id	sid	tid	token	lemma	pos
ejpam-259	1	1	5_tallafha.dvi	5_tallafha.dvi	NUM
ejpam-259	1	2	european	european	ADJ
ejpam-259	1	3	journal	journal	NOUN
ejpam-259	1	4	of	of	ADP
ejpam-259	1	5	pure	pure	ADJ
ejpam-259	1	6	and	and	CCONJ
ejpam-259	1	7	applied	apply	VERB
ejpam-259	1	8	mathematics	mathematic	NOUN
ejpam-259	1	9	vol	vol	NOUN
ejpam-259	1	10	.	.	PROPN
ejpam-259	2	1	2	2	NUM
ejpam-259	2	2	,	,	PUNCT
ejpam-259	2	3	no	no	INTJ
ejpam-259	2	4	.	.	NOUN
ejpam-259	2	5	2	2	NUM
ejpam-259	2	6	,	,	PUNCT
ejpam-259	2	7	2009	2009	NUM
ejpam-259	2	8	,	,	PUNCT
ejpam-259	2	9	(	(	PUNCT
ejpam-259	2	10	231	231	NUM
ejpam-259	2	11	-	-	SYM
ejpam-259	2	12	238	238	NUM
ejpam-259	2	13	)	)	PUNCT
ejpam-259	2	14	issn	issn	PROPN
ejpam-259	2	15	1307	1307	NUM
ejpam-259	2	16	-	-	SYM
ejpam-259	2	17	5543	5543	NUM
ejpam-259	2	18	–	–	PUNCT
ejpam-259	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-259	2	20	best	good	ADJ
ejpam-259	2	21	approximation	approximation	NOUN
ejpam-259	2	22	in	in	ADP
ejpam-259	2	23	uniformity	uniformity	NOUN
ejpam-259	2	24	type	type	NOUN
ejpam-259	2	25	spaces	space	NOUN
ejpam-259	2	26	a.	a.	NOUN
ejpam-259	2	27	tallafha∗	tallafha∗	PROPN
ejpam-259	2	28	and	and	CCONJ
ejpam-259	2	29	r.	r.	PROPN
ejpam-259	2	30	khalil	khalil	PROPN
ejpam-259	2	31	university	university	PROPN
ejpam-259	2	32	of	of	ADP
ejpam-259	2	33	jordan	jordan	PROPN
ejpam-259	2	34	,	,	PUNCT
ejpam-259	2	35	department	department	PROPN
ejpam-259	2	36	of	of	ADP
ejpam-259	2	37	mathematics	mathematic	NOUN
ejpam-259	2	38	,	,	PUNCT
ejpam-259	2	39	amman	amman	PROPN
ejpam-259	2	40	-	-	PUNCT
ejpam-259	2	41	jordan	jordan	PROPN
ejpam-259	2	42	abstract	abstract	PROPN
ejpam-259	2	43	.	.	PUNCT
ejpam-259	3	1	let	let	VERB
ejpam-259	3	2	x	x	PRON
ejpam-259	3	3	be	be	AUX
ejpam-259	3	4	a	a	DET
ejpam-259	3	5	set	set	NOUN
ejpam-259	3	6	,	,	PUNCT
ejpam-259	3	7	and	and	CCONJ
ejpam-259	3	8	γ	γ	X
ejpam-259	3	9	be	be	AUX
ejpam-259	3	10	a	a	DET
ejpam-259	3	11	collection	collection	NOUN
ejpam-259	3	12	of	of	ADP
ejpam-259	3	13	subsets	subset	NOUN
ejpam-259	3	14	of	of	ADP
ejpam-259	3	15	x	x	X
ejpam-259	3	16	×	×	NOUN
ejpam-259	3	17	x	x	X
ejpam-259	3	18	.	.	PUNCT
ejpam-259	4	1	the	the	DET
ejpam-259	4	2	object	object	NOUN
ejpam-259	4	3	of	of	ADP
ejpam-259	4	4	this	this	DET
ejpam-259	4	5	paper	paper	NOUN
ejpam-259	4	6	,	,	PUNCT
ejpam-259	4	7	is	be	AUX
ejpam-259	4	8	to	to	PART
ejpam-259	4	9	define	define	VERB
ejpam-259	4	10	a	a	DET
ejpam-259	4	11	semi	semi	ADJ
ejpam-259	4	12	-	-	ADJ
ejpam-259	4	13	linear	linear	ADJ
ejpam-259	4	14	uniform	uniform	ADJ
ejpam-259	4	15	space	space	NOUN
ejpam-259	4	16	by	by	ADP
ejpam-259	4	17	assuming	assume	VERB
ejpam-259	4	18	certain	certain	ADJ
ejpam-259	4	19	conditions	condition	NOUN
ejpam-259	4	20	on	on	ADP
ejpam-259	4	21	γ	γ	PROPN
ejpam-259	4	22	.	.	PUNCT
ejpam-259	5	1	the	the	DET
ejpam-259	5	2	structure	structure	NOUN
ejpam-259	5	3	of	of	ADP
ejpam-259	5	4	such	such	ADJ
ejpam-259	5	5	spaces	space	NOUN
ejpam-259	5	6	turned	turn	VERB
ejpam-259	5	7	to	to	PART
ejpam-259	5	8	be	be	AUX
ejpam-259	5	9	a	a	DET
ejpam-259	5	10	very	very	ADV
ejpam-259	5	11	rich	rich	ADJ
ejpam-259	5	12	structure	structure	NOUN
ejpam-259	5	13	.	.	PUNCT
ejpam-259	6	1	we	we	PRON
ejpam-259	6	2	define	define	VERB
ejpam-259	6	3	closest	close	ADJ
ejpam-259	6	4	elements	element	NOUN
ejpam-259	6	5	from	from	ADP
ejpam-259	6	6	a	a	DET
ejpam-259	6	7	given	give	VERB
ejpam-259	6	8	set	set	NOUN
ejpam-259	6	9	to	to	ADP
ejpam-259	6	10	a	a	DET
ejpam-259	6	11	given	give	VERB
ejpam-259	6	12	element	element	NOUN
ejpam-259	6	13	in	in	ADP
ejpam-259	6	14	x	x	X
ejpam-259	6	15	.	.	PUNCT
ejpam-259	7	1	then	then	ADV
ejpam-259	7	2	we	we	PRON
ejpam-259	7	3	study	study	VERB
ejpam-259	7	4	best	good	ADJ
ejpam-259	7	5	approximation	approximation	NOUN
ejpam-259	7	6	in	in	ADP
ejpam-259	7	7	semi	semi	ADJ
ejpam-259	7	8	-	-	ADJ
ejpam-259	7	9	linear	linear	ADJ
ejpam-259	7	10	uniform	uniform	ADJ
ejpam-259	7	11	spaces	space	NOUN
ejpam-259	7	12	.	.	PUNCT
ejpam-259	8	1	ams	am	NOUN
ejpam-259	8	2	subject	subject	ADJ
ejpam-259	8	3	classifications	classification	NOUN
ejpam-259	8	4	:	:	PUNCT
ejpam-259	8	5	primary	primary	ADJ
ejpam-259	8	6	:	:	PUNCT
ejpam-259	8	7	41a65	41a65	NUM
ejpam-259	8	8	,	,	PUNCT
ejpam-259	8	9	secondary	secondary	ADJ
ejpam-259	8	10	:	:	PUNCT
ejpam-259	8	11	41a99	41a99	NUM
ejpam-259	8	12	key	key	ADJ
ejpam-259	8	13	words	word	NOUN
ejpam-259	8	14	:	:	PUNCT
ejpam-259	8	15	best	good	ADJ
ejpam-259	8	16	approximation	approximation	NOUN
ejpam-259	8	17	,	,	PUNCT
ejpam-259	8	18	uniform	uniform	ADJ
ejpam-259	8	19	spaces	space	NOUN
ejpam-259	8	20	.	.	PUNCT
ejpam-259	9	1	1	1	X
ejpam-259	9	2	.	.	X
ejpam-259	9	3	introduction	introduction	NOUN
ejpam-259	9	4	let	let	VERB
ejpam-259	9	5	x	x	PRON
ejpam-259	9	6	be	be	AUX
ejpam-259	9	7	a	a	DET
ejpam-259	9	8	set	set	NOUN
ejpam-259	9	9	and	and	CCONJ
ejpam-259	9	10	dx	dx	PROPN
ejpam-259	9	11	be	be	AUX
ejpam-259	9	12	a	a	DET
ejpam-259	9	13	collection	collection	NOUN
ejpam-259	9	14	of	of	ADP
ejpam-259	9	15	subsets	subset	NOUN
ejpam-259	9	16	of	of	ADP
ejpam-259	9	17	x×x	x×x	PROPN
ejpam-259	9	18	,	,	PUNCT
ejpam-259	9	19	such	such	ADJ
ejpam-259	9	20	that	that	SCONJ
ejpam-259	9	21	each	each	DET
ejpam-259	9	22	element	element	NOUN
ejpam-259	9	23	v	v	NOUN
ejpam-259	9	24	of	of	ADP
ejpam-259	9	25	dx	dx	PROPN
ejpam-259	9	26	contains	contain	VERB
ejpam-259	9	27	the	the	DET
ejpam-259	9	28	diagonal	diagonal	ADJ
ejpam-259	9	29	∆=	∆=	NOUN
ejpam-259	9	30	{	{	PUNCT
ejpam-259	9	31	(	(	PUNCT
ejpam-259	9	32	x	x	INTJ
ejpam-259	9	33	,	,	PUNCT
ejpam-259	9	34	x	x	NOUN
ejpam-259	9	35	)	)	PUNCT
ejpam-259	9	36	:	:	PUNCT
ejpam-259	10	1	x	x	PUNCT
ejpam-259	10	2	∈	∈	PROPN
ejpam-259	10	3	x}and	x}and	NOUN
ejpam-259	10	4	v	v	X
ejpam-259	10	5	=	=	SYM
ejpam-259	10	6	v−1	v−1	PROPN
ejpam-259	10	7	=	=	SYM
ejpam-259	10	8	�	�	PROPN
ejpam-259	10	9	(	(	PUNCT
ejpam-259	10	10	y	y	PROPN
ejpam-259	10	11	,	,	PUNCT
ejpam-259	10	12	x	x	NOUN
ejpam-259	10	13	)	)	PUNCT
ejpam-259	10	14	:	:	PUNCT
ejpam-259	10	15	�	�	PROPN
ejpam-259	10	16	x	x	SYM
ejpam-259	10	17	,	,	PUNCT
ejpam-259	10	18	y	y	PROPN
ejpam-259	10	19	�	�	PROPN
ejpam-259	10	20	∈	∈	PROPN
ejpam-259	10	21	v	v	NOUN
ejpam-259	10	22	for	for	ADP
ejpam-259	10	23	all	all	PRON
ejpam-259	10	24	v	v	NOUN
ejpam-259	10	25	∈	∈	PROPN
ejpam-259	10	26	dx	dx	PROPN
ejpam-259	10	27	(	(	PUNCT
ejpam-259	10	28	symmetric	symmetric	ADJ
ejpam-259	10	29	)	)	PUNCT
ejpam-259	10	30	,	,	PUNCT
ejpam-259	10	31	dx	dx	PROPN
ejpam-259	10	32	is	be	AUX
ejpam-259	10	33	called	call	VERB
ejpam-259	10	34	the	the	DET
ejpam-259	10	35	family	family	NOUN
ejpam-259	10	36	of	of	ADP
ejpam-259	10	37	all	all	DET
ejpam-259	10	38	entourages	entourage	NOUN
ejpam-259	10	39	of	of	ADP
ejpam-259	10	40	the	the	DET
ejpam-259	10	41	diagonal	diagonal	NOUN
ejpam-259	10	42	.	.	PUNCT
ejpam-259	11	1	let	let	VERB
ejpam-259	11	2	γ	γ	X
ejpam-259	11	3	be	be	AUX
ejpam-259	11	4	a	a	DET
ejpam-259	11	5	sub	sub	NOUN
ejpam-259	11	6	collection	collection	NOUN
ejpam-259	11	7	of	of	ADP
ejpam-259	11	8	dx	dx	PROPN
ejpam-259	11	9	,	,	PUNCT
ejpam-259	11	10	then	then	ADV
ejpam-259	11	11	the	the	DET
ejpam-259	11	12	pair	pair	NOUN
ejpam-259	11	13	(	(	PUNCT
ejpam-259	11	14	x	x	X
ejpam-259	11	15	,	,	PUNCT
ejpam-259	11	16	γ	γ	X
ejpam-259	11	17	)	)	PUNCT
ejpam-259	11	18	is	be	AUX
ejpam-259	11	19	called	call	VERB
ejpam-259	11	20	a	a	DET
ejpam-259	11	21	uniform	uniform	ADJ
ejpam-259	11	22	space	space	NOUN
ejpam-259	11	23	if	if	SCONJ
ejpam-259	11	24	∗corresponding	∗corresponde	VERB
ejpam-259	11	25	author	author	NOUN
ejpam-259	11	26	.	.	PUNCT
ejpam-259	12	1	email	email	NOUN
ejpam-259	12	2	address	address	NOUN
ejpam-259	12	3	:	:	PUNCT
ejpam-259	12	4	roshdi�ju.edu.jo	roshdi�ju.edu.jo	PROPN
ejpam-259	12	5	(	(	PUNCT
ejpam-259	12	6	r.	r.	PROPN
ejpam-259	12	7	khalil	khalil	PROPN
ejpam-259	12	8	)	)	PUNCT
ejpam-259	12	9	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-259	13	1	231	231	NUM
ejpam-259	13	2	c	c	X
ejpam-259	13	3	©	©	PROPN
ejpam-259	13	4	2009	2009	NUM
ejpam-259	13	5	ejpam	ejpam	NOUN
ejpam-259	13	6	all	all	DET
ejpam-259	13	7	rights	right	NOUN
ejpam-259	13	8	reserved	reserve	VERB
ejpam-259	13	9	.	.	PUNCT
ejpam-259	14	1	a.	a.	NOUN
ejpam-259	14	2	tallafha	tallafha	PROPN
ejpam-259	14	3	and	and	CCONJ
ejpam-259	14	4	r.	r.	PROPN
ejpam-259	14	5	khalil	khalil	PROPN
ejpam-259	14	6	/	/	SYM
ejpam-259	14	7	eur	eur	PROPN
ejpam-259	14	8	.	.	PUNCT
ejpam-259	15	1	j.	j.	PROPN
ejpam-259	15	2	pure	pure	PROPN
ejpam-259	15	3	appl	appl	PROPN
ejpam-259	15	4	.	.	PROPN
ejpam-259	15	5	math	math	PROPN
ejpam-259	15	6	,	,	PUNCT
ejpam-259	15	7	2	2	NUM
ejpam-259	15	8	(	(	PUNCT
ejpam-259	15	9	2009	2009	NUM
ejpam-259	15	10	)	)	PUNCT
ejpam-259	15	11	,	,	PUNCT
ejpam-259	15	12	(	(	PUNCT
ejpam-259	15	13	231	231	NUM
ejpam-259	15	14	-	-	SYM
ejpam-259	15	15	238	238	NUM
ejpam-259	15	16	)	)	PUNCT
ejpam-259	15	17	232	232	NUM
ejpam-259	15	18	(	(	PUNCT
ejpam-259	15	19	i	i	NOUN
ejpam-259	15	20	)	)	PUNCT
ejpam-259	15	21	v1	v1	PROPN
ejpam-259	15	22	and	and	CCONJ
ejpam-259	15	23	v2	v2	NOUN
ejpam-259	15	24	are	be	AUX
ejpam-259	15	25	in	in	ADP
ejpam-259	15	26	γ	γ	PROPN
ejpam-259	15	27	then	then	ADV
ejpam-259	15	28	v1	v1	VERB
ejpam-259	15	29	∩	∩	ADJ
ejpam-259	15	30	v2	v2	PROPN
ejpam-259	15	31	∈	∈	PROPN
ejpam-259	15	32	γ	γ	X
ejpam-259	15	33	(	(	PUNCT
ejpam-259	15	34	ii	ii	NOUN
ejpam-259	15	35	)	)	PUNCT
ejpam-259	15	36	for	for	ADP
ejpam-259	15	37	every	every	DET
ejpam-259	15	38	v	v	PROPN
ejpam-259	15	39	∈	∈	PROPN
ejpam-259	15	40	γ	γ	NOUN
ejpam-259	15	41	,	,	PUNCT
ejpam-259	15	42	there	there	PRON
ejpam-259	15	43	exists	exist	VERB
ejpam-259	15	44	u	u	NOUN
ejpam-259	15	45	∈	∈	PROPN
ejpam-259	15	46	γ	γ	NOUN
ejpam-259	15	47	such	such	ADJ
ejpam-259	15	48	that	that	DET
ejpam-259	15	49	u	u	NOUN
ejpam-259	15	50	◦	◦	NOUN
ejpam-259	15	51	u	u	X
ejpam-259	15	52	⊂	⊂	PROPN
ejpam-259	15	53	v.	v.	PROPN
ejpam-259	15	54	(	(	PUNCT
ejpam-259	15	55	iii	iii	NOUN
ejpam-259	15	56	)	)	PUNCT
ejpam-259	15	57	∩	∩	NOUN
ejpam-259	15	58	v	v	ADP
ejpam-259	15	59	∈	∈	PROPN
ejpam-259	15	60	γ	γ	X
ejpam-259	15	61	v	v	NOUN
ejpam-259	15	62	=	=	PRON
ejpam-259	15	63	∆	∆	X
ejpam-259	15	64	(	(	PUNCT
ejpam-259	15	65	vi	vi	NOUN
ejpam-259	15	66	)	)	PUNCT
ejpam-259	15	67	if	if	SCONJ
ejpam-259	15	68	v	v	NOUN
ejpam-259	15	69	∈	∈	PROPN
ejpam-259	15	70	γ	γ	NOUN
ejpam-259	15	71	and	and	CCONJ
ejpam-259	15	72	v	v	NOUN
ejpam-259	15	73	⊆w	⊆w	NOUN
ejpam-259	15	74	∈	∈	PROPN
ejpam-259	15	75	dx	dx	PROPN
ejpam-259	15	76	,	,	PUNCT
ejpam-259	15	77	then	then	ADV
ejpam-259	15	78	w	w	PROPN
ejpam-259	15	79	∈	∈	PROPN
ejpam-259	15	80	γ	γ	PROPN
ejpam-259	15	81	.	.	PROPN
ejpam-259	15	82	uniform	uniform	ADJ
ejpam-259	15	83	spaces	space	NOUN
ejpam-259	15	84	had	have	AUX
ejpam-259	15	85	been	be	AUX
ejpam-259	15	86	studied	study	VERB
ejpam-259	15	87	extensively	extensively	ADV
ejpam-259	15	88	through	through	ADP
ejpam-259	15	89	years	year	NOUN
ejpam-259	15	90	.	.	PUNCT
ejpam-259	16	1	we	we	PRON
ejpam-259	16	2	refer	refer	VERB
ejpam-259	16	3	the	the	DET
ejpam-259	16	4	reader	reader	NOUN
ejpam-259	16	5	to	to	ADP
ejpam-259	16	6	[	[	X
ejpam-259	16	7	1	1	NUM
ejpam-259	16	8	]	]	PUNCT
ejpam-259	16	9	,	,	PUNCT
ejpam-259	16	10	and	and	CCONJ
ejpam-259	16	11	[	[	X
ejpam-259	16	12	2	2	NUM
ejpam-259	16	13	]	]	PUNCT
ejpam-259	16	14	,	,	PUNCT
ejpam-259	16	15	for	for	ADP
ejpam-259	16	16	the	the	DET
ejpam-259	16	17	basic	basic	ADJ
ejpam-259	16	18	structure	structure	NOUN
ejpam-259	16	19	of	of	ADP
ejpam-259	16	20	uniform	uniform	ADJ
ejpam-259	16	21	spaces	space	NOUN
ejpam-259	16	22	.	.	PUNCT
ejpam-259	17	1	the	the	DET
ejpam-259	17	2	object	object	NOUN
ejpam-259	17	3	of	of	ADP
ejpam-259	17	4	this	this	DET
ejpam-259	17	5	paper	paper	NOUN
ejpam-259	17	6	is	be	AUX
ejpam-259	17	7	to	to	PART
ejpam-259	17	8	define	define	VERB
ejpam-259	17	9	uniform	uniform	ADJ
ejpam-259	17	10	type	type	NOUN
ejpam-259	17	11	spaces	space	NOUN
ejpam-259	17	12	and	and	CCONJ
ejpam-259	17	13	a	a	DET
ejpam-259	17	14	set	set	ADJ
ejpam-259	17	15	valued	value	VERB
ejpam-259	17	16	map	map	NOUN
ejpam-259	17	17	,	,	PUNCT
ejpam-259	17	18	to	to	PART
ejpam-259	17	19	be	be	AUX
ejpam-259	17	20	called	call	VERB
ejpam-259	17	21	metric	metric	ADJ
ejpam-259	17	22	type	type	NOUN
ejpam-259	17	23	,	,	PUNCT
ejpam-259	17	24	on	on	ADP
ejpam-259	17	25	such	such	ADJ
ejpam-259	17	26	spaces	space	NOUN
ejpam-259	17	27	that	that	PRON
ejpam-259	17	28	enables	enable	VERB
ejpam-259	17	29	us	we	PRON
ejpam-259	17	30	to	to	PART
ejpam-259	17	31	study	study	VERB
ejpam-259	17	32	analytical	analytical	ADJ
ejpam-259	17	33	concepts	concept	NOUN
ejpam-259	17	34	on	on	ADP
ejpam-259	17	35	uniform	uniform	ADJ
ejpam-259	17	36	type	type	NOUN
ejpam-259	17	37	spaces	space	NOUN
ejpam-259	17	38	,	,	PUNCT
ejpam-259	17	39	namely	namely	ADV
ejpam-259	17	40	best	good	ADJ
ejpam-259	17	41	approximation	approximation	NOUN
ejpam-259	17	42	.	.	PUNCT
ejpam-259	18	1	since	since	SCONJ
ejpam-259	18	2	the	the	DET
ejpam-259	18	3	problem	problem	NOUN
ejpam-259	18	4	of	of	ADP
ejpam-259	18	5	best	good	ADJ
ejpam-259	18	6	approximation	approximation	NOUN
ejpam-259	18	7	is	be	AUX
ejpam-259	18	8	a	a	DET
