
ma-221	424	46	=	=	NOUN
ma-221	424	47	1	1	NUM
ma-221	424	48	+	+	NUM
ma-221	424	49	2	2	NUM
ma-221	424	50	+	+	SYM
ma-221	424	51	3	3	NUM
ma-221	424	52	...	...	PUNCT
ma-221	425	1	λn	λn	X
ma-221	425	2	=	=	SYM
ma-221	425	3	1	1	NUM
ma-221	425	4	+	+	NUM
ma-221	425	5	2	2	NUM
ma-221	425	6	+	+	NUM
ma-221	425	7	3	3	NUM
ma-221	425	8	+	+	CCONJ
ma-221	425	9	...	...	PUNCT
ma-221	426	1	+	+	CCONJ
ma-221	426	2	n.	n.	NOUN
ma-221	426	3	we	we	PRON
ma-221	426	4	know	know	VERB
ma-221	426	5	,	,	PUNCT
ma-221	426	6	(	(	PUNCT
ma-221	426	7	x	x	NOUN
ma-221	426	8	,	,	PUNCT
ma-221	426	9	d	d	NOUN
ma-221	426	10	)	)	PUNCT
ma-221	426	11	is	be	AUX
ma-221	426	12	a	a	DET
ma-221	426	13	complete	complete	ADJ
ma-221	426	14	metric	metric	ADJ
ma-221	426	15	space	space	NOUN
ma-221	426	16	.	.	PUNCT
ma-221	427	1	let	let	VERB
ma-221	427	2	a	a	DET
ma-221	427	3	=	=	SYM
ma-221	427	4	g3n	g3n	NOUN
ma-221	427	5	:	:	PUNCT
ma-221	427	6	n	n	CCONJ
ma-221	427	7	∈	∈	PROPN
ma-221	427	8	n	n	NOUN
ma-221	427	9	and	and	CCONJ
ma-221	427	10	b	b	X
ma-221	427	11	=	=	SYM
ma-221	427	12	g3n−1	g3n−1	PROPN
ma-221	427	13	:	:	PUNCT
ma-221	427	14	n	n	X
ma-221	427	15	∈	∈	NOUN
ma-221	427	16	n.it	n.it	NOUN
ma-221	427	17	is	be	AUX
ma-221	427	18	easy	easy	ADJ
ma-221	427	19	to	to	PART
ma-221	427	20	see	see	VERB
ma-221	427	21	that	that	SCONJ
ma-221	427	22	d(a	d(a	PROPN
ma-221	427	23	,	,	PUNCT
ma-221	427	24	b	b	NOUN
ma-221	427	25	)	)	PUNCT
ma-221	427	26	=	=	SYM
ma-221	427	27	3	3	NUM
ma-221	427	28	,	,	PUNCT
ma-221	427	29	a0	a0	PROPN
ma-221	427	30	=	=	PROPN
ma-221	427	31	a	a	PROPN
ma-221	427	32	and	and	CCONJ
ma-221	427	33	b0	b0	NOUN
ma-221	427	34	=	=	PROPN
ma-221	427	35	b.	b.	PROPN
ma-221	427	36	define	define	VERB
ma-221	427	37	a	a	DET
ma-221	427	38	mappings	mapping	NOUN
ma-221	427	39	t	t	NOUN
ma-221	427	40	:	:	PUNCT
ma-221	427	41	a	a	DET
ma-221	427	42	→	→	SYM
ma-221	427	43	b	b	NOUN
ma-221	427	44	,	,	PUNCT
ma-221	427	45	by	by	ADP
ma-221	427	46	t	t	PROPN
ma-221	427	47	(	(	PUNCT
ma-221	427	48	λ3n	λ3n	PROPN
ma-221	427	49	)	)	PUNCT
ma-221	427	50	=	=	SYM
ma-221	427	51	λ3n−1	λ3n−1	PROPN
ma-221	427	52	for	for	ADP
ma-221	427	53	all	all	DET
ma-221	427	54	n	n	PRON
ma-221	427	55	≥	≥	NOUN
ma-221	427	56	1	1	NUM
ma-221	427	57	.	.	PUNCT
ma-221	428	1	it	it	PRON
ma-221	428	2	is	be	AUX
ma-221	428	3	clear	clear	ADJ
ma-221	428	4	that	that	SCONJ
ma-221	428	5	a	a	PRON
ma-221	428	6	is	be	AUX
ma-221	428	7	approximately	approximately	ADV
ma-221	428	8	compact	compact	ADJ
ma-221	428	9	with	with	ADP
ma-221	428	10	respect	respect	NOUN
ma-221	428	11	to	to	ADP
ma-221	428	12	b	b	NOUN
ma-221	428	13	,	,	PUNCT
ma-221	428	14	(	(	PUNCT
ma-221	428	15	a	a	DET
ma-221	428	16	,	,	PUNCT
ma-221	428	17	b	b	NOUN
ma-221	428	18	)	)	PUNCT
ma-221	428	19	satisfies	satisfy	VERB
ma-221	428	20	the	the	DET
ma-221	428	21	p	p	NOUN
ma-221	428	22	-	-	PUNCT
ma-221	428	23	property	property	NOUN
ma-221	428	24	,	,	PUNCT
ma-221	428	25	t	t	PROPN
ma-221	428	26	is	be	AUX
ma-221	428	27	continuous	continuous	ADJ
ma-221	428	28	and	and	CCONJ
ma-221	428	29	t	t	PROPN
ma-221	428	30	(	(	PUNCT
ma-221	428	31	a0	a0	PROPN
ma-221	428	32	)	)	PUNCT
ma-221	428	33	⊆	⊆	NUM
ma-221	428	34	b0	b0	NOUN
ma-221	428	35	.	.	PUNCT
ma-221	429	1	we	we	PRON
ma-221	429	2	will	will	AUX
ma-221	429	3	show	show	VERB
ma-221	429	4	that	that	SCONJ
ma-221	429	5	t	t	PROPN
ma-221	429	6	is	be	AUX
ma-221	429	7	an	an	DET
ma-221	429	8	(	(	PUNCT
ma-221	429	9	θ	θ	NOUN
ma-221	429	10	,	,	PUNCT
ma-221	429	11	φ)-proximal	φ)-proximal	PUNCT
ma-221	429	12	contraction	contraction	NOUN
ma-221	429	13	with	with	ADP
ma-221	429	14	θ	θ	PROPN
