id	sid	tid	token	lemma	pos
ejpam-847	1	1	8_lee.dvi	8_lee.dvi	PROPN
ejpam-847	1	2	european	european	PROPN
ejpam-847	1	3	journal	journal	PROPN
ejpam-847	1	4	of	of	ADP
ejpam-847	1	5	pure	pure	ADJ
ejpam-847	1	6	and	and	CCONJ
ejpam-847	1	7	applied	apply	VERB
ejpam-847	1	8	mathematics	mathematic	NOUN
ejpam-847	1	9	vol	vol	NOUN
ejpam-847	1	10	.	.	PROPN
ejpam-847	1	11	5	5	NUM
ejpam-847	1	12	,	,	PUNCT
ejpam-847	1	13	no	no	INTJ
ejpam-847	1	14	.	.	NOUN
ejpam-847	1	15	4	4	NUM
ejpam-847	1	16	,	,	PUNCT
ejpam-847	1	17	2012	2012	NUM
ejpam-847	1	18	,	,	PUNCT
ejpam-847	1	19	540	540	NUM
ejpam-847	1	20	-	-	SYM
ejpam-847	1	21	553	553	NUM
ejpam-847	1	22	issn	issn	PROPN
ejpam-847	1	23	1307	1307	NUM
ejpam-847	1	24	-	-	SYM
ejpam-847	1	25	5543	5543	NUM
ejpam-847	1	26	–	–	PUNCT
ejpam-847	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-847	1	28	random	random	ADJ
ejpam-847	1	29	stability	stability	NOUN
ejpam-847	1	30	of	of	ADP
ejpam-847	1	31	a	a	DET
ejpam-847	1	32	functional	functional	ADJ
ejpam-847	1	33	equation	equation	NOUN
ejpam-847	1	34	related	relate	VERB
ejpam-847	1	35	to	to	ADP
ejpam-847	1	36	an	an	DET
ejpam-847	1	37	inner	inner	ADJ
ejpam-847	1	38	product	product	NOUN
ejpam-847	1	39	space	space	NOUN
ejpam-847	1	40	dong	dong	PROPN
ejpam-847	1	41	yun	yun	PROPN
ejpam-847	1	42	shin1	shin1	PROPN
ejpam-847	1	43	,	,	PUNCT
ejpam-847	1	44	jung	jung	PROPN
ejpam-847	1	45	rye	rye	PROPN
ejpam-847	1	46	lee2,∗	lee2,∗	PROPN
ejpam-847	1	47	,	,	PUNCT
ejpam-847	1	48	choonkil	choonkil	ADJ
ejpam-847	1	49	park3	park3	PROPN
ejpam-847	1	50	1	1	NUM
ejpam-847	1	51	department	department	NOUN
ejpam-847	1	52	of	of	ADP
ejpam-847	1	53	mathematics	mathematics	PROPN
ejpam-847	1	54	,	,	PUNCT
ejpam-847	1	55	university	university	NOUN
ejpam-847	1	56	of	of	ADP
ejpam-847	1	57	seoul	seoul	PROPN
ejpam-847	1	58	,	,	PUNCT
ejpam-847	1	59	seoul	seoul	PROPN
ejpam-847	1	60	130	130	NUM
ejpam-847	1	61	-	-	SYM
ejpam-847	1	62	743	743	NUM
ejpam-847	1	63	,	,	PUNCT
ejpam-847	1	64	korea	korea	PROPN
ejpam-847	1	65	2	2	NUM
ejpam-847	1	66	department	department	NOUN
ejpam-847	1	67	of	of	ADP
ejpam-847	1	68	mathematics	mathematic	NOUN
ejpam-847	1	69	,	,	PUNCT
ejpam-847	1	70	daejin	daejin	NOUN
ejpam-847	1	71	university	university	NOUN
ejpam-847	1	72	,	,	PUNCT
ejpam-847	1	73	kyeonggi	kyeonggi	VERB
ejpam-847	1	74	487	487	NUM
ejpam-847	1	75	-	-	SYM
ejpam-847	1	76	711	711	NUM
ejpam-847	1	77	,	,	PUNCT
ejpam-847	1	78	korea	korea	PROPN
ejpam-847	1	79	3	3	NUM
ejpam-847	1	80	research	research	PROPN
ejpam-847	1	81	institute	institute	NOUN
ejpam-847	1	82	for	for	ADP
ejpam-847	1	83	natural	natural	ADJ
ejpam-847	1	84	sciences	science	NOUN
ejpam-847	1	85	,	,	PUNCT
ejpam-847	1	86	hanyang	hanyang	NOUN
ejpam-847	1	87	university	university	PROPN
ejpam-847	1	88	,	,	PUNCT
ejpam-847	1	89	seoul	seoul	PROPN
ejpam-847	1	90	133	133	NUM
ejpam-847	1	91	-	-	SYM
ejpam-847	1	92	791	791	NUM
ejpam-847	1	93	,	,	PUNCT
ejpam-847	1	94	korea	korea	PROPN
ejpam-847	1	95	abstract	abstract	NOUN
ejpam-847	1	96	.	.	PUNCT
ejpam-847	2	1	in	in	ADP
ejpam-847	2	2	[	[	X
ejpam-847	2	3	14	14	NUM
ejpam-847	2	4	]	]	PUNCT
ejpam-847	2	5	,	,	PUNCT
ejpam-847	2	6	th.m	th.m	PROPN
ejpam-847	2	7	.	.	PUNCT
ejpam-847	3	1	rassias	rassias	PROPN
ejpam-847	3	2	introduced	introduce	VERB
ejpam-847	3	3	the	the	DET
ejpam-847	3	4	following	follow	VERB
ejpam-847	3	5	equality	equality	NOUN
ejpam-847	3	6	n	n	PRON
ejpam-847	3	7	∑	∑	PROPN
ejpam-847	3	8	i	i	PROPN
ejpam-847	3	9	,	,	PUNCT
ejpam-847	3	10	j=1	j=1	PROPN
ejpam-847	3	11	‖x	‖x	PUNCT
ejpam-847	4	1	i	i	PRON
ejpam-847	4	2	−	−	PROPN
ejpam-847	4	3	x	x	X
ejpam-847	4	4	j‖	j‖	NOUN
ejpam-847	4	5	2	2	NUM
ejpam-847	4	6	=	=	SYM
ejpam-847	4	7	2n	2n	NUM
ejpam-847	5	1	n	n	CCONJ
ejpam-847	5	2	∑	∑	PROPN
ejpam-847	5	3	i=1	i=1	PROPN
ejpam-847	5	4	‖x	‖x	PROPN
ejpam-847	5	5	i‖	i‖	PROPN
ejpam-847	5	6	2	2	NUM
ejpam-847	5	7	,	,	PUNCT
ejpam-847	5	8	n	n	CCONJ
ejpam-847	5	9	∑	∑	ADV
ejpam-847	5	10	i=1	i=1	PROPN
ejpam-847	5	11	x	x	PUNCT
ejpam-847	6	1	i	i	NOUN
ejpam-847	6	2	=	=	NOUN
ejpam-847	6	3	0	0	NUM
ejpam-847	6	4	for	for	ADP
ejpam-847	6	5	a	a	DET
ejpam-847	6	6	fixed	fix	VERB
ejpam-847	6	7	integer	integer	NOUN
ejpam-847	6	8	n	n	PRON
ejpam-847	6	9	≥	≥	NOUN
ejpam-847	6	10	3	3	NUM
ejpam-847	6	11	.	.	PUNCT
ejpam-847	6	12	for	for	ADP
ejpam-847	6	13	a	a	DET
ejpam-847	6	14	mapping	mapping	NOUN
ejpam-847	6	15	f	f	NOUN
ejpam-847	6	16	:	:	PUNCT
ejpam-847	6	17	x	x	X
ejpam-847	6	18	→	→	SYM
ejpam-847	6	19	y	y	PROPN
ejpam-847	6	20	,	,	PUNCT
ejpam-847	6	21	where	where	SCONJ
ejpam-847	6	22	x	x	PRON
ejpam-847	6	23	is	be	AUX
ejpam-847	6	24	a	a	DET
ejpam-847	6	25	vector	vector	NOUN
ejpam-847	6	26	space	space	NOUN
ejpam-847	6	27	and	and	CCONJ
ejpam-847	6	28	y	y	PROPN
ejpam-847	6	29	is	be	AUX
ejpam-847	6	30	a	a	DET
ejpam-847	6	31	complete	complete	ADJ
ejpam-847	6	32	random	random	ADJ
ejpam-847	6	33	normed	normed	ADJ
ejpam-847	6	34	space	space	NOUN
ejpam-847	6	35	,	,	PUNCT
ejpam-847	6	36	we	we	PRON
ejpam-847	6	37	consider	consider	VERB
ejpam-847	6	38	the	the	DET
ejpam-847	6	39	following	follow	VERB
ejpam-847	6	40	functional	functional	ADJ
ejpam-847	6	41	equation	equation	NOUN
ejpam-847	6	42	n	n	NOUN
ejpam-847	6	43	∑	∑	PROPN
ejpam-847	6	44	i	i	PROPN
ejpam-847	6	45	,	,	PUNCT
ejpam-847	6	46	j=1	j=1	PROPN
ejpam-847	6	47	f	f	PROPN
ejpam-847	6	48	(	(	PUNCT
ejpam-847	6	49	x	x	PROPN
ejpam-847	6	50	i	i	PRON
ejpam-847	6	51	−	−	NOUN
ejpam-847	6	52	x	x	SYM
ejpam-847	6	53	j	j	PROPN
ejpam-847	6	54	)	)	PUNCT
ejpam-847	6	55	=	=	SYM
ejpam-847	7	1	2n	2n	NUM
ejpam-847	8	1	n	n	CCONJ
ejpam-847	8	2	∑	∑	PROPN
ejpam-847	8	3	i=1	i=1	PROPN
ejpam-847	8	4	f	f	PROPN
ejpam-847	8	5	(	(	PUNCT
ejpam-847	8	6	x	x	PROPN
ejpam-847	8	7	i	i	NOUN
ejpam-847	8	8	)	)	PUNCT
ejpam-847	8	9	(	(	PUNCT
ejpam-847	8	10	1	1	X
ejpam-847	8	11	)	)	PUNCT
ejpam-847	8	12	for	for	ADP
ejpam-847	8	13	all	all	DET
ejpam-847	8	14	x1	x1	PROPN
ejpam-847	8	15	,	,	PUNCT
ejpam-847	8	16	.	.	PUNCT
ejpam-847	8	17	.	.	PUNCT
ejpam-847	8	18	.	.	PUNCT
ejpam-847	9	1	,	,	PUNCT
ejpam-847	9	2	xn	xn	PUNCT
ejpam-847	9	3	∈	∈	PROPN
ejpam-847	9	4	x	x	PUNCT
ejpam-847	9	5	with	with	ADP
ejpam-847	9	6	∑n	∑n	PROPN
ejpam-847	9	7	i=1	i=1	NOUN
ejpam-847	9	8	x	x	PUNCT
ejpam-847	10	1	i	i	NOUN
ejpam-847	10	2	=	=	NOUN
ejpam-847	10	3	0	0	X
ejpam-847	10	4	.	.	PUNCT
ejpam-847	11	1	in	in	ADP
ejpam-847	11	2	this	this	DET
ejpam-847	11	3	paper	paper	NOUN
ejpam-847	11	4	,	,	PUNCT
ejpam-847	11	5	we	we	PRON
ejpam-847	11	6	prove	prove	VERB
ejpam-847	11	7	the	the	DET
ejpam-847	11	8	hyers	hyers	PROPN
ejpam-847	11	9	-	-	PUNCT
ejpam-847	11	10	ulam	ulam	ADJ
ejpam-847	11	11	stability	stability	NOUN
ejpam-847	11	12	of	of	ADP
ejpam-847	11	13	the	the	DET
ejpam-847	11	14	functional	functional	ADJ
ejpam-847	11	15	equation	equation	NOUN
ejpam-847	11	16	(	(	PUNCT
ejpam-847	11	17	1	1	X
ejpam-847	11	18	)	)	PUNCT
ejpam-847	11	19	related	relate	VERB
ejpam-847	11	20	to	to	ADP
ejpam-847	11	21	an	an	DET
ejpam-847	11	22	inner	inner	ADJ
ejpam-847	11	23	product	product	NOUN
ejpam-847	11	24	space	space	NOUN
ejpam-847	11	25	.	.	PUNCT
ejpam-847	12	1	2010	2010	NUM
ejpam-847	12	2	mathematics	mathematic	NOUN
ejpam-847	12	3	subject	subject	NOUN
ejpam-847	12	4	classifications	classification	NOUN
ejpam-847	12	5	:	:	PUNCT
ejpam-847	12	6	39b52	39b52	NUM
ejpam-847	12	7	,	,	PUNCT
ejpam-847	12	8	46s50	46s50	NUM
ejpam-847	12	9	,	,	PUNCT
ejpam-847	12	10	46c05	46c05	NUM
ejpam-847	12	11	,	,	PUNCT
ejpam-847	12	12	47s50	47s50	NUM
ejpam-847	12	13	,	,	PUNCT
ejpam-847	12	14	26e50	26e50	NUM
ejpam-847	12	15	.	.	PUNCT
ejpam-847	13	1	key	key	ADJ
ejpam-847	13	2	words	word	NOUN
ejpam-847	13	3	and	and	CCONJ
ejpam-847	13	4	phrases	phrase	NOUN
ejpam-847	13	5	:	:	PUNCT
ejpam-847	13	6	random	random	ADJ
ejpam-847	13	7	normed	normed	ADJ
ejpam-847	13	8	space	space	NOUN
ejpam-847	13	9	,	,	PUNCT
ejpam-847	13	10	hyers	hyers	PROPN
ejpam-847	13	11	-	-	PUNCT
ejpam-847	13	12	ulam	ulam	PROPN
ejpam-847	13	13	stability	stability	NOUN
ejpam-847	13	14	,	,	PUNCT
ejpam-847	13	15	quadratic	quadratic	ADJ
ejpam-847	13	16	functional	functional	ADJ
ejpam-847	13	17	equation	equation	NOUN
ejpam-847	13	18	,	,	PUNCT
ejpam-847	13	19	inner	inner	ADJ
ejpam-847	13	20	product	product	NOUN
ejpam-847	13	21	space	space	NOUN
ejpam-847	13	22	.	.	PUNCT
ejpam-847	14	1	1	1	X
ejpam-847	14	2	.	.	X
ejpam-847	14	3	introduction	introduction	NOUN
ejpam-847	14	4	a	a	DET
ejpam-847	14	5	square	square	ADJ
ejpam-847	14	6	norm	norm	NOUN
ejpam-847	14	7	on	on	ADP
ejpam-847	14	8	an	an	DET
ejpam-847	14	9	inner	inner	ADJ
ejpam-847	14	10	product	product	NOUN
ejpam-847	14	11	space	space	NOUN
ejpam-847	14	12	satisfies	satisfy	VERB
ejpam-847	14	13	the	the	DET
ejpam-847	14	14	parallelogram	parallelogram	NOUN
ejpam-847	14	15	equality	equality	NOUN
ejpam-847	14	16	‖x	‖x	NOUN
ejpam-847	15	1	+	+	CCONJ
ejpam-847	15	2	y‖2	y‖2	X
ejpam-847	15	3	+	+	CCONJ
ejpam-847	15	4	‖x	‖x	NOUN
ejpam-847	15	5	−	−	NOUN
ejpam-847	15	6	y‖2	y‖2	X
ejpam-847	15	7	=	=	SYM
ejpam-847	16	1	2‖x‖2	2‖x‖2	PROPN
ejpam-847	16	2	+	+	NUM
ejpam-847	16	3	2‖y‖2	2‖y‖2	NOUN
ejpam-847	16	4	.	.	NOUN
ejpam-847	17	1	from	from	ADP
ejpam-847	17	2	the	the	DET
ejpam-847	17	3	above	above	ADJ
ejpam-847	17	4	equation	equation	NOUN
ejpam-847	17	5	,	,	PUNCT
ejpam-847	17	6	we	we	PRON
ejpam-847	17	7	consider	consider	VERB
ejpam-847	17	8	the	the	DET
ejpam-847	17	9	following	follow	VERB
ejpam-847	17	10	functional	functional	ADJ
ejpam-847	17	11	equation	equation	NOUN
ejpam-847	17	12	f	f	X
ejpam-847	17	13	(	(	PUNCT
ejpam-847	17	14	x	x	PROPN
ejpam-847	17	15	+	+	NUM
ejpam-847	17	16	y	y	NOUN
ejpam-847	17	17	)	)	PUNCT
ejpam-847	18	1	+	+	NOUN
ejpam-847	18	2	f	f	X
ejpam-847	18	3	(	(	PUNCT
ejpam-847	18	4	x	x	INTJ
ejpam-847	18	5	−	−	PROPN
ejpam-847	18	6	y	y	PROPN
ejpam-847	18	7	)	)	PUNCT
ejpam-847	18	8	=	=	SYM
ejpam-847	18	9	2	2	NUM
ejpam-847	18	10	f	f	NOUN
ejpam-847	18	11	(	(	PUNCT
ejpam-847	18	12	x)+	x)+	PROPN
ejpam-847	18	13	2	2	NUM
ejpam-847	18	14	f	f	NOUN
ejpam-847	18	15	(	(	PUNCT
ejpam-847	18	16	y	y	NOUN
ejpam-847	18	17	)	)	PUNCT
ejpam-847	18	18	∗corresponding	∗corresponde	VERB
ejpam-847	18	19	author	author	NOUN
ejpam-847	18	20	.	.	PUNCT
ejpam-847	19	1	email	email	NOUN
ejpam-847	19	2	addresses	address	NOUN
ejpam-847	19	3	:	:	PUNCT
ejpam-847	19	4	dyshin	dyshin	PROPN
ejpam-847	19	5	�	�	PROPN
ejpam-847	19	6	uos.a	uos.a	NUM
ejpam-847	19	7	.kr	.kr	PUNCT
ejpam-847	19	8	(	(	PUNCT
ejpam-847	19	9	d.	d.	PROPN
ejpam-847	19	10	shin	shin	PROPN
ejpam-847	19	11	)	)	PUNCT
ejpam-847	19	12	,	,	PUNCT
ejpam-847	19	13	jrlee	jrlee	PROPN
ejpam-847	19	14	�	�	PROPN
ejpam-847	19	15	daejin.a	daejin.a	PROPN
ejpam-847	19	16	.kr	.kr	PUNCT
ejpam-847	20	1	(	(	PUNCT
ejpam-847	20	2	j.	j.	PROPN
ejpam-847	20	3	lee	lee	PROPN
ejpam-847	20	4	)	)	PUNCT
ejpam-847	20	5	,	,	PUNCT
ejpam-847	20	6	baak	baak	PROPN
ejpam-847	20	7	�	�	PROPN
ejpam-847	20	8	hanyang.a	hanyang.a	PROPN
ejpam-847	20	9	.kr	.kr	PUNCT
ejpam-847	20	10	(	(	PUNCT
ejpam-847	20	11	c.	c.	PROPN
ejpam-847	20	12	park	park	PROPN
ejpam-847	20	13	)	)	PUNCT
ejpam-847	20	14	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-847	21	1	540	540	NUM
ejpam-847	21	2	c	c	X
ejpam-847	21	3	©	©	VERB
ejpam-847	21	4	2012	2012	NUM
ejpam-847	21	5	ejpam	ejpam	VERB
ejpam-847	21	6	all	all	DET
ejpam-847	21	7	rights	right	NOUN
ejpam-847	21	8	reserved	reserve	VERB
ejpam-847	21	9	.	.	PUNCT
ejpam-847	22	1	d.	d.	PROPN
ejpam-847	22	2	shin	shin	PROPN
ejpam-847	22	3	,	,	PUNCT
ejpam-847	22	4	j.	j.	PROPN
ejpam-847	22	5	lee	lee	PROPN
ejpam-847	22	6	,	,	PUNCT
ejpam-847	22	7	c.	c.	PROPN
ejpam-847	22	8	park	park	PROPN
ejpam-847	22	9	/	/	SYM
ejpam-847	22	10	eur	eur	PROPN
ejpam-847	22	11	.	.	PUNCT
ejpam-847	23	1	j.	j.	PROPN
ejpam-847	23	2	pure	pure	PROPN
ejpam-847	23	3	appl	appl	PROPN
ejpam-847	23	4	.	.	PROPN
ejpam-847	23	5	math	math	PROPN
ejpam-847	23	6	,	,	PUNCT
ejpam-847	23	7	5	5	NUM
ejpam-847	23	8	(	(	PUNCT
ejpam-847	23	9	2012	2012	NUM
ejpam-847	23	10	)	)	PUNCT
ejpam-847	23	11	,	,	PUNCT
ejpam-847	23	12	540	540	NUM
ejpam-847	23	13	-	-	SYM
ejpam-847	23	14	553	553	NUM
ejpam-847	23	15	541	541	NUM
ejpam-847	23	16	related	relate	VERB
ejpam-847	23	17	to	to	ADP
ejpam-847	23	18	an	an	DET
ejpam-847	23	19	inner	inner	ADJ
ejpam-847	23	20	product	product	NOUN
ejpam-847	23	21	space	space	NOUN
ejpam-847	23	22	.	.	PUNCT
ejpam-847	24	1	the	the	DET
ejpam-847	24	2	stability	stability	NOUN
ejpam-847	24	3	problem	problem	NOUN
ejpam-847	24	4	of	of	ADP
ejpam-847	24	5	functional	functional	ADJ
ejpam-847	24	6	equations	equation	NOUN
ejpam-847	24	7	originated	originate	VERB
ejpam-847	24	8	from	from	ADP
ejpam-847	24	9	a	a	DET
ejpam-847	24	10	question	question	NOUN
ejpam-847	24	11	of	of	ADP
ejpam-847	24	12	s.m	s.m	PROPN
ejpam-847	24	13	.	.	PUNCT
ejpam-847	24	14	ulam	ulam	PROPN
ejpam-847	25	1	[	[	X
ejpam-847	25	2	18	18	NUM
ejpam-847	25	3	]	]	PUNCT
ejpam-847	25	4	concerning	concern	VERB
ejpam-847	25	5	the	the	DET
ejpam-847	25	6	stability	stability	NOUN
ejpam-847	25	7	of	of	ADP
ejpam-847	25	8	group	group	NOUN
ejpam-847	25	9	homomorphisms	homomorphism	NOUN
ejpam-847	25	10	.	.	PUNCT
ejpam-847	26	1	d.h	d.h	PROPN
ejpam-847	26	2	.	.	PUNCT
ejpam-847	27	1	hyers	hyer	NOUN
ejpam-847	28	1	[	[	X
ejpam-847	28	2	5	5	X
ejpam-847	28	3	]	]	PUNCT
ejpam-847	28	4	gave	give	VERB
ejpam-847	28	5	a	a	DET
ejpam-847	28	6	first	first	ADJ
ejpam-847	28	7	affirmative	affirmative	ADJ
ejpam-847	28	8	partial	partial	ADJ
ejpam-847	28	9	answer	answer	NOUN
ejpam-847	28	10	to	to	ADP
ejpam-847	28	11	the	the	DET
ejpam-847	28	12	question	question	NOUN
ejpam-847	28	13	of	of	ADP
ejpam-847	28	14	ulam	ulam	NOUN
ejpam-847	28	15	for	for	ADP
ejpam-847	28	16	banach	banach	NOUN
ejpam-847	28	17	spaces	space	NOUN
ejpam-847	28	18	and	and	CCONJ
ejpam-847	28	19	hyers	hyer	NOUN
ejpam-847	28	20	’	'	PUNCT
ejpam-847	28	21	theorem	theorem	NOUN
ejpam-847	28	22	was	be	AUX
ejpam-847	28	23	generalized	generalize	VERB
ejpam-847	28	24	by	by	ADP
ejpam-847	28	25	th.m	th.m	PROPN
ejpam-847	28	26	.	.	PUNCT
ejpam-847	29	1	rassias	rassias	PROPN
ejpam-847	30	1	[	[	X
ejpam-847	30	2	13	13	NUM
ejpam-847	30	3	]	]	PUNCT
ejpam-847	30	4	for	for	ADP
ejpam-847	30	5	linear	linear	ADJ
ejpam-847	30	6	mappings	mapping	NOUN
ejpam-847	30	7	by	by	ADP
ejpam-847	30	8	considering	consider	VERB
ejpam-847	30	9	an	an	DET
ejpam-847	30	10	unbounded	unbounded	ADJ
ejpam-847	30	11	cauchy	cauchy	ADJ
ejpam-847	30	12	difference	difference	NOUN
ejpam-847	30	13	.	.	PUNCT
ejpam-847	31	1	especially	especially	ADV
ejpam-847	31	2	,	,	PUNCT
ejpam-847	31	3	the	the	DET
ejpam-847	31	4	hyers	hyers	PROPN
ejpam-847	31	5	-	-	PUNCT
ejpam-847	31	6	ulam	ulam	PROPN
ejpam-847	31	7	stability	stability	NOUN
ejpam-847	31	8	of	of	ADP
ejpam-847	31	9	the	the	DET
ejpam-847	31	10	above	above	ADJ
ejpam-847	31	11	functional	functional	ADJ
ejpam-847	31	12	equation	equation	NOUN
ejpam-847	31	13	related	relate	VERB
ejpam-847	31	14	to	to	ADP
ejpam-847	31	15	an	an	DET
ejpam-847	31	16	inner	inner	ADJ
ejpam-847	31	17	product	product	NOUN
ejpam-847	31	18	space	space	NOUN
ejpam-847	31	19	has	have	AUX
ejpam-847	31	20	been	be	AUX
ejpam-847	31	21	studied	study	VERB
ejpam-847	31	22	[	[	PUNCT
ejpam-847	31	23	see	see	VERB
ejpam-847	31	24	7	7	NUM
ejpam-847	31	25	,	,	PUNCT
ejpam-847	31	26	17	17	NUM
ejpam-847	31	27	]	]	PUNCT
ejpam-847	31	28	.	.	PUNCT
ejpam-847	32	1	a	a	DET
ejpam-847	32	2	square	square	ADJ
ejpam-847	32	3	norm	norm	NOUN
ejpam-847	32	4	on	on	ADP
ejpam-847	32	5	an	an	DET
ejpam-847	32	6	inner	inner	ADJ
ejpam-847	32	7	product	product	NOUN
ejpam-847	32	8	space	space	NOUN
ejpam-847	32	9	also	also	ADV
ejpam-847	32	10	satisfies	satisfy	VERB
ejpam-847	32	11	3	3	NUM
ejpam-847	32	12	∑	∑	PROPN
ejpam-847	32	13	i	i	PROPN
ejpam-847	32	14	,	,	PUNCT
ejpam-847	32	15	j=1	j=1	PROPN
ejpam-847	32	16	‖x	‖x	PUNCT
ejpam-847	33	1	i	i	PRON
ejpam-847	33	2	−	−	PROPN
ejpam-847	33	3	x	x	X
ejpam-847	33	4	j‖	j‖	NOUN
ejpam-847	33	5	2	2	NUM
ejpam-847	33	6	=	=	SYM
ejpam-847	33	7	6	6	NUM
ejpam-847	33	8	3	3	NUM
ejpam-847	33	9	∑	∑	PROPN
ejpam-847	33	10	i=1	i=1	PROPN
ejpam-847	33	11	‖x	‖x	PUNCT
ejpam-847	33	12	i‖	i‖	PROPN
ejpam-847	33	13	2	2	NUM
ejpam-847	33	14	for	for	ADP
ejpam-847	33	15	all	all	DET
ejpam-847	33	16	x1	x1	PROPN
ejpam-847	33	17	,	,	PUNCT
ejpam-847	33	18	x2	x2	PROPN
ejpam-847	33	19	,	,	PUNCT
ejpam-847	33	20	x3	x3	PROPN
ejpam-847	33	21	∈	∈	NOUN
ejpam-847	33	22	r	r	NOUN
ejpam-847	33	23	with	with	ADP
ejpam-847	33	24	x1	x1	PROPN
ejpam-847	33	25	+	+	NUM
ejpam-847	33	26	x2	x2	PROPN
ejpam-847	34	1	+	+	CCONJ
ejpam-847	34	2	x3	x3	ADJ
ejpam-847	34	3	=	=	SYM
ejpam-847	34	4	0	0	PUNCT
ejpam-847	35	1	[	[	X
ejpam-847	35	2	see	see	VERB
ejpam-847	35	3	14	14	NUM
ejpam-847	35	4	]	]	PUNCT
ejpam-847	35	5	.	.	PUNCT
ejpam-847	36	1	from	from	ADP
ejpam-847	36	2	the	the	DET
ejpam-847	36	3	above	above	ADJ
ejpam-847	36	4	equality	equality	NOUN
ejpam-847	36	5	we	we	PRON
ejpam-847	36	6	can	can	AUX
ejpam-847	36	7	define	define	VERB
ejpam-847	36	8	the	the	DET
ejpam-847	36	9	functional	functional	ADJ
ejpam-847	36	10	equation	equation	NOUN
ejpam-847	36	11	f	f	X
ejpam-847	36	12	(	(	PUNCT
ejpam-847	36	13	x	x	PROPN
ejpam-847	36	14	−	−	PROPN
ejpam-847	36	15	y	y	PROPN
ejpam-847	36	16	)	)	PUNCT
ejpam-847	37	1	+	+	CCONJ
ejpam-847	37	2	f	f	X
ejpam-847	37	3	(	(	PUNCT
ejpam-847	37	4	2x	2x	NUM
ejpam-847	37	5	+	+	CCONJ
ejpam-847	37	6	y	y	X
ejpam-847	37	7	)	)	PUNCT
ejpam-847	38	1	+	+	NOUN
ejpam-847	38	2	f	f	X
ejpam-847	38	3	(	(	PUNCT
ejpam-847	38	4	x	x	X
ejpam-847	38	5	+	+	PUNCT
ejpam-847	38	6	2y	2y	NUM
ejpam-847	38	7	)	)	PUNCT
ejpam-847	38	8	=	=	SYM
ejpam-847	38	9	3	3	NUM
ejpam-847	38	10	f	f	X
ejpam-847	38	11	(	(	PUNCT
ejpam-847	38	12	x)+	x)+	PROPN
ejpam-847	38	13	3	3	NUM
ejpam-847	38	14	f	f	NOUN
ejpam-847	38	15	(	(	PUNCT
ejpam-847	38	16	y	y	NOUN
ejpam-847	38	17	)	)	PUNCT
ejpam-847	38	18	+	+	CCONJ
ejpam-847	38	19	3	3	NUM
ejpam-847	38	20	f	f	X
ejpam-847	38	21	(	(	PUNCT
ejpam-847	38	22	x	x	PROPN
ejpam-847	38	23	+	+	NUM
ejpam-847	38	24	y	y	NOUN
ejpam-847	38	25	)	)	PUNCT
ejpam-847	38	26	,	,	PUNCT
ejpam-847	38	27	which	which	PRON
ejpam-847	38	28	is	be	AUX
ejpam-847	38	29	called	call	VERB
ejpam-847	38	30	a	a	DET
ejpam-847	38	31	quadratic	quadratic	ADJ
ejpam-847	38	32	functional	functional	ADJ
ejpam-847	38	33	equation	equation	NOUN
ejpam-847	38	34	.	.	PUNCT
ejpam-847	39	1	in	in	ADP
ejpam-847	39	2	fact	fact	NOUN
ejpam-847	39	3	,	,	PUNCT
ejpam-847	39	4	f	f	PROPN
ejpam-847	39	5	(	(	PUNCT
ejpam-847	39	6	x	x	X
ejpam-847	39	7	)	)	PUNCT
ejpam-847	39	8	=	=	SYM
ejpam-847	39	9	ax2	ax2	NOUN
ejpam-847	39	10	in	in	ADP
ejpam-847	39	11	r	r	NOUN
ejpam-847	39	12	satisfies	satisfie	NOUN
ejpam-847	39	13	the	the	DET
ejpam-847	39	14	quadratic	quadratic	ADJ
ejpam-847	39	15	functional	functional	ADJ
ejpam-847	39	16	equation	equation	NOUN
ejpam-847	39	17	.	.	PUNCT
ejpam-847	40	1	the	the	DET
ejpam-847	40	2	aim	aim	NOUN
ejpam-847	40	3	of	of	ADP
ejpam-847	40	4	this	this	DET
ejpam-847	40	5	paper	paper	NOUN
ejpam-847	40	6	is	be	AUX
ejpam-847	40	7	to	to	PART
ejpam-847	40	8	investigate	investigate	VERB
ejpam-847	40	9	the	the	DET
ejpam-847	40	10	hyers	hyer	NOUN
ejpam-847	40	11	-	-	PUNCT
ejpam-847	40	12	ulam	ulam	ADJ
ejpam-847	40	13	stability	stability	NOUN
ejpam-847	40	14	of	of	ADP
ejpam-847	40	15	additive	additive	ADJ
ejpam-847	40	16	-	-	PUNCT
ejpam-847	40	17	quadratic	quadratic	ADJ
ejpam-847	40	18	functional	functional	ADJ
ejpam-847	40	19	equation	equation	NOUN
ejpam-847	40	20	in	in	ADP
ejpam-847	40	21	a	a	DET
ejpam-847	40	22	random	random	ADJ
ejpam-847	40	23	normed	normed	ADJ
ejpam-847	40	24	space	space	NOUN
ejpam-847	40	25	related	relate	VERB
ejpam-847	40	26	to	to	ADP
ejpam-847	40	27	an	an	DET
ejpam-847	40	28	inner	inner	ADJ
ejpam-847	40	29	product	product	NOUN
ejpam-847	40	30	space	space	NOUN
ejpam-847	40	31	.	.	PUNCT
ejpam-847	41	1	throughout	throughout	ADP
ejpam-847	41	2	this	this	DET
ejpam-847	41	3	paper	paper	NOUN
ejpam-847	41	4	,	,	PUNCT
ejpam-847	41	5	we	we	PRON
ejpam-847	41	6	use	use	VERB
ejpam-847	41	7	the	the	DET
ejpam-847	41	8	definition	definition	NOUN
ejpam-847	41	9	of	of	ADP
ejpam-847	41	10	a	a	DET
ejpam-847	41	11	random	random	ADJ
ejpam-847	41	12	normed	normed	ADJ
ejpam-847	41	13	space	space	NOUN
ejpam-847	41	14	as	as	ADP
ejpam-847	41	15	in	in	ADP
ejpam-847	41	16	[	[	X
ejpam-847	41	17	1	1	NUM
ejpam-847	41	18	,	,	PUNCT
ejpam-847	41	19	10	10	NUM
ejpam-847	41	20	,	,	PUNCT
ejpam-847	41	21	15	15	NUM
ejpam-847	41	22	,	,	PUNCT
ejpam-847	41	23	16	16	NUM
ejpam-847	41	24	]	]	PUNCT
ejpam-847	41	25	.	.	PUNCT
ejpam-847	42	1	∆+	∆+	NUM
ejpam-847	42	2	is	be	AUX
ejpam-847	42	3	the	the	DET
ejpam-847	42	4	space	space	NOUN
ejpam-847	42	5	of	of	ADP
ejpam-847	42	6	distribution	distribution	NOUN
ejpam-847	42	7	functions	function	NOUN
ejpam-847	42	8	that	that	PRON
ejpam-847	42	9	is	be	AUX
ejpam-847	42	10	,	,	PUNCT
ejpam-847	42	11	the	the	DET
ejpam-847	42	12	space	space	NOUN
ejpam-847	42	13	of	of	ADP
ejpam-847	42	14	all	all	DET
ejpam-847	42	15	mappings	mapping	NOUN
ejpam-847	43	1	f	f	NOUN
ejpam-847	43	2	:	:	PUNCT
ejpam-847	43	3	r	r	NOUN
ejpam-847	43	4	∪	∪	ADP
ejpam-847	43	5	{	{	PUNCT
ejpam-847	43	6	−∞,∞	−∞,∞	NOUN
ejpam-847	43	7	}	}	PUNCT
ejpam-847	43	8	→	→	SYM
ejpam-847	43	9	[	[	X
ejpam-847	43	10	0,1	0,1	NUM
ejpam-847	43	11	]	]	PUNCT
ejpam-847	43	12	which	which	PRON
ejpam-847	43	13	is	be	AUX
ejpam-847	43	14	left	leave	VERB
ejpam-847	43	15	-	-	PUNCT
ejpam-847	43	16	continuous	continuous	ADJ
ejpam-847	43	17	and	and	CCONJ
ejpam-847	43	18	non	non	ADJ
ejpam-847	43	19	-	-	ADJ
ejpam-847	43	20	decreasing	decrease	VERB
ejpam-847	43	21	on	on	ADP
ejpam-847	43	22	r	r	NOUN
ejpam-847	43	23	,	,	PUNCT
ejpam-847	43	24	f(0	f(0	NOUN
ejpam-847	43	25	)	)	PUNCT
ejpam-847	43	26	=	=	SYM
ejpam-847	43	27	0	0	NUM
ejpam-847	43	28	and	and	CCONJ
ejpam-847	43	29	f(+∞	f(+∞	NOUN
ejpam-847	43	30	)	)	PUNCT
ejpam-847	43	31	=	=	SYM
ejpam-847	43	32	1	1	X
ejpam-847	43	33	.	.	X
ejpam-847	43	34	d+	d+	NOUN
ejpam-847	43	35	is	be	AUX
ejpam-847	43	36	a	a	DET
ejpam-847	43	37	subset	subset	NOUN
ejpam-847	43	38	of	of	ADP
ejpam-847	43	39	∆+	∆+	PUNCT
ejpam-847	43	40	consisting	consist	VERB
ejpam-847	43	41	of	of	ADP
ejpam-847	43	42	all	all	DET
ejpam-847	43	43	functions	function	NOUN
ejpam-847	43	44	f	f	PROPN
ejpam-847	43	45	for	for	ADP
ejpam-847	43	46	which	which	PRON
ejpam-847	43	47	l−f(+∞	l−f(+∞	VERB
ejpam-847	43	48	)	)	PUNCT
ejpam-847	43	49	=	=	SYM
ejpam-847	43	50	1	1	NUM
ejpam-847	43	51	,	,	PUNCT
ejpam-847	43	52	where	where	SCONJ
ejpam-847	43	53	l−	l−	PROPN
ejpam-847	43	54	f	f	X
ejpam-847	43	55	(	(	PUNCT
ejpam-847	43	56	x	x	X
ejpam-847	43	57	)	)	PUNCT
ejpam-847	43	58	denotes	denote	VERB
ejpam-847	43	59	the	the	DET
ejpam-847	43	60	left	left	ADJ
ejpam-847	43	61	limit	limit	NOUN
ejpam-847	43	62	of	of	ADP
ejpam-847	43	63	the	the	DET
ejpam-847	43	64	function	function	NOUN
ejpam-847	43	65	f	f	PROPN
ejpam-847	43	66	at	at	ADP
ejpam-847	43	67	the	the	DET
ejpam-847	43	68	point	point	NOUN
ejpam-847	43	69	x	x	X
ejpam-847	43	70	.	.	PUNCT
ejpam-847	44	1	the	the	DET
ejpam-847	44	2	space	space	NOUN
ejpam-847	44	3	∆+	∆+	NOUN
ejpam-847	44	4	is	be	AUX
ejpam-847	44	5	partially	partially	ADV
ejpam-847	44	6	ordered	order	VERB
ejpam-847	44	7	by	by	ADP
ejpam-847	44	8	the	the	DET
ejpam-847	44	9	usual	usual	ADJ
ejpam-847	44	10	point	point	NOUN
ejpam-847	44	11	-	-	PUNCT
ejpam-847	44	12	wise	wise	ADJ
ejpam-847	44	13	ordering	ordering	NOUN
ejpam-847	44	14	of	of	ADP
ejpam-847	44	15	functions	function	NOUN
ejpam-847	44	16	.	.	PUNCT
ejpam-847	45	1	the	the	DET
ejpam-847	45	2	maximal	maximal	ADJ
ejpam-847	45	3	element	element	NOUN
ejpam-847	45	4	for	for	ADP
ejpam-847	45	5	∆+	∆+	NUM
ejpam-847	45	6	in	in	ADP
ejpam-847	45	7	this	this	DET
ejpam-847	45	8	order	order	NOUN
ejpam-847	45	9	is	be	AUX
ejpam-847	45	10	the	the	DET
ejpam-847	45	11	distribution	distribution	NOUN
ejpam-847	45	12	function	function	NOUN
ejpam-847	45	13	ǫ0	ǫ0	NOUN
ejpam-847	45	14	given	give	VERB
ejpam-847	45	15	by	by	ADP
ejpam-847	45	16	ǫ0(t	ǫ0(t	NOUN
ejpam-847	45	17	)	)	PUNCT
ejpam-847	45	18	=	=	SYM
ejpam-847	46	1	(	(	PUNCT
ejpam-847	46	2	0	0	NUM
ejpam-847	46	3	,	,	PUNCT
ejpam-847	46	4	if	if	SCONJ
ejpam-847	46	5	t	t	NOUN
ejpam-847	46	6	≤	≤	NUM
ejpam-847	46	7	0	0	NUM
ejpam-847	46	8	,	,	PUNCT
ejpam-847	46	9	1	1	NUM
ejpam-847	46	10	,	,	PUNCT
ejpam-847	46	11	if	if	SCONJ
ejpam-847	46	12	t	t	PROPN
ejpam-847	46	13	>	>	X
ejpam-847	46	14	0	0	X
ejpam-847	46	15	.	.	PUNCT
ejpam-847	47	1	definition	definition	NOUN
ejpam-847	47	2	1	1	NUM
ejpam-847	47	3	(	(	PUNCT
ejpam-847	47	4	[	[	X
ejpam-847	47	5	15	15	NUM
ejpam-847	47	6	]	]	NUM
ejpam-847	47	7	)	)	PUNCT
ejpam-847	47	8	.	.	PUNCT
ejpam-847	48	1	a	a	DET
ejpam-847	48	2	mapping	mapping	NOUN
ejpam-847	48	3	t	t	NOUN
ejpam-847	48	4	:	:	PUNCT
ejpam-847	49	1	[	[	X
ejpam-847	49	2	0,1]×	0,1]×	ADJ
ejpam-847	49	3	[	[	X
ejpam-847	49	4	0,1	0,1	NUM
ejpam-847	49	5	]	]	PUNCT
ejpam-847	49	6	→	→	X
ejpam-847	49	7	[	[	X
ejpam-847	49	8	0,1	0,1	NUM
ejpam-847	49	9	]	]	PUNCT
ejpam-847	49	10	is	be	AUX
ejpam-847	49	11	a	a	DET
ejpam-847	49	12	continuous	continuous	ADJ
ejpam-847	49	13	triangular	triangular	NOUN
ejpam-847	49	14	norm	norm	NOUN
ejpam-847	49	15	(	(	PUNCT
ejpam-847	49	16	briefly	briefly	ADV
ejpam-847	49	17	,	,	PUNCT
ejpam-847	49	18	a	a	DET
ejpam-847	49	19	continuous	continuous	ADJ
ejpam-847	49	20	t	t	NOUN
ejpam-847	49	21	-	-	PUNCT
ejpam-847	49	22	norm	norm	NOUN
ejpam-847	49	23	)	)	PUNCT
ejpam-847	49	24	if	if	SCONJ
ejpam-847	49	25	t	t	PROPN
ejpam-847	49	26	satisfies	satisfy	VERB
ejpam-847	49	27	the	the	DET
ejpam-847	49	28	following	follow	VERB
ejpam-847	49	29	conditions	condition	NOUN
ejpam-847	49	30	:	:	PUNCT
ejpam-847	49	31	(	(	PUNCT
ejpam-847	49	32	a	a	X
ejpam-847	49	33	)	)	PUNCT
ejpam-847	49	34	t	t	PROPN
ejpam-847	49	35	is	be	AUX
ejpam-847	49	36	commutative	commutative	ADJ
ejpam-847	49	37	and	and	CCONJ
ejpam-847	49	38	associative	associative	ADJ
ejpam-847	49	39	;	;	PUNCT
ejpam-847	49	40	(	(	PUNCT
ejpam-847	49	41	b	b	X
ejpam-847	49	42	)	)	PUNCT
ejpam-847	49	43	t	t	PROPN
ejpam-847	49	44	is	be	AUX
ejpam-847	49	45	continuous	continuous	ADJ
ejpam-847	49	46	;	;	PUNCT
ejpam-847	49	47	(	(	PUNCT