ejpam-259	18	9	problem	problem	NOUN
ejpam-259	18	10	of	of	ADP
ejpam-259	18	11	nearness	nearness	NOUN
ejpam-259	18	12	between	between	ADP
ejpam-259	18	13	elements	element	NOUN
ejpam-259	18	14	and	and	CCONJ
ejpam-259	18	15	sets	set	NOUN
ejpam-259	18	16	,	,	PUNCT
ejpam-259	18	17	the	the	DET
ejpam-259	18	18	problem	problem	NOUN
ejpam-259	18	19	of	of	ADP
ejpam-259	18	20	best	good	ADJ
ejpam-259	18	21	approximation	approximation	NOUN
ejpam-259	18	22	is	be	AUX
ejpam-259	18	23	usually	usually	ADV
ejpam-259	18	24	discussed	discuss	VERB
ejpam-259	18	25	in	in	ADP
ejpam-259	18	26	metric	metric	ADJ
ejpam-259	18	27	and	and	CCONJ
ejpam-259	18	28	normed	normed	ADJ
ejpam-259	18	29	spaces	space	NOUN
ejpam-259	18	30	[	[	X
ejpam-259	18	31	3	3	NUM
ejpam-259	18	32	]	]	PUNCT
ejpam-259	18	33	,	,	PUNCT
ejpam-259	18	34	[	[	X
ejpam-259	18	35	4	4	NUM
ejpam-259	18	36	]	]	PUNCT
ejpam-259	18	37	.	.	PUNCT
ejpam-259	19	1	best	good	ADJ
ejpam-259	19	2	approximation	approximation	NOUN
ejpam-259	19	3	never	never	ADV
ejpam-259	19	4	been	be	AUX
ejpam-259	19	5	studied	study	VERB
ejpam-259	19	6	in	in	ADP
ejpam-259	19	7	spaces	space	NOUN
ejpam-259	19	8	other	other	ADJ
ejpam-259	19	9	than	than	ADP
ejpam-259	19	10	metric	metric	ADJ
ejpam-259	19	11	and	and	CCONJ
ejpam-259	19	12	normed	normed	ADJ
ejpam-259	19	13	spaces	space	NOUN
ejpam-259	19	14	we	we	PRON
ejpam-259	19	15	believe	believe	VERB
ejpam-259	19	16	that	that	SCONJ
ejpam-259	19	17	the	the	DET
ejpam-259	19	18	new	new	ADJ
ejpam-259	19	19	structure	structure	NOUN
ejpam-259	19	20	that	that	PRON
ejpam-259	19	21	we	we	PRON
ejpam-259	19	22	introduced	introduce	VERB
ejpam-259	19	23	in	in	ADP
ejpam-259	19	24	this	this	DET
ejpam-259	19	25	paper	paper	NOUN
ejpam-259	19	26	is	be	AUX
ejpam-259	19	27	very	very	ADV
ejpam-259	19	28	fruitful	fruitful	ADJ
ejpam-259	19	29	and	and	CCONJ
ejpam-259	19	30	will	will	AUX
ejpam-259	19	31	give	give	VERB
ejpam-259	19	32	rise	rise	NOUN
ejpam-259	19	33	to	to	ADP
ejpam-259	19	34	many	many	ADJ
ejpam-259	19	35	problems	problem	NOUN
ejpam-259	19	36	in	in	ADP
ejpam-259	19	37	approximation	approximation	NOUN
ejpam-259	19	38	theory	theory	NOUN
ejpam-259	19	39	in	in	ADP
ejpam-259	19	40	uniform	uniform	ADJ
ejpam-259	19	41	spaces	space	NOUN
ejpam-259	19	42	.	.	PUNCT
ejpam-259	20	1	2	2	X
ejpam-259	20	2	.	.	X
ejpam-259	20	3	uniform	uniform	ADJ
ejpam-259	20	4	type	type	NOUN
ejpam-259	20	5	spaces	space	NOUN
ejpam-259	20	6	let	let	VERB
ejpam-259	20	7	(	(	PUNCT
ejpam-259	20	8	x	x	X
ejpam-259	20	9	,	,	PUNCT
ejpam-259	20	10	γ	γ	PROPN
ejpam-259	20	11	)	)	PUNCT
ejpam-259	20	12	be	be	VERB
ejpam-259	20	13	a	a	DET
ejpam-259	20	14	uniform	uniform	ADJ
ejpam-259	20	15	space	space	NOUN
ejpam-259	20	16	.	.	PUNCT
ejpam-259	21	1	by	by	ADP
ejpam-259	21	2	a	a	DET
ejpam-259	21	3	chain	chain	NOUN
ejpam-259	21	4	in	in	ADP
ejpam-259	21	5	x	x	X
ejpam-259	21	6	×	×	NOUN
ejpam-259	21	7	x	x	INTJ
ejpam-259	21	8	we	we	PRON
ejpam-259	21	9	mean	mean	VERB
ejpam-259	21	10	a	a	DET
ejpam-259	21	11	totally	totally	ADV
ejpam-259	21	12	(	(	PUNCT
ejpam-259	21	13	or	or	CCONJ
ejpam-259	21	14	linearly	linearly	ADV
ejpam-259	21	15	)	)	PUNCT
ejpam-259	21	16	ordered	order	VERB
ejpam-259	21	17	collection	collection	NOUN
ejpam-259	21	18	of	of	ADP
ejpam-259	21	19	subsets	subset	NOUN
ejpam-259	21	20	of	of	ADP
ejpam-259	21	21	x	x	SYM
ejpam-259	21	22	×	×	PROPN
ejpam-259	21	23	x	x	X
ejpam-259	21	24	,	,	PUNCT
ejpam-259	21	25	where	where	SCONJ
ejpam-259	21	26	v1	v1	NOUN
ejpam-259	21	27	≤	≤	NUM
ejpam-259	21	28	v2	v2	PROPN
ejpam-259	21	29	means	mean	VERB
ejpam-259	21	30	v1	v1	VERB
ejpam-259	21	31	⊆	⊆	NUM
ejpam-259	21	32	v2	v2	NOUN
ejpam-259	21	33	.	.	PUNCT
ejpam-259	22	1	definition	definition	NOUN
ejpam-259	22	2	1.1	1.1	NUM
ejpam-259	22	3	.	.	PUNCT
ejpam-259	23	1	we	we	PRON
ejpam-259	23	2	call	call	VERB
ejpam-259	23	3	(	(	PUNCT
ejpam-259	23	4	x	x	INTJ
ejpam-259	23	5	,	,	PUNCT
ejpam-259	23	6	γ	γ	PROPN
ejpam-259	23	7	)	)	PUNCT
ejpam-259	23	8	a	a	DET
ejpam-259	23	9	semi	semi	ADJ
ejpam-259	23	10	-	-	ADJ
ejpam-259	23	11	linear	linear	ADJ
ejpam-259	23	12	uniform	uniform	ADJ
ejpam-259	23	13	space	space	NOUN
ejpam-259	23	14	if	if	SCONJ
ejpam-259	23	15	it	it	PRON
ejpam-259	23	16	is	be	AUX
ejpam-259	23	17	a	a	DET
ejpam-259	23	18	uniform	uniform	ADJ
ejpam-259	23	19	space	space	NOUN
ejpam-259	23	20	where	where	SCONJ
ejpam-259	23	21	γ	γ	PROPN
ejpam-259	23	22	is	be	AUX
ejpam-259	23	23	a	a	DET
ejpam-259	23	24	chain	chain	NOUN
ejpam-259	23	25	and	and	CCONJ
ejpam-259	23	26	condition	condition	NOUN
ejpam-259	23	27	(	(	PUNCT
ejpam-259	23	28	vi	vi	NOUN
ejpam-259	23	29	)	)	PUNCT
ejpam-259	23	30	is	be	AUX
ejpam-259	23	31	replaced	replace	VERB
ejpam-259	23	32	by	by	ADP
ejpam-259	23	33	⋃	⋃	NOUN
ejpam-259	23	34	v∈γ	v∈γ	NOUN
ejpam-259	23	35	v	v	ADP
ejpam-259	23	36	=	=	NOUN
ejpam-259	23	37	x	x	SYM
ejpam-259	23	38	×	×	PROPN
ejpam-259	23	39	x	x	X
ejpam-259	23	40	.	.	PUNCT
ejpam-259	24	1	an	an	DET
ejpam-259	24	2	example	example	NOUN
ejpam-259	24	3	of	of	ADP
ejpam-259	24	4	a	a	DET
ejpam-259	24	5	semi	semi	ADJ
ejpam-259	24	6	-	-	ADJ
ejpam-259	24	7	linear	linear	ADJ
ejpam-259	24	8	uniform	uniform	ADJ
ejpam-259	24	9	space	space	NOUN
ejpam-259	24	10	is	be	AUX
ejpam-259	24	11	the	the	DET
ejpam-259	24	12	following	following	NOUN
ejpam-259	24	13	.	.	PUNCT
ejpam-259	25	1	example	example	NOUN
ejpam-259	25	2	2.1	2.1	NUM
ejpam-259	25	3	.	.	PUNCT
ejpam-259	26	1	let	let	VERB
ejpam-259	26	2	vt	vt	PROPN
ejpam-259	26	3	=	=	PRON
ejpam-259	26	4	{	{	PUNCT
ejpam-259	26	5	(	(	PUNCT
ejpam-259	26	6	x	x	INTJ
ejpam-259	26	7	,	,	PUNCT
ejpam-259	26	8	y	y	PROPN
ejpam-259	26	9	)	)	PUNCT
ejpam-259	26	10	:	:	PUNCT
ejpam-259	27	1	y	y	PROPN
ejpam-259	27	2	−	−	PROPN
ejpam-259	28	1	t	t	X
ejpam-259	28	2	<	<	X
ejpam-259	28	3	x	x	X
ejpam-259	28	4	<	<	X
ejpam-259	28	5	y	y	PROPN
ejpam-259	28	6	+	+	PROPN
ejpam-259	28	7	t	t	PROPN
ejpam-259	28	8	,	,	PUNCT
ejpam-259	28	9	and	and	CCONJ
ejpam-259	28	10	−∞	−∞	ADP
ejpam-259	28	11	<	<	X
ejpam-259	28	12	y	y	X
ejpam-259	28	13	<	<	X
ejpam-259	28	14	∞	∞	PROPN
ejpam-259	28	15	}	}	PUNCT
ejpam-259	28	16	.	.	PUNCT
ejpam-259	29	1	then	then	ADV
ejpam-259	29	2	(	(	PUNCT
ejpam-259	29	3	r	r	NOUN
ejpam-259	29	4	,	,	PUNCT
ejpam-259	29	5	γ	γ	NOUN
ejpam-259	29	6	)	)	PUNCT
ejpam-259	29	7	,	,	PUNCT
ejpam-259	29	8	with	with	ADP
ejpam-259	29	9	γ	γ	X
ejpam-259	29	10	=	=	SYM
ejpam-259	29	11	{	{	PUNCT
ejpam-259	29	12	vt	vt	NOUN
ejpam-259	29	13	:	:	PUNCT
ejpam-259	29	14	0	0	NUM
ejpam-259	29	15	<	<	X
ejpam-259	29	16	t	t	X
ejpam-259	29	17	<	<	X
ejpam-259	29	18	∞	∞	PROPN
ejpam-259	29	19	}	}	PUNCT
ejpam-259	29	20	is	be	AUX
ejpam-259	29	21	a	a	DET
ejpam-259	29	22	semi	semi	ADJ
ejpam-259	29	23	-	-	ADJ
ejpam-259	29	24	linear	linear	ADJ
ejpam-259	29	25	uniform	uniform	ADJ
ejpam-259	29	26	space	space	NOUN
ejpam-259	29	27	.	.	PUNCT
ejpam-259	30	1	one	one	PRON
ejpam-259	30	2	can	can	AUX
ejpam-259	30	3	generate	generate	VERB
ejpam-259	30	4	semi	semi	ADJ
ejpam-259	30	5	-	-	ADJ
ejpam-259	30	6	linear	linear	ADJ
ejpam-259	30	7	uniform	uniform	ADJ
ejpam-259	30	8	spaces	space	NOUN
ejpam-259	30	9	as	as	SCONJ
ejpam-259	30	10	follows	follow	VERB
ejpam-259	30	11	.	.	PUNCT
ejpam-259	31	1	let	let	VERB
ejpam-259	31	2	dx	dx	PROPN
ejpam-259	31	3	be	be	AUX
ejpam-259	31	4	a	a	DET
ejpam-259	31	5	chain	chain	NOUN
ejpam-259	31	6	in	in	ADP
ejpam-259	31	7	the	the	DET
ejpam-259	31	8	power	power	NOUN
ejpam-259	31	9	set	set	NOUN
ejpam-259	31	10	of	of	ADP
ejpam-259	31	11	x	x	SYM
ejpam-259	31	12	×	×	PROPN
ejpam-259	31	13	x	x	X
ejpam-259	31	14	,	,	PUNCT
ejpam-259	31	15	such	such	ADJ
ejpam-259	31	16	that	that	SCONJ
ejpam-259	31	17	,	,	PUNCT
ejpam-259	31	18	each	each	DET
ejpam-259	31	19	element	element	NOUN
ejpam-259	31	20	of	of	ADP
ejpam-259	31	21	dx	dx	PROPN
ejpam-259	31	22	is	be	AUX
ejpam-259	31	23	symmetric	symmetric	ADJ
ejpam-259	31	24	,	,	PUNCT
ejpam-259	31	25	contains	contain	VERB
ejpam-259	31	26	△	△	NOUN
ejpam-259	31	27	,	,	PUNCT
ejpam-259	31	28	⋃	⋃	PUNCT
ejpam-259	31	29	u	u	PROPN
ejpam-259	31	30	∈	∈	PROPN
ejpam-259	31	31	dx	dx	PROPN
ejpam-259	31	32	u	u	NOUN
ejpam-259	31	33	=	=	NOUN
ejpam-259	31	34	x	x	SYM
ejpam-259	31	35	×	×	NOUN
ejpam-259	31	36	x	x	X
ejpam-259	31	37	and	and	CCONJ
ejpam-259	31	38	⋂	⋂	PROPN
ejpam-259	31	39	u	u	PROPN
ejpam-259	31	40	∈	∈	PROPN
ejpam-259	31	41	dx	dx	PROPN
ejpam-259	31	42	u	u	NOUN
ejpam-259	31	43	=	=	PROPN
ejpam-259	31	44	△	△	PROPN
ejpam-259	31	45	.	.	PUNCT
ejpam-259	32	1	then	then	ADV
ejpam-259	32	2	one	one	PRON
ejpam-259	32	3	can	can	AUX
ejpam-259	32	4	easily	easily	ADV
ejpam-259	32	5	see	see	VERB
ejpam-259	32	6	that	that	PRON
ejpam-259	32	7	(	(	PUNCT
ejpam-259	32	8	x	x	X
ejpam-259	32	9	,	,	PUNCT
ejpam-259	32	10	dx	dx	PROPN
ejpam-259	32	11	)	)	PUNCT
ejpam-259	32	12	is	be	AUX
ejpam-259	32	13	a	a	DET
ejpam-259	32	14	semia	semia	NOUN
ejpam-259	32	15	.	.	PUNCT
ejpam-259	33	1	tallafha	tallafha	NOUN
ejpam-259	33	2	and	and	CCONJ
ejpam-259	33	3	r.	r.	PROPN
ejpam-259	33	4	khalil	khalil	PROPN
ejpam-259	33	5	/	/	SYM
ejpam-259	33	6	eur	eur	PROPN
ejpam-259	33	7	.	.	PUNCT
ejpam-259	34	1	j.	j.	PROPN
ejpam-259	34	2	pure	pure	PROPN
ejpam-259	34	3	appl	appl	PROPN
ejpam-259	34	4	.	.	PROPN
ejpam-259	34	5	math	math	PROPN
ejpam-259	34	6	,	,	PUNCT
ejpam-259	34	7	2	2	NUM
ejpam-259	34	8	(	(	PUNCT
ejpam-259	34	9	2009	2009	NUM
ejpam-259	34	10	)	)	PUNCT
ejpam-259	34	11	,	,	PUNCT
ejpam-259	34	12	(	(	PUNCT
ejpam-259	34	13	231	231	NUM
ejpam-259	34	14	-	-	SYM
ejpam-259	34	15	238	238	NUM
ejpam-259	34	16	)	)	PUNCT
ejpam-259	34	17	233	233	NUM
ejpam-259	34	18	linear	linear	ADJ
ejpam-259	34	19	uniform	uniform	ADJ
ejpam-259	34	20	space	space	NOUN
ejpam-259	34	21	.	.	PUNCT
ejpam-259	35	1	we	we	PRON
ejpam-259	35	2	should	should	AUX
ejpam-259	35	3	remark	remark	VERB
ejpam-259	35	4	that	that	SCONJ
ejpam-259	35	5	the	the	DET
ejpam-259	35	6	topology	topology	NOUN
ejpam-259	35	7	in	in	ADP
ejpam-259	35	8	metric	metric	ADJ
ejpam-259	35	9	and	and	CCONJ
ejpam-259	35	10	normed	normed	ADJ
ejpam-259	35	11	spaces	space	NOUN
ejpam-259	35	12	can	can	AUX
ejpam-259	35	13	be	be	AUX
ejpam-259	35	14	generated	generate	VERB
ejpam-259	35	15	by	by	ADP
ejpam-259	35	16	semi	semi	ADJ
ejpam-259	35	17	-	-	ADJ
ejpam-259	35	18	linear	linear	ADJ
ejpam-259	35	19	uniformities	uniformity	NOUN
ejpam-259	35	20	.	.	PUNCT
ejpam-259	36	1	throughout	throughout	ADP
ejpam-259	36	2	the	the	DET
ejpam-259	36	3	rest	rest	NOUN
ejpam-259	36	4	of	of	ADP
ejpam-259	36	5	this	this	DET
ejpam-259	36	6	paper	paper	NOUN
ejpam-259	36	7	,	,	PUNCT
ejpam-259	36	8	(	(	PUNCT
ejpam-259	36	9	x	x	X
ejpam-259	36	10	,	,	PUNCT
ejpam-259	36	11	γ	γ	NOUN
ejpam-259	36	12	)	)	PUNCT
ejpam-259	36	13	will	will	AUX
ejpam-259	36	14	be	be	AUX
ejpam-259	36	15	assumed	assume	VERB
ejpam-259	36	16	semi	semi	ADJ
ejpam-259	36	17	-	-	ADJ
ejpam-259	36	18	linear	linear	ADJ
ejpam-259	36	19	uniform	uniform	ADJ
ejpam-259	36	20	space	space	NOUN
ejpam-259	36	21	.	.	PUNCT
ejpam-259	37	1	now	now	ADV
ejpam-259	37	2	we	we	PRON
ejpam-259	37	3	introduce	introduce	VERB
ejpam-259	37	4	one	one	NUM
ejpam-259	37	5	of	of	ADP
ejpam-259	37	6	the	the	DET
ejpam-259	37	7	main	main	ADJ
ejpam-259	37	8	concepts	concept	NOUN
ejpam-259	37	9	in	in	ADP
ejpam-259	37	10	this	this	DET
ejpam-259	37	11	paper	paper	NOUN
ejpam-259	37	12	.	.	PUNCT
ejpam-259	38	1	let	let	VERB
ejpam-259	38	2	(	(	PUNCT
ejpam-259	38	3	x	x	X
ejpam-259	38	4	,	,	PUNCT
ejpam-259	38	5	γ	γ	PROPN
ejpam-259	38	6	)	)	PUNCT
ejpam-259	38	7	be	be	VERB
ejpam-259	38	8	a	a	DET
ejpam-259	38	9	semi	semi	ADJ
ejpam-259	38	10	-	-	ADJ
ejpam-259	38	11	linear	linear	ADJ
ejpam-259	38	12	uniform	uniform	ADJ
ejpam-259	38	13	space	space	NOUN
ejpam-259	38	14	.	.	PUNCT
ejpam-259	39	1	for	for	ADP
ejpam-259	39	2	x	x	PRON
ejpam-259	39	3	,	,	PUNCT
ejpam-259	39	4	y	y	PROPN
ejpam-259	39	5	∈	∈	PROPN
ejpam-259	39	6	x	x	PUNCT
ejpam-259	39	7	,	,	PUNCT
ejpam-259	39	8	let	let	VERB
ejpam-259	39	9	c	c	NOUN
ejpam-259	39	10	(	(	PUNCT
ejpam-259	39	11	x	x	NOUN
ejpam-259	39	12	,	,	PUNCT
ejpam-259	39	13	y	y	PROPN
ejpam-259	39	14	)	)	PUNCT
ejpam-259	39	15	=	=	VERB
ejpam-259	40	1	∩{v	∩{v	PROPN
ejpam-259	40	2	∈	∈	PROPN
ejpam-259	40	3	γ	γ	X
ejpam-259	40	4	:	:	PUNCT
ejpam-259	40	5	(	(	PUNCT
ejpam-259	40	6	x	x	X
ejpam-259	40	7	,	,	PUNCT
ejpam-259	40	8	y	y	PROPN
ejpam-259	40	9	)	)	PUNCT
ejpam-259	40	10	∈	∈	PROPN
ejpam-259	40	11	v	v	NOUN
ejpam-259	40	12	}	}	PUNCT
ejpam-259	40	13	,	,	PUNCT
ejpam-259	40	14	and	and	CCONJ
ejpam-259	40	15	σ	σ	PROPN
ejpam-259	40	16	=	=	PROPN
ejpam-259	40	17	�	�	PROPN
ejpam-259	40	18	c	c	PROPN
ejpam-259	40	19	(	(	PUNCT
ejpam-259	40	20	x	x	PROPN
ejpam-259	40	21	,	,	PUNCT
ejpam-259	40	22	y	y	PROPN
ejpam-259	40	23	)	)	PUNCT
ejpam-259	40	24	:	:	PUNCT
ejpam-259	41	1	x	x	X
ejpam-259	41	2	,	,	PUNCT
ejpam-259	41	3	y	y	PROPN
ejpam-259	41	4	∈	∈	PROPN
ejpam-259	41	5	x	x	X
ejpam-259	41	6	.	.	PUNCT
ejpam-259	42	1	clearly	clearly	ADV
ejpam-259	42	2	c	c	X
ejpam-259	42	3	(	(	PUNCT
ejpam-259	42	4	x	x	INTJ
ejpam-259	42	5	,	,	PUNCT
ejpam-259	42	6	y	y	PROPN
ejpam-259	42	7	)	)	PUNCT
ejpam-259	42	8	=	=	SYM
ejpam-259	43	1	∩{v−1	∩{v−1	PUNCT
ejpam-259	43	2	∈	∈	PROPN
ejpam-259	43	3	γ	γ	X
ejpam-259	43	4	:	:	PUNCT
