id	sid	tid	token	lemma	pos
ejpam-677	1	1	6_677_rassias.dvi	6_677_rassias.dvi	NUM
ejpam-677	1	2	european	european	ADJ
ejpam-677	1	3	journal	journal	NOUN
ejpam-677	1	4	of	of	ADP
ejpam-677	1	5	pure	pure	ADJ
ejpam-677	1	6	and	and	CCONJ
ejpam-677	1	7	applied	apply	VERB
ejpam-677	1	8	mathematics	mathematic	NOUN
ejpam-677	1	9	vol	vol	NOUN
ejpam-677	1	10	.	.	PROPN
ejpam-677	1	11	4	4	NUM
ejpam-677	1	12	,	,	PUNCT
ejpam-677	1	13	no	no	INTJ
ejpam-677	1	14	.	.	NOUN
ejpam-677	1	15	1	1	NUM
ejpam-677	1	16	,	,	PUNCT
ejpam-677	1	17	2011	2011	NUM
ejpam-677	1	18	,	,	PUNCT
ejpam-677	1	19	50	50	NUM
ejpam-677	1	20	-	-	SYM
ejpam-677	1	21	58	58	NUM
ejpam-677	1	22	issn	issn	PROPN
ejpam-677	1	23	1307	1307	NUM
ejpam-677	1	24	-	-	SYM
ejpam-677	1	25	5543	5543	NUM
ejpam-677	1	26	–	–	PUNCT
ejpam-677	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-677	1	28	generalised	generalise	VERB
ejpam-677	1	29	hyers	hyers	PROPN
ejpam-677	1	30	-	-	PUNCT
ejpam-677	1	31	ulam	ulam	PROPN
ejpam-677	1	32	product	product	NOUN
ejpam-677	1	33	-	-	PUNCT
ejpam-677	1	34	sum	sum	NOUN
ejpam-677	1	35	stability	stability	NOUN
ejpam-677	1	36	of	of	ADP
ejpam-677	1	37	a	a	DET
ejpam-677	1	38	cauchy	cauchy	ADJ
ejpam-677	1	39	type	type	NOUN
ejpam-677	1	40	additive	additive	ADJ
ejpam-677	1	41	functional	functional	ADJ
ejpam-677	1	42	equation	equation	NOUN
ejpam-677	1	43	matina	matina	PROPN
ejpam-677	1	44	j.	j.	PROPN
ejpam-677	1	45	rassias	rassias	PROPN
ejpam-677	1	46	department	department	PROPN
ejpam-677	1	47	of	of	ADP
ejpam-677	1	48	statistics	statistic	NOUN
ejpam-677	1	49	,	,	PUNCT
ejpam-677	1	50	university	university	NOUN
ejpam-677	1	51	of	of	ADP
ejpam-677	1	52	glasgow	glasgow	PROPN
ejpam-677	1	53	,	,	PUNCT
ejpam-677	1	54	mathematics	mathematic	NOUN
ejpam-677	1	55	building	building	NOUN
ejpam-677	1	56	,	,	PUNCT
ejpam-677	1	57	office	office	NOUN
ejpam-677	1	58	no	no	NOUN
ejpam-677	1	59	.	.	NOUN
ejpam-677	1	60	208	208	NUM
ejpam-677	1	61	,	,	PUNCT
ejpam-677	1	62	university	university	NOUN
ejpam-677	1	63	gardens	garden	NOUN
ejpam-677	1	64	,	,	PUNCT
ejpam-677	1	65	glasgow	glasgow	NOUN
ejpam-677	1	66	g12	g12	PROPN
ejpam-677	1	67	8qw	8qw	PROPN
ejpam-677	1	68	,	,	PUNCT
ejpam-677	1	69	u.k	u.k	PROPN
ejpam-677	1	70	.	.	PROPN
ejpam-677	1	71	abstract	abstract	PROPN
ejpam-677	1	72	.	.	PUNCT
ejpam-677	2	1	in	in	ADP
ejpam-677	2	2	1940	1940	NUM
ejpam-677	2	3	(	(	PUNCT
ejpam-677	2	4	and	and	CCONJ
ejpam-677	2	5	1964	1964	NUM
ejpam-677	2	6	)	)	PUNCT
ejpam-677	2	7	s.m	s.m	PROPN
ejpam-677	2	8	.	.	PUNCT
ejpam-677	3	1	ulam	ulam	PROPN
ejpam-677	3	2	proposed	propose	VERB
ejpam-677	3	3	the	the	DET
ejpam-677	3	4	well	well	ADV
ejpam-677	3	5	-	-	PUNCT
ejpam-677	3	6	known	know	VERB
ejpam-677	3	7	ulam	ulam	NOUN
ejpam-677	3	8	stability	stability	PROPN
ejpam-677	3	9	problem	problem	NOUN
ejpam-677	3	10	.	.	PUNCT
ejpam-677	4	1	in	in	ADP
ejpam-677	4	2	1941	1941	NUM
ejpam-677	4	3	d.h	d.h	PROPN
ejpam-677	4	4	.	.	PUNCT
ejpam-677	5	1	hyers	hyer	NOUN
ejpam-677	5	2	solved	solve	VERB
ejpam-677	5	3	the	the	DET
ejpam-677	5	4	hyers	hyers	PROPN
ejpam-677	5	5	-	-	PUNCT
ejpam-677	5	6	ulam	ulam	PROPN
ejpam-677	5	7	problem	problem	NOUN
ejpam-677	5	8	for	for	ADP
ejpam-677	5	9	linear	linear	NOUN
ejpam-677	5	10	mappings	mapping	NOUN
ejpam-677	5	11	.	.	PUNCT
ejpam-677	6	1	in	in	ADP
ejpam-677	6	2	2008	2008	NUM
ejpam-677	6	3	,	,	PUNCT
ejpam-677	6	4	j.	j.	PROPN
ejpam-677	6	5	m.	m.	PROPN
ejpam-677	6	6	rassias	rassias	PROPN
ejpam-677	6	7	introduced	introduce	VERB
ejpam-677	6	8	the	the	DET
ejpam-677	6	9	generalised	generalise	VERB
ejpam-677	6	10	hyers	hyer	NOUN
ejpam-677	6	11	-	-	PUNCT
ejpam-677	6	12	ulam	ulam	ADJ
ejpam-677	6	13	“	"	PUNCT
ejpam-677	6	14	product	product	NOUN
ejpam-677	6	15	-	-	PUNCT
ejpam-677	6	16	sum	sum	NOUN
ejpam-677	6	17	”	"	PUNCT
ejpam-677	6	18	stability	stability	NOUN
ejpam-677	6	19	.	.	PUNCT
ejpam-677	7	1	in	in	ADP
ejpam-677	7	2	this	this	DET
ejpam-677	7	3	paper	paper	NOUN
ejpam-677	7	4	we	we	PRON
ejpam-677	7	5	introduce	introduce	VERB
ejpam-677	7	6	a	a	DET
ejpam-677	7	7	cauchy	cauchy	ADJ
ejpam-677	7	8	type	type	NOUN
ejpam-677	7	9	additive	additive	ADJ
ejpam-677	7	10	functional	functional	ADJ
ejpam-677	7	11	equation	equation	NOUN
ejpam-677	7	12	and	and	CCONJ
ejpam-677	7	13	investigate	investigate	VERB
ejpam-677	7	14	the	the	DET
ejpam-677	7	15	generalised	generalise	VERB
ejpam-677	7	16	hyers	hyer	NOUN
ejpam-677	7	17	-	-	PUNCT
ejpam-677	7	18	ulam	ulam	ADJ
ejpam-677	7	19	“	"	PUNCT
ejpam-677	7	20	product	product	NOUN
ejpam-677	7	21	-	-	PUNCT
ejpam-677	7	22	sum	sum	NOUN
ejpam-677	7	23	”	"	PUNCT
ejpam-677	7	24	stability	stability	NOUN
ejpam-677	7	25	of	of	ADP
ejpam-677	7	26	this	this	DET
ejpam-677	7	27	equation	equation	NOUN
ejpam-677	7	28	.	.	PUNCT
ejpam-677	8	1	2000	2000	NUM
ejpam-677	8	2	mathematics	mathematic	NOUN
ejpam-677	8	3	subject	subject	NOUN
ejpam-677	8	4	classifications	classification	NOUN
ejpam-677	8	5	:	:	PUNCT
ejpam-677	8	6	primary	primary	ADJ
ejpam-677	8	7	39b	39b	NOUN
ejpam-677	8	8	.	.	PUNCT
ejpam-677	9	1	secondary	secondary	ADJ
ejpam-677	9	2	26d	26d	NUM
ejpam-677	9	3	.	.	PUNCT
ejpam-677	10	1	key	key	ADJ
ejpam-677	10	2	words	word	NOUN
ejpam-677	10	3	and	and	CCONJ
ejpam-677	10	4	phrases	phrase	NOUN
ejpam-677	10	5	:	:	PUNCT
ejpam-677	10	6	generalised	generalise	VERB
ejpam-677	10	7	“	"	PUNCT
ejpam-677	10	8	product	product	NOUN
ejpam-677	10	9	-	-	PUNCT
ejpam-677	10	10	sum	sum	NOUN
ejpam-677	10	11	”	"	PUNCT
ejpam-677	10	12	hyers	hyers	PROPN
ejpam-677	10	13	-	-	PUNCT
ejpam-677	10	14	ulam	ulam	PROPN
ejpam-677	10	15	stability	stability	NOUN
ejpam-677	10	16	,	,	PUNCT
ejpam-677	10	17	cauchy	cauchy	ADJ
ejpam-677	10	18	type	type	NOUN
ejpam-677	10	19	additive	additive	ADJ
ejpam-677	10	20	functional	functional	ADJ
ejpam-677	10	21	equation	equation	NOUN
ejpam-677	10	22	.	.	PUNCT
ejpam-677	11	1	1	1	X
ejpam-677	11	2	.	.	X
ejpam-677	11	3	introduction	introduction	NOUN
ejpam-677	11	4	and	and	CCONJ
ejpam-677	11	5	preliminaries	preliminary	NOUN
ejpam-677	11	6	in	in	ADP
ejpam-677	11	7	1940	1940	NUM
ejpam-677	11	8	(	(	PUNCT
ejpam-677	11	9	and	and	CCONJ
ejpam-677	11	10	1964	1964	NUM
ejpam-677	11	11	)	)	PUNCT
ejpam-677	11	12	stanislaw	stanislaw	NOUN
ejpam-677	11	13	m.	m.	NOUN
ejpam-677	11	14	ulam	ulam	PROPN
ejpam-677	12	1	[	[	X
ejpam-677	12	2	9	9	NUM
ejpam-677	12	3	]	]	PUNCT
ejpam-677	12	4	proposed	propose	VERB
ejpam-677	12	5	the	the	DET
ejpam-677	12	6	following	follow	VERB
ejpam-677	12	7	stability	stability	NOUN
ejpam-677	12	8	problem	problem	NOUN
ejpam-677	12	9	,	,	PUNCT
ejpam-677	12	10	well	well	ADV
ejpam-677	12	11	-	-	PUNCT
ejpam-677	12	12	known	know	VERB
ejpam-677	12	13	as	as	ADP
ejpam-677	12	14	ulam	ulam	PROPN
ejpam-677	12	15	stability	stability	PROPN
ejpam-677	12	16	problem	problem	NOUN
ejpam-677	12	17	:	:	PUNCT
ejpam-677	12	18	“	"	PUNCT
ejpam-677	12	19	when	when	SCONJ
ejpam-677	12	20	is	be	AUX
ejpam-677	12	21	true	true	ADJ
ejpam-677	12	22	that	that	SCONJ
ejpam-677	12	23	by	by	ADP
ejpam-677	12	24	slightly	slightly	ADV
ejpam-677	12	25	changing	change	VERB
ejpam-677	12	26	the	the	DET
ejpam-677	12	27	hypotheses	hypothesis	NOUN
ejpam-677	12	28	of	of	ADP
ejpam-677	12	29	a	a	DET
ejpam-677	12	30	theorem	theorem	ADJ
ejpam-677	12	31	one	one	PRON
ejpam-677	12	32	can	can	AUX
ejpam-677	12	33	still	still	ADV
ejpam-677	12	34	assert	assert	VERB
ejpam-677	12	35	that	that	SCONJ
ejpam-677	12	36	the	the	DET
ejpam-677	12	37	thesis	thesis	NOUN
ejpam-677	12	38	of	of	ADP
ejpam-677	12	39	the	the	DET