ejpam-847	49	48	c	c	X
ejpam-847	49	49	)	)	PUNCT
ejpam-847	49	50	t	t	NOUN
ejpam-847	49	51	(	(	PUNCT
ejpam-847	49	52	a	a	DET
ejpam-847	49	53	,	,	PUNCT
ejpam-847	49	54	1	1	NUM
ejpam-847	49	55	)	)	PUNCT
ejpam-847	49	56	=	=	NOUN
ejpam-847	49	57	a	a	PRON
ejpam-847	49	58	for	for	ADP
ejpam-847	49	59	all	all	DET
ejpam-847	49	60	a	a	DET
ejpam-847	49	61	∈	∈	NOUN
ejpam-847	50	1	[	[	X
ejpam-847	50	2	0,1	0,1	NUM
ejpam-847	50	3	]	]	PUNCT
ejpam-847	50	4	;	;	PUNCT
ejpam-847	50	5	(	(	PUNCT
ejpam-847	50	6	d	d	X
ejpam-847	50	7	)	)	PUNCT
ejpam-847	50	8	t	t	PROPN
ejpam-847	50	9	(	(	PUNCT
ejpam-847	50	10	a	a	PRON
ejpam-847	50	11	,	,	PUNCT
ejpam-847	50	12	b	b	NOUN
ejpam-847	50	13	)	)	PUNCT
ejpam-847	50	14	≤	≤	NOUN
ejpam-847	50	15	t	t	NOUN
ejpam-847	50	16	(	(	PUNCT
ejpam-847	50	17	c	c	X
ejpam-847	50	18	,	,	PUNCT
ejpam-847	50	19	d	d	NOUN
ejpam-847	50	20	)	)	PUNCT
ejpam-847	50	21	whenever	whenever	SCONJ
ejpam-847	50	22	a	a	DET
ejpam-847	50	23	≤	≤	NUM
ejpam-847	50	24	c	c	NOUN
ejpam-847	50	25	and	and	CCONJ
ejpam-847	50	26	b	b	NOUN
ejpam-847	50	27	≤	≤	NUM
ejpam-847	50	28	d	d	NOUN
ejpam-847	50	29	for	for	ADP
ejpam-847	50	30	all	all	DET
ejpam-847	50	31	a	a	DET
ejpam-847	50	32	,	,	PUNCT
ejpam-847	50	33	b	b	NOUN
ejpam-847	50	34	,	,	PUNCT
ejpam-847	50	35	c	c	NOUN
ejpam-847	50	36	,	,	PUNCT
ejpam-847	50	37	d	d	PROPN
ejpam-847	50	38	∈	∈	PROPN
ejpam-847	51	1	[	[	X
ejpam-847	51	2	0,1	0,1	NUM
ejpam-847	51	3	]	]	PUNCT
ejpam-847	51	4	.	.	PUNCT
ejpam-847	52	1	recall	recall	VERB
ejpam-847	52	2	that	that	SCONJ
ejpam-847	52	3	if	if	SCONJ
ejpam-847	52	4	t	t	PROPN
ejpam-847	52	5	is	be	AUX
ejpam-847	52	6	a	a	DET
ejpam-847	52	7	t	t	NOUN
ejpam-847	52	8	-	-	PUNCT
ejpam-847	52	9	norm	norm	NOUN
ejpam-847	52	10	and	and	CCONJ
ejpam-847	52	11	{	{	PUNCT
ejpam-847	52	12	xn	xn	X
ejpam-847	52	13	}	}	PUNCT
ejpam-847	52	14	is	be	AUX
ejpam-847	52	15	a	a	DET
ejpam-847	52	16	sequence	sequence	NOUN
ejpam-847	52	17	of	of	ADP
ejpam-847	52	18	numbers	number	NOUN
ejpam-847	52	19	in	in	ADP
ejpam-847	52	20	[	[	X
ejpam-847	52	21	0,1	0,1	NUM
ejpam-847	52	22	]	]	PUNCT
ejpam-847	52	23	,	,	PUNCT
ejpam-847	52	24	then	then	ADV
ejpam-847	52	25	t	t	PROPN
ejpam-847	52	26	n	n	CCONJ
ejpam-847	52	27	i=1	i=1	PROPN
ejpam-847	52	28	x	x	VERB
ejpam-847	53	1	i	i	PRON
ejpam-847	53	2	is	be	AUX
ejpam-847	53	3	defined	define	VERB
ejpam-847	53	4	recurrently	recurrently	ADV
ejpam-847	53	5	by	by	ADP
ejpam-847	53	6	t	t	PROPN
ejpam-847	53	7	1	1	NUM
ejpam-847	53	8	i=1	i=1	PROPN
ejpam-847	53	9	x	x	PUNCT
ejpam-847	54	1	i	i	NOUN
ejpam-847	54	2	=	=	PUNCT
ejpam-847	54	3	x1	x1	PROPN
ejpam-847	54	4	and	and	CCONJ
ejpam-847	54	5	t	t	PROPN
ejpam-847	54	6	n	n	CCONJ
ejpam-847	54	7	i=1	i=1	PROPN
ejpam-847	54	8	x	x	PUNCT
ejpam-847	55	1	i	i	PROPN
ejpam-847	55	2	=	=	SYM
ejpam-847	55	3	t	t	PROPN
ejpam-847	55	4	(	(	PUNCT
ejpam-847	55	5	t	t	PROPN
ejpam-847	55	6	n−1	n−1	PROPN
ejpam-847	55	7	i=1	i=1	PROPN
ejpam-847	55	8	x	x	PROPN
ejpam-847	55	9	i	i	PROPN
ejpam-847	55	10	,	,	PUNCT
ejpam-847	55	11	xn	xn	PROPN
ejpam-847	55	12	)	)	PUNCT
ejpam-847	55	13	for	for	ADP
ejpam-847	55	14	n	n	X
ejpam-847	55	15	≥	≥	NUM
ejpam-847	55	16	2	2	NUM
ejpam-847	55	17	[	[	X
ejpam-847	55	18	see	see	VERB
ejpam-847	55	19	3	3	NUM
ejpam-847	55	20	]	]	PUNCT
ejpam-847	55	21	.	.	PUNCT
ejpam-847	56	1	t∞	t∞	NOUN
ejpam-847	56	2	i=1	i=1	X
ejpam-847	57	1	x	x	X
ejpam-847	57	2	i	i	PRON
ejpam-847	57	3	is	be	AUX
ejpam-847	57	4	defined	define	VERB
ejpam-847	57	5	as	as	ADP
ejpam-847	57	6	limm→∞	limm→∞	PROPN
ejpam-847	57	7	t	t	PROPN
ejpam-847	57	8	m	m	VERB
ejpam-847	57	9	i=1	i=1	PROPN
ejpam-847	57	10	x	x	PUNCT
ejpam-847	57	11	i.	i.	PROPN
ejpam-847	57	12	d.	d.	PROPN
ejpam-847	57	13	shin	shin	PROPN
ejpam-847	57	14	,	,	PUNCT
ejpam-847	57	15	j.	j.	PROPN
ejpam-847	57	16	lee	lee	PROPN
ejpam-847	57	17	,	,	PUNCT
ejpam-847	57	18	c.	c.	PROPN
ejpam-847	57	19	park	park	PROPN
ejpam-847	57	20	/	/	SYM
ejpam-847	57	21	eur	eur	PROPN
ejpam-847	57	22	.	.	PUNCT
ejpam-847	58	1	j.	j.	PROPN
ejpam-847	58	2	pure	pure	PROPN
ejpam-847	58	3	appl	appl	PROPN
ejpam-847	58	4	.	.	PROPN
ejpam-847	58	5	math	math	PROPN
ejpam-847	58	6	,	,	PUNCT
ejpam-847	58	7	5	5	NUM
ejpam-847	58	8	(	(	PUNCT
ejpam-847	58	9	2012	2012	NUM
ejpam-847	58	10	)	)	PUNCT
ejpam-847	58	11	,	,	PUNCT
ejpam-847	58	12	540	540	NUM
ejpam-847	58	13	-	-	SYM
ejpam-847	58	14	553	553	NUM
ejpam-847	58	15	542	542	NUM
ejpam-847	58	16	definition	definition	NOUN
ejpam-847	58	17	2	2	NUM
ejpam-847	58	18	(	(	PUNCT
ejpam-847	58	19	[	[	X
ejpam-847	58	20	16	16	NUM
ejpam-847	58	21	]	]	PUNCT
ejpam-847	58	22	)	)	PUNCT
ejpam-847	58	23	.	.	PUNCT
ejpam-847	59	1	a	a	DET
ejpam-847	59	2	random	random	ADJ
ejpam-847	59	3	normed	normed	ADJ
ejpam-847	59	4	space	space	NOUN
ejpam-847	59	5	(	(	PUNCT
ejpam-847	59	6	briefly	briefly	ADV
ejpam-847	59	7	,	,	PUNCT
ejpam-847	59	8	rn	rn	NOUN
ejpam-847	59	9	-	-	NOUN
ejpam-847	59	10	space	space	NOUN
ejpam-847	59	11	)	)	PUNCT
ejpam-847	59	12	is	be	AUX
ejpam-847	59	13	a	a	DET
ejpam-847	59	14	triple	triple	ADJ
ejpam-847	59	15	(	(	PUNCT
ejpam-847	59	16	x	x	SYM
ejpam-847	59	17	,	,	PUNCT
ejpam-847	59	18	µ	µ	NOUN
ejpam-847	59	19	,	,	PUNCT
ejpam-847	59	20	t	t	PROPN
ejpam-847	59	21	)	)	PUNCT
ejpam-847	59	22	,	,	PUNCT
ejpam-847	59	23	where	where	SCONJ
ejpam-847	59	24	x	x	PRON
ejpam-847	59	25	is	be	AUX
ejpam-847	59	26	a	a	DET
ejpam-847	59	27	vector	vector	NOUN
ejpam-847	59	28	space	space	NOUN
ejpam-847	59	29	,	,	PUNCT
ejpam-847	59	30	t	t	PROPN
ejpam-847	59	31	is	be	AUX
ejpam-847	59	32	a	a	DET
ejpam-847	59	33	continuous	continuous	ADJ
ejpam-847	59	34	t	t	NOUN
ejpam-847	59	35	-	-	PUNCT
ejpam-847	59	36	norm	norm	NOUN
ejpam-847	59	37	and	and	CCONJ
ejpam-847	59	38	µ	µ	NOUN
ejpam-847	59	39	is	be	AUX
ejpam-847	59	40	a	a	DET
ejpam-847	59	41	mapping	mapping	NOUN
ejpam-847	59	42	from	from	ADP
ejpam-847	59	43	x	x	PUNCT
ejpam-847	59	44	into	into	ADP
ejpam-847	59	45	d+	d+	NOUN
ejpam-847	59	46	satisfies	satisfy	VERB
ejpam-847	59	47	the	the	DET
ejpam-847	59	48	following	follow	VERB
ejpam-847	59	49	conditions	condition	NOUN
ejpam-847	59	50	:	:	PUNCT
ejpam-847	59	51	(	(	PUNCT
ejpam-847	59	52	rn1	rn1	NOUN
ejpam-847	59	53	)	)	PUNCT
ejpam-847	59	54	µx(t	µx(t	NUM
ejpam-847	59	55	)	)	PUNCT
ejpam-847	59	56	=	=	SYM
ejpam-847	60	1	ǫ0(t	ǫ0(t	X
ejpam-847	60	2	)	)	PUNCT
ejpam-847	60	3	for	for	ADP
ejpam-847	60	4	all	all	DET
ejpam-847	60	5	t	t	PROPN
ejpam-847	60	6	>	>	X
ejpam-847	60	7	0	0	PUNCT
ejpam-847	61	1	if	if	SCONJ
ejpam-847	61	2	and	and	CCONJ
ejpam-847	61	3	only	only	ADV
ejpam-847	61	4	if	if	SCONJ
ejpam-847	61	5	x	x	SYM
ejpam-847	61	6	=	=	SYM
ejpam-847	61	7	0	0	NUM
ejpam-847	61	8	;	;	PUNCT
ejpam-847	61	9	(	(	PUNCT
ejpam-847	61	10	rn2	rn2	NOUN
ejpam-847	61	11	)	)	PUNCT
ejpam-847	61	12	µαx(t	µαx(t	PROPN
ejpam-847	61	13	)	)	PUNCT
ejpam-847	61	14	=	=	PRON
ejpam-847	61	15	µx	µx	NOUN
ejpam-847	61	16	(	(	PUNCT
ejpam-847	61	17	t	t	PROPN
ejpam-847	61	18	|α|	|α|	PROPN
ejpam-847	61	19	)	)	PUNCT
ejpam-847	61	20	for	for	ADP
ejpam-847	61	21	all	all	DET
ejpam-847	61	22	x	x	SYM
ejpam-847	61	23	∈	∈	PROPN
ejpam-847	61	24	x	x	X
ejpam-847	61	25	,	,	PUNCT
ejpam-847	61	26	α	α	PROPN
ejpam-847	61	27	6=	6=	ADP
ejpam-847	61	28	0	0	NUM
ejpam-847	61	29	;	;	PUNCT
ejpam-847	61	30	(	(	PUNCT
ejpam-847	61	31	rn3	rn3	NOUN
ejpam-847	61	32	)	)	PUNCT
ejpam-847	61	33	µx+y(t	µx+y(t	PROPN
ejpam-847	62	1	+	+	CCONJ
ejpam-847	62	2	s)≥	s)≥	PROPN
ejpam-847	62	3	t	t	PROPN
ejpam-847	62	4	(	(	PUNCT
ejpam-847	62	5	µx(t),µy	µx(t),µy	ADV
ejpam-847	62	6	(	(	PUNCT
ejpam-847	62	7	s	s	NOUN
ejpam-847	62	8	)	)	PUNCT
ejpam-847	62	9	)	)	PUNCT
ejpam-847	62	10	for	for	ADP
ejpam-847	62	11	all	all	PRON
ejpam-847	62	12	x	x	SYM
ejpam-847	62	13	,	,	PUNCT
ejpam-847	62	14	y	y	PROPN
ejpam-847	62	15	∈	∈	PROPN
ejpam-847	62	16	x	x	X
ejpam-847	62	17	and	and	CCONJ
ejpam-847	62	18	t	t	PROPN
ejpam-847	62	19	,	,	PUNCT
ejpam-847	62	20	s	s	VERB
ejpam-847	62	21	≥	≥	NOUN
ejpam-847	62	22	0	0	NUM
ejpam-847	62	23	.	.	PUNCT
ejpam-847	63	1	a	a	DET
ejpam-847	63	2	sequence	sequence	NOUN
ejpam-847	63	3	{	{	PUNCT
ejpam-847	63	4	xn	xn	NOUN
ejpam-847	63	5	}	}	PUNCT
ejpam-847	63	6	in	in	ADP
ejpam-847	63	7	an	an	DET
ejpam-847	63	8	rn	rn	NOUN
ejpam-847	63	9	-	-	NOUN
ejpam-847	63	10	space	space	NOUN
ejpam-847	63	11	(	(	PUNCT
ejpam-847	63	12	x	x	X
ejpam-847	63	13	,	,	PUNCT
ejpam-847	63	14	µ	µ	NOUN
ejpam-847	63	15	,	,	PUNCT
ejpam-847	63	16	t	t	PROPN
ejpam-847	63	17	)	)	PUNCT
ejpam-847	63	18	is	be	AUX
ejpam-847	63	19	said	say	VERB
ejpam-847	63	20	to	to	PART
ejpam-847	63	21	be	be	AUX
ejpam-847	63	22	convergent	convergent	ADJ
ejpam-847	63	23	to	to	ADP
ejpam-847	63	24	x	x	PUNCT
ejpam-847	63	25	in	in	ADP
ejpam-847	63	26	x	x	PUNCT
ejpam-847	63	27	if	if	SCONJ
ejpam-847	63	28	,	,	PUNCT
ejpam-847	63	29	for	for	ADP
ejpam-847	63	30	every	every	DET
ejpam-847	63	31	ε	ε	PROPN
ejpam-847	63	32	>	>	X
ejpam-847	63	33	0	0	PUNCT
ejpam-847	63	34	and	and	CCONJ
ejpam-847	63	35	λ	λ	X
ejpam-847	63	36	>	>	X
ejpam-847	63	37	0	0	NUM
ejpam-847	63	38	,	,	PUNCT
ejpam-847	63	39	there	there	PRON
ejpam-847	63	40	exists	exist	VERB
ejpam-847	63	41	a	a	DET
ejpam-847	63	42	positive	positive	ADJ
ejpam-847	63	43	integer	integer	NOUN
ejpam-847	63	44	n	n	CCONJ
ejpam-847	63	45	such	such	ADJ
ejpam-847	63	46	that	that	SCONJ
ejpam-847	63	47	µxn−x(ε	µxn−x(ε	PROPN
ejpam-847	63	48	)	)	PUNCT
ejpam-847	63	49	>	>	X
ejpam-847	64	1	1	1	NUM
ejpam-847	64	2	−	−	PROPN
ejpam-847	64	3	λ	λ	INTJ
ejpam-847	64	4	whenever	whenever	SCONJ
ejpam-847	64	5	n≥	n≥	PROPN
ejpam-847	64	6	n	n	X
ejpam-847	64	7	.	.	PUNCT
ejpam-847	65	1	an	an	DET
ejpam-847	65	2	rn	rn	NOUN
ejpam-847	65	3	-	-	NOUN
ejpam-847	65	4	space	space	NOUN
ejpam-847	65	5	(	(	PUNCT
ejpam-847	65	6	x	x	X
ejpam-847	65	7	,	,	PUNCT
ejpam-847	65	8	µ	µ	NOUN
ejpam-847	65	9	,	,	PUNCT
ejpam-847	65	10	t	t	PROPN
ejpam-847	65	11	)	)	PUNCT
ejpam-847	65	12	is	be	AUX
ejpam-847	65	13	said	say	VERB
ejpam-847	65	14	to	to	PART
ejpam-847	65	15	be	be	AUX
ejpam-847	65	16	complete	complete	ADJ
ejpam-847	65	17	if	if	SCONJ
ejpam-847	65	18	and	and	CCONJ
ejpam-847	65	19	only	only	ADV
ejpam-847	65	20	if	if	SCONJ
ejpam-847	65	21	every	every	DET
ejpam-847	65	22	cauchy	cauchy	ADJ
ejpam-847	65	23	sequence	sequence	NOUN
ejpam-847	65	24	in	in	ADP
ejpam-847	65	25	x	x	PROPN
ejpam-847	65	26	is	be	AUX
ejpam-847	65	27	convergent	convergent	ADJ
ejpam-847	65	28	to	to	ADP
ejpam-847	65	29	a	a	DET
ejpam-847	65	30	point	point	NOUN
ejpam-847	65	31	in	in	ADP
ejpam-847	65	32	x	x	X
ejpam-847	65	33	.	.	PUNCT
ejpam-847	66	1	the	the	DET
ejpam-847	66	2	hyers	hyers	PROPN
ejpam-847	66	3	-	-	PUNCT
ejpam-847	66	4	ulam	ulam	PROPN
ejpam-847	66	5	stability	stability	NOUN
ejpam-847	66	6	of	of	ADP
ejpam-847	66	7	functional	functional	ADJ
ejpam-847	66	8	equations	equation	NOUN
ejpam-847	66	9	in	in	ADP
ejpam-847	66	10	random	random	ADJ
ejpam-847	66	11	normed	normed	ADJ
ejpam-847	66	12	spaces	space	NOUN
ejpam-847	66	13	and	and	CCONJ
ejpam-847	66	14	fuzzy	fuzzy	ADJ
ejpam-847	66	15	normed	normed	ADJ
ejpam-847	66	16	spaces	space	NOUN
ejpam-847	66	17	has	have	AUX
ejpam-847	66	18	been	be	AUX
ejpam-847	66	19	studied	study	VERB
ejpam-847	66	20	[	[	PUNCT
ejpam-847	66	21	see	see	VERB
ejpam-847	66	22	3	3	NUM
ejpam-847	66	23	,	,	PUNCT
ejpam-847	66	24	4	4	NUM
ejpam-847	66	25	,	,	PUNCT
ejpam-847	66	26	6	6	NUM
ejpam-847	66	27	,	,	PUNCT
ejpam-847	66	28	8	8	NUM
ejpam-847	66	29	,	,	PUNCT
ejpam-847	66	30	9	9	NUM
ejpam-847	66	31	,	,	PUNCT
ejpam-847	66	32	11	11	NUM
ejpam-847	66	33	,	,	PUNCT
ejpam-847	66	34	12	12	NUM
ejpam-847	66	35	]	]	PUNCT
ejpam-847	66	36	.	.	PUNCT
ejpam-847	67	1	let	let	VERB
ejpam-847	67	2	v	v	X
ejpam-847	67	3	,	,	PUNCT
ejpam-847	67	4	w	w	NOUN
ejpam-847	67	5	be	be	AUX
ejpam-847	67	6	vector	vector	NOUN
ejpam-847	67	7	spaces	space	NOUN
ejpam-847	67	8	.	.	PUNCT
ejpam-847	68	1	it	it	PRON
ejpam-847	68	2	is	be	AUX
ejpam-847	68	3	shown	show	VERB
ejpam-847	68	4	that	that	SCONJ
ejpam-847	68	5	if	if	SCONJ
ejpam-847	68	6	a	a	DET
ejpam-847	68	7	mapping	mapping	NOUN
ejpam-847	68	8	f	f	X
ejpam-847	68	9	:	:	PUNCT
ejpam-847	68	10	v	v	X
ejpam-847	68	11	→	→	SYM
ejpam-847	68	12	w	w	ADP
ejpam-847	68	13	satisfies	satisfie	NOUN
ejpam-847	68	14	the	the	DET
ejpam-847	68	15	functional	functional	ADJ
ejpam-847	68	16	equation	equation	NOUN
ejpam-847	68	17	(	(	PUNCT
ejpam-847	68	18	1	1	NUM
ejpam-847	68	19	)	)	PUNCT
ejpam-847	68	20	,	,	PUNCT
ejpam-847	68	21	then	then	ADV
ejpam-847	68	22	the	the	DET
ejpam-847	68	23	mapping	mapping	NOUN
ejpam-847	68	24	f	f	X
ejpam-847	68	25	is	be	AUX
ejpam-847	68	26	the	the	DET
ejpam-847	68	27	sum	sum	NOUN
ejpam-847	68	28	of	of	ADP
ejpam-847	68	29	an	an	DET
ejpam-847	68	30	additive	additive	ADJ
ejpam-847	68	31	mapping	mapping	NOUN
ejpam-847	68	32	and	and	CCONJ
ejpam-847	68	33	a	a	DET
ejpam-847	68	34	quadratic	quadratic	ADJ
ejpam-847	68	35	mapping	mapping	NOUN
ejpam-847	68	36	[	[	X
ejpam-847	68	37	see	see	VERB
ejpam-847	68	38	2	2	NUM
ejpam-847	68	39	]	]	PUNCT
ejpam-847	68	40	.	.	PUNCT
ejpam-847	69	1	in	in	ADP
ejpam-847	69	2	this	this	DET
ejpam-847	69	3	paper	paper	NOUN
ejpam-847	69	4	,	,	PUNCT
ejpam-847	69	5	we	we	PRON
ejpam-847	69	6	investigate	investigate	VERB
ejpam-847	69	7	the	the	DET
ejpam-847	69	8	hyers	hyers	PROPN
ejpam-847	69	9	-	-	PUNCT
ejpam-847	69	10	ulam	ulam	ADJ
ejpam-847	69	11	stability	stability	NOUN
ejpam-847	69	12	of	of	ADP
ejpam-847	69	13	the	the	DET
ejpam-847	69	14	functional	functional	ADJ
ejpam-847	69	15	equation	equation	NOUN
ejpam-847	69	16	(	(	PUNCT
ejpam-847	69	17	1	1	NUM
ejpam-847	69	18	)	)	PUNCT
ejpam-847	69	19	in	in	ADP
ejpam-847	69	20	rn	rn	NOUN
ejpam-847	69	21	-	-	NOUN
ejpam-847	69	22	spaces	space	NOUN
ejpam-847	69	23	.	.	PUNCT
ejpam-847	70	1	throughout	throughout	ADP
ejpam-847	70	2	this	this	DET
ejpam-847	70	3	paper	paper	NOUN
ejpam-847	70	4	,	,	PUNCT
ejpam-847	70	5	assume	assume	VERB
ejpam-847	70	6	that	that	SCONJ
ejpam-847	70	7	x	x	PRON
ejpam-847	70	8	is	be	AUX
ejpam-847	70	9	a	a	DET
ejpam-847	70	10	vector	vector	NOUN
ejpam-847	70	11	space	space	NOUN
ejpam-847	70	12	and	and	CCONJ
ejpam-847	70	13	that	that	SCONJ
ejpam-847	70	14	(	(	PUNCT
ejpam-847	70	15	y,µ	y,µ	NOUN
ejpam-847	70	16	,	,	PUNCT
ejpam-847	70	17	t	t	PROPN
ejpam-847	70	18	)	)	PUNCT
ejpam-847	70	19	is	be	AUX
ejpam-847	70	20	a	a	DET
ejpam-847	70	21	complete	complete	ADJ
ejpam-847	70	22	rn	rn	NOUN
ejpam-847	70	23	-	-	NOUN
ejpam-847	70	24	space	space	NOUN
ejpam-847	70	25	.	.	PUNCT
ejpam-847	71	1	2	2	X
ejpam-847	71	2	.	.	X
ejpam-847	71	3	hyers	hyer	NOUN
ejpam-847	71	4	-	-	PUNCT
ejpam-847	71	5	ulam	ulam	PROPN
ejpam-847	71	6	stability	stability	NOUN
ejpam-847	71	7	of	of	ADP
ejpam-847	71	8	the	the	DET
ejpam-847	71	9	functional	functional	ADJ
ejpam-847	71	10	equation	equation	NOUN
ejpam-847	71	11	(	(	PUNCT
ejpam-847	71	12	1	1	NUM
ejpam-847	71	13	):	):	PUNCT
ejpam-847	71	14	an	an	DET
ejpam-847	71	15	odd	odd	ADJ
ejpam-847	71	16	case	case	NOUN
ejpam-847	71	17	we	we	PRON
ejpam-847	71	18	investigate	investigate	VERB
ejpam-847	71	19	the	the	DET
ejpam-847	71	20	functional	functional	ADJ
ejpam-847	71	21	equation	equation	NOUN
ejpam-847	71	22	(	(	PUNCT
ejpam-847	71	23	1	1	NUM
ejpam-847	71	24	)	)	PUNCT
ejpam-847	71	25	for	for	ADP
ejpam-847	71	26	an	an	DET
ejpam-847	71	27	odd	odd	ADJ
ejpam-847	71	28	mapping	mapping	NOUN
ejpam-847	71	29	in	in	ADP
ejpam-847	71	30	rn	rn	NOUN
ejpam-847	71	31	-	-	NOUN
ejpam-847	71	32	spaces	space	NOUN
ejpam-847	71	33	.	.	PUNCT
ejpam-847	72	1	for	for	ADP
ejpam-847	72	2	a	a	DET
ejpam-847	72	3	given	give	VERB
ejpam-847	72	4	mapping	mapping	NOUN
ejpam-847	72	5	f	f	NOUN
ejpam-847	72	6	:	:	PUNCT
ejpam-847	72	7	x	x	X
ejpam-847	72	8	→	→	SYM
ejpam-847	72	9	y	y	PROPN
ejpam-847	72	10	,	,	PUNCT
ejpam-847	72	11	we	we	PRON
ejpam-847	72	12	define	define	VERB
ejpam-847	72	13	d	d	PROPN
ejpam-847	72	14	f	f	PROPN
ejpam-847	72	15	(	(	PUNCT
ejpam-847	72	16	x1	x1	PROPN
ejpam-847	72	17	,	,	PUNCT
ejpam-847	72	18	.	.	PUNCT
ejpam-847	72	19	.	.	PUNCT
ejpam-847	72	20	.	.	PUNCT
ejpam-847	73	1	,	,	PUNCT
ejpam-847	73	2	xn	xn	X
ejpam-847	73	3	)	)	PUNCT
ejpam-847	73	4	:	:	PUNCT
ejpam-847	74	1	=	=	SYM
ejpam-847	74	2	n	n	CCONJ
ejpam-847	74	3	∑	∑	PROPN
ejpam-847	74	4	i	i	PROPN
ejpam-847	74	5	,	,	PUNCT
ejpam-847	74	6	j=1	j=1	PROPN
ejpam-847	74	7	f	f	PROPN
ejpam-847	74	8	(	(	PUNCT
ejpam-847	74	9	x	x	PROPN
ejpam-847	74	10	i	i	PRON
ejpam-847	74	11	−	−	NOUN
ejpam-847	74	12	x	x	SYM
ejpam-847	74	13	j)−	j)−	NOUN
ejpam-847	74	14	2n	2n	NUM
ejpam-847	75	1	n	n	CCONJ
ejpam-847	75	2	∑	∑	PROPN
ejpam-847	75	3	i=1	i=1	PROPN
ejpam-847	75	4	f	f	PROPN
ejpam-847	75	5	(	(	PUNCT
ejpam-847	75	6	x	x	PROPN
ejpam-847	75	7	i	i	NOUN
ejpam-847	75	8	)	)	PUNCT
ejpam-847	75	9	for	for	ADP
ejpam-847	75	10	all	all	DET
ejpam-847	75	11	x1	x1	PROPN
ejpam-847	75	12	,	,	PUNCT
ejpam-847	75	13	.	.	PUNCT
ejpam-847	75	14	.	.	PUNCT
ejpam-847	75	15	.	.	PUNCT
ejpam-847	76	1	,	,	PUNCT
ejpam-847	76	2	xn	xn	PUNCT
ejpam-847	76	3	∈	∈	PROPN
ejpam-847	76	4	x	x	PUNCT
ejpam-847	76	5	with	with	ADP
ejpam-847	76	6	∑n	∑n	PROPN
ejpam-847	76	7	i=1	i=1	NOUN
ejpam-847	76	8	x	x	PUNCT
ejpam-847	77	1	i	i	NOUN
ejpam-847	77	2	=	=	NOUN
ejpam-847	77	3	0	0	X
ejpam-847	77	4	.	.	PUNCT
ejpam-847	78	1	for	for	ADP
ejpam-847	78	2	an	an	DET
ejpam-847	78	3	odd	odd	ADJ
ejpam-847	78	4	mapping	mapping	NOUN
ejpam-847	78	5	f	f	NOUN
ejpam-847	78	6	:	:	PUNCT
ejpam-847	78	7	x	x	X
ejpam-847	78	8	→	→	SYM
ejpam-847	78	9	y	y	PROPN
ejpam-847	78	10	,	,	PUNCT
ejpam-847	78	11	we	we	PRON
ejpam-847	78	12	note	note	VERB
ejpam-847	78	13	that	that	SCONJ
ejpam-847	78	14	if	if	SCONJ
ejpam-847	78	15	f	f	PROPN
ejpam-847	78	16	satisfies	satisfy	VERB
ejpam-847	78	17	d	d	X
ejpam-847	78	18	f	f	X
ejpam-847	78	19	(	(	PUNCT
ejpam-847	78	20	x1	x1	PROPN
ejpam-847	78	21	,	,	PUNCT
ejpam-847	78	22	x2	x2	PROPN
ejpam-847	78	23	,	,	PUNCT
ejpam-847	78	24	.	.	PUNCT
ejpam-847	78	25	.	.	PUNCT
ejpam-847	79	1	.	.	PUNCT
ejpam-847	80	1	,	,	PUNCT
ejpam-847	80	2	xn	xn	X
ejpam-847	80	3	)	)	PUNCT
ejpam-847	80	4	=	=	SYM
ejpam-847	80	5	0	0	NUM
ejpam-847	80	6	for	for	ADP
ejpam-847	80	7	all	all	DET
ejpam-847	80	8	x1	x1	PROPN
ejpam-847	80	9	,	,	PUNCT
ejpam-847	80	10	.	.	PUNCT
ejpam-847	80	11	.	.	PUNCT
ejpam-847	81	1	.	.	PUNCT
ejpam-847	82	1	,	,	PUNCT
ejpam-847	82	2	xn	xn	PUNCT
ejpam-847	82	3	∈	∈	PROPN
ejpam-847	82	4	x	x	PUNCT
ejpam-847	82	5	with	with	ADP
ejpam-847	82	6	∑n	∑n	PROPN
ejpam-847	82	7	i=1	i=1	NOUN
ejpam-847	82	8	x	x	PUNCT
ejpam-847	83	1	i	i	NOUN
ejpam-847	83	2	=	=	NOUN
ejpam-847	83	3	0	0	PUNCT
ejpam-847	83	4	then	then	ADV
ejpam-847	83	5	the	the	DET
ejpam-847	83	6	mapping	mapping	NOUN
ejpam-847	83	7	f	f	PROPN
ejpam-847	83	8	is	be	AUX
ejpam-847	83	9	additive	additive	ADJ
ejpam-847	83	10	.	.	PUNCT
ejpam-847	84	1	we	we	PRON
ejpam-847	84	2	prove	prove	VERB
ejpam-847	84	3	the	the	DET
ejpam-847	84	4	hyers	hyers	PROPN
ejpam-847	84	5	-	-	PUNCT
ejpam-847	84	6	ulam	ulam	ADJ
ejpam-847	84	7	stability	stability	NOUN
ejpam-847	84	8	of	of	ADP
ejpam-847	84	9	the	the	DET
ejpam-847	84	10	functional	functional	ADJ
ejpam-847	84	11	equation	equation	NOUN
ejpam-847	84	12	(	(	PUNCT
ejpam-847	84	13	1	1	NUM
ejpam-847	84	14	)	)	PUNCT
ejpam-847	84	15	of	of	ADP
ejpam-847	84	16	an	an	DET
ejpam-847	84	17	odd	odd	ADJ
ejpam-847	84	18	mapping	mapping	NOUN
ejpam-847	84	19	in	in	ADP
ejpam-847	84	20	rn	rn	NOUN
ejpam-847	84	21	-	-	NOUN
ejpam-847	84	22	spaces	space	NOUN
ejpam-847	84	23	.	.	PUNCT
ejpam-847	85	1	theorem	theorem	NOUN
ejpam-847	85	2	1	1	NUM
ejpam-847	85	3	.	.	PUNCT
ejpam-847	86	1	let	let	VERB
ejpam-847	86	2	f	f	NOUN
ejpam-847	86	3	:	:	PUNCT
ejpam-847	86	4	x	x	X
ejpam-847	86	5	→	→	SYM
ejpam-847	86	6	y	y	X
ejpam-847	86	7	be	be	AUX
ejpam-847	86	8	an	an	DET
ejpam-847	86	9	odd	odd	ADJ
ejpam-847	86	10	mapping	mapping	NOUN
ejpam-847	86	11	for	for	ADP
ejpam-847	86	12	which	which	PRON
ejpam-847	86	13	there	there	PRON
ejpam-847	86	14	is	be	VERB
ejpam-847	86	15	a	a	DET
ejpam-847	86	16	ρ	ρ	NOUN
ejpam-847	86	17	:	:	PUNCT
ejpam-847	86	18	x	x	PROPN
ejpam-847	86	19	n→	n→	X
ejpam-847	86	20	d+	d+	X
ejpam-847	86	21	(	(	PUNCT
ejpam-847	86	22	ρ(x1	ρ(x1	NOUN
ejpam-847	86	23	,	,	PUNCT
ejpam-847	86	24	x2	x2	PROPN
ejpam-847	86	25	,	,	PUNCT
ejpam-847	86	26	.	.	PUNCT
ejpam-847	86	27	.	.	PUNCT
ejpam-847	87	1	.	.	PUNCT
ejpam-847	88	1	,	,	PUNCT
ejpam-847	88	2	xn	xn	X
ejpam-847	88	3	)	)	PUNCT
ejpam-847	88	4	is	be	AUX
ejpam-847	88	5	denoted	denote	VERB
ejpam-847	88	6	by	by	ADP
ejpam-847	88	7	ρ(x1,x2,	ρ(x1,x2,	PROPN
ejpam-847	88	8	...	...	PUNCT
ejpam-847	88	9	,xn	,xn	PUNCT
ejpam-847	88	10	)	)	PUNCT
ejpam-847	88	11	)	)	PUNCT
ejpam-847	89	1	such	such	ADJ
ejpam-847	89	2	that	that	SCONJ
ejpam-847	89	3	µd	µd	PROPN
ejpam-847	89	4	f	f	PROPN
ejpam-847	89	5	(	(	PUNCT
ejpam-847	89	6	x1,x2,	x1,x2,	PROPN
ejpam-847	89	7	...	...	PUNCT
ejpam-847	89	8	,xn	,xn	PUNCT
ejpam-847	89	9	)	)	PUNCT
ejpam-847	89	10	(	(	PUNCT
ejpam-847	89	11	t	t	PROPN
ejpam-847	89	12	)	)	PUNCT
ejpam-847	89	13	≥	≥	NOUN
ejpam-847	89	14	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	89	15	...	...	PUNCT
ejpam-847	89	16	,xn	,xn	PUNCT
ejpam-847	89	17	)	)	PUNCT
ejpam-847	89	18	(	(	PUNCT
ejpam-847	89	19	t	t	NOUN
ejpam-847	89	20	)	)	PUNCT
ejpam-847	89	21	(	(	PUNCT
ejpam-847	89	22	2	2	X
ejpam-847	89	23	)	)	PUNCT
ejpam-847	89	24	for	for	ADP
ejpam-847	89	25	all	all	PRON
ejpam-847	89	26	(	(	PUNCT
ejpam-847	89	27	x1	x1	PROPN
ejpam-847	89	28	,	,	PUNCT
ejpam-847	89	29	x2	x2	PROPN
ejpam-847	89	30	,	,	PUNCT
ejpam-847	89	31	.	.	PUNCT
ejpam-847	89	32	.	.	PUNCT
ejpam-847	89	33	.	.	PUNCT
ejpam-847	89	34	,	,	PUNCT
ejpam-847	89	35	xn	xn	X
ejpam-847	89	36	)	)	PUNCT
ejpam-847	89	37	∈	∈	PROPN
ejpam-847	89	38	x	x	SYM
ejpam-847	89	39	n	n	NOUN
ejpam-847	89	40	and	and	CCONJ
ejpam-847	89	41	all	all	PRON
ejpam-847	89	42	t	t	NOUN
ejpam-847	89	43	>	>	X
ejpam-847	89	44	0	0	X
ejpam-847	89	45	.	.	PUNCT
ejpam-847	90	1	if	if	SCONJ
ejpam-847	90	2	t∞k=1ρ	t∞k=1ρ	PROPN
ejpam-847	90	3	�	�	PROPN
ejpam-847	90	4	x	x	SYM
ejpam-847	90	5	2k+l	2k+l	NUM
ejpam-847	90	6	,	,	PUNCT
ejpam-847	90	7	x	x	X
ejpam-847	90	8	2k+l	2k+l	NUM
ejpam-847	90	9	,	,	PUNCT
ejpam-847	90	10	−	−	PROPN
ejpam-847	90	11	x	x	SYM
ejpam-847	90	12	2k+l−1	2k+l−1	NUM
ejpam-847	90	13	,	,	PUNCT
ejpam-847	90	14	0,	0,	NUM
ejpam-847	90	15	...	...	PUNCT
ejpam-847	90	16	,0	,0	PUNCT
ejpam-847	90	17	�	�	PROPN
ejpam-847	90	18	�	�	PROPN
ejpam-847	90	19	nt	nt	PROPN
ejpam-847	90	20	22k+l−2	22k+l−2	NUM
ejpam-847	90	21	�	�	PROPN
ejpam-847	90	22	=	=	SYM
ejpam-847	90	23	1	1	NUM
ejpam-847	90	24	(	(	PUNCT
ejpam-847	90	25	3	3	NUM
ejpam-847	90	26	)	)	PUNCT
ejpam-847	90	27	d.	d.	PROPN
ejpam-847	90	28	shin	shin	PROPN
ejpam-847	90	29	,	,	PUNCT
ejpam-847	90	30	j.	j.	PROPN
ejpam-847	90	31	lee	lee	PROPN
ejpam-847	90	32	,	,	PUNCT
ejpam-847	90	33	c.	c.	PROPN
ejpam-847	90	34	park	park	PROPN
ejpam-847	90	35	/	/	SYM
ejpam-847	90	36	eur	eur	PROPN
ejpam-847	90	37	.	.	PUNCT
ejpam-847	91	1	j.	j.	PROPN
ejpam-847	91	2	pure	pure	PROPN
ejpam-847	91	3	appl	appl	PROPN
ejpam-847	91	4	.	.	PROPN
ejpam-847	91	5	math	math	PROPN
ejpam-847	91	6	,	,	PUNCT
ejpam-847	91	7	5	5	NUM
ejpam-847	91	8	(	(	PUNCT
ejpam-847	91	9	2012	2012	NUM
ejpam-847	91	10	)	)	PUNCT
ejpam-847	91	11	,	,	PUNCT
ejpam-847	91	12	540	540	NUM
ejpam-847	91	13	-	-	SYM
ejpam-847	91	14	553	553	NUM
ejpam-847	91	15	543	543	NUM
ejpam-847	91	16	and	and	CCONJ
ejpam-847	91	17	lim	lim	PROPN
ejpam-847	91	18	m→∞	m→∞	NUM
ejpam-847	91	19	ρ	ρ	PROPN
ejpam-847	91	20	�	�	PROPN
ejpam-847	91	21	x	x	PUNCT
ejpam-847	91	22	2	2	NUM
ejpam-847	91	23	m	m	NOUN
ejpam-847	91	24	,	,	PUNCT
ejpam-847	91	25	y	y	PROPN
ejpam-847	91	26	2	2	NUM
ejpam-847	91	27	m	m	NOUN
ejpam-847	91	28	,	,	PUNCT
ejpam-847	91	29	−	−	PROPN
ejpam-847	91	30	x+y	x+y	NUM
ejpam-847	91	31	2	2	NUM
ejpam-847	91	32	m	m	NOUN
ejpam-847	91	33	,	,	PUNCT
ejpam-847	91	34	0,	0,	NUM
ejpam-847	91	35	...	...	PUNCT
ejpam-847	91	36	,0	,0	PUNCT
ejpam-847	91	37	�	�	PROPN
ejpam-847	91	38	�	�	PROPN
ejpam-847	91	39	nt	not	PART
ejpam-847	91	40	2m−1	2m−1	PROPN
ejpam-847	91	41	�	�	PROPN
ejpam-847	91	42	=	=	SYM
ejpam-847	91	43	1	1	NUM
ejpam-847	91	44	(	(	PUNCT
ejpam-847	91	45	4	4	NUM
ejpam-847	91	46	)	)	PUNCT
ejpam-847	91	47	for	for	ADP
ejpam-847	91	48	all	all	PRON
ejpam-847	91	49	x	x	SYM
ejpam-847	91	50	,	,	PUNCT
ejpam-847	91	51	y	y	PROPN
ejpam-847	91	52	∈	∈	PROPN
ejpam-847	91	53	x	x	INTJ
ejpam-847	91	54	,	,	PUNCT
ejpam-847	91	55	all	all	PRON
ejpam-847	91	56	t	t	X
ejpam-847	91	57	>	>	X
ejpam-847	91	58	0	0	PUNCT
ejpam-847	92	1	and	and	CCONJ
ejpam-847	92	2	all	all	DET
ejpam-847	92	3	l	l	NOUN
ejpam-847	92	4	=	=	PUNCT
ejpam-847	92	5	0,1,2	0,1,2	NUM
ejpam-847	92	6	,	,	PUNCT
ejpam-847	92	7	.	.	PUNCT
ejpam-847	92	8	.	.	PUNCT
ejpam-847	93	1	.	.	PUNCT
ejpam-847	94	1	,	,	PUNCT
ejpam-847	94	2	then	then	ADV
ejpam-847	94	3	there	there	PRON
ejpam-847	94	4	exists	exist	VERB
ejpam-847	94	5	a	a	DET
ejpam-847	94	6	unique	unique	ADJ
ejpam-847	94	7	additive	additive	NOUN
ejpam-847	94	8	mapping	mapping	NOUN
ejpam-847	94	9	a	a	DET
ejpam-847	94	10	:	:	PUNCT
ejpam-847	94	11	x	x	X
ejpam-847	94	12	→	→	SYM
ejpam-847	94	13	y	y	PROPN
ejpam-847	94	14	such	such	ADJ
ejpam-847	94	15	that	that	SCONJ
ejpam-847	94	16	µ	µ	PROPN
ejpam-847	94	17	f	f	X
ejpam-847	94	18	(	(	PUNCT
ejpam-847	94	19	x)−a(x)(t)≥	x)−a(x)(t)≥	PROPN
ejpam-847	94	20	t∞k=1ρ	t∞k=1ρ	PROPN
ejpam-847	94	21	�	�	PROPN
ejpam-847	94	22	x	x	SYM
ejpam-847	94	23	2k	2k	NUM
ejpam-847	94	24	,	,	PUNCT
ejpam-847	94	25	x	x	PROPN
ejpam-847	94	26	2k	2k	NUM
ejpam-847	94	27	,	,	PUNCT
ejpam-847	94	28	−	−	PROPN
ejpam-847	94	29	x	x	SYM
ejpam-847	94	30	2k−1	2k−1	NUM
ejpam-847	94	31	,	,	PUNCT
ejpam-847	94	32	0,	0,	NUM
ejpam-847	94	33	...	...	PUNCT
ejpam-847	94	34	,0	,0	PUNCT
ejpam-847	94	35	�	�	PROPN
ejpam-847	94	36	�	�	PROPN
ejpam-847	94	37	nt	not	PART
ejpam-847	94	38	22k−2	22k−2	PROPN
ejpam-847	94	39	�	�	PROPN
ejpam-847	94	40	(	(	PUNCT
ejpam-847	94	41	5	5	NUM
ejpam-847	94	42	)	)	PUNCT
ejpam-847	94	43	for	for	ADP
ejpam-847	94	44	all	all	PRON
ejpam-847	94	45	x	x	SYM
ejpam-847	94	46	∈	∈	ADJ
ejpam-847	94	47	x	x	X
ejpam-847	94	48	and	and	CCONJ
ejpam-847	94	49	all	all	DET
ejpam-847	94	50	t	t	NOUN
ejpam-847	94	51	>	>	X
ejpam-847	94	52	0	0	X
ejpam-847	94	53	.	.	PUNCT
ejpam-847	95	1	proof	proof	NOUN
ejpam-847	95	2	.	.	PUNCT