ma-221	429	15	∈	∈	PROPN
ma-221	429	16	θ	θ	PROPN
ma-221	429	17	and	and	CCONJ
ma-221	429	18	φ	φ	PROPN
ma-221	429	19	∈	∈	PROPN
ma-221	429	20	φ	φ	PROPN
ma-221	429	21	that	that	PRON
ma-221	429	22	is	be	AUX
ma-221	429	23	θ(t	θ(t	PROPN
ma-221	429	24	)	)	PUNCT
ma-221	429	25	=	=	SYM
ma-221	429	26	et	et	NOUN
ma-221	429	27	and	and	CCONJ
ma-221	429	28	φ(t	φ(t	PROPN
ma-221	429	29	)	)	PUNCT
ma-221	429	30	=	=	PUNCT
ma-221	429	31	t	t	PROPN
ma-221	429	32	1	1	NUM
ma-221	429	33	2	2	NUM
ma-221	429	34	.	.	PUNCT
ma-221	430	1	observe	observe	VERB
ma-221	430	2	that	that	SCONJ
ma-221	430	3	,	,	PUNCT
ma-221	430	4	with	with	ADP
ma-221	430	5	out	out	ADP
ma-221	430	6	of	of	ADP
ma-221	430	7	generality	generality	NOUN
ma-221	430	8	,	,	PUNCT
ma-221	430	9	we	we	PRON
ma-221	430	10	may	may	AUX
ma-221	430	11	assume	assume	VERB
ma-221	430	12	that	that	SCONJ
ma-221	430	13	n	n	PROPN
ma-221	430	14	<	<	X
ma-221	430	15	m	m	PROPN
ma-221	430	16	,	,	PUNCT
ma-221	430	17	and	and	CCONJ
ma-221	430	18	since	since	SCONJ
ma-221	430	19	λ3n−1	λ3n−1	PROPN
ma-221	430	20	=	=	NOUN
ma-221	430	21	1	1	NUM
ma-221	430	22	+	+	NUM
ma-221	430	23	2	2	NUM
ma-221	430	24	+	+	NUM
ma-221	430	25	3	3	NUM
ma-221	430	26	+	+	NUM
ma-221	430	27	...	...	PUNCT
ma-221	431	1	+	+	NUM
ma-221	431	2	3n	3n	NUM
ma-221	431	3	−	−	NUM
ma-221	431	4	1	1	NUM
ma-221	431	5	,	,	PUNCT
ma-221	431	6	λ3m−1	λ3m−1	PROPN
ma-221	431	7	=	=	PUNCT
ma-221	432	1	1	1	NUM
ma-221	432	2	+	+	NUM
ma-221	432	3	2	2	NUM
ma-221	432	4	+	+	NUM
ma-221	432	5	3	3	NUM
ma-221	432	6	+	+	CCONJ
ma-221	432	7	...	...	PUNCT
ma-221	433	1	+	+	CCONJ
ma-221	433	2	3	3	NUM
ma-221	433	3	m	m	NOUN
ma-221	433	4	−	−	NOUN
ma-221	433	5	1	1	NUM
ma-221	433	6	,	,	PUNCT
ma-221	433	7	λ3n	λ3n	VERB
ma-221	433	8	=	=	NOUN
ma-221	433	9	1	1	NUM
ma-221	434	1	+	+	NUM
ma-221	434	2	2	2	NUM
ma-221	434	3	+	+	NUM
ma-221	434	4	3	3	NUM
ma-221	434	5	+	+	NUM
ma-221	434	6	...	...	PUNCT
ma-221	435	1	+	+	NUM
ma-221	435	2	3n	3n	NUM
ma-221	435	3	−	−	NOUN
ma-221	435	4	1	1	NUM
ma-221	435	5	+	+	NUM
ma-221	435	6	3n	3n	NUM
ma-221	435	7	,	,	PUNCT
ma-221	435	8	λ3	λ3	PROPN
ma-221	435	9	m	m	NOUN
ma-221	435	10	=	=	SYM
ma-221	435	11	1	1	NUM
ma-221	435	12	+	+	NUM
ma-221	435	13	2	2	NUM
ma-221	435	14	+	+	NUM
ma-221	435	15	3	3	NUM
ma-221	435	16	+	+	CCONJ
ma-221	435	17	...	...	PUNCT
ma-221	435	18	+	+	CCONJ
ma-221	435	19	3	3	NUM
ma-221	435	20	m	m	NOUN
ma-221	435	21	−	−	NOUN
ma-221	435	22	1	1	NUM
ma-221	435	23	+	+	SYM
ma-221	435	24	3	3	NUM
ma-221	435	25	m.	m.	NOUN
ma-221	435	26	it	it	PRON
ma-221	435	27	follow	follow	VERB
ma-221	435	28	that	that	SCONJ
ma-221	435	29	,	,	PUNCT
ma-221	435	30	d(t	d(t	PROPN
ma-221	435	31	(	(	PUNCT
ma-221	435	32	λ3n	λ3n	PROPN
ma-221	435	33	)	)	PUNCT
ma-221	435	34	,	,	PUNCT
ma-221	435	35	t	t	PROPN
ma-221	435	36	(	(	PUNCT
ma-221	435	37	λ3	λ3	PROPN
ma-221	435	38	m	m	NOUN
ma-221	435	39	)	)	PUNCT
ma-221	435	40	)	)	PUNCT
ma-221	436	1	=	=	PUNCT
ma-221	436	2	|λ3n−1	|λ3n−1	ADJ
ma-221	436	3	−	−	PUNCT
ma-221	437	1	λ3m−1|	λ3m−1|	NOUN
ma-221	437	2	=	=	SYM
ma-221	437	3	3n	3n	NOUN
ma-221	437	4	+	+	CCONJ
ma-221	437	5	(	(	PUNCT
ma-221	437	6	3n	3n	NUM
ma-221	437	7	+	+	CCONJ
ma-221	437	8	1	1	NUM
ma-221	437	9	)	)	PUNCT
ma-221	437	10	+	+	CCONJ
ma-221	437	11	...	...	PUNCT
ma-221	438	1	+	+	CCONJ
ma-221	438	2	(	(	PUNCT
ma-221	438	3	3	3	NUM
ma-221	438	4	m	m	NOUN
ma-221	438	5	−	−	NOUN
ma-221	438	6	1	1	NUM
ma-221	438	7	)	)	PUNCT
ma-221	438	8	,	,	PUNCT
ma-221	438	9	d(λ3n	d(λ3n	PROPN
ma-221	438	10	,	,	PUNCT
ma-221	438	11	λ2	λ2	PROPN
ma-221	438	12	m	m	NOUN
ma-221	438	13	)	)	PUNCT
ma-221	438	14	=	=	SYM