ejpam-677	12	40	theorem	theorem	NOUN
ejpam-677	12	41	remains	remain	VERB
ejpam-677	12	42	true	true	ADJ
ejpam-677	12	43	or	or	CCONJ
ejpam-677	12	44	approximately	approximately	ADV
ejpam-677	12	45	true	true	ADJ
ejpam-677	12	46	?	?	PUNCT
ejpam-677	12	47	”	"	PUNCT
ejpam-677	13	1	in	in	ADP
ejpam-677	13	2	particular	particular	ADJ
ejpam-677	13	3	he	he	PRON
ejpam-677	13	4	stated	state	VERB
ejpam-677	13	5	the	the	DET
ejpam-677	13	6	stability	stability	NOUN
ejpam-677	13	7	question	question	NOUN
ejpam-677	13	8	:	:	PUNCT
ejpam-677	13	9	“	"	PUNCT
ejpam-677	13	10	let	let	VERB
ejpam-677	13	11	g1	g1	PROPN
ejpam-677	13	12	be	be	AUX
ejpam-677	13	13	a	a	DET
ejpam-677	13	14	group	group	NOUN
ejpam-677	13	15	and	and	CCONJ
ejpam-677	13	16	g2	g2	PROPN
ejpam-677	13	17	a	a	DET
ejpam-677	13	18	metric	metric	ADJ
ejpam-677	13	19	group	group	NOUN
ejpam-677	13	20	with	with	ADP
ejpam-677	13	21	the	the	DET
ejpam-677	13	22	metric	metric	ADJ
ejpam-677	13	23	ρ	ρ	PROPN
ejpam-677	13	24	(	(	PUNCT
ejpam-677	13	25	.	.	PUNCT
ejpam-677	13	26	,	,	PUNCT
ejpam-677	13	27	.	.	PUNCT
ejpam-677	13	28	)	)	PUNCT
ejpam-677	13	29	.	.	PUNCT
ejpam-677	14	1	given	give	VERB
ejpam-677	14	2	a	a	DET
ejpam-677	14	3	constant	constant	ADJ
ejpam-677	14	4	δ	δ	NOUN
ejpam-677	14	5	>	>	X
ejpam-677	14	6	0	0	PROPN
ejpam-677	14	7	,	,	PUNCT
ejpam-677	14	8	does	do	AUX
ejpam-677	14	9	there	there	PRON
ejpam-677	14	10	exist	exist	VERB
ejpam-677	14	11	a	a	DET
ejpam-677	14	12	constant	constant	ADJ
ejpam-677	14	13	c	c	NOUN
ejpam-677	14	14	>	>	X
ejpam-677	14	15	0	0	NUM
ejpam-677	15	1	such	such	ADJ
ejpam-677	15	2	that	that	SCONJ
ejpam-677	15	3	if	if	SCONJ
ejpam-677	15	4	a	a	DET
ejpam-677	15	5	mapping	mapping	NOUN
ejpam-677	15	6	f	f	NOUN
ejpam-677	15	7	:	:	PUNCT
ejpam-677	15	8	g1	g1	PROPN
ejpam-677	15	9	→	→	SYM
ejpam-677	15	10	g2	g2	PROPN
ejpam-677	15	11	satisfies	satisfy	VERB
ejpam-677	15	12	ρ	ρ	PROPN
ejpam-677	15	13	(	(	PUNCT
ejpam-677	15	14	f	f	PROPN
ejpam-677	15	15	(	(	PUNCT
ejpam-677	15	16	x	x	PROPN
ejpam-677	15	17	y	y	PROPN
ejpam-677	15	18	)	)	PUNCT
ejpam-677	15	19	,	,	PUNCT
ejpam-677	15	20	f	f	PROPN
ejpam-677	15	21	(	(	PUNCT
ejpam-677	15	22	x	x	X
ejpam-677	15	23	)	)	PUNCT
ejpam-677	15	24	f	f	PROPN
ejpam-677	15	25	(	(	PUNCT
ejpam-677	15	26	y	y	NOUN
ejpam-677	15	27	)	)	PUNCT
ejpam-677	15	28	)	)	PUNCT
ejpam-677	16	1	<	<	X
ejpam-677	16	2	c	c	NOUN
ejpam-677	16	3	for	for	ADP
ejpam-677	16	4	all	all	DET
ejpam-677	16	5	x	x	SYM
ejpam-677	16	6	,	,	PUNCT
ejpam-677	16	7	y	y	PROPN
ejpam-677	16	8	∈	∈	PROPN
ejpam-677	16	9	g1	g1	PROPN
ejpam-677	16	10	,	,	PUNCT
ejpam-677	16	11	then	then	ADV
ejpam-677	16	12	a	a	DET
ejpam-677	16	13	unique	unique	ADJ
ejpam-677	16	14	homomorphism	homomorphism	NOUN
ejpam-677	16	15	h	h	NOUN
ejpam-677	16	16	:	:	PUNCT
ejpam-677	16	17	g1→	g1→	NOUN
ejpam-677	16	18	g2	g2	PROPN
ejpam-677	16	19	exists	exist	VERB
ejpam-677	16	20	with	with	ADP
ejpam-677	16	21	ρ	ρ	PROPN
ejpam-677	16	22	(	(	PUNCT
ejpam-677	16	23	f	f	PROPN
ejpam-677	16	24	(	(	PUNCT
ejpam-677	16	25	x),h(x	x),h(x	PROPN
ejpam-677	16	26	)	)	PUNCT
ejpam-677	16	27	)	)	PUNCT
ejpam-677	16	28	<	<	X
ejpam-677	16	29	δ	δ	PROPN
ejpam-677	16	30	for	for	ADP
ejpam-677	16	31	all	all	DET
ejpam-677	16	32	x	x	SYM
ejpam-677	16	33	∈	∈	PROPN
ejpam-677	16	34	g1	g1	PROPN
ejpam-677	16	35	?	?	PUNCT
ejpam-677	16	36	”	"	PUNCT
ejpam-677	17	1	in	in	ADP
ejpam-677	17	2	1941	1941	NUM
ejpam-677	17	3	d.h	d.h	PROPN
ejpam-677	17	4	.	.	PUNCT
ejpam-677	17	5	hyers	hyer	NOUN
ejpam-677	18	1	[	[	X
ejpam-677	18	2	2	2	X
ejpam-677	18	3	]	]	PUNCT
ejpam-677	18	4	solved	solve	VERB
ejpam-677	18	5	this	this	DET
ejpam-677	18	6	problem	problem	NOUN
ejpam-677	18	7	for	for	ADP
ejpam-677	18	8	linear	linear	ADJ
ejpam-677	18	9	mappings	mapping	NOUN
ejpam-677	18	10	as	as	SCONJ
ejpam-677	18	11	follows	follow	VERB
ejpam-677	18	12	:	:	PUNCT
ejpam-677	18	13	email	email	NOUN
ejpam-677	18	14	address	address	NOUN
ejpam-677	18	15	:	:	PUNCT
ejpam-677	18	16	rassias.matina	rassias.matina	NOUN
ejpam-677	18	17	�	�	NOUN
ejpam-677	18	18	gmail	gmail	NOUN
ejpam-677	18	19	.	.	PUNCT
ejpam-677	19	1	om	om	PROPN
ejpam-677	19	2	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-677	20	1	50	50	NUM
ejpam-677	20	2	c	c	NOUN
ejpam-677	20	3	©	©	PROPN
ejpam-677	20	4	2010	2010	NUM
ejpam-677	20	5	ejpam	ejpam	NOUN
ejpam-677	20	6	all	all	DET
ejpam-677	20	7	rights	right	NOUN
ejpam-677	20	8	reserved	reserve	VERB
ejpam-677	20	9	.	.	PUNCT
ejpam-677	21	1	m.	m.	NOUN
ejpam-677	21	2	rassias	rassias	PROPN
ejpam-677	21	3	/	/	SYM
ejpam-677	21	4	eur	eur	PROPN
ejpam-677	21	5	.	.	PUNCT
ejpam-677	22	1	j.	j.	PROPN
ejpam-677	22	2	pure	pure	PROPN
ejpam-677	22	3	appl	appl	PROPN
ejpam-677	22	4	.	.	PROPN
ejpam-677	22	5	math	math	PROPN
ejpam-677	22	6	,	,	PUNCT
ejpam-677	22	7	4	4	NUM
ejpam-677	22	8	(	(	PUNCT
ejpam-677	22	9	2011	2011	NUM
ejpam-677	22	10	)	)	PUNCT
ejpam-677	22	11	,	,	PUNCT
ejpam-677	22	12	50	50	NUM
ejpam-677	22	13	-	-	SYM
ejpam-677	22	14	58	58	NUM
ejpam-677	22	15	51	51	NUM
ejpam-677	22	16	theorem	theorem	NOUN
ejpam-677	22	17	1	1	NUM
ejpam-677	22	18	(	(	PUNCT
ejpam-677	22	19	d.h	d.h	PROPN
ejpam-677	22	20	.	.	PROPN
ejpam-677	22	21	hyers	hyer	NOUN
ejpam-677	22	22	,	,	PUNCT
ejpam-677	22	23	1941	1941	NUM
ejpam-677	22	24	:	:	PUNCT
ejpam-677	23	1	[	[	X
ejpam-677	23	2	2	2	NUM
ejpam-677	23	3	]	]	PUNCT
ejpam-677	23	4	)	)	PUNCT
ejpam-677	23	5	.	.	PUNCT
ejpam-677	24	1	if	if	SCONJ
ejpam-677	24	2	a	a	DET
ejpam-677	24	3	mapping	mapping	NOUN
ejpam-677	24	4	f	f	X
ejpam-677	24	5	:	:	PUNCT
ejpam-677	24	6	e	e	X
ejpam-677	24	7	→	→	SYM
ejpam-677	24	8	e′	e′	X
ejpam-677	24	9	satisfies	satisfy	VERB
ejpam-677	24	10	the	the	DET
ejpam-677	24	11	approximately	approximately	ADV
ejpam-677	24	12	additive	additive	ADJ
ejpam-677	24	13	inequality	inequality	NOUN
ejpam-677	24	14	||	||	PUNCT
ejpam-677	25	1	f	f	PROPN
ejpam-677	25	2	(	(	PUNCT
ejpam-677	25	3	x	x	X
ejpam-677	25	4	+	+	PUNCT
ejpam-677	25	5	y)−	y)−	PROPN
ejpam-677	25	6	f	f	NOUN
ejpam-677	25	7	(	(	PUNCT
ejpam-677	25	8	x)−	x)−	PROPN
ejpam-677	25	9	f	f	PROPN
ejpam-677	25	10	(	(	PUNCT
ejpam-677	25	11	y)||	y)||	INTJ
ejpam-677	25	12	≤	≤	ADJ
ejpam-677	25	13	ǫ	ǫ	X
ejpam-677	25	14	,	,	PUNCT
ejpam-677	25	15	for	for	ADP
ejpam-677	25	16	some	some	DET
ejpam-677	25	17	fixed	fixed	ADJ
ejpam-677	25	18	ǫ	ǫ	NOUN
ejpam-677	25	19	>	>	X
ejpam-677	25	20	0	0	PUNCT
ejpam-677	25	21	and	and	CCONJ
ejpam-677	25	22	all	all	PRON
ejpam-677	25	23	x	x	NOUN
ejpam-677	25	24	,	,	PUNCT
ejpam-677	25	25	y	y	PROPN
ejpam-677	25	26	∈	∈	PROPN
ejpam-677	25	27	e	e	NOUN
ejpam-677	25	28	,	,	PUNCT
ejpam-677	25	29	where	where	SCONJ
ejpam-677	25	30	e	e	NOUN
ejpam-677	25	31	and	and	CCONJ
ejpam-677	25	32	e′	e′	NOUN
ejpam-677	25	33	are	be	AUX
ejpam-677	25	34	banach	banach	NOUN
ejpam-677	25	35	spaces	space	NOUN
ejpam-677	25	36	,	,	PUNCT
ejpam-677	25	37	then	then	ADV
ejpam-677	25	38	there	there	PRON
ejpam-677	25	39	exists	exist	VERB
ejpam-677	25	40	a	a	DET
ejpam-677	25	41	unique	unique	ADJ
ejpam-677	25	42	additive	additive	NOUN
ejpam-677	25	43	mapping	mapping	NOUN
ejpam-677	25	44	a	a	PRON
ejpam-677	25	45	:	:	PUNCT
ejpam-677	25	46	e→	e→	PROPN
ejpam-677	25	47	e′	e′	PROPN
ejpam-677	25	48	,	,	PUNCT
ejpam-677	25	49	satisfying	satisfy	VERB
ejpam-677	25	50	the	the	DET
ejpam-677	25	51	formula	formula	NOUN
ejpam-677	25	52	a(x	a(x	NOUN
ejpam-677	25	53	)	)	PUNCT
ejpam-677	25	54	=	=	VERB
ejpam-677	26	1	lim	lim	PROPN
ejpam-677	26	2	n→∞2−n	n→∞2−n	PROPN
ejpam-677	26	3	f	f	PROPN
ejpam-677	26	4	(	(	PUNCT
ejpam-677	26	5	2n	2n	NUM
ejpam-677	26	6	x	x	NOUN