ejpam-847	96	1	putting	put	VERB
ejpam-847	96	2	x1	x1	PROPN
ejpam-847	97	1	=	=	PUNCT
ejpam-847	97	2	x2	x2	NOUN
ejpam-847	97	3	=	=	PUNCT
ejpam-847	97	4	x	x	SYM
ejpam-847	97	5	2	2	NUM
ejpam-847	97	6	,	,	PUNCT
ejpam-847	97	7	x3	x3	ADJ
ejpam-847	97	8	=	=	PUNCT
ejpam-847	97	9	−x	−x	NOUN
ejpam-847	97	10	,	,	PUNCT
ejpam-847	97	11	x4	x4	PROPN
ejpam-847	97	12	=	=	PRON
ejpam-847	97	13	.	.	PUNCT
ejpam-847	97	14	.	.	PUNCT
ejpam-847	97	15	.	.	PUNCT
ejpam-847	98	1	=	=	PUNCT
ejpam-847	98	2	xn	xn	PUNCT
ejpam-847	99	1	=	=	SYM
ejpam-847	99	2	0	0	NUM
ejpam-847	99	3	in	in	ADP
ejpam-847	99	4	(	(	PUNCT
ejpam-847	99	5	2	2	NUM
ejpam-847	99	6	)	)	PUNCT
ejpam-847	99	7	,	,	PUNCT
ejpam-847	99	8	we	we	PRON
ejpam-847	99	9	get	get	VERB
ejpam-847	99	10	µ2n	µ2n	PROPN
ejpam-847	99	11	�	�	PROPN
ejpam-847	99	12	f	f	PROPN
ejpam-847	99	13	(	(	PUNCT
ejpam-847	99	14	x)−2	x)−2	X
ejpam-847	99	15	f	f	PROPN
ejpam-847	99	16	�	�	PROPN
ejpam-847	99	17	x	x	SYM
ejpam-847	99	18	2	2	NUM
ejpam-847	99	19	�	�	PROPN
ejpam-847	99	20	�	�	PROPN
ejpam-847	99	21	(	(	PUNCT
ejpam-847	99	22	t	t	PROPN
ejpam-847	99	23	)	)	PUNCT
ejpam-847	99	24	≥	≥	NOUN
ejpam-847	99	25	ρ	ρ	NUM
ejpam-847	99	26	�	�	PROPN
ejpam-847	99	27	x	x	SYM
ejpam-847	99	28	2	2	NUM
ejpam-847	99	29	,	,	PUNCT
ejpam-847	99	30	x	x	PROPN
ejpam-847	99	31	2	2	NUM
ejpam-847	99	32	,	,	PUNCT
ejpam-847	99	33	−x	−x	NOUN
ejpam-847	99	34	,	,	PUNCT
ejpam-847	99	35	0,	0,	NUM
ejpam-847	99	36	...	...	PUNCT
ejpam-847	99	37	,0	,0	PUNCT
ejpam-847	99	38	�	�	PROPN
ejpam-847	99	39	(	(	PUNCT
ejpam-847	99	40	t	t	PROPN
ejpam-847	99	41	)	)	PUNCT
ejpam-847	99	42	which	which	PRON
ejpam-847	99	43	is	be	AUX
ejpam-847	99	44	equivalent	equivalent	ADJ
ejpam-847	99	45	to	to	ADP
ejpam-847	99	46	µ	µ	PROPN
ejpam-847	99	47	f	f	X
ejpam-847	99	48	(	(	PUNCT
ejpam-847	99	49	x)−2	x)−2	PROPN
ejpam-847	99	50	f	f	PROPN
ejpam-847	99	51	�	�	PROPN
ejpam-847	99	52	x	x	PROPN
ejpam-847	99	53	2	2	NUM
ejpam-847	99	54	�	�	PROPN
ejpam-847	99	55	(	(	PUNCT
ejpam-847	99	56	t	t	PROPN
ejpam-847	99	57	)	)	PUNCT
ejpam-847	99	58	≥	≥	NOUN
ejpam-847	99	59	ρ	ρ	NUM
ejpam-847	99	60	�	�	PROPN
ejpam-847	99	61	x	x	SYM
ejpam-847	99	62	2	2	NUM
ejpam-847	99	63	,	,	PUNCT
ejpam-847	99	64	x	x	PROPN
ejpam-847	99	65	2	2	NUM
ejpam-847	99	66	,	,	PUNCT
ejpam-847	99	67	−x	−x	NOUN
ejpam-847	99	68	,	,	PUNCT
ejpam-847	99	69	0,	0,	NUM
ejpam-847	99	70	...	...	PUNCT
ejpam-847	99	71	,0	,0	PUNCT
ejpam-847	99	72	�	�	PROPN
ejpam-847	99	73	(	(	PUNCT
ejpam-847	99	74	2nt	2nt	NOUN
ejpam-847	99	75	)	)	PUNCT
ejpam-847	99	76	for	for	ADP
ejpam-847	99	77	all	all	DET
ejpam-847	99	78	x	x	SYM
ejpam-847	99	79	∈	∈	ADJ
ejpam-847	99	80	x	x	X
ejpam-847	99	81	and	and	CCONJ
ejpam-847	99	82	all	all	DET
ejpam-847	99	83	t	t	NOUN
ejpam-847	99	84	>	>	X
ejpam-847	99	85	0	0	X
ejpam-847	99	86	.	.	PUNCT
ejpam-847	100	1	replacing	replace	VERB
ejpam-847	100	2	x	x	PUNCT
ejpam-847	100	3	and	and	CCONJ
ejpam-847	100	4	t	t	X
ejpam-847	100	5	by	by	ADP
ejpam-847	100	6	x	x	SYM
ejpam-847	100	7	2k−1	2k−1	NUM
ejpam-847	100	8	and	and	CCONJ
ejpam-847	100	9	t	t	NOUN
ejpam-847	100	10	22k−1	22k−1	NUM
ejpam-847	100	11	,	,	PUNCT
ejpam-847	100	12	respectively	respectively	ADV
ejpam-847	100	13	in	in	ADP
ejpam-847	100	14	the	the	DET
ejpam-847	100	15	above	above	ADJ
ejpam-847	100	16	inequality	inequality	NOUN
ejpam-847	100	17	,	,	PUNCT
ejpam-847	100	18	we	we	PRON
ejpam-847	100	19	get	get	VERB
ejpam-847	100	20	µ	µ	PRON
ejpam-847	100	21	2k−1	2k−1	NUM
ejpam-847	100	22	f	f	PROPN
ejpam-847	100	23	�	�	PROPN
ejpam-847	100	24	x	x	SYM
ejpam-847	100	25	2k−1	2k−1	NUM
ejpam-847	100	26	�	�	PROPN
ejpam-847	100	27	−2k	−2k	PROPN
ejpam-847	100	28	f	f	PROPN
ejpam-847	100	29	�	�	PROPN
ejpam-847	100	30	x	x	ADP
ejpam-847	100	31	2k	2k	PROPN
ejpam-847	100	32	�	�	PROPN
ejpam-847	100	33	�	�	PROPN
ejpam-847	100	34	t	t	PROPN
ejpam-847	100	35	2k	2k	PROPN
ejpam-847	100	36	�	�	PROPN
ejpam-847	100	37	≥	≥	PROPN
ejpam-847	100	38	ρ	ρ	NUM
ejpam-847	100	39	�	�	PROPN
ejpam-847	100	40	x	x	SYM
ejpam-847	100	41	2k	2k	NUM
ejpam-847	100	42	,	,	PUNCT
ejpam-847	100	43	x	x	PROPN
ejpam-847	100	44	2k	2k	NUM
ejpam-847	100	45	,	,	PUNCT
ejpam-847	100	46	−	−	PROPN
ejpam-847	100	47	x	x	SYM
ejpam-847	100	48	2k−1	2k−1	NUM
ejpam-847	100	49	,	,	PUNCT
ejpam-847	100	50	0,	0,	NUM
ejpam-847	100	51	...	...	PUNCT
ejpam-847	100	52	,0	,0	PUNCT
ejpam-847	100	53	�	�	PROPN
ejpam-847	100	54	�	�	PROPN
ejpam-847	100	55	nt	not	PART
ejpam-847	100	56	22k−2	22k−2	PROPN
ejpam-847	100	57	�	�	PROPN
ejpam-847	100	58	for	for	ADP
ejpam-847	100	59	all	all	DET
ejpam-847	100	60	x	x	SYM
ejpam-847	100	61	∈	∈	ADJ
ejpam-847	100	62	x	x	X
ejpam-847	100	63	and	and	CCONJ
ejpam-847	100	64	all	all	DET
ejpam-847	100	65	t	t	PROPN
ejpam-847	100	66	>	>	X
ejpam-847	100	67	0	0	X
ejpam-847	100	68	.	.	PUNCT
ejpam-847	101	1	since	since	SCONJ
ejpam-847	101	2	µx(s)≤	µx(s)≤	ADJ
ejpam-847	101	3	µx(t	µx(t	NOUN
ejpam-847	101	4	)	)	PUNCT
ejpam-847	101	5	for	for	ADP
ejpam-847	101	6	all	all	DET
ejpam-847	101	7	s	s	NOUN
ejpam-847	101	8	and	and	CCONJ
ejpam-847	101	9	t	t	X
ejpam-847	101	10	with	with	ADP
ejpam-847	101	11	0	0	NUM
ejpam-847	101	12	<	<	X
ejpam-847	101	13	s	s	PART
ejpam-847	101	14	≤	≤	PROPN
ejpam-847	101	15	t	t	PROPN
ejpam-847	101	16	,	,	PUNCT
ejpam-847	101	17	we	we	PRON
ejpam-847	101	18	obtain	obtain	VERB
ejpam-847	101	19	µ	µ	DET
ejpam-847	101	20	f	f	X
ejpam-847	101	21	(	(	PUNCT
ejpam-847	101	22	x)−2	x)−2	NOUN
ejpam-847	101	23	m	m	VERB
ejpam-847	101	24	f	f	PROPN
ejpam-847	101	25	�	�	PROPN
ejpam-847	101	26	x	x	PUNCT
ejpam-847	101	27	2	2	NUM
ejpam-847	101	28	m	m	PROPN
ejpam-847	101	29	�	�	PROPN
ejpam-847	101	30	(	(	PUNCT
ejpam-847	101	31	t	t	PROPN
ejpam-847	101	32	)	)	PUNCT
ejpam-847	101	33	=	=	NOUN
ejpam-847	101	34	µ∑m	µ∑m	PROPN
ejpam-847	101	35	k=1	k=1	PROPN
ejpam-847	101	36	�	�	PROPN
ejpam-847	101	37	2k−1	2k−1	NUM
ejpam-847	101	38	f	f	PROPN
ejpam-847	101	39	�	�	PROPN
ejpam-847	101	40	x	x	SYM
ejpam-847	101	41	2k−1	2k−1	NUM
ejpam-847	101	42	�	�	PROPN
ejpam-847	101	43	−2k	−2k	PROPN
ejpam-847	101	44	f	f	PROPN
ejpam-847	101	45	�	�	PROPN
ejpam-847	101	46	x	x	PROPN
ejpam-847	101	47	2k	2k	PROPN
ejpam-847	101	48	�	�	PROPN
ejpam-847	101	49	�	�	PROPN
ejpam-847	101	50	(	(	PUNCT
ejpam-847	101	51	t	t	PROPN
ejpam-847	101	52	)	)	PUNCT
ejpam-847	101	53	≥µ∑m	≥µ∑m	PROPN
ejpam-847	101	54	k=1	k=1	X
ejpam-847	101	55	�	�	PROPN
ejpam-847	101	56	2k−1	2k−1	NUM
ejpam-847	101	57	f	f	PROPN
ejpam-847	101	58	�	�	PROPN
ejpam-847	101	59	x	x	SYM
ejpam-847	101	60	2k−1	2k−1	NUM
ejpam-847	101	61	�	�	PROPN
ejpam-847	102	1	−2k	−2k	PROPN
ejpam-847	102	2	f	f	PROPN
ejpam-847	102	3	�	�	PROPN
ejpam-847	102	4	x	x	PROPN
ejpam-847	102	5	2k	2k	PROPN
ejpam-847	102	6	�	�	PROPN
ejpam-847	102	7	�	�	PROPN
ejpam-847	102	8	m	m	PROPN
ejpam-847	102	9	∑	∑	PROPN
ejpam-847	102	10	k=1	k=1	PROPN
ejpam-847	102	11	t	t	PROPN
ejpam-847	102	12	2k	2k	PROPN
ejpam-847	102	13	!	!	PUNCT
ejpam-847	103	1	≥t	≥t	VERB
ejpam-847	103	2	m	m	VERB
ejpam-847	103	3	k=1	k=1	PROPN
ejpam-847	103	4	ρ	ρ	NUM
ejpam-847	103	5	�	�	PROPN
ejpam-847	103	6	x	x	SYM
ejpam-847	103	7	2k	2k	NUM
ejpam-847	103	8	,	,	PUNCT
ejpam-847	103	9	x	x	PROPN
ejpam-847	103	10	2k	2k	NUM
ejpam-847	103	11	,	,	PUNCT
ejpam-847	103	12	−	−	PROPN
ejpam-847	103	13	x	x	SYM
ejpam-847	103	14	2k−1	2k−1	NUM
ejpam-847	103	15	,	,	PUNCT
ejpam-847	103	16	0,	0,	NUM
ejpam-847	103	17	...	...	PUNCT
ejpam-847	103	18	,0	,0	PUNCT
ejpam-847	103	19	�	�	PROPN
ejpam-847	103	20	�	�	PROPN
ejpam-847	103	21	nt	not	PART
ejpam-847	104	1	22k−2	22k−2	NUM
ejpam-847	104	2	�	�	PROPN
ejpam-847	104	3	replacing	replace	VERB
ejpam-847	104	4	x	x	PUNCT
ejpam-847	104	5	by	by	ADP
ejpam-847	104	6	x	x	SYM
ejpam-847	104	7	2l	2l	NUM
ejpam-847	104	8	in	in	ADP
ejpam-847	104	9	the	the	DET
ejpam-847	104	10	above	above	ADJ
ejpam-847	104	11	inequality	inequality	NOUN
ejpam-847	104	12	,	,	PUNCT
ejpam-847	104	13	we	we	PRON
ejpam-847	104	14	get	get	VERB
ejpam-847	104	15	µ	µ	PRON
ejpam-847	104	16	f	f	X
ejpam-847	104	17	�	�	PROPN
ejpam-847	104	18	x	x	SYM
ejpam-847	104	19	2l	2l	NUM
ejpam-847	104	20	�	�	PROPN
ejpam-847	104	21	−2	−2	PROPN
ejpam-847	104	22	m	m	PROPN
ejpam-847	104	23	f	f	PROPN
ejpam-847	104	24	�	�	PROPN
ejpam-847	104	25	x	x	PROPN
ejpam-847	104	26	2m+l	2m+l	NUM
ejpam-847	104	27	�	�	PROPN
ejpam-847	104	28	(	(	PUNCT
ejpam-847	104	29	t	t	PROPN
ejpam-847	104	30	)	)	PUNCT
ejpam-847	104	31	≥	≥	NOUN
ejpam-847	104	32	t	t	PROPN
ejpam-847	104	33	m	m	VERB
ejpam-847	104	34	k=1	k=1	PROPN
ejpam-847	104	35	ρ	ρ	NUM
ejpam-847	104	36	�	�	PROPN
ejpam-847	104	37	x	x	SYM
ejpam-847	104	38	2k+l	2k+l	NUM
ejpam-847	104	39	,	,	PUNCT
ejpam-847	104	40	x	x	X
ejpam-847	104	41	2k+l	2k+l	NUM
ejpam-847	104	42	,	,	PUNCT
ejpam-847	104	43	−	−	PROPN
ejpam-847	104	44	x	x	SYM
ejpam-847	104	45	2k+l−1	2k+l−1	NUM
ejpam-847	104	46	,	,	PUNCT
ejpam-847	104	47	0,	0,	NUM
ejpam-847	104	48	...	...	PUNCT
ejpam-847	104	49	,0	,0	PUNCT
ejpam-847	104	50	�	�	PROPN
ejpam-847	104	51	�	�	PROPN
ejpam-847	104	52	nt	not	PART
ejpam-847	104	53	22k−2	22k−2	PROPN
ejpam-847	104	54	�	�	PROPN
ejpam-847	104	55	which	which	PRON
ejpam-847	104	56	is	be	AUX
ejpam-847	104	57	equivalent	equivalent	ADJ
ejpam-847	104	58	to	to	ADP
ejpam-847	104	59	µ	µ	PROPN
ejpam-847	104	60	2l	2l	PROPN
ejpam-847	104	61	f	f	PROPN
ejpam-847	104	62	�	�	PROPN
ejpam-847	104	63	x	x	SYM
ejpam-847	104	64	2l	2l	NUM
ejpam-847	104	65	�	�	PROPN
ejpam-847	104	66	−2m+l	−2m+l	PROPN
ejpam-847	104	67	f	f	PROPN
ejpam-847	104	68	�	�	PROPN
ejpam-847	104	69	x	x	PROPN
ejpam-847	104	70	2m+l	2m+l	NUM
ejpam-847	104	71	�	�	PROPN
ejpam-847	104	72	(	(	PUNCT
ejpam-847	104	73	t	t	PROPN
ejpam-847	104	74	)	)	PUNCT
ejpam-847	104	75	≥	≥	NOUN
ejpam-847	104	76	t	t	PROPN
ejpam-847	104	77	m	m	VERB
ejpam-847	104	78	k=1	k=1	PROPN
ejpam-847	104	79	ρ	ρ	NUM
ejpam-847	104	80	�	�	PROPN
ejpam-847	104	81	x	x	SYM
ejpam-847	104	82	2k+l	2k+l	NUM
ejpam-847	104	83	,	,	PUNCT
ejpam-847	104	84	x	x	X
ejpam-847	104	85	2k+l	2k+l	NUM
ejpam-847	104	86	,	,	PUNCT
ejpam-847	104	87	−	−	PROPN
ejpam-847	104	88	x	x	SYM
ejpam-847	104	89	2k+l−1	2k+l−1	NUM
ejpam-847	104	90	,	,	PUNCT
ejpam-847	104	91	0,	0,	NUM
ejpam-847	104	92	...	...	PUNCT
ejpam-847	104	93	,0	,0	PUNCT
ejpam-847	104	94	�	�	PROPN
ejpam-847	104	95	�	�	PROPN
ejpam-847	104	96	nt	nt	PROPN
ejpam-847	104	97	22k+l−2	22k+l−2	NUM
ejpam-847	104	98	�	�	PROPN
ejpam-847	104	99	(	(	PUNCT
ejpam-847	104	100	6	6	NUM
ejpam-847	104	101	)	)	PUNCT
ejpam-847	104	102	for	for	ADP
ejpam-847	104	103	all	all	DET
ejpam-847	104	104	x	x	SYM
ejpam-847	104	105	∈	∈	PROPN
ejpam-847	104	106	x	x	X
ejpam-847	104	107	,	,	PUNCT
ejpam-847	104	108	all	all	PRON
ejpam-847	104	109	t	t	X
ejpam-847	104	110	>	>	X
ejpam-847	104	111	0	0	PUNCT
ejpam-847	105	1	and	and	CCONJ
ejpam-847	105	2	all	all	DET
ejpam-847	105	3	l	l	NOUN
ejpam-847	105	4	=	=	PUNCT
ejpam-847	105	5	0,1,2	0,1,2	NUM
ejpam-847	105	6	,	,	PUNCT
ejpam-847	105	7	.	.	PUNCT
ejpam-847	105	8	.	.	PUNCT
ejpam-847	106	1	..	..	PUNCT
ejpam-847	107	1	d.	d.	PROPN
ejpam-847	107	2	shin	shin	PROPN
ejpam-847	107	3	,	,	PUNCT
ejpam-847	107	4	j.	j.	PROPN
ejpam-847	107	5	lee	lee	PROPN
ejpam-847	107	6	,	,	PUNCT
ejpam-847	107	7	c.	c.	PROPN
ejpam-847	107	8	park	park	PROPN
ejpam-847	107	9	/	/	SYM
ejpam-847	107	10	eur	eur	PROPN
ejpam-847	107	11	.	.	PUNCT
ejpam-847	108	1	j.	j.	PROPN
ejpam-847	108	2	pure	pure	PROPN
ejpam-847	108	3	appl	appl	PROPN
ejpam-847	108	4	.	.	PROPN
ejpam-847	108	5	math	math	PROPN
ejpam-847	108	6	,	,	PUNCT
ejpam-847	108	7	5	5	NUM
ejpam-847	108	8	(	(	PUNCT
ejpam-847	108	9	2012	2012	NUM
ejpam-847	108	10	)	)	PUNCT
ejpam-847	108	11	,	,	PUNCT
ejpam-847	108	12	540	540	NUM
ejpam-847	108	13	-	-	SYM
ejpam-847	108	14	553	553	NUM
ejpam-847	108	15	544	544	NUM
ejpam-847	108	16	since	since	SCONJ
ejpam-847	108	17	the	the	DET
ejpam-847	108	18	right	right	ADJ
ejpam-847	108	19	hand	hand	NOUN
ejpam-847	108	20	side	side	NOUN
ejpam-847	108	21	of	of	ADP
ejpam-847	108	22	the	the	DET
ejpam-847	108	23	inequality	inequality	NOUN
ejpam-847	108	24	(	(	PUNCT
ejpam-847	108	25	6	6	NUM
ejpam-847	108	26	)	)	PUNCT
ejpam-847	108	27	tends	tend	VERB
ejpam-847	108	28	to	to	ADP
ejpam-847	108	29	1	1	NUM
ejpam-847	108	30	as	as	ADP
ejpam-847	108	31	m→∞	m→∞	NUM
ejpam-847	108	32	by	by	ADP
ejpam-847	108	33	(	(	PUNCT
ejpam-847	108	34	3	3	NUM
ejpam-847	108	35	)	)	PUNCT
ejpam-847	108	36	,	,	PUNCT
ejpam-847	108	37	the	the	DET
ejpam-847	108	38	sequence	sequence	NOUN
ejpam-847	108	39	{	{	PUNCT
ejpam-847	108	40	2	2	NUM
ejpam-847	108	41	m	m	PROPN
ejpam-847	108	42	f	f	PROPN
ejpam-847	108	43	�	�	PROPN
ejpam-847	108	44	x	x	PUNCT
ejpam-847	108	45	2	2	NUM
ejpam-847	108	46	m	m	PROPN
ejpam-847	108	47	�	�	PROPN
ejpam-847	108	48	}	}	PUNCT
ejpam-847	108	49	is	be	AUX
ejpam-847	108	50	a	a	DET
ejpam-847	108	51	cauchy	cauchy	ADJ
ejpam-847	108	52	sequence	sequence	NOUN
ejpam-847	108	53	.	.	PUNCT
ejpam-847	109	1	thus	thus	ADV
ejpam-847	109	2	we	we	PRON
ejpam-847	109	3	define	define	VERB
ejpam-847	109	4	a(x	a(x	NOUN
ejpam-847	109	5	)	)	PUNCT
ejpam-847	109	6	:	:	PUNCT
ejpam-847	109	7	=	=	SYM
ejpam-847	109	8	limm→∞	limm→∞	PROPN
ejpam-847	109	9	2	2	NUM
ejpam-847	109	10	m	m	PROPN
ejpam-847	109	11	f	f	PROPN
ejpam-847	109	12	�	�	PROPN
ejpam-847	109	13	x	x	PUNCT
ejpam-847	109	14	2	2	NUM
ejpam-847	109	15	m	m	NOUN
ejpam-847	109	16	�	�	NOUN
ejpam-847	109	17	for	for	ADP
ejpam-847	109	18	all	all	DET
ejpam-847	109	19	x	x	SYM
ejpam-847	109	20	∈	∈	PROPN
ejpam-847	109	21	x	x	X
ejpam-847	109	22	,	,	PUNCT
ejpam-847	109	23	which	which	PRON
ejpam-847	109	24	is	be	AUX
ejpam-847	109	25	an	an	DET
ejpam-847	109	26	odd	odd	ADJ
ejpam-847	109	27	mapping	mapping	NOUN
ejpam-847	109	28	.	.	PUNCT
ejpam-847	110	1	now	now	ADV
ejpam-847	110	2	we	we	PRON
ejpam-847	110	3	show	show	VERB
ejpam-847	110	4	that	that	SCONJ
ejpam-847	110	5	a	a	PRON
ejpam-847	110	6	is	be	AUX
ejpam-847	110	7	an	an	DET
ejpam-847	110	8	additive	additive	ADJ
ejpam-847	110	9	mapping	mapping	NOUN
ejpam-847	110	10	.	.	PUNCT
ejpam-847	111	1	by	by	ADP
ejpam-847	111	2	(	(	PUNCT
ejpam-847	111	3	2	2	NUM
ejpam-847	111	4	)	)	PUNCT
ejpam-847	111	5	,	,	PUNCT
ejpam-847	111	6	we	we	PRON
ejpam-847	111	7	get	get	VERB
ejpam-847	111	8	µ2	µ2	PROPN
ejpam-847	111	9	m	m	PROPN
ejpam-847	111	10	�	�	PROPN
ejpam-847	111	11	f	f	PROPN
ejpam-847	111	12	�	�	PROPN
ejpam-847	111	13	x+y	x+y	NUM
ejpam-847	111	14	2	2	NUM
ejpam-847	111	15	m	m	PROPN
ejpam-847	111	16	�	�	NOUN
ejpam-847	111	17	−	−	PROPN
ejpam-847	111	18	f	f	PROPN
ejpam-847	111	19	�	�	PROPN
ejpam-847	111	20	x	x	PUNCT
ejpam-847	111	21	2	2	NUM
ejpam-847	111	22	m	m	PROPN
ejpam-847	111	23	�	�	NOUN
ejpam-847	111	24	−	−	PROPN
ejpam-847	111	25	f	f	PROPN
ejpam-847	111	26	�	�	PROPN
ejpam-847	111	27	y	y	PROPN
ejpam-847	111	28	2	2	NUM
ejpam-847	111	29	m	m	PROPN
ejpam-847	111	30	�	�	PROPN
ejpam-847	111	31	�	�	PROPN
ejpam-847	111	32	(	(	PUNCT
ejpam-847	111	33	t	t	PROPN
ejpam-847	111	34	)	)	PUNCT
ejpam-847	111	35	≥	≥	NOUN
ejpam-847	111	36	ρ	ρ	NUM
ejpam-847	111	37	�	�	PROPN
ejpam-847	111	38	x	x	PUNCT
ejpam-847	111	39	2	2	NUM
ejpam-847	111	40	m	m	NOUN
ejpam-847	111	41	,	,	PUNCT
ejpam-847	111	42	y	y	PROPN
ejpam-847	111	43	2	2	NUM
ejpam-847	111	44	m	m	NOUN
ejpam-847	111	45	,	,	PUNCT
ejpam-847	111	46	−	−	PROPN
ejpam-847	111	47	�	�	PROPN
ejpam-847	111	48	x+y	x+y	NUM
ejpam-847	111	49	2	2	NUM
ejpam-847	111	50	m	m	PROPN
ejpam-847	111	51	�	�	PROPN
ejpam-847	111	52	,	,	PUNCT
ejpam-847	111	53	0,	0,	NUM
ejpam-847	111	54	...	...	PUNCT
ejpam-847	111	55	,0	,0	PUNCT
ejpam-847	111	56	�	�	PROPN
ejpam-847	111	57	�	�	PROPN
ejpam-847	111	58	nt	not	PART
ejpam-847	111	59	2m−1	2m−1	PROPN
ejpam-847	111	60	�	�	PROPN
ejpam-847	111	61	.	.	PUNCT
ejpam-847	112	1	taking	take	VERB
ejpam-847	112	2	the	the	DET
ejpam-847	112	3	limit	limit	NOUN
ejpam-847	112	4	as	as	ADP
ejpam-847	112	5	m	m	PROPN
ejpam-847	112	6	→	→	SYM
ejpam-847	112	7	∞	∞	PROPN
ejpam-847	112	8	in	in	ADP
ejpam-847	112	9	the	the	DET
ejpam-847	112	10	above	above	ADJ
ejpam-847	112	11	inequality	inequality	NOUN
ejpam-847	112	12	,	,	PUNCT
ejpam-847	112	13	by	by	ADP
ejpam-847	112	14	(	(	PUNCT
ejpam-847	112	15	4	4	NUM
ejpam-847	112	16	)	)	PUNCT
ejpam-847	112	17	,	,	PUNCT
ejpam-847	112	18	the	the	DET
ejpam-847	112	19	mapping	mapping	NOUN
ejpam-847	112	20	a	a	PRON
ejpam-847	112	21	is	be	AUX
ejpam-847	112	22	additive	additive	ADJ
ejpam-847	112	23	.	.	PUNCT
ejpam-847	113	1	by	by	ADP
ejpam-847	113	2	letting	let	VERB
ejpam-847	113	3	l	l	NOUN
ejpam-847	113	4	=	=	SYM
ejpam-847	113	5	0	0	PUNCT
ejpam-847	113	6	and	and	CCONJ
ejpam-847	113	7	taking	take	VERB
ejpam-847	113	8	the	the	DET
ejpam-847	113	9	limit	limit	NOUN
ejpam-847	113	10	as	as	ADP
ejpam-847	113	11	m→∞	m→∞	NOUN
ejpam-847	113	12	in	in	ADP
ejpam-847	113	13	(	(	PUNCT
ejpam-847	113	14	6	6	NUM
ejpam-847	113	15	)	)	PUNCT
ejpam-847	113	16	,	,	PUNCT
ejpam-847	113	17	we	we	PRON
ejpam-847	113	18	get	get	VERB
ejpam-847	113	19	(	(	PUNCT
ejpam-847	113	20	5	5	NUM
ejpam-847	113	21	)	)	PUNCT
ejpam-847	113	22	.	.	PUNCT
ejpam-847	114	1	finally	finally	ADV
ejpam-847	114	2	,	,	PUNCT
ejpam-847	114	3	to	to	PART
ejpam-847	114	4	prove	prove	VERB
ejpam-847	114	5	the	the	DET
ejpam-847	114	6	uniqueness	uniqueness	NOUN
ejpam-847	114	7	of	of	ADP
ejpam-847	114	8	the	the	DET
ejpam-847	114	9	additive	additive	ADJ
ejpam-847	114	10	mapping	mapping	NOUN
ejpam-847	114	11	a	a	DET
ejpam-847	114	12	subject	subject	NOUN
ejpam-847	114	13	to	to	ADP
ejpam-847	114	14	(	(	PUNCT
ejpam-847	114	15	5	5	NUM
ejpam-847	114	16	)	)	PUNCT
ejpam-847	114	17	,	,	PUNCT
ejpam-847	114	18	let	let	VERB
ejpam-847	114	19	us	we	PRON
ejpam-847	114	20	assume	assume	VERB
ejpam-847	114	21	that	that	SCONJ
ejpam-847	114	22	there	there	PRON
ejpam-847	114	23	exists	exist	VERB
ejpam-847	114	24	another	another	DET
ejpam-847	114	25	additive	additive	ADJ
ejpam-847	114	26	mapping	mapping	NOUN
ejpam-847	114	27	b	b	NOUN
ejpam-847	114	28	which	which	PRON
ejpam-847	114	29	satisfies	satisfy	VERB
ejpam-847	114	30	(	(	PUNCT
ejpam-847	114	31	5	5	NUM
ejpam-847	114	32	)	)	PUNCT
ejpam-847	114	33	.	.	PUNCT
ejpam-847	115	1	since	since	SCONJ
ejpam-847	115	2	µa(x)−b(x)(2	µa(x)−b(x)(2	ADP
ejpam-847	115	3	t	t	PROPN
ejpam-847	115	4	)	)	PUNCT
ejpam-847	115	5	=	=	SYM
ejpam-847	115	6	µa(x)−2	µa(x)−2	NOUN
ejpam-847	115	7	m	m	PROPN
ejpam-847	115	8	f	f	PROPN
ejpam-847	115	9	�	�	PROPN
ejpam-847	115	10	x	x	PUNCT
ejpam-847	115	11	2	2	NUM
ejpam-847	115	12	m	m	NOUN
ejpam-847	115	13	�	�	PROPN
ejpam-847	115	14	+2	+2	PROPN
ejpam-847	115	15	m	m	PROPN
ejpam-847	115	16	f	f	PROPN
ejpam-847	115	17	�	�	PROPN
ejpam-847	115	18	x	x	PUNCT
ejpam-847	115	19	2	2	NUM
ejpam-847	115	20	m	m	PROPN
ejpam-847	115	21	�	�	PROPN
ejpam-847	115	22	−b(x)(2	−b(x)(2	PROPN
ejpam-847	115	23	t	t	PROPN
ejpam-847	115	24	)	)	PUNCT
ejpam-847	115	25	≥t	≥t	X
ejpam-847	115	26	�	�	PROPN
ejpam-847	115	27	µa(x)−2	µa(x)−2	NOUN
ejpam-847	115	28	m	m	PROPN
ejpam-847	115	29	f	f	PROPN
ejpam-847	115	30	�	�	PROPN
ejpam-847	115	31	x	x	PUNCT
ejpam-847	115	32	2	2	NUM
ejpam-847	115	33	m	m	PROPN
ejpam-847	115	34	�	�	NOUN
ejpam-847	115	35	(	(	PUNCT
ejpam-847	115	36	t),µ2	t),µ2	NOUN
ejpam-847	115	37	m	m	PROPN
ejpam-847	115	38	f	f	PROPN
ejpam-847	115	39	�	�	PROPN
ejpam-847	115	40	x	x	PUNCT
ejpam-847	115	41	2	2	NUM
ejpam-847	115	42	m	m	PROPN
ejpam-847	115	43	�	�	PROPN
ejpam-847	115	44	−b(x)(t	−b(x)(t	NOUN
ejpam-847	115	45	)	)	PUNCT
ejpam-847	115	46	�	�	PROPN
ejpam-847	115	47	and	and	CCONJ
ejpam-847	115	48	lim	lim	PROPN
ejpam-847	115	49	m→∞	m→∞	NUM
ejpam-847	115	50	µa(x)−2	µa(x)−2	NOUN
ejpam-847	115	51	m	m	PROPN
ejpam-847	115	52	f	f	PROPN
ejpam-847	115	53	�	�	PROPN
ejpam-847	115	54	x	x	PUNCT
ejpam-847	115	55	2	2	NUM
ejpam-847	115	56	m	m	PROPN
ejpam-847	115	57	�	�	NOUN
ejpam-847	115	58	=	=	SYM
ejpam-847	115	59	lim	lim	PROPN
ejpam-847	115	60	m→∞	m→∞	NUM
ejpam-847	115	61	µb(x)−2	µb(x)−2	NOUN
ejpam-847	115	62	m	m	PROPN
ejpam-847	115	63	f	f	PROPN
ejpam-847	115	64	�	�	PROPN
ejpam-847	115	65	x	x	PUNCT
ejpam-847	115	66	2	2	NUM
ejpam-847	115	67	m	m	PROPN
ejpam-847	115	68	�	�	NOUN
ejpam-847	115	69	=	=	NOUN
ejpam-847	115	70	1	1	NUM
ejpam-847	115	71	for	for	ADP
ejpam-847	115	72	all	all	DET
ejpam-847	115	73	x	x	SYM
ejpam-847	115	74	∈	∈	ADJ
ejpam-847	115	75	x	x	X
ejpam-847	115	76	and	and	CCONJ
ejpam-847	115	77	all	all	PRON
ejpam-847	115	78	t	t	PROPN
ejpam-847	115	79	>	>	X
ejpam-847	115	80	0	0	NUM
ejpam-847	115	81	,	,	PUNCT
ejpam-847	115	82	we	we	PRON
ejpam-847	115	83	get	get	VERB
ejpam-847	115	84	lim	lim	PROPN
ejpam-847	115	85	m→∞	m→∞	PROPN
ejpam-847	115	86	t	t	PROPN
ejpam-847	115	87	�	�	PROPN
ejpam-847	115	88	µa(x)−2	µa(x)−2	NOUN
ejpam-847	115	89	m	m	PROPN
ejpam-847	115	90	f	f	PROPN
ejpam-847	115	91	�	�	PROPN
ejpam-847	115	92	x	x	PUNCT
ejpam-847	115	93	2	2	NUM
ejpam-847	115	94	m	m	PROPN
ejpam-847	115	95	�	�	NOUN
ejpam-847	115	96	(	(	PUNCT
ejpam-847	115	97	t),µ2	t),µ2	NOUN
ejpam-847	115	98	m	m	PROPN
ejpam-847	115	99	f	f	PROPN
ejpam-847	115	100	�	�	PROPN
ejpam-847	115	101	x	x	PUNCT
ejpam-847	115	102	2	2	NUM
ejpam-847	115	103	m	m	PROPN
ejpam-847	115	104	�	�	PROPN
ejpam-847	115	105	−b(x)(t	−b(x)(t	NOUN
ejpam-847	115	106	)	)	PUNCT
ejpam-847	115	107	�	�	PROPN
ejpam-847	116	1	=	=	SYM
ejpam-847	116	2	1	1	X
ejpam-847	116	3	.	.	PUNCT
ejpam-847	117	1	thus	thus	ADV
ejpam-847	117	2	we	we	PRON
ejpam-847	117	3	have	have	AUX
ejpam-847	117	4	a=	a=	VERB
ejpam-847	117	5	b.	b.	PROPN
ejpam-847	117	6	corollary	corollary	NOUN
ejpam-847	117	7	1	1	NUM
ejpam-847	117	8	.	.	PUNCT
ejpam-847	118	1	let	let	VERB
ejpam-847	118	2	θ	θ	PROPN
ejpam-847	118	3	≥	≥	X
ejpam-847	118	4	0	0	NUM
ejpam-847	118	5	and	and	CCONJ
ejpam-847	118	6	let	let	VERB
ejpam-847	118	7	p	p	PRON
ejpam-847	118	8	be	be	AUX
ejpam-847	118	9	a	a	DET
ejpam-847	118	10	constant	constant	ADJ
ejpam-847	118	11	with	with	ADP
ejpam-847	118	12	p	p	PROPN
ejpam-847	118	13	>	>	X
ejpam-847	118	14	1	1	NUM
ejpam-847	118	15	.	.	PUNCT
ejpam-847	119	1	for	for	ADP
ejpam-847	119	2	a	a	DET
ejpam-847	119	3	normed	normed	ADJ
ejpam-847	119	4	vector	vector	NOUN
ejpam-847	119	5	space	space	NOUN
ejpam-847	119	6	x	x	PUNCT
ejpam-847	119	7	and	and	CCONJ
ejpam-847	119	8	complete	complete	ADJ
ejpam-847	119	9	rn	rn	NOUN
ejpam-847	119	10	-	-	NOUN
ejpam-847	119	11	space	space	NOUN
ejpam-847	119	12	y	y	PROPN
ejpam-847	119	13	,	,	PUNCT
ejpam-847	119	14	let	let	VERB
ejpam-847	119	15	f	f	PRON
ejpam-847	119	16	:	:	PUNCT
ejpam-847	119	17	x	x	X
ejpam-847	119	18	→	→	SYM
ejpam-847	119	19	y	y	X
ejpam-847	119	20	be	be	AUX
ejpam-847	119	21	an	an	DET
ejpam-847	119	22	odd	odd	ADJ
ejpam-847	119	23	mapping	mapping	NOUN
ejpam-847	119	24	satisfying	satisfy	VERB
ejpam-847	119	25	µd	µd	DET
ejpam-847	119	26	f	f	PROPN
ejpam-847	119	27	(	(	PUNCT
ejpam-847	119	28	x1	x1	INTJ
ejpam-847	119	29	,	,	PUNCT
ejpam-847	119	30	x2,	x2,	PROPN
ejpam-847	119	31	...	...	PUNCT
ejpam-847	119	32	,xn	,xn	PUNCT
ejpam-847	119	33	)	)	PUNCT
ejpam-847	119	34	(	(	PUNCT
ejpam-847	119	35	t	t	PROPN
ejpam-847	119	36	)	)	PUNCT
ejpam-847	119	37	≥	≥	PROPN
ejpam-847	119	38	t	t	PROPN
ejpam-847	119	39	t	t	NOUN
ejpam-847	120	1	+	+	CCONJ
ejpam-847	120	2	θ	θ	PROPN
ejpam-847	120	3	∑n	∑n	PROPN
ejpam-847	120	4	i=1	i=1	PROPN
ejpam-847	120	5	||x	||x	PROPN
ejpam-847	120	6	i||p	i||p	PROPN
ejpam-847	120	7	for	for	ADP
ejpam-847	120	8	all	all	PRON
ejpam-847	120	9	(	(	PUNCT
ejpam-847	120	10	x1	x1	PROPN
ejpam-847	120	11	,	,	PUNCT
ejpam-847	120	12	x2	x2	PROPN
ejpam-847	120	13	,	,	PUNCT
ejpam-847	120	14	.	.	PUNCT
ejpam-847	120	15	.	.	PUNCT
ejpam-847	120	16	.	.	PUNCT
ejpam-847	121	1	,	,	PUNCT
ejpam-847	121	2	xn	xn	X
ejpam-847	121	3	)	)	PUNCT
ejpam-847	121	4	∈	∈	PROPN
ejpam-847	121	5	x	x	PUNCT
ejpam-847	121	6	with	with	ADP
ejpam-847	121	7	∑n	∑n	PROPN
ejpam-847	121	8	i=1	i=1	NOUN
ejpam-847	121	9	x	x	PUNCT
ejpam-847	122	1	i	i	NOUN
ejpam-847	122	2	=	=	NOUN
ejpam-847	122	3	0	0	NUM
ejpam-847	122	4	and	and	CCONJ
ejpam-847	122	5	all	all	DET
ejpam-847	122	6	t	t	NOUN
ejpam-847	122	7	>	>	X
ejpam-847	122	8	0	0	X
ejpam-847	122	9	.	.	PUNCT
ejpam-847	123	1	if	if	SCONJ
ejpam-847	123	2	t∞k=1	t∞k=1	NOUN
ejpam-847	123	3	�	�	PROPN
ejpam-847	123	4	2(k+l)pnt	2(k+l)pnt	NOUN
ejpam-847	123	5	2(k+l)pnt	2(k+l)pnt	NOUN
ejpam-847	123	6	+	+	CCONJ
ejpam-847	123	7	22k+l−2(2	22k+l−2(2	NUM
ejpam-847	123	8	+	+	SYM
ejpam-847	123	9	2p)θ	2p)θ	NUM
ejpam-847	123	10	||x	||x	NOUN
ejpam-847	123	11	||p	||p	NOUN
ejpam-847	123	12	�	�	NOUN
ejpam-847	124	1	=	=	NOUN
ejpam-847	124	2	1	1	NUM
ejpam-847	124	3	for	for	ADP
ejpam-847	124	4	all	all	DET
ejpam-847	124	5	x	x	SYM
ejpam-847	124	6	∈	∈	PROPN
ejpam-847	124	7	x	x	X
ejpam-847	124	8	,	,	PUNCT
ejpam-847	124	9	all	all	PRON
ejpam-847	124	10	t	t	X
ejpam-847	124	11	>	>	X
ejpam-847	124	12	0	0	PUNCT
ejpam-847	124	13	and	and	CCONJ
ejpam-847	124	14	all	all	DET
ejpam-847	124	15	l	l	NOUN
ejpam-847	124	16	=	=	PUNCT
ejpam-847	124	17	0,1,2	0,1,2	NUM
ejpam-847	124	18	,	,	PUNCT
ejpam-847	124	19	.	.	PUNCT
ejpam-847	124	20	.	.	PUNCT
ejpam-847	125	1	.	.	PUNCT
ejpam-847	126	1	,	,	PUNCT
ejpam-847	126	2	then	then	ADV
ejpam-847	126	3	there	there	PRON
ejpam-847	126	4	exists	exist	VERB
ejpam-847	126	5	a	a	DET
ejpam-847	126	6	unique	unique	ADJ
ejpam-847	126	7	additive	additive	NOUN
ejpam-847	126	8	mapping	mapping	NOUN
ejpam-847	126	9	a	a	DET
ejpam-847	126	10	:	:	PUNCT
ejpam-847	126	11	x	x	X
ejpam-847	126	12	→	→	SYM
ejpam-847	126	13	y	y	PROPN
ejpam-847	126	14	such	such	ADJ
ejpam-847	127	1	that	that	SCONJ
ejpam-847	127	2	µ	µ	PROPN
ejpam-847	127	3	f	f	X
ejpam-847	127	4	(	(	PUNCT
ejpam-847	127	5	x)−a(x)(t	x)−a(x)(t	PROPN
ejpam-847	127	6	)	)	PUNCT
ejpam-847	127	7	≥	≥	NOUN
ejpam-847	127	8	t∞k=1	t∞k=1	NOUN
ejpam-847	127	9	�	�	PROPN
ejpam-847	127	10	2kpnt	2kpnt	NUM
ejpam-847	127	11	2kpnt	2kpnt	PROPN
ejpam-847	127	12	+	+	CCONJ
ejpam-847	127	13	22k−2(2	22k−2(2	PROPN
ejpam-847	127	14	+	+	SYM
ejpam-847	127	15	2p)θ	2p)θ	NUM
ejpam-847	127	16	||x	||x	NOUN
ejpam-847	127	17	||p	||p	NOUN
ejpam-847	127	18	�	�	PROPN
ejpam-847	127	19	for	for	ADP
ejpam-847	127	20	all	all	DET
ejpam-847	127	21	x	x	SYM
ejpam-847	127	22	∈	∈	ADJ
ejpam-847	127	23	x	x	X
ejpam-847	127	24	and	and	CCONJ
ejpam-847	127	25	all	all	DET
ejpam-847	127	26	t	t	PROPN
ejpam-847	127	27	>	>	X
ejpam-847	127	28	0	0	X
ejpam-847	127	29	.	.	PUNCT
ejpam-847	127	30	d.	d.	PROPN