ma-221	438	15	|λ2n	|λ2n	NOUN
ma-221	438	16	−	−	NOUN
ma-221	438	17	λ2m|	λ2m|	X
ma-221	438	18	=	=	SYM
ma-221	438	19	3n	3n	NOUN
ma-221	438	20	+	+	CCONJ
ma-221	438	21	(	(	PUNCT
ma-221	438	22	3n	3n	NUM
ma-221	438	23	+	+	CCONJ
ma-221	438	24	1	1	NUM
ma-221	438	25	)	)	PUNCT
ma-221	438	26	+	+	CCONJ
ma-221	438	27	...	...	PUNCT
ma-221	439	1	+	+	CCONJ
ma-221	439	2	(	(	PUNCT
ma-221	439	3	3	3	NUM
ma-221	439	4	m	m	NOUN
ma-221	439	5	)	)	PUNCT
ma-221	439	6	,	,	PUNCT
ma-221	439	7	and	and	CCONJ
ma-221	439	8	d(t	d(t	PROPN
ma-221	439	9	(	(	PUNCT
ma-221	439	10	λ2n	λ2n	NOUN
ma-221	439	11	)	)	PUNCT
ma-221	439	12	,	,	PUNCT
ma-221	439	13	t	t	PROPN
ma-221	439	14	(	(	PUNCT
ma-221	439	15	λ3m))−	λ3m))−	X
ma-221	439	16	d(λ3n	d(λ3n	ADP
ma-221	439	17	,	,	PUNCT
ma-221	439	18	λ3	λ3	PROPN
ma-221	439	19	m	m	NOUN
ma-221	439	20	)	)	PUNCT
ma-221	440	1	=	=	SYM
ma-221	440	2	|λ3n−1	|λ3n−1	ADJ
ma-221	440	3	−	−	X
ma-221	441	1	λ3m−1|	λ3m−1|	PRON
ma-221	441	2	−	−	PROPN
ma-221	441	3	|λ3n	|λ3n	NUM
ma-221	441	4	−	−	PROPN
ma-221	441	5	λ3m|	λ3m|	NOUN
ma-221	441	6	=	=	SYM
ma-221	441	7	3n	3n	NUM
ma-221	441	8	−	−	NUM
ma-221	441	9	3	3	NUM
ma-221	441	10	m.	m.	NOUN
ma-221	441	11	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	441	12	eur	eur	PROPN
ma-221	441	13	.	.	PUNCT
ma-221	442	1	j.	j.	PROPN
ma-221	442	2	math	math	PROPN
ma-221	442	3	.	.	PUNCT
ma-221	443	1	anal	anal	PROPN
ma-221	443	2	.	.	PUNCT
ma-221	444	1	10.28924	10.28924	NUM
ma-221	444	2	/	/	SYM
ma-221	444	3	ada	ada	PROPN
ma-221	444	4	/	/	SYM
ma-221	444	5	ma.4.13	ma.4.13	PROPN
ma-221	444	6	13so	13so	NOUN
ma-221	445	1	that	that	SCONJ
ma-221	445	2	,	,	PUNCT
ma-221	445	3	ed(t	ed(t	X
ma-221	445	4	(	(	PUNCT
ma-221	445	5	λ3n),t	λ3n),t	X
ma-221	445	6	(	(	PUNCT
ma-221	445	7	λ3m))−d(λ3n	λ3m))−d(λ3n	PROPN
ma-221	445	8	,	,	PUNCT
ma-221	445	9	λ3	λ3	PROPN
ma-221	445	10	m	m	NOUN
ma-221	445	11	)	)	PUNCT
ma-221	445	12	)	)	PUNCT
ma-221	445	13	=	=	PRON
ma-221	445	14	ed(t	ed(t	X
ma-221	445	15	(	(	PUNCT
ma-221	445	16	λ3n),t	λ3n),t	X
ma-221	445	17	(	(	PUNCT
ma-221	445	18	λ3	λ3	PROPN
ma-221	445	19	m	m	NOUN
ma-221	445	20	)	)	PUNCT
ma-221	445	21	)	)	PUNCT
ma-221	446	1	ed(λ3n	ed(λ3n	PROPN
ma-221	446	2	,	,	PUNCT
ma-221	446	3	λ3	λ3	PROPN
ma-221	446	4	m	m	NOUN
ma-221	446	5	)	)	PUNCT
ma-221	447	1	=	=	PUNCT
ma-221	448	1	e3n−3	e3n−3	PROPN
ma-221	448	2	m	m	PROPN
ma-221	448	3	)	)	PUNCT
ma-221	449	1	=	=	SYM
ma-221	449	2	e−3(m−n	e−3(m−n	NOUN
ma-221	449	3	)	)	PUNCT
ma-221	449	4	)	)	PUNCT
ma-221	450	1	≤	≤	NUM
ma-221	451	1	e−3	e−3	PROPN
ma-221	451	2	=	=	SYM
ma-221	451	3	1	1	NUM
ma-221	451	4	e3	e3	NOUN
ma-221	451	5	.	.	PUNCT
ma-221	452	1	so	so	ADV
ma-221	452	2	that	that	SCONJ
ma-221	452	3	,	,	PUNCT
ma-221	452	4	ed(t	ed(t	X
ma-221	452	5	(	(	PUNCT
ma-221	452	6	λ3n),t	λ3n),t	X
ma-221	452	7	(	(	PUNCT
ma-221	452	8	λ3	λ3	PROPN
ma-221	452	9	m	m	PROPN
ma-221	452	10	)	)	PUNCT
ma-221	452	11	)	)	PUNCT
ma-221	453	1	+	+	CCONJ
ma-221	453	2	1	1	NUM
ma-221	453	3	=	=	NOUN
ma-221	453	4	θ(d(t	θ(d(t	NOUN
ma-221	453	5	(	(	PUNCT
ma-221	453	6	λ3n	λ3n	PROPN
ma-221	453	7	)	)	PUNCT
ma-221	453	8	,	,	PUNCT
ma-221	453	9	t	t	PROPN
ma-221	453	10	(	(	PUNCT
ma-221	453	11	λ3	λ3	PROPN
ma-221	453	12	m	m	PROPN
ma-221	453	13	)	)	PUNCT
ma-221	453	14	)	)	PUNCT
ma-221	453	15	)	)	PUNCT
ma-221	453	16	≤	≤	PUNCT
ma-221	454	1	ed(λ3n	ed(λ3n	PROPN
ma-221	454	2	,	,	PUNCT
ma-221	454	3	λ3	λ3	PROPN
ma-221	454	4	m	m	NOUN
ma-221	454	5	)	)	PUNCT
ma-221	454	6	1	1	NUM
ma-221	454	7	e3	e3	VERB
ma-221	454	8	+	+	CCONJ
ma-221	454	9	1	1	NUM
ma-221	454	10	≤	≤	ADJ
ma-221	454	11	ed(λ3n	ed(λ3n	PROPN
ma-221	454	12	,	,	PUNCT