ejpam-677	26	7	)	)	PUNCT
ejpam-677	26	8	,	,	PUNCT
ejpam-677	26	9	and	and	CCONJ
ejpam-677	26	10	inequality	inequality	NOUN
ejpam-677	26	11	||	||	PUNCT
ejpam-677	27	1	f	f	X
ejpam-677	27	2	(	(	PUNCT
ejpam-677	27	3	x)−	x)−	PROPN
ejpam-677	27	4	a(x)||	a(x)||	NOUN
ejpam-677	27	5	≤	≤	NOUN
ejpam-677	27	6	ǫ	ǫ	NOUN
ejpam-677	27	7	for	for	ADP
ejpam-677	27	8	some	some	DET
ejpam-677	27	9	fixed	fixed	ADJ
ejpam-677	27	10	ǫ	ǫ	NOUN
ejpam-677	27	11	>	>	X
ejpam-677	27	12	0	0	PUNCT
ejpam-677	27	13	and	and	CCONJ
ejpam-677	27	14	all	all	DET
ejpam-677	27	15	x	x	SYM
ejpam-677	27	16	∈	∈	PROPN
ejpam-677	27	17	e.	e.	NOUN
ejpam-677	28	1	no	no	DET
ejpam-677	28	2	continuity	continuity	NOUN
ejpam-677	28	3	conditions	condition	NOUN
ejpam-677	28	4	are	be	AUX
ejpam-677	28	5	required	require	VERB
ejpam-677	28	6	for	for	ADP
ejpam-677	28	7	this	this	DET
ejpam-677	28	8	result	result	NOUN
ejpam-677	28	9	.	.	PUNCT
ejpam-677	29	1	theorem	theorem	ADJ
ejpam-677	29	2	2	2	NUM
ejpam-677	29	3	(	(	PUNCT
ejpam-677	29	4	t.	t.	PROPN
ejpam-677	29	5	aoki	aoki	PROPN
ejpam-677	29	6	,	,	PUNCT
ejpam-677	29	7	1950	1950	NUM
ejpam-677	29	8	:	:	PUNCT
ejpam-677	30	1	[	[	X
ejpam-677	30	2	1	1	NUM
ejpam-677	30	3	]	]	PUNCT
ejpam-677	30	4	)	)	PUNCT
ejpam-677	30	5	.	.	PUNCT
ejpam-677	31	1	let	let	VERB
ejpam-677	31	2	f	f	NOUN
ejpam-677	31	3	:	:	PUNCT
ejpam-677	31	4	e	e	X
ejpam-677	31	5	→	→	SYM
ejpam-677	31	6	e′	e′	X
ejpam-677	31	7	be	be	AUX
ejpam-677	31	8	a	a	DET
ejpam-677	31	9	mapping	mapping	NOUN
ejpam-677	31	10	from	from	ADP
ejpam-677	31	11	a	a	DET
ejpam-677	31	12	normed	normed	ADJ
ejpam-677	31	13	vector	vector	NOUN
ejpam-677	31	14	space	space	NOUN
ejpam-677	31	15	e	e	NOUN
ejpam-677	31	16	into	into	ADP
ejpam-677	31	17	a	a	DET
ejpam-677	31	18	banach	banach	NOUN
ejpam-677	31	19	space	space	NOUN
ejpam-677	31	20	e′	e′	NOUN
ejpam-677	31	21	subject	subject	NOUN
ejpam-677	31	22	to	to	ADP
ejpam-677	31	23	the	the	DET
ejpam-677	31	24	inequality	inequality	NOUN
ejpam-677	31	25	||	||	PUNCT
ejpam-677	32	1	f	f	PROPN
ejpam-677	32	2	(	(	PUNCT
ejpam-677	32	3	x	x	X
ejpam-677	32	4	+	+	PUNCT
ejpam-677	32	5	y)−	y)−	PROPN
ejpam-677	32	6	f	f	NOUN
ejpam-677	32	7	(	(	PUNCT
ejpam-677	32	8	x)−	x)−	PROPN
ejpam-677	32	9	f	f	PROPN
ejpam-677	32	10	(	(	PUNCT
ejpam-677	32	11	y)||	y)||	ADJ
ejpam-677	32	12	≤	≤	NUM
ejpam-677	32	13	ǫ(||x	ǫ(||x	ADJ
ejpam-677	32	14	||p+	||p+	NOUN
ejpam-677	32	15	||y||p	||y||p	NOUN
ejpam-677	32	16	)	)	PUNCT
ejpam-677	32	17	,	,	PUNCT
ejpam-677	32	18	(	(	PUNCT
ejpam-677	32	19	1	1	X
ejpam-677	32	20	)	)	PUNCT
ejpam-677	32	21	for	for	ADP
ejpam-677	32	22	all	all	DET
ejpam-677	32	23	x	x	SYM
ejpam-677	32	24	,	,	PUNCT
ejpam-677	32	25	y	y	PROPN
ejpam-677	32	26	∈	∈	PROPN
ejpam-677	32	27	e	e	NOUN
ejpam-677	32	28	,	,	PUNCT
ejpam-677	32	29	where	where	SCONJ
ejpam-677	32	30	ǫ	ǫ	PRON
ejpam-677	32	31	>	>	X
ejpam-677	32	32	0	0	PUNCT
ejpam-677	32	33	and	and	CCONJ
ejpam-677	32	34	p	p	X
ejpam-677	32	35	<	<	X
ejpam-677	32	36	1	1	NUM
ejpam-677	32	37	constants	constant	NOUN
ejpam-677	32	38	.	.	PUNCT
ejpam-677	33	1	then	then	ADV
ejpam-677	33	2	the	the	DET
ejpam-677	33	3	limit	limit	NOUN
ejpam-677	33	4	a(x	a(x	NOUN
ejpam-677	33	5	)	)	PUNCT
ejpam-677	33	6	=	=	VERB
ejpam-677	34	1	lim	lim	PROPN
ejpam-677	34	2	n→∞2−n	n→∞2−n	PROPN
ejpam-677	34	3	f	f	PROPN
ejpam-677	34	4	(	(	PUNCT
ejpam-677	34	5	2n	2n	NUM
ejpam-677	34	6	x	x	NOUN
ejpam-677	34	7	)	)	PUNCT
ejpam-677	34	8	,	,	PUNCT
ejpam-677	34	9	exists	exist	VERB
ejpam-677	34	10	for	for	ADP
ejpam-677	34	11	all	all	DET
ejpam-677	34	12	x	x	SYM
ejpam-677	34	13	∈	∈	PROPN
ejpam-677	34	14	e	e	NOUN
ejpam-677	34	15	and	and	CCONJ
ejpam-677	34	16	a	a	DET
ejpam-677	34	17	:	:	PUNCT
ejpam-677	34	18	e→	e→	NOUN
ejpam-677	34	19	e′	e′	PROPN
ejpam-677	34	20	is	be	AUX
ejpam-677	34	21	the	the	DET
ejpam-677	34	22	unique	unique	ADJ
ejpam-677	34	23	additive	additive	ADJ
ejpam-677	34	24	mapping	mapping	NOUN
ejpam-677	34	25	which	which	PRON
ejpam-677	34	26	satisfies	satisfy	VERB
ejpam-677	34	27	||	||	PUNCT
ejpam-677	35	1	f	f	X
ejpam-677	35	2	(	(	PUNCT
ejpam-677	35	3	x)−	x)−	PROPN
ejpam-677	35	4	a(x)||	a(x)||	NOUN
ejpam-677	35	5	≤	≤	ADJ
ejpam-677	35	6	2ǫ	2ǫ	NOUN
ejpam-677	35	7	2−	2−	NUM
ejpam-677	35	8	2p	2p	NOUN
ejpam-677	35	9	||x	||x	NOUN
ejpam-677	35	10	||p	||p	NOUN
ejpam-677	35	11	(	(	PUNCT
ejpam-677	35	12	2	2	NUM
ejpam-677	35	13	)	)	PUNCT
ejpam-677	35	14	for	for	ADP
ejpam-677	35	15	all	all	DET
ejpam-677	35	16	x	x	SYM
ejpam-677	35	17	∈	∈	PROPN
ejpam-677	35	18	e.	e.	NOUN
ejpam-677	35	19	if	if	SCONJ
ejpam-677	35	20	p	p	PROPN
ejpam-677	35	21	<	<	X
ejpam-677	35	22	0	0	X
ejpam-677	36	1	then	then	ADV
ejpam-677	36	2	the	the	DET
ejpam-677	36	3	inequality	inequality	NOUN
ejpam-677	36	4	(	(	PUNCT
ejpam-677	36	5	1	1	X
ejpam-677	36	6	)	)	PUNCT
ejpam-677	36	7	holds	hold	VERB
ejpam-677	36	8	for	for	ADP
ejpam-677	36	9	x	x	SYM
ejpam-677	36	10	,	,	PUNCT
ejpam-677	36	11	y	y	PROPN
ejpam-677	36	12	6=	6=	PROPN
ejpam-677	36	13	0	0	NUM
ejpam-677	36	14	and	and	CCONJ
ejpam-677	36	15	(	(	PUNCT
ejpam-677	36	16	2	2	NUM
ejpam-677	36	17	)	)	PUNCT
ejpam-677	36	18	for	for	ADP
ejpam-677	36	19	x	x	SYM
ejpam-677	36	20	6=	6=	ADP
ejpam-677	36	21	0	0	NUM
ejpam-677	36	22	.	.	PUNCT
ejpam-677	37	1	theorem	theorem	ADJ
ejpam-677	37	2	3	3	NUM
ejpam-677	37	3	(	(	PUNCT
ejpam-677	37	4	th	th	NOUN
ejpam-677	37	5	.	.	PUNCT
ejpam-677	37	6	m.	m.	NOUN
ejpam-677	37	7	rassias	rassias	PROPN
ejpam-677	37	8	,	,	PUNCT
ejpam-677	37	9	1978	1978	NUM
ejpam-677	37	10	:	:	PUNCT
ejpam-677	38	1	[	[	X
ejpam-677	38	2	6	6	NUM
ejpam-677	38	3	]	]	PUNCT
ejpam-677	38	4	)	)	PUNCT
ejpam-677	38	5	.	.	PUNCT
ejpam-677	39	1	let	let	VERB
ejpam-677	39	2	f	f	NOUN
ejpam-677	39	3	:	:	PUNCT
ejpam-677	39	4	e	e	X
ejpam-677	39	5	→	→	SYM
ejpam-677	39	6	e′	e′	X
ejpam-677	39	7	be	be	AUX
ejpam-677	39	8	a	a	DET
ejpam-677	39	9	mapping	mapping	NOUN
ejpam-677	39	10	from	from	ADP
ejpam-677	39	11	a	a	DET
ejpam-677	39	12	normed	normed	ADJ
ejpam-677	39	13	vector	vector	NOUN
ejpam-677	39	14	space	space	NOUN
ejpam-677	39	15	e	e	NOUN
ejpam-677	39	16	into	into	ADP
ejpam-677	39	17	a	a	DET
ejpam-677	39	18	banach	banach	NOUN
ejpam-677	39	19	space	space	NOUN
ejpam-677	39	20	e′	e′	NOUN
ejpam-677	39	21	subject	subject	NOUN
ejpam-677	39	22	to	to	ADP
ejpam-677	39	23	the	the	DET
ejpam-677	39	24	inequality	inequality	NOUN
ejpam-677	39	25	||	||	PUNCT
ejpam-677	40	1	f	f	PROPN
ejpam-677	40	2	(	(	PUNCT
ejpam-677	40	3	x	x	X
ejpam-677	40	4	+	+	PUNCT
ejpam-677	40	5	y)−	y)−	PROPN
ejpam-677	40	6	f	f	NOUN
ejpam-677	40	7	(	(	PUNCT
ejpam-677	40	8	x)−	x)−	PROPN
ejpam-677	40	9	f	f	PROPN
ejpam-677	40	10	(	(	PUNCT
ejpam-677	40	11	y)||	y)||	ADJ
ejpam-677	40	12	≤	≤	NUM
ejpam-677	40	13	ǫ(||x	ǫ(||x	ADJ
ejpam-677	40	14	||p+	||p+	NOUN
ejpam-677	40	15	||y||p	||y||p	NOUN
ejpam-677	40	16	)	)	PUNCT
ejpam-677	40	17	,	,	PUNCT
ejpam-677	40	18	(	(	PUNCT
ejpam-677	40	19	3	3	X
ejpam-677	40	20	)	)	PUNCT
ejpam-677	40	21	for	for	ADP
ejpam-677	40	22	all	all	DET
ejpam-677	40	23	x	x	SYM
ejpam-677	40	24	,	,	PUNCT
ejpam-677	40	25	y	y	PROPN
ejpam-677	40	26	∈	∈	PROPN
ejpam-677	40	27	e	e	NOUN
ejpam-677	40	28	,	,	PUNCT
ejpam-677	40	29	where	where	SCONJ
ejpam-677	40	30	ǫ	ǫ	PRON
ejpam-677	40	31	>	>	X
ejpam-677	40	32	0	0	PUNCT
ejpam-677	40	33	and	and	CCONJ
ejpam-677	40	34	p	p	X
ejpam-677	40	35	<	<	X
ejpam-677	40	36	1	1	NUM
ejpam-677	40	37	constants	constant	NOUN
ejpam-677	40	38	.	.	PUNCT
ejpam-677	41	1	then	then	ADV
ejpam-677	41	2	the	the	DET
ejpam-677	41	3	limit	limit	NOUN
ejpam-677	41	4	a(x	a(x	NOUN
ejpam-677	41	5	)	)	PUNCT