ejpam-259	43	5	(	(	PUNCT
ejpam-259	43	6	x	x	X
ejpam-259	43	7	,	,	PUNCT
ejpam-259	43	8	y	y	PROPN
ejpam-259	43	9	)	)	PUNCT
ejpam-259	43	10	∈	∈	PROPN
ejpam-259	43	11	v	v	NOUN
ejpam-259	43	12	}	}	PUNCT
ejpam-259	43	13	.	.	PUNCT
ejpam-259	44	1	definition	definition	NOUN
ejpam-259	44	2	3.1	3.1	NUM
ejpam-259	44	3	.	.	PUNCT
ejpam-259	45	1	let	let	AUX
ejpam-259	45	2	(	(	PUNCT
ejpam-259	45	3	x	x	X
ejpam-259	45	4	,	,	PUNCT
ejpam-259	45	5	γ	γ	PROPN
ejpam-259	45	6	)	)	PUNCT
ejpam-259	45	7	be	be	VERB
ejpam-259	45	8	a	a	DET
ejpam-259	45	9	semi	semi	ADJ
ejpam-259	45	10	-	-	ADJ
ejpam-259	45	11	linear	linear	ADJ
ejpam-259	45	12	uniform	uniform	ADJ
ejpam-259	45	13	space	space	NOUN
ejpam-259	45	14	.	.	PUNCT
ejpam-259	46	1	we	we	PRON
ejpam-259	46	2	define	define	VERB
ejpam-259	46	3	the	the	DET
ejpam-259	46	4	set	set	NOUN
ejpam-259	46	5	valued	value	VERB
ejpam-259	46	6	map	map	NOUN
ejpam-259	46	7	:	:	PUNCT
ejpam-259	46	8	ρ	ρ	NOUN
ejpam-259	46	9	:	:	PUNCT
ejpam-259	46	10	x	x	SYM
ejpam-259	46	11	×	×	NOUN
ejpam-259	46	12	x	x	INTJ
ejpam-259	46	13	→	→	SYM
ejpam-259	46	14	σ	σ	PROPN
ejpam-259	46	15	,	,	PUNCT
ejpam-259	46	16	ρ(x	ρ(x	PROPN
ejpam-259	46	17	,	,	PUNCT
ejpam-259	46	18	y	y	PROPN
ejpam-259	46	19	)	)	PUNCT
ejpam-259	47	1	=	=	SYM
ejpam-259	47	2	c	c	X
ejpam-259	47	3	(	(	PUNCT
ejpam-259	47	4	x	x	X
ejpam-259	47	5	,	,	PUNCT
ejpam-259	47	6	y).the	y).the	PROPN
ejpam-259	47	7	map	map	NOUN
ejpam-259	47	8	ρ	ρ	NOUN
ejpam-259	47	9	will	will	AUX
ejpam-259	47	10	be	be	AUX
ejpam-259	47	11	called	call	VERB
ejpam-259	47	12	a	a	DET
ejpam-259	47	13	set	set	VERB
ejpam-259	47	14	metric	metric	NOUN
ejpam-259	47	15	on	on	ADP
ejpam-259	47	16	(	(	PUNCT
ejpam-259	47	17	x	x	INTJ
ejpam-259	47	18	,	,	PUNCT
ejpam-259	47	19	γ	γ	PROPN
ejpam-259	47	20	)	)	PUNCT
ejpam-259	47	21	.	.	PUNCT
ejpam-259	48	1	the	the	DET
ejpam-259	48	2	proof	proof	NOUN
ejpam-259	48	3	of	of	ADP
ejpam-259	48	4	the	the	DET
ejpam-259	48	5	following	following	ADJ
ejpam-259	48	6	result	result	NOUN
ejpam-259	48	7	is	be	AUX
ejpam-259	48	8	immediate	immediate	ADJ
ejpam-259	48	9	and	and	CCONJ
ejpam-259	48	10	will	will	AUX
ejpam-259	48	11	be	be	AUX
ejpam-259	48	12	omitted	omit	VERB
ejpam-259	48	13	.	.	PUNCT
ejpam-259	49	1	proposition	proposition	NOUN
ejpam-259	49	2	4.1	4.1	NUM
ejpam-259	49	3	.	.	PUNCT
ejpam-259	50	1	for	for	ADP
ejpam-259	50	2	a	a	DET
ejpam-259	50	3	semi	semi	ADJ
ejpam-259	50	4	-	-	ADJ
ejpam-259	50	5	linear	linear	ADJ
ejpam-259	50	6	uniform	uniform	ADJ
ejpam-259	50	7	space	space	NOUN
ejpam-259	50	8	,	,	PUNCT
ejpam-259	50	9	we	we	PRON
ejpam-259	50	10	have	have	VERB
ejpam-259	50	11	the	the	DET
ejpam-259	50	12	followings	following	NOUN
ejpam-259	50	13	.	.	PUNCT
ejpam-259	51	1	(	(	PUNCT
ejpam-259	51	2	i	i	NOUN
ejpam-259	51	3	)	)	PUNCT
ejpam-259	51	4	ρ(x	ρ(x	PROPN
ejpam-259	51	5	,	,	PUNCT
ejpam-259	51	6	y	y	PROPN
ejpam-259	51	7	)	)	PUNCT
ejpam-259	51	8	=	=	PUNCT
ejpam-259	52	1	∆	∆	PROPN
ejpam-259	52	2	if	if	SCONJ
ejpam-259	52	3	and	and	CCONJ
ejpam-259	52	4	only	only	ADV
ejpam-259	52	5	if	if	SCONJ
ejpam-259	52	6	x	x	X
ejpam-259	52	7	=	=	SYM
ejpam-259	52	8	y.	y.	PROPN
ejpam-259	52	9	(	(	PUNCT
ejpam-259	52	10	ii	ii	PROPN
ejpam-259	52	11	)	)	PUNCT
ejpam-259	52	12	ρ(x	ρ(x	PROPN
ejpam-259	52	13	,	,	PUNCT
ejpam-259	52	14	y	y	NOUN
ejpam-259	52	15	)	)	PUNCT
ejpam-259	52	16	=	=	SYM
ejpam-259	52	17	ρ(y	ρ(y	NOUN
ejpam-259	52	18	,	,	PUNCT
ejpam-259	52	19	x	x	NOUN
ejpam-259	52	20	)	)	PUNCT
ejpam-259	52	21	.	.	PUNCT
ejpam-259	53	1	now	now	ADV
ejpam-259	53	2	we	we	PRON
ejpam-259	53	3	have	have	VERB
ejpam-259	53	4	the	the	DET
ejpam-259	53	5	following	follow	VERB
ejpam-259	53	6	natural	natural	ADJ
ejpam-259	53	7	questions	question	NOUN
ejpam-259	53	8	.	.	PUNCT
ejpam-259	54	1	question1	question1	ADJ
ejpam-259	54	2	:	:	PUNCT
ejpam-259	54	3	is	be	AUX
ejpam-259	54	4	ρ(x	ρ(x	PROPN
ejpam-259	54	5	,	,	PUNCT
ejpam-259	54	6	y)⊆	y)⊆	PROPN
ejpam-259	54	7	ρ(x	ρ(x	PROPN
ejpam-259	54	8	,	,	PUNCT
ejpam-259	54	9	z)∩ρ(z	z)∩ρ(z	PROPN
ejpam-259	54	10	,	,	PUNCT
ejpam-259	54	11	y	y	PROPN
ejpam-259	54	12	)	)	PUNCT
ejpam-259	54	13	?	?	PUNCT
ejpam-259	54	14	.	.	PUNCT
ejpam-259	55	1	in	in	ADP
ejpam-259	55	2	metric	metric	ADJ
ejpam-259	55	3	spaces	space	NOUN
ejpam-259	55	4	,	,	PUNCT
ejpam-259	55	5	it	it	PRON
ejpam-259	55	6	is	be	AUX
ejpam-259	55	7	known	know	VERB
ejpam-259	55	8	that	that	SCONJ
ejpam-259	55	9	if	if	SCONJ
ejpam-259	55	10	d(x	d(x	PROPN
ejpam-259	55	11	,	,	PUNCT
ejpam-259	55	12	y	y	NOUN
ejpam-259	55	13	)	)	PUNCT
ejpam-259	55	14	=	=	SYM
ejpam-259	55	15	d(x	d(x	PROPN
ejpam-259	55	16	,	,	PUNCT
ejpam-259	55	17	z	z	X
ejpam-259	55	18	)	)	PUNCT
ejpam-259	55	19	then	then	ADV
ejpam-259	55	20	y	y	PROPN
ejpam-259	55	21	need	need	VERB
ejpam-259	55	22	not	not	PART
ejpam-259	55	23	equal	equal	ADJ
ejpam-259	55	24	z	z	X
ejpam-259	55	25	..	..	PUNCT
ejpam-259	55	26	in	in	ADP
ejpam-259	55	27	semi	semi	ADJ
ejpam-259	55	28	-	-	ADJ
ejpam-259	55	29	linear	linear	ADJ
ejpam-259	55	30	type	type	NOUN
ejpam-259	55	31	spaces	space	NOUN
ejpam-259	55	32	,	,	PUNCT
ejpam-259	55	33	the	the	DET
ejpam-259	55	34	story	story	NOUN
ejpam-259	55	35	is	be	AUX
ejpam-259	55	36	different	different	ADJ
ejpam-259	55	37	.	.	PUNCT
ejpam-259	56	1	so	so	ADV
ejpam-259	56	2	we	we	PRON
ejpam-259	56	3	pose	pose	VERB
ejpam-259	56	4	the	the	DET
ejpam-259	56	5	following	following	ADJ
ejpam-259	56	6	question	question	NOUN
ejpam-259	56	7	.	.	PUNCT
ejpam-259	57	1	question	question	NOUN
ejpam-259	57	2	2	2	NUM
ejpam-259	57	3	.	.	PUNCT
ejpam-259	58	1	if	if	SCONJ
ejpam-259	58	2	ρ(x	ρ(x	PROPN
ejpam-259	58	3	,	,	PUNCT
ejpam-259	58	4	z	z	NOUN
ejpam-259	58	5	)	)	PUNCT
ejpam-259	58	6	=	=	SYM
ejpam-259	59	1	ρ(x	ρ(x	NOUN
ejpam-259	59	2	,	,	PUNCT
ejpam-259	59	3	w),for	w),for	ADP
ejpam-259	59	4	some	some	DET
ejpam-259	59	5	x	x	SYM
ejpam-259	59	6	∈	∈	PROPN
ejpam-259	59	7	x	x	X
ejpam-259	59	8	.	.	PUNCT
ejpam-259	60	1	must	must	AUX
ejpam-259	60	2	w	w	NOUN
ejpam-259	60	3	=	=	NOUN
ejpam-259	60	4	z	z	NOUN
ejpam-259	60	5	?	?	PUNCT
ejpam-259	60	6	.	.	PUNCT
ejpam-259	61	1	using	use	VERB
ejpam-259	61	2	the	the	DET
ejpam-259	61	3	concept	concept	NOUN
ejpam-259	61	4	of	of	ADP
ejpam-259	61	5	set	set	NOUN
ejpam-259	61	6	metric	metric	ADJ
ejpam-259	61	7	,	,	PUNCT
ejpam-259	61	8	we	we	PRON
ejpam-259	61	9	introduce	introduce	VERB
ejpam-259	61	10	the	the	DET
ejpam-259	61	11	following	follow	VERB
ejpam-259	61	12	concepts	concept	NOUN
ejpam-259	61	13	.	.	PUNCT
ejpam-259	62	1	definition	definition	NOUN
ejpam-259	62	2	5.1	5.1	NUM
ejpam-259	62	3	.	.	PUNCT
ejpam-259	63	1	for	for	SCONJ
ejpam-259	63	2	x	x	SYM
ejpam-259	63	3	∈	∈	PROPN
ejpam-259	63	4	x	x	X
ejpam-259	63	5	and	and	CCONJ
ejpam-259	63	6	e	e	X
ejpam-259	63	7	⊂	⊂	PROPN
ejpam-259	63	8	x	x	X
ejpam-259	63	9	,	,	PUNCT
ejpam-259	63	10	we	we	PRON
ejpam-259	63	11	define	define	VERB
ejpam-259	63	12	ρ(x	ρ(x	PROPN
ejpam-259	63	13	,	,	PUNCT
ejpam-259	63	14	e	e	NOUN
ejpam-259	63	15	)	)	PUNCT
ejpam-259	63	16	=	=	NOUN
ejpam-259	63	17	∩	∩	NOUN
ejpam-259	63	18	y	y	PROPN
ejpam-259	63	19	∈	∈	PROPN
ejpam-259	63	20	e	e	PROPN
ejpam-259	63	21	ρ(x	ρ(x	PROPN
ejpam-259	63	22	,	,	PUNCT
ejpam-259	63	23	y	y	PROPN
ejpam-259	63	24	)	)	PUNCT
ejpam-259	63	25	.	.	PUNCT
ejpam-259	64	1	clearly	clearly	ADV
ejpam-259	64	2	,	,	PUNCT
ejpam-259	64	3	if	if	SCONJ
ejpam-259	64	4	x	x	SYM
ejpam-259	64	5	∈	∈	PROPN
ejpam-259	64	6	e	e	NOUN
ejpam-259	64	7	,	,	PUNCT
ejpam-259	64	8	then	then	ADV
ejpam-259	64	9	ρ(x	ρ(x	PROPN
ejpam-259	64	10	,	,	PUNCT
ejpam-259	64	11	e	e	NOUN
ejpam-259	64	12	)	)	PUNCT
ejpam-259	64	13	=	=	PUNCT
ejpam-259	64	14	∆.	∆.	ADJ
ejpam-259	64	15	definition	definition	NOUN
ejpam-259	64	16	6.1	6.1	NUM
ejpam-259	64	17	.	.	PUNCT
ejpam-259	65	1	for	for	SCONJ
ejpam-259	65	2	x	x	SYM
ejpam-259	65	3	∈	∈	PROPN
ejpam-259	65	4	x	x	X
ejpam-259	65	5	and	and	CCONJ
ejpam-259	65	6	v	v	ADP
ejpam-259	65	7	∈	∈	PROPN
ejpam-259	65	8	γ	γ	X
ejpam-259	65	9	,	,	PUNCT
ejpam-259	65	10	we	we	PRON
ejpam-259	65	11	define	define	VERB
ejpam-259	65	12	the	the	DET
ejpam-259	65	13	open	open	ADJ
ejpam-259	65	14	ball	ball	NOUN
ejpam-259	65	15	of	of	ADP
ejpam-259	65	16	center	center	NOUN
ejpam-259	65	17	x	x	PUNCT
ejpam-259	65	18	and	and	CCONJ
ejpam-259	65	19	radius	radius	NOUN
ejpam-259	65	20	v	v	NOUN
ejpam-259	65	21	to	to	PART
ejpam-259	65	22	be	be	AUX
ejpam-259	65	23	b(x	b(x	VERB
ejpam-259	65	24	,	,	PUNCT
ejpam-259	65	25	v	v	NOUN
ejpam-259	65	26	)	)	PUNCT
ejpam-259	65	27	=	=	SYM
ejpam-259	65	28	{	{	PUNCT
ejpam-259	65	29	y	y	NOUN
ejpam-259	65	30	:	:	PUNCT
ejpam-259	65	31	(	(	PUNCT
ejpam-259	65	32	x	x	X
ejpam-259	65	33	,	,	PUNCT
ejpam-259	65	34	y	y	PROPN
ejpam-259	65	35	)	)	PUNCT
ejpam-259	65	36	∈	∈	PROPN
ejpam-259	65	37	v	v	ADP
ejpam-259	65	38	}	}	PUNCT
ejpam-259	65	39	.	.	PUNCT
ejpam-259	66	1	equivalently	equivalently	ADV
ejpam-259	66	2	b(x	b(x	NOUN
ejpam-259	66	3	,	,	PUNCT
ejpam-259	66	4	v	v	NOUN
ejpam-259	66	5	)	)	PUNCT
ejpam-259	66	6	=	=	SYM
ejpam-259	66	7	{	{	PUNCT
ejpam-259	66	8	y	y	NOUN
ejpam-259	66	9	:	:	PUNCT
ejpam-259	66	10	ρ(x	ρ(x	PROPN
ejpam-259	66	11	,	,	PUNCT
ejpam-259	66	12	y	y	PROPN
ejpam-259	66	13	)	)	PUNCT
ejpam-259	66	14	⊆	⊆	NUM
ejpam-259	66	15	v	v	NOUN
ejpam-259	66	16	}	}	PUNCT
ejpam-259	66	17	.clearly	.clearly	ADV
ejpam-259	66	18	if	if	SCONJ
ejpam-259	66	19	y	y	PROPN
ejpam-259	66	20	∈	∈	PROPN
ejpam-259	66	21	b(x	b(x	PROPN
ejpam-259	66	22	,	,	PUNCT
ejpam-259	66	23	v	v	NOUN
ejpam-259	66	24	)	)	PUNCT
ejpam-259	66	25	,	,	PUNCT
ejpam-259	66	26	then	then	ADV
ejpam-259	66	27	there	there	PRON
ejpam-259	66	28	is	be	VERB
ejpam-259	66	29	a	a	DET
ejpam-259	66	30	w	w	PROPN
ejpam-259	66	31	∈	∈	PROPN
ejpam-259	66	32	γ	γ	NOUN
ejpam-259	66	33	such	such	ADJ
ejpam-259	66	34	that	that	SCONJ
ejpam-259	66	35	b(y	b(y	PROPN
ejpam-259	66	36	,	,	PUNCT
ejpam-259	66	37	w	w	NOUN
ejpam-259	66	38	)	)	PUNCT
ejpam-259	66	39	⊆	⊆	NUM
ejpam-259	66	40	b(x	b(x	NOUN
ejpam-259	66	41	,	,	PUNCT
ejpam-259	66	42	v	v	NOUN
ejpam-259	66	43	)	)	PUNCT
ejpam-259	66	44	.	.	PUNCT
ejpam-259	67	1	a.	a.	NOUN
ejpam-259	67	2	tallafha	tallafha	NOUN
ejpam-259	67	3	and	and	CCONJ
ejpam-259	67	4	r.	r.	PROPN
ejpam-259	67	5	khalil	khalil	PROPN
ejpam-259	67	6	/	/	SYM
ejpam-259	67	7	eur	eur	PROPN
ejpam-259	67	8	.	.	PUNCT
ejpam-259	68	1	j.	j.	PROPN
ejpam-259	68	2	pure	pure	PROPN
ejpam-259	68	3	appl	appl	PROPN
ejpam-259	68	4	.	.	PROPN
ejpam-259	68	5	math	math	PROPN
ejpam-259	68	6	,	,	PUNCT
ejpam-259	68	7	2	2	NUM
ejpam-259	68	8	(	(	PUNCT
ejpam-259	68	9	2009	2009	NUM
ejpam-259	68	10	)	)	PUNCT
ejpam-259	68	11	,	,	PUNCT
ejpam-259	68	12	(	(	PUNCT
ejpam-259	68	13	231	231	NUM
ejpam-259	68	14	-	-	SYM
ejpam-259	68	15	238	238	NUM
ejpam-259	68	16	)	)	PUNCT
ejpam-259	68	17	234	234	NUM
ejpam-259	68	18	definition	definition	NOUN
ejpam-259	68	19	7.1	7.1	NUM
ejpam-259	68	20	.	.	PUNCT
ejpam-259	69	1	b	b	X
ejpam-259	69	2	⊆	⊆	NUM
ejpam-259	69	3	x	x	PUNCT
ejpam-259	69	4	is	be	AUX
ejpam-259	69	5	called	call	VERB
ejpam-259	69	6	bounded	bound	VERB
ejpam-259	69	7	if	if	SCONJ
ejpam-259	69	8	b	b	PROPN
ejpam-259	69	9	⊆	⊆	NUM
ejpam-259	69	10	b(x	b(x	NOUN
ejpam-259	69	11	,	,	PUNCT
ejpam-259	69	12	v	v	NOUN
ejpam-259	69	13	)	)	PUNCT
ejpam-259	69	14	,	,	PUNCT
ejpam-259	69	15	for	for	ADP
ejpam-259	69	16	some	some	DET
ejpam-259	69	17	v	v	ADP
ejpam-259	69	18	∈	∈	PROPN
ejpam-259	69	19	γ	γ	X
ejpam-259	69	20	,	,	PUNCT
ejpam-259	69	21	x	x	SYM
ejpam-259	69	22	∈	∈	PROPN
ejpam-259	69	23	x	x	X
ejpam-259	69	24	.	.	PUNCT
ejpam-259	70	1	definition	definition	NOUN
ejpam-259	70	2	8.1	8.1	NUM
ejpam-259	70	3	.	.	PUNCT
ejpam-259	71	1	let	let	AUX
ejpam-259	71	2	(	(	PUNCT
ejpam-259	71	3	xn	xn	X
ejpam-259	71	4	)	)	PUNCT
ejpam-259	71	5	be	be	VERB
ejpam-259	71	6	a	a	DET
ejpam-259	71	7	sequence	sequence	NOUN
ejpam-259	71	8	in	in	ADP
ejpam-259	71	9	x	x	PROPN
ejpam-259	71	10	.we	.we	PUNCT
ejpam-259	71	11	say	say	VERB
ejpam-259	71	12	xn	xn	PROPN
ejpam-259	71	13	converges	converge	VERB
ejpam-259	71	14	to	to	ADP
ejpam-259	71	15	x	x	PUNCT
ejpam-259	71	16	in	in	ADP
ejpam-259	71	17	x	x	X
ejpam-259	71	18	,	,	PUNCT
ejpam-259	71	19	and	and	CCONJ
ejpam-259	71	20	we	we	PRON
ejpam-259	71	21	write	write	VERB
ejpam-259	71	22	xn	xn	PROPN
ejpam-259	72	1	→	→	SYM
ejpam-259	72	2	x	x	SYM
ejpam-259	72	3	,	,	PUNCT
ejpam-259	72	4	if	if	SCONJ
ejpam-259	72	5	for	for	ADP
ejpam-259	72	6	every	every	DET
ejpam-259	72	7	v	v	NOUN
ejpam-259	72	8	∈	∈	PROPN
ejpam-259	72	9	γ	γ	NOUN
ejpam-259	72	10	there	there	PRON
ejpam-259	72	11	exists	exist	VERB
ejpam-259	72	12	k	k	X
ejpam-259	72	13	such	such	ADJ
ejpam-259	72	14	that	that	PRON
ejpam-259	72	15	(	(	PUNCT
ejpam-259	72	16	xn	xn	PROPN
ejpam-259	72	17	,	,	PUNCT
ejpam-259	72	18	x	x	X
ejpam-259	72	19	)	)	PUNCT
ejpam-259	72	20	∈	∈	NOUN
ejpam-259	72	21	v	v	NOUN
ejpam-259	72	22	for	for	ADP
ejpam-259	72	23	every	every	DET
ejpam-259	72	24	n	n	PRON
ejpam-259	72	25	≥	≥	NOUN
ejpam-259	72	26	k.	k.	X
ejpam-259	73	1	clearly	clearly	ADV
ejpam-259	73	2	if	if	SCONJ
ejpam-259	73	3	xn→	xn→	PROPN
ejpam-259	73	4	x	x	X
ejpam-259	73	5	,	,	PUNCT
ejpam-259	73	6	then	then	ADV
ejpam-259	73	7	for	for	ADP
ejpam-259	73	8	every	every	DET
ejpam-259	73	9	j	j	PROPN
ejpam-259	73	10	,	,	PUNCT
ejpam-259	73	11	∞	∞	PROPN
ejpam-259	73	12	⋂	⋂	PROPN