ma-221	454	13	λ3	λ3	PROPN
ma-221	454	14	m	m	NOUN
ma-221	454	15	)	)	PUNCT
ma-221	455	1	+	+	CCONJ
ma-221	455	2	2	2	NUM
ma-221	455	3	2	2	NUM
ma-221	455	4	=	=	SYM
ma-221	455	5	φ	φ	PROPN
ma-221	456	1	[	[	X
ma-221	456	2	θ(d(λ3n	θ(d(λ3n	PROPN
ma-221	456	3	,	,	PUNCT
ma-221	456	4	λ3	λ3	PROPN
ma-221	456	5	m	m	PROPN
ma-221	456	6	)	)	PUNCT
ma-221	456	7	)	)	PUNCT
ma-221	456	8	]	]	PUNCT
ma-221	456	9	.	.	PUNCT
ma-221	457	1	consequently	consequently	ADV
ma-221	457	2	,	,	PUNCT
ma-221	457	3	t	t	PROPN
ma-221	457	4	is	be	AUX
ma-221	457	5	an	an	DET
ma-221	457	6	generalized	generalized	ADJ
ma-221	457	7	(	(	PUNCT
ma-221	457	8	θ	θ	NOUN
ma-221	457	9	,	,	PUNCT
ma-221	457	10	φ)-proximal	φ)-proximal	ADJ
ma-221	457	11	contraction	contraction	NOUN
ma-221	457	12	of	of	ADP
ma-221	457	13	the	the	DET
ma-221	457	14	second	second	ADJ
ma-221	457	15	kind	kind	NOUN
ma-221	457	16	with	with	ADP
ma-221	457	17	a	a	DET
ma-221	457	18	=	=	SYM
ma-221	457	19	1	1	NUM
ma-221	457	20	,	,	PUNCT
ma-221	457	21	b	b	NOUN
ma-221	457	22	=	=	SYM
ma-221	457	23	c	c	NOUN
ma-221	457	24	=	=	SYM
ma-221	457	25	h	h	NOUN
ma-221	457	26	=	=	NOUN
ma-221	457	27	0	0	NUM
ma-221	457	28	.	.	PUNCT
ma-221	458	1	thus	thus	ADV
ma-221	458	2	,	,	PUNCT
ma-221	458	3	all	all	DET
ma-221	458	4	the	the	DET
ma-221	458	5	conditions	condition	NOUN
ma-221	458	6	of	of	ADP
ma-221	458	7	theorem	theorem	ADJ
ma-221	458	8	3.4	3.4	NUM
ma-221	458	9	are	be	AUX
ma-221	458	10	satisfied	satisfied	ADJ
ma-221	458	11	.	.	PUNCT
ma-221	459	1	hence	hence	ADV
ma-221	459	2	,	,	PUNCT
ma-221	459	3	t	t	PROPN
ma-221	459	4	has	have	VERB
ma-221	459	5	a	a	DET
ma-221	459	6	unique	unique	ADJ
ma-221	459	7	bestproximity	bestproximity	NOUN
ma-221	459	8	point	point	NOUN
ma-221	459	9	and	and	CCONJ
ma-221	459	10	there	there	PRON
ma-221	459	11	exist	exist	VERB
ma-221	459	12	λ3	λ3	PROPN
ma-221	459	13	∈	∈	PROPN
ma-221	459	14	a	a	DET
ma-221	459	15	such	such	ADJ
ma-221	459	16	that	that	DET
ma-221	459	17	d(λ3	d(λ3	NOUN
ma-221	459	18	,	,	PUNCT
ma-221	459	19	tλ3	tλ3	NOUN
ma-221	459	20	)	)	PUNCT
ma-221	459	21	=	=	SYM
ma-221	459	22	d(λ3	d(λ3	NOUN
ma-221	459	23	,	,	PUNCT
ma-221	459	24	λ2	λ2	NOUN
ma-221	459	25	)	)	PUNCT
ma-221	459	26	=	=	SYM
ma-221	459	27	3	3	NUM
ma-221	459	28	=	=	SYM
ma-221	459	29	d(a	d(a	PROPN
ma-221	459	30	,	,	PUNCT
ma-221	459	31	b	b	NOUN
ma-221	459	32	)	)	PUNCT
ma-221	459	33	conflict	conflict	NOUN
ma-221	459	34	of	of	ADP
ma-221	459	35	interestthe	interestthe	ADJ
ma-221	459	36	authors	author	NOUN
ma-221	459	37	declare	declare	VERB
ma-221	459	38	that	that	SCONJ
ma-221	459	39	they	they	PRON
ma-221	459	40	have	have	VERB
ma-221	459	41	no	no	DET
ma-221	459	42	competing	compete	VERB
ma-221	459	43	interests	interest	NOUN
ma-221	459	44	.	.	PUNCT
ma-221	460	1	authors	author	NOUN
ma-221	460	2	’	'	PUNCT
ma-221	460	3	contributionsthe	contributionsthe	DET
ma-221	460	4	authors	author	NOUN
ma-221	460	5	equally	equally	ADV
ma-221	460	6	conceived	conceive	VERB
ma-221	460	7	of	of	ADP
ma-221	460	8	the	the	DET
ma-221	460	9	study	study	NOUN
ma-221	460	10	,	,	PUNCT
ma-221	460	11	participated	participate	VERB
ma-221	460	12	in	in	ADP
ma-221	460	13	its	its	PRON
ma-221	460	14	design	design	NOUN
ma-221	460	15	and	and	CCONJ
ma-221	460	16	coordination	coordination	NOUN
ma-221	460	17	,	,	PUNCT
ma-221	460	18	drafted	draft	VERB
ma-221	460	19	themanuscript	themanuscript	NOUN
ma-221	460	20	,	,	PUNCT
ma-221	460	21	participated	participate	VERB
ma-221	460	22	in	in	ADP
ma-221	460	23	the	the	DET
ma-221	460	24	sequence	sequence	NOUN
ma-221	460	25	alignment	alignment	NOUN