ejpam-677	41	6	=	=	VERB
ejpam-677	42	1	lim	lim	PROPN
ejpam-677	42	2	n→∞2−n	n→∞2−n	PROPN
ejpam-677	42	3	f	f	PROPN
ejpam-677	42	4	(	(	PUNCT
ejpam-677	42	5	2n	2n	NUM
ejpam-677	42	6	x	x	NOUN
ejpam-677	42	7	)	)	PUNCT
ejpam-677	42	8	,	,	PUNCT
ejpam-677	42	9	exists	exist	VERB
ejpam-677	42	10	for	for	ADP
ejpam-677	42	11	all	all	DET
ejpam-677	42	12	x	x	SYM
ejpam-677	42	13	∈	∈	PROPN
ejpam-677	42	14	e	e	NOUN
ejpam-677	42	15	and	and	CCONJ
ejpam-677	42	16	a	a	DET
ejpam-677	42	17	:	:	PUNCT
ejpam-677	42	18	e→	e→	NOUN
ejpam-677	42	19	e′	e′	PROPN
ejpam-677	42	20	is	be	AUX
ejpam-677	42	21	the	the	DET
ejpam-677	42	22	unique	unique	ADJ
ejpam-677	42	23	additive	additive	ADJ
ejpam-677	42	24	mapping	mapping	NOUN
ejpam-677	42	25	which	which	PRON
ejpam-677	42	26	satisfies	satisfy	VERB
ejpam-677	42	27	||	||	PUNCT
ejpam-677	43	1	f	f	X
ejpam-677	43	2	(	(	PUNCT
ejpam-677	43	3	x)−	x)−	PROPN
ejpam-677	43	4	a(x)||	a(x)||	NOUN
ejpam-677	43	5	≤	≤	ADJ
ejpam-677	43	6	2ǫ	2ǫ	NOUN
ejpam-677	43	7	2−	2−	NUM
ejpam-677	43	8	2p	2p	NOUN
ejpam-677	43	9	||x	||x	NOUN
ejpam-677	43	10	||p	||p	NOUN
ejpam-677	43	11	(	(	PUNCT
ejpam-677	43	12	4	4	NUM
ejpam-677	43	13	)	)	PUNCT
ejpam-677	43	14	for	for	ADP
ejpam-677	43	15	all	all	DET
ejpam-677	43	16	x	x	SYM
ejpam-677	43	17	∈	∈	PROPN
ejpam-677	43	18	e.	e.	NOUN
ejpam-677	43	19	if	if	SCONJ
ejpam-677	43	20	p	p	PROPN
ejpam-677	43	21	<	<	X
ejpam-677	43	22	0	0	X
ejpam-677	44	1	then	then	ADV
ejpam-677	44	2	the	the	DET
ejpam-677	44	3	inequality	inequality	NOUN
ejpam-677	44	4	(	(	PUNCT
ejpam-677	44	5	3	3	X
ejpam-677	44	6	)	)	PUNCT
ejpam-677	44	7	holds	hold	VERB
ejpam-677	44	8	for	for	ADP
ejpam-677	44	9	x	x	SYM
ejpam-677	44	10	,	,	PUNCT
ejpam-677	44	11	y	y	PROPN
ejpam-677	44	12	6=	6=	PROPN
ejpam-677	44	13	0	0	NUM
ejpam-677	44	14	and	and	CCONJ
ejpam-677	44	15	(	(	PUNCT
ejpam-677	44	16	4	4	NUM
ejpam-677	44	17	)	)	PUNCT
ejpam-677	44	18	for	for	ADP
ejpam-677	44	19	x	x	SYM
ejpam-677	44	20	6=	6=	ADP
ejpam-677	44	21	0	0	NUM
ejpam-677	44	22	.	.	PUNCT
ejpam-677	45	1	if	if	SCONJ
ejpam-677	45	2	,	,	PUNCT
ejpam-677	45	3	moreover	moreover	ADV
ejpam-677	45	4	,	,	PUNCT
ejpam-677	45	5	f	f	PROPN
ejpam-677	45	6	(	(	PUNCT
ejpam-677	45	7	t	t	NOUN
ejpam-677	45	8	x	x	VERB
ejpam-677	45	9	)	)	PUNCT
ejpam-677	45	10	is	be	AUX
ejpam-677	45	11	continuous	continuous	ADJ
ejpam-677	45	12	in	in	ADP
ejpam-677	45	13	t	t	PROPN
ejpam-677	45	14	∈	∈	NOUN
ejpam-677	45	15	r	r	NOUN
ejpam-677	45	16	for	for	ADP
ejpam-677	45	17	each	each	DET
ejpam-677	45	18	fixed	fix	VERB
ejpam-677	45	19	x	x	SYM
ejpam-677	45	20	∈	∈	PROPN
ejpam-677	45	21	e	e	NOUN
ejpam-677	45	22	,	,	PUNCT
ejpam-677	45	23	then	then	ADV
ejpam-677	45	24	a(t	a(t	PROPN
ejpam-677	45	25	x	x	PRON
ejpam-677	45	26	)	)	PUNCT
ejpam-677	45	27	=	=	SYM
ejpam-677	45	28	ta(x	ta(x	NOUN
ejpam-677	45	29	)	)	PUNCT
ejpam-677	45	30	for	for	ADP
ejpam-677	45	31	all	all	DET
ejpam-677	45	32	x	x	SYM
ejpam-677	45	33	∈	∈	PROPN
ejpam-677	45	34	e	e	NOUN
ejpam-677	45	35	and	and	CCONJ
ejpam-677	45	36	t	t	PROPN
ejpam-677	45	37	∈	∈	PROPN
ejpam-677	45	38	r.	r.	PROPN
ejpam-677	45	39	a	a	PRON
ejpam-677	45	40	:	:	PUNCT
ejpam-677	45	41	e→	e→	PROPN
ejpam-677	45	42	e′	e′	PROPN
ejpam-677	45	43	is	be	AUX
ejpam-677	45	44	a	a	DET
ejpam-677	45	45	unique	unique	ADJ
ejpam-677	45	46	linear	linear	NOUN
ejpam-677	45	47	additive	additive	ADJ
ejpam-677	45	48	mapping	mapping	NOUN
ejpam-677	45	49	satisfying	satisfy	VERB
ejpam-677	45	50	equation	equation	NOUN
ejpam-677	45	51	a(x	a(x	NOUN
ejpam-677	45	52	+	+	PROPN
ejpam-677	45	53	y	y	NOUN
ejpam-677	45	54	)	)	PUNCT
ejpam-677	45	55	=	=	SYM
ejpam-677	45	56	a(x)+	a(x)+	ADP
ejpam-677	45	57	a(y	a(y	PROPN
ejpam-677	45	58	)	)	PUNCT
ejpam-677	45	59	.	.	PUNCT
ejpam-677	46	1	m.	m.	NOUN
ejpam-677	46	2	rassias	rassias	PROPN
ejpam-677	46	3	/	/	SYM
ejpam-677	46	4	eur	eur	PROPN
ejpam-677	46	5	.	.	PUNCT
ejpam-677	47	1	j.	j.	PROPN
ejpam-677	47	2	pure	pure	PROPN
ejpam-677	47	3	appl	appl	PROPN
ejpam-677	47	4	.	.	PROPN
ejpam-677	47	5	math	math	PROPN
ejpam-677	47	6	,	,	PUNCT
ejpam-677	47	7	4	4	NUM
ejpam-677	47	8	(	(	PUNCT
ejpam-677	47	9	2011	2011	NUM
ejpam-677	47	10	)	)	PUNCT
ejpam-677	47	11	,	,	PUNCT
ejpam-677	47	12	50	50	NUM
ejpam-677	47	13	-	-	SYM
ejpam-677	47	14	58	58	NUM
ejpam-677	47	15	52	52	NUM
ejpam-677	47	16	theorem	theorem	NOUN
ejpam-677	47	17	4	4	NUM
ejpam-677	47	18	(	(	PUNCT
ejpam-677	47	19	j.	j.	PROPN
ejpam-677	47	20	m.	m.	PROPN
ejpam-677	47	21	rassias	rassias	PROPN
ejpam-677	47	22	,	,	PUNCT
ejpam-677	47	23	1982	1982	NUM
ejpam-677	47	24	-	-	SYM
ejpam-677	47	25	1989	1989	NUM
ejpam-677	47	26	:	:	PUNCT
ejpam-677	48	1	[	[	X
ejpam-677	48	2	3	3	NUM
ejpam-677	48	3	,	,	PUNCT
ejpam-677	48	4	4	4	NUM
ejpam-677	48	5	,	,	PUNCT
ejpam-677	48	6	5	5	NUM
ejpam-677	48	7	]	]	NUM
ejpam-677	48	8	)	)	PUNCT
ejpam-677	48	9	.	.	PUNCT
ejpam-677	49	1	let	let	VERB
ejpam-677	49	2	x	x	PRON
ejpam-677	49	3	be	be	AUX
ejpam-677	49	4	a	a	DET
ejpam-677	49	5	real	real	ADV
ejpam-677	49	6	normed	normed	ADJ
ejpam-677	49	7	linear	linear	ADJ
ejpam-677	49	8	space	space	NOUN
ejpam-677	49	9	and	and	CCONJ
ejpam-677	49	10	y	y	PROPN
ejpam-677	49	11	a	a	DET
ejpam-677	49	12	real	real	ADJ
ejpam-677	49	13	banach	banach	NOUN
ejpam-677	49	14	space	space	NOUN
ejpam-677	49	15	.	.	PUNCT
ejpam-677	50	1	assume	assume	VERB
ejpam-677	50	2	that	that	SCONJ
ejpam-677	50	3	f	f	X
ejpam-677	50	4	:	:	PUNCT
ejpam-677	50	5	x	x	X
ejpam-677	50	6	→	→	SYM
ejpam-677	50	7	y	y	PROPN
ejpam-677	50	8	is	be	AUX
ejpam-677	50	9	a	a	DET
ejpam-677	50	10	mapping	mapping	NOUN
ejpam-677	50	11	for	for	ADP
ejpam-677	50	12	which	which	PRON
ejpam-677	50	13	there	there	PRON
ejpam-677	50	14	exist	exist	VERB
ejpam-677	50	15	constants	constant	NOUN
ejpam-677	50	16	θ	θ	PROPN
ejpam-677	50	17	≥	≥	NUM
ejpam-677	50	18	0	0	NUM
ejpam-677	50	19	and	and	CCONJ
ejpam-677	50	20	p	p	X
ejpam-677	50	21	,	,	PUNCT
ejpam-677	50	22	q	q	PROPN
ejpam-677	50	23	∈	∈	NOUN
ejpam-677	50	24	r	r	NOUN
ejpam-677	50	25	such	such	ADJ
ejpam-677	50	26	that	that	DET
ejpam-677	50	27	r	r	NOUN
ejpam-677	50	28	=	=	SYM
ejpam-677	50	29	p+	p+	PROPN
ejpam-677	50	30	q	q	X
ejpam-677	50	31	6=	6=	NUM
ejpam-677	50	32	1	1	NUM
ejpam-677	50	33	and	and	CCONJ
ejpam-677	50	34	f	f	PROPN
ejpam-677	50	35	satisfies	satisfy	VERB
ejpam-677	50	36	the	the	DET
ejpam-677	50	37	functional	functional	ADJ
ejpam-677	50	38	inequality	inequality	NOUN
ejpam-677	50	39	||	||	PUNCT
ejpam-677	51	1	f	f	PROPN
ejpam-677	51	2	(	(	PUNCT
ejpam-677	51	3	x	x	X
ejpam-677	51	4	+	+	PUNCT
ejpam-677	51	5	y)−	y)−	PROPN
ejpam-677	51	6	f	f	NOUN
ejpam-677	51	7	(	(	PUNCT
ejpam-677	51	8	x)−	x)−	PROPN
ejpam-677	51	9	f	f	PROPN
ejpam-677	51	10	(	(	PUNCT
ejpam-677	51	11	y)||	y)||	ADJ
ejpam-677	51	12	≤	≤	NUM
ejpam-677	51	13	θ	θ	PROPN
ejpam-677	51	14	||x	||x	NOUN
ejpam-677	51	15	||p||y||q	||p||y||q	ADV
ejpam-677	51	16	,	,	PUNCT
ejpam-677	51	17	for	for	ADP
ejpam-677	51	18	all	all	PRON
ejpam-677	51	19	x	x	SYM
ejpam-677	51	20	,	,	PUNCT
ejpam-677	51	21	y	y	PROPN
ejpam-677	51	22	∈	∈	PROPN
ejpam-677	51	23	x	x	X
ejpam-677	51	24	.	.	PUNCT
ejpam-677	52	1	then	then	ADV
ejpam-677	52	2	the	the	DET
ejpam-677	52	3	limit	limit	NOUN
ejpam-677	52	4	a(x	a(x	NOUN
ejpam-677	52	5	)	)	PUNCT
ejpam-677	52	6	=	=	VERB
ejpam-677	53	1	lim	lim	PROPN
ejpam-677	53	2	n→∞2−n	n→∞2−n	PROPN
ejpam-677	53	3	f	f	PROPN
ejpam-677	53	4	(	(	PUNCT
ejpam-677	53	5	2n	2n	NUM
ejpam-677	53	6	x	x	NOUN
ejpam-677	53	7	)	)	PUNCT
ejpam-677	53	8	,	,	PUNCT
ejpam-677	53	9	exists	exist	VERB
ejpam-677	53	10	for	for	ADP
ejpam-677	53	11	all	all	PRON