ejpam-259	73	13	n=	n=	ADJ
ejpam-259	73	14	j	j	PROPN
ejpam-259	73	15	ρ(x	ρ(x	PROPN
ejpam-259	73	16	,	,	PUNCT
ejpam-259	73	17	xn	xn	PROPN
ejpam-259	73	18	)	)	PUNCT
ejpam-259	73	19	=	=	PUNCT
ejpam-259	73	20	∆.unfortunately	∆.unfortunately	ADV
ejpam-259	73	21	the	the	DET
ejpam-259	73	22	converse	converse	NOUN
ejpam-259	73	23	is	be	AUX
ejpam-259	73	24	not	not	PART
ejpam-259	73	25	true	true	ADJ
ejpam-259	73	26	.	.	PUNCT
ejpam-259	74	1	but	but	CCONJ
ejpam-259	74	2	we	we	PRON
ejpam-259	74	3	have	have	VERB
ejpam-259	74	4	lemma	lemma	PROPN
ejpam-259	74	5	9.1	9.1	NUM
ejpam-259	74	6	.	.	PUNCT
ejpam-259	75	1	let	let	AUX
ejpam-259	75	2	(	(	PUNCT
ejpam-259	75	3	xn	xn	X
ejpam-259	75	4	)	)	PUNCT
ejpam-259	75	5	be	be	VERB
ejpam-259	75	6	a	a	DET
ejpam-259	75	7	sequence	sequence	NOUN
ejpam-259	75	8	in	in	ADP
ejpam-259	75	9	x	x	X
ejpam-259	75	10	.	.	PUNCT
ejpam-259	76	1	if	if	SCONJ
ejpam-259	76	2	∞	∞	PROPN
ejpam-259	76	3	⋂	⋂	PROPN
ejpam-259	76	4	n=	n=	ADJ
ejpam-259	76	5	j	j	PROPN
ejpam-259	76	6	ρ(x	ρ(x	PROPN
ejpam-259	76	7	,	,	PUNCT
ejpam-259	76	8	xn	xn	PROPN
ejpam-259	76	9	)	)	PUNCT
ejpam-259	76	10	=	=	SYM
ejpam-259	76	11	∆	∆	NOUN
ejpam-259	76	12	,	,	PUNCT
ejpam-259	76	13	for	for	ADP
ejpam-259	76	14	every	every	DET
ejpam-259	76	15	j	j	PROPN
ejpam-259	76	16	,	,	PUNCT
ejpam-259	76	17	then	then	ADV
ejpam-259	76	18	there	there	PRON
ejpam-259	76	19	exist	exist	VERB
ejpam-259	76	20	a	a	DET
ejpam-259	76	21	subsequence	subsequence	NOUN
ejpam-259	76	22	xnk	xnk	NOUN
ejpam-259	77	1	→	→	SYM
ejpam-259	77	2	x	x	X
ejpam-259	77	3	.	.	PUNCT
ejpam-259	78	1	proof	proof	NOUN
ejpam-259	78	2	.	.	PUNCT
ejpam-259	79	1	we	we	PRON
ejpam-259	79	2	may	may	AUX
ejpam-259	79	3	assume	assume	VERB
ejpam-259	79	4	that	that	SCONJ
ejpam-259	79	5	,	,	PUNCT
ejpam-259	79	6	for	for	ADP
ejpam-259	79	7	every	every	DET
ejpam-259	79	8	j	j	NOUN
ejpam-259	79	9	there	there	PRON
ejpam-259	79	10	is	be	VERB
ejpam-259	79	11	n	n	PRON
ejpam-259	79	12	j	j	PROPN
ejpam-259	79	13	≥	≥	NUM
ejpam-259	79	14	j	j	PROPN
ejpam-259	79	15	such	such	ADJ
ejpam-259	79	16	that	that	PRON
ejpam-259	79	17	ρ(x	ρ(x	NOUN
ejpam-259	79	18	,	,	PUNCT
ejpam-259	79	19	xn	xn	PROPN
ejpam-259	79	20	j	j	PROPN
ejpam-259	79	21	)	)	PUNCT
ejpam-259	79	22	6=	6=	NUM
ejpam-259	80	1	∆	∆	PROPN
ejpam-259	80	2	,	,	PUNCT
ejpam-259	80	3	also	also	ADV
ejpam-259	80	4	we	we	PRON
ejpam-259	80	5	may	may	AUX
ejpam-259	80	6	assume	assume	VERB
ejpam-259	80	7	ρ(x	ρ(x	PROPN
ejpam-259	80	8	,	,	PUNCT
ejpam-259	80	9	xn	xn	PROPN
ejpam-259	80	10	j	j	PROPN
ejpam-259	80	11	)	)	PUNCT
ejpam-259	80	12	is	be	AUX
ejpam-259	80	13	a	a	DET
ejpam-259	80	14	decreasing	decrease	VERB
ejpam-259	80	15	sequence	sequence	NOUN
ejpam-259	80	16	and	and	CCONJ
ejpam-259	81	1	∞	∞	NUM
ejpam-259	81	2	⋂	⋂	PROPN
ejpam-259	81	3	j	j	PROPN
ejpam-259	81	4	=	=	SYM
ejpam-259	81	5	1	1	NUM
ejpam-259	81	6	ρ(x	ρ(x	PROPN
ejpam-259	81	7	,	,	PUNCT
ejpam-259	81	8	xn	xn	PROPN
ejpam-259	81	9	j	j	PROPN
ejpam-259	81	10	)	)	PUNCT
ejpam-259	82	1	=	=	SYM
ejpam-259	82	2	∆.let	∆.let	VERB
ejpam-259	82	3	v	v	ADP
ejpam-259	82	4	∈	∈	PROPN
ejpam-259	82	5	γ	γ	X
ejpam-259	82	6	,	,	PUNCT
ejpam-259	82	7	then	then	ADV
ejpam-259	82	8	there	there	PRON
ejpam-259	82	9	exist	exist	VERB
ejpam-259	82	10	j	j	PROPN
ejpam-259	82	11	1	1	NUM
ejpam-259	82	12	such	such	ADJ
ejpam-259	82	13	that	that	DET
ejpam-259	82	14	ρ(x	ρ(x	NOUN
ejpam-259	82	15	,	,	PUNCT
ejpam-259	82	16	xn	xn	PROPN
ejpam-259	82	17	j	j	PROPN
ejpam-259	82	18	)	)	PUNCT
ejpam-259	82	19	⊆	⊆	NUM
ejpam-259	82	20	v	v	NOUN
ejpam-259	82	21	for	for	ADP
ejpam-259	82	22	all	all	DET
ejpam-259	82	23	j	j	PROPN
ejpam-259	82	24	≥	≥	X
ejpam-259	82	25	j	j	PROPN
ejpam-259	82	26	1	1	NUM
ejpam-259	82	27	,	,	PUNCT
ejpam-259	82	28	hence	hence	ADV
ejpam-259	82	29	xn	xn	PROPN
ejpam-259	83	1	j	j	PROPN
ejpam-259	83	2	→	→	PUNCT
ejpam-259	83	3	x	x	X
ejpam-259	83	4	.	.	PUNCT
ejpam-259	84	1	definition	definition	NOUN
ejpam-259	84	2	10.1	10.1	NUM
ejpam-259	84	3	.	.	PUNCT
ejpam-259	85	1	let	let	AUX
ejpam-259	85	2	(	(	PUNCT
ejpam-259	85	3	xn	xn	X
ejpam-259	85	4	)	)	PUNCT
ejpam-259	85	5	be	be	VERB
ejpam-259	85	6	a	a	DET
ejpam-259	85	7	sequence	sequence	NOUN
ejpam-259	85	8	in	in	ADP
ejpam-259	85	9	x	x	X
ejpam-259	85	10	,	,	PUNCT
ejpam-259	85	11	(	(	PUNCT
ejpam-259	85	12	xn	xn	X
ejpam-259	85	13	)	)	PUNCT
ejpam-259	85	14	is	be	AUX
ejpam-259	85	15	called	call	VERB
ejpam-259	85	16	cauchy	cauchy	PROPN
ejpam-259	85	17	if	if	SCONJ
ejpam-259	85	18	for	for	SCONJ
ejpam-259	85	19	every	every	DET
ejpam-259	85	20	v	v	NOUN
ejpam-259	85	21	∈	∈	PROPN
ejpam-259	85	22	γ	γ	NOUN
ejpam-259	85	23	there	there	PRON
ejpam-259	85	24	exists	exist	VERB
ejpam-259	85	25	k	k	X
ejpam-259	85	26	such	such	ADJ
ejpam-259	85	27	that	that	PRON
ejpam-259	85	28	(	(	PUNCT
ejpam-259	85	29	xn	xn	PROPN
ejpam-259	85	30	,	,	PUNCT
ejpam-259	85	31	xm	xm	NUM
ejpam-259	85	32	)	)	PUNCT
ejpam-259	85	33	∈	∈	PROPN
ejpam-259	85	34	v	v	NOUN
ejpam-259	85	35	for	for	ADP
ejpam-259	85	36	every	every	DET
ejpam-259	85	37	n	n	CCONJ
ejpam-259	85	38	,	,	PUNCT
ejpam-259	85	39	m	m	VERB
ejpam-259	85	40	≥	≥	NOUN
ejpam-259	85	41	k.	k.	INTJ
ejpam-259	86	1	now	now	ADV
ejpam-259	86	2	it	it	PRON
ejpam-259	86	3	is	be	AUX
ejpam-259	86	4	easy	easy	ADJ
ejpam-259	86	5	to	to	PART
ejpam-259	86	6	prove	prove	VERB
ejpam-259	86	7	the	the	DET
ejpam-259	86	8	following	follow	VERB
ejpam-259	86	9	corollary	corollary	ADJ
ejpam-259	86	10	corollary	corollary	NOUN
ejpam-259	86	11	11.1	11.1	NUM
ejpam-259	86	12	.	.	PUNCT
ejpam-259	87	1	let	let	AUX
ejpam-259	87	2	(	(	PUNCT
ejpam-259	87	3	xn	xn	X
ejpam-259	87	4	)	)	PUNCT
ejpam-259	87	5	be	be	VERB
ejpam-259	87	6	a	a	DET
ejpam-259	87	7	cauchy	cauchy	ADJ
ejpam-259	87	8	sequence	sequence	NOUN
ejpam-259	87	9	in	in	ADP
ejpam-259	87	10	x	x	X
ejpam-259	87	11	.	.	PUNCT
ejpam-259	88	1	then	then	ADV
ejpam-259	88	2	xn	xn	PUNCT
ejpam-259	88	3	→	→	SYM
ejpam-259	88	4	x	x	SYM
ejpam-259	88	5	,	,	PUNCT
ejpam-259	88	6	iff	iff	PROPN
ejpam-259	88	7	for	for	ADP
ejpam-259	88	8	every	every	DET
ejpam-259	88	9	j	j	PROPN
ejpam-259	88	10	∈	∈	PROPN
ejpam-259	88	11	n	n	CCONJ
ejpam-259	88	12	,	,	PUNCT
ejpam-259	88	13	∞	∞	PROPN
ejpam-259	88	14	⋂	⋂	PROPN
ejpam-259	88	15	n	n	PROPN
ejpam-259	88	16	=	=	SYM
ejpam-259	88	17	j	j	PROPN
ejpam-259	88	18	ρ(x	ρ(x	PROPN
ejpam-259	88	19	,	,	PUNCT
ejpam-259	88	20	xn	xn	PROPN
ejpam-259	88	21	)	)	PUNCT
ejpam-259	89	1	=	=	PUNCT
ejpam-259	90	1	∆.	∆.	ADV
ejpam-259	90	2	now	now	ADV
ejpam-259	90	3	,	,	PUNCT
ejpam-259	90	4	we	we	PRON
ejpam-259	90	5	prove	prove	VERB
ejpam-259	90	6	:	:	PUNCT
ejpam-259	90	7	lemma	lemma	PROPN
ejpam-259	90	8	12.1	12.1	NUM
ejpam-259	90	9	.	.	PUNCT
ejpam-259	91	1	let	let	AUX
ejpam-259	91	2	(	(	PUNCT
ejpam-259	91	3	xn	xn	X
ejpam-259	91	4	)	)	PUNCT
ejpam-259	91	5	be	be	VERB
ejpam-259	91	6	a	a	DET
ejpam-259	91	7	sequence	sequence	NOUN
ejpam-259	91	8	in	in	ADP
ejpam-259	91	9	(	(	PUNCT
ejpam-259	91	10	x	x	INTJ
ejpam-259	91	11	,	,	PUNCT
ejpam-259	91	12	γ	γ	PROPN
ejpam-259	91	13	)	)	PUNCT
ejpam-259	91	14	.	.	PUNCT
ejpam-259	92	1	then	then	ADV
ejpam-259	92	2	.	.	PUNCT
ejpam-259	93	1	(	(	PUNCT
ejpam-259	93	2	i	i	NOUN
ejpam-259	93	3	)	)	PUNCT
ejpam-259	93	4	every	every	DET
ejpam-259	93	5	convergent	convergent	NOUN
ejpam-259	93	6	sequence	sequence	NOUN
ejpam-259	93	7	is	be	AUX
ejpam-259	93	8	cauchy	cauchy	NOUN
ejpam-259	93	9	.	.	PUNCT
ejpam-259	94	1	(	(	PUNCT
ejpam-259	94	2	ii	ii	NOUN
ejpam-259	94	3	)	)	PUNCT
ejpam-259	94	4	every	every	DET
ejpam-259	94	5	cauchy	cauchy	ADJ
ejpam-259	94	6	sequence	sequence	NOUN
ejpam-259	94	7	is	be	AUX
ejpam-259	94	8	bounded	bound	VERB
ejpam-259	94	9	.	.	PUNCT
ejpam-259	95	1	proof	proof	NOUN
ejpam-259	95	2	.	.	PUNCT
ejpam-259	96	1	(	(	PUNCT
ejpam-259	96	2	i	i	NOUN
ejpam-259	96	3	)	)	PUNCT
ejpam-259	96	4	let	let	AUX
ejpam-259	96	5	(	(	PUNCT
ejpam-259	96	6	xn	xn	X
ejpam-259	96	7	)	)	PUNCT
ejpam-259	96	8	converges	converge	VERB
ejpam-259	96	9	to	to	ADP
ejpam-259	96	10	x	x	PUNCT
ejpam-259	96	11	in	in	ADP
ejpam-259	96	12	x	x	X
ejpam-259	96	13	,	,	PUNCT
ejpam-259	96	14	and	and	CCONJ
ejpam-259	96	15	v	v	ADP
ejpam-259	96	16	∈	∈	PROPN
ejpam-259	96	17	γ.let	γ.let	NOUN
ejpam-259	96	18	u	u	PROPN
ejpam-259	96	19	∈	∈	PROPN
ejpam-259	96	20	γ	γ	NOUN
ejpam-259	96	21	such	such	ADJ
ejpam-259	96	22	that	that	DET
ejpam-259	96	23	u	u	NOUN
ejpam-259	96	24	◦	◦	NOUN
ejpam-259	96	25	u	u	X
ejpam-259	96	26	⊂	⊂	X
ejpam-259	97	1	v.	v.	CCONJ
ejpam-259	97	2	from	from	ADP
ejpam-259	97	3	the	the	DET
ejpam-259	97	4	definition	definition	NOUN
ejpam-259	97	5	of	of	ADP
ejpam-259	97	6	convergence	convergence	NOUN
ejpam-259	97	7	,	,	PUNCT
ejpam-259	97	8	there	there	PRON
ejpam-259	97	9	exists	exist	VERB
ejpam-259	97	10	k	k	X
ejpam-259	97	11	such	such	ADJ
ejpam-259	97	12	that	that	SCONJ
ejpam-259	97	13	(	(	PUNCT
ejpam-259	97	14	x	x	X
ejpam-259	97	15	,	,	PUNCT
ejpam-259	97	16	xn	xn	X
ejpam-259	97	17	)	)	PUNCT
ejpam-259	97	18	∈	∈	NOUN
ejpam-259	97	19	u	u	NOUN
ejpam-259	97	20	for	for	ADP
ejpam-259	97	21	all	all	DET
ejpam-259	97	22	n	n	PROPN
ejpam-259	97	23	>	>	X
ejpam-259	97	24	k.	k.	PROPN
ejpam-259	98	1	since	since	SCONJ
ejpam-259	98	2	u	u	NOUN
ejpam-259	98	3	is	be	AUX
ejpam-259	98	4	symmetric	symmetric	ADJ
ejpam-259	98	5	,	,	PUNCT
ejpam-259	98	6	(	(	PUNCT
ejpam-259	98	7	xm	xm	PROPN
ejpam-259	98	8	,	,	PUNCT
ejpam-259	98	9	x	x	NOUN
ejpam-259	98	10	)	)	PUNCT
ejpam-259	98	11	∈	∈	PROPN
ejpam-259	98	12	u	u	NOUN
ejpam-259	98	13	for	for	ADP
ejpam-259	98	14	all	all	DET
ejpam-259	98	15	m	m	PROPN
ejpam-259	98	16	>	>	X
ejpam-259	98	17	k.	k.	PROPN
ejpam-259	99	1	hence	hence	ADV
ejpam-259	99	2	(	(	PUNCT
ejpam-259	99	3	xn	xn	PROPN
ejpam-259	99	4	,	,	PUNCT
ejpam-259	99	5	x	x	NOUN
ejpam-259	99	6	)	)	PUNCT
ejpam-259	99	7	◦	◦	NOUN
ejpam-259	99	8	(	(	PUNCT
ejpam-259	99	9	x	x	X
ejpam-259	99	10	,	,	PUNCT
ejpam-259	99	11	xm	xm	PROPN
ejpam-259	99	12	)	)	PUNCT
ejpam-259	99	13	=	=	PRON
ejpam-259	100	1	(	(	PUNCT
ejpam-259	100	2	xn	xn	PROPN
ejpam-259	100	3	,	,	PUNCT
ejpam-259	100	4	xm	xm	NUM
ejpam-259	100	5	)	)	PUNCT
ejpam-259	100	6	∈	∈	PROPN
ejpam-259	100	7	u	u	NOUN
ejpam-259	100	8	◦	◦	NOUN
ejpam-259	100	9	u	u	X
ejpam-259	100	10	⊂	⊂	X
ejpam-259	100	11	v	v	NOUN
ejpam-259	100	12	for	for	ADP
ejpam-259	100	13	all	all	DET
ejpam-259	100	14	n	n	CCONJ
ejpam-259	100	15	,	,	PUNCT
ejpam-259	100	16	m	m	VERB
ejpam-259	100	17	>	>	X
ejpam-259	100	18	k	k	NOUN
ejpam-259	100	19	,	,	PUNCT
ejpam-259	100	20	and	and	CCONJ
ejpam-259	100	21	(	(	PUNCT
ejpam-259	100	22	xn	xn	X
ejpam-259	100	23	)	)	PUNCT
ejpam-259	100	24	is	be	AUX
ejpam-259	100	25	cauchy	cauchy	PROPN
ejpam-259	100	26	.	.	PUNCT
ejpam-259	101	1	(	(	PUNCT
ejpam-259	101	2	ii	ii	NOUN
ejpam-259	101	3	)	)	PUNCT
ejpam-259	101	4	let	let	VERB
ejpam-259	101	5	xn	xn	PROPN
ejpam-259	101	6	be	be	AUX
ejpam-259	101	7	cauchy	cauchy	NOUN
ejpam-259	101	8	,	,	PUNCT
ejpam-259	101	9	and	and	CCONJ
ejpam-259	101	10	v	v	ADP
ejpam-259	101	11	∈	∈	PROPN
ejpam-259	101	12	γ	γ	X
ejpam-259	101	13	.	.	PUNCT
ejpam-259	101	14	then	then	ADV
ejpam-259	101	15	there	there	PRON
ejpam-259	101	16	exists	exist	VERB
ejpam-259	101	17	k	k	X
ejpam-259	101	18	such	such	ADJ
ejpam-259	101	19	that	that	PRON
ejpam-259	101	20	(	(	PUNCT
ejpam-259	101	21	xn	xn	PROPN
ejpam-259	101	22	,	,	PUNCT
ejpam-259	101	23	xm	xm	NUM
ejpam-259	101	24	)	)	PUNCT
ejpam-259	101	25	∈	∈	PROPN
ejpam-259	101	26	v	v	NOUN
ejpam-259	101	27	for	for	ADP
ejpam-259	101	28	every	every	DET
ejpam-259	101	29	n	n	CCONJ
ejpam-259	101	30	,	,	PUNCT
ejpam-259	101	31	m	m	VERB
ejpam-259	101	32	≥	≥	PROPN
ejpam-259	101	33	k.	k.	INTJ
ejpam-259	101	34	let	let	VERB
ejpam-259	101	35	u	u	PRON
ejpam-259	101	36	∈	∈	PROPN
ejpam-259	101	37	γ	γ	NOUN
ejpam-259	101	38	be	be	AUX
ejpam-259	101	39	such	such	ADJ
ejpam-259	101	40	that	that	SCONJ
ejpam-259	101	41	�	�	PROPN
ejpam-259	101	42	�	�	PROPN
ejpam-259	101	43	xk	xk	PROPN
ejpam-259	101	44	,	,	PUNCT
ejpam-259	101	45	x1	x1	PROPN
ejpam-259	101	46	�	�	PROPN
ejpam-259	101	47	,	,	PUNCT
ejpam-259	101	48	�	�	PROPN
ejpam-259	101	49	xk	xk	PROPN
ejpam-259	101	50	,	,	PUNCT
ejpam-259	101	51	x2	x2	PROPN
ejpam-259	101	52	�	�	PROPN
ejpam-259	101	53	,	,	PUNCT
ejpam-259	101	54	...	...	PUNCT
ejpam-259	101	55	,	,	PUNCT
ejpam-259	101	56	�	�	PROPN
ejpam-259	101	57	xk	xk	PROPN
ejpam-259	101	58	,	,	PUNCT
ejpam-259	101	59	xk−1	xk−1	PROPN
ejpam-259	101	60	�	�	PROPN
ejpam-259	101	61	⊆	⊆	NUM
ejpam-259	101	62	u	u	NOUN
ejpam-259	101	63	.	.	PUNCT
ejpam-259	102	1	then	then	ADV
ejpam-259	102	2	a.	a.	PROPN
ejpam-259	102	3	tallafha	tallafha	PROPN
ejpam-259	102	4	and	and	CCONJ
ejpam-259	102	5	r.	r.	PROPN
ejpam-259	102	6	khalil	khalil	PROPN
ejpam-259	102	7	/	/	SYM
ejpam-259	102	8	eur	eur	PROPN
ejpam-259	102	9	.	.	PUNCT
ejpam-259	103	1	j.	j.	PROPN
ejpam-259	103	2	pure	pure	PROPN
ejpam-259	103	3	appl	appl	PROPN
ejpam-259	103	4	.	.	PROPN
ejpam-259	103	5	math	math	PROPN
ejpam-259	103	6	,	,	PUNCT
ejpam-259	103	7	2	2	NUM
ejpam-259	103	8	(	(	PUNCT
ejpam-259	103	9	2009	2009	NUM
ejpam-259	103	10	)	)	PUNCT
ejpam-259	103	11	,	,	PUNCT
ejpam-259	103	12	(	(	PUNCT
ejpam-259	103	13	231	231	NUM
ejpam-259	103	14	-	-	SYM
ejpam-259	103	15	238	238	NUM
ejpam-259	103	16	)	)	PUNCT
ejpam-259	103	17	235	235	NUM