ma-221	460	26	,	,	PUNCT
ma-221	460	27	and	and	CCONJ
ma-221	460	28	read	read	VERB
ma-221	460	29	and	and	CCONJ
ma-221	460	30	approved	approve	VERB
ma-221	460	31	the	the	DET
ma-221	460	32	final	final	ADJ
ma-221	460	33	manuscript	manuscript	NOUN
ma-221	460	34	.	.	PUNCT
ma-221	461	1	references	reference	NOUN
ma-221	461	2	[	[	X
ma-221	461	3	1	1	NUM
ma-221	461	4	]	]	X
ma-221	461	5	h.	h.	PROPN
ma-221	461	6	aydi	aydi	PROPN
ma-221	461	7	,	,	PUNCT
ma-221	461	8	h.	h.	PROPN
ma-221	461	9	lakzian	lakzian	PROPN
ma-221	461	10	,	,	PUNCT
ma-221	461	11	z.d	z.d	PROPN
ma-221	461	12	.	.	PROPN
ma-221	461	13	mitrović	mitrović	PROPN
ma-221	461	14	,	,	PUNCT
ma-221	461	15	s.	s.	PROPN
ma-221	461	16	radenović	radenović	PROPN
ma-221	461	17	,	,	PUNCT
ma-221	461	18	best	good	ADJ
ma-221	461	19	proximity	proximity	NOUN
ma-221	461	20	points	point	NOUN
ma-221	461	21	of	of	ADP
ma-221	461	22	mt	mt	PROPN
ma-221	461	23	-	-	PUNCT
ma-221	461	24	cyclic	cyclic	ADJ
ma-221	461	25	contractions	contraction	NOUN
ma-221	461	26	with	with	ADP
ma-221	461	27	property	property	NOUN
ma-221	461	28	uc	uc	PROPN
ma-221	461	29	,	,	PUNCT
ma-221	461	30	numer	numer	PROPN
ma-221	461	31	.	.	PUNCT
ma-221	462	1	funct	funct	PROPN
ma-221	462	2	.	.	PUNCT
ma-221	463	1	anal	anal	PROPN
ma-221	463	2	.	.	PUNCT
ma-221	464	1	optim	optim	PROPN
ma-221	464	2	.	.	PUNCT
ma-221	465	1	41	41	NUM
ma-221	465	2	(	(	PUNCT
ma-221	465	3	2020	2020	NUM
ma-221	465	4	)	)	PUNCT
ma-221	465	5	871	871	NUM
ma-221	465	6	-	-	SYM
ma-221	465	7	882.[2	882.[2	NUM
ma-221	465	8	]	]	X
ma-221	465	9	s.	s.	PROPN
ma-221	465	10	banach	banach	PROPN
ma-221	465	11	,	,	PUNCT
ma-221	465	12	sur	sur	PROPN
ma-221	465	13	les	les	X
ma-221	465	14	opérations	opération	NOUN
ma-221	465	15	dans	dan	NOUN
ma-221	465	16	les	les	X
ma-221	465	17	ensembles	ensemble	NOUN
ma-221	465	18	abstraits	abstrait	NOUN
ma-221	465	19	et	et	PROPN
ma-221	465	20	leur	leur	X
ma-221	465	21	application	application	PROPN
ma-221	465	22	aux	aux	PROPN
ma-221	465	23	équations	équations	PROPN
ma-221	465	24	intégrales	intégrale	NOUN
ma-221	465	25	,	,	PUNCT
ma-221	465	26	fund.math	fund.math	PROPN
ma-221	465	27	.	.	NOUN
ma-221	465	28	3	3	NUM
ma-221	465	29	(	(	PUNCT
ma-221	465	30	1922	1922	NUM
ma-221	465	31	)	)	PUNCT
ma-221	465	32	133	133	NUM
ma-221	465	33	-	-	SYM
ma-221	465	34	181.[3	181.[3	NUM
ma-221	465	35	]	]	X
ma-221	465	36	s.s	s.s	PROPN
ma-221	465	37	.	.	PROPN
ma-221	465	38	basha	basha	PROPN
ma-221	465	39	,	,	PUNCT
ma-221	465	40	p.	p.	PROPN
ma-221	465	41	veeramani	veeramani	PROPN
ma-221	465	42	,	,	PUNCT
ma-221	465	43	best	good	ADJ
ma-221	465	44	proximity	proximity	NOUN
ma-221	465	45	pair	pair	NOUN
ma-221	465	46	theorems	theorem	NOUN
ma-221	465	47	for	for	ADP
ma-221	465	48	multifunctions	multifunction	NOUN
ma-221	465	49	with	with	ADP
ma-221	465	50	open	open	ADJ
ma-221	465	51	fibres	fibre	NOUN
ma-221	465	52	,	,	PUNCT
ma-221	465	53	j.	j.	PROPN
ma-221	465	54	approx	approx	PROPN
ma-221	465	55	.	.	PUNCT
ma-221	466	1	theory	theory	NOUN
ma-221	466	2	103(2000	103(2000	NUM
ma-221	466	3	)	)	PUNCT
ma-221	466	4	119–129.[4	119–129.[4	NUM
ma-221	466	5	]	]	X
ma-221	466	6	i.	i.	NOUN
ma-221	466	7	beg	beg	PROPN
ma-221	466	8	,	,	PUNCT
ma-221	466	9	g.	g.	PROPN
ma-221	466	10	mani	mani	PROPN
ma-221	466	11	,	,	PUNCT
ma-221	466	12	a.j	a.j	PROPN
ma-221	466	13	.	.	PROPN
ma-221	466	14	gnanaprakasam	gnanaprakasam	PROPN
ma-221	466	15	,	,	PUNCT
ma-221	466	16	best	good	ADJ
ma-221	466	17	proximity	proximity	NOUN
ma-221	466	18	point	point	NOUN
ma-221	466	19	of	of	ADP
ma-221	466	20	generalized	generalized	ADJ