ejpam-847	127	31	shin	shin	PROPN
ejpam-847	127	32	,	,	PUNCT
ejpam-847	127	33	j.	j.	PROPN
ejpam-847	127	34	lee	lee	PROPN
ejpam-847	127	35	,	,	PUNCT
ejpam-847	127	36	c.	c.	PROPN
ejpam-847	127	37	park	park	PROPN
ejpam-847	127	38	/	/	SYM
ejpam-847	127	39	eur	eur	PROPN
ejpam-847	127	40	.	.	PUNCT
ejpam-847	128	1	j.	j.	PROPN
ejpam-847	128	2	pure	pure	PROPN
ejpam-847	128	3	appl	appl	PROPN
ejpam-847	128	4	.	.	PROPN
ejpam-847	128	5	math	math	PROPN
ejpam-847	128	6	,	,	PUNCT
ejpam-847	128	7	5	5	NUM
ejpam-847	128	8	(	(	PUNCT
ejpam-847	128	9	2012	2012	NUM
ejpam-847	128	10	)	)	PUNCT
ejpam-847	128	11	,	,	PUNCT
ejpam-847	128	12	540	540	NUM
ejpam-847	128	13	-	-	SYM
ejpam-847	128	14	553	553	NUM
ejpam-847	128	15	545	545	NUM
ejpam-847	128	16	proof	proof	NOUN
ejpam-847	128	17	.	.	PUNCT
ejpam-847	129	1	if	if	SCONJ
ejpam-847	129	2	we	we	PRON
ejpam-847	129	3	define	define	VERB
ejpam-847	129	4	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	129	5	...	...	PUNCT
ejpam-847	129	6	,xn	,xn	PUNCT
ejpam-847	129	7	)	)	PUNCT
ejpam-847	129	8	(	(	PUNCT
ejpam-847	129	9	t	t	NOUN
ejpam-847	129	10	)	)	PUNCT
ejpam-847	129	11	=	=	SYM
ejpam-847	129	12	t	t	PROPN
ejpam-847	129	13	t	t	NOUN
ejpam-847	130	1	+	+	CCONJ
ejpam-847	130	2	θ	θ	PROPN
ejpam-847	130	3	∑n	∑n	PROPN
ejpam-847	130	4	i=1	i=1	PROPN
ejpam-847	130	5	||x	||x	PROPN
ejpam-847	130	6	i||p	i||p	PROPN
ejpam-847	130	7	and	and	CCONJ
ejpam-847	130	8	apply	apply	VERB
ejpam-847	130	9	theorem	theorem	NOUN
ejpam-847	130	10	1	1	NUM
ejpam-847	130	11	,	,	PUNCT
ejpam-847	130	12	then	then	ADV
ejpam-847	130	13	we	we	PRON
ejpam-847	130	14	get	get	VERB
ejpam-847	130	15	the	the	DET
ejpam-847	130	16	desired	desire	VERB
ejpam-847	130	17	result	result	NOUN
ejpam-847	130	18	.	.	PUNCT
ejpam-847	131	1	theorem	theorem	NOUN
ejpam-847	131	2	2	2	NUM
ejpam-847	131	3	.	.	PUNCT
ejpam-847	132	1	let	let	VERB
ejpam-847	132	2	f	f	NOUN
ejpam-847	132	3	:	:	PUNCT
ejpam-847	132	4	x	x	X
ejpam-847	132	5	→	→	SYM
ejpam-847	132	6	y	y	X
ejpam-847	132	7	be	be	AUX
ejpam-847	132	8	an	an	DET
ejpam-847	132	9	odd	odd	ADJ
ejpam-847	132	10	mapping	mapping	NOUN
ejpam-847	132	11	for	for	ADP
ejpam-847	132	12	which	which	PRON
ejpam-847	132	13	there	there	PRON
ejpam-847	132	14	is	be	VERB
ejpam-847	132	15	a	a	DET
ejpam-847	132	16	ρ	ρ	NOUN
ejpam-847	132	17	:	:	PUNCT
ejpam-847	132	18	x	x	SYM
ejpam-847	132	19	n→	n→	X
ejpam-847	132	20	d+	d+	X
ejpam-847	132	21	satisfying	satisfy	VERB
ejpam-847	132	22	(	(	PUNCT
ejpam-847	132	23	2	2	NUM
ejpam-847	132	24	)	)	PUNCT
ejpam-847	132	25	.	.	PUNCT
ejpam-847	133	1	if	if	SCONJ
ejpam-847	133	2	t∞k=1ρ(2k+l−2	t∞k=1ρ(2k+l−2	ADV
ejpam-847	133	3	x	x	SYM
ejpam-847	133	4	,	,	PUNCT
ejpam-847	133	5	2k+l−2	2k+l−2	NUM
ejpam-847	133	6	x	x	NUM
ejpam-847	133	7	,	,	PUNCT
ejpam-847	133	8	−2k+l−1	−2k+l−1	X
ejpam-847	133	9	x	x	SYM
ejpam-847	133	10	,	,	PUNCT
ejpam-847	133	11	0,	0,	NUM
ejpam-847	133	12	...	...	PUNCT
ejpam-847	133	13	,0	,0	PUNCT
ejpam-847	133	14	)	)	PUNCT
ejpam-847	133	15	�	�	PROPN
ejpam-847	133	16	2l+1nt	2l+1nt	NUM
ejpam-847	133	17	�	�	PROPN
ejpam-847	133	18	=	=	SYM
ejpam-847	133	19	1	1	NUM
ejpam-847	133	20	(	(	PUNCT
ejpam-847	133	21	7	7	NUM
ejpam-847	133	22	)	)	PUNCT
ejpam-847	133	23	and	and	CCONJ
ejpam-847	133	24	lim	lim	PROPN
ejpam-847	133	25	m→∞	m→∞	PROPN
ejpam-847	134	1	ρ(2	ρ(2	PROPN
ejpam-847	134	2	m	m	PROPN
ejpam-847	134	3	x	x	NOUN
ejpam-847	134	4	,	,	PUNCT
ejpam-847	134	5	2	2	NUM
ejpam-847	134	6	m	m	NOUN
ejpam-847	134	7	y,−2m(x+y),0,	y,−2m(x+y),0,	NOUN
ejpam-847	134	8	...	...	PUNCT
ejpam-847	134	9	,0	,0	PUNCT
ejpam-847	134	10	)	)	PUNCT
ejpam-847	134	11	�	�	PROPN
ejpam-847	134	12	2m+1nt	2m+1nt	NUM
ejpam-847	134	13	�	�	PROPN
ejpam-847	134	14	=	=	SYM
ejpam-847	134	15	1	1	NUM
ejpam-847	134	16	(	(	PUNCT
ejpam-847	134	17	8)	8)	NUM
ejpam-847	134	18	for	for	ADP
ejpam-847	134	19	all	all	DET
ejpam-847	134	20	x	x	SYM
ejpam-847	134	21	,	,	PUNCT
ejpam-847	134	22	y	y	PROPN
ejpam-847	134	23	∈	∈	PROPN
ejpam-847	134	24	x	x	INTJ
ejpam-847	134	25	,	,	PUNCT
ejpam-847	134	26	all	all	PRON
ejpam-847	134	27	t	t	X
ejpam-847	134	28	>	>	X
ejpam-847	134	29	0	0	PUNCT
ejpam-847	134	30	and	and	CCONJ
ejpam-847	134	31	all	all	DET
ejpam-847	134	32	l	l	NOUN
ejpam-847	134	33	=	=	PUNCT
ejpam-847	134	34	0,1,2	0,1,2	NUM
ejpam-847	134	35	,	,	PUNCT
ejpam-847	134	36	.	.	PUNCT
ejpam-847	134	37	.	.	PUNCT
ejpam-847	135	1	.	.	PUNCT
ejpam-847	136	1	,	,	PUNCT
ejpam-847	136	2	then	then	ADV
ejpam-847	136	3	there	there	PRON
ejpam-847	136	4	exists	exist	VERB
ejpam-847	136	5	a	a	DET
ejpam-847	136	6	unique	unique	ADJ
ejpam-847	136	7	additive	additive	NOUN
ejpam-847	136	8	mapping	mapping	NOUN
ejpam-847	136	9	a	a	DET
ejpam-847	136	10	:	:	PUNCT
ejpam-847	136	11	x	x	X
ejpam-847	136	12	→	→	SYM
ejpam-847	136	13	y	y	PROPN
ejpam-847	136	14	such	such	ADJ
ejpam-847	136	15	that	that	SCONJ
ejpam-847	136	16	µ	µ	PROPN
ejpam-847	136	17	f	f	X
ejpam-847	136	18	(	(	PUNCT
ejpam-847	136	19	x)−a(x)(t	x)−a(x)(t	PROPN
ejpam-847	136	20	)	)	PUNCT
ejpam-847	136	21	≥	≥	NOUN
ejpam-847	136	22	t∞k=1ρ(2k−2	t∞k=1ρ(2k−2	NOUN
ejpam-847	136	23	x	x	X
ejpam-847	136	24	,	,	PUNCT
ejpam-847	136	25	2k−2	2k−2	PROPN
ejpam-847	136	26	x	x	X
ejpam-847	136	27	,	,	PUNCT
ejpam-847	136	28	−2k−1	−2k−1	PROPN
ejpam-847	136	29	x	x	PROPN
ejpam-847	136	30	,	,	PUNCT
ejpam-847	136	31	0,	0,	NUM
ejpam-847	136	32	...	...	PUNCT
ejpam-847	136	33	,0	,0	PUNCT
ejpam-847	136	34	)	)	PUNCT
ejpam-847	136	35	(	(	PUNCT
ejpam-847	136	36	2nt	2nt	NOUN
ejpam-847	136	37	)	)	PUNCT
ejpam-847	136	38	(	(	PUNCT
ejpam-847	136	39	9	9	NUM
ejpam-847	136	40	)	)	PUNCT
ejpam-847	136	41	for	for	ADP
ejpam-847	136	42	all	all	DET
ejpam-847	136	43	x	x	SYM
ejpam-847	136	44	∈	∈	ADJ
ejpam-847	136	45	x	x	X
ejpam-847	136	46	and	and	CCONJ
ejpam-847	136	47	all	all	DET
ejpam-847	136	48	t	t	NOUN
ejpam-847	136	49	>	>	X
ejpam-847	136	50	0	0	X
ejpam-847	136	51	.	.	PUNCT
ejpam-847	137	1	proof	proof	NOUN
ejpam-847	137	2	.	.	PUNCT
ejpam-847	138	1	putting	put	VERB
ejpam-847	138	2	x1	x1	PROPN
ejpam-847	139	1	=	=	PUNCT
ejpam-847	139	2	x2	x2	NOUN
ejpam-847	139	3	=	=	PUNCT
ejpam-847	140	1	x	x	X
ejpam-847	140	2	,	,	PUNCT
ejpam-847	140	3	x3	x3	ADJ
ejpam-847	140	4	=	=	PUNCT
ejpam-847	140	5	−2x	−2x	NOUN
ejpam-847	140	6	,	,	PUNCT
ejpam-847	140	7	x4	x4	PROPN
ejpam-847	140	8	=	=	X
ejpam-847	140	9	.	.	PUNCT
ejpam-847	140	10	.	.	PUNCT
ejpam-847	140	11	.	.	PUNCT
ejpam-847	141	1	=	=	PUNCT
ejpam-847	141	2	xn	xn	PUNCT
ejpam-847	142	1	=	=	SYM
ejpam-847	142	2	0	0	NUM
ejpam-847	142	3	in	in	ADP
ejpam-847	142	4	(	(	PUNCT
ejpam-847	142	5	2	2	NUM
ejpam-847	142	6	)	)	PUNCT
ejpam-847	142	7	,	,	PUNCT
ejpam-847	142	8	we	we	PRON
ejpam-847	142	9	get	get	VERB
ejpam-847	142	10	µ2n	µ2n	NOUN
ejpam-847	142	11	(	(	PUNCT
ejpam-847	142	12	f	f	PROPN
ejpam-847	142	13	(	(	PUNCT
ejpam-847	142	14	2x)−2	2x)−2	NUM
ejpam-847	142	15	f	f	X
ejpam-847	142	16	(	(	PUNCT
ejpam-847	142	17	x	x	NOUN
ejpam-847	142	18	)	)	PUNCT
ejpam-847	142	19	)	)	PUNCT
ejpam-847	142	20	(	(	PUNCT
ejpam-847	142	21	t	t	PROPN
ejpam-847	142	22	)	)	PUNCT
ejpam-847	142	23	≥	≥	NOUN
ejpam-847	142	24	ρ(x	ρ(x	NOUN
ejpam-847	142	25	,	,	PUNCT
ejpam-847	142	26	x	x	X
ejpam-847	142	27	,	,	PUNCT
ejpam-847	142	28	−2x	−2x	PROPN
ejpam-847	142	29	,	,	PUNCT
ejpam-847	142	30	0,	0,	NUM
ejpam-847	142	31	...	...	PUNCT
ejpam-847	142	32	,0)(t	,0)(t	PUNCT
ejpam-847	142	33	)	)	PUNCT
ejpam-847	142	34	which	which	PRON
ejpam-847	142	35	is	be	AUX
ejpam-847	142	36	equivalent	equivalent	ADJ
ejpam-847	142	37	to	to	ADP
ejpam-847	142	38	µ	µ	PROPN
ejpam-847	142	39	f	f	X
ejpam-847	142	40	(	(	PUNCT
ejpam-847	142	41	x)−	x)−	PROPN
ejpam-847	142	42	1	1	NUM
ejpam-847	142	43	2	2	NUM
ejpam-847	142	44	f	f	X
ejpam-847	142	45	(	(	PUNCT
ejpam-847	142	46	2x)(t	2x)(t	NUM
ejpam-847	142	47	)	)	PUNCT
ejpam-847	142	48	≥	≥	NOUN
ejpam-847	142	49	ρ	ρ	NUM
ejpam-847	142	50	�	�	PROPN
ejpam-847	142	51	x	x	SYM
ejpam-847	142	52	2	2	NUM
ejpam-847	142	53	,	,	PUNCT
ejpam-847	142	54	x	x	PROPN
ejpam-847	142	55	2	2	NUM
ejpam-847	142	56	,	,	PUNCT
ejpam-847	142	57	−x	−x	NOUN
ejpam-847	142	58	,	,	PUNCT
ejpam-847	142	59	0,	0,	NUM
ejpam-847	142	60	...	...	PUNCT
ejpam-847	142	61	,0	,0	PUNCT
ejpam-847	142	62	�	�	PROPN
ejpam-847	142	63	(	(	PUNCT
ejpam-847	142	64	4nt	4nt	NOUN
ejpam-847	142	65	)	)	PUNCT
ejpam-847	142	66	for	for	ADP
ejpam-847	142	67	all	all	DET
ejpam-847	142	68	x	x	SYM
ejpam-847	142	69	∈	∈	ADJ
ejpam-847	142	70	x	x	X
ejpam-847	142	71	and	and	CCONJ
ejpam-847	142	72	all	all	DET
ejpam-847	142	73	t	t	NOUN
ejpam-847	142	74	>	>	X
ejpam-847	142	75	0	0	X
ejpam-847	142	76	.	.	PUNCT
ejpam-847	143	1	replacing	replace	VERB
ejpam-847	143	2	x	x	PUNCT
ejpam-847	143	3	and	and	CCONJ
ejpam-847	143	4	t	t	X
ejpam-847	143	5	by	by	ADP
ejpam-847	143	6	2k−1x	2k−1x	NOUN
ejpam-847	143	7	and	and	CCONJ
ejpam-847	143	8	2	2	NUM
ejpam-847	143	9	t	t	NOUN
ejpam-847	143	10	,	,	PUNCT
ejpam-847	143	11	respectively	respectively	ADV
ejpam-847	143	12	,	,	PUNCT
ejpam-847	143	13	in	in	ADP
ejpam-847	143	14	the	the	DET
ejpam-847	143	15	above	above	ADJ
ejpam-847	143	16	inequality	inequality	NOUN
ejpam-847	143	17	,	,	PUNCT
ejpam-847	143	18	we	we	PRON
ejpam-847	143	19	get	get	VERB
ejpam-847	143	20	µ	µ	PRON
ejpam-847	143	21	1	1	NUM
ejpam-847	143	22	2k−1	2k−1	NUM
ejpam-847	143	23	f	f	NOUN
ejpam-847	143	24	(	(	PUNCT
ejpam-847	143	25	2k−1	2k−1	NUM
ejpam-847	143	26	x)−	x)−	PROPN
ejpam-847	143	27	1	1	NUM
ejpam-847	143	28	2k	2k	NOUN
ejpam-847	143	29	f	f	X
ejpam-847	143	30	(	(	PUNCT
ejpam-847	143	31	2k	2k	NUM
ejpam-847	143	32	x	x	SYM
ejpam-847	143	33	)	)	PUNCT
ejpam-847	143	34	�	�	PROPN
ejpam-847	143	35	t	t	PROPN
ejpam-847	143	36	2k	2k	PROPN
ejpam-847	143	37	�	�	PROPN
ejpam-847	143	38	≥	≥	PROPN
ejpam-847	143	39	ρ(2k−2	ρ(2k−2	PROPN
ejpam-847	143	40	x	x	SYM
ejpam-847	143	41	,	,	PUNCT
ejpam-847	143	42	2k−2	2k−2	PROPN
ejpam-847	143	43	x	x	X
ejpam-847	143	44	,	,	PUNCT
ejpam-847	143	45	−2k−1	−2k−1	PROPN
ejpam-847	143	46	x	x	PROPN
ejpam-847	143	47	,	,	PUNCT
ejpam-847	143	48	0,	0,	NUM
ejpam-847	143	49	...	...	PUNCT
ejpam-847	143	50	,0)(2nt	,0)(2nt	PUNCT
ejpam-847	143	51	)	)	PUNCT
ejpam-847	143	52	for	for	ADP
ejpam-847	143	53	all	all	DET
ejpam-847	143	54	x	x	SYM
ejpam-847	143	55	∈	∈	ADJ
ejpam-847	143	56	x	x	X
ejpam-847	143	57	and	and	CCONJ
ejpam-847	143	58	all	all	DET
ejpam-847	143	59	t	t	PROPN
ejpam-847	143	60	>	>	X
ejpam-847	143	61	0	0	X
ejpam-847	143	62	.	.	PUNCT
ejpam-847	144	1	since	since	SCONJ
ejpam-847	144	2	µx(s)≤	µx(s)≤	ADJ
ejpam-847	144	3	µx(t	µx(t	NOUN
ejpam-847	144	4	)	)	PUNCT
ejpam-847	144	5	for	for	ADP
ejpam-847	144	6	all	all	DET
ejpam-847	144	7	s	s	NOUN
ejpam-847	144	8	and	and	CCONJ
ejpam-847	144	9	t	t	X
ejpam-847	144	10	with	with	ADP
ejpam-847	144	11	0	0	NUM
ejpam-847	144	12	<	<	X
ejpam-847	144	13	s	s	PART
ejpam-847	144	14	≤	≤	PROPN
ejpam-847	144	15	t	t	PROPN
ejpam-847	144	16	,	,	PUNCT
ejpam-847	144	17	we	we	PRON
ejpam-847	144	18	obtain	obtain	VERB
ejpam-847	144	19	µ	µ	DET
ejpam-847	144	20	f	f	X
ejpam-847	144	21	(	(	PUNCT
ejpam-847	144	22	x)−	x)−	PROPN
ejpam-847	144	23	1	1	NUM
ejpam-847	144	24	2	2	NUM
ejpam-847	144	25	m	m	NOUN
ejpam-847	144	26	f	f	NOUN
ejpam-847	144	27	(	(	PUNCT
ejpam-847	144	28	2	2	NUM
ejpam-847	144	29	m	m	NOUN
ejpam-847	144	30	x)(t	x)(t	NUM
ejpam-847	144	31	)	)	PUNCT
ejpam-847	145	1	=	=	X
ejpam-847	145	2	µ∑m	µ∑m	PRON
ejpam-847	145	3	k=1	k=1	PROPN
ejpam-847	145	4	�	�	PROPN
ejpam-847	145	5	1	1	NUM
ejpam-847	145	6	2k−1	2k−1	NUM
ejpam-847	145	7	f	f	NOUN
ejpam-847	145	8	(	(	PUNCT
ejpam-847	145	9	2k−1	2k−1	NUM
ejpam-847	145	10	x)−	x)−	PROPN
ejpam-847	145	11	1	1	NUM
ejpam-847	145	12	2k	2k	NOUN
ejpam-847	145	13	f	f	X
ejpam-847	145	14	(	(	PUNCT
ejpam-847	145	15	2k	2k	NUM
ejpam-847	145	16	x	x	SYM
ejpam-847	145	17	)	)	PUNCT
ejpam-847	145	18	�	�	PROPN
ejpam-847	145	19	(	(	PUNCT
ejpam-847	145	20	t	t	PROPN
ejpam-847	145	21	)	)	PUNCT
ejpam-847	145	22	≥µ∑m	≥µ∑m	PROPN
ejpam-847	145	23	k=1	k=1	X
ejpam-847	145	24	�	�	PROPN
ejpam-847	145	25	1	1	NUM
ejpam-847	145	26	2k−1	2k−1	NUM
ejpam-847	145	27	f	f	NOUN
ejpam-847	145	28	(	(	PUNCT
ejpam-847	145	29	2k−1	2k−1	NUM
ejpam-847	145	30	x)−	x)−	PROPN
ejpam-847	145	31	1	1	NUM
ejpam-847	145	32	2k	2k	NOUN
ejpam-847	145	33	f	f	X
ejpam-847	145	34	(	(	PUNCT
ejpam-847	145	35	2k	2k	NUM
ejpam-847	145	36	x	x	SYM
ejpam-847	145	37	)	)	PUNCT
ejpam-847	145	38	�	�	PROPN
ejpam-847	145	39	m	m	VERB
ejpam-847	145	40	∑	∑	PROPN
ejpam-847	145	41	k=1	k=1	PROPN
ejpam-847	145	42	t	t	PROPN
ejpam-847	145	43	2k	2k	PROPN
ejpam-847	145	44	!	!	PUNCT
ejpam-847	146	1	≥t	≥t	VERB
ejpam-847	146	2	m	m	VERB
ejpam-847	146	3	k=1	k=1	X
ejpam-847	146	4	ρ(2k−2	ρ(2k−2	PROPN
ejpam-847	146	5	x	x	SYM
ejpam-847	146	6	,	,	PUNCT
ejpam-847	146	7	2k−2	2k−2	PROPN
ejpam-847	146	8	x	x	X
ejpam-847	146	9	,	,	PUNCT
ejpam-847	146	10	−2k−1	−2k−1	PROPN
ejpam-847	146	11	x	x	PROPN
ejpam-847	146	12	,	,	PUNCT
ejpam-847	146	13	0,	0,	NUM
ejpam-847	146	14	...	...	PUNCT
ejpam-847	146	15	,0	,0	PUNCT
ejpam-847	146	16	)	)	PUNCT
ejpam-847	147	1	(	(	PUNCT
ejpam-847	147	2	2nt	2nt	NOUN
ejpam-847	147	3	)	)	PUNCT
ejpam-847	147	4	replacing	replace	VERB
ejpam-847	147	5	x	x	PUNCT
ejpam-847	147	6	by	by	ADP
ejpam-847	147	7	2l	2l	NOUN
ejpam-847	147	8	x	x	PUNCT
ejpam-847	147	9	in	in	ADP
ejpam-847	147	10	the	the	DET
ejpam-847	147	11	above	above	ADJ
ejpam-847	147	12	inequality	inequality	NOUN
ejpam-847	147	13	,	,	PUNCT
ejpam-847	147	14	we	we	PRON
ejpam-847	147	15	get	get	VERB
ejpam-847	147	16	µ	µ	PRON
ejpam-847	147	17	f	f	X
ejpam-847	147	18	(	(	PUNCT
ejpam-847	147	19	2l	2l	X
ejpam-847	147	20	x)−	x)−	PROPN
ejpam-847	147	21	1	1	NUM
ejpam-847	147	22	2	2	NUM
ejpam-847	147	23	m	m	NOUN
ejpam-847	147	24	f	f	NOUN
ejpam-847	147	25	(	(	PUNCT
ejpam-847	147	26	2m+l	2m+l	NUM
ejpam-847	147	27	x)(t	x)(t	NUM
ejpam-847	147	28	)	)	PUNCT
ejpam-847	147	29	≥	≥	PROPN
ejpam-847	147	30	t	t	NOUN
ejpam-847	147	31	m	m	VERB
ejpam-847	147	32	k=1ρ(2k+l−2	k=1ρ(2k+l−2	NOUN
ejpam-847	147	33	x	x	SYM
ejpam-847	147	34	,	,	PUNCT
ejpam-847	147	35	2k+l−2	2k+l−2	NUM
ejpam-847	147	36	x	x	NUM
ejpam-847	147	37	,	,	PUNCT
ejpam-847	147	38	−2k+l−1	−2k+l−1	X
ejpam-847	147	39	x	x	SYM
ejpam-847	147	40	,	,	PUNCT
ejpam-847	147	41	0,	0,	NUM
ejpam-847	147	42	...	...	PUNCT
ejpam-847	147	43	,0	,0	PUNCT
ejpam-847	147	44	)	)	PUNCT
ejpam-847	147	45	(	(	PUNCT
ejpam-847	147	46	2nt	2nt	NOUN
ejpam-847	147	47	)	)	PUNCT
ejpam-847	147	48	d.	d.	PROPN
ejpam-847	147	49	shin	shin	PROPN
ejpam-847	147	50	,	,	PUNCT
ejpam-847	147	51	j.	j.	PROPN
ejpam-847	147	52	lee	lee	PROPN
ejpam-847	147	53	,	,	PUNCT
ejpam-847	147	54	c.	c.	PROPN
ejpam-847	147	55	park	park	PROPN
ejpam-847	147	56	/	/	SYM
ejpam-847	147	57	eur	eur	PROPN
ejpam-847	147	58	.	.	PUNCT
ejpam-847	148	1	j.	j.	PROPN
ejpam-847	148	2	pure	pure	PROPN
ejpam-847	148	3	appl	appl	PROPN
ejpam-847	148	4	.	.	PROPN
ejpam-847	148	5	math	math	PROPN
ejpam-847	148	6	,	,	PUNCT
ejpam-847	148	7	5	5	NUM
ejpam-847	148	8	(	(	PUNCT
ejpam-847	148	9	2012	2012	NUM
ejpam-847	148	10	)	)	PUNCT
ejpam-847	148	11	,	,	PUNCT
ejpam-847	148	12	540	540	NUM
ejpam-847	148	13	-	-	SYM
ejpam-847	148	14	553	553	NUM
ejpam-847	148	15	546	546	NUM
ejpam-847	148	16	which	which	PRON
ejpam-847	148	17	is	be	AUX
ejpam-847	148	18	equivalent	equivalent	ADJ
ejpam-847	148	19	to	to	ADP
ejpam-847	148	20	µ	µ	PRON
ejpam-847	148	21	1	1	NUM
ejpam-847	148	22	2l	2l	NUM
ejpam-847	148	23	f	f	NOUN
ejpam-847	148	24	(	(	PUNCT
ejpam-847	148	25	2l	2l	X
ejpam-847	148	26	x)−	x)−	PROPN
ejpam-847	148	27	1	1	NUM
ejpam-847	148	28	2m+l	2m+l	NUM
ejpam-847	148	29	f	f	NOUN
ejpam-847	148	30	(	(	PUNCT
ejpam-847	148	31	2m+l	2m+l	NUM
ejpam-847	148	32	x)(t)≥	x)(t)≥	PROPN
ejpam-847	149	1	t	t	PROPN
ejpam-847	149	2	m	m	VERB
ejpam-847	149	3	k=1ρ(2k+l−2	k=1ρ(2k+l−2	NOUN
ejpam-847	149	4	x	x	SYM
ejpam-847	149	5	,	,	PUNCT
ejpam-847	149	6	2k+l−2	2k+l−2	NUM
ejpam-847	149	7	x	x	NUM
ejpam-847	149	8	,	,	PUNCT
ejpam-847	149	9	−2k+l−1	−2k+l−1	X
ejpam-847	149	10	x	x	SYM
ejpam-847	149	11	,	,	PUNCT
ejpam-847	149	12	0,	0,	NUM
ejpam-847	149	13	...	...	PUNCT
ejpam-847	149	14	,0	,0	PUNCT
ejpam-847	149	15	)	)	PUNCT
ejpam-847	149	16	�	�	PROPN
ejpam-847	149	17	2l+1nt	2l+1nt	NUM
ejpam-847	149	18	�	�	PROPN
ejpam-847	149	19	(	(	PUNCT
ejpam-847	149	20	10	10	NUM
ejpam-847	149	21	)	)	PUNCT
ejpam-847	149	22	for	for	ADP
ejpam-847	149	23	all	all	DET
ejpam-847	149	24	x	x	SYM
ejpam-847	149	25	∈	∈	PROPN
ejpam-847	149	26	x	x	X
ejpam-847	149	27	,	,	PUNCT
ejpam-847	149	28	all	all	PRON
ejpam-847	149	29	t	t	X
ejpam-847	149	30	>	>	X
ejpam-847	149	31	0	0	PUNCT
ejpam-847	150	1	and	and	CCONJ
ejpam-847	150	2	all	all	DET
ejpam-847	150	3	l	l	NOUN
ejpam-847	150	4	=	=	PUNCT
ejpam-847	150	5	0,1,2	0,1,2	NUM
ejpam-847	150	6	,	,	PUNCT
ejpam-847	150	7	.	.	PUNCT
ejpam-847	150	8	.	.	PUNCT
ejpam-847	151	1	..	..	PUNCT
ejpam-847	152	1	since	since	SCONJ
ejpam-847	152	2	the	the	DET
ejpam-847	152	3	right	right	ADJ
ejpam-847	152	4	hand	hand	NOUN
ejpam-847	152	5	side	side	NOUN
ejpam-847	152	6	of	of	ADP
ejpam-847	152	7	the	the	DET
ejpam-847	152	8	inequality	inequality	NOUN
ejpam-847	152	9	(	(	PUNCT
ejpam-847	152	10	10	10	NUM
ejpam-847	152	11	)	)	PUNCT
ejpam-847	152	12	tends	tend	VERB
ejpam-847	152	13	to	to	ADP
ejpam-847	152	14	1	1	NUM
ejpam-847	152	15	as	as	ADP
ejpam-847	152	16	m→∞	m→∞	NUM
ejpam-847	152	17	by	by	ADP
ejpam-847	152	18	(	(	PUNCT
ejpam-847	152	19	7	7	NUM
ejpam-847	152	20	)	)	PUNCT
ejpam-847	152	21	,	,	PUNCT
ejpam-847	152	22	the	the	DET
ejpam-847	152	23	sequence	sequence	NOUN
ejpam-847	152	24	{	{	PUNCT
ejpam-847	152	25	1	1	NUM
ejpam-847	152	26	2	2	NUM
ejpam-847	152	27	m	m	NOUN
ejpam-847	152	28	f	f	NOUN
ejpam-847	152	29	(	(	PUNCT
ejpam-847	152	30	2mx	2mx	ADJ
ejpam-847	152	31	)	)	PUNCT
ejpam-847	152	32	}	}	PUNCT
ejpam-847	152	33	is	be	AUX
ejpam-847	152	34	a	a	DET
ejpam-847	152	35	cauchy	cauchy	ADJ
ejpam-847	152	36	sequence	sequence	NOUN
ejpam-847	152	37	.	.	PUNCT
ejpam-847	153	1	thus	thus	ADV
ejpam-847	153	2	we	we	PRON
ejpam-847	153	3	define	define	VERB
ejpam-847	153	4	a(x	a(x	NOUN
ejpam-847	153	5	)	)	PUNCT
ejpam-847	153	6	:	:	PUNCT
ejpam-847	153	7	=	=	SYM
ejpam-847	153	8	limm→∞	limm→∞	NOUN
ejpam-847	153	9	1	1	NUM
ejpam-847	153	10	2	2	NUM
ejpam-847	153	11	m	m	NOUN
ejpam-847	153	12	f	f	NOUN
ejpam-847	153	13	(	(	PUNCT
ejpam-847	153	14	2mx	2mx	ADJ
ejpam-847	153	15	)	)	PUNCT
ejpam-847	153	16	for	for	ADP
ejpam-847	153	17	all	all	DET
ejpam-847	153	18	x	x	SYM
ejpam-847	153	19	∈	∈	PROPN
ejpam-847	153	20	x	x	X
ejpam-847	153	21	,	,	PUNCT
ejpam-847	153	22	which	which	PRON
ejpam-847	153	23	is	be	AUX
ejpam-847	153	24	an	an	DET
ejpam-847	153	25	odd	odd	ADJ
ejpam-847	153	26	mapping	mapping	NOUN
ejpam-847	153	27	.	.	PUNCT
ejpam-847	154	1	now	now	ADV
ejpam-847	154	2	we	we	PRON
ejpam-847	154	3	show	show	VERB
ejpam-847	154	4	that	that	SCONJ
ejpam-847	154	5	a	a	PRON
ejpam-847	154	6	is	be	AUX
ejpam-847	154	7	an	an	DET
ejpam-847	154	8	additive	additive	ADJ
ejpam-847	154	9	mapping	mapping	NOUN
ejpam-847	154	10	.	.	PUNCT
ejpam-847	155	1	by	by	ADP
ejpam-847	155	2	(	(	PUNCT
ejpam-847	155	3	2	2	NUM
ejpam-847	155	4	)	)	PUNCT
ejpam-847	155	5	,	,	PUNCT
ejpam-847	155	6	we	we	PRON
ejpam-847	155	7	get	get	VERB
ejpam-847	155	8	µ	µ	PRON
ejpam-847	155	9	1	1	NUM
ejpam-847	155	10	2	2	NUM
ejpam-847	155	11	m	m	NOUN
ejpam-847	155	12	(	(	PUNCT
ejpam-847	155	13	f	f	X
ejpam-847	155	14	(	(	PUNCT
ejpam-847	155	15	2	2	NUM
ejpam-847	155	16	m(x+y))−	m(x+y))−	NOUN
ejpam-847	155	17	f	f	NOUN
ejpam-847	155	18	(	(	PUNCT
ejpam-847	155	19	2	2	NUM
ejpam-847	155	20	m	m	NOUN
ejpam-847	155	21	x)−	x)−	PROPN
ejpam-847	155	22	f	f	X
ejpam-847	155	23	(	(	PUNCT
ejpam-847	155	24	2	2	NUM
ejpam-847	155	25	m	m	NOUN
ejpam-847	155	26	y))(t	y))(t	PROPN
ejpam-847	155	27	)	)	PUNCT
ejpam-847	155	28	≥	≥	NOUN
ejpam-847	156	1	ρ(2	ρ(2	NOUN
ejpam-847	156	2	m	m	NOUN
ejpam-847	156	3	x	x	NOUN
ejpam-847	156	4	,	,	PUNCT
ejpam-847	156	5	2	2	NUM
ejpam-847	156	6	m	m	NOUN
ejpam-847	156	7	y,−2m(x+y),0,	y,−2m(x+y),0,	NOUN
ejpam-847	156	8	...	...	PUNCT
ejpam-847	156	9	,0)(2	,0)(2	PUNCT
ejpam-847	156	10	m+1nt	m+1nt	NOUN
ejpam-847	156	11	)	)	PUNCT
ejpam-847	156	12	.	.	PUNCT
ejpam-847	157	1	taking	take	VERB
ejpam-847	157	2	the	the	DET
ejpam-847	157	3	limit	limit	NOUN
ejpam-847	157	4	as	as	ADP
ejpam-847	157	5	m→∞	m→∞	NOUN
ejpam-847	157	6	in	in	ADP
ejpam-847	157	7	the	the	DET
ejpam-847	157	8	above	above	ADJ
ejpam-847	157	9	inequality	inequality	NOUN
ejpam-847	157	10	,	,	PUNCT
ejpam-847	157	11	by	by	ADP
ejpam-847	157	12	(	(	PUNCT
ejpam-847	157	13	8)	8)	NUM
ejpam-847	157	14	the	the	DET
ejpam-847	157	15	mapping	mapping	NOUN
ejpam-847	157	16	a	a	PRON
ejpam-847	157	17	is	be	AUX
ejpam-847	157	18	additive	additive	ADJ
ejpam-847	157	19	.	.	PUNCT
ejpam-847	158	1	by	by	ADP
ejpam-847	158	2	letting	let	VERB
ejpam-847	158	3	l	l	NOUN
ejpam-847	158	4	=	=	SYM
ejpam-847	158	5	0	0	PUNCT
ejpam-847	158	6	an	an	DET
ejpam-847	158	7	taking	take	VERB
ejpam-847	158	8	the	the	DET
ejpam-847	158	9	limit	limit	NOUN
ejpam-847	158	10	as	as	ADP
ejpam-847	158	11	m→∞	m→∞	NOUN
ejpam-847	158	12	in	in	ADP
ejpam-847	158	13	(	(	PUNCT
ejpam-847	158	14	10	10	NUM
ejpam-847	158	15	)	)	PUNCT
ejpam-847	158	16	,	,	PUNCT
ejpam-847	158	17	we	we	PRON
ejpam-847	158	18	get	get	VERB
ejpam-847	158	19	(	(	PUNCT
ejpam-847	158	20	9	9	NUM
ejpam-847	158	21	)	)	PUNCT
ejpam-847	158	22	.	.	PUNCT
ejpam-847	159	1	the	the	DET
ejpam-847	159	2	rest	rest	NOUN
ejpam-847	159	3	of	of	ADP
ejpam-847	159	4	the	the	DET
ejpam-847	159	5	proof	proof	NOUN
ejpam-847	159	6	is	be	AUX
ejpam-847	159	7	the	the	DET
ejpam-847	159	8	same	same	ADJ
ejpam-847	159	9	as	as	ADP
ejpam-847	159	10	in	in	ADP
ejpam-847	159	11	the	the	DET
ejpam-847	159	12	proof	proof	NOUN
ejpam-847	159	13	of	of	ADP
ejpam-847	159	14	theorem	theorem	ADJ
ejpam-847	159	15	1	1	NUM
ejpam-847	159	16	.	.	PUNCT
ejpam-847	159	17	corollary	corollary	ADJ
ejpam-847	159	18	2	2	NUM
ejpam-847	159	19	.	.	PUNCT
ejpam-847	160	1	let	let	VERB
ejpam-847	160	2	θ	θ	PROPN
ejpam-847	160	3	≥	≥	X
ejpam-847	160	4	0	0	NUM
ejpam-847	160	5	and	and	CCONJ
ejpam-847	160	6	let	let	VERB
ejpam-847	160	7	p	p	PRON
ejpam-847	160	8	be	be	AUX
ejpam-847	160	9	a	a	DET
ejpam-847	160	10	constant	constant	ADJ
ejpam-847	160	11	with	with	ADP
ejpam-847	160	12	0	0	NUM
ejpam-847	160	13	<	<	X
ejpam-847	160	14	p	p	X
ejpam-847	160	15	<	<	X
ejpam-847	160	16	1	1	NUM
ejpam-847	160	17	.	.	PUNCT
ejpam-847	161	1	for	for	ADP
ejpam-847	161	2	a	a	DET
ejpam-847	161	3	normed	normed	ADJ
ejpam-847	161	4	vector	vector	NOUN
ejpam-847	161	5	space	space	NOUN
ejpam-847	161	6	x	x	PUNCT
ejpam-847	161	7	and	and	CCONJ
ejpam-847	161	8	complete	complete	ADJ
ejpam-847	161	9	rn	rn	NOUN
ejpam-847	161	10	-	-	NOUN
ejpam-847	161	11	space	space	NOUN
ejpam-847	161	12	y	y	PROPN
ejpam-847	161	13	,	,	PUNCT
ejpam-847	161	14	let	let	VERB
ejpam-847	161	15	f	f	PRON
ejpam-847	161	16	:	:	PUNCT
ejpam-847	161	17	x	x	X
ejpam-847	161	18	→	→	SYM
ejpam-847	161	19	y	y	X
ejpam-847	161	20	be	be	AUX
ejpam-847	161	21	an	an	DET
ejpam-847	161	22	odd	odd	ADJ
ejpam-847	161	23	mapping	mapping	NOUN
ejpam-847	161	24	satisfying	satisfy	VERB
ejpam-847	161	25	µd	µd	DET
ejpam-847	161	26	f	f	PROPN
ejpam-847	161	27	(	(	PUNCT
ejpam-847	161	28	x1	x1	INTJ
ejpam-847	161	29	,	,	PUNCT
ejpam-847	161	30	x2,	x2,	PROPN
ejpam-847	161	31	...	...	PUNCT
ejpam-847	161	32	,xn	,xn	PUNCT
ejpam-847	161	33	)	)	PUNCT
ejpam-847	161	34	(	(	PUNCT
ejpam-847	161	35	t	t	PROPN
ejpam-847	161	36	)	)	PUNCT
ejpam-847	161	37	≥	≥	PROPN
ejpam-847	161	38	t	t	PROPN
ejpam-847	161	39	t	t	NOUN
ejpam-847	162	1	+	+	CCONJ
ejpam-847	162	2	θ	θ	PROPN
ejpam-847	162	3	∑n	∑n	PROPN
ejpam-847	162	4	i=1	i=1	PROPN
ejpam-847	162	5	||x	||x	PROPN
ejpam-847	162	6	i||p	i||p	PROPN
ejpam-847	162	7	for	for	ADP
ejpam-847	162	8	all	all	PRON
ejpam-847	162	9	(	(	PUNCT
ejpam-847	162	10	x1	x1	PROPN
ejpam-847	162	11	,	,	PUNCT
ejpam-847	162	12	x2	x2	PROPN
ejpam-847	162	13	,	,	PUNCT
ejpam-847	162	14	.	.	PUNCT
ejpam-847	162	15	.	.	PUNCT
ejpam-847	162	16	.	.	PUNCT
ejpam-847	163	1	,	,	PUNCT
ejpam-847	163	2	xn	xn	X
ejpam-847	163	3	)	)	PUNCT
ejpam-847	163	4	∈	∈	PROPN
ejpam-847	163	5	x	x	PUNCT
ejpam-847	163	6	with	with	ADP
ejpam-847	163	7	∑n	∑n	PROPN
ejpam-847	163	8	i=1	i=1	NOUN
ejpam-847	163	9	x	x	PUNCT
ejpam-847	164	1	i	i	NOUN
ejpam-847	164	2	=	=	NOUN
ejpam-847	164	3	0	0	NUM
ejpam-847	164	4	and	and	CCONJ
ejpam-847	164	5	all	all	DET
ejpam-847	164	6	t	t	NOUN
ejpam-847	164	7	>	>	X
ejpam-847	164	8	0	0	X
ejpam-847	164	9	.	.	PUNCT
ejpam-847	165	1	if	if	SCONJ
ejpam-847	165	2	t∞k=1	t∞k=1	PROPN
ejpam-847	165	3	�	�	PROPN
ejpam-847	165	4	2l+1nt	2l+1nt	NUM
ejpam-847	165	5	2l+1nt	2l+1nt	NUM
ejpam-847	166	1	+	+	CCONJ
ejpam-847	166	2	2(k+l−1)p(21−p	2(k+l−1)p(21−p	NUM
ejpam-847	167	1	+	+	CCONJ
ejpam-847	167	2	1)θ	1)θ	NUM
ejpam-847	167	3	||x	||x	PROPN
ejpam-847	167	4	||p	||p	NOUN
ejpam-847	167	5	�	�	NOUN
ejpam-847	167	6	=	=	NOUN
ejpam-847	167	7	1	1	NUM
ejpam-847	167	8	for	for	ADP
ejpam-847	167	9	all	all	DET
ejpam-847	167	10	x	x	SYM
ejpam-847	167	11	∈	∈	PROPN
ejpam-847	167	12	x	x	X
ejpam-847	167	13	,	,	PUNCT
ejpam-847	167	14	all	all	PRON
ejpam-847	167	15	t	t	X
ejpam-847	167	16	>	>	X
ejpam-847	167	17	0	0	PUNCT
ejpam-847	167	18	and	and	CCONJ
ejpam-847	167	19	all	all	DET
ejpam-847	167	20	l	l	NOUN
ejpam-847	167	21	=	=	PUNCT
ejpam-847	167	22	0,1,2	0,1,2	NUM
ejpam-847	167	23	,	,	PUNCT
ejpam-847	167	24	.	.	PUNCT
ejpam-847	167	25	.	.	PUNCT
ejpam-847	168	1	.	.	PUNCT
ejpam-847	169	1	,	,	PUNCT
ejpam-847	169	2	then	then	ADV
ejpam-847	169	3	there	there	PRON
ejpam-847	169	4	exists	exist	VERB
ejpam-847	169	5	a	a	DET
ejpam-847	169	6	unique	unique	ADJ
ejpam-847	169	7	additive	additive	NOUN
ejpam-847	169	8	mapping	mapping	NOUN
ejpam-847	169	9	a	a	DET
ejpam-847	169	10	:	:	PUNCT
ejpam-847	169	11	x	x	X
ejpam-847	169	12	→	→	SYM
ejpam-847	169	13	y	y	PROPN
ejpam-847	169	14	such	such	ADJ
ejpam-847	169	15	that	that	SCONJ
ejpam-847	169	16	µ	µ	PROPN
ejpam-847	169	17	f	f	X
ejpam-847	169	18	(	(	PUNCT
ejpam-847	169	19	x)−a(x)(t	x)−a(x)(t	PROPN
ejpam-847	169	20	)	)	PUNCT
ejpam-847	169	21	≥	≥	NOUN
ejpam-847	169	22	t∞k=1	t∞k=1	NOUN
ejpam-847	169	23	�	�	PROPN
ejpam-847	170	1	2nt	2nt	NOUN
ejpam-847	170	2	2nt	2nt	PROPN
ejpam-847	171	1	+	+	CCONJ
ejpam-847	172	1	2(k−1)p(21−p	2(k−1)p(21−p	NUM