ejpam-259	103	18	�	�	PROPN
ejpam-259	103	19	x1	x1	PROPN
ejpam-259	103	20	,	,	PUNCT
ejpam-259	103	21	x2	x2	PROPN
ejpam-259	103	22	,	,	PUNCT
ejpam-259	103	23	...	...	PUNCT
ejpam-259	104	1	⊆	⊆	NUM
ejpam-259	104	2	b	b	PROPN
ejpam-259	104	3	�	�	PROPN
ejpam-259	104	4	xk	xk	PROPN
ejpam-259	104	5	,	,	PUNCT
ejpam-259	104	6	w	w	PROPN
ejpam-259	104	7	�	�	PROPN
ejpam-259	104	8	,	,	PUNCT
ejpam-259	104	9	where	where	SCONJ
ejpam-259	104	10	w	w	NOUN
ejpam-259	104	11	=	=	SYM
ejpam-259	104	12	u	u	NOUN
ejpam-259	104	13	∪v	∪v	NOUN
ejpam-259	104	14	.	.	PUNCT
ejpam-259	104	15	lemma	lemma	PROPN
ejpam-259	104	16	13.1	13.1	NUM
ejpam-259	104	17	.	.	PUNCT
ejpam-259	105	1	let	let	AUX
ejpam-259	105	2	(	(	PUNCT
ejpam-259	105	3	xn	xn	X
ejpam-259	105	4	)	)	PUNCT
ejpam-259	105	5	be	be	VERB
ejpam-259	105	6	a	a	DET
ejpam-259	105	7	sequence	sequence	NOUN
ejpam-259	105	8	in	in	ADP
ejpam-259	105	9	x	x	X
ejpam-259	105	10	.	.	PUNCT
ejpam-259	106	1	if	if	SCONJ
ejpam-259	106	2	(	(	PUNCT
ejpam-259	106	3	xn	xn	X
ejpam-259	106	4	)	)	PUNCT
ejpam-259	106	5	converges	converge	VERB
ejpam-259	106	6	then	then	ADV
ejpam-259	106	7	the	the	DET
ejpam-259	106	8	limit	limit	NOUN
ejpam-259	106	9	is	be	AUX
ejpam-259	106	10	unique	unique	ADJ
ejpam-259	106	11	.	.	PUNCT
ejpam-259	107	1	proof	proof	NOUN
ejpam-259	107	2	.	.	PUNCT
ejpam-259	108	1	if	if	SCONJ
ejpam-259	108	2	possible	possible	ADJ
ejpam-259	108	3	assume	assume	VERB
ejpam-259	108	4	that	that	SCONJ
ejpam-259	108	5	xn	xn	PROPN
ejpam-259	108	6	→	→	SYM
ejpam-259	108	7	x	x	X
ejpam-259	108	8	and	and	CCONJ
ejpam-259	108	9	xn	xn	PROPN
ejpam-259	108	10	→	→	SYM
ejpam-259	108	11	y.	y.	NOUN
ejpam-259	108	12	let	let	VERB
ejpam-259	108	13	v	v	ADP
ejpam-259	108	14	∈	∈	PROPN
ejpam-259	108	15	γ	γ	NOUN
ejpam-259	108	16	be	be	AUX
ejpam-259	108	17	arbitrary	arbitrary	ADJ
ejpam-259	108	18	.	.	PUNCT
ejpam-259	109	1	condition	condition	NOUN
ejpam-259	109	2	(	(	PUNCT
ejpam-259	109	3	iii	iii	NOUN
ejpam-259	109	4	)	)	PUNCT
ejpam-259	109	5	of	of	ADP
ejpam-259	109	6	uniform	uniform	ADJ
ejpam-259	109	7	spaces	space	NOUN
ejpam-259	109	8	implies	imply	VERB
ejpam-259	109	9	the	the	DET
ejpam-259	109	10	existence	existence	NOUN
ejpam-259	109	11	of	of	ADP
ejpam-259	109	12	some	some	DET
ejpam-259	109	13	w	w	NOUN
ejpam-259	109	14	∈	∈	PROPN
ejpam-259	109	15	γ	γ	NOUN
ejpam-259	109	16	such	such	ADJ
ejpam-259	109	17	that	that	DET
ejpam-259	109	18	w	w	PROPN
ejpam-259	109	19	◦	◦	PROPN
ejpam-259	109	20	w	w	NOUN
ejpam-259	109	21	⊂	⊂	PROPN
ejpam-259	109	22	v.	v.	CCONJ
ejpam-259	109	23	from	from	ADP
ejpam-259	109	24	the	the	DET
ejpam-259	109	25	definition	definition	NOUN
ejpam-259	109	26	of	of	ADP
ejpam-259	109	27	convergence	convergence	NOUN
ejpam-259	109	28	,	,	PUNCT
ejpam-259	109	29	there	there	PRON
ejpam-259	109	30	exists	exist	VERB
ejpam-259	109	31	n	n	CCONJ
ejpam-259	109	32	◦	◦	VERB
ejpam-259	109	33	such	such	ADJ
ejpam-259	109	34	that	that	SCONJ
ejpam-259	109	35	(	(	PUNCT
ejpam-259	109	36	x	x	X
ejpam-259	109	37	,	,	PUNCT
ejpam-259	109	38	xn	xn	PROPN
ejpam-259	109	39	)	)	PUNCT
ejpam-259	109	40	and	and	CCONJ
ejpam-259	109	41	(	(	PUNCT
ejpam-259	109	42	y	y	PROPN
ejpam-259	109	43	,	,	PUNCT
ejpam-259	109	44	xn	xn	PROPN
ejpam-259	109	45	)	)	PUNCT
ejpam-259	109	46	are	be	AUX
ejpam-259	109	47	in	in	ADP
ejpam-259	109	48	w.	w.	NOUN
ejpam-259	109	49	hence	hence	ADV
ejpam-259	109	50	(	(	PUNCT
ejpam-259	109	51	x	x	X
ejpam-259	109	52	,	,	PUNCT
ejpam-259	109	53	y	y	NOUN
ejpam-259	109	54	)	)	PUNCT
ejpam-259	109	55	∈w	∈w	VERB
ejpam-259	109	56	◦	◦	NOUN
ejpam-259	109	57	w	w	NOUN
ejpam-259	109	58	⊂	⊂	PROPN
ejpam-259	109	59	v.	v.	CCONJ
ejpam-259	110	1	thus	thus	ADV
ejpam-259	110	2	,	,	PUNCT
ejpam-259	110	3	since	since	SCONJ
ejpam-259	110	4	v	v	NOUN
ejpam-259	110	5	was	be	AUX
ejpam-259	110	6	arbitrary	arbitrary	ADJ
ejpam-259	110	7	,	,	PUNCT
ejpam-259	110	8	(	(	PUNCT
ejpam-259	110	9	x	x	X
ejpam-259	110	10	,	,	PUNCT
ejpam-259	110	11	y	y	PROPN
ejpam-259	110	12	)	)	PUNCT
ejpam-259	110	13	∈∆	∈∆	NOUN
ejpam-259	110	14	,	,	PUNCT
ejpam-259	110	15	and	and	CCONJ
ejpam-259	111	1	so	so	ADV
ejpam-259	111	2	x	x	X
ejpam-259	111	3	=	=	PUNCT
ejpam-259	111	4	y.	y.	PROPN
ejpam-259	111	5	now	now	ADV
ejpam-259	111	6	,	,	PUNCT
ejpam-259	111	7	a	a	DET
ejpam-259	111	8	set	set	NOUN
ejpam-259	111	9	e	e	NOUN
ejpam-259	111	10	will	will	AUX
ejpam-259	111	11	be	be	AUX
ejpam-259	111	12	called	call	VERB
ejpam-259	111	13	open	open	ADJ
ejpam-259	111	14	if	if	SCONJ
ejpam-259	111	15	for	for	ADP
ejpam-259	111	16	every	every	DET
ejpam-259	111	17	point	point	NOUN
ejpam-259	111	18	x	x	PUNCT
ejpam-259	111	19	in	in	ADP
ejpam-259	111	20	e	e	X
ejpam-259	111	21	there	there	PRON
ejpam-259	111	22	exists	exist	VERB
ejpam-259	111	23	v	v	ADP
ejpam-259	111	24	∈	∈	PROPN
ejpam-259	111	25	γ	γ	NOUN
ejpam-259	111	26	,	,	PUNCT
ejpam-259	111	27	such	such	ADJ
ejpam-259	111	28	that	that	SCONJ
ejpam-259	111	29	b(x	b(x	NOUN
ejpam-259	111	30	,	,	PUNCT
ejpam-259	111	31	v	v	NOUN
ejpam-259	111	32	)	)	PUNCT
ejpam-259	112	1	⊆	⊆	NUM
ejpam-259	112	2	e.	e.	PROPN
ejpam-259	112	3	the	the	DET
ejpam-259	112	4	set	set	PROPN
ejpam-259	112	5	e	e	PROPN
ejpam-259	112	6	is	be	AUX
ejpam-259	112	7	called	call	VERB
ejpam-259	112	8	closed	closed	ADJ
ejpam-259	112	9	if	if	SCONJ
ejpam-259	112	10	e	e	PROPN
ejpam-259	112	11	c	c	PROPN
ejpam-259	112	12	is	be	AUX
ejpam-259	112	13	open	open	ADJ
ejpam-259	112	14	.	.	PUNCT
ejpam-259	113	1	a	a	DET
ejpam-259	113	2	point	point	NOUN
ejpam-259	113	3	x	x	PUNCT
ejpam-259	113	4	is	be	AUX
ejpam-259	113	5	called	call	VERB
ejpam-259	113	6	a	a	DET
ejpam-259	113	7	limit	limit	NOUN
ejpam-259	113	8	point	point	NOUN
ejpam-259	113	9	of	of	ADP
ejpam-259	113	10	e	e	NOUN
ejpam-259	113	11	if	if	SCONJ
ejpam-259	113	12	there	there	PRON
ejpam-259	113	13	is	be	VERB
ejpam-259	113	14	a	a	DET
ejpam-259	113	15	sequence	sequence	NOUN
ejpam-259	113	16	(	(	PUNCT
ejpam-259	113	17	xn	xn	X
ejpam-259	113	18	)	)	PUNCT
ejpam-259	113	19	in	in	ADP
ejpam-259	113	20	e	e	ADP
ejpam-259	113	21	such	such	ADJ
ejpam-259	113	22	that	that	SCONJ
ejpam-259	113	23	xn→	xn→	PROPN
ejpam-259	114	1	x	x	X
ejpam-259	114	2	.	.	PUNCT
ejpam-259	115	1	the	the	DET
ejpam-259	115	2	set	set	NOUN
ejpam-259	115	3	of	of	ADP
ejpam-259	115	4	limit	limit	NOUN
ejpam-259	115	5	points	point	NOUN
ejpam-259	115	6	of	of	ADP
ejpam-259	115	7	the	the	DET
ejpam-259	115	8	set	set	NOUN
ejpam-259	115	9	e	e	NOUN
ejpam-259	115	10	will	will	AUX
ejpam-259	115	11	be	be	AUX
ejpam-259	115	12	denoted	denote	VERB
ejpam-259	115	13	by	by	ADP
ejpam-259	115	14	e	e	NOUN
ejpam-259	115	15	ℓ.	ℓ.	NOUN
ejpam-259	115	16	for	for	ADP
ejpam-259	115	17	any	any	DET
ejpam-259	115	18	set	set	NOUN
ejpam-259	115	19	e	e	NOUN
ejpam-259	115	20	in	in	ADP
ejpam-259	115	21	x	x	SYM
ejpam-259	115	22	,	,	PUNCT
ejpam-259	115	23	we	we	PRON
ejpam-259	115	24	let	let	VERB
ejpam-259	115	25	_	_	PUNCT
ejpam-259	116	1	e	e	NOUN
ejpam-259	117	1	=	=	PUNCT
ejpam-259	117	2	e	e	X
ejpam-259	117	3	∪	∪	VERB
ejpam-259	117	4	e	e	NOUN
ejpam-259	117	5	ℓ.	ℓ.	NOUN
ejpam-259	117	6	the	the	DET
ejpam-259	117	7	proof	proof	NOUN
ejpam-259	117	8	of	of	ADP
ejpam-259	117	9	the	the	DET
ejpam-259	117	10	following	follow	VERB
ejpam-259	117	11	lemma	lemma	PROPN
ejpam-259	117	12	is	be	AUX
ejpam-259	117	13	similar	similar	ADJ
ejpam-259	117	14	to	to	ADP
ejpam-259	117	15	that	that	PRON
ejpam-259	117	16	in	in	ADP
ejpam-259	117	17	metric	metric	ADJ
ejpam-259	117	18	spaces	space	NOUN
ejpam-259	117	19	and	and	CCONJ
ejpam-259	117	20	will	will	AUX
ejpam-259	117	21	be	be	AUX
ejpam-259	117	22	omitted	omit	VERB
ejpam-259	117	23	.	.	PUNCT
ejpam-259	118	1	lemma	lemma	PROPN
ejpam-259	118	2	14.1	14.1	NUM
ejpam-259	118	3	.	.	PUNCT
ejpam-259	119	1	a	a	DET
ejpam-259	119	2	set	set	NOUN
ejpam-259	119	3	e	e	NOUN
ejpam-259	119	4	is	be	AUX
ejpam-259	119	5	closed	close	VERB
ejpam-259	119	6	if	if	SCONJ
ejpam-259	119	7	and	and	CCONJ
ejpam-259	119	8	only	only	ADV
ejpam-259	119	9	if	if	SCONJ
ejpam-259	119	10	e	e	PROPN
ejpam-259	119	11	ℓ	ℓ	PROPN
ejpam-259	119	12	⊑	⊑	PROPN
ejpam-259	119	13	e.	e.	PROPN
ejpam-259	119	14	question	question	PROPN
ejpam-259	119	15	3	3	NUM
ejpam-259	119	16	.	.	PUNCT
ejpam-259	120	1	if	if	SCONJ
ejpam-259	120	2	ρ(x	ρ(x	PROPN
ejpam-259	120	3	,	,	PUNCT
ejpam-259	120	4	e	e	NOUN
ejpam-259	120	5	)	)	PUNCT
ejpam-259	120	6	=	=	SYM
ejpam-259	120	7	∆	∆	X
ejpam-259	120	8	,	,	PUNCT
ejpam-259	120	9	must	must	AUX
ejpam-259	120	10	x	x	SYM
ejpam-259	120	11	∈	∈	PROPN
ejpam-259	120	12	e	e	NOUN
ejpam-259	120	13	ℓ	ℓ	PROPN
ejpam-259	120	14	?	?	PUNCT
ejpam-259	120	15	proposition	proposition	NOUN
ejpam-259	120	16	15.1	15.1	NUM
ejpam-259	120	17	.	.	PUNCT
ejpam-259	121	1	if	if	SCONJ
ejpam-259	121	2	x	x	SYM
ejpam-259	121	3	∈	∈	PROPN
ejpam-259	121	4	e	e	NOUN
ejpam-259	121	5	ℓ	ℓ	PROPN
ejpam-259	121	6	,	,	PUNCT
ejpam-259	121	7	then	then	ADV
ejpam-259	121	8	ρ(x	ρ(x	PROPN
ejpam-259	121	9	,	,	PUNCT
ejpam-259	121	10	e	e	NOUN
ejpam-259	121	11	)	)	PUNCT
ejpam-259	121	12	=	=	PUNCT
ejpam-259	121	13	∆.	∆.	NOUN
ejpam-259	121	14	proof	proof	NOUN
ejpam-259	121	15	.	.	PUNCT
ejpam-259	122	1	let	let	VERB
ejpam-259	122	2	x	x	SYM
ejpam-259	122	3	∈	∈	PROPN
ejpam-259	122	4	e	e	NOUN
ejpam-259	122	5	ℓ.	ℓ.	NOUN
ejpam-259	122	6	then	then	ADV
ejpam-259	122	7	there	there	PRON
ejpam-259	122	8	exists	exist	VERB
ejpam-259	122	9	(	(	PUNCT
ejpam-259	122	10	xn	xn	X
ejpam-259	122	11	)	)	PUNCT
ejpam-259	122	12	in	in	ADP
ejpam-259	122	13	e	e	ADP
ejpam-259	122	14	such	such	ADJ
ejpam-259	122	15	that	that	PRON
ejpam-259	122	16	xn	xn	PUNCT
ejpam-259	123	1	→	→	SYM
ejpam-259	123	2	x	x	X
ejpam-259	123	3	.	.	PUNCT
ejpam-259	124	1	hence	hence	ADV
ejpam-259	124	2	,	,	PUNCT
ejpam-259	124	3	∞	∞	PROPN
ejpam-259	124	4	∩	∩	X
ejpam-259	124	5	n	n	CCONJ
ejpam-259	124	6	=	=	SYM
ejpam-259	124	7	1	1	NUM
ejpam-259	124	8	ρ(x	ρ(x	PROPN
ejpam-259	124	9	,	,	PUNCT
ejpam-259	124	10	xn	xn	PROPN
ejpam-259	124	11	)	)	PUNCT
ejpam-259	124	12	=	=	PUNCT
ejpam-259	125	1	∆.	∆.	NOUN
ejpam-259	125	2	but	but	CCONJ
ejpam-259	125	3	ρ(x	ρ(x	PROPN
ejpam-259	125	4	,	,	PUNCT
ejpam-259	125	5	e	e	X
ejpam-259	125	6	)	)	PUNCT
ejpam-259	126	1	=	=	NOUN
ejpam-259	126	2	∩	∩	NOUN
ejpam-259	126	3	y	y	PROPN
ejpam-259	126	4	∈	∈	PROPN
ejpam-259	126	5	e	e	PROPN
ejpam-259	126	6	ρ(x	ρ(x	PROPN
ejpam-259	126	7	,	,	PUNCT
ejpam-259	126	8	y)⊂	y)⊂	PROPN
ejpam-259	126	9	∞	∞	PROPN
ejpam-259	126	10	∩	∩	NOUN
ejpam-259	126	11	n	n	PROPN
ejpam-259	126	12	=	=	SYM
ejpam-259	126	13	1	1	NUM
ejpam-259	126	14	ρ(x	ρ(x	PROPN
ejpam-259	126	15	,	,	PUNCT
ejpam-259	126	16	xn	xn	PROPN
ejpam-259	126	17	)	)	PUNCT
ejpam-259	126	18	=	=	PUNCT
ejpam-259	127	1	∆.	∆.	NOUN
ejpam-259	127	2	so	so	ADV
ejpam-259	127	3	ρ(x	ρ(x	PROPN
ejpam-259	127	4	,	,	PUNCT
ejpam-259	127	5	e	e	X
ejpam-259	127	6	)	)	PUNCT
ejpam-259	127	7	=	=	PUNCT
ejpam-259	128	1	∆.	∆.	ADP
ejpam-259	128	2	a	a	DET
ejpam-259	128	3	nice	nice	ADJ
ejpam-259	128	4	property	property	NOUN
ejpam-259	128	5	of	of	ADP
ejpam-259	128	6	semi	semi	ADJ
ejpam-259	128	7	-	-	ADJ
ejpam-259	128	8	linear	linear	ADJ
ejpam-259	128	9	uniform	uniform	ADJ
ejpam-259	128	10	spaces	space	NOUN
ejpam-259	128	11	is	be	AUX
ejpam-259	128	12	:	:	PUNCT
ejpam-259	128	13	theorem	theorem	VERB
ejpam-259	128	14	16.1	16.1	NUM
ejpam-259	128	15	.	.	PUNCT
ejpam-259	129	1	open	open	ADJ
ejpam-259	129	2	balls	ball	NOUN
ejpam-259	129	3	separate	separate	ADJ
ejpam-259	129	4	points	point	NOUN
ejpam-259	129	5	in	in	ADP
ejpam-259	129	6	(	(	PUNCT
ejpam-259	129	7	x	x	INTJ
ejpam-259	129	8	,	,	PUNCT
ejpam-259	129	9	γ	γ	NOUN
ejpam-259	129	10	)	)	PUNCT
ejpam-259	129	11	.	.	PUNCT
ejpam-259	130	1	proof	proof	NOUN
ejpam-259	130	2	.	.	PUNCT
ejpam-259	131	1	let	let	VERB
ejpam-259	131	2	x	x	PRON
ejpam-259	131	3	,	,	PUNCT
ejpam-259	131	4	y	y	PROPN
ejpam-259	131	5	be	be	VERB
ejpam-259	131	6	any	any	DET
ejpam-259	131	7	two	two	NUM
ejpam-259	131	8	elements	element	NOUN
ejpam-259	131	9	in	in	ADP
ejpam-259	131	10	(	(	PUNCT
ejpam-259	131	11	x	x	INTJ
ejpam-259	131	12	,	,	PUNCT
ejpam-259	131	13	γ	γ	PROPN
ejpam-259	131	14	)	)	PUNCT
ejpam-259	131	15	such	such	ADJ
ejpam-259	131	16	that	that	SCONJ
ejpam-259	131	17	x	x	PRON
ejpam-259	132	1	6=	6=	ADP
ejpam-259	132	2	y.	y.	NOUN
ejpam-259	132	3	if	if	SCONJ
ejpam-259	132	4	possible	possible	ADJ
ejpam-259	132	5	assume	assume	VERB
ejpam-259	132	6	that	that	SCONJ
ejpam-259	132	7	b(x	b(x	NOUN
ejpam-259	132	8	,	,	PUNCT
ejpam-259	132	9	u	u	NOUN
ejpam-259	132	10	)	)	PUNCT
ejpam-259	132	11	∩	∩	PROPN
ejpam-259	132	12	b(y	b(y	PROPN
ejpam-259	132	13	,	,	PUNCT
ejpam-259	132	14	u	u	NOUN
ejpam-259	132	15	)	)	PUNCT
ejpam-259	132	16	6=	6=	ADP
ejpam-259	132	17	φ	φ	PROPN
ejpam-259	132	18	for	for	ADP
ejpam-259	132	19	all	all	DET
ejpam-259	132	20	u	u	PROPN
ejpam-259	132	21	∈	∈	PROPN
ejpam-259	132	22	γ	γ	X
ejpam-259	132	23	.	.	PUNCT
ejpam-259	133	1	let	let	VERB
ejpam-259	133	2	v	v	PART
ejpam-259	133	3	be	be	AUX
ejpam-259	133	4	any	any	DET
ejpam-259	133	5	element	element	NOUN
ejpam-259	133	6	in	in	ADP
ejpam-259	133	7	γ	γ	PROPN
ejpam-259	133	8	.	.	PUNCT
ejpam-259	134	1	since	since	SCONJ
ejpam-259	134	2	x	x	PRON
ejpam-259	134	3	is	be	AUX
ejpam-259	134	4	a	a	DET
ejpam-259	134	5	uniform	uniform	ADJ