ma-221	466	21	f	f	ADJ
ma-221	466	22	-	-	ADJ
ma-221	466	23	proximal	proximal	ADJ
ma-221	466	24	non	non	ADJ
ma-221	466	25	-	-	ADJ
ma-221	466	26	self	self	ADJ
ma-221	466	27	contractions	contraction	NOUN
ma-221	466	28	,	,	PUNCT
ma-221	466	29	j.	j.	PROPN
ma-221	466	30	fixedpoint	fixedpoint	PROPN
ma-221	466	31	theory	theory	NOUN
ma-221	466	32	appl	appl	PROPN
ma-221	466	33	.	.	PUNCT
ma-221	467	1	23	23	NUM
ma-221	467	2	(	(	PUNCT
ma-221	467	3	2021	2021	NUM
ma-221	467	4	)	)	PUNCT
ma-221	467	5	,	,	PUNCT
ma-221	467	6	1	1	NUM
ma-221	467	7	-	-	NUM
ma-221	467	8	11[5	11[5	NUM
ma-221	467	9	]	]	PUNCT
ma-221	467	10	a.	a.	PROPN
ma-221	467	11	eldred	eldred	PROPN
ma-221	467	12	,	,	PUNCT
ma-221	467	13	w.	w.	PROPN
ma-221	467	14	kirk	kirk	PROPN
ma-221	467	15	,	,	PUNCT
ma-221	467	16	p.	p.	PROPN
ma-221	467	17	veeramani	veeramani	PROPN
ma-221	467	18	,	,	PUNCT
ma-221	467	19	proximal	proximal	ADJ
ma-221	467	20	normal	normal	ADJ
ma-221	467	21	structure	structure	NOUN
ma-221	467	22	and	and	CCONJ
ma-221	467	23	relatively	relatively	ADV
ma-221	467	24	nonexpansive	nonexpansive	ADJ
ma-221	467	25	mappings	mapping	NOUN
ma-221	467	26	,	,	PUNCT
ma-221	467	27	stud	stud	NOUN
ma-221	467	28	.	.	PUNCT
ma-221	468	1	math	math	NOUN
ma-221	468	2	.	.	PUNCT
ma-221	469	1	171(2005	171(2005	NUM
ma-221	469	2	)	)	PUNCT
ma-221	469	3	283	283	NUM
ma-221	469	4	-	-	SYM
ma-221	469	5	293.[6	293.[6	NUM
ma-221	469	6	]	]	PUNCT
ma-221	469	7	m.	m.	NOUN
ma-221	469	8	jleli	jleli	PROPN
ma-221	469	9	,	,	PUNCT
ma-221	469	10	b.	b.	PROPN
ma-221	469	11	samet	samet	PROPN
ma-221	469	12	,	,	PUNCT
ma-221	469	13	a	a	DET
ma-221	469	14	new	new	ADJ
ma-221	469	15	generalization	generalization	NOUN
ma-221	469	16	of	of	ADP
ma-221	469	17	the	the	DET
ma-221	469	18	banach	banach	NOUN
ma-221	469	19	contraction	contraction	NOUN
ma-221	469	20	principle	principle	NOUN
ma-221	469	21	,	,	PUNCT
ma-221	469	22	j.	j.	PROPN
ma-221	469	23	ineq	ineq	PROPN
ma-221	469	24	.	.	PUNCT
ma-221	470	1	appl	appl	PROPN
ma-221	470	2	.	.	PROPN
ma-221	471	1	2014	2014	NUM
ma-221	471	2	(	(	PUNCT
ma-221	471	3	2014	2014	NUM
ma-221	471	4	)	)	PUNCT
ma-221	472	1	38.[7	38.[7	NUM
ma-221	472	2	]	]	X
ma-221	472	3	a.	a.	NOUN
ma-221	472	4	kari	kari	PROPN
ma-221	472	5	,	,	PUNCT
ma-221	472	6	m.	m.	NOUN
ma-221	472	7	rossafi	rossafi	PROPN
ma-221	472	8	,	,	PUNCT
ma-221	472	9	e.	e.	PROPN
ma-221	472	10	marhrani	marhrani	PROPN
ma-221	472	11	,	,	PUNCT
ma-221	472	12	m.	m.	NOUN
ma-221	472	13	aamri	aamri	PROPN
ma-221	472	14	,	,	PUNCT
ma-221	472	15	fixed	fix	VERB
ma-221	472	16	-	-	PUNCT
ma-221	472	17	point	point	NOUN
ma-221	472	18	theorem	theorem	NOUN
ma-221	472	19	for	for	ADP
ma-221	472	20	nonlinear	nonlinear	ADJ
ma-221	472	21	f	f	NOUN
ma-221	472	22	-	-	PUNCT
ma-221	472	23	contraction	contraction	NOUN
ma-221	472	24	via	via	ADP
ma-221	472	25	w	w	NOUN
ma-221	472	26	-	-	PUNCT
ma-221	472	27	distance	distance	NOUN
ma-221	472	28	,	,	PUNCT
ma-221	472	29	adv.math	adv.math	NOUN
ma-221	472	30	.	.	PUNCT
ma-221	473	1	phys	phy	NOUN
ma-221	473	2	.	.	PUNCT
ma-221	474	1	2020	2020	NUM
ma-221	474	2	(	(	PUNCT
ma-221	474	3	2020	2020	NUM
ma-221	474	4	)	)	PUNCT
ma-221	474	5	6617517	6617517	NUM
ma-221	474	6	.	.	PUNCT
ma-221	475	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	475	2	eur	eur	PROPN
ma-221	475	3	.	.	PUNCT
ma-221	476	1	j.	j.	PROPN
ma-221	476	2	math	math	PROPN
ma-221	476	3	.	.	PUNCT
ma-221	477	1	anal	anal	PROPN
ma-221	477	2	.	.	PUNCT
ma-221	478	1	10.28924	10.28924	NUM
ma-221	478	2	/	/	SYM
ma-221	478	3	ada	ada	PROPN
ma-221	478	4	/	/	SYM
ma-221	478	5	ma.4.13	ma.4.13	PROPN