ejpam-847	172	2	+	+	CCONJ
ejpam-847	172	3	1)θ	1)θ	NUM
ejpam-847	172	4	||x	||x	PROPN
ejpam-847	172	5	||p	||p	NOUN
ejpam-847	172	6	�	�	PROPN
ejpam-847	172	7	for	for	ADP
ejpam-847	172	8	all	all	DET
ejpam-847	172	9	x	x	SYM
ejpam-847	172	10	∈	∈	ADJ
ejpam-847	172	11	x	x	X
ejpam-847	172	12	and	and	CCONJ
ejpam-847	172	13	all	all	DET
ejpam-847	172	14	t	t	NOUN
ejpam-847	172	15	>	>	X
ejpam-847	172	16	0	0	X
ejpam-847	172	17	.	.	PUNCT
ejpam-847	173	1	proof	proof	NOUN
ejpam-847	173	2	.	.	PUNCT
ejpam-847	174	1	if	if	SCONJ
ejpam-847	174	2	we	we	PRON
ejpam-847	174	3	define	define	VERB
ejpam-847	174	4	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	174	5	...	...	PUNCT
ejpam-847	174	6	,xn	,xn	PUNCT
ejpam-847	174	7	)	)	PUNCT
ejpam-847	174	8	(	(	PUNCT
ejpam-847	174	9	t	t	NOUN
ejpam-847	174	10	)	)	PUNCT
ejpam-847	174	11	=	=	SYM
ejpam-847	174	12	t	t	PROPN
ejpam-847	174	13	t	t	NOUN
ejpam-847	175	1	+	+	CCONJ
ejpam-847	175	2	θ	θ	PROPN
ejpam-847	175	3	∑n	∑n	PROPN
ejpam-847	175	4	i=1	i=1	PROPN
ejpam-847	175	5	||x	||x	PROPN
ejpam-847	175	6	i||p	i||p	PROPN
ejpam-847	175	7	and	and	CCONJ
ejpam-847	175	8	apply	apply	VERB
ejpam-847	175	9	theorem	theorem	NOUN
ejpam-847	175	10	2	2	NUM
ejpam-847	175	11	,	,	PUNCT
ejpam-847	175	12	then	then	ADV
ejpam-847	175	13	we	we	PRON
ejpam-847	175	14	get	get	VERB
ejpam-847	175	15	the	the	DET
ejpam-847	175	16	desired	desire	VERB
ejpam-847	175	17	result	result	NOUN
ejpam-847	175	18	.	.	PUNCT
ejpam-847	176	1	d.	d.	PROPN
ejpam-847	176	2	shin	shin	PROPN
ejpam-847	176	3	,	,	PUNCT
ejpam-847	176	4	j.	j.	PROPN
ejpam-847	176	5	lee	lee	PROPN
ejpam-847	176	6	,	,	PUNCT
ejpam-847	176	7	c.	c.	PROPN
ejpam-847	176	8	park	park	PROPN
ejpam-847	176	9	/	/	SYM
ejpam-847	176	10	eur	eur	PROPN
ejpam-847	176	11	.	.	PUNCT
ejpam-847	177	1	j.	j.	PROPN
ejpam-847	177	2	pure	pure	PROPN
ejpam-847	177	3	appl	appl	PROPN
ejpam-847	177	4	.	.	PROPN
ejpam-847	177	5	math	math	PROPN
ejpam-847	177	6	,	,	PUNCT
ejpam-847	177	7	5	5	NUM
ejpam-847	177	8	(	(	PUNCT
ejpam-847	177	9	2012	2012	NUM
ejpam-847	177	10	)	)	PUNCT
ejpam-847	177	11	,	,	PUNCT
ejpam-847	177	12	540	540	NUM
ejpam-847	177	13	-	-	SYM
ejpam-847	177	14	553	553	NUM
ejpam-847	177	15	547	547	NUM
ejpam-847	177	16	3	3	NUM
ejpam-847	177	17	.	.	PUNCT
ejpam-847	178	1	hyers	hyer	NOUN
ejpam-847	178	2	-	-	PUNCT
ejpam-847	178	3	ulam	ulam	PROPN
ejpam-847	178	4	stability	stability	NOUN
ejpam-847	178	5	of	of	ADP
ejpam-847	178	6	the	the	DET
ejpam-847	178	7	functional	functional	ADJ
ejpam-847	178	8	equation	equation	NOUN
ejpam-847	178	9	(	(	PUNCT
ejpam-847	178	10	1	1	NUM
ejpam-847	178	11	):	):	PUNCT
ejpam-847	178	12	an	an	DET
ejpam-847	178	13	even	even	ADJ
ejpam-847	178	14	case	case	NOUN
ejpam-847	178	15	we	we	PRON
ejpam-847	178	16	prove	prove	VERB
ejpam-847	178	17	the	the	DET
ejpam-847	178	18	hyers	hyers	PROPN
ejpam-847	178	19	-	-	PUNCT
ejpam-847	178	20	ulam	ulam	ADJ
ejpam-847	178	21	stability	stability	NOUN
ejpam-847	178	22	of	of	ADP
ejpam-847	178	23	the	the	DET
ejpam-847	178	24	functional	functional	ADJ
ejpam-847	178	25	equation	equation	NOUN
ejpam-847	178	26	(	(	PUNCT
ejpam-847	178	27	1	1	NUM
ejpam-847	178	28	)	)	PUNCT
ejpam-847	178	29	of	of	ADP
ejpam-847	178	30	an	an	DET
ejpam-847	178	31	even	even	ADV
ejpam-847	178	32	mapping	mapping	NOUN
ejpam-847	178	33	in	in	ADP
ejpam-847	178	34	rn	rn	NOUN
ejpam-847	178	35	-	-	NOUN
ejpam-847	178	36	spaces	space	NOUN
ejpam-847	178	37	.	.	PUNCT
ejpam-847	179	1	for	for	ADP
ejpam-847	179	2	an	an	DET
ejpam-847	179	3	even	even	ADV
ejpam-847	179	4	mapping	mapping	NOUN
ejpam-847	179	5	f	f	NOUN
ejpam-847	179	6	:	:	PUNCT
ejpam-847	179	7	x	x	X
ejpam-847	179	8	→	→	SYM
ejpam-847	179	9	y	y	PROPN
ejpam-847	179	10	with	with	ADP
ejpam-847	179	11	f	f	PROPN
ejpam-847	179	12	(	(	PUNCT
ejpam-847	179	13	0	0	NUM
ejpam-847	179	14	)	)	PUNCT
ejpam-847	179	15	=	=	SYM
ejpam-847	179	16	0	0	NUM
ejpam-847	179	17	,	,	PUNCT
ejpam-847	179	18	we	we	PRON
ejpam-847	179	19	note	note	VERB
ejpam-847	179	20	that	that	SCONJ
ejpam-847	179	21	if	if	SCONJ
ejpam-847	179	22	f	f	PROPN
ejpam-847	179	23	satisfies	satisfy	VERB
ejpam-847	179	24	d	d	X
ejpam-847	179	25	f	f	X
ejpam-847	179	26	(	(	PUNCT
ejpam-847	179	27	x1	x1	PROPN
ejpam-847	179	28	,	,	PUNCT
ejpam-847	179	29	x2	x2	PROPN
ejpam-847	179	30	,	,	PUNCT
ejpam-847	179	31	.	.	PUNCT
ejpam-847	179	32	.	.	PUNCT
ejpam-847	179	33	.	.	PUNCT
ejpam-847	180	1	,	,	PUNCT
ejpam-847	180	2	xn	xn	X
ejpam-847	180	3	)	)	PUNCT
ejpam-847	180	4	=	=	SYM
ejpam-847	180	5	0	0	NUM
ejpam-847	180	6	for	for	ADP
ejpam-847	180	7	all	all	DET
ejpam-847	180	8	x1	x1	PROPN
ejpam-847	180	9	,	,	PUNCT
ejpam-847	180	10	.	.	PUNCT
ejpam-847	180	11	.	.	PUNCT
ejpam-847	181	1	.	.	PUNCT
ejpam-847	182	1	,	,	PUNCT
ejpam-847	182	2	xn	xn	PUNCT
ejpam-847	182	3	∈	∈	PROPN
ejpam-847	182	4	x	x	PUNCT
ejpam-847	182	5	with	with	ADP
ejpam-847	182	6	∑n	∑n	PROPN
ejpam-847	182	7	i=1	i=1	NOUN
ejpam-847	182	8	x	x	PUNCT
ejpam-847	183	1	i	i	NOUN
ejpam-847	183	2	=	=	NOUN
ejpam-847	183	3	0	0	PUNCT
ejpam-847	183	4	then	then	ADV
ejpam-847	183	5	the	the	DET
ejpam-847	183	6	mapping	mapping	NOUN
ejpam-847	183	7	f	f	X
ejpam-847	183	8	is	be	AUX
ejpam-847	183	9	quadratic	quadratic	ADJ
ejpam-847	183	10	.	.	PUNCT
ejpam-847	184	1	theorem	theorem	NOUN
ejpam-847	184	2	3	3	X
ejpam-847	184	3	.	.	PUNCT
ejpam-847	185	1	let	let	VERB
ejpam-847	185	2	f	f	NOUN
ejpam-847	185	3	:	:	PUNCT
ejpam-847	185	4	x	x	X
ejpam-847	185	5	→	→	SYM
ejpam-847	185	6	y	y	X
ejpam-847	185	7	be	be	AUX
ejpam-847	185	8	an	an	DET
ejpam-847	185	9	even	even	ADV
ejpam-847	185	10	mapping	mapping	NOUN
ejpam-847	185	11	with	with	ADP
ejpam-847	185	12	f	f	PROPN
ejpam-847	185	13	(	(	PUNCT
ejpam-847	185	14	0	0	NUM
ejpam-847	185	15	)	)	PUNCT
ejpam-847	185	16	=	=	SYM
ejpam-847	185	17	0	0	NUM
ejpam-847	185	18	for	for	ADP
ejpam-847	185	19	which	which	PRON
ejpam-847	185	20	there	there	PRON
ejpam-847	185	21	is	be	VERB
ejpam-847	185	22	a	a	DET
ejpam-847	185	23	ρ	ρ	NOUN
ejpam-847	185	24	:	:	PUNCT
ejpam-847	185	25	x	x	SYM
ejpam-847	185	26	n→	n→	X
ejpam-847	185	27	d+	d+	X
ejpam-847	185	28	satisfying	satisfy	VERB
ejpam-847	185	29	(	(	PUNCT
ejpam-847	185	30	2	2	NUM
ejpam-847	185	31	)	)	PUNCT
ejpam-847	185	32	.	.	PUNCT
ejpam-847	186	1	if	if	SCONJ
ejpam-847	186	2	t∞k=1ρ	t∞k=1ρ	PROPN
ejpam-847	186	3	�	�	PROPN
ejpam-847	186	4	x	x	SYM
ejpam-847	186	5	2k+l	2k+l	NUM
ejpam-847	186	6	,	,	PUNCT
ejpam-847	186	7	−	−	PROPN
ejpam-847	186	8	x	x	SYM
ejpam-847	186	9	2k+l	2k+l	NUM
ejpam-847	186	10	,	,	PUNCT
ejpam-847	186	11	0,	0,	NUM
ejpam-847	186	12	...	...	PUNCT
ejpam-847	186	13	,0	,0	PUNCT
ejpam-847	186	14	�	�	PROPN
ejpam-847	186	15	�	�	PROPN
ejpam-847	186	16	t	t	PROPN
ejpam-847	186	17	23k+2l−3	23k+2l−3	NUM
ejpam-847	186	18	�	�	PROPN
ejpam-847	186	19	=	=	SYM
ejpam-847	186	20	1	1	NUM
ejpam-847	186	21	(	(	PUNCT
ejpam-847	186	22	11	11	NUM
ejpam-847	186	23	)	)	PUNCT
ejpam-847	186	24	and	and	CCONJ
ejpam-847	186	25	lim	lim	PROPN
ejpam-847	186	26	m→∞	m→∞	NUM
ejpam-847	186	27	ρ	ρ	PROPN
ejpam-847	186	28	�	�	PROPN
ejpam-847	186	29	x	x	PUNCT
ejpam-847	186	30	2	2	NUM
ejpam-847	186	31	m	m	NOUN
ejpam-847	186	32	,	,	PUNCT
ejpam-847	186	33	y	y	PROPN
ejpam-847	186	34	2	2	NUM
ejpam-847	186	35	m	m	NOUN
ejpam-847	186	36	,	,	PUNCT
ejpam-847	186	37	−	−	PROPN
ejpam-847	186	38	x+y	x+y	NUM
ejpam-847	186	39	2	2	NUM
ejpam-847	186	40	m	m	NOUN
ejpam-847	186	41	,	,	PUNCT
ejpam-847	186	42	0,	0,	NUM
ejpam-847	186	43	...	...	PUNCT
ejpam-847	186	44	,0	,0	PUNCT
ejpam-847	186	45	�	�	PROPN
ejpam-847	186	46	�	�	PROPN
ejpam-847	186	47	t	t	PROPN
ejpam-847	186	48	22m−1	22m−1	PROPN
ejpam-847	186	49	�	�	PROPN
ejpam-847	186	50	=	=	SYM
ejpam-847	186	51	1	1	NUM
ejpam-847	186	52	(	(	PUNCT
ejpam-847	186	53	12	12	NUM
ejpam-847	186	54	)	)	PUNCT
ejpam-847	186	55	for	for	ADP
ejpam-847	186	56	all	all	PRON
ejpam-847	186	57	x	x	SYM
ejpam-847	186	58	,	,	PUNCT
ejpam-847	186	59	y	y	PROPN
ejpam-847	186	60	∈	∈	PROPN
ejpam-847	186	61	x	x	INTJ
ejpam-847	186	62	,	,	PUNCT
ejpam-847	186	63	all	all	PRON
ejpam-847	186	64	t	t	X
ejpam-847	186	65	>	>	X
ejpam-847	186	66	0	0	PUNCT
ejpam-847	186	67	and	and	CCONJ
ejpam-847	186	68	all	all	DET
ejpam-847	186	69	l	l	NOUN
ejpam-847	186	70	=	=	PUNCT
ejpam-847	186	71	0,1,2	0,1,2	NUM
ejpam-847	186	72	,	,	PUNCT
ejpam-847	186	73	.	.	PUNCT
ejpam-847	186	74	.	.	PUNCT
ejpam-847	187	1	.	.	PUNCT
ejpam-847	188	1	,	,	PUNCT
ejpam-847	188	2	then	then	ADV
ejpam-847	188	3	there	there	PRON
ejpam-847	188	4	exists	exist	VERB
ejpam-847	188	5	a	a	DET
ejpam-847	188	6	unique	unique	ADJ
ejpam-847	188	7	quadratic	quadratic	ADJ
ejpam-847	188	8	mapping	mapping	NOUN
ejpam-847	188	9	q	q	NOUN
ejpam-847	188	10	:	:	PUNCT
ejpam-847	188	11	x	x	X
ejpam-847	188	12	→	→	SYM
ejpam-847	188	13	y	y	PROPN
ejpam-847	188	14	such	such	ADJ
ejpam-847	188	15	that	that	SCONJ
ejpam-847	188	16	µ	µ	PROPN
ejpam-847	188	17	f	f	X
ejpam-847	188	18	(	(	PUNCT
ejpam-847	188	19	x)−q(x)(t	x)−q(x)(t	PROPN
ejpam-847	188	20	)	)	PUNCT
ejpam-847	188	21	≥	≥	NOUN
ejpam-847	188	22	t∞k=1ρ	t∞k=1ρ	VERB
ejpam-847	188	23	�	�	PROPN
ejpam-847	188	24	x	x	SYM
ejpam-847	188	25	2k	2k	NUM
ejpam-847	188	26	,	,	PUNCT
ejpam-847	188	27	−	−	PROPN
ejpam-847	188	28	x	x	SYM
ejpam-847	188	29	2k	2k	NUM
ejpam-847	188	30	,	,	PUNCT
ejpam-847	188	31	0,	0,	NUM
ejpam-847	188	32	...	...	PUNCT
ejpam-847	188	33	,0	,0	PUNCT
ejpam-847	188	34	�	�	PROPN
ejpam-847	188	35	�	�	PROPN
ejpam-847	188	36	t	t	PROPN
ejpam-847	188	37	23k−3	23k−3	PROPN
ejpam-847	188	38	�	�	PROPN
ejpam-847	188	39	(	(	PUNCT
ejpam-847	188	40	13	13	NUM
ejpam-847	188	41	)	)	PUNCT
ejpam-847	188	42	for	for	ADP
ejpam-847	188	43	all	all	DET
ejpam-847	188	44	x	x	SYM
ejpam-847	188	45	∈	∈	ADJ
ejpam-847	188	46	x	x	X
ejpam-847	188	47	and	and	CCONJ
ejpam-847	188	48	all	all	DET
ejpam-847	188	49	t	t	NOUN
ejpam-847	188	50	>	>	X
ejpam-847	188	51	0	0	X
ejpam-847	188	52	.	.	PUNCT
ejpam-847	189	1	proof	proof	NOUN
ejpam-847	189	2	.	.	PUNCT
ejpam-847	190	1	putting	put	VERB
ejpam-847	190	2	x1	x1	NOUN
ejpam-847	190	3	=	=	PUNCT
ejpam-847	190	4	x	x	X
ejpam-847	190	5	,	,	PUNCT
ejpam-847	190	6	x2	x2	NOUN
ejpam-847	190	7	=	=	PUNCT
ejpam-847	190	8	−x	−x	NOUN
ejpam-847	190	9	,	,	PUNCT
ejpam-847	190	10	x3	x3	PROPN
ejpam-847	190	11	=	=	PUNCT
ejpam-847	190	12	.	.	PUNCT
ejpam-847	190	13	.	.	PUNCT
ejpam-847	190	14	.	.	PUNCT
ejpam-847	191	1	=	=	PUNCT
ejpam-847	191	2	xn	xn	PUNCT
ejpam-847	192	1	=	=	SYM
ejpam-847	192	2	0	0	NUM
ejpam-847	192	3	in	in	ADP
ejpam-847	192	4	(	(	PUNCT
ejpam-847	192	5	2	2	NUM
ejpam-847	192	6	)	)	PUNCT
ejpam-847	192	7	,	,	PUNCT
ejpam-847	192	8	we	we	PRON
ejpam-847	192	9	get	get	VERB
ejpam-847	192	10	µ2	µ2	PROPN
ejpam-847	192	11	(	(	PUNCT
ejpam-847	192	12	f	f	PROPN
ejpam-847	192	13	(	(	PUNCT
ejpam-847	192	14	2x)−4	2x)−4	NUM
ejpam-847	192	15	f	f	PROPN
ejpam-847	192	16	(	(	PUNCT
ejpam-847	192	17	x	x	NOUN
ejpam-847	192	18	)	)	PUNCT
ejpam-847	192	19	)	)	PUNCT
ejpam-847	193	1	(	(	PUNCT
ejpam-847	193	2	t	t	PROPN
ejpam-847	193	3	)	)	PUNCT
ejpam-847	193	4	≥	≥	NOUN
ejpam-847	193	5	ρ(x	ρ(x	NOUN
ejpam-847	193	6	,	,	PUNCT
ejpam-847	193	7	−x	−x	INTJ
ejpam-847	193	8	,	,	PUNCT
ejpam-847	193	9	0,	0,	NUM
ejpam-847	193	10	...	...	PUNCT
ejpam-847	193	11	,0)(t	,0)(t	PUNCT
ejpam-847	193	12	)	)	PUNCT
ejpam-847	193	13	which	which	PRON
ejpam-847	193	14	is	be	AUX
ejpam-847	193	15	equivalent	equivalent	ADJ
ejpam-847	193	16	to	to	ADP
ejpam-847	193	17	µ	µ	PROPN
ejpam-847	193	18	f	f	X
ejpam-847	193	19	(	(	PUNCT
ejpam-847	193	20	x)−4	x)−4	PROPN
ejpam-847	193	21	f	f	PROPN
ejpam-847	193	22	�	�	PROPN
ejpam-847	193	23	x	x	SYM
ejpam-847	193	24	2	2	NUM
ejpam-847	193	25	�	�	PROPN
ejpam-847	193	26	(	(	PUNCT
ejpam-847	193	27	t	t	PROPN
ejpam-847	193	28	)	)	PUNCT
ejpam-847	193	29	≥	≥	NOUN
ejpam-847	193	30	ρ	ρ	NUM
ejpam-847	193	31	�	�	PROPN
ejpam-847	193	32	x	x	SYM
ejpam-847	193	33	2	2	NUM
ejpam-847	193	34	,	,	PUNCT
ejpam-847	193	35	−	−	PROPN
ejpam-847	193	36	x	x	SYM
ejpam-847	193	37	2	2	NUM
ejpam-847	193	38	,	,	PUNCT
ejpam-847	193	39	0,	0,	NUM
ejpam-847	193	40	...	...	PUNCT
ejpam-847	193	41	,0	,0	PUNCT
ejpam-847	193	42	�	�	PROPN
ejpam-847	193	43	(	(	PUNCT
ejpam-847	193	44	2	2	NUM
ejpam-847	193	45	t	t	NOUN
ejpam-847	193	46	)	)	PUNCT
ejpam-847	193	47	for	for	ADP
ejpam-847	193	48	all	all	DET
ejpam-847	193	49	x	x	SYM
ejpam-847	193	50	∈	∈	ADJ
ejpam-847	193	51	x	x	X
ejpam-847	193	52	and	and	CCONJ
ejpam-847	193	53	all	all	DET
ejpam-847	193	54	t	t	NOUN
ejpam-847	193	55	>	>	X
ejpam-847	193	56	0	0	X
ejpam-847	193	57	.	.	PUNCT
ejpam-847	194	1	replacing	replace	VERB
ejpam-847	194	2	x	x	PUNCT
ejpam-847	194	3	and	and	CCONJ
ejpam-847	194	4	t	t	X
ejpam-847	194	5	by	by	ADP
ejpam-847	194	6	x	x	SYM
ejpam-847	194	7	2k−1	2k−1	NUM
ejpam-847	194	8	and	and	CCONJ
ejpam-847	194	9	t	t	PROPN
ejpam-847	194	10	23k−2	23k−2	NUM
ejpam-847	194	11	,	,	PUNCT
ejpam-847	194	12	respectively	respectively	ADV
ejpam-847	194	13	in	in	ADP
ejpam-847	194	14	the	the	DET
ejpam-847	194	15	above	above	ADJ
ejpam-847	194	16	inequality	inequality	NOUN
ejpam-847	194	17	,	,	PUNCT
ejpam-847	194	18	we	we	PRON
ejpam-847	194	19	get	get	VERB
ejpam-847	194	20	µ	µ	NUM
ejpam-847	194	21	4k−1	4k−1	NUM
ejpam-847	194	22	f	f	PROPN
ejpam-847	194	23	�	�	PROPN
ejpam-847	194	24	x	x	SYM
ejpam-847	194	25	2k−1	2k−1	NUM
ejpam-847	194	26	�	�	PROPN
ejpam-847	194	27	−4k	−4k	PROPN
ejpam-847	194	28	f	f	PROPN
ejpam-847	194	29	�	�	PROPN
ejpam-847	194	30	x	x	ADP
ejpam-847	194	31	2k	2k	NUM
ejpam-847	194	32	�	�	PROPN
ejpam-847	194	33	�	�	PROPN
ejpam-847	194	34	t	t	PROPN
ejpam-847	194	35	2k	2k	PROPN
ejpam-847	194	36	�	�	PROPN
ejpam-847	194	37	≥	≥	PROPN
ejpam-847	194	38	ρ	ρ	NUM
ejpam-847	194	39	�	�	PROPN
ejpam-847	194	40	x	x	SYM
ejpam-847	194	41	2k	2k	NUM
ejpam-847	194	42	,	,	PUNCT
ejpam-847	194	43	−	−	PROPN
ejpam-847	194	44	x	x	SYM
ejpam-847	194	45	2k	2k	NUM
ejpam-847	194	46	0,	0,	NUM
ejpam-847	194	47	...	...	PUNCT
ejpam-847	194	48	,0	,0	PUNCT
ejpam-847	194	49	�	�	PROPN
ejpam-847	194	50	�	�	PROPN
ejpam-847	194	51	t	t	PROPN
ejpam-847	194	52	23k−3	23k−3	PROPN
ejpam-847	194	53	�	�	PROPN
ejpam-847	194	54	for	for	ADP
ejpam-847	194	55	all	all	DET
ejpam-847	194	56	x	x	SYM
ejpam-847	194	57	∈	∈	ADJ
ejpam-847	194	58	x	x	X
ejpam-847	194	59	and	and	CCONJ
ejpam-847	194	60	all	all	DET
ejpam-847	194	61	t	t	PROPN
ejpam-847	194	62	>	>	X
ejpam-847	194	63	0	0	X
ejpam-847	194	64	.	.	PUNCT
ejpam-847	195	1	since	since	SCONJ
ejpam-847	195	2	µx(s)≤	µx(s)≤	ADJ
ejpam-847	195	3	µx(t	µx(t	NOUN
ejpam-847	195	4	)	)	PUNCT
ejpam-847	195	5	for	for	ADP
ejpam-847	195	6	all	all	DET
ejpam-847	195	7	s	s	NOUN
ejpam-847	195	8	and	and	CCONJ
ejpam-847	195	9	t	t	X
ejpam-847	195	10	with	with	ADP
ejpam-847	195	11	0	0	NUM
ejpam-847	195	12	<	<	X
ejpam-847	195	13	s	s	PART
ejpam-847	195	14	≤	≤	PROPN
ejpam-847	195	15	t	t	PROPN
ejpam-847	195	16	,	,	PUNCT
ejpam-847	195	17	we	we	PRON
ejpam-847	195	18	obtain	obtain	VERB
ejpam-847	195	19	µ	µ	PRON
ejpam-847	195	20	f	f	X
ejpam-847	195	21	(	(	PUNCT
ejpam-847	195	22	x)−4	x)−4	X
ejpam-847	195	23	m	m	PROPN
ejpam-847	195	24	f	f	PROPN
ejpam-847	195	25	�	�	PROPN
ejpam-847	195	26	x	x	PUNCT
ejpam-847	195	27	2	2	NUM
ejpam-847	195	28	m	m	PROPN
ejpam-847	195	29	�	�	PROPN
ejpam-847	195	30	(	(	PUNCT
ejpam-847	195	31	t	t	PROPN
ejpam-847	195	32	)	)	PUNCT
ejpam-847	195	33	=	=	NOUN
ejpam-847	195	34	µ∑m	µ∑m	DET
ejpam-847	195	35	k=1	k=1	PROPN
ejpam-847	195	36	�	�	PROPN
ejpam-847	195	37	4k−1	4k−1	NUM
ejpam-847	195	38	f	f	PROPN
ejpam-847	195	39	�	�	PROPN
ejpam-847	195	40	x	x	SYM
ejpam-847	195	41	2k−1	2k−1	NUM
ejpam-847	195	42	�	�	PROPN
ejpam-847	195	43	−4k	−4k	PROPN
ejpam-847	195	44	f	f	PROPN
ejpam-847	195	45	�	�	PROPN
ejpam-847	195	46	x	x	ADP
ejpam-847	195	47	2k	2k	PROPN
ejpam-847	195	48	�	�	PROPN
ejpam-847	195	49	�	�	PROPN
ejpam-847	195	50	(	(	PUNCT
ejpam-847	195	51	t	t	PROPN
ejpam-847	195	52	)	)	PUNCT
ejpam-847	195	53	≥µ∑m	≥µ∑m	PROPN
ejpam-847	195	54	k=1	k=1	X
ejpam-847	195	55	�	�	PROPN
ejpam-847	195	56	4k−1	4k−1	NUM
ejpam-847	195	57	f	f	PROPN
ejpam-847	195	58	�	�	PROPN
ejpam-847	195	59	x	x	SYM
ejpam-847	195	60	2k−1	2k−1	NUM
ejpam-847	195	61	�	�	PROPN
ejpam-847	195	62	−4k	−4k	PROPN
ejpam-847	195	63	f	f	PROPN
ejpam-847	195	64	�	�	PROPN
ejpam-847	195	65	x	x	ADP
ejpam-847	195	66	2k	2k	PROPN
ejpam-847	195	67	�	�	PROPN
ejpam-847	195	68	�	�	PROPN
ejpam-847	195	69	m	m	PROPN
ejpam-847	195	70	∑	∑	PROPN
ejpam-847	195	71	k=1	k=1	PROPN
ejpam-847	195	72	t	t	PROPN
ejpam-847	195	73	2k	2k	PROPN
ejpam-847	195	74	!	!	PUNCT
ejpam-847	196	1	≥t	≥t	X
ejpam-847	196	2	m	m	VERB
ejpam-847	196	3	k=1ρ	k=1ρ	PROPN
ejpam-847	196	4	�	�	PROPN
ejpam-847	196	5	x	x	SYM
ejpam-847	196	6	2k	2k	NUM
ejpam-847	196	7	,	,	PUNCT
ejpam-847	196	8	−	−	PROPN
ejpam-847	196	9	x	x	SYM
ejpam-847	196	10	2k	2k	NUM
ejpam-847	196	11	,	,	PUNCT
ejpam-847	196	12	0,	0,	NUM
ejpam-847	196	13	...	...	PUNCT
ejpam-847	196	14	,0	,0	PUNCT
ejpam-847	196	15	�	�	PROPN
ejpam-847	196	16	�	�	PROPN
ejpam-847	196	17	t	t	PROPN
ejpam-847	196	18	23k−3	23k−3	PROPN
ejpam-847	196	19	�	�	PROPN
ejpam-847	196	20	d.	d.	PROPN
ejpam-847	196	21	shin	shin	PROPN
ejpam-847	196	22	,	,	PUNCT
ejpam-847	196	23	j.	j.	PROPN
ejpam-847	196	24	lee	lee	PROPN
ejpam-847	196	25	,	,	PUNCT
ejpam-847	196	26	c.	c.	PROPN
ejpam-847	196	27	park	park	PROPN
ejpam-847	196	28	/	/	SYM
ejpam-847	196	29	eur	eur	PROPN
ejpam-847	196	30	.	.	PUNCT
ejpam-847	197	1	j.	j.	PROPN
ejpam-847	197	2	pure	pure	PROPN
ejpam-847	197	3	appl	appl	PROPN
ejpam-847	197	4	.	.	PROPN
ejpam-847	197	5	math	math	PROPN
ejpam-847	197	6	,	,	PUNCT
ejpam-847	197	7	5	5	NUM
ejpam-847	197	8	(	(	PUNCT
ejpam-847	197	9	2012	2012	NUM
ejpam-847	197	10	)	)	PUNCT
ejpam-847	197	11	,	,	PUNCT
ejpam-847	197	12	540	540	NUM
ejpam-847	197	13	-	-	SYM
ejpam-847	197	14	553	553	NUM
ejpam-847	197	15	548	548	NUM
ejpam-847	197	16	replacing	replace	VERB
ejpam-847	197	17	x	x	PUNCT
ejpam-847	197	18	by	by	ADP
ejpam-847	197	19	x	x	SYM
ejpam-847	197	20	2l	2l	NUM
ejpam-847	197	21	in	in	ADP
ejpam-847	197	22	the	the	DET
ejpam-847	197	23	above	above	ADJ
ejpam-847	197	24	inequality	inequality	NOUN
ejpam-847	197	25	,	,	PUNCT
ejpam-847	197	26	we	we	PRON
ejpam-847	197	27	get	get	VERB
ejpam-847	197	28	µ	µ	PRON
ejpam-847	197	29	f	f	X
ejpam-847	197	30	�	�	PROPN
ejpam-847	197	31	x	x	SYM
ejpam-847	197	32	2l	2l	NUM
ejpam-847	197	33	�	�	PROPN
ejpam-847	197	34	−4	−4	NOUN
ejpam-847	197	35	m	m	PROPN
ejpam-847	197	36	f	f	PROPN
ejpam-847	197	37	�	�	PROPN
ejpam-847	197	38	x	x	PROPN
ejpam-847	197	39	2m+l	2m+l	NUM
ejpam-847	197	40	�	�	PROPN
ejpam-847	197	41	(	(	PUNCT
ejpam-847	197	42	t	t	PROPN
ejpam-847	197	43	)	)	PUNCT
ejpam-847	197	44	≥	≥	PROPN
ejpam-847	197	45	t	t	PROPN
ejpam-847	197	46	m	m	AUX
ejpam-847	197	47	k=1ρ	k=1ρ	PROPN
ejpam-847	197	48	�	�	PROPN
ejpam-847	197	49	x	x	SYM
ejpam-847	197	50	2k+l	2k+l	NUM
ejpam-847	197	51	,	,	PUNCT
ejpam-847	197	52	−	−	PROPN
ejpam-847	197	53	x	x	SYM
ejpam-847	197	54	2k+l	2k+l	NUM
ejpam-847	197	55	,	,	PUNCT
ejpam-847	197	56	0,	0,	NUM
ejpam-847	197	57	...	...	PUNCT
ejpam-847	197	58	,0	,0	PUNCT
ejpam-847	197	59	�	�	PROPN
ejpam-847	197	60	�	�	PROPN
ejpam-847	197	61	t	t	PROPN
ejpam-847	197	62	23k−3	23k−3	PROPN
ejpam-847	197	63	�	�	PROPN
ejpam-847	197	64	which	which	PRON
ejpam-847	197	65	is	be	AUX
ejpam-847	197	66	equivalent	equivalent	ADJ
ejpam-847	197	67	to	to	ADP
ejpam-847	197	68	µ	µ	PROPN
ejpam-847	197	69	4l	4l	NOUN
ejpam-847	197	70	f	f	PROPN
ejpam-847	197	71	�	�	PROPN
ejpam-847	197	72	x	x	SYM
ejpam-847	197	73	2l	2l	NUM
ejpam-847	197	74	�	�	PROPN
ejpam-847	197	75	−4m+l	−4m+l	NOUN
ejpam-847	197	76	f	f	PROPN
ejpam-847	197	77	�	�	PROPN
ejpam-847	197	78	x	x	PROPN
ejpam-847	197	79	2m+l	2m+l	NUM
ejpam-847	197	80	�	�	PROPN
ejpam-847	197	81	(	(	PUNCT
ejpam-847	197	82	t	t	PROPN
ejpam-847	197	83	)	)	PUNCT
ejpam-847	197	84	≥	≥	NOUN
ejpam-847	197	85	t	t	PROPN
ejpam-847	197	86	m	m	VERB
ejpam-847	197	87	k=1	k=1	PROPN
ejpam-847	197	88	ρ	ρ	NUM
ejpam-847	197	89	�	�	PROPN
ejpam-847	197	90	x	x	SYM
ejpam-847	197	91	2k+l	2k+l	NUM
ejpam-847	197	92	,	,	PUNCT
ejpam-847	197	93	−	−	PROPN
ejpam-847	197	94	x	x	SYM
ejpam-847	197	95	2k+l	2k+l	NUM
ejpam-847	197	96	,	,	PUNCT
ejpam-847	197	97	0,	0,	NUM
ejpam-847	197	98	...	...	PUNCT
ejpam-847	197	99	,0	,0	PUNCT
ejpam-847	197	100	�	�	PROPN
ejpam-847	197	101	�	�	PROPN
ejpam-847	197	102	t	t	PROPN
ejpam-847	197	103	23k+2l−3	23k+2l−3	NUM
ejpam-847	197	104	�	�	PROPN
ejpam-847	197	105	(	(	PUNCT
ejpam-847	197	106	14	14	NUM
ejpam-847	197	107	)	)	PUNCT
ejpam-847	197	108	for	for	ADP
ejpam-847	197	109	all	all	DET
ejpam-847	197	110	x	x	SYM
ejpam-847	197	111	∈	∈	PROPN
ejpam-847	197	112	x	x	X
ejpam-847	197	113	,	,	PUNCT
ejpam-847	197	114	all	all	PRON
ejpam-847	197	115	t	t	X
ejpam-847	197	116	>	>	X
ejpam-847	197	117	0	0	PUNCT
ejpam-847	198	1	and	and	CCONJ
ejpam-847	198	2	all	all	DET
ejpam-847	198	3	l	l	NOUN
ejpam-847	198	4	=	=	PUNCT
ejpam-847	198	5	0,1,2	0,1,2	NUM
ejpam-847	198	6	,	,	PUNCT
ejpam-847	198	7	.	.	PUNCT
ejpam-847	198	8	.	.	PUNCT
ejpam-847	199	1	..	..	PUNCT
ejpam-847	200	1	since	since	SCONJ
ejpam-847	200	2	the	the	DET
ejpam-847	200	3	right	right	ADJ
ejpam-847	200	4	hand	hand	NOUN
ejpam-847	200	5	side	side	NOUN
ejpam-847	200	6	of	of	ADP
ejpam-847	200	7	the	the	DET
ejpam-847	200	8	inequality	inequality	NOUN
ejpam-847	200	9	(	(	PUNCT
ejpam-847	200	10	14	14	NUM
ejpam-847	200	11	)	)	PUNCT
ejpam-847	200	12	tends	tend	VERB
ejpam-847	200	13	to	to	ADP
ejpam-847	200	14	1	1	NUM
ejpam-847	200	15	as	as	ADP
ejpam-847	200	16	m→∞	m→∞	NUM
ejpam-847	200	17	by	by	ADP
ejpam-847	200	18	(	(	PUNCT
ejpam-847	200	19	11	11	NUM
ejpam-847	200	20	)	)	PUNCT
ejpam-847	200	21	,	,	PUNCT
ejpam-847	200	22	the	the	DET
ejpam-847	200	23	sequence	sequence	NOUN
ejpam-847	200	24	{	{	PUNCT
ejpam-847	200	25	4	4	NUM
ejpam-847	200	26	m	m	PROPN
ejpam-847	200	27	f	f	PROPN
ejpam-847	200	28	�	�	PROPN
ejpam-847	200	29	x	x	PUNCT
ejpam-847	200	30	2	2	NUM
ejpam-847	200	31	m	m	PROPN
ejpam-847	200	32	�	�	PROPN
ejpam-847	200	33	}	}	PUNCT
ejpam-847	200	34	is	be	AUX
ejpam-847	200	35	a	a	DET
ejpam-847	200	36	cauchy	cauchy	ADJ
ejpam-847	200	37	sequence	sequence	NOUN
ejpam-847	200	38	.	.	PUNCT
ejpam-847	201	1	thus	thus	ADV
ejpam-847	201	2	we	we	PRON
ejpam-847	201	3	define	define	VERB
ejpam-847	201	4	q(x	q(x	NOUN
ejpam-847	201	5	)	)	PUNCT
ejpam-847	201	6	:	:	PUNCT
ejpam-847	201	7	=	=	SYM
ejpam-847	201	8	limm→∞	limm→∞	PROPN
ejpam-847	201	9	4	4	NUM
ejpam-847	201	10	m	m	PROPN
ejpam-847	201	11	f	f	PROPN
ejpam-847	201	12	�	�	PROPN
ejpam-847	201	13	x	x	PUNCT
ejpam-847	201	14	2	2	NUM
ejpam-847	201	15	m	m	NOUN
ejpam-847	201	16	�	�	NOUN
ejpam-847	201	17	for	for	ADP
ejpam-847	201	18	all	all	DET
ejpam-847	201	19	x	x	SYM
ejpam-847	201	20	∈	∈	PROPN
ejpam-847	201	21	x	x	X
ejpam-847	201	22	,	,	PUNCT
ejpam-847	201	23	which	which	PRON
ejpam-847	201	24	is	be	AUX
ejpam-847	201	25	an	an	DET
ejpam-847	201	26	even	even	ADJ
ejpam-847	201	27	mapping	mapping	NOUN
ejpam-847	201	28	.	.	PUNCT
ejpam-847	202	1	now	now	ADV
ejpam-847	202	2	we	we	PRON
ejpam-847	202	3	show	show	VERB
ejpam-847	202	4	that	that	SCONJ
ejpam-847	202	5	q	q	NOUN
ejpam-847	202	6	is	be	AUX
ejpam-847	202	7	an	an	DET
ejpam-847	202	8	quadratic	quadratic	ADJ
ejpam-847	202	9	mapping	mapping	NOUN
ejpam-847	202	10	.	.	PUNCT
ejpam-847	203	1	by	by	ADP
ejpam-847	203	2	(	(	PUNCT
ejpam-847	203	3	2	2	NUM
ejpam-847	203	4	)	)	PUNCT
ejpam-847	203	5	,	,	PUNCT
ejpam-847	203	6	we	we	PRON
ejpam-847	203	7	get	get	VERB
ejpam-847	203	8	µ	µ	PRON
ejpam-847	203	9	4	4	NUM
ejpam-847	203	10	m	m	NOUN
ejpam-847	203	11	�	�	PROPN
ejpam-847	204	1	f	f	PROPN
ejpam-847	204	2	�	�	PROPN
ejpam-847	204	3	x−y	x−y	PROPN
ejpam-847	204	4	2	2	NUM
ejpam-847	204	5	m	m	PROPN
ejpam-847	204	6	�	�	NOUN
ejpam-847	204	7	+	+	CCONJ
ejpam-847	204	8	f	f	PROPN
ejpam-847	204	9	�	�	PROPN
ejpam-847	204	10	2x+y	2x+y	NUM
ejpam-847	204	11	2	2	NUM
ejpam-847	204	12	m	m	PROPN
ejpam-847	204	13	�	�	NOUN
ejpam-847	204	14	+	+	CCONJ
ejpam-847	204	15	f	f	PROPN
ejpam-847	204	16	�	�	PROPN
ejpam-847	205	1	x+2y	x+2y	PROPN
ejpam-847	205	2	2	2	NUM
ejpam-847	205	3	m	m	NOUN
ejpam-847	205	4	�	�	NOUN
ejpam-847	205	5	−3	−3	PROPN
ejpam-847	205	6	f	f	PROPN
ejpam-847	205	7	�	�	PROPN
ejpam-847	205	8	x+y	x+y	NUM
ejpam-847	205	9	2	2	NUM
ejpam-847	205	10	m	m	PROPN
ejpam-847	205	11	�	�	PROPN
ejpam-847	205	12	−3	−3	PROPN
ejpam-847	205	13	f	f	PROPN
ejpam-847	205	14	�	�	PROPN
ejpam-847	205	15	x	x	PUNCT
ejpam-847	205	16	2	2	NUM
ejpam-847	205	17	m	m	PROPN
ejpam-847	205	18	�	�	NOUN
ejpam-847	205	19	−3	−3	PROPN
ejpam-847	205	20	f	f	PROPN
ejpam-847	205	21	�	�	PROPN
ejpam-847	205	22	y	y	PROPN
ejpam-847	205	23	2	2	NUM
ejpam-847	205	24	m	m	PROPN
ejpam-847	205	25	�	�	PROPN
ejpam-847	205	26	�	�	PROPN
ejpam-847	205	27	(	(	PUNCT
ejpam-847	205	28	t	t	PROPN
ejpam-847	205	29	)	)	PUNCT
ejpam-847	205	30	≥ρ	≥ρ	NOUN
ejpam-847	205	31	�	�	PROPN
ejpam-847	205	32	x	x	PUNCT
ejpam-847	205	33	2	2	NUM
ejpam-847	205	34	m	m	NOUN
ejpam-847	205	35	,	,	PUNCT
ejpam-847	205	36	y	y	PROPN
ejpam-847	205	37	2	2	NUM
ejpam-847	205	38	m	m	NOUN
ejpam-847	205	39	,	,	PUNCT
ejpam-847	205	40	−	−	PROPN
ejpam-847	205	41	x+y	x+y	NUM
ejpam-847	205	42	2	2	NUM
ejpam-847	205	43	m	m	NOUN
ejpam-847	205	44	,	,	PUNCT
ejpam-847	205	45	0,	0,	NUM
ejpam-847	205	46	...	...	PUNCT
ejpam-847	205	47	,0	,0	PUNCT
ejpam-847	205	48	�	�	PROPN
ejpam-847	205	49	�	�	PROPN
ejpam-847	205	50	t	t	PROPN
ejpam-847	205	51	22m−1	22m−1	PROPN
ejpam-847	205	52	�	�	PROPN
ejpam-847	205	53	.	.	PUNCT
ejpam-847	206	1	taking	take	VERB
ejpam-847	206	2	the	the	DET
ejpam-847	206	3	limit	limit	NOUN
ejpam-847	206	4	as	as	ADP
ejpam-847	206	5	m	m	PROPN
ejpam-847	206	6	→	→	SYM
ejpam-847	206	7	∞	∞	PROPN
ejpam-847	206	8	in	in	ADP
ejpam-847	206	9	the	the	DET
ejpam-847	206	10	above	above	ADJ
ejpam-847	206	11	inequality	inequality	NOUN
ejpam-847	206	12	,	,	PUNCT
ejpam-847	206	13	by	by	ADP
ejpam-847	206	14	(	(	PUNCT
ejpam-847	206	15	12	12	NUM
ejpam-847	206	16	)	)	PUNCT
ejpam-847	206	17	,	,	PUNCT
ejpam-847	206	18	the	the	DET
ejpam-847	206	19	mapping	mapping	NOUN
ejpam-847	206	20	q	q	NOUN
ejpam-847	206	21	is	be	AUX
ejpam-847	206	22	quadratic	quadratic	ADJ
ejpam-847	206	23	.	.	PUNCT
ejpam-847	207	1	moreover	moreover	ADV
ejpam-847	207	2	,	,	PUNCT
ejpam-847	207	3	letting	let	VERB