ejpam-259	134	6	space	space	NOUN
ejpam-259	134	7	,	,	PUNCT
ejpam-259	134	8	then	then	ADV
ejpam-259	134	9	there	there	PRON
ejpam-259	134	10	exists	exist	VERB
ejpam-259	134	11	w	w	PROPN
ejpam-259	134	12	∈	∈	PROPN
ejpam-259	134	13	γ	γ	NOUN
ejpam-259	134	14	such	such	ADJ
ejpam-259	134	15	that	that	PRON
ejpam-259	134	16	w	w	PROPN
ejpam-259	134	17	◦	◦	PROPN
ejpam-259	134	18	w	w	NOUN
ejpam-259	134	19	⊂	⊂	PROPN
ejpam-259	134	20	v.	v.	ADP
ejpam-259	134	21	by	by	ADP
ejpam-259	134	22	assumption	assumption	NOUN
ejpam-259	134	23	b(x	b(x	NOUN
ejpam-259	134	24	,	,	PUNCT
ejpam-259	134	25	w	w	NOUN
ejpam-259	134	26	)	)	PUNCT
ejpam-259	134	27	∩	∩	ADJ
ejpam-259	134	28	b(y	b(y	PROPN
ejpam-259	134	29	,	,	PUNCT
ejpam-259	134	30	w	w	NOUN
ejpam-259	134	31	)	)	PUNCT
ejpam-259	134	32	6=	6=	ADP
ejpam-259	135	1	φ	φ	PROPN
ejpam-259	135	2	.	.	PUNCT
ejpam-259	136	1	hence	hence	ADV
ejpam-259	136	2	,	,	PUNCT
ejpam-259	136	3	there	there	PRON
ejpam-259	136	4	exists	exist	VERB
ejpam-259	136	5	z	z	NOUN
ejpam-259	136	6	∈	∈	PROPN
ejpam-259	136	7	x	x	PUNCT
ejpam-259	136	8	such	such	ADJ
ejpam-259	136	9	that	that	SCONJ
ejpam-259	136	10	(	(	PUNCT
ejpam-259	136	11	x	x	X
ejpam-259	136	12	,	,	PUNCT
ejpam-259	136	13	z	z	NOUN
ejpam-259	136	14	)	)	PUNCT
ejpam-259	136	15	,	,	PUNCT
ejpam-259	136	16	(	(	PUNCT
ejpam-259	136	17	z	z	X
ejpam-259	136	18	,	,	PUNCT
ejpam-259	136	19	y	y	NOUN
ejpam-259	136	20	)	)	PUNCT
ejpam-259	136	21	∈	∈	PROPN
ejpam-259	136	22	w.	w.	PROPN
ejpam-259	136	23	consequently	consequently	ADV
ejpam-259	136	24	,	,	PUNCT
ejpam-259	136	25	(	(	PUNCT
ejpam-259	136	26	x	x	X
ejpam-259	136	27	,	,	PUNCT
ejpam-259	136	28	y	y	PROPN
ejpam-259	136	29	)	)	PUNCT
ejpam-259	136	30	∈	∈	PROPN
ejpam-259	136	31	w	w	PROPN
ejpam-259	136	32	◦	◦	PROPN
ejpam-259	136	33	w	w	NOUN
ejpam-259	136	34	⊂	⊂	PROPN
ejpam-259	136	35	v.	v.	CCONJ
ejpam-259	136	36	so	so	ADV
ejpam-259	136	37	(	(	PUNCT
ejpam-259	136	38	x	x	X
ejpam-259	136	39	,	,	PUNCT
ejpam-259	136	40	y	y	PROPN
ejpam-259	136	41	)	)	PUNCT
ejpam-259	136	42	∈	∈	NOUN
ejpam-259	136	43	v	v	NOUN
ejpam-259	136	44	for	for	ADP
ejpam-259	136	45	all	all	PRON
ejpam-259	136	46	v	v	ADP
ejpam-259	136	47	∈	∈	PROPN
ejpam-259	136	48	γ	γ	X
ejpam-259	136	49	.	.	PUNCT
ejpam-259	137	1	but	but	CCONJ
ejpam-259	137	2	this	this	PRON
ejpam-259	137	3	implies	imply	VERB
ejpam-259	137	4	that	that	SCONJ
ejpam-259	137	5	(	(	PUNCT
ejpam-259	137	6	x	x	X
ejpam-259	137	7	,	,	PUNCT
ejpam-259	137	8	y	y	PROPN
ejpam-259	137	9	)	)	PUNCT
ejpam-259	137	10	∈	∈	PROPN
ejpam-259	137	11	∆	∆	PROPN
ejpam-259	137	12	,	,	PUNCT
ejpam-259	137	13	which	which	PRON
ejpam-259	137	14	in	in	ADP
ejpam-259	137	15	turn	turn	NOUN
ejpam-259	137	16	implies	imply	VERB
ejpam-259	137	17	that	that	SCONJ
ejpam-259	137	18	x	x	X
ejpam-259	137	19	=	=	PUNCT
ejpam-259	137	20	y.	y.	NOUN
ejpam-259	137	21	this	this	PRON
ejpam-259	137	22	contradicts	contradict	VERB
ejpam-259	137	23	the	the	DET
ejpam-259	137	24	assumption	assumption	NOUN
ejpam-259	137	25	.	.	PUNCT
ejpam-259	138	1	so	so	ADV
ejpam-259	138	2	there	there	PRON
ejpam-259	138	3	must	must	AUX
ejpam-259	138	4	exist	exist	VERB
ejpam-259	138	5	a.	a.	NOUN
ejpam-259	138	6	tallafha	tallafha	NOUN
ejpam-259	138	7	and	and	CCONJ
ejpam-259	138	8	r.	r.	PROPN
ejpam-259	138	9	khalil	khalil	PROPN
ejpam-259	138	10	/	/	SYM
ejpam-259	138	11	eur	eur	PROPN
ejpam-259	138	12	.	.	PUNCT
ejpam-259	139	1	j.	j.	PROPN
ejpam-259	139	2	pure	pure	PROPN
ejpam-259	139	3	appl	appl	PROPN
ejpam-259	139	4	.	.	PROPN
ejpam-259	139	5	math	math	PROPN
ejpam-259	139	6	,	,	PUNCT
ejpam-259	139	7	2	2	NUM
ejpam-259	139	8	(	(	PUNCT
ejpam-259	139	9	2009	2009	NUM
ejpam-259	139	10	)	)	PUNCT
ejpam-259	139	11	,	,	PUNCT
ejpam-259	139	12	(	(	PUNCT
ejpam-259	139	13	231	231	NUM
ejpam-259	139	14	-	-	SYM
ejpam-259	139	15	238	238	NUM
ejpam-259	139	16	)	)	PUNCT
ejpam-259	139	17	236	236	NUM
ejpam-259	139	18	some	some	DET
ejpam-259	139	19	w	w	NOUN
ejpam-259	139	20	∈	∈	PROPN
ejpam-259	139	21	γ	γ	NOUN
ejpam-259	139	22	such	such	ADJ
ejpam-259	139	23	that	that	DET
ejpam-259	139	24	b(x	b(x	NOUN
ejpam-259	139	25	,	,	PUNCT
ejpam-259	139	26	w	w	NOUN
ejpam-259	139	27	)	)	PUNCT
ejpam-259	139	28	∩	∩	ADJ
ejpam-259	139	29	b(y	b(y	PROPN
ejpam-259	139	30	,	,	PUNCT
ejpam-259	139	31	w	w	NOUN
ejpam-259	139	32	)	)	PUNCT
ejpam-259	139	33	=	=	SYM
ejpam-259	140	1	φ	φ	PROPN
ejpam-259	140	2	.	.	PUNCT
ejpam-259	141	1	now	now	ADV
ejpam-259	141	2	,	,	PUNCT
ejpam-259	141	3	let	let	VERB
ejpam-259	141	4	us	we	PRON
ejpam-259	141	5	define	define	VERB
ejpam-259	141	6	a	a	DET
ejpam-259	141	7	set	set	NOUN
ejpam-259	141	8	e	e	X
ejpam-259	141	9	⊂	⊂	PROPN
ejpam-259	141	10	(	(	PUNCT
ejpam-259	141	11	x	x	INTJ
ejpam-259	141	12	,	,	PUNCT
ejpam-259	141	13	γ	γ	PROPN
ejpam-259	141	14	)	)	PUNCT
ejpam-259	141	15	to	to	PART
ejpam-259	141	16	be	be	AUX
ejpam-259	141	17	compact	compact	ADJ
ejpam-259	141	18	,	,	PUNCT
ejpam-259	141	19	if	if	SCONJ
ejpam-259	141	20	every	every	DET
ejpam-259	141	21	sequence	sequence	NOUN
ejpam-259	141	22	in	in	ADP
ejpam-259	141	23	e	e	PROPN
ejpam-259	141	24	has	have	VERB
ejpam-259	141	25	a	a	DET
ejpam-259	141	26	convergent	convergent	ADJ
ejpam-259	141	27	subsequence	subsequence	NOUN
ejpam-259	141	28	in	in	ADP
ejpam-259	141	29	e.	e.	PROPN
ejpam-259	141	30	clearly	clearly	ADV
ejpam-259	141	31	,	,	PUNCT
ejpam-259	141	32	every	every	DET
ejpam-259	141	33	finite	finite	NOUN
ejpam-259	141	34	set	set	NOUN
ejpam-259	141	35	is	be	AUX
ejpam-259	141	36	compact	compact	ADJ
ejpam-259	141	37	,	,	PUNCT
ejpam-259	141	38	and	and	CCONJ
ejpam-259	141	39	every	every	DET
ejpam-259	141	40	compact	compact	ADJ
ejpam-259	141	41	set	set	NOUN
ejpam-259	141	42	is	be	AUX
ejpam-259	141	43	closed	closed	ADJ
ejpam-259	141	44	.	.	PUNCT
ejpam-259	142	1	3	3	X
ejpam-259	142	2	.	.	X
ejpam-259	142	3	proximinality	proximinality	NOUN
ejpam-259	142	4	in	in	ADP
ejpam-259	142	5	semi	semi	ADJ
ejpam-259	142	6	-	-	ADJ
ejpam-259	142	7	linear	linear	ADJ
ejpam-259	142	8	uniform	uniform	NOUN
ejpam-259	142	9	spaces	space	VERB
ejpam-259	142	10	what	what	PRON
ejpam-259	142	11	is	be	AUX
ejpam-259	142	12	nice	nice	ADJ
ejpam-259	142	13	about	about	ADP
ejpam-259	142	14	semi	semi	ADJ
ejpam-259	142	15	-	-	ADJ
ejpam-259	142	16	linear	linear	ADJ
ejpam-259	142	17	uniform	uniform	ADJ
ejpam-259	142	18	spaces	space	NOUN
ejpam-259	142	19	is	be	AUX
ejpam-259	142	20	that	that	DET
ejpam-259	142	21	theory	theory	NOUN
ejpam-259	142	22	of	of	ADP
ejpam-259	142	23	best	good	ADJ
ejpam-259	142	24	approximation	approximation	NOUN
ejpam-259	142	25	can	can	AUX
ejpam-259	142	26	be	be	AUX
ejpam-259	142	27	studied	study	VERB
ejpam-259	142	28	in	in	ADP
ejpam-259	142	29	such	such	ADJ
ejpam-259	142	30	spaces	space	NOUN
ejpam-259	142	31	without	without	ADP
ejpam-259	142	32	tools	tool	NOUN
ejpam-259	142	33	that	that	PRON
ejpam-259	142	34	metric	metric	ADJ
ejpam-259	142	35	structure	structure	NOUN
ejpam-259	142	36	usually	usually	ADV
ejpam-259	142	37	offers	offer	VERB
ejpam-259	142	38	.	.	PUNCT
ejpam-259	143	1	in	in	ADP
ejpam-259	143	2	this	this	DET
ejpam-259	143	3	section	section	NOUN
ejpam-259	143	4	we	we	PRON
ejpam-259	143	5	present	present	VERB
ejpam-259	143	6	some	some	DET
ejpam-259	143	7	results	result	NOUN
ejpam-259	143	8	in	in	ADP
ejpam-259	143	9	approximation	approximation	NOUN
ejpam-259	143	10	theory	theory	NOUN
ejpam-259	143	11	in	in	ADP
ejpam-259	143	12	semi	semi	ADJ
ejpam-259	143	13	-	-	ADJ
ejpam-259	143	14	linear	linear	ADJ
ejpam-259	143	15	uniform	uniform	ADJ
ejpam-259	143	16	spaces	space	NOUN
ejpam-259	143	17	.	.	PUNCT
ejpam-259	144	1	definition	definition	NOUN
ejpam-259	144	2	1.2	1.2	NUM
ejpam-259	144	3	.	.	PUNCT
ejpam-259	145	1	let	let	AUX
ejpam-259	145	2	(	(	PUNCT
ejpam-259	145	3	x	x	X
ejpam-259	145	4	,	,	PUNCT
ejpam-259	145	5	γ	γ	NOUN
ejpam-259	145	6	)	)	PUNCT
ejpam-259	145	7	be	be	VERB
ejpam-259	145	8	semi	semi	ADJ
ejpam-259	145	9	-	-	ADJ
ejpam-259	145	10	linear	linear	ADJ
ejpam-259	145	11	uniform	uniform	ADJ
ejpam-259	145	12	space	space	NOUN
ejpam-259	145	13	,	,	PUNCT
ejpam-259	145	14	and	and	CCONJ
ejpam-259	145	15	e	e	X
ejpam-259	145	16	⊂	⊂	PROPN
ejpam-259	145	17	x	x	X
ejpam-259	145	18	.	.	PUNCT
ejpam-259	146	1	the	the	DET
ejpam-259	146	2	set	set	PROPN
ejpam-259	146	3	e	e	NOUN
ejpam-259	146	4	is	be	AUX
ejpam-259	146	5	called	call	VERB
ejpam-259	146	6	proximinal	proximinal	ADJ
ejpam-259	146	7	if	if	SCONJ
ejpam-259	146	8	for	for	ADP
ejpam-259	146	9	any	any	DET
ejpam-259	146	10	x	x	SYM
ejpam-259	146	11	∈	∈	PROPN
ejpam-259	146	12	x	x	X
ejpam-259	146	13	,	,	PUNCT
ejpam-259	146	14	there	there	PRON
ejpam-259	146	15	exists	exist	VERB
ejpam-259	146	16	some	some	DET
ejpam-259	146	17	e	e	NOUN
ejpam-259	146	18	∈	∈	NOUN
ejpam-259	146	19	e	e	NOUN
ejpam-259	146	20	such	such	ADJ
ejpam-259	146	21	that	that	PRON
ejpam-259	146	22	ρ(x	ρ(x	NOUN
ejpam-259	146	23	,	,	PUNCT
ejpam-259	146	24	e	e	NOUN
ejpam-259	146	25	)	)	PUNCT
ejpam-259	147	1	=	=	SYM
ejpam-259	147	2	ρ(x	ρ(x	PROPN
ejpam-259	147	3	,	,	PUNCT
ejpam-259	147	4	e	e	NOUN
ejpam-259	147	5	)	)	PUNCT
ejpam-259	147	6	.	.	PUNCT
ejpam-259	148	1	proposition	proposition	NOUN
ejpam-259	148	2	2.2	2.2	NUM
ejpam-259	148	3	.	.	PUNCT
ejpam-259	149	1	if	if	SCONJ
ejpam-259	149	2	e	e	PROPN
ejpam-259	149	3	⊂	⊂	PROPN
ejpam-259	149	4	x	x	X
ejpam-259	149	5	is	be	AUX
ejpam-259	149	6	proximinal	proximinal	ADJ
ejpam-259	149	7	,	,	PUNCT
ejpam-259	149	8	then	then	ADV
ejpam-259	149	9	e	e	PROPN
ejpam-259	149	10	is	be	AUX
ejpam-259	149	11	closed	closed	ADJ
ejpam-259	149	12	.	.	PUNCT
ejpam-259	150	1	proof	proof	NOUN
ejpam-259	150	2	.	.	PUNCT
ejpam-259	151	1	let	let	VERB
ejpam-259	151	2	x	x	SYM
ejpam-259	151	3	∈	∈	PROPN
ejpam-259	151	4	e	e	NOUN
ejpam-259	151	5	ℓ.	ℓ.	NOUN
ejpam-259	151	6	by	by	ADP
ejpam-259	151	7	proposition	proposition	NOUN
ejpam-259	151	8	11.1	11.1	NUM
ejpam-259	151	9	,	,	PUNCT
ejpam-259	151	10	ρ(x	ρ(x	PROPN
ejpam-259	151	11	,	,	PUNCT
ejpam-259	151	12	e	e	NOUN
ejpam-259	151	13	)	)	PUNCT
ejpam-259	151	14	=	=	SYM
ejpam-259	151	15	∆	∆	NOUN
ejpam-259	151	16	,	,	PUNCT
ejpam-259	151	17	then	then	ADV
ejpam-259	151	18	by	by	ADP
ejpam-259	151	19	assumption	assumption	NOUN
ejpam-259	151	20	of	of	ADP
ejpam-259	151	21	proximinality	proximinality	NOUN
ejpam-259	151	22	,	,	PUNCT
ejpam-259	151	23	there	there	PRON
ejpam-259	151	24	exists	exist	VERB
ejpam-259	151	25	some	some	DET
ejpam-259	151	26	e	e	NOUN
ejpam-259	151	27	∈	∈	NOUN
ejpam-259	151	28	e	e	NOUN
ejpam-259	151	29	such	such	ADJ
ejpam-259	151	30	that	that	PRON
ejpam-259	151	31	ρ(x	ρ(x	NOUN
ejpam-259	151	32	,	,	PUNCT
ejpam-259	151	33	e	e	NOUN
ejpam-259	151	34	)	)	PUNCT
ejpam-259	151	35	=	=	SYM
ejpam-259	152	1	ρ(x	ρ(x	PROPN
ejpam-259	152	2	,	,	PUNCT
ejpam-259	152	3	e	e	NOUN
ejpam-259	152	4	)	)	PUNCT
ejpam-259	152	5	=	=	PUNCT
ejpam-259	153	1	∆.	∆.	NOUN
ejpam-259	153	2	so	so	ADV
ejpam-259	153	3	x	x	PRON
ejpam-259	153	4	must	must	AUX
ejpam-259	153	5	equal	equal	VERB
ejpam-259	153	6	e	e	NOUN
ejpam-259	153	7	and	and	CCONJ
ejpam-259	153	8	e	e	PROPN
ejpam-259	153	9	is	be	AUX
ejpam-259	153	10	closed	closed	ADJ
ejpam-259	153	11	.	.	PUNCT
ejpam-259	154	1	compact	compact	ADJ
ejpam-259	154	2	sets	set	NOUN
ejpam-259	154	3	are	be	AUX
ejpam-259	154	4	nice	nice	ADJ
ejpam-259	154	5	proximinal	proximinal	ADJ
ejpam-259	154	6	sets	set	NOUN
ejpam-259	154	7	in	in	ADP
ejpam-259	154	8	normed	normed	ADJ
ejpam-259	154	9	spaces	space	NOUN
ejpam-259	154	10	[	[	X
ejpam-259	154	11	4	4	NUM
ejpam-259	154	12	]	]	PUNCT
ejpam-259	154	13	.	.	PUNCT
ejpam-259	155	1	but	but	CCONJ
ejpam-259	155	2	what	what	PRON
ejpam-259	155	3	about	about	ADP
ejpam-259	155	4	proximinality	proximinality	NOUN
ejpam-259	155	5	of	of	ADP
ejpam-259	155	6	compact	compact	ADJ
ejpam-259	155	7	sets	set	NOUN
ejpam-259	155	8	in	in	ADP
ejpam-259	155	9	semi	semi	ADJ
ejpam-259	155	10	-	-	ADJ
ejpam-259	155	11	linear	linear	ADJ
ejpam-259	155	12	spaces	space	NOUN
ejpam-259	155	13	.	.	PUNCT
ejpam-259	156	1	question	question	NOUN
ejpam-259	156	2	4	4	NUM
ejpam-259	156	3	.	.	PUNCT
ejpam-259	157	1	if	if	SCONJ
ejpam-259	157	2	e	e	NOUN
ejpam-259	157	3	is	be	AUX
ejpam-259	157	4	compact	compact	ADJ
ejpam-259	157	5	,	,	PUNCT
ejpam-259	157	6	must	must	AUX
ejpam-259	157	7	e	e	NOUN
ejpam-259	157	8	be	be	AUX
ejpam-259	157	9	proximinal	proximinal	ADJ
ejpam-259	157	10	?	?	PUNCT
ejpam-259	157	11	.	.	PUNCT
ejpam-259	158	1	every	every	DET
ejpam-259	158	2	finite	finite	NOUN
ejpam-259	158	3	set	set	NOUN
ejpam-259	158	4	is	be	AUX
ejpam-259	158	5	compact	compact	ADJ
ejpam-259	158	6	,	,	PUNCT
ejpam-259	158	7	so	so	ADV
ejpam-259	158	8	the	the	DET
ejpam-259	158	9	following	follow	VERB
ejpam-259	158	10	is	be	AUX
ejpam-259	158	11	a	a	DET
ejpam-259	158	12	partial	partial	ADJ
ejpam-259	158	13	answer	answer	NOUN
ejpam-259	158	14	to	to	ADP
ejpam-259	158	15	our	our	PRON
ejpam-259	158	16	question	question	NOUN
ejpam-259	158	17	.	.	PUNCT
ejpam-259	159	1	theorem	theorem	VERB
ejpam-259	159	2	2.3	2.3	NUM
ejpam-259	159	3	.	.	PUNCT
ejpam-259	160	1	let	let	AUX
ejpam-259	160	2	(	(	PUNCT
ejpam-259	160	3	x	x	X
ejpam-259	160	4	,	,	PUNCT
ejpam-259	160	5	γ	γ	PROPN
ejpam-259	160	6	)	)	PUNCT
ejpam-259	160	7	be	be	VERB
ejpam-259	160	8	a	a	DET