ma-221	478	6	14	14	NUM
ma-221	479	1	[	[	X
ma-221	479	2	8	8	NUM
ma-221	479	3	]	]	PUNCT
ma-221	479	4	a.	a.	NOUN
ma-221	479	5	kari	kari	PROPN
ma-221	479	6	,	,	PUNCT
ma-221	479	7	m.	m.	NOUN
ma-221	479	8	rossafi	rossafi	PROPN
ma-221	479	9	,	,	PUNCT
ma-221	479	10	e.	e.	PROPN
ma-221	479	11	marhrani	marhrani	PROPN
ma-221	479	12	,	,	PUNCT
ma-221	479	13	m.	m.	NOUN
ma-221	479	14	aamri	aamri	PROPN
ma-221	479	15	,	,	PUNCT
ma-221	480	1	θ	θ	PROPN
ma-221	480	2	−	−	PROPN
ma-221	480	3	φ−contraction	φ−contraction	NOUN
ma-221	480	4	on	on	ADP
ma-221	480	5	(	(	PUNCT
ma-221	480	6	α	α	X
ma-221	480	7	,	,	PUNCT
ma-221	480	8	η)−complete	η)−complete	VERB
ma-221	480	9	rectangular	rectangular	ADJ
ma-221	480	10	b−metric	b−metric	ADJ
ma-221	480	11	spaces	space	NOUN
ma-221	480	12	,	,	PUNCT
ma-221	480	13	int	int	NOUN
ma-221	480	14	.	.	PUNCT
ma-221	481	1	j.	j.	PROPN
ma-221	481	2	math	math	PROPN
ma-221	481	3	.	.	PUNCT
ma-221	482	1	math	math	NOUN
ma-221	482	2	.	.	PUNCT
ma-221	483	1	sci	sci	PROPN
ma-221	483	2	.	.	PROPN
ma-221	483	3	2020	2020	NUM
ma-221	483	4	(	(	PUNCT
ma-221	483	5	2020	2020	NUM
ma-221	483	6	)	)	PUNCT
ma-221	483	7	5689458.[9	5689458.[9	NUM
ma-221	483	8	]	]	X
ma-221	483	9	v.	v.	CCONJ
ma-221	483	10	parvaneh	parvaneh	PROPN
ma-221	483	11	,	,	PUNCT
ma-221	483	12	m.	m.	NOUN
ma-221	483	13	reza	reza	PROPN
ma-221	483	14	haddadi	haddadi	PROPN
ma-221	483	15	,	,	PUNCT
ma-221	483	16	h.	h.	PROPN
ma-221	483	17	aydi	aydi	PROPN
ma-221	483	18	,	,	PUNCT
ma-221	483	19	on	on	ADP
ma-221	483	20	best	good	ADJ
ma-221	483	21	proximity	proximity	NOUN
ma-221	483	22	point	point	NOUN
ma-221	483	23	results	result	NOUN
ma-221	483	24	for	for	ADP
ma-221	483	25	some	some	DET
ma-221	483	26	type	type	NOUN
ma-221	483	27	of	of	ADP
ma-221	483	28	mappings	mapping	NOUN
ma-221	483	29	,	,	PUNCT
ma-221	483	30	j.	j.	PROPN
ma-221	483	31	funct	funct	PROPN
ma-221	483	32	.	.	PUNCT
ma-221	484	1	spaces2020	spaces2020	PROPN
ma-221	484	2	(	(	PUNCT
ma-221	484	3	2020	2020	NUM
ma-221	484	4	)	)	PUNCT
ma-221	485	1	6298138.[10	6298138.[10	NUM
ma-221	485	2	]	]	PUNCT
ma-221	485	3	v.s.	v.s.	X
ma-221	485	4	raj	raj	PROPN
ma-221	485	5	,	,	PUNCT
ma-221	485	6	a	a	DET
ma-221	485	7	best	good	ADJ
ma-221	485	8	proximity	proximity	NOUN
ma-221	485	9	point	point	NOUN
ma-221	485	10	theorem	theorem	NOUN
ma-221	485	11	for	for	ADP
ma-221	485	12	weakly	weakly	ADJ
ma-221	485	13	contractive	contractive	ADJ
ma-221	485	14	non	non	ADJ
ma-221	485	15	-	-	ADJ
ma-221	485	16	self	self	NOUN
ma-221	485	17	-	-	PUNCT
ma-221	485	18	mappings	mapping	NOUN
ma-221	485	19	,	,	PUNCT
ma-221	485	20	nonlinear	nonlinear	ADJ
ma-221	485	21	anal	anal	NOUN
ma-221	485	22	.	.	PUNCT
ma-221	486	1	tma	tma	PROPN
ma-221	486	2	.	.	PROPN
ma-221	486	3	74	74	NUM
ma-221	486	4	(	(	PUNCT
ma-221	486	5	2011)4804	2011)4804	NOUN
ma-221	486	6	-	-	SYM
ma-221	486	7	4808.[11	4808.[11	NUM
ma-221	486	8	]	]	PUNCT
ma-221	486	9	m.	m.	NOUN
ma-221	486	10	rossafi	rossafi	PROPN
ma-221	486	11	,	,	PUNCT
ma-221	486	12	a.	a.	NOUN
ma-221	486	13	kari	kari	PROPN
ma-221	486	14	,	,	PUNCT
ma-221	486	15	some	some	DET
ma-221	486	16	fixed	fix	VERB
ma-221	486	17	point	point	NOUN
ma-221	486	18	theorems	theorem	NOUN
ma-221	486	19	for	for	ADP
ma-221	486	20	f	f	PROPN
ma-221	486	21	-expansive	-expansive	ADJ
ma-221	486	22	mapping	mapping	NOUN
ma-221	486	23	in	in	ADP
ma-221	486	24	generalized	generalized	ADJ
ma-221	486	25	metric	metric	ADJ
ma-221	486	26	spaces	space	NOUN
ma-221	486	27	,	,	PUNCT
ma-221	486	28	open	open	ADJ
ma-221	486	29	j.	j.	PROPN
ma-221	486	30	math.anal	math.anal	PROPN