ejpam-847	207	4	l	l	NOUN
ejpam-847	207	5	=	=	SYM
ejpam-847	207	6	0	0	PUNCT
ejpam-847	207	7	and	and	CCONJ
ejpam-847	207	8	taking	take	VERB
ejpam-847	207	9	the	the	DET
ejpam-847	207	10	limit	limit	NOUN
ejpam-847	207	11	as	as	ADP
ejpam-847	207	12	m→∞	m→∞	NOUN
ejpam-847	207	13	in	in	ADP
ejpam-847	207	14	(	(	PUNCT
ejpam-847	207	15	14	14	NUM
ejpam-847	207	16	)	)	PUNCT
ejpam-847	207	17	,	,	PUNCT
ejpam-847	207	18	we	we	PRON
ejpam-847	207	19	get	get	VERB
ejpam-847	207	20	(	(	PUNCT
ejpam-847	207	21	13	13	NUM
ejpam-847	207	22	)	)	PUNCT
ejpam-847	207	23	.	.	PUNCT
ejpam-847	208	1	the	the	DET
ejpam-847	208	2	rest	rest	NOUN
ejpam-847	208	3	of	of	ADP
ejpam-847	208	4	the	the	DET
ejpam-847	208	5	proof	proof	NOUN
ejpam-847	208	6	is	be	AUX
ejpam-847	208	7	the	the	DET
ejpam-847	208	8	same	same	ADJ
ejpam-847	208	9	as	as	ADP
ejpam-847	208	10	in	in	ADP
ejpam-847	208	11	the	the	DET
ejpam-847	208	12	proof	proof	NOUN
ejpam-847	208	13	of	of	ADP
ejpam-847	208	14	theorem	theorem	ADJ
ejpam-847	208	15	1	1	NUM
ejpam-847	208	16	.	.	PUNCT
ejpam-847	208	17	corollary	corollary	ADJ
ejpam-847	208	18	3	3	X
ejpam-847	208	19	.	.	PUNCT
ejpam-847	209	1	let	let	VERB
ejpam-847	209	2	θ	θ	PROPN
ejpam-847	209	3	≥	≥	X
ejpam-847	209	4	0	0	NUM
ejpam-847	209	5	and	and	CCONJ
ejpam-847	209	6	let	let	VERB
ejpam-847	209	7	p	p	PRON
ejpam-847	209	8	be	be	AUX
ejpam-847	209	9	a	a	DET
ejpam-847	209	10	constant	constant	ADJ
ejpam-847	209	11	with	with	ADP
ejpam-847	209	12	p	p	PROPN
ejpam-847	209	13	>	>	X
ejpam-847	209	14	2	2	NUM
ejpam-847	209	15	.	.	PUNCT
ejpam-847	210	1	for	for	ADP
ejpam-847	210	2	a	a	DET
ejpam-847	210	3	normed	normed	ADJ
ejpam-847	210	4	vector	vector	NOUN
ejpam-847	210	5	space	space	NOUN
ejpam-847	210	6	x	x	PUNCT
ejpam-847	210	7	and	and	CCONJ
ejpam-847	210	8	complete	complete	ADJ
ejpam-847	210	9	rn	rn	NOUN
ejpam-847	210	10	-	-	NOUN
ejpam-847	210	11	space	space	NOUN
ejpam-847	210	12	y	y	PROPN
ejpam-847	210	13	,	,	PUNCT
ejpam-847	210	14	let	let	VERB
ejpam-847	210	15	f	f	PRON
ejpam-847	210	16	:	:	PUNCT
ejpam-847	210	17	x	x	X
ejpam-847	210	18	→	→	SYM
ejpam-847	210	19	y	y	X
ejpam-847	210	20	be	be	AUX
ejpam-847	210	21	an	an	DET
ejpam-847	210	22	even	even	ADV
ejpam-847	210	23	mapping	mapping	NOUN
ejpam-847	210	24	satisfying	satisfy	VERB
ejpam-847	210	25	µd	µd	DET
ejpam-847	210	26	f	f	PROPN
ejpam-847	210	27	(	(	PUNCT
ejpam-847	210	28	x1	x1	INTJ
ejpam-847	210	29	,	,	PUNCT
ejpam-847	210	30	x2,	x2,	PROPN
ejpam-847	210	31	...	...	PUNCT
ejpam-847	210	32	,xn	,xn	PUNCT
ejpam-847	210	33	)	)	PUNCT
ejpam-847	210	34	(	(	PUNCT
ejpam-847	210	35	t	t	PROPN
ejpam-847	210	36	)	)	PUNCT
ejpam-847	210	37	≥	≥	PROPN
ejpam-847	210	38	t	t	PROPN
ejpam-847	210	39	t	t	NOUN
ejpam-847	211	1	+	+	CCONJ
ejpam-847	211	2	θ	θ	PROPN
ejpam-847	211	3	∑n	∑n	PROPN
ejpam-847	211	4	i=1	i=1	PROPN
ejpam-847	211	5	||x	||x	PROPN
ejpam-847	211	6	i||p	i||p	PROPN
ejpam-847	211	7	for	for	ADP
ejpam-847	211	8	all	all	PRON
ejpam-847	211	9	(	(	PUNCT
ejpam-847	211	10	x1	x1	PROPN
ejpam-847	211	11	,	,	PUNCT
ejpam-847	211	12	x2	x2	PROPN
ejpam-847	211	13	,	,	PUNCT
ejpam-847	211	14	.	.	PUNCT
ejpam-847	211	15	.	.	PUNCT
ejpam-847	211	16	.	.	PUNCT
ejpam-847	212	1	,	,	PUNCT
ejpam-847	212	2	xn	xn	X
ejpam-847	212	3	)	)	PUNCT
ejpam-847	212	4	∈	∈	PROPN
ejpam-847	212	5	x	x	PUNCT
ejpam-847	212	6	with	with	ADP
ejpam-847	212	7	∑n	∑n	PROPN
ejpam-847	212	8	i=1	i=1	NOUN
ejpam-847	212	9	x	x	PUNCT
ejpam-847	213	1	i	i	NOUN
ejpam-847	213	2	=	=	NOUN
ejpam-847	213	3	0	0	NUM
ejpam-847	213	4	and	and	CCONJ
ejpam-847	213	5	all	all	DET
ejpam-847	213	6	t	t	NOUN
ejpam-847	213	7	>	>	X
ejpam-847	213	8	0	0	X
ejpam-847	213	9	.	.	PUNCT
ejpam-847	214	1	if	if	SCONJ
ejpam-847	214	2	t∞k=1	t∞k=1	PROPN
ejpam-847	214	3	�	�	PROPN
ejpam-847	214	4	2(k+l)p	2(k+l)p	NUM
ejpam-847	214	5	t	t	PROPN
ejpam-847	214	6	2(k+l)p	2(k+l)p	NUM
ejpam-847	214	7	t	t	NOUN
ejpam-847	214	8	+	+	CCONJ
ejpam-847	214	9	23k+2l−2θ	23k+2l−2θ	PROPN
ejpam-847	214	10	||x	||x	PROPN
ejpam-847	214	11	||p	||p	NOUN
ejpam-847	214	12	�	�	NOUN
ejpam-847	214	13	=	=	NOUN
ejpam-847	214	14	1	1	NUM
ejpam-847	214	15	for	for	ADP
ejpam-847	214	16	all	all	DET
ejpam-847	214	17	x	x	SYM
ejpam-847	214	18	∈	∈	PROPN
ejpam-847	214	19	x	x	X
ejpam-847	214	20	,	,	PUNCT
ejpam-847	214	21	all	all	PRON
ejpam-847	214	22	t	t	X
ejpam-847	214	23	>	>	X
ejpam-847	214	24	0	0	PUNCT
ejpam-847	215	1	and	and	CCONJ
ejpam-847	215	2	all	all	DET
ejpam-847	215	3	l	l	NOUN
ejpam-847	215	4	=	=	PUNCT
ejpam-847	215	5	0,1,2	0,1,2	NUM
ejpam-847	215	6	,	,	PUNCT
ejpam-847	215	7	.	.	PUNCT
ejpam-847	215	8	.	.	PUNCT
ejpam-847	216	1	.	.	PUNCT
ejpam-847	217	1	,	,	PUNCT
ejpam-847	217	2	then	then	ADV
ejpam-847	217	3	there	there	PRON
ejpam-847	217	4	exists	exist	VERB
ejpam-847	217	5	a	a	DET
ejpam-847	217	6	unique	unique	ADJ
ejpam-847	217	7	quadratic	quadratic	ADJ
ejpam-847	217	8	mapping	mapping	NOUN
ejpam-847	217	9	q	q	NOUN
ejpam-847	217	10	:	:	PUNCT
ejpam-847	217	11	x	x	X
ejpam-847	217	12	→	→	SYM
ejpam-847	217	13	y	y	PROPN
ejpam-847	217	14	such	such	ADJ
ejpam-847	217	15	that	that	SCONJ
ejpam-847	218	1	µ	µ	PROPN
ejpam-847	218	2	f	f	X
ejpam-847	218	3	(	(	PUNCT
ejpam-847	218	4	x)−q(x)(t	x)−q(x)(t	PROPN
ejpam-847	218	5	)	)	PUNCT
ejpam-847	218	6	≥	≥	NOUN
ejpam-847	218	7	t∞k=1	t∞k=1	NOUN
ejpam-847	218	8	�	�	PROPN
ejpam-847	218	9	2kp	2kp	PROPN
ejpam-847	218	10	t	t	PROPN
ejpam-847	218	11	2kp	2kp	PROPN
ejpam-847	218	12	t	t	PROPN
ejpam-847	219	1	+	+	CCONJ
ejpam-847	219	2	23k−2θ	23k−2θ	NUM
ejpam-847	219	3	||x	||x	NOUN
ejpam-847	219	4	||p	||p	NOUN
ejpam-847	219	5	�	�	PROPN
ejpam-847	219	6	for	for	ADP
ejpam-847	219	7	all	all	DET
ejpam-847	219	8	x	x	SYM
ejpam-847	219	9	∈	∈	ADJ
ejpam-847	219	10	x	x	X
ejpam-847	219	11	and	and	CCONJ
ejpam-847	219	12	all	all	DET
ejpam-847	219	13	t	t	NOUN
ejpam-847	219	14	>	>	X
ejpam-847	219	15	0	0	X
ejpam-847	219	16	.	.	PUNCT
ejpam-847	220	1	proof	proof	NOUN
ejpam-847	220	2	.	.	PUNCT
ejpam-847	221	1	if	if	SCONJ
ejpam-847	221	2	we	we	PRON
ejpam-847	221	3	define	define	VERB
ejpam-847	221	4	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	221	5	...	...	PUNCT
ejpam-847	221	6	,xn	,xn	PUNCT
ejpam-847	221	7	)	)	PUNCT
ejpam-847	221	8	(	(	PUNCT
ejpam-847	221	9	t	t	NOUN
ejpam-847	221	10	)	)	PUNCT
ejpam-847	221	11	=	=	SYM
ejpam-847	221	12	t	t	PROPN
ejpam-847	221	13	t	t	NOUN
ejpam-847	222	1	+	+	CCONJ
ejpam-847	222	2	θ	θ	PROPN
ejpam-847	222	3	∑n	∑n	PROPN
ejpam-847	222	4	i=1	i=1	PROPN
ejpam-847	222	5	||x	||x	PROPN
ejpam-847	222	6	i||p	i||p	PROPN
ejpam-847	222	7	and	and	CCONJ
ejpam-847	222	8	apply	apply	VERB
ejpam-847	222	9	theorem	theorem	NOUN
ejpam-847	222	10	3	3	NUM
ejpam-847	222	11	,	,	PUNCT
ejpam-847	222	12	then	then	ADV
ejpam-847	222	13	we	we	PRON
ejpam-847	222	14	get	get	VERB
ejpam-847	222	15	the	the	DET
ejpam-847	222	16	desired	desire	VERB
ejpam-847	222	17	result	result	NOUN
ejpam-847	222	18	.	.	PUNCT
ejpam-847	223	1	d.	d.	PROPN
ejpam-847	223	2	shin	shin	PROPN
ejpam-847	223	3	,	,	PUNCT
ejpam-847	223	4	j.	j.	PROPN
ejpam-847	223	5	lee	lee	PROPN
ejpam-847	223	6	,	,	PUNCT
ejpam-847	223	7	c.	c.	PROPN
ejpam-847	223	8	park	park	PROPN
ejpam-847	223	9	/	/	SYM
ejpam-847	223	10	eur	eur	PROPN
ejpam-847	223	11	.	.	PUNCT
ejpam-847	224	1	j.	j.	PROPN
ejpam-847	224	2	pure	pure	PROPN
ejpam-847	224	3	appl	appl	PROPN
ejpam-847	224	4	.	.	PROPN
ejpam-847	224	5	math	math	PROPN
ejpam-847	224	6	,	,	PUNCT
ejpam-847	224	7	5	5	NUM
ejpam-847	224	8	(	(	PUNCT
ejpam-847	224	9	2012	2012	NUM
ejpam-847	224	10	)	)	PUNCT
ejpam-847	224	11	,	,	PUNCT
ejpam-847	224	12	540	540	NUM
ejpam-847	224	13	-	-	SYM
ejpam-847	224	14	553	553	NUM
ejpam-847	224	15	549	549	NUM
ejpam-847	224	16	theorem	theorem	NOUN
ejpam-847	224	17	4	4	NUM
ejpam-847	224	18	.	.	PUNCT
ejpam-847	225	1	let	let	VERB
ejpam-847	225	2	f	f	NOUN
ejpam-847	225	3	:	:	PUNCT
ejpam-847	225	4	x	x	X
ejpam-847	225	5	→	→	SYM
ejpam-847	225	6	y	y	X
ejpam-847	225	7	be	be	AUX
ejpam-847	225	8	an	an	DET
ejpam-847	225	9	even	even	ADV
ejpam-847	225	10	mapping	mapping	NOUN
ejpam-847	225	11	with	with	ADP
ejpam-847	225	12	f	f	PROPN
ejpam-847	225	13	(	(	PUNCT
ejpam-847	225	14	0	0	NUM
ejpam-847	225	15	)	)	PUNCT
ejpam-847	225	16	=	=	SYM
ejpam-847	225	17	0	0	NUM
ejpam-847	225	18	for	for	ADP
ejpam-847	225	19	which	which	PRON
ejpam-847	225	20	there	there	PRON
ejpam-847	225	21	is	be	VERB
ejpam-847	225	22	a	a	DET
ejpam-847	225	23	ρ	ρ	NOUN
ejpam-847	225	24	:	:	PUNCT
ejpam-847	225	25	x	x	SYM
ejpam-847	225	26	n→	n→	X
ejpam-847	225	27	d+	d+	X
ejpam-847	225	28	satisfying	satisfy	VERB
ejpam-847	225	29	(	(	PUNCT
ejpam-847	225	30	2	2	NUM
ejpam-847	225	31	)	)	PUNCT
ejpam-847	225	32	.	.	PUNCT
ejpam-847	226	1	if	if	SCONJ
ejpam-847	226	2	t∞k=1ρ(2k+l−1	t∞k=1ρ(2k+l−1	NOUN
ejpam-847	226	3	x	x	SYM
ejpam-847	226	4	,	,	PUNCT
ejpam-847	226	5	−2k+l−1	−2k+l−1	NOUN
ejpam-847	226	6	x	x	SYM
ejpam-847	226	7	,	,	PUNCT
ejpam-847	226	8	0,	0,	NUM
ejpam-847	226	9	...	...	PUNCT
ejpam-847	226	10	,0	,0	PUNCT
ejpam-847	226	11	)	)	PUNCT
ejpam-847	226	12	�	�	PROPN
ejpam-847	226	13	2k+2l−1	2k+2l−1	NUM
ejpam-847	226	14	t	t	NOUN
ejpam-847	226	15	�	�	NOUN
ejpam-847	226	16	=	=	SYM
ejpam-847	226	17	1	1	NUM
ejpam-847	226	18	(	(	PUNCT
ejpam-847	226	19	15	15	NUM
ejpam-847	226	20	)	)	PUNCT
ejpam-847	226	21	and	and	CCONJ
ejpam-847	226	22	lim	lim	PROPN
ejpam-847	226	23	m→∞	m→∞	PROPN
ejpam-847	227	1	ρ(2	ρ(2	PROPN
ejpam-847	227	2	m	m	PROPN
ejpam-847	227	3	x	x	NOUN
ejpam-847	227	4	,	,	PUNCT
ejpam-847	227	5	2	2	NUM
ejpam-847	227	6	m	m	NOUN
ejpam-847	227	7	y,−2m(x+y),0,	y,−2m(x+y),0,	NOUN
ejpam-847	227	8	...	...	PUNCT
ejpam-847	227	9	,0	,0	PUNCT
ejpam-847	227	10	)	)	PUNCT
ejpam-847	227	11	�	�	PROPN
ejpam-847	227	12	2m+1	2m+1	PROPN
ejpam-847	227	13	t	t	PROPN
ejpam-847	227	14	�	�	PROPN
ejpam-847	227	15	=	=	SYM
ejpam-847	227	16	1	1	NUM
ejpam-847	227	17	(	(	PUNCT
ejpam-847	227	18	16	16	NUM
ejpam-847	227	19	)	)	PUNCT
ejpam-847	227	20	for	for	ADP
ejpam-847	227	21	all	all	PRON
ejpam-847	227	22	x	x	SYM
ejpam-847	227	23	,	,	PUNCT
ejpam-847	227	24	y	y	PROPN
ejpam-847	227	25	∈	∈	PROPN
ejpam-847	227	26	x	x	INTJ
ejpam-847	227	27	,	,	PUNCT
ejpam-847	227	28	all	all	PRON
ejpam-847	227	29	t	t	X
ejpam-847	227	30	>	>	X
ejpam-847	227	31	0	0	PUNCT
ejpam-847	227	32	and	and	CCONJ
ejpam-847	227	33	all	all	DET
ejpam-847	227	34	l	l	NOUN
ejpam-847	227	35	=	=	PUNCT
ejpam-847	227	36	0,1,2	0,1,2	NUM
ejpam-847	227	37	,	,	PUNCT
ejpam-847	227	38	.	.	PUNCT
ejpam-847	227	39	.	.	PUNCT
ejpam-847	228	1	.	.	PUNCT
ejpam-847	229	1	,	,	PUNCT
ejpam-847	229	2	then	then	ADV
ejpam-847	229	3	there	there	PRON
ejpam-847	229	4	exists	exist	VERB
ejpam-847	229	5	a	a	DET
ejpam-847	229	6	unique	unique	ADJ
ejpam-847	229	7	quadratic	quadratic	ADJ
ejpam-847	229	8	mapping	mapping	NOUN
ejpam-847	229	9	q	q	NOUN
ejpam-847	229	10	:	:	PUNCT
ejpam-847	229	11	x	x	X
ejpam-847	229	12	→	→	SYM
ejpam-847	229	13	y	y	PROPN
ejpam-847	229	14	such	such	ADJ
ejpam-847	229	15	that	that	SCONJ
ejpam-847	229	16	µ	µ	PROPN
ejpam-847	229	17	f	f	X
ejpam-847	229	18	(	(	PUNCT
ejpam-847	229	19	x)−q(x)(t	x)−q(x)(t	PROPN
ejpam-847	229	20	)	)	PUNCT
ejpam-847	229	21	≥	≥	NOUN
ejpam-847	229	22	t∞k=1ρ(2k	t∞k=1ρ(2k	PROPN
ejpam-847	229	23	x	x	SYM
ejpam-847	229	24	,	,	PUNCT
ejpam-847	229	25	−2k	−2k	PROPN
ejpam-847	229	26	x	x	SYM
ejpam-847	229	27	,	,	PUNCT
ejpam-847	229	28	0,	0,	NUM
ejpam-847	229	29	...	...	PUNCT
ejpam-847	229	30	,0	,0	PUNCT
ejpam-847	229	31	)	)	PUNCT
ejpam-847	229	32	�	�	PROPN
ejpam-847	229	33	2k−1	2k−1	NUM
ejpam-847	229	34	t	t	PROPN
ejpam-847	229	35	�	�	PROPN
ejpam-847	229	36	(	(	PUNCT
ejpam-847	229	37	17	17	NUM
ejpam-847	229	38	)	)	PUNCT
ejpam-847	229	39	for	for	ADP
ejpam-847	229	40	all	all	PRON
ejpam-847	229	41	x	x	SYM
ejpam-847	229	42	∈	∈	ADJ
ejpam-847	229	43	x	x	X
ejpam-847	229	44	and	and	CCONJ
ejpam-847	229	45	all	all	DET
ejpam-847	229	46	t	t	NOUN
ejpam-847	229	47	>	>	X
ejpam-847	229	48	0	0	X
ejpam-847	229	49	.	.	PUNCT
ejpam-847	230	1	proof	proof	NOUN
ejpam-847	230	2	.	.	PUNCT
ejpam-847	231	1	letting	let	VERB
ejpam-847	231	2	x1	x1	NOUN
ejpam-847	232	1	=	=	PUNCT
ejpam-847	232	2	x	x	X
ejpam-847	232	3	,	,	PUNCT
ejpam-847	232	4	x2	x2	PROPN
ejpam-847	232	5	=	=	NOUN
ejpam-847	232	6	−x	−x	NOUN
ejpam-847	232	7	,	,	PUNCT
ejpam-847	232	8	x3	x3	PROPN
ejpam-847	232	9	=	=	PUNCT
ejpam-847	232	10	.	.	PUNCT
ejpam-847	232	11	.	.	PUNCT
ejpam-847	232	12	.	.	PUNCT
ejpam-847	233	1	=	=	PUNCT
ejpam-847	233	2	xn	xn	PUNCT
ejpam-847	234	1	=	=	SYM
ejpam-847	234	2	0	0	NUM
ejpam-847	234	3	in	in	ADP
ejpam-847	234	4	(	(	PUNCT
ejpam-847	234	5	2	2	NUM
ejpam-847	234	6	)	)	PUNCT
ejpam-847	234	7	,	,	PUNCT
ejpam-847	234	8	we	we	PRON
ejpam-847	234	9	get	get	VERB
ejpam-847	234	10	µ2	µ2	PROPN
ejpam-847	234	11	(	(	PUNCT
ejpam-847	234	12	f	f	PROPN
ejpam-847	234	13	(	(	PUNCT
ejpam-847	234	14	2x)−4	2x)−4	NUM
ejpam-847	234	15	f	f	PROPN
ejpam-847	234	16	(	(	PUNCT
ejpam-847	234	17	x	x	NOUN
ejpam-847	234	18	)	)	PUNCT
ejpam-847	234	19	)	)	PUNCT
ejpam-847	235	1	(	(	PUNCT
ejpam-847	235	2	t	t	PROPN
ejpam-847	235	3	)	)	PUNCT
ejpam-847	235	4	≥	≥	NOUN
ejpam-847	235	5	ρ(x	ρ(x	NOUN
ejpam-847	235	6	,	,	PUNCT
ejpam-847	235	7	−x	−x	INTJ
ejpam-847	235	8	,	,	PUNCT
ejpam-847	235	9	0,	0,	NUM
ejpam-847	235	10	...	...	PUNCT
ejpam-847	235	11	,0)(t	,0)(t	PUNCT
ejpam-847	235	12	)	)	PUNCT
ejpam-847	235	13	which	which	PRON
ejpam-847	235	14	is	be	AUX
ejpam-847	235	15	equivalent	equivalent	ADJ
ejpam-847	235	16	to	to	ADP
ejpam-847	235	17	µ	µ	PROPN
ejpam-847	235	18	f	f	X
ejpam-847	235	19	(	(	PUNCT
ejpam-847	235	20	x)−	x)−	PROPN
ejpam-847	235	21	1	1	NUM
ejpam-847	235	22	4	4	NUM
ejpam-847	235	23	f	f	NOUN
ejpam-847	235	24	(	(	PUNCT
ejpam-847	235	25	2x	2x	NUM
ejpam-847	235	26	)	)	PUNCT
ejpam-847	235	27	�	�	PROPN
ejpam-847	235	28	t	t	PROPN
ejpam-847	235	29	4	4	NUM
ejpam-847	235	30	�	�	PROPN
ejpam-847	235	31	≥	≥	PRON
ejpam-847	235	32	ρ(x	ρ(x	PROPN
ejpam-847	235	33	,	,	PUNCT
ejpam-847	235	34	−x	−x	INTJ
ejpam-847	235	35	,	,	PUNCT
ejpam-847	235	36	0,	0,	NUM
ejpam-847	235	37	...	...	PUNCT
ejpam-847	235	38	,0)(2	,0)(2	PUNCT
ejpam-847	235	39	t	t	PROPN
ejpam-847	235	40	)	)	PUNCT
ejpam-847	235	41	for	for	ADP
ejpam-847	235	42	all	all	DET
ejpam-847	235	43	x	x	SYM
ejpam-847	235	44	∈	∈	ADJ
ejpam-847	235	45	x	x	X
ejpam-847	235	46	and	and	CCONJ
ejpam-847	235	47	all	all	DET
ejpam-847	235	48	t	t	NOUN
ejpam-847	235	49	>	>	X
ejpam-847	235	50	0	0	X
ejpam-847	235	51	.	.	PUNCT
ejpam-847	236	1	replacing	replace	VERB
ejpam-847	236	2	x	x	PUNCT
ejpam-847	236	3	and	and	CCONJ
ejpam-847	236	4	t	t	PROPN
ejpam-847	236	5	by	by	ADP
ejpam-847	236	6	2k−1	2k−1	NUM
ejpam-847	236	7	x	x	NOUN
ejpam-847	236	8	and	and	CCONJ
ejpam-847	236	9	2k−2	2k−2	PROPN
ejpam-847	236	10	t	t	PROPN
ejpam-847	236	11	,	,	PUNCT
ejpam-847	236	12	respectively	respectively	ADV
ejpam-847	236	13	in	in	ADP
ejpam-847	236	14	the	the	DET
ejpam-847	236	15	above	above	ADJ
ejpam-847	236	16	inequality	inequality	NOUN
ejpam-847	236	17	,	,	PUNCT
ejpam-847	236	18	we	we	PRON
ejpam-847	236	19	get	get	VERB
ejpam-847	236	20	µ	µ	PRON
ejpam-847	236	21	1	1	NUM
ejpam-847	236	22	4k−1	4k−1	NUM
ejpam-847	236	23	f	f	NOUN
ejpam-847	236	24	(	(	PUNCT
ejpam-847	236	25	2k−1	2k−1	NUM
ejpam-847	236	26	x)−	x)−	PROPN
ejpam-847	236	27	1	1	NUM
ejpam-847	236	28	4k	4k	NOUN
ejpam-847	236	29	f	f	PROPN
ejpam-847	236	30	(	(	PUNCT
ejpam-847	236	31	2k	2k	NUM
ejpam-847	236	32	x	x	SYM
ejpam-847	236	33	)	)	PUNCT
ejpam-847	236	34	�	�	PROPN
ejpam-847	236	35	t	t	PROPN
ejpam-847	236	36	2k	2k	PROPN
ejpam-847	236	37	�	�	PROPN
ejpam-847	236	38	≥	≥	NUM
ejpam-847	236	39	ρ(2k−1	ρ(2k−1	PROPN
ejpam-847	236	40	x	x	SYM
ejpam-847	236	41	,	,	PUNCT
ejpam-847	236	42	−2k−1	−2k−1	PROPN
ejpam-847	236	43	x	x	PROPN
ejpam-847	236	44	,	,	PUNCT
ejpam-847	236	45	0,	0,	NUM
ejpam-847	236	46	...	...	PUNCT
ejpam-847	236	47	,0)(2	,0)(2	PUNCT
ejpam-847	237	1	k−1	k−1	PROPN
ejpam-847	237	2	t	t	PROPN
ejpam-847	237	3	)	)	PUNCT
ejpam-847	237	4	for	for	ADP
ejpam-847	237	5	all	all	PRON
ejpam-847	237	6	x	x	SYM
ejpam-847	237	7	∈	∈	ADJ
ejpam-847	237	8	x	x	X
ejpam-847	237	9	and	and	CCONJ
ejpam-847	237	10	all	all	DET
ejpam-847	237	11	t	t	PROPN
ejpam-847	237	12	>	>	X
ejpam-847	238	1	0	0	X
ejpam-847	238	2	.	.	PUNCT
ejpam-847	239	1	since	since	SCONJ
ejpam-847	239	2	µx(s)≤	µx(s)≤	ADJ
ejpam-847	239	3	µx(t	µx(t	NOUN
ejpam-847	239	4	)	)	PUNCT
ejpam-847	239	5	for	for	ADP
ejpam-847	239	6	all	all	DET
ejpam-847	239	7	s	s	NOUN
ejpam-847	239	8	and	and	CCONJ
ejpam-847	239	9	t	t	X
ejpam-847	239	10	with	with	ADP
ejpam-847	239	11	0	0	NUM
ejpam-847	239	12	<	<	X
ejpam-847	239	13	s	s	PART
ejpam-847	239	14	≤	≤	PROPN
ejpam-847	239	15	t	t	PROPN
ejpam-847	239	16	,	,	PUNCT
ejpam-847	239	17	we	we	PRON
ejpam-847	239	18	obtain	obtain	VERB
ejpam-847	239	19	µ	µ	PRON
ejpam-847	239	20	f	f	X
ejpam-847	239	21	(	(	PUNCT
ejpam-847	239	22	x)−	x)−	PROPN
ejpam-847	239	23	1	1	NUM
ejpam-847	239	24	4	4	NUM
ejpam-847	239	25	m	m	PROPN
ejpam-847	239	26	f	f	NOUN
ejpam-847	239	27	(	(	PUNCT
ejpam-847	239	28	2	2	NUM
ejpam-847	239	29	m	m	NOUN
ejpam-847	239	30	x)(t	x)(t	NUM
ejpam-847	239	31	)	)	PUNCT
ejpam-847	240	1	=	=	X
ejpam-847	240	2	µ∑m	µ∑m	PRON
ejpam-847	240	3	k=1	k=1	PROPN
ejpam-847	240	4	�	�	PROPN
ejpam-847	240	5	1	1	NUM
ejpam-847	240	6	4k−1	4k−1	NUM
ejpam-847	240	7	f	f	NOUN
ejpam-847	240	8	(	(	PUNCT
ejpam-847	240	9	2k−1	2k−1	NUM
ejpam-847	240	10	x)−	x)−	PROPN
ejpam-847	240	11	1	1	NUM
ejpam-847	240	12	4k	4k	NOUN
ejpam-847	240	13	f	f	PROPN
ejpam-847	240	14	(	(	PUNCT
ejpam-847	240	15	2k	2k	PROPN
ejpam-847	240	16	x	x	SYM
ejpam-847	240	17	)	)	PUNCT
ejpam-847	240	18	�	�	PROPN
ejpam-847	240	19	(	(	PUNCT
ejpam-847	240	20	t	t	PROPN
ejpam-847	240	21	)	)	PUNCT
ejpam-847	240	22	≥µ∑m	≥µ∑m	PROPN
ejpam-847	240	23	k=1	k=1	X
ejpam-847	240	24	�	�	PROPN
ejpam-847	240	25	1	1	NUM
ejpam-847	240	26	4k−1	4k−1	NUM
ejpam-847	240	27	f	f	NOUN
ejpam-847	240	28	(	(	PUNCT
ejpam-847	240	29	2k−1	2k−1	NUM
ejpam-847	240	30	x)−	x)−	PROPN
ejpam-847	240	31	1	1	NUM
ejpam-847	240	32	4k	4k	NOUN
ejpam-847	240	33	f	f	PROPN
ejpam-847	240	34	(	(	PUNCT
ejpam-847	240	35	2k	2k	NUM
ejpam-847	240	36	x	x	SYM
ejpam-847	240	37	)	)	PUNCT
ejpam-847	240	38	�	�	PROPN
ejpam-847	240	39	m	m	VERB
ejpam-847	240	40	∑	∑	PROPN
ejpam-847	240	41	k=1	k=1	PROPN
ejpam-847	240	42	t	t	PROPN
ejpam-847	240	43	2k	2k	PROPN
ejpam-847	240	44	!	!	PUNCT
ejpam-847	241	1	≥t	≥t	VERB
ejpam-847	241	2	m	m	VERB
ejpam-847	241	3	k=1	k=1	X
ejpam-847	241	4	ρ(2k−1	ρ(2k−1	PROPN
ejpam-847	241	5	x	x	SYM
ejpam-847	241	6	,	,	PUNCT
ejpam-847	241	7	−2k−1	−2k−1	PROPN
ejpam-847	241	8	x	x	PROPN
ejpam-847	241	9	,	,	PUNCT
ejpam-847	241	10	0,	0,	NUM
ejpam-847	241	11	...	...	PUNCT
ejpam-847	241	12	,0	,0	PUNCT
ejpam-847	241	13	)	)	PUNCT
ejpam-847	241	14	�	�	PROPN
ejpam-847	241	15	2k−1	2k−1	NUM
ejpam-847	241	16	t	t	NOUN
ejpam-847	241	17	�	�	NOUN
ejpam-847	241	18	replacing	replace	VERB
ejpam-847	241	19	x	x	PUNCT
ejpam-847	241	20	by	by	ADP
ejpam-847	241	21	2l	2l	NOUN
ejpam-847	241	22	x	x	PUNCT
ejpam-847	241	23	in	in	ADP
ejpam-847	241	24	the	the	DET
ejpam-847	241	25	above	above	ADJ
ejpam-847	241	26	inequality	inequality	NOUN
ejpam-847	241	27	,	,	PUNCT
ejpam-847	241	28	we	we	PRON
ejpam-847	241	29	get	get	VERB
ejpam-847	241	30	µ	µ	PRON
ejpam-847	241	31	f	f	X
ejpam-847	241	32	(	(	PUNCT
ejpam-847	241	33	2l	2l	X
ejpam-847	241	34	x)−	x)−	PROPN
ejpam-847	241	35	1	1	NUM
ejpam-847	241	36	4	4	NUM
ejpam-847	241	37	m	m	NOUN
ejpam-847	241	38	f	f	NOUN
ejpam-847	241	39	(	(	PUNCT
ejpam-847	241	40	2m+l	2m+l	NUM
ejpam-847	241	41	x)(t	x)(t	NUM
ejpam-847	241	42	)	)	PUNCT
ejpam-847	241	43	≥	≥	PROPN
ejpam-847	241	44	t	t	NOUN
ejpam-847	241	45	m	m	NOUN
ejpam-847	241	46	k=1ρ(2k+l−1	k=1ρ(2k+l−1	PROPN
ejpam-847	241	47	x	x	SYM
ejpam-847	241	48	,	,	PUNCT
ejpam-847	241	49	−2k+l−1	−2k+l−1	X
ejpam-847	241	50	x	x	SYM
ejpam-847	241	51	,	,	PUNCT
ejpam-847	241	52	0,	0,	NUM
ejpam-847	241	53	...	...	PUNCT
ejpam-847	241	54	,0	,0	PUNCT
ejpam-847	241	55	)	)	PUNCT
ejpam-847	241	56	�	�	PROPN
ejpam-847	241	57	2k−1	2k−1	NUM
ejpam-847	241	58	t	t	NOUN
ejpam-847	241	59	�	�	NOUN
ejpam-847	241	60	which	which	PRON
ejpam-847	241	61	is	be	AUX
ejpam-847	241	62	equivalent	equivalent	ADJ
ejpam-847	241	63	to	to	ADP
ejpam-847	241	64	µ	µ	PROPN
ejpam-847	241	65	1	1	NUM
ejpam-847	241	66	4l	4l	NOUN
ejpam-847	241	67	f	f	PROPN
ejpam-847	241	68	(	(	PUNCT
ejpam-847	241	69	2l	2l	X
ejpam-847	241	70	x)−	x)−	PROPN
ejpam-847	241	71	1	1	NUM
ejpam-847	241	72	4m+l	4m+l	NUM
ejpam-847	241	73	f	f	NOUN
ejpam-847	241	74	(	(	PUNCT
ejpam-847	241	75	2m+l	2m+l	NUM
ejpam-847	241	76	x	x	NOUN
ejpam-847	241	77	)	)	PUNCT
ejpam-847	241	78	(	(	PUNCT
ejpam-847	241	79	t	t	PROPN
ejpam-847	241	80	)	)	PUNCT
ejpam-847	241	81	≥	≥	NOUN
ejpam-847	242	1	t	t	PROPN
ejpam-847	242	2	m	m	NOUN
ejpam-847	242	3	k=1ρ(2k+l−1	k=1ρ(2k+l−1	PROPN
ejpam-847	242	4	x	x	SYM
ejpam-847	242	5	,	,	PUNCT
ejpam-847	242	6	−2k+l−1	−2k+l−1	X
ejpam-847	242	7	x	x	SYM
ejpam-847	242	8	,	,	PUNCT
ejpam-847	242	9	0,	0,	NUM
ejpam-847	242	10	...	...	PUNCT
ejpam-847	242	11	,0	,0	PUNCT
ejpam-847	242	12	)	)	PUNCT
ejpam-847	242	13	�	�	PROPN
ejpam-847	242	14	2k+2l−1	2k+2l−1	NUM
ejpam-847	242	15	t	t	PROPN
ejpam-847	242	16	�	�	PROPN
ejpam-847	242	17	(	(	PUNCT
ejpam-847	242	18	18	18	NUM
ejpam-847	242	19	)	)	PUNCT
ejpam-847	242	20	for	for	ADP
ejpam-847	242	21	all	all	PRON
ejpam-847	242	22	x	x	SYM
ejpam-847	242	23	∈	∈	PROPN
ejpam-847	242	24	x	x	X
ejpam-847	242	25	,	,	PUNCT
ejpam-847	242	26	all	all	PRON
ejpam-847	242	27	t	t	X
ejpam-847	242	28	>	>	X
ejpam-847	242	29	0	0	PUNCT
ejpam-847	243	1	and	and	CCONJ
ejpam-847	243	2	all	all	DET
ejpam-847	243	3	l	l	NOUN
ejpam-847	243	4	=	=	PUNCT
ejpam-847	243	5	0,1,2	0,1,2	NUM
ejpam-847	243	6	,	,	PUNCT
ejpam-847	243	7	.	.	PUNCT
ejpam-847	243	8	.	.	PUNCT
ejpam-847	244	1	..	..	PUNCT
ejpam-847	245	1	d.	d.	PROPN
ejpam-847	245	2	shin	shin	PROPN
ejpam-847	245	3	,	,	PUNCT
ejpam-847	245	4	j.	j.	PROPN
ejpam-847	245	5	lee	lee	PROPN
ejpam-847	245	6	,	,	PUNCT
ejpam-847	245	7	c.	c.	PROPN
ejpam-847	245	8	park	park	PROPN
ejpam-847	245	9	/	/	SYM
ejpam-847	245	10	eur	eur	PROPN
ejpam-847	245	11	.	.	PUNCT
ejpam-847	246	1	j.	j.	PROPN
ejpam-847	246	2	pure	pure	PROPN
ejpam-847	246	3	appl	appl	PROPN
ejpam-847	246	4	.	.	PROPN
ejpam-847	246	5	math	math	PROPN
ejpam-847	246	6	,	,	PUNCT
ejpam-847	246	7	5	5	NUM
ejpam-847	246	8	(	(	PUNCT
ejpam-847	246	9	2012	2012	NUM
ejpam-847	246	10	)	)	PUNCT
ejpam-847	246	11	,	,	PUNCT
ejpam-847	246	12	540	540	NUM
ejpam-847	246	13	-	-	SYM
ejpam-847	246	14	553	553	NUM
ejpam-847	246	15	550	550	NUM
ejpam-847	246	16	since	since	SCONJ
ejpam-847	246	17	the	the	DET
ejpam-847	246	18	right	right	ADJ
ejpam-847	246	19	hand	hand	NOUN
ejpam-847	246	20	side	side	NOUN
ejpam-847	246	21	of	of	ADP
ejpam-847	246	22	the	the	DET
ejpam-847	246	23	inequality	inequality	NOUN
ejpam-847	246	24	(	(	PUNCT
ejpam-847	246	25	18	18	NUM
ejpam-847	246	26	)	)	PUNCT
ejpam-847	246	27	tends	tend	VERB
ejpam-847	246	28	to	to	ADP
ejpam-847	246	29	1	1	NUM
ejpam-847	246	30	as	as	ADP
ejpam-847	246	31	m→∞	m→∞	NUM
ejpam-847	246	32	by	by	ADP
ejpam-847	246	33	(	(	PUNCT
ejpam-847	246	34	15	15	NUM
ejpam-847	246	35	)	)	PUNCT
ejpam-847	246	36	,	,	PUNCT
ejpam-847	246	37	the	the	DET
ejpam-847	246	38	sequence	sequence	NOUN
ejpam-847	246	39	{	{	PUNCT
ejpam-847	246	40	1	1	NUM
ejpam-847	246	41	4	4	NUM
ejpam-847	246	42	m	m	NOUN
ejpam-847	246	43	f	f	NOUN
ejpam-847	246	44	(	(	PUNCT
ejpam-847	246	45	2mx	2mx	ADJ
ejpam-847	246	46	)	)	PUNCT
ejpam-847	246	47	}	}	PUNCT
ejpam-847	246	48	is	be	AUX
ejpam-847	246	49	a	a	DET
ejpam-847	246	50	cauchy	cauchy	ADJ
ejpam-847	246	51	sequence	sequence	NOUN
ejpam-847	246	52	.	.	PUNCT
ejpam-847	247	1	thus	thus	ADV
ejpam-847	247	2	we	we	PRON
ejpam-847	247	3	define	define	VERB
ejpam-847	247	4	q(x	q(x	NOUN
ejpam-847	247	5	)	)	PUNCT
ejpam-847	247	6	:	:	PUNCT
ejpam-847	247	7	=	=	SYM
ejpam-847	247	8	limm→∞	limm→∞	NOUN
ejpam-847	247	9	1	1	NUM
ejpam-847	247	10	4	4	NUM
ejpam-847	247	11	m	m	NOUN
ejpam-847	247	12	f	f	NOUN
ejpam-847	247	13	(	(	PUNCT
ejpam-847	247	14	2mx	2mx	ADJ
ejpam-847	247	15	)	)	PUNCT
ejpam-847	247	16	for	for	ADP
ejpam-847	247	17	all	all	DET
ejpam-847	247	18	x	x	SYM
ejpam-847	247	19	∈	∈	PROPN
ejpam-847	247	20	x	x	X
ejpam-847	247	21	,	,	PUNCT
ejpam-847	247	22	which	which	PRON
ejpam-847	247	23	is	be	AUX
ejpam-847	247	24	an	an	DET
ejpam-847	247	25	even	even	ADJ
ejpam-847	247	26	mapping	mapping	NOUN
ejpam-847	247	27	.	.	PUNCT
ejpam-847	248	1	now	now	ADV
ejpam-847	248	2	we	we	PRON
ejpam-847	248	3	show	show	VERB
ejpam-847	248	4	that	that	SCONJ
ejpam-847	248	5	q	q	NOUN
ejpam-847	248	6	is	be	AUX
ejpam-847	248	7	a	a	DET
ejpam-847	248	8	quadratic	quadratic	ADJ
ejpam-847	248	9	mapping	mapping	NOUN
ejpam-847	248	10	.	.	PUNCT
ejpam-847	249	1	by	by	ADP
ejpam-847	249	2	(	(	PUNCT
ejpam-847	249	3	2	2	NUM
ejpam-847	249	4	)	)	PUNCT
ejpam-847	249	5	,	,	PUNCT
ejpam-847	249	6	we	we	PRON
ejpam-847	249	7	get	get	VERB
ejpam-847	249	8	µ	µ	PRON
ejpam-847	249	9	1	1	NUM
ejpam-847	249	10	4	4	NUM
ejpam-847	249	11	m	m	NOUN
ejpam-847	249	12	(	(	PUNCT
ejpam-847	249	13	f	f	X
ejpam-847	249	14	(	(	PUNCT
ejpam-847	249	15	2	2	NUM
ejpam-847	249	16	m(x−y))+	m(x−y))+	PROPN
ejpam-847	249	17	f	f	PROPN
ejpam-847	249	18	(	(	PUNCT
ejpam-847	249	19	2m(2x+y))+	2m(2x+y))+	NOUN
ejpam-847	249	20	f	f	PROPN
ejpam-847	249	21	(	(	PUNCT
ejpam-847	249	22	2m(x+2y))−3	2m(x+2y))−3	NUM
ejpam-847	249	23	f	f	X
ejpam-847	249	24	(	(	PUNCT
ejpam-847	249	25	2m(x+y))−3	2m(x+y))−3	NOUN
ejpam-847	249	26	f	f	X
ejpam-847	249	27	(	(	PUNCT
ejpam-847	249	28	2	2	NUM
ejpam-847	249	29	m	m	NOUN
ejpam-847	249	30	x)−3	x)−3	ADJ
ejpam-847	249	31	f	f	X
ejpam-847	249	32	(	(	PUNCT
ejpam-847	249	33	2	2	NUM
ejpam-847	249	34	m	m	NOUN
ejpam-847	249	35	y))(t	y))(t	NOUN
ejpam-847	249	36	)	)	PUNCT
ejpam-847	250	1	≥ρ(2	≥ρ(2	NOUN
ejpam-847	250	2	m	m	NOUN
ejpam-847	250	3	x	x	SYM
ejpam-847	250	4	,	,	PUNCT
ejpam-847	250	5	2	2	NUM
ejpam-847	250	6	m	m	NOUN
ejpam-847	250	7	y,−2m(x+y),0,	y,−2m(x+y),0,	NOUN
ejpam-847	250	8	...	...	PUNCT
ejpam-847	250	9	,0)(2	,0)(2	PUNCT
ejpam-847	250	10	m+1	m+1	NUM
ejpam-847	250	11	t	t	NOUN
ejpam-847	250	12	)	)	PUNCT
ejpam-847	250	13	.	.	PUNCT
ejpam-847	251	1	taking	take	VERB
ejpam-847	251	2	the	the	DET
ejpam-847	251	3	limit	limit	NOUN