ejpam-259	160	9	semi	semi	ADJ
ejpam-259	160	10	-	-	ADJ
ejpam-259	160	11	linear	linear	ADJ
ejpam-259	160	12	uniform	uniform	ADJ
ejpam-259	160	13	space	space	NOUN
ejpam-259	160	14	.	.	PUNCT
ejpam-259	161	1	then	then	ADV
ejpam-259	161	2	every	every	DET
ejpam-259	161	3	finite	finite	NOUN
ejpam-259	161	4	set	set	NOUN
ejpam-259	161	5	is	be	AUX
ejpam-259	161	6	proximinal	proximinal	ADJ
ejpam-259	161	7	.	.	PUNCT
ejpam-259	162	1	proof	proof	NOUN
ejpam-259	162	2	.	.	PUNCT
ejpam-259	163	1	since	since	SCONJ
ejpam-259	163	2	e	e	PROPN
ejpam-259	163	3	is	be	AUX
ejpam-259	163	4	finite	finite	ADJ
ejpam-259	163	5	,	,	PUNCT
ejpam-259	163	6	then	then	ADV
ejpam-259	163	7	e	e	X
ejpam-259	163	8	=	=	PRON
ejpam-259	163	9	{	{	PUNCT
ejpam-259	163	10	e1	e1	PROPN
ejpam-259	163	11	,	,	PUNCT
ejpam-259	163	12	e2	e2	PROPN
ejpam-259	163	13	,	,	PUNCT
ejpam-259	163	14	...	...	PUNCT
ejpam-259	163	15	en}.let	en}.let	NOUN
ejpam-259	163	16	x	x	SYM
ejpam-259	163	17	∈	∈	PROPN
ejpam-259	163	18	x	x	X
ejpam-259	163	19	.	.	PUNCT
ejpam-259	164	1	then	then	ADV
ejpam-259	164	2	ρ(x	ρ(x	NOUN
ejpam-259	164	3	,	,	PUNCT
ejpam-259	164	4	e	e	NOUN
ejpam-259	164	5	)	)	PUNCT
ejpam-259	164	6	=	=	SYM
ejpam-259	164	7	n	n	NOUN
ejpam-259	164	8	∩	∩	NOUN
ejpam-259	164	9	i	i	NOUN
ejpam-259	164	10	=	=	NOUN
ejpam-259	164	11	1	1	NUM
ejpam-259	164	12	ρ(x	ρ(x	PROPN
ejpam-259	164	13	,	,	PUNCT
ejpam-259	164	14	ei	ei	NOUN
ejpam-259	164	15	)	)	PUNCT
ejpam-259	164	16	.	.	PUNCT
ejpam-259	165	1	the	the	DET
ejpam-259	165	2	chain	chain	NOUN
ejpam-259	165	3	property	property	NOUN
ejpam-259	165	4	of	of	ADP
ejpam-259	165	5	semi	semi	ADJ
ejpam-259	165	6	-	-	ADJ
ejpam-259	165	7	linear	linear	ADJ
ejpam-259	165	8	uniform	uniform	ADJ
ejpam-259	165	9	spaces	space	NOUN
ejpam-259	165	10	implies	imply	VERB
ejpam-259	165	11	that	that	SCONJ
ejpam-259	165	12	any	any	DET
ejpam-259	165	13	two	two	NUM
ejpam-259	165	14	elements	element	NOUN
ejpam-259	165	15	ρ(x	ρ(x	PROPN
ejpam-259	165	16	,	,	PUNCT
ejpam-259	165	17	ei	ei	NOUN
ejpam-259	165	18	)	)	PUNCT
ejpam-259	165	19	,	,	PUNCT
ejpam-259	165	20	ρ(x	ρ(x	PROPN
ejpam-259	165	21	,	,	PUNCT
ejpam-259	165	22	ek	ek	NOUN
ejpam-259	165	23	)	)	PUNCT
ejpam-259	165	24	one	one	NUM
ejpam-259	165	25	of	of	ADP
ejpam-259	165	26	them	they	PRON
ejpam-259	165	27	must	must	AUX
ejpam-259	165	28	be	be	AUX
ejpam-259	165	29	contained	contain	VERB
ejpam-259	165	30	in	in	ADP
ejpam-259	165	31	the	the	DET
ejpam-259	165	32	other	other	ADJ
ejpam-259	165	33	.	.	PUNCT
ejpam-259	166	1	thus	thus	ADV
ejpam-259	166	2	{	{	PUNCT
ejpam-259	166	3	ρ(x	ρ(x	NOUN
ejpam-259	166	4	,	,	PUNCT
ejpam-259	166	5	e1	e1	PROPN
ejpam-259	166	6	)	)	PUNCT
ejpam-259	166	7	,	,	PUNCT
ejpam-259	166	8	...	...	PUNCT
ejpam-259	167	1	ρ(x	ρ(x	NOUN
ejpam-259	167	2	,	,	PUNCT
ejpam-259	167	3	en	en	ADJ
ejpam-259	167	4	)	)	PUNCT
ejpam-259	167	5	}	}	PUNCT
ejpam-259	167	6	a.	a.	NOUN
ejpam-259	167	7	tallafha	tallafha	NOUN
ejpam-259	167	8	and	and	CCONJ
ejpam-259	167	9	r.	r.	PROPN
ejpam-259	167	10	khalil	khalil	PROPN
ejpam-259	167	11	/	/	SYM
ejpam-259	167	12	eur	eur	PROPN
ejpam-259	167	13	.	.	PUNCT
ejpam-259	168	1	j.	j.	PROPN
ejpam-259	168	2	pure	pure	PROPN
ejpam-259	168	3	appl	appl	PROPN
ejpam-259	168	4	.	.	PROPN
ejpam-259	168	5	math	math	PROPN
ejpam-259	168	6	,	,	PUNCT
ejpam-259	168	7	2	2	NUM
ejpam-259	168	8	(	(	PUNCT
ejpam-259	168	9	2009	2009	NUM
ejpam-259	168	10	)	)	PUNCT
ejpam-259	168	11	,	,	PUNCT
ejpam-259	168	12	(	(	PUNCT
ejpam-259	168	13	231	231	NUM
ejpam-259	168	14	-	-	SYM
ejpam-259	168	15	238	238	NUM
ejpam-259	168	16	)	)	PUNCT
ejpam-259	168	17	237	237	NUM
ejpam-259	168	18	is	be	AUX
ejpam-259	168	19	a	a	DET
ejpam-259	168	20	finite	finite	ADJ
ejpam-259	168	21	chain	chain	NOUN
ejpam-259	168	22	.	.	PUNCT
ejpam-259	169	1	consequently	consequently	ADV
ejpam-259	169	2	,	,	PUNCT
ejpam-259	169	3	n	n	PRON
ejpam-259	169	4	∩	∩	NOUN
ejpam-259	169	5	i	i	NOUN
ejpam-259	169	6	=	=	NOUN
ejpam-259	169	7	1	1	NUM
ejpam-259	169	8	ρ(x	ρ(x	PROPN
ejpam-259	169	9	,	,	PUNCT
ejpam-259	169	10	ei	ei	NOUN
ejpam-259	169	11	)	)	PUNCT
ejpam-259	169	12	=	=	SYM
ejpam-259	169	13	ρ(x	ρ(x	PROPN
ejpam-259	169	14	,	,	PUNCT
ejpam-259	169	15	ek	ek	NOUN
ejpam-259	169	16	)	)	PUNCT
ejpam-259	169	17	for	for	ADP
ejpam-259	169	18	some	some	DET
ejpam-259	169	19	k	k	NOUN
ejpam-259	169	20	,	,	PUNCT
ejpam-259	169	21	with	with	ADP
ejpam-259	169	22	1	1	NUM
ejpam-259	169	23	≤	≤	NUM
ejpam-259	169	24	k	k	PROPN
ejpam-259	169	25	≤	≤	PROPN
ejpam-259	169	26	n.	n.	NOUN
ejpam-259	169	27	hence	hence	ADV
ejpam-259	169	28	e	e	PROPN
ejpam-259	169	29	is	be	AUX
ejpam-259	169	30	proximinal	proximinal	ADJ
ejpam-259	169	31	.	.	PUNCT
ejpam-259	170	1	corollary	corollary	ADJ
ejpam-259	170	2	2.4	2.4	NUM
ejpam-259	170	3	.	.	PUNCT
ejpam-259	171	1	if	if	SCONJ
ejpam-259	171	2	e1	e1	PROPN
ejpam-259	171	3	,	,	PUNCT
ejpam-259	171	4	e2	e2	PROPN
ejpam-259	171	5	,	,	PUNCT
ejpam-259	171	6	...	...	PUNCT
ejpam-259	171	7	,	,	PUNCT
ejpam-259	171	8	en	en	X
ejpam-259	171	9	are	be	AUX
ejpam-259	171	10	proximinal	proximinal	ADJ
ejpam-259	171	11	in	in	ADP
ejpam-259	171	12	(	(	PUNCT
ejpam-259	171	13	x	x	INTJ
ejpam-259	171	14	,	,	PUNCT
ejpam-259	171	15	γ	γ	PROPN
ejpam-259	171	16	)	)	PUNCT
ejpam-259	171	17	,	,	PUNCT
ejpam-259	171	18	then	then	ADV
ejpam-259	171	19	n	n	CCONJ
ejpam-259	171	20	⋃	⋃	ADP
ejpam-259	171	21	i	i	NOUN
ejpam-259	171	22	=	=	NOUN
ejpam-259	171	23	1	1	NUM
ejpam-259	171	24	ei	ei	NOUN
ejpam-259	171	25	is	be	AUX
ejpam-259	171	26	proximinal	proximinal	ADJ
ejpam-259	171	27	too	too	ADV
ejpam-259	171	28	.	.	PUNCT
ejpam-259	172	1	proof	proof	NOUN
ejpam-259	172	2	.	.	PUNCT
ejpam-259	173	1	let	let	VERB
ejpam-259	173	2	x	x	PUNCT
ejpam-259	173	3	∈	∈	PROPN
ejpam-259	173	4	x	x	X
ejpam-259	173	5	.	.	PUNCT
ejpam-259	174	1	then	then	ADV
ejpam-259	174	2	ρ(x	ρ(x	NOUN
ejpam-259	174	3	,	,	PUNCT
ejpam-259	174	4	n	n	CCONJ
ejpam-259	174	5	⋃	⋃	PROPN
ejpam-259	174	6	i	i	NOUN
ejpam-259	174	7	=	=	NOUN
ejpam-259	174	8	1	1	NUM
ejpam-259	174	9	ei	ei	NOUN
ejpam-259	174	10	)	)	PUNCT
ejpam-259	174	11	=	=	NOUN
ejpam-259	174	12	∩	∩	NOUN
ejpam-259	174	13	y	y	PROPN
ejpam-259	174	14	∈	∈	PROPN
ejpam-259	175	1	n	n	PRON
ejpam-259	175	2	⋃	⋃	ADP
ejpam-259	175	3	i=1	i=1	X
ejpam-259	175	4	ei	ei	ADP
ejpam-259	175	5	ρ(x	ρ(x	PROPN
ejpam-259	175	6	,	,	PUNCT
ejpam-259	175	7	y	y	PROPN
ejpam-259	175	8	)	)	PUNCT
ejpam-259	175	9	=	=	SYM
ejpam-259	176	1	n	n	NUM
ejpam-259	176	2	⋂	⋂	PROPN
ejpam-259	176	3	i	i	NOUN
ejpam-259	176	4	=	=	NOUN
ejpam-259	176	5	1	1	NUM
ejpam-259	176	6	(	(	PUNCT
ejpam-259	176	7	∩	∩	NOUN
ejpam-259	176	8	y	y	PROPN
ejpam-259	176	9	∈ei	∈ei	SYM
ejpam-259	176	10	ρ(x	ρ(x	PROPN
ejpam-259	176	11	,	,	PUNCT
ejpam-259	176	12	y	y	NOUN
ejpam-259	176	13	)	)	PUNCT
ejpam-259	176	14	)	)	PUNCT
ejpam-259	176	15	)	)	PUNCT
ejpam-259	176	16	.	.	PUNCT
ejpam-259	177	1	since	since	SCONJ
ejpam-259	177	2	ei	ei	PROPN
ejpam-259	177	3	all	all	PRON
ejpam-259	177	4	are	be	AUX
ejpam-259	177	5	proximinal	proximinal	ADJ
ejpam-259	177	6	,	,	PUNCT
ejpam-259	177	7	then	then	ADV
ejpam-259	177	8	∩	∩	NOUN
ejpam-259	177	9	y	y	PROPN
ejpam-259	177	10	∈ei	∈ei	SYM
ejpam-259	177	11	ρ(x	ρ(x	PROPN
ejpam-259	177	12	,	,	PUNCT
ejpam-259	177	13	y	y	NOUN
ejpam-259	177	14	)	)	PUNCT
ejpam-259	177	15	=	=	PUNCT
ejpam-259	178	1	ρ(x	ρ(x	PROPN
ejpam-259	178	2	,	,	PUNCT
ejpam-259	178	3	ei	ei	NOUN
ejpam-259	178	4	)	)	PUNCT
ejpam-259	178	5	for	for	ADP
ejpam-259	178	6	some	some	DET
ejpam-259	178	7	ei	ei	ADP
ejpam-259	178	8	∈	∈	NOUN
ejpam-259	178	9	ei	ei	X
ejpam-259	178	10	.	.	PUNCT
ejpam-259	179	1	hence	hence	ADV
ejpam-259	179	2	ρ(x	ρ(x	PROPN
ejpam-259	179	3	,	,	PUNCT
ejpam-259	179	4	n	n	CCONJ
ejpam-259	179	5	⋃	⋃	PROPN
ejpam-259	179	6	i	i	NOUN
ejpam-259	179	7	=	=	NOUN
ejpam-259	179	8	1	1	NUM
ejpam-259	179	9	ei	ei	NOUN
ejpam-259	179	10	)	)	PUNCT
ejpam-259	179	11	=	=	SYM
ejpam-259	180	1	n	n	CCONJ
ejpam-259	180	2	⋂	⋂	PROPN
ejpam-259	181	1	i	i	NOUN
ejpam-259	181	2	=	=	NOUN
ejpam-259	181	3	1	1	NUM
ejpam-259	181	4	ρ(x	ρ(x	PROPN
ejpam-259	181	5	,	,	PUNCT
ejpam-259	181	6	ei	ei	NOUN
ejpam-259	181	7	)	)	PUNCT
ejpam-259	181	8	=	=	SYM
ejpam-259	181	9	ρ(x	ρ(x	PROPN
ejpam-259	181	10	,	,	PUNCT
ejpam-259	181	11	ek	ek	NOUN
ejpam-259	181	12	)	)	PUNCT
ejpam-259	181	13	for	for	ADP
ejpam-259	181	14	some	some	DET
ejpam-259	181	15	k	k	PROPN
ejpam-259	181	16	∈	∈	PROPN
ejpam-259	181	17	{	{	PUNCT
ejpam-259	181	18	1	1	NUM
ejpam-259	181	19	,	,	PUNCT
ejpam-259	181	20	2	2	NUM
ejpam-259	181	21	,	,	PUNCT
ejpam-259	181	22	...	...	PUNCT
ejpam-259	181	23	,	,	PUNCT
ejpam-259	181	24	n	n	CCONJ
ejpam-259	181	25	}	}	PUNCT
ejpam-259	181	26	.	.	PUNCT
ejpam-259	182	1	so	so	ADV
ejpam-259	182	2	ρ(x	ρ(x	PROPN
ejpam-259	182	3	,	,	PUNCT
ejpam-259	182	4	n	n	CCONJ
ejpam-259	182	5	⋃	⋃	PROPN
ejpam-259	182	6	i	i	NOUN
ejpam-259	182	7	=	=	NOUN
ejpam-259	182	8	1	1	NUM
ejpam-259	182	9	ei	ei	NOUN
ejpam-259	182	10	)	)	PUNCT
ejpam-259	182	11	=	=	SYM
ejpam-259	183	1	n	n	CCONJ
ejpam-259	183	2	⋂	⋂	PROPN
ejpam-259	184	1	i	i	NOUN
ejpam-259	184	2	=	=	NOUN
ejpam-259	184	3	1	1	NUM
ejpam-259	184	4	ρ(x	ρ(x	PROPN
ejpam-259	184	5	,	,	PUNCT
ejpam-259	184	6	ei	ei	NOUN
ejpam-259	184	7	)	)	PUNCT
ejpam-259	184	8	=	=	SYM
ejpam-259	184	9	ρ(x	ρ(x	PROPN
ejpam-259	184	10	,	,	PUNCT
ejpam-259	184	11	ek	ek	PROPN
ejpam-259	184	12	)	)	PUNCT
ejpam-259	184	13	,	,	PUNCT
ejpam-259	184	14	ek	ek	PROPN
ejpam-259	184	15	∈	∈	PROPN
ejpam-259	184	16	n	n	CCONJ
ejpam-259	184	17	⋃	⋃	PROPN
ejpam-259	184	18	i	i	NOUN
ejpam-259	184	19	=	=	NOUN
ejpam-259	184	20	1	1	NUM
ejpam-259	184	21	ei	ei	NOUN
ejpam-259	184	22	.	.	PUNCT
ejpam-259	185	1	also	also	ADV
ejpam-259	185	2	every	every	DET
ejpam-259	185	3	sequence	sequence	NOUN
ejpam-259	185	4	with	with	ADP
ejpam-259	185	5	it	it	PRON
ejpam-259	185	6	’s	’s	ADJ
ejpam-259	185	7	limit	limit	NOUN
ejpam-259	185	8	is	be	AUX
ejpam-259	185	9	compact	compact	ADJ
ejpam-259	185	10	,	,	PUNCT
ejpam-259	185	11	so	so	ADV
ejpam-259	185	12	we	we	PRON
ejpam-259	185	13	have	have	VERB
ejpam-259	185	14	another	another	DET
ejpam-259	185	15	partial	partial	ADJ
ejpam-259	185	16	answer	answer	NOUN
ejpam-259	185	17	to	to	ADP
ejpam-259	185	18	our	our	PRON
ejpam-259	185	19	question	question	NOUN
ejpam-259	185	20	.	.	PUNCT
ejpam-259	186	1	theorem	theorem	VERB
ejpam-259	186	2	2.5	2.5	NUM
ejpam-259	186	3	.	.	PUNCT
ejpam-259	187	1	let	let	AUX
ejpam-259	187	2	(	(	PUNCT
ejpam-259	187	3	x	x	X
ejpam-259	187	4	,	,	PUNCT
ejpam-259	187	5	γ	γ	PROPN
ejpam-259	187	6	)	)	PUNCT
ejpam-259	187	7	be	be	VERB
ejpam-259	187	8	a	a	DET
ejpam-259	187	9	semi	semi	ADJ
ejpam-259	187	10	-	-	ADJ
ejpam-259	187	11	linear	linear	ADJ
ejpam-259	187	12	uniform	uniform	ADJ
ejpam-259	187	13	space	space	NOUN
ejpam-259	187	14	and	and	CCONJ
ejpam-259	187	15	(	(	PUNCT
ejpam-259	187	16	yn	yn	NOUN
ejpam-259	187	17	)	)	PUNCT
ejpam-259	187	18	be	be	VERB
ejpam-259	187	19	a	a	DET
ejpam-259	187	20	convergent	convergent	NOUN
ejpam-259	187	21	sequence	sequence	NOUN
ejpam-259	187	22	in	in	ADP
ejpam-259	187	23	x	x	X
ejpam-259	187	24	.	.	PUNCT
ejpam-259	188	1	then	then	ADV
ejpam-259	188	2	e	e	PROPN
ejpam-259	188	3	=	=	SYM
ejpam-259	188	4	¦	¦	PROPN
ejpam-259	188	5	y	y	PROPN
ejpam-259	188	6	,	,	PUNCT
ejpam-259	188	7	y	y	PROPN
ejpam-259	188	8	1	1	NUM
ejpam-259	188	9	,	,	PUNCT
ejpam-259	188	10	y	y	PROPN
ejpam-259	188	11	2	2	NUM
ejpam-259	188	12	,	,	PUNCT
ejpam-259	188	13	...	...	PUNCT
ejpam-259	189	1	©	©	PROPN
ejpam-259	189	2	is	be	AUX
ejpam-259	189	3	proximinal	proximinal	ADJ
ejpam-259	189	4	,	,	PUNCT
ejpam-259	189	5	where	where	SCONJ
ejpam-259	189	6	y	y	PROPN
ejpam-259	189	7	=	=	PROPN
ejpam-259	189	8	lim	lim	PROPN
ejpam-259	189	9	yn	yn	PROPN
ejpam-259	189	10	.	.	PUNCT
ejpam-259	190	1	proof.let	proof.let	X
ejpam-259	191	1	x	x	PUNCT
ejpam-259	191	2	∈	∈	PROPN
ejpam-259	191	3	x	x	PUNCT
ejpam-259	191	4	\	\	X
ejpam-259	191	5	e	e	X
ejpam-259	191	6	(	(	PUNCT
ejpam-259	191	7	if	if	SCONJ
ejpam-259	191	8	x	x	SYM
ejpam-259	191	9	∈	∈	PROPN
ejpam-259	191	10	e	e	NOUN
ejpam-259	191	11	then	then	ADV
ejpam-259	191	12	ρ(x	ρ(x	PROPN
ejpam-259	191	13	,	,	PUNCT
ejpam-259	191	14	e	e	NOUN
ejpam-259	191	15	)	)	PUNCT
ejpam-259	191	16	=	=	SYM
ejpam-259	192	1	ρ(x	ρ(x	NOUN
ejpam-259	192	2	,	,	PUNCT
ejpam-259	192	3	x	x	NOUN
ejpam-259	192	4	)	)	PUNCT
ejpam-259	192	5	)	)	PUNCT
ejpam-259	192	6	.	.	PUNCT
ejpam-259	193	1	so	so	ADV
ejpam-259	193	2	we	we	PRON
ejpam-259	193	3	may	may	AUX
ejpam-259	193	4	assume	assume	VERB
ejpam-259	193	5	ρ(x	ρ(x	PROPN
ejpam-259	193	6	,	,	PUNCT
ejpam-259	193	7	yn	yn	PROPN
ejpam-259	193	8	)	)	PUNCT
ejpam-259	193	9	6=∆	6=∆	NOUN
ejpam-259	193	10	for	for	ADP
ejpam-259	193	11	all	all	DET
ejpam-259	193	12	n.	n.	NOUN
ejpam-259	193	13	now	now	ADV
ejpam-259	193	14	if	if	SCONJ
ejpam-259	193	15	there	there	PRON
ejpam-259	193	16	exist	exist	VERB
ejpam-259	193	17	n	n	PRON
ejpam-259	193	18	0	0	NUM
ejpam-259	193	19	such	such	ADJ
ejpam-259	193	20	that	that	DET
ejpam-259	193	21	ρ(x	ρ(x	NOUN
ejpam-259	193	22	,	,	PUNCT
ejpam-259	193	23	yn0	yn0	PROPN
ejpam-259	193	24	)	)	PUNCT
ejpam-259	194	1	⊆	⊆	NUM
ejpam-259	194	2	ρ(x	ρ(x	PROPN
ejpam-259	194	3	,	,	PUNCT
ejpam-259	194	4	yn	yn	PROPN
ejpam-259	194	5	)	)	PUNCT
ejpam-259	194	6	.	.	PUNCT