ejpam-677	53	12	x	x	SYM
ejpam-677	53	13	∈	∈	PROPN
ejpam-677	53	14	x	x	X
ejpam-677	53	15	and	and	CCONJ
ejpam-677	53	16	a	a	PRON
ejpam-677	53	17	:	:	PUNCT
ejpam-677	53	18	x	x	X
ejpam-677	53	19	→	→	SYM
ejpam-677	53	20	y	y	PROPN
ejpam-677	53	21	is	be	AUX
ejpam-677	53	22	the	the	DET
ejpam-677	53	23	unique	unique	ADJ
ejpam-677	53	24	additive	additive	ADJ
ejpam-677	53	25	mapping	mapping	NOUN
ejpam-677	53	26	which	which	PRON
ejpam-677	53	27	satisfies	satisfy	VERB
ejpam-677	53	28	||	||	PUNCT
ejpam-677	54	1	f	f	X
ejpam-677	54	2	(	(	PUNCT
ejpam-677	54	3	x)−	x)−	PROPN
ejpam-677	54	4	a(x)||	a(x)||	NOUN
ejpam-677	54	5	≤	≤	PROPN
ejpam-677	54	6	θ	θ	PROPN
ejpam-677	54	7	|2r	|2r	NUM
ejpam-677	55	1	−	−	PROPN
ejpam-677	55	2	2|	2|	NUM
ejpam-677	55	3	||x	||x	NOUN
ejpam-677	55	4	||	||	PUNCT
ejpam-677	56	1	r	r	NOUN
ejpam-677	56	2	for	for	ADP
ejpam-677	56	3	all	all	DET
ejpam-677	56	4	x	x	SYM
ejpam-677	56	5	∈	∈	NOUN
ejpam-677	56	6	x	x	X
ejpam-677	56	7	.	.	PUNCT
ejpam-677	57	1	if	if	SCONJ
ejpam-677	57	2	,	,	PUNCT
ejpam-677	57	3	moreover	moreover	ADV
ejpam-677	57	4	,	,	PUNCT
ejpam-677	57	5	f	f	PROPN
ejpam-677	57	6	(	(	PUNCT
ejpam-677	57	7	t	t	NOUN
ejpam-677	57	8	x	x	VERB
ejpam-677	57	9	)	)	PUNCT
ejpam-677	57	10	is	be	AUX
ejpam-677	57	11	continuous	continuous	ADJ
ejpam-677	57	12	in	in	ADP
ejpam-677	57	13	t	t	PROPN
ejpam-677	57	14	∈	∈	NOUN
ejpam-677	57	15	r	r	NOUN
ejpam-677	57	16	for	for	ADP
ejpam-677	57	17	each	each	DET
ejpam-677	57	18	fixed	fix	VERB
ejpam-677	57	19	x	x	SYM
ejpam-677	57	20	∈	∈	PROPN
ejpam-677	57	21	x	x	X
ejpam-677	57	22	,	,	PUNCT
ejpam-677	57	23	then	then	ADV
ejpam-677	57	24	a(t	a(t	VERB
ejpam-677	57	25	x	x	PRON
ejpam-677	57	26	)	)	PUNCT
ejpam-677	57	27	=	=	SYM
ejpam-677	57	28	ta(x	ta(x	NOUN
ejpam-677	57	29	)	)	PUNCT
ejpam-677	57	30	for	for	ADP
ejpam-677	57	31	all	all	PRON
ejpam-677	57	32	x	x	SYM
ejpam-677	57	33	∈	∈	PROPN
ejpam-677	57	34	x	x	X
ejpam-677	57	35	and	and	CCONJ
ejpam-677	57	36	t	t	PROPN
ejpam-677	57	37	∈	∈	PROPN
ejpam-677	57	38	r.	r.	PROPN
ejpam-677	57	39	a	a	PRON
ejpam-677	57	40	:	:	PUNCT
ejpam-677	57	41	x	x	SYM
ejpam-677	57	42	→	→	SYM
ejpam-677	57	43	y	y	PROPN
ejpam-677	57	44	is	be	AUX
ejpam-677	57	45	a	a	DET
ejpam-677	57	46	unique	unique	ADJ
ejpam-677	57	47	linear	linear	NOUN
ejpam-677	57	48	additive	additive	ADJ
ejpam-677	57	49	mapping	mapping	NOUN
ejpam-677	57	50	satisfying	satisfy	VERB
ejpam-677	57	51	equation	equation	NOUN
ejpam-677	57	52	a(x	a(x	NOUN
ejpam-677	57	53	+	+	PROPN
ejpam-677	57	54	y	y	NOUN
ejpam-677	57	55	)	)	PUNCT
ejpam-677	57	56	=	=	SYM
ejpam-677	57	57	a(x)+	a(x)+	ADP
ejpam-677	57	58	a(y	a(y	PROPN
ejpam-677	57	59	)	)	PUNCT
ejpam-677	57	60	.	.	PUNCT
ejpam-677	58	1	for	for	ADP
ejpam-677	58	2	the	the	DET
ejpam-677	58	3	theorem	theorem	NOUN
ejpam-677	58	4	that	that	PRON
ejpam-677	58	5	follows	follow	VERB
ejpam-677	58	6	,	,	PUNCT
ejpam-677	58	7	let	let	VERB
ejpam-677	58	8	(	(	PUNCT
ejpam-677	58	9	e,⊥	e,⊥	NOUN
ejpam-677	58	10	)	)	PUNCT
ejpam-677	58	11	denote	denote	VERB
ejpam-677	58	12	an	an	DET
ejpam-677	58	13	orthogonality	orthogonality	NOUN
ejpam-677	58	14	normed	normed	ADJ
ejpam-677	58	15	space	space	NOUN
ejpam-677	58	16	with	with	ADP
ejpam-677	58	17	norm	norm	NOUN
ejpam-677	58	18	||.||e	||.||e	PROPN
ejpam-677	58	19	and	and	CCONJ
ejpam-677	58	20	(	(	PUNCT
ejpam-677	58	21	f	f	X
ejpam-677	58	22	,	,	PUNCT
ejpam-677	58	23	||.||f	||.||f	PROPN
ejpam-677	58	24	)	)	PUNCT
ejpam-677	58	25	is	be	AUX
ejpam-677	58	26	a	a	DET
ejpam-677	58	27	banach	banach	NOUN
ejpam-677	58	28	space	space	NOUN
ejpam-677	58	29	.	.	PUNCT
ejpam-677	59	1	theorem	theorem	ADJ
ejpam-677	59	2	5	5	NUM
ejpam-677	59	3	(	(	PUNCT
ejpam-677	59	4	ravi	ravi	NOUN
ejpam-677	59	5	,	,	PUNCT
ejpam-677	59	6	k.	k.	PROPN
ejpam-677	59	7	,	,	PUNCT
ejpam-677	59	8	arunkumar	arunkumar	PROPN
ejpam-677	59	9	,	,	PUNCT
ejpam-677	59	10	m.	m.	NOUN
ejpam-677	59	11	and	and	CCONJ
ejpam-677	59	12	rassias	rassias	PROPN
ejpam-677	59	13	,	,	PUNCT
ejpam-677	59	14	j.	j.	PROPN
ejpam-677	59	15	m.	m.	PROPN
ejpam-677	59	16	,	,	PUNCT
ejpam-677	59	17	2008	2008	NUM
ejpam-677	59	18	:	:	PUNCT
ejpam-677	60	1	[	[	X
ejpam-677	60	2	7	7	NUM
ejpam-677	60	3	]	]	PUNCT
ejpam-677	60	4	)	)	PUNCT
ejpam-677	60	5	.	.	PUNCT
ejpam-677	61	1	let	let	VERB
ejpam-677	61	2	f	f	NOUN
ejpam-677	61	3	:	:	PUNCT
ejpam-677	61	4	e	e	X
ejpam-677	61	5	→	→	PUNCT
ejpam-677	61	6	f	f	X
ejpam-677	61	7	be	be	AUX
ejpam-677	61	8	a	a	DET
ejpam-677	61	9	mapping	mapping	NOUN
ejpam-677	61	10	which	which	PRON
ejpam-677	61	11	satisfies	satisfy	VERB
ejpam-677	61	12	the	the	DET
ejpam-677	61	13	inequality	inequality	NOUN
ejpam-677	61	14	||	||	PUNCT
ejpam-677	62	1	f	f	PROPN
ejpam-677	62	2	(	(	PUNCT
ejpam-677	62	3	mx	mx	PROPN
ejpam-677	62	4	+	+	CCONJ
ejpam-677	62	5	y	y	NOUN
ejpam-677	62	6	)	)	PUNCT
ejpam-677	63	1	+	+	NOUN
ejpam-677	63	2	f	f	X
ejpam-677	63	3	(	(	PUNCT
ejpam-677	63	4	mx	mx	PROPN
ejpam-677	63	5	−	−	PROPN
ejpam-677	63	6	y)−	y)−	PROPN
ejpam-677	63	7	2	2	NUM
ejpam-677	63	8	f	f	NOUN
ejpam-677	63	9	(	(	PUNCT
ejpam-677	63	10	x	x	PROPN
ejpam-677	63	11	+	+	PUNCT
ejpam-677	63	12	y)−	y)−	PROPN
ejpam-677	63	13	2	2	NUM
ejpam-677	63	14	f	f	NOUN
ejpam-677	63	15	(	(	PUNCT
ejpam-677	63	16	x	x	X
ejpam-677	63	17	−	−	PROPN
ejpam-677	63	18	y)−	y)−	PROPN
ejpam-677	63	19	2(m2−	2(m2−	NUM
ejpam-677	63	20	2	2	NUM
ejpam-677	63	21	)	)	PUNCT
ejpam-677	63	22	f	f	NOUN
ejpam-677	63	23	(	(	PUNCT
ejpam-677	63	24	x)+	x)+	PROPN
ejpam-677	63	25	2	2	NUM
ejpam-677	63	26	f	f	NOUN
ejpam-677	63	27	(	(	PUNCT
ejpam-677	63	28	y)||f	y)||f	PROPN
ejpam-677	63	29	≤	≤	PROPN
ejpam-677	63	30	ǫ	ǫ	NUM
ejpam-677	63	31	�	�	NOUN
ejpam-677	63	32	||x	||x	NOUN
ejpam-677	63	33	||pe||y||pe	||pe||y||pe	ADJ
ejpam-677	63	34	+	+	SYM
ejpam-677	63	35	�	�	NOUN
ejpam-677	63	36	||x	||x	NOUN
ejpam-677	63	37	||2p	||2p	PUNCT
ejpam-677	63	38	e	e	X
ejpam-677	63	39	+	+	CCONJ
ejpam-677	63	40	||y||2p	||y||2p	PROPN
ejpam-677	63	41	e	e	PROPN
ejpam-677	63	42	�	�	PROPN
ejpam-677	63	43	(	(	PUNCT
ejpam-677	63	44	5	5	NUM
ejpam-677	63	45	)	)	PUNCT
ejpam-677	63	46	for	for	ADP
ejpam-677	63	47	all	all	PRON
ejpam-677	63	48	x	x	SYM
ejpam-677	63	49	,	,	PUNCT
ejpam-677	63	50	y	y	PROPN
ejpam-677	63	51	∈	∈	PROPN
ejpam-677	63	52	e	e	X
ejpam-677	63	53	with	with	ADP
ejpam-677	63	54	x	x	PROPN
ejpam-677	63	55	⊥	⊥	PROPN
ejpam-677	63	56	y	y	PROPN
ejpam-677	63	57	,	,	PUNCT
ejpam-677	63	58	where	where	SCONJ
ejpam-677	63	59	ǫ	ǫ	PRON
ejpam-677	63	60	and	and	CCONJ
ejpam-677	63	61	p	p	NOUN
ejpam-677	63	62	are	be	AUX
ejpam-677	63	63	constants	constant	NOUN
ejpam-677	63	64	with	with	ADP
ejpam-677	63	65	ǫ	ǫ	PRON
ejpam-677	63	66	,	,	PUNCT
ejpam-677	63	67	p	p	X
ejpam-677	63	68	>	>	X
ejpam-677	63	69	0	0	PUNCT
ejpam-677	63	70	and	and	CCONJ
ejpam-677	63	71	either	either	CCONJ
ejpam-677	63	72	m	m	VERB
ejpam-677	63	73	>	>	X
ejpam-677	63	74	1	1	NUM
ejpam-677	63	75	;	;	PUNCT
ejpam-677	63	76	p	p	X
ejpam-677	63	77	<	<	X
ejpam-677	63	78	1	1	NUM
ejpam-677	63	79	or	or	CCONJ
ejpam-677	63	80	m	m	PRON
ejpam-677	63	81	<	<	X
ejpam-677	63	82	1	1	NUM
ejpam-677	63	83	;	;	PUNCT
ejpam-677	63	84	p	p	X
ejpam-677	63	85	>	>	X
ejpam-677	63	86	1	1	NUM
ejpam-677	63	87	with	with	ADP
ejpam-677	63	88	m	m	PROPN
ejpam-677	63	89	6=	6=	NUM
ejpam-677	63	90	0	0	NUM
ejpam-677	63	91	;	;	PUNCT
ejpam-677	63	92	m	m	PROPN
ejpam-677	63	93	6=	6=	NUM
ejpam-677	63	94	±1	±1	ADJ
ejpam-677	63	95	;	;	PUNCT
ejpam-677	63	96	m	m	PROPN
ejpam-677	63	97	6=	6=	NOUN
ejpam-677	63	98	±p2	±p2	NOUN
ejpam-677	63	99	and	and	CCONJ
ejpam-677	63	100	−1	−1	NOUN