ejpam-847	251	4	as	as	ADP
ejpam-847	251	5	m	m	PROPN
ejpam-847	251	6	→	→	SYM
ejpam-847	251	7	∞	∞	PROPN
ejpam-847	251	8	in	in	ADP
ejpam-847	251	9	the	the	DET
ejpam-847	251	10	above	above	ADJ
ejpam-847	251	11	inequality	inequality	NOUN
ejpam-847	251	12	,	,	PUNCT
ejpam-847	251	13	by	by	ADP
ejpam-847	251	14	(	(	PUNCT
ejpam-847	251	15	16	16	NUM
ejpam-847	251	16	)	)	PUNCT
ejpam-847	251	17	,	,	PUNCT
ejpam-847	251	18	the	the	DET
ejpam-847	251	19	mapping	mapping	NOUN
ejpam-847	251	20	q	q	NOUN
ejpam-847	251	21	is	be	AUX
ejpam-847	251	22	quadratic	quadratic	ADJ
ejpam-847	251	23	.	.	PUNCT
ejpam-847	252	1	moreover	moreover	ADV
ejpam-847	252	2	,	,	PUNCT
ejpam-847	252	3	letting	let	VERB
ejpam-847	252	4	l	l	NOUN
ejpam-847	252	5	=	=	SYM
ejpam-847	252	6	0	0	PUNCT
ejpam-847	252	7	and	and	CCONJ
ejpam-847	252	8	taking	take	VERB
ejpam-847	252	9	the	the	DET
ejpam-847	252	10	limit	limit	NOUN
ejpam-847	252	11	as	as	ADP
ejpam-847	252	12	m→∞	m→∞	NOUN
ejpam-847	252	13	in	in	ADP
ejpam-847	252	14	(	(	PUNCT
ejpam-847	252	15	18	18	NUM
ejpam-847	252	16	)	)	PUNCT
ejpam-847	252	17	,	,	PUNCT
ejpam-847	252	18	we	we	PRON
ejpam-847	252	19	get	get	VERB
ejpam-847	252	20	(	(	PUNCT
ejpam-847	252	21	17	17	NUM
ejpam-847	252	22	)	)	PUNCT
ejpam-847	252	23	.	.	PUNCT
ejpam-847	253	1	the	the	DET
ejpam-847	253	2	rest	rest	NOUN
ejpam-847	253	3	of	of	ADP
ejpam-847	253	4	the	the	DET
ejpam-847	253	5	proof	proof	NOUN
ejpam-847	253	6	is	be	AUX
ejpam-847	253	7	the	the	DET
ejpam-847	253	8	same	same	ADJ
ejpam-847	253	9	as	as	ADP
ejpam-847	253	10	in	in	ADP
ejpam-847	253	11	the	the	DET
ejpam-847	253	12	proof	proof	NOUN
ejpam-847	253	13	of	of	ADP
ejpam-847	253	14	theorem	theorem	ADJ
ejpam-847	253	15	3	3	NUM
ejpam-847	253	16	.	.	PUNCT
ejpam-847	253	17	corollary	corollary	ADJ
ejpam-847	253	18	4	4	NUM
ejpam-847	253	19	.	.	PUNCT
ejpam-847	254	1	let	let	VERB
ejpam-847	254	2	θ	θ	PROPN
ejpam-847	254	3	≥	≥	X
ejpam-847	254	4	0	0	NUM
ejpam-847	254	5	and	and	CCONJ
ejpam-847	254	6	let	let	VERB
ejpam-847	254	7	p	p	PRON
ejpam-847	254	8	be	be	AUX
ejpam-847	254	9	a	a	DET
ejpam-847	254	10	constant	constant	ADJ
ejpam-847	254	11	with	with	ADP
ejpam-847	254	12	0	0	NUM
ejpam-847	254	13	<	<	X
ejpam-847	254	14	p	p	X
ejpam-847	254	15	<	<	X
ejpam-847	254	16	2	2	NUM
ejpam-847	254	17	.	.	PUNCT
ejpam-847	255	1	for	for	ADP
ejpam-847	255	2	a	a	DET
ejpam-847	255	3	normed	normed	ADJ
ejpam-847	255	4	vector	vector	NOUN
ejpam-847	255	5	space	space	NOUN
ejpam-847	255	6	x	x	PUNCT
ejpam-847	255	7	and	and	CCONJ
ejpam-847	255	8	complete	complete	ADJ
ejpam-847	255	9	rn	rn	NOUN
ejpam-847	255	10	-	-	NOUN
ejpam-847	255	11	space	space	NOUN
ejpam-847	255	12	y	y	PROPN
ejpam-847	255	13	,	,	PUNCT
ejpam-847	255	14	let	let	VERB
ejpam-847	255	15	f	f	PRON
ejpam-847	255	16	:	:	PUNCT
ejpam-847	255	17	x	x	X
ejpam-847	255	18	→	→	SYM
ejpam-847	255	19	y	y	X
ejpam-847	255	20	be	be	AUX
ejpam-847	255	21	an	an	DET
ejpam-847	255	22	even	even	ADV
ejpam-847	255	23	mapping	mapping	NOUN
ejpam-847	255	24	satisfying	satisfy	VERB
ejpam-847	255	25	µd	µd	DET
ejpam-847	255	26	f	f	PROPN
ejpam-847	255	27	(	(	PUNCT
ejpam-847	255	28	x1	x1	INTJ
ejpam-847	255	29	,	,	PUNCT
ejpam-847	255	30	x2,	x2,	PROPN
ejpam-847	255	31	...	...	PUNCT
ejpam-847	255	32	,xn	,xn	PUNCT
ejpam-847	255	33	)	)	PUNCT
ejpam-847	255	34	(	(	PUNCT
ejpam-847	255	35	t	t	PROPN
ejpam-847	255	36	)	)	PUNCT
ejpam-847	255	37	≥	≥	PROPN
ejpam-847	255	38	t	t	PROPN
ejpam-847	255	39	t	t	NOUN
ejpam-847	256	1	+	+	CCONJ
ejpam-847	256	2	θ	θ	PROPN
ejpam-847	256	3	∑n	∑n	PROPN
ejpam-847	256	4	i=1	i=1	PROPN
ejpam-847	256	5	||x	||x	PROPN
ejpam-847	256	6	i||p	i||p	PROPN
ejpam-847	256	7	for	for	ADP
ejpam-847	256	8	all	all	PRON
ejpam-847	256	9	(	(	PUNCT
ejpam-847	256	10	x1	x1	PROPN
ejpam-847	256	11	,	,	PUNCT
ejpam-847	256	12	x2	x2	PROPN
ejpam-847	256	13	,	,	PUNCT
ejpam-847	256	14	.	.	PUNCT
ejpam-847	256	15	.	.	PUNCT
ejpam-847	256	16	.	.	PUNCT
ejpam-847	257	1	,	,	PUNCT
ejpam-847	257	2	xn	xn	X
ejpam-847	257	3	)	)	PUNCT
ejpam-847	257	4	∈	∈	PROPN
ejpam-847	257	5	x	x	PUNCT
ejpam-847	257	6	with	with	ADP
ejpam-847	257	7	∑n	∑n	PROPN
ejpam-847	257	8	i=1	i=1	NOUN
ejpam-847	257	9	x	x	PUNCT
ejpam-847	258	1	i	i	NOUN
ejpam-847	258	2	=	=	NOUN
ejpam-847	258	3	0	0	NUM
ejpam-847	258	4	and	and	CCONJ
ejpam-847	258	5	all	all	DET
ejpam-847	258	6	t	t	NOUN
ejpam-847	258	7	>	>	X
ejpam-847	258	8	0	0	X
ejpam-847	258	9	.	.	PUNCT
ejpam-847	259	1	if	if	SCONJ
ejpam-847	259	2	t∞k=1	t∞k=1	PROPN
ejpam-847	259	3	�	�	PROPN
ejpam-847	259	4	2k+2l−2	2k+2l−2	NUM
ejpam-847	259	5	t	t	NOUN
ejpam-847	259	6	2k+2l−2	2k+2l−2	NUM
ejpam-847	259	7	t	t	NOUN
ejpam-847	259	8	+	+	CCONJ
ejpam-847	259	9	2(k+l)pθ	2(k+l)pθ	NUM
ejpam-847	259	10	||x	||x	ADJ
ejpam-847	259	11	||p	||p	NOUN
ejpam-847	259	12	�	�	NOUN
ejpam-847	259	13	=	=	NOUN
ejpam-847	259	14	1	1	NUM
ejpam-847	259	15	for	for	ADP
ejpam-847	259	16	all	all	DET
ejpam-847	259	17	x	x	SYM
ejpam-847	259	18	∈	∈	PROPN
ejpam-847	259	19	x	x	X
ejpam-847	259	20	,	,	PUNCT
ejpam-847	259	21	all	all	PRON
ejpam-847	259	22	t	t	X
ejpam-847	259	23	>	>	X
ejpam-847	259	24	0	0	PUNCT
ejpam-847	259	25	and	and	CCONJ
ejpam-847	259	26	all	all	DET
ejpam-847	259	27	l	l	NOUN
ejpam-847	259	28	=	=	PUNCT
ejpam-847	259	29	0,1,2	0,1,2	NUM
ejpam-847	259	30	,	,	PUNCT
ejpam-847	259	31	.	.	PUNCT
ejpam-847	259	32	.	.	PUNCT
ejpam-847	260	1	.	.	PUNCT
ejpam-847	261	1	,	,	PUNCT
ejpam-847	261	2	then	then	ADV
ejpam-847	261	3	there	there	PRON
ejpam-847	261	4	exists	exist	VERB
ejpam-847	261	5	a	a	DET
ejpam-847	261	6	unique	unique	ADJ
ejpam-847	261	7	quadratic	quadratic	ADJ
ejpam-847	261	8	mapping	mapping	NOUN
ejpam-847	261	9	q	q	NOUN
ejpam-847	261	10	:	:	PUNCT
ejpam-847	261	11	x	x	X
ejpam-847	261	12	→	→	SYM
ejpam-847	261	13	y	y	PROPN
ejpam-847	261	14	such	such	ADJ
ejpam-847	261	15	that	that	SCONJ
ejpam-847	261	16	µ	µ	PROPN
ejpam-847	261	17	f	f	X
ejpam-847	261	18	(	(	PUNCT
ejpam-847	261	19	x)−q(x)(t)≥	x)−q(x)(t)≥	PROPN
ejpam-847	261	20	lim	lim	PROPN
ejpam-847	261	21	m→∞	m→∞	PROPN
ejpam-847	261	22	t	t	INTJ
ejpam-847	261	23	m	m	VERB
ejpam-847	261	24	k=1	k=1	PROPN
ejpam-847	261	25	�	�	PROPN
ejpam-847	261	26	2k−2	2k−2	PROPN
ejpam-847	261	27	t	t	PROPN
ejpam-847	261	28	2k−2	2k−2	PROPN
ejpam-847	261	29	t	t	PROPN
ejpam-847	261	30	+	+	CCONJ
ejpam-847	261	31	2kpθ	2kpθ	NUM
ejpam-847	261	32	||x	||x	NOUN
ejpam-847	261	33	||p	||p	NOUN
ejpam-847	261	34	�	�	NOUN
ejpam-847	261	35	for	for	ADP
ejpam-847	261	36	all	all	DET
ejpam-847	261	37	x	x	SYM
ejpam-847	261	38	∈	∈	ADJ
ejpam-847	261	39	x	x	X
ejpam-847	261	40	and	and	CCONJ
ejpam-847	261	41	all	all	DET
ejpam-847	261	42	t	t	NOUN
ejpam-847	261	43	>	>	X
ejpam-847	261	44	0	0	X
ejpam-847	261	45	.	.	PUNCT
ejpam-847	262	1	proof	proof	NOUN
ejpam-847	262	2	.	.	PUNCT
ejpam-847	263	1	if	if	SCONJ
ejpam-847	263	2	we	we	PRON
ejpam-847	263	3	define	define	VERB
ejpam-847	263	4	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	263	5	...	...	PUNCT
ejpam-847	263	6	,xn	,xn	PUNCT
ejpam-847	263	7	)	)	PUNCT
ejpam-847	263	8	(	(	PUNCT
ejpam-847	263	9	t	t	NOUN
ejpam-847	263	10	)	)	PUNCT
ejpam-847	263	11	=	=	SYM
ejpam-847	263	12	t	t	PROPN
ejpam-847	263	13	t	t	NOUN
ejpam-847	264	1	+	+	CCONJ
ejpam-847	264	2	θ	θ	PROPN
ejpam-847	264	3	∑n	∑n	PROPN
ejpam-847	264	4	i=1	i=1	PROPN
ejpam-847	264	5	||x	||x	PROPN
ejpam-847	264	6	i||p	i||p	PROPN
ejpam-847	265	1	and	and	CCONJ
ejpam-847	265	2	apply	apply	VERB
ejpam-847	265	3	theorem	theorem	NOUN
ejpam-847	265	4	4	4	NUM
ejpam-847	265	5	,	,	PUNCT
ejpam-847	265	6	then	then	ADV
ejpam-847	265	7	we	we	PRON
ejpam-847	265	8	get	get	VERB
ejpam-847	265	9	the	the	DET
ejpam-847	265	10	desired	desire	VERB
ejpam-847	265	11	result	result	NOUN
ejpam-847	265	12	.	.	PUNCT
ejpam-847	266	1	4	4	X
ejpam-847	266	2	.	.	X
ejpam-847	266	3	hyers	hyer	NOUN
ejpam-847	266	4	-	-	PUNCT
ejpam-847	266	5	ulam	ulam	PROPN
ejpam-847	266	6	stability	stability	NOUN
ejpam-847	266	7	of	of	ADP
ejpam-847	266	8	the	the	DET
ejpam-847	266	9	functional	functional	ADJ
ejpam-847	266	10	equation	equation	NOUN
ejpam-847	266	11	(	(	PUNCT
ejpam-847	266	12	1	1	X
ejpam-847	266	13	)	)	PUNCT
ejpam-847	266	14	we	we	PRON
ejpam-847	266	15	note	note	VERB
ejpam-847	266	16	that	that	SCONJ
ejpam-847	266	17	if	if	SCONJ
ejpam-847	266	18	a	a	DET
ejpam-847	266	19	mapping	mapping	NOUN
ejpam-847	266	20	f	f	X
ejpam-847	266	21	:	:	PUNCT
ejpam-847	266	22	x	x	X
ejpam-847	266	23	→	→	SYM
ejpam-847	266	24	y	y	PROPN
ejpam-847	266	25	satisfies	satisfy	VERB
ejpam-847	266	26	the	the	DET
ejpam-847	266	27	functional	functional	ADJ
ejpam-847	266	28	equation	equation	NOUN
ejpam-847	266	29	(	(	PUNCT
ejpam-847	266	30	1	1	NUM
ejpam-847	266	31	)	)	PUNCT
ejpam-847	266	32	,	,	PUNCT
ejpam-847	266	33	then	then	ADV
ejpam-847	266	34	the	the	DET
ejpam-847	266	35	mapping	mapping	NOUN
ejpam-847	266	36	f	f	X
ejpam-847	266	37	is	be	AUX
ejpam-847	266	38	realized	realize	VERB
ejpam-847	266	39	as	as	ADP
ejpam-847	266	40	the	the	DET
ejpam-847	266	41	sum	sum	NOUN
ejpam-847	266	42	of	of	ADP
ejpam-847	266	43	an	an	DET
ejpam-847	266	44	additive	additive	ADJ
ejpam-847	266	45	mapping	mapping	NOUN
ejpam-847	266	46	and	and	CCONJ
ejpam-847	266	47	a	a	DET
ejpam-847	266	48	quadratic	quadratic	ADJ
ejpam-847	266	49	mapping	mapping	NOUN
ejpam-847	266	50	[	[	X
ejpam-847	266	51	see	see	ADJ
ejpam-847	266	52	2	2	NUM
ejpam-847	266	53	,	,	PUNCT
ejpam-847	266	54	lemma	lemma	PROPN
ejpam-847	266	55	2.1	2.1	NUM
ejpam-847	266	56	]	]	PUNCT
ejpam-847	266	57	.	.	PUNCT
ejpam-847	267	1	here	here	ADV
ejpam-847	267	2	,	,	PUNCT
ejpam-847	267	3	we	we	PRON
ejpam-847	267	4	let	let	VERB
ejpam-847	267	5	g(x	g(x	NOUN
ejpam-847	267	6	)	)	PUNCT
ejpam-847	267	7	:	:	PUNCT
ejpam-847	268	1	=	=	SYM
ejpam-847	268	2	1	1	NUM
ejpam-847	268	3	2	2	NUM
ejpam-847	268	4	(	(	PUNCT
ejpam-847	268	5	f	f	PROPN
ejpam-847	268	6	(	(	PUNCT
ejpam-847	268	7	x)−	x)−	PROPN
ejpam-847	268	8	f	f	PROPN
ejpam-847	268	9	(	(	PUNCT
ejpam-847	268	10	−x	−x	NOUN
ejpam-847	268	11	)	)	PUNCT
ejpam-847	268	12	)	)	PUNCT
ejpam-847	268	13	and	and	CCONJ
ejpam-847	268	14	h(x	h(x	PROPN
ejpam-847	268	15	)	)	PUNCT
ejpam-847	268	16	:	:	PUNCT
ejpam-847	269	1	=	=	SYM
ejpam-847	269	2	1	1	NUM
ejpam-847	269	3	2	2	NUM
ejpam-847	269	4	(	(	PUNCT
ejpam-847	269	5	f	f	X
ejpam-847	269	6	(	(	PUNCT
ejpam-847	269	7	x	x	X
ejpam-847	269	8	)	)	PUNCT
ejpam-847	270	1	+	+	NUM
ejpam-847	270	2	f	f	X
ejpam-847	270	3	(	(	PUNCT
ejpam-847	270	4	−x	−x	NOUN
ejpam-847	270	5	)	)	PUNCT
ejpam-847	270	6	)	)	PUNCT
ejpam-847	270	7	for	for	ADP
ejpam-847	270	8	all	all	DET
ejpam-847	270	9	x	x	SYM
ejpam-847	270	10	∈	∈	NOUN
ejpam-847	270	11	x	x	X
ejpam-847	270	12	.	.	PUNCT
ejpam-847	271	1	then	then	ADV
ejpam-847	271	2	g(x	g(x	NOUN
ejpam-847	271	3	)	)	PUNCT
ejpam-847	271	4	is	be	AUX
ejpam-847	271	5	an	an	DET
ejpam-847	271	6	odd	odd	ADJ
ejpam-847	271	7	mapping	mapping	NOUN
ejpam-847	271	8	and	and	CCONJ
ejpam-847	271	9	h(x	h(x	PROPN
ejpam-847	271	10	)	)	PUNCT
ejpam-847	272	1	is	be	AUX
ejpam-847	272	2	an	an	DET
ejpam-847	272	3	even	even	ADV
ejpam-847	272	4	mapping	map	VERB
ejpam-847	272	5	satisfying	satisfying	ADJ
ejpam-847	272	6	f	f	X
ejpam-847	272	7	(	(	PUNCT
ejpam-847	272	8	x	x	NOUN
ejpam-847	272	9	)	)	PUNCT
ejpam-847	272	10	=	=	SYM
ejpam-847	272	11	g(x)+h(x	g(x)+h(x	PROPN
ejpam-847	272	12	)	)	PUNCT
ejpam-847	272	13	.	.	PUNCT
ejpam-847	273	1	moreover	moreover	ADV
ejpam-847	273	2	,	,	PUNCT
ejpam-847	273	3	we	we	PRON
ejpam-847	273	4	get	get	VERB
ejpam-847	273	5	the	the	DET
ejpam-847	273	6	following	following	NOUN
ejpam-847	273	7	:	:	PUNCT
ejpam-847	273	8	dg(x1	dg(x1	NOUN
ejpam-847	273	9	,	,	PUNCT
ejpam-847	273	10	x2	x2	NOUN
ejpam-847	273	11	,	,	PUNCT
ejpam-847	273	12	.	.	PUNCT
ejpam-847	273	13	.	.	PUNCT
ejpam-847	274	1	.	.	PUNCT
ejpam-847	275	1	,	,	PUNCT
ejpam-847	275	2	xn	xn	X
ejpam-847	275	3	)	)	PUNCT
ejpam-847	275	4	=	=	SYM
ejpam-847	275	5	1	1	NUM
ejpam-847	275	6	2	2	NUM
ejpam-847	275	7	{	{	PUNCT
ejpam-847	275	8	d	d	X
ejpam-847	275	9	f	f	X
ejpam-847	275	10	(	(	PUNCT
ejpam-847	275	11	x1	x1	PROPN
ejpam-847	275	12	,	,	PUNCT
ejpam-847	275	13	x2	x2	PROPN
ejpam-847	275	14	,	,	PUNCT
ejpam-847	275	15	.	.	PUNCT
ejpam-847	275	16	.	.	PUNCT
ejpam-847	275	17	.	.	PUNCT
ejpam-847	276	1	,	,	PUNCT
ejpam-847	276	2	xn)−	xn)−	PUNCT
ejpam-847	277	1	d	d	X
ejpam-847	277	2	f	f	X
ejpam-847	277	3	(	(	PUNCT
ejpam-847	277	4	−x1,−x2	−x1,−x2	NOUN
ejpam-847	277	5	,	,	PUNCT
ejpam-847	277	6	.	.	PUNCT
ejpam-847	277	7	.	.	PUNCT
ejpam-847	277	8	.	.	PUNCT
ejpam-847	278	1	,	,	PUNCT
ejpam-847	278	2	−xn	−xn	PROPN
ejpam-847	278	3	)	)	PUNCT
ejpam-847	278	4	}	}	PUNCT
ejpam-847	278	5	d.	d.	PROPN
ejpam-847	278	6	shin	shin	PROPN
ejpam-847	278	7	,	,	PUNCT
ejpam-847	278	8	j.	j.	PROPN
ejpam-847	278	9	lee	lee	PROPN
ejpam-847	278	10	,	,	PUNCT
ejpam-847	278	11	c.	c.	PROPN
ejpam-847	278	12	park	park	PROPN
ejpam-847	278	13	/	/	SYM
ejpam-847	278	14	eur	eur	PROPN
ejpam-847	278	15	.	.	PUNCT
ejpam-847	279	1	j.	j.	PROPN
ejpam-847	279	2	pure	pure	PROPN
ejpam-847	279	3	appl	appl	PROPN
ejpam-847	279	4	.	.	PROPN
ejpam-847	279	5	math	math	PROPN
ejpam-847	279	6	,	,	PUNCT
ejpam-847	279	7	5	5	NUM
ejpam-847	279	8	(	(	PUNCT
ejpam-847	279	9	2012	2012	NUM
ejpam-847	279	10	)	)	PUNCT
ejpam-847	279	11	,	,	PUNCT
ejpam-847	279	12	540	540	NUM
ejpam-847	279	13	-	-	SYM
ejpam-847	279	14	553	553	NUM
ejpam-847	279	15	551	551	NUM
ejpam-847	279	16	dh(x1	dh(x1	NOUN
ejpam-847	279	17	,	,	PUNCT
ejpam-847	279	18	x2	x2	PROPN
ejpam-847	279	19	,	,	PUNCT
ejpam-847	279	20	.	.	PUNCT
ejpam-847	279	21	.	.	PUNCT
ejpam-847	279	22	.	.	PUNCT
ejpam-847	280	1	,	,	PUNCT
ejpam-847	280	2	xn	xn	X
ejpam-847	280	3	)	)	PUNCT
ejpam-847	280	4	=	=	SYM
ejpam-847	280	5	1	1	NUM
ejpam-847	280	6	2	2	NUM
ejpam-847	280	7	{	{	PUNCT
ejpam-847	280	8	d	d	X
ejpam-847	280	9	f	f	X
ejpam-847	280	10	(	(	PUNCT
ejpam-847	280	11	x1	x1	PROPN
ejpam-847	280	12	,	,	PUNCT
ejpam-847	280	13	x2	x2	PROPN
ejpam-847	280	14	,	,	PUNCT
ejpam-847	280	15	.	.	PUNCT
ejpam-847	280	16	.	.	PUNCT
ejpam-847	280	17	.	.	PUNCT
ejpam-847	281	1	,	,	PUNCT
ejpam-847	281	2	xn	xn	X
ejpam-847	281	3	)	)	PUNCT
ejpam-847	282	1	+	+	CCONJ
ejpam-847	283	1	d	d	X
ejpam-847	283	2	f	f	X
ejpam-847	283	3	(	(	PUNCT
ejpam-847	283	4	−x1,−x2	−x1,−x2	NOUN
ejpam-847	283	5	,	,	PUNCT
ejpam-847	283	6	.	.	PUNCT
ejpam-847	283	7	.	.	PUNCT
ejpam-847	283	8	.	.	PUNCT
ejpam-847	284	1	,	,	PUNCT
ejpam-847	284	2	−xn	−xn	PROPN
ejpam-847	284	3	)	)	PUNCT
ejpam-847	284	4	}	}	PUNCT
ejpam-847	284	5	for	for	ADP
ejpam-847	284	6	all	all	DET
ejpam-847	284	7	x1	x1	PROPN
ejpam-847	284	8	,	,	PUNCT
ejpam-847	284	9	x2	x2	PROPN
ejpam-847	284	10	,	,	PUNCT
ejpam-847	284	11	.	.	PUNCT
ejpam-847	284	12	.	.	PUNCT
ejpam-847	285	1	.	.	PUNCT
ejpam-847	286	1	,	,	PUNCT
ejpam-847	286	2	xn	xn	PUNCT
ejpam-847	286	3	∈	∈	PROPN
ejpam-847	286	4	x	x	X
ejpam-847	286	5	.	.	PUNCT
ejpam-847	287	1	note	note	VERB
ejpam-847	287	2	that	that	SCONJ
ejpam-847	288	1	d	d	PROPN
ejpam-847	288	2	f	f	X
ejpam-847	288	3	(	(	PUNCT
ejpam-847	288	4	x1	x1	PROPN
ejpam-847	288	5	,	,	PUNCT
ejpam-847	288	6	.	.	PUNCT
ejpam-847	288	7	.	.	PUNCT
ejpam-847	288	8	.	.	PUNCT
ejpam-847	289	1	,	,	PUNCT
ejpam-847	289	2	xn	xn	X
ejpam-847	289	3	)	)	PUNCT
ejpam-847	289	4	=	=	SYM
ejpam-847	289	5	0	0	NUM
ejpam-847	289	6	implies	imply	VERB
ejpam-847	289	7	that	that	SCONJ
ejpam-847	289	8	dg(x1	dg(x1	NOUN
ejpam-847	289	9	,	,	PUNCT
ejpam-847	289	10	.	.	PUNCT
ejpam-847	289	11	.	.	PUNCT
ejpam-847	290	1	.	.	PUNCT
ejpam-847	291	1	,	,	PUNCT
ejpam-847	291	2	xn	xn	X
ejpam-847	291	3	)	)	PUNCT
ejpam-847	291	4	=	=	SYM
ejpam-847	291	5	0	0	NUM
ejpam-847	291	6	and	and	CCONJ
ejpam-847	291	7	dh(x1	dh(x1	NOUN
ejpam-847	291	8	,	,	PUNCT
ejpam-847	291	9	.	.	PUNCT
ejpam-847	291	10	.	.	PUNCT
ejpam-847	292	1	.	.	PUNCT
ejpam-847	293	1	,	,	PUNCT
ejpam-847	293	2	xn	xn	X
ejpam-847	293	3	)	)	PUNCT
ejpam-847	294	1	=	=	SYM
ejpam-847	294	2	0	0	X
ejpam-847	294	3	.	.	PUNCT
ejpam-847	294	4	theorem	theorem	NOUN
ejpam-847	294	5	5	5	NUM
ejpam-847	294	6	.	.	PUNCT
ejpam-847	295	1	let	let	VERB
ejpam-847	295	2	f	f	NOUN
ejpam-847	295	3	:	:	PUNCT
ejpam-847	295	4	x	x	X
ejpam-847	295	5	→	→	SYM
ejpam-847	295	6	y	y	X
ejpam-847	295	7	be	be	AUX
ejpam-847	295	8	a	a	DET
ejpam-847	295	9	mapping	mapping	NOUN
ejpam-847	295	10	with	with	ADP
ejpam-847	295	11	f	f	PROPN
ejpam-847	295	12	(	(	PUNCT
ejpam-847	295	13	0	0	NUM
ejpam-847	295	14	)	)	PUNCT
ejpam-847	295	15	=	=	SYM
ejpam-847	295	16	0	0	NUM
ejpam-847	295	17	for	for	ADP
ejpam-847	295	18	which	which	PRON
ejpam-847	295	19	there	there	PRON
ejpam-847	295	20	is	be	VERB
ejpam-847	295	21	a	a	DET
ejpam-847	295	22	ρ	ρ	NOUN
ejpam-847	295	23	:	:	PUNCT
ejpam-847	295	24	x	x	SYM
ejpam-847	295	25	n→	n→	X
ejpam-847	295	26	d+	d+	NOUN
ejpam-847	295	27	such	such	ADJ
ejpam-847	295	28	that	that	SCONJ
ejpam-847	295	29	µd	µd	PRON
ejpam-847	295	30	f	f	PROPN
ejpam-847	295	31	(	(	PUNCT
ejpam-847	295	32	x1,x2,	x1,x2,	PROPN
ejpam-847	295	33	...	...	PUNCT
ejpam-847	295	34	,xn)+d	,xn)+d	PUNCT
ejpam-847	295	35	f	f	PROPN
ejpam-847	295	36	(	(	PUNCT
ejpam-847	295	37	−x1,−x2,	−x1,−x2,	NOUN
ejpam-847	295	38	...	...	PUNCT
ejpam-847	295	39	,−xn	,−xn	PUNCT
ejpam-847	295	40	)	)	PUNCT
ejpam-847	295	41	(	(	PUNCT
ejpam-847	295	42	2	2	NUM
ejpam-847	295	43	t	t	NOUN
ejpam-847	295	44	)	)	PUNCT
ejpam-847	295	45	≥	≥	NOUN
ejpam-847	295	46	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	295	47	...	...	PUNCT
ejpam-847	295	48	,xn	,xn	PUNCT
ejpam-847	295	49	)	)	PUNCT
ejpam-847	295	50	(	(	PUNCT
ejpam-847	295	51	t	t	NOUN
ejpam-847	295	52	)	)	PUNCT
ejpam-847	295	53	(	(	PUNCT
ejpam-847	295	54	19	19	NUM
ejpam-847	295	55	)	)	PUNCT
ejpam-847	295	56	µd	µd	PRON
ejpam-847	295	57	f	f	PROPN
ejpam-847	295	58	(	(	PUNCT
ejpam-847	295	59	x1,x2,	x1,x2,	PROPN
ejpam-847	295	60	...	...	PUNCT
ejpam-847	295	61	,xn)−d	,xn)−d	PUNCT
ejpam-847	295	62	f	f	PROPN
ejpam-847	295	63	(	(	PUNCT
ejpam-847	295	64	−x1,−x2,	−x1,−x2,	NOUN
ejpam-847	295	65	...	...	PUNCT
ejpam-847	295	66	,−xn	,−xn	PUNCT
ejpam-847	295	67	)	)	PUNCT
ejpam-847	295	68	(	(	PUNCT
ejpam-847	295	69	2	2	NUM
ejpam-847	295	70	t	t	NOUN
ejpam-847	295	71	)	)	PUNCT
ejpam-847	295	72	≥	≥	NOUN
ejpam-847	295	73	ρ(x1,x2,	ρ(x1,x2,	NUM
ejpam-847	295	74	...	...	PUNCT
ejpam-847	295	75	,xn	,xn	PUNCT
ejpam-847	295	76	)	)	PUNCT
ejpam-847	295	77	(	(	PUNCT
ejpam-847	295	78	t	t	NOUN
ejpam-847	295	79	)	)	PUNCT
ejpam-847	295	80	(	(	PUNCT
ejpam-847	295	81	20	20	NUM
ejpam-847	295	82	)	)	PUNCT
ejpam-847	295	83	for	for	ADP
ejpam-847	295	84	all	all	PRON
ejpam-847	295	85	(	(	PUNCT
ejpam-847	295	86	x1	x1	PROPN
ejpam-847	295	87	,	,	PUNCT
ejpam-847	295	88	x2	x2	PROPN
ejpam-847	295	89	,	,	PUNCT
ejpam-847	295	90	.	.	PUNCT
ejpam-847	295	91	.	.	PUNCT
ejpam-847	296	1	.	.	PUNCT
ejpam-847	297	1	,	,	PUNCT
ejpam-847	297	2	xn	xn	X
ejpam-847	297	3	)	)	PUNCT
ejpam-847	297	4	∈	∈	PROPN
ejpam-847	297	5	x	x	SYM
ejpam-847	297	6	n	n	NOUN
ejpam-847	297	7	and	and	CCONJ
ejpam-847	297	8	all	all	PRON
ejpam-847	297	9	t	t	NOUN
ejpam-847	297	10	>	>	X
ejpam-847	297	11	0	0	X
ejpam-847	297	12	.	.	PUNCT
ejpam-847	298	1	if	if	SCONJ
ejpam-847	298	2	ρ	ρ	PROPN
ejpam-847	298	3	satisfies	satisfie	NOUN
ejpam-847	298	4	(	(	PUNCT
ejpam-847	298	5	3	3	NUM
ejpam-847	298	6	)	)	PUNCT
ejpam-847	298	7	,	,	PUNCT
ejpam-847	298	8	(	(	PUNCT
ejpam-847	298	9	11	11	NUM
ejpam-847	298	10	)	)	PUNCT
ejpam-847	298	11	and	and	CCONJ
ejpam-847	298	12	(	(	PUNCT
ejpam-847	298	13	12	12	NUM
ejpam-847	298	14	)	)	PUNCT
ejpam-847	298	15	,	,	PUNCT
ejpam-847	298	16	then	then	ADV
ejpam-847	298	17	there	there	PRON
ejpam-847	298	18	exist	exist	VERB
ejpam-847	298	19	an	an	DET
ejpam-847	298	20	additive	additive	ADJ
ejpam-847	298	21	mapping	mapping	NOUN
ejpam-847	298	22	a	a	DET
ejpam-847	298	23	:	:	PUNCT
ejpam-847	298	24	x	x	X
ejpam-847	298	25	→	→	SYM
ejpam-847	298	26	y	y	PROPN
ejpam-847	298	27	and	and	CCONJ
ejpam-847	298	28	a	a	DET
ejpam-847	298	29	quadratic	quadratic	ADJ
ejpam-847	298	30	mapping	mapping	NOUN
ejpam-847	298	31	q	q	NOUN
ejpam-847	298	32	:	:	PUNCT
ejpam-847	298	33	x	x	X
ejpam-847	298	34	→	→	SYM
ejpam-847	298	35	y	y	PROPN
ejpam-847	298	36	such	such	ADJ
ejpam-847	298	37	that	that	SCONJ
ejpam-847	298	38	µ	µ	PROPN
ejpam-847	298	39	f	f	X
ejpam-847	298	40	(	(	PUNCT
ejpam-847	298	41	x)−a(x)−q(x)(2	x)−a(x)−q(x)(2	PROPN
ejpam-847	298	42	t	t	NOUN
ejpam-847	298	43	)	)	PUNCT
ejpam-847	298	44	≥t	≥t	PROPN
ejpam-847	298	45	�	�	PROPN
ejpam-847	298	46	t∞k=1ρ	t∞k=1ρ	PROPN
ejpam-847	298	47	�	�	PROPN
ejpam-847	298	48	x	x	PUNCT
ejpam-847	298	49	2k	2k	NUM
ejpam-847	298	50	,	,	PUNCT
ejpam-847	298	51	x	x	PROPN
ejpam-847	298	52	2k	2k	NUM
ejpam-847	298	53	,	,	PUNCT
ejpam-847	298	54	−	−	PROPN
ejpam-847	298	55	x	x	SYM
ejpam-847	298	56	2k−1	2k−1	NUM
ejpam-847	298	57	,	,	PUNCT
ejpam-847	298	58	0,	0,	NUM
ejpam-847	298	59	...	...	PUNCT
ejpam-847	298	60	,0	,0	PUNCT
ejpam-847	298	61	�	�	PROPN
ejpam-847	298	62	�	�	PROPN
ejpam-847	298	63	nt	not	PART
ejpam-847	298	64	22k−2	22k−2	PROPN
ejpam-847	298	65	�	�	PROPN
ejpam-847	298	66	,	,	PUNCT
ejpam-847	298	67	t∞k=1ρ	t∞k=1ρ	VERB
ejpam-847	298	68	�	�	PROPN
ejpam-847	298	69	x	x	SYM
ejpam-847	298	70	2k	2k	NUM
ejpam-847	298	71	,	,	PUNCT
ejpam-847	298	72	−	−	PROPN
ejpam-847	298	73	x	x	SYM
ejpam-847	298	74	2k	2k	NUM
ejpam-847	298	75	,	,	PUNCT
ejpam-847	298	76	0,	0,	NUM
ejpam-847	298	77	...	...	PUNCT
ejpam-847	298	78	,0	,0	PUNCT
ejpam-847	298	79	�	�	PROPN
ejpam-847	298	80	�	�	PROPN
ejpam-847	298	81	t	t	PROPN
ejpam-847	298	82	23k−3	23k−3	PROPN
ejpam-847	298	83	�	�	PROPN
ejpam-847	298	84	�	�	PROPN
ejpam-847	298	85	for	for	ADP
ejpam-847	298	86	all	all	DET
ejpam-847	298	87	x	x	SYM
ejpam-847	298	88	∈	∈	ADJ
ejpam-847	298	89	x	x	X
ejpam-847	298	90	and	and	CCONJ
ejpam-847	298	91	all	all	DET
ejpam-847	298	92	t	t	NOUN
ejpam-847	298	93	>	>	X
ejpam-847	298	94	0	0	X
ejpam-847	298	95	.	.	PUNCT
ejpam-847	299	1	proof	proof	NOUN
ejpam-847	299	2	.	.	PUNCT
ejpam-847	300	1	consider	consider	VERB
ejpam-847	300	2	an	an	DET
ejpam-847	300	3	odd	odd	ADJ
ejpam-847	300	4	mapping	mapping	NOUN
ejpam-847	300	5	g(x	g(x	NOUN
ejpam-847	300	6	)	)	PUNCT
ejpam-847	300	7	:	:	PUNCT
ejpam-847	301	1	=	=	SYM
ejpam-847	301	2	1	1	NUM
ejpam-847	301	3	2	2	NUM
ejpam-847	301	4	(	(	PUNCT
ejpam-847	301	5	f	f	PROPN
ejpam-847	301	6	(	(	PUNCT
ejpam-847	301	7	x)−	x)−	PROPN
ejpam-847	301	8	f	f	PROPN
ejpam-847	301	9	(	(	PUNCT
ejpam-847	301	10	−x	−x	NOUN
ejpam-847	301	11	)	)	PUNCT
ejpam-847	301	12	)	)	PUNCT
ejpam-847	301	13	and	and	CCONJ
ejpam-847	301	14	an	an	DET
ejpam-847	301	15	even	even	ADV
ejpam-847	301	16	mapping	map	VERB
ejpam-847	301	17	h(x	h(x	PROPN
ejpam-847	301	18	)	)	PUNCT
ejpam-847	301	19	:	:	PUNCT
ejpam-847	302	1	=	=	SYM
ejpam-847	302	2	1	1	NUM
ejpam-847	302	3	2	2	NUM
ejpam-847	302	4	(	(	PUNCT
ejpam-847	302	5	f	f	X
ejpam-847	302	6	(	(	PUNCT
ejpam-847	302	7	x	x	X
ejpam-847	302	8	)	)	PUNCT
ejpam-847	303	1	+	+	NUM
ejpam-847	303	2	f	f	X
ejpam-847	303	3	(	(	PUNCT
ejpam-847	303	4	−x	−x	NOUN
ejpam-847	303	5	)	)	PUNCT
ejpam-847	303	6	)	)	PUNCT
ejpam-847	303	7	for	for	ADP
ejpam-847	303	8	all	all	DET
ejpam-847	303	9	x	x	SYM
ejpam-847	303	10	∈	∈	PROPN
ejpam-847	303	11	x	x	PUNCT
ejpam-847	303	12	with	with	ADP
ejpam-847	303	13	f	f	PROPN
ejpam-847	303	14	(	(	PUNCT
ejpam-847	303	15	x	x	NOUN
ejpam-847	303	16	)	)	PUNCT
ejpam-847	303	17	=	=	SYM
ejpam-847	303	18	g(x	g(x	NOUN
ejpam-847	303	19	)	)	PUNCT
ejpam-847	304	1	+	+	CCONJ
ejpam-847	304	2	h(x	h(x	PROPN
ejpam-847	304	3	)	)	PUNCT
ejpam-847	304	4	.	.	PUNCT
ejpam-847	305	1	by	by	ADP
ejpam-847	305	2	theorem	theorem	NOUN
ejpam-847	305	3	1	1	NUM
ejpam-847	305	4	,	,	PUNCT
ejpam-847	305	5	there	there	PRON
ejpam-847	305	6	exists	exist	VERB
ejpam-847	305	7	a	a	DET
ejpam-847	305	8	unique	unique	ADJ
ejpam-847	305	9	additive	additive	NOUN
ejpam-847	305	10	mapping	mapping	NOUN
ejpam-847	305	11	a	a	PRON
ejpam-847	305	12	:	:	PUNCT
ejpam-847	305	13	x	x	X
ejpam-847	305	14	→	→	SYM
ejpam-847	305	15	y	y	NUM
ejpam-847	305	16	such	such	ADJ
ejpam-847	305	17	that	that	DET
ejpam-847	305	18	µg(x)−a(x)(t	µg(x)−a(x)(t	PROPN
ejpam-847	305	19	)	)	PUNCT
ejpam-847	305	20	≥	≥	NOUN
ejpam-847	305	21	t∞k=1ρ	t∞k=1ρ	VERB
ejpam-847	305	22	�	�	PROPN
ejpam-847	305	23	x	x	SYM
ejpam-847	305	24	2k	2k	NUM
ejpam-847	305	25	,	,	PUNCT
ejpam-847	305	26	x	x	PROPN
ejpam-847	305	27	2k	2k	NUM
ejpam-847	305	28	,	,	PUNCT
ejpam-847	305	29	−	−	PROPN
ejpam-847	305	30	x	x	SYM
ejpam-847	305	31	2k−1	2k−1	NUM
ejpam-847	305	32	,	,	PUNCT
ejpam-847	305	33	0,	0,	NUM
ejpam-847	305	34	...	...	PUNCT
ejpam-847	305	35	,0	,0	PUNCT
ejpam-847	305	36	�	�	PROPN
ejpam-847	305	37	�	�	PROPN
ejpam-847	305	38	nt	not	PART
ejpam-847	305	39	22k−3	22k−3	PROPN
ejpam-847	305	40	�	�	PROPN
ejpam-847	305	41	for	for	ADP
ejpam-847	305	42	all	all	DET
ejpam-847	305	43	x	x	SYM
ejpam-847	305	44	∈	∈	ADJ
ejpam-847	305	45	x	x	X
ejpam-847	305	46	and	and	CCONJ
ejpam-847	305	47	all	all	DET
ejpam-847	305	48	t	t	NOUN
ejpam-847	305	49	>	>	X
ejpam-847	305	50	0	0	X
ejpam-847	305	51	.	.	PUNCT
ejpam-847	305	52	and	and	CCONJ
ejpam-847	305	53	by	by	ADP
ejpam-847	305	54	theorem	theorem	NOUN
ejpam-847	305	55	3	3	NUM
ejpam-847	305	56	,	,	PUNCT
ejpam-847	305	57	there	there	PRON
ejpam-847	305	58	exists	exist	VERB
ejpam-847	305	59	a	a	DET
ejpam-847	305	60	unique	unique	ADJ
ejpam-847	305	61	quadratic	quadratic	ADJ
ejpam-847	305	62	mapping	mapping	NOUN
ejpam-847	305	63	q	q	NOUN
ejpam-847	305	64	:	:	PUNCT
ejpam-847	305	65	x	x	X
ejpam-847	305	66	→	→	SYM
ejpam-847	305	67	y	y	NUM
ejpam-847	305	68	such	such	ADJ
ejpam-847	305	69	that	that	SCONJ
ejpam-847	305	70	µh(x)−q(x)(t	µh(x)−q(x)(t	PROPN
ejpam-847	305	71	)	)	PUNCT
ejpam-847	305	72	≥	≥	NOUN
ejpam-847	305	73	t∞k=1ρ	t∞k=1ρ	VERB
ejpam-847	305	74	�	�	PROPN
ejpam-847	305	75	x	x	SYM
ejpam-847	305	76	2k	2k	NUM
ejpam-847	305	77	,	,	PUNCT
ejpam-847	305	78	−	−	PROPN
ejpam-847	305	79	x	x	SYM
ejpam-847	305	80	2k	2k	NUM
ejpam-847	305	81	,	,	PUNCT
ejpam-847	305	82	0,	0,	NUM
ejpam-847	305	83	...	...	PUNCT
ejpam-847	305	84	,0	,0	PUNCT
ejpam-847	305	85	�	�	PROPN
ejpam-847	305	86	�	�	PROPN
ejpam-847	305	87	t	t	PROPN
ejpam-847	305	88	23k−3	23k−3	PROPN
ejpam-847	305	89	�	�	PROPN
ejpam-847	305	90	for	for	ADP
ejpam-847	305	91	all	all	DET
ejpam-847	305	92	x	x	SYM
ejpam-847	305	93	∈	∈	ADJ