ejpam-259	195	1	for	for	ADP
ejpam-259	195	2	all	all	DET
ejpam-259	195	3	n	n	PROPN
ejpam-259	195	4	,	,	PUNCT
ejpam-259	195	5	then	then	ADV
ejpam-259	195	6	ρ(x	ρ(x	PROPN
ejpam-259	195	7	,	,	PUNCT
ejpam-259	195	8	e	e	NOUN
ejpam-259	195	9	)	)	PUNCT
ejpam-259	195	10	=	=	SYM
ejpam-259	195	11	ρ(x	ρ(x	NOUN
ejpam-259	195	12	,	,	PUNCT
ejpam-259	195	13	yn0	yn0	PROPN
ejpam-259	195	14	)	)	PUNCT
ejpam-259	195	15	∩ρ(x	∩ρ(x	PROPN
ejpam-259	195	16	,	,	PUNCT
ejpam-259	195	17	y	y	PROPN
ejpam-259	195	18	)	)	PUNCT
ejpam-259	195	19	and	and	CCONJ
ejpam-259	195	20	by	by	ADP
ejpam-259	195	21	theorem	theorem	NOUN
ejpam-259	195	22	2.3	2.3	NUM
ejpam-259	195	23	we	we	PRON
ejpam-259	195	24	are	be	AUX
ejpam-259	195	25	done	do	VERB
ejpam-259	195	26	.	.	PUNCT
ejpam-259	196	1	if	if	SCONJ
ejpam-259	196	2	not	not	PART
ejpam-259	196	3	,	,	PUNCT
ejpam-259	196	4	then	then	ADV
ejpam-259	196	5	for	for	ADP
ejpam-259	196	6	all	all	PRON
ejpam-259	196	7	n	n	PRON
ejpam-259	196	8	there	there	ADV
ejpam-259	196	9	exist	exist	VERB
ejpam-259	196	10	mn	mn	PROPN
ejpam-259	197	1	such	such	ADJ
ejpam-259	197	2	that	that	SCONJ
ejpam-259	197	3	mn	mn	PROPN
ejpam-259	197	4	<	<	X
ejpam-259	197	5	mn+1and	mn+1and	PROPN
ejpam-259	197	6	ρ(x	ρ(x	NOUN
ejpam-259	197	7	,	,	PUNCT
ejpam-259	197	8	ymn	ymn	ADV
ejpam-259	197	9	)	)	PUNCT
ejpam-259	197	10	$	$	SYM
ejpam-259	197	11	ρ(x	ρ(x	PROPN
ejpam-259	197	12	,	,	PUNCT
ejpam-259	197	13	yn	yn	NOUN
ejpam-259	197	14	)	)	PUNCT
ejpam-259	197	15	∩	∩	PROPN
ejpam-259	197	16	ρ(x	ρ(x	PROPN
ejpam-259	197	17	,	,	PUNCT
ejpam-259	197	18	ymn−1	ymn−1	PROPN
ejpam-259	197	19	)	)	PUNCT
ejpam-259	197	20	.so	.so	PUNCT
ejpam-259	198	1	∞	∞	NUM
ejpam-259	198	2	⋂	⋂	PROPN
ejpam-259	198	3	n	n	PROPN
ejpam-259	198	4	=	=	SYM
ejpam-259	198	5	1	1	NUM
ejpam-259	198	6	ρ(x	ρ(x	PROPN
ejpam-259	198	7	,	,	PUNCT
ejpam-259	198	8	ymn	ymn	ADV
ejpam-259	198	9	)	)	PUNCT
ejpam-259	199	1	=	=	SYM
ejpam-259	200	1	∞	∞	NUM
ejpam-259	200	2	⋂	⋂	PROPN
ejpam-259	200	3	n	n	CCONJ
ejpam-259	200	4	=	=	SYM
ejpam-259	200	5	1	1	NUM
ejpam-259	200	6	ρ(x	ρ(x	NOUN
ejpam-259	200	7	,	,	PUNCT
ejpam-259	200	8	yn).now	yn).now	ADV
ejpam-259	200	9	we	we	PRON
ejpam-259	200	10	want	want	VERB
ejpam-259	200	11	to	to	PART
ejpam-259	200	12	show	show	VERB
ejpam-259	200	13	that	that	SCONJ
ejpam-259	200	14	ρ(x	ρ(x	NOUN
ejpam-259	200	15	,	,	PUNCT
ejpam-259	200	16	y	y	PROPN
ejpam-259	200	17	)	)	PUNCT
ejpam-259	201	1	⊆	⊆	NUM
ejpam-259	201	2	∞	∞	NUM
ejpam-259	201	3	⋂	⋂	PROPN
ejpam-259	201	4	n	n	PROPN
ejpam-259	201	5	=	=	SYM
ejpam-259	201	6	1	1	NUM
ejpam-259	201	7	ρ(x	ρ(x	PROPN
ejpam-259	201	8	,	,	PUNCT
ejpam-259	201	9	ymn	ymn	ADV
ejpam-259	201	10	)	)	PUNCT
ejpam-259	201	11	.let	.let	PUNCT
ejpam-259	202	1	u	u	PROPN
ejpam-259	202	2	∈	∈	PROPN
ejpam-259	202	3	γ	γ	NOUN
ejpam-259	202	4	be	be	VERB
ejpam-259	202	5	such	such	ADJ
ejpam-259	202	6	that	that	SCONJ
ejpam-259	202	7	(	(	PUNCT
ejpam-259	202	8	x	x	X
ejpam-259	202	9	,	,	PUNCT
ejpam-259	202	10	ymn	ymn	NOUN
ejpam-259	202	11	)	)	PUNCT
ejpam-259	202	12	∈	∈	PROPN
ejpam-259	202	13	u	u	NOUN
ejpam-259	202	14	,	,	PUNCT
ejpam-259	202	15	for	for	ADP
ejpam-259	202	16	some	some	DET
ejpam-259	202	17	mn	mn	PROPN
ejpam-259	202	18	∈	∈	PROPN
ejpam-259	202	19	n	n	CCONJ
ejpam-259	202	20	,	,	PUNCT
ejpam-259	202	21	therefor	therefor	ADP
ejpam-259	202	22	(	(	PUNCT
ejpam-259	202	23	x	x	INTJ
ejpam-259	202	24	,	,	PUNCT
ejpam-259	202	25	ymk	ymk	INTJ
ejpam-259	202	26	)	)	PUNCT
ejpam-259	202	27	∈	∈	PROPN
ejpam-259	202	28	u	u	NOUN
ejpam-259	202	29	for	for	ADP
ejpam-259	202	30	all	all	DET
ejpam-259	202	31	j	j	PROPN
ejpam-259	202	32	≥	≥	PROPN
ejpam-259	202	33	n.	n.	NOUN
ejpam-259	202	34	let	let	VERB
ejpam-259	202	35	wj	wj	PROPN
ejpam-259	202	36	∈	∈	PROPN
ejpam-259	202	37	γ	γ	X
ejpam-259	202	38	be	be	AUX
ejpam-259	202	39	such	such	ADJ
ejpam-259	202	40	that	that	SCONJ
ejpam-259	202	41	b	b	PROPN
ejpam-259	202	42	�	�	PROPN
ejpam-259	202	43	x	x	SYM
ejpam-259	202	44	,	,	PUNCT
ejpam-259	202	45	2wj	2wj	ADJ
ejpam-259	202	46	�	�	PROPN
ejpam-259	202	47	×	×	PROPN
ejpam-259	202	48	b	b	PROPN
ejpam-259	202	49	�	�	PROPN
ejpam-259	202	50	ymk	ymk	PROPN
ejpam-259	202	51	,	,	PUNCT
ejpam-259	202	52	2wj	2wj	ADJ
ejpam-259	202	53	�	�	PROPN
ejpam-259	202	54	⊆	⊆	NUM
ejpam-259	202	55	u	u	NOUN
ejpam-259	202	56	,	,	PUNCT
ejpam-259	202	57	also	also	ADV
ejpam-259	202	58	we	we	PRON
ejpam-259	202	59	may	may	AUX
ejpam-259	202	60	assume	assume	VERB
ejpam-259	202	61	that	that	SCONJ
ejpam-259	202	62	wj	wj	PROPN
ejpam-259	202	63	⊇	⊇	PROPN
ejpam-259	202	64	wj+1.if	wj+1.if	PROPN
ejpam-259	202	65	∞	∞	PROPN
ejpam-259	202	66	⋂	⋂	PROPN
ejpam-259	202	67	j	j	PROPN
ejpam-259	202	68	=	=	SYM
ejpam-259	202	69	1	1	NUM
ejpam-259	202	70	wj	wj	NOUN
ejpam-259	202	71	=	=	PUNCT
ejpam-259	202	72	∆,let	∆,let	PROPN
ejpam-259	202	73	�	�	PROPN
ejpam-259	202	74	t	t	PROPN
ejpam-259	202	75	j	j	PROPN
ejpam-259	202	76	,	,	PUNCT
ejpam-259	202	77	s	s	PROPN
ejpam-259	202	78	j	j	PROPN
ejpam-259	202	79	�	�	PROPN
ejpam-259	202	80	∈	∈	PROPN
ejpam-259	202	81	b	b	PROPN
ejpam-259	202	82	�	�	PROPN
ejpam-259	202	83	x	x	SYM
ejpam-259	202	84	,	,	PUNCT
ejpam-259	202	85	wj	wj	PROPN
ejpam-259	202	86	�	�	PROPN
ejpam-259	202	87	×	×	PROPN
ejpam-259	202	88	b	b	PROPN
ejpam-259	202	89	�	�	PROPN
ejpam-259	202	90	ymk	ymk	PROPN
ejpam-259	202	91	,	,	PUNCT
ejpam-259	202	92	wj	wj	PROPN
ejpam-259	202	93	�	�	PROPN
ejpam-259	202	94	,	,	PUNCT
ejpam-259	202	95	then	then	ADV
ejpam-259	202	96	t	t	PROPN
ejpam-259	202	97	j	j	PROPN
ejpam-259	202	98	→	→	PUNCT
ejpam-259	202	99	x	x	X
ejpam-259	202	100	and	and	CCONJ
ejpam-259	202	101	s	s	PROPN
ejpam-259	202	102	j	j	PROPN
ejpam-259	202	103	→	→	SYM
ejpam-259	202	104	y	y	PROPN
ejpam-259	202	105	(	(	PUNCT
ejpam-259	202	106	lim	lim	PROPN
ejpam-259	202	107	s	s	PART
ejpam-259	202	108	j	j	PROPN
ejpam-259	202	109	=	=	PROPN
ejpam-259	202	110	lim	lim	PROPN
ejpam-259	202	111	ymk	ymk	PROPN
ejpam-259	202	112	)	)	PUNCT
ejpam-259	202	113	.	.	PUNCT
ejpam-259	203	1	a.	a.	NOUN
ejpam-259	203	2	tallafha	tallafha	NOUN
ejpam-259	203	3	and	and	CCONJ
ejpam-259	203	4	r.	r.	PROPN
ejpam-259	203	5	khalil	khalil	PROPN
ejpam-259	203	6	/	/	SYM
ejpam-259	203	7	eur	eur	PROPN
ejpam-259	203	8	.	.	PUNCT
ejpam-259	204	1	j.	j.	PROPN
ejpam-259	204	2	pure	pure	PROPN
ejpam-259	204	3	appl	appl	PROPN
ejpam-259	204	4	.	.	PROPN
ejpam-259	204	5	math	math	PROPN
ejpam-259	204	6	,	,	PUNCT
ejpam-259	204	7	2	2	NUM
ejpam-259	204	8	(	(	PUNCT
ejpam-259	204	9	2009	2009	NUM
ejpam-259	204	10	)	)	PUNCT
ejpam-259	204	11	,	,	PUNCT
ejpam-259	204	12	(	(	PUNCT
ejpam-259	204	13	231	231	NUM
ejpam-259	204	14	-	-	SYM
ejpam-259	204	15	238	238	NUM
ejpam-259	204	16	)	)	PUNCT
ejpam-259	204	17	238	238	NUM
ejpam-259	204	18	therefor	therefor	ADP
ejpam-259	204	19	�	�	PROPN
ejpam-259	204	20	x	x	SYM
ejpam-259	204	21	,	,	PUNCT
ejpam-259	204	22	y	y	PROPN
ejpam-259	204	23	�	�	PROPN
ejpam-259	204	24	∈	∈	PROPN
ejpam-259	204	25	−−−−−−−−	−−−−−−−−	PROPN
ejpam-259	204	26	b	b	X
ejpam-259	204	27	�	�	PROPN
ejpam-259	204	28	x	x	SYM
ejpam-259	204	29	,	,	PUNCT
ejpam-259	204	30	wj	wj	PROPN
ejpam-259	204	31	�	�	PROPN
ejpam-259	204	32	×	×	PROPN
ejpam-259	204	33	−−−−−−−−−−	−−−−−−−−−−	CCONJ
ejpam-259	204	34	b	b	PROPN
ejpam-259	204	35	�	�	PROPN
ejpam-259	204	36	ymk	ymk	PROPN
ejpam-259	204	37	,	,	PUNCT
ejpam-259	204	38	wj	wj	PROPN
ejpam-259	204	39	�	�	PROPN
ejpam-259	204	40	⊆	⊆	NUM
ejpam-259	204	41	b	b	PROPN
ejpam-259	204	42	�	�	PROPN
ejpam-259	204	43	x	x	SYM
ejpam-259	204	44	,	,	PUNCT
ejpam-259	204	45	2wj	2wj	ADJ
ejpam-259	204	46	�	�	PROPN
ejpam-259	204	47	×	×	PROPN
ejpam-259	204	48	b	b	PROPN
ejpam-259	204	49	�	�	PROPN
ejpam-259	204	50	ymk	ymk	PROPN
ejpam-259	204	51	,	,	PUNCT
ejpam-259	204	52	2wj	2wj	ADJ
ejpam-259	204	53	�	�	PROPN
ejpam-259	204	54	⊆	⊆	NUM
ejpam-259	204	55	u	u	NOUN
ejpam-259	204	56	.	.	PUNCT
ejpam-259	205	1	now	now	ADV
ejpam-259	205	2	if	if	SCONJ
ejpam-259	205	3	∞	∞	PROPN
ejpam-259	205	4	⋂	⋂	PROPN
ejpam-259	205	5	j	j	PROPN
ejpam-259	205	6	=	=	SYM
ejpam-259	205	7	1	1	NUM
ejpam-259	205	8	wj	wj	NOUN
ejpam-259	205	9	6=	6=	ADP
ejpam-259	205	10	∆	∆	PROPN
ejpam-259	205	11	,	,	PUNCT
ejpam-259	205	12	then	then	ADV
ejpam-259	205	13	there	there	PRON
ejpam-259	205	14	exist	exist	VERB
ejpam-259	205	15	w	w	PROPN
ejpam-259	205	16	∈	∈	PROPN
ejpam-259	205	17	γ	γ	NOUN
ejpam-259	205	18	such	such	ADJ
ejpam-259	205	19	that	that	PRON
ejpam-259	205	20	w	w	ADP
ejpam-259	205	21	⊆	⊆	NUM
ejpam-259	205	22	∞	∞	NUM
ejpam-259	205	23	⋂	⋂	PROPN
ejpam-259	205	24	j	j	PROPN
ejpam-259	205	25	=	=	SYM
ejpam-259	205	26	1	1	NUM
ejpam-259	205	27	wj	wj	NOUN
ejpam-259	205	28	,	,	PUNCT
ejpam-259	205	29	so	so	ADV
ejpam-259	205	30	b	b	X
ejpam-259	205	31	(	(	PUNCT
ejpam-259	205	32	x	x	INTJ
ejpam-259	205	33	,	,	PUNCT
ejpam-259	205	34	w)×	w)×	PROPN
ejpam-259	205	35	b	b	PROPN
ejpam-259	205	36	�	�	PROPN
ejpam-259	205	37	ymk	ymk	PROPN
ejpam-259	205	38	,	,	PUNCT
ejpam-259	205	39	w	w	PROPN
ejpam-259	205	40	�	�	PROPN
ejpam-259	205	41	⊆	⊆	NUM
ejpam-259	205	42	u	u	NOUN
ejpam-259	205	43	for	for	ADP
ejpam-259	205	44	all	all	DET
ejpam-259	205	45	j.since	j.since	NOUN
ejpam-259	205	46	ymk	ymk	PROPN
ejpam-259	205	47	−→	−→	PROPN
ejpam-259	205	48	y	y	PROPN
ejpam-259	205	49	,	,	PUNCT
ejpam-259	205	50	there	there	PRON
ejpam-259	205	51	exist	exist	VERB
ejpam-259	205	52	n	n	DET
ejpam-259	205	53	such	such	ADJ
ejpam-259	205	54	that	that	DET
ejpam-259	205	55	�	�	PROPN
ejpam-259	205	56	ymn	ymn	PROPN
ejpam-259	205	57	,	,	PUNCT
ejpam-259	205	58	y	y	PROPN
ejpam-259	205	59	�	�	PROPN
ejpam-259	205	60	∈w	∈w	PROPN
ejpam-259	205	61	,	,	PUNCT
ejpam-259	205	62	so	so	ADV
ejpam-259	205	63	�	�	PROPN
ejpam-259	205	64	x	x	SYM
ejpam-259	205	65	,	,	PUNCT
ejpam-259	205	66	y	y	PROPN
ejpam-259	205	67	�	�	PROPN
ejpam-259	205	68	∈	∈	PROPN
ejpam-259	205	69	u	u	PROPN
ejpam-259	205	70	,	,	PUNCT
ejpam-259	205	71	and	and	CCONJ
ejpam-259	205	72	the	the	DET
ejpam-259	205	73	result	result	NOUN
ejpam-259	205	74	follows	follow	VERB
ejpam-259	205	75	.	.	PUNCT
ejpam-259	206	1	references	reference	NOUN
ejpam-259	206	2	[	[	X
ejpam-259	206	3	1	1	NUM
ejpam-259	206	4	]	]	X
ejpam-259	206	5	engelking	engelke	VERB
ejpam-259	206	6	,	,	PUNCT
ejpam-259	206	7	r.	r.	PROPN
ejpam-259	206	8	outline	outline	NOUN
ejpam-259	206	9	of	of	ADP
ejpam-259	206	10	general	general	ADJ
ejpam-259	206	11	topology	topology	NOUN
ejpam-259	206	12	,	,	PUNCT
ejpam-259	206	13	north	north	NOUN
ejpam-259	206	14	-	-	PUNCT
ejpam-259	206	15	holand	holand	PROPN
ejpam-259	206	16	,	,	PUNCT
ejpam-259	206	17	amsterdam	amsterdam	PROPN
ejpam-259	206	18	,	,	PUNCT
ejpam-259	206	19	1968	1968	NUM
ejpam-259	206	20	.	.	PUNCT
ejpam-259	207	1	[	[	X
ejpam-259	207	2	2	2	NUM
ejpam-259	207	3	]	]	X
ejpam-259	207	4	james	james	PROPN
ejpam-259	207	5	,	,	PUNCT
ejpam-259	207	6	i.m	i.m	PROPN
ejpam-259	207	7	.	.	PUNCT
ejpam-259	207	8	topological	topological	PROPN
ejpam-259	207	9	and	and	CCONJ
ejpam-259	207	10	uniform	uniform	ADJ
ejpam-259	207	11	spaces	space	NOUN
ejpam-259	207	12	.	.	PUNCT
ejpam-259	208	1	undergraduate	undergraduate	ADJ
ejpam-259	208	2	texts	text	NOUN
ejpam-259	208	3	in	in	ADP
ejpam-259	208	4	mathematics	mathematic	NOUN
ejpam-259	208	5	.	.	PUNCT
ejpam-259	209	1	springer	springer	NOUN
ejpam-259	209	2	-	-	PUNCT
ejpam-259	209	3	verlag	verlag	PROPN
ejpam-259	209	4	1987	1987	NUM
ejpam-259	209	5	.	.	PUNCT
ejpam-259	210	1	[	[	X
ejpam-259	210	2	3	3	NUM
ejpam-259	210	3	]	]	X
ejpam-259	210	4	light	light	NOUN
ejpam-259	210	5	,	,	PUNCT
ejpam-259	210	6	w.	w.	PROPN
ejpam-259	210	7	and	and	CCONJ
ejpam-259	210	8	cheney	cheney	PROPN
ejpam-259	210	9	,	,	PUNCT
ejpam-259	210	10	e.	e.	PROPN
ejpam-259	210	11	approximation	approximation	PROPN
ejpam-259	210	12	theory	theory	NOUN
ejpam-259	210	13	in	in	ADP
ejpam-259	210	14	tensor	tensor	NOUN
ejpam-259	210	15	product	product	NOUN
ejpam-259	210	16	spaces	space	VERB
ejpam-259	210	17	.	.	PUNCT
ejpam-259	211	1	lecture	lecture	NOUN
ejpam-259	211	2	notes	note	NOUN
ejpam-259	211	3	in	in	ADP
ejpam-259	211	4	math	math	NOUN
ejpam-259	211	5	.	.	PUNCT
ejpam-259	212	1	1169	1169	NUM
ejpam-259	212	2	.	.	PUNCT
ejpam-259	213	1	springer	springer	PROPN
ejpam-259	213	2	verlag	verlag	PROPN
ejpam-259	213	3	,	,	PUNCT
ejpam-259	213	4	new	new	PROPN
ejpam-259	213	5	york	york	PROPN
ejpam-259	213	6	,	,	PUNCT
ejpam-259	213	7	1985	1985	NUM
ejpam-259	213	8	.	.	PUNCT
ejpam-259	214	1	[	[	X
ejpam-259	214	2	4	4	NUM
ejpam-259	214	3	]	]	X
ejpam-259	214	4	singer	singer	NOUN
ejpam-259	214	5	,	,	PUNCT
ejpam-259	214	6	i.	i.	PROPN
ejpam-259	214	7	best	good	ADJ
ejpam-259	214	8	approximation	approximation	NOUN
ejpam-259	214	9	in	in	ADP
ejpam-259	214	10	normed	normed	ADJ
ejpam-259	214	11	linear	linear	PROPN
ejpam-259	214	12	spaces	space	NOUN
ejpam-259	214	13	by	by	ADP
ejpam-259	214	14	elements	element	NOUN
ejpam-259	214	15	of	of	ADP
ejpam-259	214	16	linear	linear	ADJ
ejpam-259	214	17	subspaces	subspace	NOUN
ejpam-259	214	18	.	.	PUNCT
ejpam-259	214	19	springer	springer	NOUN
ejpam-259	214	20	-	-	PUNCT
ejpam-259	214	21	verlag	verlag	PROPN
ejpam-259	214	22	,	,	PUNCT
ejpam-259	214	23	new	new	PROPN
ejpam-259	214	24	york	york	PROPN
ejpam-259	214	25	,	,	PUNCT
ejpam-259	214	26	1970	1970	NUM
ejpam-259	214	27	.	.	PUNCT