ejpam-677	63	101	6=	6=	ADP
ejpam-677	63	102	|m|p−1	|m|p−1	CCONJ
ejpam-677	63	103	<	<	X
ejpam-677	63	104	1	1	NUM
ejpam-677	63	105	.	.	PUNCT
ejpam-677	64	1	then	then	ADV
ejpam-677	64	2	the	the	DET
ejpam-677	64	3	limit	limit	NOUN
ejpam-677	64	4	q(x	q(x	PROPN
ejpam-677	64	5	)	)	PUNCT
ejpam-677	64	6	=	=	VERB
ejpam-677	64	7	lim	lim	PROPN
ejpam-677	64	8	n→∞	n→∞	X
ejpam-677	65	1	f	f	PROPN
ejpam-677	65	2	(	(	PUNCT
ejpam-677	65	3	mn	mn	PROPN
ejpam-677	65	4	x	x	PROPN
ejpam-677	65	5	)	)	PUNCT
ejpam-677	65	6	m2n	m2n	NOUN
ejpam-677	65	7	exists	exist	VERB
ejpam-677	65	8	for	for	ADP
ejpam-677	65	9	all	all	DET
ejpam-677	65	10	x	x	SYM
ejpam-677	65	11	∈	∈	PROPN
ejpam-677	65	12	e	e	NOUN
ejpam-677	65	13	and	and	CCONJ
ejpam-677	65	14	q	q	NOUN
ejpam-677	65	15	:	:	PUNCT
ejpam-677	65	16	e→	e→	NOUN
ejpam-677	65	17	f	f	PROPN
ejpam-677	65	18	is	be	AUX
ejpam-677	65	19	the	the	DET
ejpam-677	65	20	unique	unique	ADJ
ejpam-677	65	21	orthogonally	orthogonally	ADV
ejpam-677	65	22	euler	euler	NOUN
ejpam-677	65	23	-	-	PUNCT
ejpam-677	65	24	lagrange	lagrange	NOUN
ejpam-677	65	25	quadratic	quadratic	ADJ
ejpam-677	65	26	mapping	mapping	NOUN
ejpam-677	65	27	such	such	ADJ
ejpam-677	65	28	that	that	PRON
ejpam-677	65	29	||	||	NOUN
ejpam-677	66	1	f	f	X
ejpam-677	66	2	(	(	PUNCT
ejpam-677	66	3	x)−q(x)||f	x)−q(x)||f	X
ejpam-677	66	4	≤	≤	ADV
ejpam-677	66	5	ǫ	ǫ	DET
ejpam-677	66	6	2|m2	2|m2	NOUN
ejpam-677	66	7	−m2p|	−m2p|	NOUN
ejpam-677	66	8	||x	||x	NOUN
ejpam-677	66	9	||	||	NOUN
ejpam-677	66	10	2p	2p	NUM
ejpam-677	66	11	e	e	NOUN
ejpam-677	66	12	for	for	ADP
ejpam-677	66	13	all	all	DET
ejpam-677	66	14	x	x	SYM
ejpam-677	66	15	∈	∈	PROPN
ejpam-677	66	16	e.	e.	PROPN
ejpam-677	66	17	note	note	VERB
ejpam-677	66	18	that	that	SCONJ
ejpam-677	66	19	the	the	DET
ejpam-677	66	20	mixed	mixed	ADJ
ejpam-677	66	21	type	type	NOUN
ejpam-677	66	22	product	product	NOUN
ejpam-677	66	23	-	-	PUNCT
ejpam-677	66	24	sum	sum	NOUN
ejpam-677	66	25	function	function	NOUN
ejpam-677	66	26	(	(	PUNCT
ejpam-677	66	27	x	x	INTJ
ejpam-677	66	28	,	,	PUNCT
ejpam-677	66	29	y)→	y)→	PROPN
ejpam-677	66	30	ǫ	ǫ	SYM
ejpam-677	66	31	�	�	NOUN
ejpam-677	66	32	||x	||x	NOUN
ejpam-677	66	33	||pe||y||pe	||pe||y||pe	ADJ
ejpam-677	66	34	+	+	SYM
ejpam-677	66	35	�	�	NOUN
ejpam-677	66	36	||x	||x	NOUN
ejpam-677	66	37	||2p	||2p	PUNCT
ejpam-677	66	38	e	e	X
ejpam-677	66	39	+	+	CCONJ
ejpam-677	66	40	||y||2p	||y||2p	PROPN
ejpam-677	66	41	e	e	PROPN
ejpam-677	66	42	�	�	PROPN
ejpam-677	66	43	�	�	PROPN
ejpam-677	66	44	was	be	AUX
ejpam-677	66	45	introduced	introduce	VERB
ejpam-677	66	46	by	by	ADP
ejpam-677	66	47	j.	j.	PROPN
ejpam-677	66	48	m.	m.	PROPN
ejpam-677	66	49	rassias	rassias	PROPN
ejpam-677	66	50	(	(	PUNCT
ejpam-677	66	51	[	[	X
ejpam-677	66	52	7	7	NUM
ejpam-677	66	53	,	,	PUNCT
ejpam-677	66	54	8	8	NUM
ejpam-677	66	55	]	]	NUM
ejpam-677	66	56	)	)	PUNCT
ejpam-677	66	57	.	.	PUNCT
ejpam-677	67	1	in	in	ADP
ejpam-677	67	2	this	this	DET
ejpam-677	67	3	paper	paper	NOUN
ejpam-677	67	4	we	we	PRON
ejpam-677	67	5	introduce	introduce	VERB
ejpam-677	67	6	a	a	DET
ejpam-677	67	7	cauchy	cauchy	ADJ
ejpam-677	67	8	type	type	NOUN
ejpam-677	67	9	additive	additive	ADJ
ejpam-677	67	10	functional	functional	ADJ
ejpam-677	67	11	equation	equation	NOUN
ejpam-677	67	12	and	and	CCONJ
ejpam-677	67	13	investigate	investigate	VERB
ejpam-677	67	14	the	the	DET
ejpam-677	67	15	generalised	generalise	VERB
ejpam-677	67	16	hyers	hyer	NOUN
ejpam-677	67	17	-	-	PUNCT
ejpam-677	67	18	ulam	ulam	ADJ
ejpam-677	67	19	“	"	PUNCT
ejpam-677	67	20	product	product	NOUN
ejpam-677	67	21	-	-	PUNCT
ejpam-677	67	22	sum	sum	NOUN
ejpam-677	67	23	”	"	PUNCT
ejpam-677	67	24	stability	stability	NOUN
ejpam-677	67	25	of	of	ADP
ejpam-677	67	26	this	this	DET
ejpam-677	67	27	equation	equation	NOUN
ejpam-677	67	28	.	.	PUNCT
ejpam-677	68	1	m.	m.	NOUN
ejpam-677	68	2	rassias	rassias	PROPN
ejpam-677	68	3	/	/	SYM
ejpam-677	68	4	eur	eur	PROPN
ejpam-677	68	5	.	.	PUNCT
ejpam-677	69	1	j.	j.	PROPN
ejpam-677	69	2	pure	pure	PROPN
ejpam-677	69	3	appl	appl	PROPN
ejpam-677	69	4	.	.	PROPN
ejpam-677	69	5	math	math	PROPN
ejpam-677	69	6	,	,	PUNCT
ejpam-677	69	7	4	4	NUM
ejpam-677	69	8	(	(	PUNCT
ejpam-677	69	9	2011	2011	NUM
ejpam-677	69	10	)	)	PUNCT
ejpam-677	69	11	,	,	PUNCT
ejpam-677	69	12	50	50	NUM
ejpam-677	69	13	-	-	SYM
ejpam-677	69	14	58	58	NUM
ejpam-677	69	15	53	53	NUM
ejpam-677	69	16	2	2	NUM
ejpam-677	69	17	.	.	PUNCT
ejpam-677	69	18	cauchy	cauchy	ADJ
ejpam-677	69	19	type	type	NOUN
ejpam-677	69	20	additive	additive	ADJ
ejpam-677	69	21	functional	functional	ADJ
ejpam-677	69	22	equation	equation	NOUN
ejpam-677	69	23	let	let	VERB
ejpam-677	69	24	x	x	PRON
ejpam-677	69	25	be	be	AUX
ejpam-677	69	26	a	a	DET
ejpam-677	69	27	real	real	ADV
ejpam-677	69	28	normed	normed	ADJ
ejpam-677	69	29	linear	linear	ADJ
ejpam-677	69	30	space	space	NOUN
ejpam-677	69	31	and	and	CCONJ
ejpam-677	69	32	y	y	PROPN
ejpam-677	69	33	a	a	DET
ejpam-677	69	34	real	real	ADJ
ejpam-677	69	35	banach	banach	NOUN
ejpam-677	69	36	space	space	NOUN
ejpam-677	69	37	.	.	PUNCT
ejpam-677	70	1	definition	definition	NOUN
ejpam-677	70	2	1	1	NUM
ejpam-677	70	3	.	.	PUNCT
ejpam-677	71	1	a	a	DET
ejpam-677	71	2	mapping	mapping	NOUN
ejpam-677	71	3	f	f	NOUN
ejpam-677	71	4	:	:	PUNCT
ejpam-677	71	5	x	x	X
ejpam-677	71	6	→	→	SYM
ejpam-677	71	7	y	y	PROPN
ejpam-677	71	8	is	be	AUX
ejpam-677	71	9	called	call	VERB
ejpam-677	71	10	approximately	approximately	ADV
ejpam-677	71	11	cauchy	cauchy	ADJ
ejpam-677	71	12	type	type	NOUN
ejpam-677	71	13	additive	additive	NOUN
ejpam-677	71	14	,	,	PUNCT
ejpam-677	71	15	if	if	SCONJ
ejpam-677	71	16	the	the	DET
ejpam-677	71	17	approximately	approximately	ADV
ejpam-677	71	18	cauchy	cauchy	ADJ
ejpam-677	71	19	additive	additive	ADJ
ejpam-677	71	20	functional	functional	ADJ
ejpam-677	71	21	inequality	inequality	NOUN
ejpam-677	71	22	||	||	PUNCT
ejpam-677	72	1	f	f	PROPN
ejpam-677	72	2	(	(	PUNCT
ejpam-677	72	3	x	x	PROPN
ejpam-677	72	4	+	+	NUM
ejpam-677	72	5	y	y	NOUN
ejpam-677	72	6	)	)	PUNCT
ejpam-677	73	1	+	+	NOUN
ejpam-677	73	2	f	f	X
ejpam-677	73	3	(	(	PUNCT
ejpam-677	73	4	x	x	INTJ
ejpam-677	73	5	−	−	PROPN
ejpam-677	73	6	y	y	PROPN
ejpam-677	73	7	)	)	PUNCT
ejpam-677	74	1	+	+	CCONJ
ejpam-677	74	2	f	f	X
ejpam-677	74	3	(	(	PUNCT
ejpam-677	74	4	y	y	PROPN
ejpam-677	74	5	−	−	PROPN
ejpam-677	74	6	x)−	x)−	PROPN
ejpam-677	74	7	f	f	PROPN
ejpam-677	74	8	(	(	PUNCT
ejpam-677	74	9	x)−	x)−	PROPN
ejpam-677	74	10	f	f	PROPN
ejpam-677	74	11	(	(	PUNCT
ejpam-677	74	12	y)||	y)||	INTJ
ejpam-677	74	13	≤	≤	NUM
ejpam-677	74	14	ǫ	ǫ	NUM
ejpam-677	74	15	�	�	NOUN
ejpam-677	74	16	||x	||x	NOUN
ejpam-677	74	17	||	||	NOUN
ejpam-677	74	18	α2	α2	PROPN
ejpam-677	74	19	||y||	||y||	PROPN
ejpam-677	74	20	α2	α2	PROPN
ejpam-677	74	21	+	+	CCONJ
ejpam-677	74	22	||x	||x	NOUN
ejpam-677	74	23	||α+	||α+	PROPN
ejpam-677	74	24	||y||α	||y||α	PROPN
ejpam-677	74	25	�	�	PROPN
ejpam-677	74	26	(	(	PUNCT
ejpam-677	74	27	6	6	NUM
ejpam-677	74	28	)	)	PUNCT
ejpam-677	74	29	holds	hold	VERB
ejpam-677	74	30	for	for	ADP
ejpam-677	74	31	every	every	DET
ejpam-677	74	32	x	x	X
ejpam-677	74	33	,	,	PUNCT
ejpam-677	74	34	y	y	PROPN
ejpam-677	74	35	∈	∈	PROPN
ejpam-677	74	36	x	x	PUNCT
ejpam-677	74	37	with	with	ADP
ejpam-677	74	38	ǫ	ǫ	PRON
ejpam-677	74	39	≥	≥	NOUN
ejpam-677	74	40	0	0	NUM
ejpam-677	74	41	and	and	CCONJ
ejpam-677	74	42	α	α	PRON
ejpam-677	74	43	6=	6=	ADP
ejpam-677	74	44	1	1	NUM
ejpam-677	74	45	.	.	PUNCT
ejpam-677	75	1	lemma	lemma	PROPN
ejpam-677	75	2	1	1	X
ejpam-677	75	3	.	.	PUNCT