ma-221	486	31	.	.	PROPN
ma-221	486	32	5	5	NUM
ma-221	486	33	(	(	PUNCT
ma-221	486	34	2021	2021	NUM
ma-221	486	35	)	)	PUNCT
ma-221	486	36	17	17	NUM
ma-221	486	37	-	-	SYM
ma-221	486	38	30.[12	30.[12	NUM
ma-221	486	39	]	]	PUNCT
ma-221	486	40	m.	m.	NOUN
ma-221	486	41	rossafi	rossafi	PROPN
ma-221	486	42	,	,	PUNCT
ma-221	486	43	a.	a.	NOUN
ma-221	486	44	kari	kari	PROPN
ma-221	486	45	,	,	PUNCT
ma-221	486	46	best	good	ADJ
ma-221	486	47	proximity	proximity	NOUN
ma-221	486	48	point	point	NOUN
ma-221	486	49	theorems	theorem	NOUN
ma-221	486	50	for	for	ADP
ma-221	486	51	α	α	NOUN
ma-221	486	52	-	-	ADJ
ma-221	486	53	proximal	proximal	ADJ
ma-221	486	54	θ	θ	PROPN
ma-221	486	55	,	,	PUNCT
ma-221	486	56	φ	φ	PROPN
ma-221	486	57	-	-	PUNCT
ma-221	486	58	non	non	ADJ
ma-221	486	59	-	-	ADJ
ma-221	486	60	self	self	NOUN
ma-221	486	61	mappings	mapping	NOUN
ma-221	486	62	,	,	PUNCT
ma-221	486	63	asian	asian	PROPN
ma-221	486	64	j.	j.	PROPN
ma-221	486	65	math	math	PROPN
ma-221	486	66	.	.	PUNCT
ma-221	487	1	appl	appl	PROPN
ma-221	487	2	.	.	PUNCT
ma-221	488	1	2022(2022	2022(2022	NUM
ma-221	488	2	)	)	PUNCT
ma-221	489	1	3.[13	3.[13	NUM
ma-221	489	2	]	]	PUNCT
ma-221	489	3	j.	j.	PROPN
ma-221	489	4	zhang	zhang	PROPN
ma-221	489	5	,	,	PUNCT
ma-221	489	6	y.	y.	PROPN
ma-221	489	7	su	su	PROPN
ma-221	489	8	,	,	PUNCT
ma-221	489	9	q.	q.	PROPN
ma-221	489	10	cheng	cheng	PROPN
ma-221	489	11	,	,	PUNCT
ma-221	489	12	a	a	DET
ma-221	489	13	note	note	NOUN
ma-221	489	14	on	on	ADP
ma-221	489	15	’	'	PUNCT
ma-221	489	16	a	a	DET
ma-221	489	17	best	good	ADJ
ma-221	489	18	proximity	proximity	NOUN
ma-221	489	19	point	point	NOUN
ma-221	489	20	theorem	theorem	NOUN
ma-221	489	21	for	for	ADP
ma-221	489	22	geraghty	geraghty	NOUN
ma-221	489	23	-	-	PUNCT
ma-221	489	24	contractions	contraction	NOUN
ma-221	489	25	’	'	PUNCT
ma-221	489	26	,	,	PUNCT
ma-221	489	27	fixed	fix	VERB
ma-221	489	28	point	point	NOUN
ma-221	489	29	theoryappl	theoryappl	ADJ
ma-221	489	30	.	.	PUNCT
ma-221	490	1	2013	2013	NUM
ma-221	490	2	(	(	PUNCT
ma-221	490	3	2013	2013	NUM
ma-221	490	4	)	)	PUNCT
ma-221	490	5	99.[14	99.[14	PROPN
ma-221	490	6	]	]	X
ma-221	490	7	d.	d.	PROPN
ma-221	490	8	zheng	zheng	PROPN
ma-221	490	9	,	,	PUNCT
ma-221	490	10	z.	z.	PROPN
ma-221	490	11	cai	cai	PROPN
ma-221	490	12	,	,	PUNCT
ma-221	490	13	p.	p.	PROPN
ma-221	490	14	wang	wang	PROPN
ma-221	490	15	,	,	PUNCT
ma-221	490	16	new	new	ADJ
ma-221	490	17	fixed	fix	VERB
ma-221	490	18	point	point	NOUN
ma-221	490	19	theorems	theorem	NOUN
ma-221	490	20	for	for	ADP
ma-221	490	21	theta	theta	NOUN
ma-221	490	22	-	-	PUNCT
ma-221	490	23	phi	phi	NOUN
ma-221	490	24	contraction	contraction	NOUN
ma-221	490	25	in	in	ADP
ma-221	490	26	complete	complete	ADJ
ma-221	490	27	metric	metric	ADJ
ma-221	490	28	spaces	space	NOUN
ma-221	490	29	,	,	PUNCT
ma-221	490	30	j.	j.	PROPN
ma-221	490	31	nonlinearsci	nonlinearsci	PROPN
ma-221	490	32	.	.	PUNCT
ma-221	491	1	appl	appl	PROPN
ma-221	491	2	.	.	PUNCT
ma-221	492	1	10	10	NUM
ma-221	492	2	(	(	PUNCT
ma-221	492	3	2017	2017	NUM
ma-221	492	4	)	)	PUNCT
ma-221	492	5	2662	2662	NUM
ma-221	492	6	-	-	SYM
ma-221	492	7	2670	2670	NUM
ma-221	492	8	.	.	PUNCT
ma-221	493	1	https://doi.org/10.28924/ada/ma.4.13	https://doi.org/10.28924/ada/ma.4.13	PROPN
ma-221	493	2	1	1	NUM
ma-221	493	3	.	.	PUNCT
ma-221	493	4	introduction	introduction	NOUN
ma-221	493	5	2	2	NUM
ma-221	493	6	.	.	PUNCT
ma-221	493	7	preliminaries	preliminary	NOUN
ma-221	493	8	3	3	NUM
ma-221	493	9	.	.	X
ma-221	493	10	main	main	ADJ
ma-221	493	11	result	result	NOUN
ma-221	493	12	references	reference	NOUN