ejpam-847	305	94	x	x	X
ejpam-847	305	95	and	and	CCONJ
ejpam-847	305	96	all	all	DET
ejpam-847	305	97	t	t	PROPN
ejpam-847	305	98	>	>	X
ejpam-847	305	99	0	0	X
ejpam-847	305	100	.	.	PUNCT
ejpam-847	306	1	since	since	SCONJ
ejpam-847	306	2	f	f	PROPN
ejpam-847	306	3	(	(	PUNCT
ejpam-847	306	4	x	x	X
ejpam-847	306	5	)	)	PUNCT
ejpam-847	306	6	=	=	SYM
ejpam-847	306	7	g(x)+	g(x)+	VERB
ejpam-847	306	8	h(x	h(x	PROPN
ejpam-847	306	9	)	)	PUNCT
ejpam-847	306	10	,	,	PUNCT
ejpam-847	306	11	we	we	PRON
ejpam-847	306	12	obtain	obtain	VERB
ejpam-847	306	13	µ	µ	PRON
ejpam-847	306	14	f	f	X
ejpam-847	306	15	(	(	PUNCT
ejpam-847	306	16	x)−a(x)−q(x	x)−a(x)−q(x	PROPN
ejpam-847	306	17	)	)	PUNCT
ejpam-847	306	18	(	(	PUNCT
ejpam-847	306	19	2	2	NUM
ejpam-847	306	20	t	t	NOUN
ejpam-847	306	21	)	)	PUNCT
ejpam-847	306	22	=	=	PUNCT
ejpam-847	307	1	µg(x)−a(x)+h(x)−q(x)(2	µg(x)−a(x)+h(x)−q(x)(2	SYM
ejpam-847	307	2	t	t	PROPN
ejpam-847	307	3	)	)	PUNCT
ejpam-847	307	4	≥	≥	NOUN
ejpam-847	307	5	t	t	PROPN
ejpam-847	307	6	(	(	PUNCT
ejpam-847	307	7	µg(x)−a(x)(t),µh(x)−q(x)(t	µg(x)−a(x)(t),µh(x)−q(x)(t	PROPN
ejpam-847	307	8	)	)	PUNCT
ejpam-847	307	9	)	)	PUNCT
ejpam-847	307	10	≥	≥	PROPN
ejpam-847	307	11	t	t	PROPN
ejpam-847	307	12	�	�	PROPN
ejpam-847	307	13	t∞k=1ρ	t∞k=1ρ	PROPN
ejpam-847	307	14	�	�	PROPN
ejpam-847	307	15	x	x	PUNCT
ejpam-847	307	16	2k	2k	NUM
ejpam-847	307	17	,	,	PUNCT
ejpam-847	307	18	x	x	PROPN
ejpam-847	307	19	2k	2k	NUM
ejpam-847	307	20	,	,	PUNCT
ejpam-847	307	21	−	−	PROPN
ejpam-847	307	22	x	x	SYM
ejpam-847	307	23	2k−1	2k−1	NUM
ejpam-847	307	24	,	,	PUNCT
ejpam-847	307	25	0,	0,	NUM
ejpam-847	307	26	...	...	PUNCT
ejpam-847	307	27	,0	,0	PUNCT
ejpam-847	307	28	�	�	PROPN
ejpam-847	307	29	�	�	PROPN
ejpam-847	307	30	nt	not	PART
ejpam-847	307	31	22k−2	22k−2	PROPN
ejpam-847	307	32	�	�	PROPN
ejpam-847	307	33	,	,	PUNCT
ejpam-847	307	34	t∞k=1ρ	t∞k=1ρ	VERB
ejpam-847	307	35	�	�	PROPN
ejpam-847	307	36	x	x	SYM
ejpam-847	307	37	2k	2k	NUM
ejpam-847	307	38	,	,	PUNCT
ejpam-847	307	39	−	−	PROPN
ejpam-847	307	40	x	x	SYM
ejpam-847	307	41	2k	2k	NUM
ejpam-847	307	42	,	,	PUNCT
ejpam-847	307	43	0,	0,	NUM
ejpam-847	307	44	...	...	PUNCT
ejpam-847	307	45	,0	,0	PUNCT
ejpam-847	307	46	�	�	PROPN
ejpam-847	307	47	�	�	PROPN
ejpam-847	307	48	t	t	PROPN
ejpam-847	307	49	23k−3	23k−3	PROPN
ejpam-847	307	50	�	�	PROPN
ejpam-847	307	51	�	�	PROPN
ejpam-847	307	52	for	for	ADP
ejpam-847	307	53	all	all	DET
ejpam-847	307	54	x	x	SYM
ejpam-847	307	55	∈	∈	ADJ
ejpam-847	307	56	x	x	X
ejpam-847	307	57	and	and	CCONJ
ejpam-847	307	58	all	all	DET
ejpam-847	307	59	t	t	PROPN
ejpam-847	307	60	>	>	X
ejpam-847	307	61	0	0	NUM
ejpam-847	307	62	,	,	PUNCT
ejpam-847	307	63	as	as	SCONJ
ejpam-847	307	64	desired	desire	VERB
ejpam-847	307	65	.	.	PUNCT
ejpam-847	308	1	similarly	similarly	ADV
ejpam-847	308	2	,	,	PUNCT
ejpam-847	308	3	we	we	PRON
ejpam-847	308	4	can	can	AUX
ejpam-847	308	5	obtain	obtain	VERB
ejpam-847	308	6	the	the	DET
ejpam-847	308	7	following	following	NOUN
ejpam-847	308	8	.	.	PUNCT
ejpam-847	309	1	we	we	PRON
ejpam-847	309	2	will	will	AUX
ejpam-847	309	3	omit	omit	VERB
ejpam-847	309	4	the	the	DET
ejpam-847	309	5	proof	proof	NOUN
ejpam-847	309	6	.	.	PUNCT
ejpam-847	310	1	theorem	theorem	ADJ
ejpam-847	310	2	6	6	NUM
ejpam-847	310	3	.	.	PUNCT
ejpam-847	311	1	let	let	VERB
ejpam-847	311	2	f	f	NOUN
ejpam-847	311	3	:	:	PUNCT
ejpam-847	311	4	x	x	X
ejpam-847	311	5	→	→	SYM
ejpam-847	311	6	y	y	X
ejpam-847	311	7	be	be	AUX
ejpam-847	311	8	a	a	DET
ejpam-847	311	9	mapping	mapping	NOUN
ejpam-847	311	10	with	with	ADP
ejpam-847	311	11	f	f	PROPN
ejpam-847	311	12	(	(	PUNCT
ejpam-847	311	13	0	0	NUM
ejpam-847	311	14	)	)	PUNCT
ejpam-847	311	15	=	=	SYM
ejpam-847	311	16	0	0	NUM
ejpam-847	311	17	for	for	ADP
ejpam-847	311	18	which	which	PRON
ejpam-847	311	19	there	there	PRON
ejpam-847	311	20	is	be	VERB
ejpam-847	311	21	a	a	DET
ejpam-847	311	22	ρ	ρ	NOUN
ejpam-847	311	23	:	:	PUNCT
ejpam-847	311	24	x	x	SYM
ejpam-847	311	25	n	n	PRON
ejpam-847	311	26	→	→	SYM
ejpam-847	311	27	d+	d+	X
ejpam-847	311	28	satisfying	satisfy	VERB
ejpam-847	311	29	(	(	PUNCT
ejpam-847	311	30	19	19	NUM
ejpam-847	311	31	)	)	PUNCT
ejpam-847	311	32	and	and	CCONJ
ejpam-847	311	33	(	(	PUNCT
ejpam-847	311	34	20	20	NUM
ejpam-847	311	35	)	)	PUNCT
ejpam-847	311	36	.	.	PUNCT
ejpam-847	312	1	if	if	SCONJ
ejpam-847	312	2	ρ	ρ	PROPN
ejpam-847	312	3	satisfies	satisfie	NOUN
ejpam-847	312	4	(	(	PUNCT
ejpam-847	312	5	7	7	NUM
ejpam-847	312	6	)	)	PUNCT
ejpam-847	312	7	,	,	PUNCT
ejpam-847	312	8	(	(	PUNCT
ejpam-847	312	9	15	15	NUM
ejpam-847	312	10	)	)	PUNCT
ejpam-847	312	11	and	and	CCONJ
ejpam-847	312	12	(	(	PUNCT
ejpam-847	312	13	16	16	NUM
ejpam-847	312	14	)	)	PUNCT
ejpam-847	312	15	,	,	PUNCT
ejpam-847	312	16	then	then	ADV
ejpam-847	312	17	there	there	PRON
ejpam-847	312	18	exist	exist	VERB
ejpam-847	312	19	an	an	DET
ejpam-847	312	20	additive	additive	ADJ
ejpam-847	312	21	mapping	mapping	NOUN
ejpam-847	312	22	a	a	DET
ejpam-847	312	23	:	:	PUNCT
ejpam-847	312	24	x	x	X
ejpam-847	312	25	→	→	SYM
ejpam-847	312	26	y	y	PROPN
ejpam-847	312	27	and	and	CCONJ
ejpam-847	312	28	a	a	DET
ejpam-847	312	29	quadratic	quadratic	ADJ
ejpam-847	312	30	mapping	mapping	NOUN
ejpam-847	312	31	q	q	NOUN
ejpam-847	312	32	:	:	PUNCT
ejpam-847	312	33	x	x	X
ejpam-847	312	34	→	→	SYM
ejpam-847	312	35	y	y	PROPN
ejpam-847	312	36	such	such	ADJ
ejpam-847	312	37	that	that	SCONJ
ejpam-847	312	38	µ	µ	PROPN
ejpam-847	312	39	f	f	X
ejpam-847	312	40	(	(	PUNCT
ejpam-847	312	41	x)−a(x)−q(x)(2	x)−a(x)−q(x)(2	PROPN
ejpam-847	312	42	t	t	NOUN
ejpam-847	312	43	)	)	PUNCT
ejpam-847	312	44	≥t	≥t	X
ejpam-847	312	45	�	�	PROPN
ejpam-847	312	46	t∞k=1ρ(2k−2	t∞k=1ρ(2k−2	PROPN
ejpam-847	312	47	x	x	X
ejpam-847	312	48	,	,	PUNCT
ejpam-847	312	49	2k−2	2k−2	PROPN
ejpam-847	312	50	x	x	X
ejpam-847	312	51	,	,	PUNCT
ejpam-847	312	52	−2k−1	−2k−1	PROPN
ejpam-847	312	53	x	x	PROPN
ejpam-847	312	54	,	,	PUNCT
ejpam-847	312	55	0,	0,	NUM
ejpam-847	312	56	...	...	PUNCT
ejpam-847	312	57	,0)(2nt	,0)(2nt	PUNCT
ejpam-847	312	58	)	)	PUNCT
ejpam-847	312	59	,	,	PUNCT
ejpam-847	312	60	t∞k=1ρ(2k	t∞k=1ρ(2k	PROPN
ejpam-847	312	61	x	x	SYM
ejpam-847	312	62	,	,	PUNCT
ejpam-847	312	63	−2k	−2k	PROPN
ejpam-847	312	64	x	x	SYM
ejpam-847	312	65	,	,	PUNCT
ejpam-847	312	66	0,	0,	NUM
ejpam-847	312	67	...	...	PUNCT
ejpam-847	312	68	,0	,0	PUNCT
ejpam-847	312	69	)	)	PUNCT
ejpam-847	312	70	�	�	PROPN
ejpam-847	312	71	2k−1	2k−1	NUM
ejpam-847	312	72	t	t	PROPN
ejpam-847	312	73	�	�	PROPN
ejpam-847	312	74	�	�	PROPN
ejpam-847	312	75	for	for	ADP
ejpam-847	312	76	all	all	DET
ejpam-847	312	77	x	x	SYM
ejpam-847	312	78	∈	∈	ADJ
ejpam-847	312	79	x	x	X
ejpam-847	312	80	and	and	CCONJ
ejpam-847	312	81	all	all	DET
ejpam-847	312	82	t	t	PROPN
ejpam-847	312	83	>	>	X
ejpam-847	312	84	0	0	X
ejpam-847	312	85	.	.	PUNCT
ejpam-847	313	1	references	reference	NOUN
ejpam-847	313	2	552	552	NUM
ejpam-847	313	3	acknowledgements	acknowledgement	NOUN
ejpam-847	313	4	d.	d.	PROPN
ejpam-847	313	5	y.	y.	PROPN
ejpam-847	313	6	shin	shin	PROPN
ejpam-847	313	7	was	be	AUX
ejpam-847	313	8	supported	support	VERB
ejpam-847	313	9	by	by	ADP
ejpam-847	313	10	basic	basic	ADJ
ejpam-847	313	11	science	science	NOUN
ejpam-847	313	12	research	research	NOUN
ejpam-847	313	13	program	program	NOUN
ejpam-847	313	14	through	through	ADP
ejpam-847	313	15	the	the	DET
ejpam-847	313	16	national	national	PROPN
ejpam-847	313	17	research	research	PROPN
ejpam-847	313	18	foundation	foundation	PROPN
ejpam-847	313	19	of	of	ADP
ejpam-847	313	20	korea	korea	PROPN
ejpam-847	313	21	funded	fund	VERB
ejpam-847	313	22	by	by	ADP
ejpam-847	313	23	the	the	DET
ejpam-847	313	24	ministry	ministry	PROPN
ejpam-847	313	25	of	of	ADP
ejpam-847	313	26	education	education	PROPN
ejpam-847	313	27	,	,	PUNCT
ejpam-847	313	28	science	science	NOUN
ejpam-847	313	29	and	and	CCONJ
ejpam-847	313	30	technology	technology	NOUN
ejpam-847	313	31	(	(	PUNCT
ejpam-847	313	32	nrf-2010	nrf-2010	NOUN
ejpam-847	313	33	-	-	PUNCT
ejpam-847	313	34	0021792	0021792	NUM
ejpam-847	313	35	)	)	PUNCT
ejpam-847	313	36	.	.	PUNCT
ejpam-847	314	1	references	reference	NOUN
ejpam-847	314	2	[	[	X
ejpam-847	314	3	1	1	NUM
ejpam-847	314	4	]	]	X
ejpam-847	314	5	s.s	s.s	PROPN
ejpam-847	314	6	.	.	PROPN
ejpam-847	314	7	chang	chang	PROPN
ejpam-847	314	8	,	,	PUNCT
ejpam-847	314	9	y.	y.	PROPN
ejpam-847	314	10	cho	cho	PROPN
ejpam-847	314	11	,	,	PUNCT
ejpam-847	314	12	and	and	CCONJ
ejpam-847	314	13	s.	s.	PROPN
ejpam-847	314	14	kang	kang	PROPN
ejpam-847	314	15	.	.	PUNCT
ejpam-847	315	1	nonlinear	nonlinear	ADJ
ejpam-847	315	2	operator	operator	NOUN
ejpam-847	315	3	theory	theory	NOUN
ejpam-847	315	4	in	in	ADP
ejpam-847	315	5	probabilistic	probabilistic	ADJ
ejpam-847	315	6	metric	metric	ADJ
ejpam-847	315	7	spaces	space	NOUN
ejpam-847	315	8	.	.	PUNCT
ejpam-847	316	1	nova	nova	PROPN
ejpam-847	316	2	science	science	PROPN
ejpam-847	316	3	publishers	publishers	PROPN
ejpam-847	316	4	inc	inc	PROPN
ejpam-847	316	5	.	.	PROPN
ejpam-847	316	6	,	,	PUNCT
ejpam-847	316	7	new	new	PROPN
ejpam-847	316	8	york	york	PROPN
ejpam-847	316	9	,	,	PUNCT
ejpam-847	316	10	2001	2001	NUM
ejpam-847	316	11	.	.	PUNCT
ejpam-847	317	1	[	[	X
ejpam-847	317	2	2	2	NUM
ejpam-847	317	3	]	]	X
ejpam-847	317	4	y.	y.	PROPN
ejpam-847	317	5	cho	cho	PROPN
ejpam-847	317	6	,	,	PUNCT
ejpam-847	317	7	c.	c.	PROPN
ejpam-847	317	8	park	park	PROPN
ejpam-847	317	9	,	,	PUNCT
ejpam-847	317	10	and	and	CCONJ
ejpam-847	317	11	t.m	t.m	PROPN
ejpam-847	317	12	.	.	PROPN
ejpam-847	317	13	rassias	rassias	PROPN
ejpam-847	317	14	.	.	PUNCT
ejpam-847	318	1	inner	inner	ADJ
ejpam-847	318	2	product	product	NOUN
ejpam-847	318	3	spaces	space	NOUN
ejpam-847	318	4	and	and	CCONJ
ejpam-847	318	5	functional	functional	ADJ
ejpam-847	318	6	equations	equation	NOUN
ejpam-847	318	7	.	.	PUNCT
ejpam-847	319	1	journal	journal	NOUN
ejpam-847	319	2	of	of	ADP
ejpam-847	319	3	computational	computational	ADJ
ejpam-847	319	4	analysis	analysis	NOUN
ejpam-847	319	5	and	and	CCONJ
ejpam-847	319	6	applications	application	NOUN
ejpam-847	319	7	,	,	PUNCT
ejpam-847	319	8	13(2):296–304	13(2):296–304	PROPN
ejpam-847	319	9	,	,	PUNCT
ejpam-847	319	10	2011	2011	NUM
ejpam-847	319	11	.	.	PUNCT
ejpam-847	320	1	[	[	X
ejpam-847	320	2	3	3	NUM
ejpam-847	320	3	]	]	X
ejpam-847	320	4	m.e	m.e	PROPN
ejpam-847	320	5	.	.	PROPN
ejpam-847	320	6	gordji	gordji	PROPN
ejpam-847	320	7	,	,	PUNCT
ejpam-847	320	8	j.m	j.m	PROPN
ejpam-847	320	9	.	.	PROPN
ejpam-847	320	10	rassias	rassias	PROPN
ejpam-847	320	11	,	,	PUNCT
ejpam-847	320	12	and	and	CCONJ
ejpam-847	320	13	m.b	m.b	PROPN
ejpam-847	320	14	.	.	PROPN
ejpam-847	320	15	savadkouhi	savadkouhi	PROPN
ejpam-847	320	16	.	.	PUNCT
ejpam-847	321	1	approximation	approximation	NOUN
ejpam-847	321	2	of	of	ADP
ejpam-847	321	3	the	the	DET
ejpam-847	321	4	quadratic	quadratic	ADJ
ejpam-847	321	5	and	and	CCONJ
ejpam-847	321	6	cubic	cubic	ADJ
ejpam-847	321	7	functional	functional	ADJ
ejpam-847	321	8	equation	equation	NOUN
ejpam-847	321	9	in	in	ADP
ejpam-847	321	10	rn	rn	NOUN
ejpam-847	321	11	-	-	NOUN
ejpam-847	321	12	spaces	space	NOUN
ejpam-847	321	13	.	.	PUNCT
ejpam-847	322	1	european	european	ADJ
ejpam-847	322	2	journal	journal	PROPN
ejpam-847	322	3	of	of	ADP
ejpam-847	322	4	pure	pure	ADJ
ejpam-847	322	5	and	and	CCONJ
ejpam-847	322	6	applied	applied	ADJ
ejpam-847	322	7	mathematics	mathematic	NOUN
ejpam-847	322	8	,	,	PUNCT
ejpam-847	322	9	2:494–507	2:494–507	NOUN
ejpam-847	322	10	,	,	PUNCT
ejpam-847	322	11	2009	2009	NUM
ejpam-847	322	12	.	.	PUNCT
ejpam-847	323	1	[	[	X
ejpam-847	323	2	4	4	NUM
ejpam-847	323	3	]	]	X
ejpam-847	323	4	m.e	m.e	PROPN
ejpam-847	323	5	.	.	PROPN
ejpam-847	323	6	gordji	gordji	PROPN
ejpam-847	323	7	,	,	PUNCT
ejpam-847	323	8	m.b	m.b	PROPN
ejpam-847	323	9	.	.	PROPN
ejpam-847	323	10	savadkouhi	savadkouhi	PROPN
ejpam-847	323	11	,	,	PUNCT
ejpam-847	323	12	and	and	CCONJ
ejpam-847	323	13	j.m	j.m	PROPN
ejpam-847	323	14	.	.	PROPN
ejpam-847	323	15	rassias	rassias	PROPN
ejpam-847	323	16	.	.	PUNCT
ejpam-847	324	1	stability	stability	NOUN
ejpam-847	324	2	of	of	ADP
ejpam-847	324	3	a	a	DET
ejpam-847	324	4	mixed	mixed	ADJ
ejpam-847	324	5	type	type	NOUN
ejpam-847	324	6	additive	additive	ADJ
ejpam-847	324	7	and	and	CCONJ
ejpam-847	324	8	quadratic	quadratic	ADJ
ejpam-847	324	9	functional	functional	ADJ
ejpam-847	324	10	equation	equation	NOUN
ejpam-847	324	11	in	in	ADP
ejpam-847	324	12	random	random	ADJ
ejpam-847	324	13	normed	normed	ADJ
ejpam-847	324	14	spaces	space	NOUN
ejpam-847	324	15	.	.	PUNCT
ejpam-847	325	1	journal	journal	NOUN
ejpam-847	325	2	of	of	ADP
ejpam-847	325	3	concrete	concrete	ADJ
ejpam-847	325	4	and	and	CCONJ
ejpam-847	325	5	applicable	applicable	ADJ
ejpam-847	325	6	mathematics	mathematic	NOUN
ejpam-847	325	7	,	,	PUNCT
ejpam-847	325	8	10(1	10(1	PROPN
ejpam-847	325	9	-	-	SYM
ejpam-847	325	10	2):117–129	2):117–129	NUM
ejpam-847	325	11	,	,	PUNCT
ejpam-847	325	12	2012	2012	NUM
ejpam-847	325	13	.	.	PUNCT
ejpam-847	326	1	[	[	X
ejpam-847	326	2	5	5	NUM
ejpam-847	326	3	]	]	X
ejpam-847	326	4	d.h	d.h	PROPN
ejpam-847	326	5	.	.	PROPN
ejpam-847	326	6	hyers	hyer	NOUN
ejpam-847	326	7	.	.	PUNCT
ejpam-847	327	1	on	on	ADP
ejpam-847	327	2	the	the	DET
ejpam-847	327	3	stability	stability	NOUN
ejpam-847	327	4	of	of	ADP
ejpam-847	327	5	the	the	DET
ejpam-847	327	6	linear	linear	ADJ
ejpam-847	327	7	functional	functional	ADJ
ejpam-847	327	8	equation	equation	NOUN
ejpam-847	327	9	.	.	PUNCT
ejpam-847	328	1	proceedings	proceeding	NOUN
ejpam-847	328	2	of	of	ADP
ejpam-847	328	3	the	the	DET
ejpam-847	328	4	national	national	PROPN
ejpam-847	328	5	academy	academy	PROPN
ejpam-847	328	6	of	of	ADP
ejpam-847	328	7	sciences	sciences	PROPN
ejpam-847	328	8	of	of	ADP
ejpam-847	328	9	the	the	DET
ejpam-847	328	10	united	united	PROPN
ejpam-847	328	11	states	states	PROPN
ejpam-847	328	12	of	of	ADP
ejpam-847	328	13	america	america	PROPN
ejpam-847	328	14	,	,	PUNCT
ejpam-847	328	15	27:222–224	27:222–224	NUM
ejpam-847	328	16	,	,	PUNCT
ejpam-847	328	17	1941	1941	NUM
ejpam-847	328	18	.	.	PUNCT
ejpam-847	329	1	[	[	X
ejpam-847	329	2	6	6	NUM
ejpam-847	329	3	]	]	PUNCT
ejpam-847	329	4	s.	s.	PROPN
ejpam-847	329	5	jang	jang	PROPN
ejpam-847	329	6	and	and	CCONJ
ejpam-847	329	7	c.	c.	PROPN
ejpam-847	329	8	park	park	PROPN
ejpam-847	329	9	.	.	PUNCT
ejpam-847	330	1	fuzzy	fuzzy	ADJ
ejpam-847	330	2	stability	stability	NOUN
ejpam-847	330	3	of	of	ADP
ejpam-847	330	4	a	a	DET
ejpam-847	330	5	functional	functional	ADJ
ejpam-847	330	6	equation	equation	NOUN
ejpam-847	330	7	related	relate	VERB
ejpam-847	330	8	to	to	ADP
ejpam-847	330	9	inner	inner	ADJ
ejpam-847	330	10	product	product	NOUN
ejpam-847	330	11	spaces	space	NOUN
ejpam-847	330	12	.	.	PUNCT
ejpam-847	331	1	hacettepe	hacettepe	PROPN
ejpam-847	331	2	journal	journal	PROPN
ejpam-847	331	3	of	of	ADP
ejpam-847	331	4	mathematics	mathematic	NOUN
ejpam-847	331	5	and	and	CCONJ
ejpam-847	331	6	statistics	statistic	NOUN
ejpam-847	331	7	,	,	PUNCT
ejpam-847	331	8	40(5):711–723	40(5):711–723	NOUN
ejpam-847	331	9	,	,	PUNCT
ejpam-847	331	10	2011	2011	NUM
ejpam-847	331	11	.	.	PUNCT
ejpam-847	332	1	[	[	X
ejpam-847	332	2	7	7	X
ejpam-847	332	3	]	]	X
ejpam-847	332	4	p.l	p.l	PROPN
ejpam-847	332	5	.	.	PROPN
ejpam-847	332	6	kannappan	kannappan	PROPN
ejpam-847	332	7	.	.	PUNCT
ejpam-847	333	1	quadratic	quadratic	ADJ
ejpam-847	333	2	functional	functional	ADJ
ejpam-847	333	3	equation	equation	NOUN
ejpam-847	333	4	and	and	CCONJ
ejpam-847	333	5	inner	inner	ADJ
ejpam-847	333	6	product	product	NOUN
ejpam-847	333	7	spaces	space	VERB
ejpam-847	333	8	.	.	PUNCT
ejpam-847	334	1	results	result	NOUN
ejpam-847	334	2	in	in	ADP
ejpam-847	334	3	mathematics	mathematic	NOUN
ejpam-847	334	4	,	,	PUNCT
ejpam-847	334	5	27:368–372	27:368–372	PROPN
ejpam-847	334	6	,	,	PUNCT
ejpam-847	334	7	1995	1995	NUM
ejpam-847	334	8	.	.	PUNCT
ejpam-847	335	1	[	[	X
ejpam-847	335	2	8	8	NUM
ejpam-847	335	3	]	]	X
ejpam-847	335	4	d.	d.	PROPN
ejpam-847	335	5	mihȩt	mihȩt	PROPN
ejpam-847	335	6	.	.	PUNCT
ejpam-847	336	1	the	the	DET
ejpam-847	336	2	fixed	fix	VERB
ejpam-847	336	3	point	point	NOUN
ejpam-847	336	4	method	method	NOUN
ejpam-847	336	5	for	for	ADP
ejpam-847	336	6	fuzzy	fuzzy	ADJ
ejpam-847	336	7	stability	stability	NOUN
ejpam-847	336	8	of	of	ADP
ejpam-847	336	9	the	the	DET
ejpam-847	336	10	jensen	jensen	PROPN
ejpam-847	336	11	functional	functional	ADJ
ejpam-847	336	12	equation	equation	NOUN
ejpam-847	336	13	.	.	PUNCT
ejpam-847	337	1	fuzzy	fuzzy	ADJ
ejpam-847	337	2	sets	set	NOUN
ejpam-847	337	3	and	and	CCONJ
ejpam-847	337	4	systems	system	NOUN
ejpam-847	337	5	,	,	PUNCT
ejpam-847	337	6	160:1663–1667	160:1663–1667	NUM
ejpam-847	337	7	,	,	PUNCT
ejpam-847	337	8	2009	2009	NUM
ejpam-847	337	9	.	.	PUNCT
ejpam-847	338	1	[	[	X
ejpam-847	338	2	9	9	NUM
ejpam-847	338	3	]	]	PUNCT
ejpam-847	338	4	d.	d.	PROPN
ejpam-847	338	5	mihȩt	mihȩt	PROPN
ejpam-847	338	6	.	.	PUNCT
ejpam-847	339	1	the	the	DET
ejpam-847	339	2	probabilistic	probabilistic	ADJ
ejpam-847	339	3	stability	stability	NOUN
ejpam-847	339	4	for	for	ADP
ejpam-847	339	5	a	a	DET
ejpam-847	339	6	functional	functional	ADJ
ejpam-847	339	7	equation	equation	NOUN
ejpam-847	339	8	in	in	ADP
ejpam-847	339	9	a	a	DET
ejpam-847	339	10	single	single	ADJ
ejpam-847	339	11	variable	variable	NOUN
ejpam-847	339	12	.	.	PUNCT
ejpam-847	340	1	acta	acta	PROPN
ejpam-847	340	2	mathematica	mathematica	PROPN
ejpam-847	340	3	hungaria	hungaria	PROPN
ejpam-847	340	4	,	,	PUNCT
ejpam-847	340	5	123:249–256	123:249–256	NUM
ejpam-847	340	6	,	,	PUNCT
ejpam-847	340	7	2009	2009	NUM
ejpam-847	340	8	.	.	PUNCT
ejpam-847	341	1	[	[	X
ejpam-847	341	2	10	10	NUM
ejpam-847	341	3	]	]	X
ejpam-847	341	4	d.	d.	PROPN
ejpam-847	341	5	mihȩt	mihȩt	ADV
ejpam-847	341	6	and	and	CCONJ
ejpam-847	341	7	v.	v.	ADP
ejpam-847	341	8	radu	radu	PROPN
ejpam-847	341	9	.	.	PUNCT
ejpam-847	342	1	on	on	ADP
ejpam-847	342	2	the	the	DET
ejpam-847	342	3	stability	stability	NOUN
ejpam-847	342	4	of	of	ADP
ejpam-847	342	5	the	the	DET
ejpam-847	342	6	additive	additive	ADJ
ejpam-847	342	7	cauchy	cauchy	ADJ
ejpam-847	342	8	functional	functional	ADJ
ejpam-847	342	9	equation	equation	NOUN
ejpam-847	342	10	in	in	ADP
ejpam-847	342	11	random	random	ADJ
ejpam-847	342	12	normed	normed	ADJ
ejpam-847	342	13	spaces	space	NOUN
ejpam-847	342	14	.	.	PUNCT
ejpam-847	343	1	journal	journal	PROPN
ejpam-847	343	2	of	of	ADP
ejpam-847	343	3	mathematical	mathematical	ADJ
ejpam-847	343	4	analysis	analysis	NOUN
ejpam-847	343	5	and	and	CCONJ
ejpam-847	343	6	applications	application	NOUN
ejpam-847	343	7	,	,	PUNCT
ejpam-847	343	8	343:567	343:567	NUM
ejpam-847	343	9	–	–	PUNCT
ejpam-847	343	10	572	572	NUM
ejpam-847	343	11	,	,	PUNCT
ejpam-847	343	12	2008	2008	NUM
ejpam-847	343	13	.	.	PUNCT
ejpam-847	344	1	[	[	X
ejpam-847	344	2	11	11	NUM
ejpam-847	344	3	]	]	X
ejpam-847	344	4	a.k	a.k	PROPN
ejpam-847	344	5	.	.	PROPN
ejpam-847	344	6	mirmostafaee	mirmostafaee	PROPN
ejpam-847	344	7	,	,	PUNCT
ejpam-847	344	8	m.	m.	NOUN
ejpam-847	344	9	mirzavaziri	mirzavaziri	PROPN
ejpam-847	344	10	,	,	PUNCT
ejpam-847	344	11	and	and	CCONJ
ejpam-847	344	12	m.s	m.s	PROPN
ejpam-847	344	13	.	.	PROPN
ejpam-847	344	14	moslehian	moslehian	PROPN
ejpam-847	344	15	.	.	PUNCT
ejpam-847	345	1	fuzzy	fuzzy	ADJ
ejpam-847	345	2	stability	stability	NOUN
ejpam-847	345	3	of	of	ADP
ejpam-847	345	4	the	the	DET
ejpam-847	345	5	jensen	jensen	PROPN
ejpam-847	345	6	functional	functional	ADJ
ejpam-847	345	7	equation	equation	NOUN
ejpam-847	345	8	.	.	PUNCT
ejpam-847	346	1	fuzzy	fuzzy	ADJ
ejpam-847	346	2	sets	set	NOUN
ejpam-847	346	3	and	and	CCONJ
ejpam-847	346	4	systems	system	NOUN
ejpam-847	346	5	,	,	PUNCT
ejpam-847	346	6	159:730–738	159:730–738	NUM
ejpam-847	346	7	,	,	PUNCT
ejpam-847	346	8	2008	2008	NUM
ejpam-847	346	9	.	.	PUNCT
ejpam-847	347	1	[	[	X
ejpam-847	347	2	12	12	NUM
ejpam-847	347	3	]	]	X
ejpam-847	347	4	a.k	a.k	PROPN
ejpam-847	347	5	.	.	PROPN
ejpam-847	347	6	mirmostafaee	mirmostafaee	PROPN
ejpam-847	347	7	and	and	CCONJ
ejpam-847	347	8	m.s	m.s	PROPN
ejpam-847	347	9	.	.	PROPN
ejpam-847	347	10	moslehian	moslehian	PROPN
ejpam-847	347	11	.	.	PUNCT
ejpam-847	348	1	fuzzy	fuzzy	ADJ
ejpam-847	348	2	approximately	approximately	ADV
ejpam-847	348	3	cubic	cubic	ADJ
ejpam-847	348	4	mappings	mapping	NOUN
ejpam-847	348	5	.	.	PUNCT
ejpam-847	349	1	information	information	NOUN
ejpam-847	349	2	sciences	sciences	PROPN
ejpam-847	349	3	,	,	PUNCT
ejpam-847	349	4	178:3791–3798	178:3791–3798	NUM
ejpam-847	349	5	,	,	PUNCT
ejpam-847	349	6	2008	2008	NUM
ejpam-847	349	7	.	.	PUNCT
ejpam-847	350	1	[	[	X
ejpam-847	350	2	13	13	NUM
ejpam-847	350	3	]	]	X
ejpam-847	350	4	t.m	t.m	PROPN
ejpam-847	350	5	.	.	PROPN
ejpam-847	350	6	rassias	rassias	PROPN
ejpam-847	350	7	.	.	PUNCT
ejpam-847	351	1	on	on	ADP
ejpam-847	351	2	the	the	DET
ejpam-847	351	3	stability	stability	NOUN
ejpam-847	351	4	of	of	ADP
ejpam-847	351	5	the	the	DET
ejpam-847	351	6	linear	linear	ADJ
ejpam-847	351	7	mapping	mapping	NOUN
ejpam-847	351	8	in	in	ADP
ejpam-847	351	9	banach	banach	NOUN
ejpam-847	351	10	spaces	space	NOUN
ejpam-847	351	11	.	.	PUNCT
ejpam-847	352	1	proceedings	proceeding	NOUN
ejpam-847	352	2	of	of	ADP
ejpam-847	352	3	the	the	DET
ejpam-847	352	4	american	american	PROPN
ejpam-847	352	5	mathematical	mathematical	PROPN
ejpam-847	352	6	society	society	NOUN
ejpam-847	352	7	,	,	PUNCT
ejpam-847	352	8	72:297–300	72:297–300	PROPN
ejpam-847	352	9	,	,	PUNCT
ejpam-847	352	10	1978	1978	NUM
ejpam-847	352	11	.	.	PUNCT
ejpam-847	353	1	references	reference	NOUN
ejpam-847	353	2	553	553	NUM
ejpam-847	353	3	[	[	X
ejpam-847	353	4	14	14	NUM
ejpam-847	353	5	]	]	X
ejpam-847	353	6	t.m	t.m	PROPN
ejpam-847	353	7	.	.	PROPN
ejpam-847	353	8	rassias	rassias	PROPN
ejpam-847	353	9	.	.	PUNCT
ejpam-847	354	1	on	on	ADP
ejpam-847	354	2	characterizations	characterization	NOUN
ejpam-847	354	3	of	of	ADP
ejpam-847	354	4	inner	inner	ADJ
ejpam-847	354	5	product	product	NOUN
ejpam-847	354	6	spaces	space	NOUN
ejpam-847	354	7	and	and	CCONJ
ejpam-847	354	8	generalizations	generalization	NOUN
ejpam-847	354	9	of	of	ADP
ejpam-847	354	10	the	the	DET
ejpam-847	354	11	h.	h.	PROPN
ejpam-847	354	12	bohr	bohr	PROPN
ejpam-847	354	13	inequality	inequality	PROPN
ejpam-847	354	14	.	.	PUNCT
ejpam-847	355	1	in	in	ADP
ejpam-847	355	2	t.m	t.m	PROPN
ejpam-847	355	3	.	.	PROPN
ejpam-847	355	4	rassias	rassias	PROPN
ejpam-847	355	5	et	et	PROPN
ejpam-847	355	6	al	al	PROPN
ejpam-847	355	7	.	.	PROPN
ejpam-847	355	8	,	,	PUNCT
ejpam-847	355	9	editor	editor	NOUN
ejpam-847	355	10	,	,	PUNCT
ejpam-847	355	11	topics	topic	NOUN
ejpam-847	355	12	in	in	ADP
ejpam-847	355	13	mathematical	mathematical	ADJ
ejpam-847	355	14	analysis	analysis	NOUN
ejpam-847	355	15	.	.	PUNCT
ejpam-847	355	16	,	,	PUNCT
ejpam-847	355	17	pages	page	NOUN
ejpam-847	355	18	803–819	803–819	NUM
ejpam-847	355	19	.	.	PUNCT
ejpam-847	355	20	world	world	NOUN
ejpam-847	355	21	scientific	scientific	ADJ
ejpam-847	355	22	publ	publ	NOUN
ejpam-847	355	23	.	.	PUNCT
ejpam-847	356	1	co.	co.	PROPN
ejpam-847	356	2	,	,	PUNCT
ejpam-847	356	3	singapore	singapore	PROPN
ejpam-847	356	4	,	,	PUNCT
ejpam-847	356	5	1989	1989	NUM
ejpam-847	356	6	.	.	PUNCT
ejpam-847	357	1	[	[	X
ejpam-847	357	2	15	15	NUM
ejpam-847	357	3	]	]	X
ejpam-847	357	4	b.	b.	PROPN
ejpam-847	357	5	schweizer	schweizer	PROPN
ejpam-847	357	6	and	and	CCONJ
ejpam-847	357	7	a.	a.	NOUN
ejpam-847	357	8	sklar	sklar	PROPN
ejpam-847	357	9	.	.	PUNCT
ejpam-847	358	1	probabilistic	probabilistic	ADJ
ejpam-847	358	2	metric	metric	ADJ
ejpam-847	358	3	spaces	space	NOUN
ejpam-847	358	4	.	.	PUNCT
ejpam-847	359	1	elsevier	elsevier	NOUN
ejpam-847	359	2	,	,	PUNCT
ejpam-847	359	3	north	north	PROPN
ejpam-847	359	4	holand	holand	PROPN
ejpam-847	359	5	,	,	PUNCT
ejpam-847	359	6	new	new	PROPN
ejpam-847	359	7	york	york	PROPN
ejpam-847	359	8	,	,	PUNCT
ejpam-847	359	9	1983	1983	NUM
ejpam-847	359	10	.	.	PUNCT
ejpam-847	360	1	[	[	X
ejpam-847	360	2	16	16	NUM
ejpam-847	360	3	]	]	X
ejpam-847	360	4	a.n	a.n	PROPN
ejpam-847	360	5	.	.	PROPN
ejpam-847	360	6	sherstnev	sherstnev	PROPN
ejpam-847	360	7	.	.	PUNCT
ejpam-847	361	1	on	on	ADP
ejpam-847	361	2	the	the	DET
ejpam-847	361	3	notion	notion	NOUN
ejpam-847	361	4	of	of	ADP
ejpam-847	361	5	a	a	DET
ejpam-847	361	6	random	random	ADJ
ejpam-847	361	7	normed	normed	ADJ
ejpam-847	361	8	space	space	NOUN
ejpam-847	361	9	.	.	PUNCT
ejpam-847	362	1	doklady	doklady	PROPN
ejpam-847	362	2	akademii	akademii	PROPN
ejpam-847	362	3	nauk	nauk	NOUN
ejpam-847	362	4	sssr	sssr	NOUN
ejpam-847	362	5	,	,	PUNCT
ejpam-847	362	6	149:280–283	149:280–283	NUM
ejpam-847	362	7	,	,	PUNCT
ejpam-847	362	8	1963	1963	NUM
ejpam-847	362	9	.	.	PUNCT
ejpam-847	363	1	[	[	X
ejpam-847	363	2	17	17	NUM
ejpam-847	363	3	]	]	X
ejpam-847	363	4	f.	f.	PROPN
ejpam-847	363	5	skof	skof	PROPN
ejpam-847	363	6	.	.	PUNCT
ejpam-847	364	1	propriet	propriet	PROPN
ejpam-847	364	2	locali	locali	PROPN
ejpam-847	364	3	e	e	PROPN
ejpam-847	364	4	approssimazione	approssimazione	PROPN
ejpam-847	364	5	di	di	PROPN
ejpam-847	364	6	operatori	operatori	PROPN
ejpam-847	364	7	.	.	PROPN
ejpam-847	364	8	rendiconti	rendiconti	PROPN
ejpam-847	364	9	del	del	PROPN
ejpam-847	364	10	seminario	seminario	NOUN
ejpam-847	364	11	matematico	matematico	NOUN
ejpam-847	364	12	e	e	PROPN
ejpam-847	364	13	fisico	fisico	PROPN
ejpam-847	364	14	di	di	X
ejpam-847	364	15	milano	milano	PROPN
ejpam-847	364	16	,	,	PUNCT
ejpam-847	364	17	53:113–129	53:113–129	PROPN
ejpam-847	364	18	,	,	PUNCT
ejpam-847	364	19	1983	1983	NUM
ejpam-847	364	20	.	.	PUNCT
ejpam-847	365	1	[	[	X
ejpam-847	365	2	18	18	NUM
ejpam-847	365	3	]	]	X
ejpam-847	365	4	s.m	s.m	PROPN
ejpam-847	365	5	.	.	PUNCT
ejpam-847	365	6	ulam	ulam	PROPN
ejpam-847	365	7	.	.	PUNCT
ejpam-847	365	8	problems	problem	NOUN
ejpam-847	365	9	in	in	ADP
ejpam-847	365	10	modern	modern	ADJ
ejpam-847	365	11	mathematics	mathematic	NOUN
ejpam-847	365	12	.	.	PUNCT
ejpam-847	366	1	wiley	wiley	PROPN
ejpam-847	366	2	,	,	PUNCT
ejpam-847	366	3	new	new	PROPN
ejpam-847	366	4	york	york	PROPN
ejpam-847	366	5	,	,	PUNCT
ejpam-847	366	6	1960	1960	NUM
ejpam-847	366	7	.	.	PUNCT