ejpam-677	76	1	mapping	map	VERB
ejpam-677	76	2	a	a	PRON
ejpam-677	76	3	:	:	PUNCT
ejpam-677	76	4	x	x	X
ejpam-677	76	5	→	→	SYM
ejpam-677	76	6	y	y	PROPN
ejpam-677	76	7	satisfies	satisfy	VERB
ejpam-677	76	8	the	the	DET
ejpam-677	76	9	cauchy	cauchy	NOUN
ejpam-677	76	10	-	-	PUNCT
ejpam-677	76	11	type	type	NOUN
ejpam-677	76	12	additive	additive	ADJ
ejpam-677	76	13	equation	equation	NOUN
ejpam-677	76	14	a(x	a(x	NOUN
ejpam-677	76	15	+	+	PROPN
ejpam-677	76	16	y	y	NOUN
ejpam-677	76	17	)	)	PUNCT
ejpam-677	77	1	+	+	CCONJ
ejpam-677	77	2	a(x	a(x	PROPN
ejpam-677	77	3	−	−	PROPN
ejpam-677	77	4	y	y	NOUN
ejpam-677	77	5	)	)	PUNCT
ejpam-677	78	1	+	+	CCONJ
ejpam-677	78	2	a(y	a(y	PROPN
ejpam-677	78	3	−	−	NOUN
ejpam-677	78	4	x	x	NOUN
ejpam-677	78	5	)	)	PUNCT
ejpam-677	78	6	=	=	SYM
ejpam-677	78	7	a(x)+	a(x)+	ADP
ejpam-677	78	8	a(y	a(y	PROPN
ejpam-677	78	9	)	)	PUNCT
ejpam-677	78	10	for	for	ADP
ejpam-677	78	11	all	all	DET
ejpam-677	78	12	x	x	SYM
ejpam-677	78	13	,	,	PUNCT
ejpam-677	78	14	y	y	PROPN
ejpam-677	78	15	∈	∈	PROPN
ejpam-677	78	16	x	x	INTJ
ejpam-677	78	17	if	if	SCONJ
ejpam-677	78	18	and	and	CCONJ
ejpam-677	78	19	only	only	ADV
ejpam-677	78	20	if	if	SCONJ
ejpam-677	78	21	there	there	PRON
ejpam-677	78	22	exists	exist	VERB
ejpam-677	78	23	a	a	DET
ejpam-677	78	24	mapping	mapping	NOUN
ejpam-677	78	25	c	c	NOUN
ejpam-677	79	1	:	:	PUNCT
ejpam-677	79	2	x	x	X
ejpam-677	79	3	→	→	SYM
ejpam-677	79	4	y	y	PROPN
ejpam-677	79	5	satisfying	satisfy	VERB
ejpam-677	79	6	the	the	DET
ejpam-677	79	7	cauchy	cauchy	ADJ
ejpam-677	79	8	additive	additive	ADJ
ejpam-677	79	9	equation	equation	NOUN
ejpam-677	79	10	c(x	c(x	NOUN
ejpam-677	79	11	+	+	CCONJ
ejpam-677	79	12	y	y	NOUN
ejpam-677	79	13	)	)	PUNCT
ejpam-677	79	14	=	=	SYM
ejpam-677	79	15	c(x)+	c(x)+	PROPN
ejpam-677	79	16	c(y	c(y	PROPN
ejpam-677	79	17	)	)	PUNCT
ejpam-677	79	18	for	for	ADP
ejpam-677	79	19	all	all	DET
ejpam-677	79	20	x	x	SYM
ejpam-677	79	21	,	,	PUNCT
ejpam-677	79	22	y	y	PROPN
ejpam-677	79	23	∈	∈	PROPN
ejpam-677	79	24	x	x	PUNCT
ejpam-677	79	25	such	such	ADJ
ejpam-677	79	26	that	that	SCONJ
ejpam-677	79	27	a(x	a(x	NOUN
ejpam-677	79	28	)	)	PUNCT
ejpam-677	79	29	=	=	SYM
ejpam-677	79	30	c(x	c(x	NOUN
ejpam-677	79	31	)	)	PUNCT
ejpam-677	79	32	for	for	ADP
ejpam-677	79	33	all	all	DET
ejpam-677	79	34	x	x	SYM
ejpam-677	79	35	∈	∈	PROPN
ejpam-677	79	36	x	x	X
ejpam-677	79	37	.	.	PUNCT
ejpam-677	80	1	proof	proof	NOUN
ejpam-677	80	2	.	.	PUNCT
ejpam-677	81	1	(	(	PUNCT
ejpam-677	81	2	⇒	⇒	PROPN
ejpam-677	81	3	)	)	PUNCT
ejpam-677	81	4	let	let	VERB
ejpam-677	81	5	mapping	mapping	NOUN
ejpam-677	81	6	a	a	PRON
ejpam-677	81	7	:	:	PUNCT
ejpam-677	81	8	x	x	X
ejpam-677	81	9	→	→	SYM
ejpam-677	81	10	y	y	PROPN
ejpam-677	81	11	satisfy	satisfy	VERB
ejpam-677	81	12	the	the	DET
ejpam-677	81	13	cauchy	cauchy	NOUN
ejpam-677	81	14	-	-	PUNCT
ejpam-677	81	15	type	type	NOUN
ejpam-677	81	16	additive	additive	ADJ
ejpam-677	81	17	equation	equation	NOUN
ejpam-677	81	18	a(x	a(x	NOUN
ejpam-677	81	19	+	+	PROPN
ejpam-677	81	20	y	y	NOUN
ejpam-677	81	21	)	)	PUNCT
ejpam-677	82	1	+	+	CCONJ
ejpam-677	82	2	a(x	a(x	PROPN
ejpam-677	82	3	−	−	PROPN
ejpam-677	82	4	y	y	NOUN
ejpam-677	82	5	)	)	PUNCT
ejpam-677	83	1	+	+	CCONJ
ejpam-677	83	2	a(y	a(y	PROPN
ejpam-677	83	3	−	−	NOUN
ejpam-677	83	4	x	x	NOUN
ejpam-677	83	5	)	)	PUNCT
ejpam-677	83	6	=	=	SYM
ejpam-677	83	7	a(x)+	a(x)+	ADP
ejpam-677	83	8	a(y	a(y	PROPN
ejpam-677	83	9	)	)	PUNCT
ejpam-677	83	10	(	(	PUNCT
ejpam-677	83	11	7	7	X
ejpam-677	83	12	)	)	PUNCT
ejpam-677	83	13	for	for	ADP
ejpam-677	83	14	all	all	DET
ejpam-677	83	15	x	x	SYM
ejpam-677	83	16	,	,	PUNCT
ejpam-677	83	17	y	y	PROPN
ejpam-677	83	18	∈	∈	PROPN
ejpam-677	83	19	x	x	X
ejpam-677	83	20	.	.	PUNCT
ejpam-677	84	1	assume	assume	VERB
ejpam-677	84	2	that	that	SCONJ
ejpam-677	84	3	there	there	PRON
ejpam-677	84	4	exists	exist	VERB
ejpam-677	84	5	a	a	DET
ejpam-677	84	6	mapping	mapping	NOUN
ejpam-677	84	7	c	c	NOUN
ejpam-677	84	8	:	:	PUNCT
ejpam-677	84	9	x	x	X
ejpam-677	84	10	→	→	PUNCT
ejpam-677	84	11	y	y	NUM
ejpam-677	84	12	such	such	ADJ
ejpam-677	84	13	that	that	SCONJ
ejpam-677	84	14	a(x	a(x	NOUN
ejpam-677	84	15	)	)	PUNCT
ejpam-677	84	16	=	=	SYM
ejpam-677	84	17	c(x	c(x	NOUN
ejpam-677	84	18	)	)	PUNCT
ejpam-677	84	19	for	for	ADP
ejpam-677	84	20	all	all	DET
ejpam-677	84	21	x	x	SYM
ejpam-677	84	22	∈	∈	NOUN
ejpam-677	84	23	x	x	X
ejpam-677	84	24	.	.	PUNCT
ejpam-677	85	1	observe	observe	VERB
ejpam-677	85	2	that	that	SCONJ
ejpam-677	85	3	for	for	ADP
ejpam-677	85	4	x	x	SYM
ejpam-677	85	5	=	=	PUNCT
ejpam-677	85	6	y	y	PROPN
ejpam-677	85	7	=	=	SYM
ejpam-677	85	8	0	0	PROPN
ejpam-677	86	1	and	and	CCONJ
ejpam-677	86	2	x	x	SYM
ejpam-677	87	1	=	=	SYM
ejpam-677	87	2	x	x	X
ejpam-677	87	3	,	,	PUNCT
ejpam-677	87	4	y	y	PROPN
ejpam-677	87	5	=	=	PUNCT
ejpam-677	87	6	x	x	PUNCT
ejpam-677	87	7	from	from	ADP
ejpam-677	87	8	(	(	PUNCT
ejpam-677	87	9	7	7	X
ejpam-677	87	10	)	)	PUNCT
ejpam-677	87	11	we	we	PRON
ejpam-677	87	12	obtain	obtain	VERB
ejpam-677	87	13	respectively	respectively	ADV
ejpam-677	87	14	c(0	c(0	NOUN
ejpam-677	87	15	)	)	PUNCT
ejpam-677	87	16	=	=	SYM
ejpam-677	87	17	a(0	a(0	PROPN
ejpam-677	87	18	)	)	PUNCT
ejpam-677	87	19	=	=	SYM
ejpam-677	87	20	0	0	NUM
ejpam-677	87	21	and	and	CCONJ
ejpam-677	87	22	c(−x	c(−x	X
ejpam-677	87	23	)	)	PUNCT
ejpam-677	87	24	=	=	SYM
ejpam-677	87	25	a(−x	a(−x	NOUN
ejpam-677	87	26	)	)	PUNCT
ejpam-677	87	27	=	=	SYM
ejpam-677	87	28	−a(x	−a(x	PROPN
ejpam-677	87	29	)	)	PUNCT
ejpam-677	87	30	=	=	SYM
ejpam-677	87	31	−c(x	−c(x	NOUN
ejpam-677	87	32	)	)	PUNCT
ejpam-677	87	33	,	,	PUNCT
ejpam-677	87	34	for	for	ADP
ejpam-677	87	35	x	x	SYM
ejpam-677	87	36	∈	∈	PROPN
ejpam-677	87	37	x	x	X
ejpam-677	87	38	.	.	PUNCT
ejpam-677	88	1	(	(	PUNCT
ejpam-677	88	2	8)	8)	NUM
ejpam-677	88	3	from	from	ADP
ejpam-677	88	4	(	(	PUNCT
ejpam-677	88	5	7	7	NUM
ejpam-677	88	6	)	)	PUNCT
ejpam-677	88	7	and	and	CCONJ
ejpam-677	88	8	(	(	PUNCT
ejpam-677	88	9	8)	8)	NUM
ejpam-677	88	10	it	it	PRON
ejpam-677	88	11	is	be	AUX
ejpam-677	88	12	obvious	obvious	ADJ
ejpam-677	88	13	that	that	SCONJ
ejpam-677	88	14	c(x	c(x	NOUN
ejpam-677	88	15	+	+	CCONJ
ejpam-677	88	16	y	y	NOUN
ejpam-677	88	17	)	)	PUNCT
ejpam-677	88	18	+	+	CCONJ
ejpam-677	88	19	c(x	c(x	NOUN
ejpam-677	88	20	−	−	PROPN
ejpam-677	88	21	y	y	NOUN
ejpam-677	88	22	)	)	PUNCT
ejpam-677	88	23	+	+	NUM
ejpam-677	88	24	c(y	c(y	PROPN
ejpam-677	88	25	−	−	NOUN
ejpam-677	88	26	x	x	SYM
ejpam-677	88	27	)	)	PUNCT
ejpam-677	88	28	=	=	SYM
ejpam-677	88	29	c(x)+	c(x)+	PROPN
ejpam-677	88	30	c(y	c(y	PROPN
ejpam-677	88	31	)	)	PUNCT
ejpam-677	88	32	,	,	PUNCT
ejpam-677	88	33	or	or	CCONJ
ejpam-677	88	34	c(x	c(x	NOUN
ejpam-677	88	35	+	+	CCONJ
ejpam-677	88	36	y	y	NOUN
ejpam-677	88	37	)	)	PUNCT
ejpam-677	88	38	+	+	CCONJ
ejpam-677	88	39	c(x	c(x	NOUN
ejpam-677	88	40	−	−	PROPN
ejpam-677	88	41	y	y	NOUN
ejpam-677	88	42	)	)	PUNCT
ejpam-677	89	1	+	+	CCONJ
ejpam-677	89	2	c(−(x	c(−(x	ADP
ejpam-677	89	3	−	−	PROPN
ejpam-677	89	4	y	y	NOUN
ejpam-677	89	5	)	)	PUNCT
ejpam-677	89	6	)	)	PUNCT
ejpam-677	90	1	=	=	SYM
ejpam-677	90	2	c(x)+	c(x)+	PROPN
ejpam-677	90	3	c(y	c(y	PROPN
ejpam-677	90	4	)	)	PUNCT
ejpam-677	90	5	,	,	PUNCT
ejpam-677	90	6	or	or	CCONJ
ejpam-677	90	7	c(x	c(x	NOUN
ejpam-677	90	8	+	+	CCONJ
ejpam-677	90	9	y	y	NOUN
ejpam-677	90	10	)	)	PUNCT
ejpam-677	90	11	=	=	SYM
ejpam-677	90	12	c(x)+	c(x)+	PROPN
ejpam-677	90	13	c(y	c(y	PROPN
ejpam-677	90	14	)	)	PUNCT
ejpam-677	90	15	.	.	PUNCT
ejpam-677	91	1	hence	hence	ADV
ejpam-677	91	2	,	,	PUNCT
ejpam-677	91	3	c	c	PROPN
ejpam-677	91	4	satisfies	satisfy	VERB
ejpam-677	91	5	the	the	DET
ejpam-677	91	6	cauchy	cauchy	ADJ
ejpam-677	91	7	additive	additive	NOUN
ejpam-677	91	8	equation	equation	NOUN
ejpam-677	91	9	.	.	PUNCT
ejpam-677	92	1	(	(	PUNCT
ejpam-677	92	2	⇐	⇐	NOUN
ejpam-677	92	3	)	)	PUNCT
ejpam-677	92	4	let	let	VERB
ejpam-677	92	5	mapping	mapping	NOUN
ejpam-677	92	6	c	c	NOUN
ejpam-677	92	7	:	:	PUNCT
ejpam-677	92	8	x	x	X
ejpam-677	92	9	→	→	SYM
ejpam-677	92	10	y	y	PROPN
ejpam-677	92	11	satisfy	satisfy	VERB
ejpam-677	92	12	the	the	DET
ejpam-677	92	13	cauchy	cauchy	ADJ
ejpam-677	92	14	additive	additive	ADJ
ejpam-677	92	15	equation	equation	NOUN
ejpam-677	92	16	c(x	c(x	NOUN
ejpam-677	92	17	+	+	CCONJ
ejpam-677	92	18	y	y	NOUN
ejpam-677	92	19	)	)	PUNCT
ejpam-677	92	20	=	=	SYM
ejpam-677	92	21	c(x)+	c(x)+	PROPN
ejpam-677	92	22	c(y	c(y	PROPN
ejpam-677	92	23	)	)	PUNCT
ejpam-677	92	24	(	(	PUNCT
ejpam-677	92	25	9	9	X
ejpam-677	92	26	)	)	PUNCT
ejpam-677	92	27	for	for	ADP
ejpam-677	92	28	all	all	DET
ejpam-677	92	29	x	x	SYM
ejpam-677	92	30	,	,	PUNCT
ejpam-677	92	31	y	y	PROPN
ejpam-677	92	32	∈	∈	PROPN
ejpam-677	92	33	x	x	X
ejpam-677	92	34	.	.	PUNCT
ejpam-677	93	1	assume	assume	VERB
ejpam-677	93	2	that	that	SCONJ
ejpam-677	93	3	there	there	PRON
ejpam-677	93	4	exists	exist	VERB
ejpam-677	93	5	a	a	DET
ejpam-677	93	6	mapping	mapping	NOUN
ejpam-677	93	7	a	a	DET
ejpam-677	93	8	:	:	PUNCT
ejpam-677	93	9	x	x	X
ejpam-677	93	10	→	→	SYM
ejpam-677	93	11	y	y	NUM
ejpam-677	93	12	such	such	ADJ
ejpam-677	93	13	that	that	SCONJ
ejpam-677	93	14	a(x	a(x	NOUN
ejpam-677	93	15	)	)	PUNCT
ejpam-677	93	16	=	=	SYM
ejpam-677	93	17	c(x	c(x	NOUN
ejpam-677	93	18	)	)	PUNCT
ejpam-677	93	19	for	for	ADP
ejpam-677	93	20	all	all	DET
ejpam-677	93	21	x	x	SYM
ejpam-677	93	22	∈	∈	NOUN
ejpam-677	93	23	x	x	X
ejpam-677	93	24	.	.	PUNCT
ejpam-677	94	1	observe	observe	VERB
ejpam-677	94	2	that	that	SCONJ
ejpam-677	94	3	for	for	ADP
ejpam-677	94	4	x	x	SYM
ejpam-677	94	5	=	=	PUNCT
ejpam-677	94	6	y	y	PROPN
ejpam-677	94	7	=	=	SYM
ejpam-677	94	8	0	0	PROPN
ejpam-677	94	9	,	,	PUNCT
ejpam-677	94	10	from	from	ADP
ejpam-677	94	11	(	(	PUNCT
ejpam-677	94	12	9	9	X
ejpam-677	94	13	)	)	PUNCT
ejpam-677	94	14	we	we	PRON
ejpam-677	94	15	obtain	obtain	VERB
ejpam-677	94	16	a(0	a(0	PROPN
ejpam-677	94	17	)	)	PUNCT
ejpam-677	94	18	=	=	SYM
ejpam-677	94	19	c(0	c(0	NOUN
ejpam-677	94	20	)	)	PUNCT
ejpam-677	94	21	=	=	SYM
ejpam-677	94	22	0	0	X
ejpam-677	94	23	.	.	PUNCT
ejpam-677	94	24	(	(	PUNCT
ejpam-677	94	25	10	10	NUM
ejpam-677	94	26	)	)	PUNCT
ejpam-677	94	27	m.	m.	NOUN
ejpam-677	94	28	rassias	rassias	PROPN
ejpam-677	94	29	/	/	SYM
ejpam-677	94	30	eur	eur	PROPN
ejpam-677	94	31	.	.	PUNCT
ejpam-677	95	1	j.	j.	PROPN
ejpam-677	95	2	pure	pure	PROPN
ejpam-677	95	3	appl	appl	PROPN
ejpam-677	95	4	.	.	PROPN
ejpam-677	95	5	math	math	PROPN
ejpam-677	95	6	,	,	PUNCT
ejpam-677	95	7	4	4	NUM
ejpam-677	95	8	(	(	PUNCT
ejpam-677	95	9	2011	2011	NUM
ejpam-677	95	10	)	)	PUNCT
ejpam-677	95	11	,	,	PUNCT
ejpam-677	95	12	50	50	NUM
ejpam-677	95	13	-	-	SYM
ejpam-677	95	14	58	58	NUM
ejpam-677	95	15	54	54	NUM
ejpam-677	95	16	thus	thus	ADV
ejpam-677	95	17	,	,	PUNCT
ejpam-677	95	18	from	from	ADP
ejpam-677	95	19	(	(	PUNCT
ejpam-677	95	20	9	9	NUM
ejpam-677	95	21	)	)	PUNCT
ejpam-677	95	22	and	and	CCONJ
ejpam-677	95	23	(	(	PUNCT
ejpam-677	95	24	10	10	NUM
ejpam-677	95	25	)	)	PUNCT
ejpam-677	95	26	one	one	NOUN
ejpam-677	95	27	gets	get	VERB
ejpam-677	95	28	a(x)+	a(x)+	ADP
ejpam-677	95	29	a(y	a(y	PROPN
ejpam-677	95	30	)	)	PUNCT
ejpam-677	95	31	=	=	SYM
ejpam-677	95	32	c(x)+	c(x)+	PROPN
ejpam-677	95	33	c(y	c(y	PROPN
ejpam-677	95	34	)	)	PUNCT
ejpam-677	95	35	=	=	PUNCT
ejpam-677	96	1	c(x	c(x	NOUN
ejpam-677	96	2	+	+	CCONJ
ejpam-677	96	3	y	y	NOUN
ejpam-677	96	4	)	)	PUNCT
ejpam-677	97	1	=	=	PUNCT
ejpam-677	97	2	a(x	a(x	NOUN
ejpam-677	97	3	+	+	NUM
ejpam-677	97	4	y	y	NOUN
ejpam-677	97	5	)	)	PUNCT
ejpam-677	97	6	=	=	PUNCT
ejpam-677	98	1	a(x	a(x	PROPN
ejpam-677	98	2	+	+	NUM
ejpam-677	98	3	y	y	NOUN
ejpam-677	98	4	)	)	PUNCT
ejpam-677	98	5	+	+	CCONJ
ejpam-677	98	6	a(0	a(0	PROPN
ejpam-677	98	7	)	)	PUNCT
ejpam-677	98	8	=	=	PUNCT
ejpam-677	99	1	a(x	a(x	PROPN
ejpam-677	99	2	+	+	NUM
ejpam-677	99	3	y	y	NOUN
ejpam-677	99	4	)	)	PUNCT
ejpam-677	100	1	+	+	CCONJ
ejpam-677	100	2	a((x	a((x	NOUN
ejpam-677	100	3	−	−	PROPN
ejpam-677	100	4	y	y	PROPN
ejpam-677	100	5	)	)	PUNCT
ejpam-677	101	1	+	+	CCONJ
ejpam-677	101	2	(	(	PUNCT
ejpam-677	101	3	y	y	PROPN
ejpam-677	101	4	−	−	PROPN
ejpam-677	101	5	x	x	NOUN
ejpam-677	101	6	)	)	PUNCT
ejpam-677	101	7	)	)	PUNCT
ejpam-677	102	1	=	=	PUNCT
ejpam-677	102	2	a(x	a(x	NOUN
ejpam-677	102	3	+	+	NUM
ejpam-677	102	4	y	y	NOUN
ejpam-677	102	5	)	)	PUNCT
ejpam-677	103	1	+	+	CCONJ
ejpam-677	103	2	a(x	a(x	PROPN
ejpam-677	103	3	−	−	PROPN
ejpam-677	103	4	y	y	NOUN
ejpam-677	103	5	)	)	PUNCT
ejpam-677	104	1	+	+	CCONJ
ejpam-677	104	2	a(y	a(y	PROPN
ejpam-677	104	3	−	−	NOUN
ejpam-677	104	4	x	x	NOUN
ejpam-677	104	5	)	)	PUNCT
ejpam-677	104	6	.	.	PUNCT
ejpam-677	105	1	hence	hence	ADV
ejpam-677	105	2	,	,	PUNCT
ejpam-677	105	3	a	a	DET
ejpam-677	105	4	satisfies	satisfie	NOUN
ejpam-677	105	5	the	the	DET
ejpam-677	105	6	cauchy	cauchy	ADJ
ejpam-677	105	7	type	type	NOUN
ejpam-677	105	8	additive	additive	ADJ
ejpam-677	105	9	equation	equation	NOUN
ejpam-677	105	10	.	.	PUNCT
ejpam-677	106	1	thus	thus	ADV
ejpam-677	106	2	the	the	DET
ejpam-677	106	3	proof	proof	NOUN
ejpam-677	106	4	of	of	ADP
ejpam-677	106	5	lemma	lemma	PROPN
ejpam-677	106	6	1	1	NUM
ejpam-677	106	7	is	be	AUX
ejpam-677	106	8	complete	complete	ADJ
ejpam-677	106	9	.	.	PUNCT
ejpam-677	107	1	theorem	theorem	ADJ
ejpam-677	107	2	6	6	NUM
ejpam-677	107	3	.	.	PUNCT
ejpam-677	108	1	assume	assume	VERB
ejpam-677	108	2	that	that	SCONJ
ejpam-677	108	3	f	f	X
ejpam-677	108	4	:	:	PUNCT
ejpam-677	108	5	x	x	X
ejpam-677	108	6	→	→	SYM
ejpam-677	108	7	y	y	PROPN
ejpam-677	108	8	is	be	AUX
ejpam-677	108	9	an	an	DET
ejpam-677	108	10	approximately	approximately	ADV
ejpam-677	108	11	cauchy	cauchy	ADJ
ejpam-677	108	12	type	type	NOUN
ejpam-677	108	13	additive	additive	ADJ
ejpam-677	108	14	mapping	mapping	NOUN
ejpam-677	108	15	satisfying	satisfy	VERB
ejpam-677	108	16	(	(	PUNCT
ejpam-677	108	17	6	6	NUM
ejpam-677	108	18	)	)	PUNCT
ejpam-677	108	19	.	.	PUNCT
ejpam-677	109	1	then	then	ADV
ejpam-677	109	2	,	,	PUNCT
ejpam-677	109	3	there	there	PRON
ejpam-677	109	4	exists	exist	VERB
ejpam-677	109	5	a	a	DET
ejpam-677	109	6	unique	unique	ADJ
ejpam-677	109	7	cauchy	cauchy	ADJ
ejpam-677	109	8	type	type	NOUN
ejpam-677	109	9	additive	additive	NOUN
ejpam-677	109	10	mapping	mapping	NOUN
ejpam-677	109	11	a	a	DET
ejpam-677	109	12	:	:	PUNCT
ejpam-677	109	13	x	x	X
ejpam-677	109	14	→	→	SYM
ejpam-677	109	15	y	y	PROPN
ejpam-677	109	16	which	which	PRON
ejpam-677	109	17	satisfies	satisfy	VERB
ejpam-677	109	18	the	the	DET
ejpam-677	109	19	formula	formula	NOUN
ejpam-677	109	20	a(x	a(x	NOUN
ejpam-677	109	21	)	)	PUNCT
ejpam-677	109	22	=	=	SYM
ejpam-677	109	23	lim	lim	PROPN
ejpam-677	109	24	n→∞	n→∞	X
ejpam-677	109	25	fn(x	fn(x	NOUN
ejpam-677	109	26	)	)	PUNCT
ejpam-677	109	27	,	,	PUNCT
ejpam-677	109	28	where	where	SCONJ
ejpam-677	109	29	fn(x	fn(x	X
ejpam-677	109	30	)	)	PUNCT
ejpam-677	109	31	=	=	SYM
ejpam-677	109	32	�	�	PROPN
ejpam-677	109	33	2−n	2−n	NUM
ejpam-677	109	34	f	f	PROPN
ejpam-677	109	35	(	(	PUNCT
ejpam-677	109	36	2n	2n	NUM
ejpam-677	109	37	x	x	NOUN
ejpam-677	109	38	)	)	PUNCT
ejpam-677	109	39	,	,	PUNCT
ejpam-677	109	40	−∞<α<1	−∞<α<1	NOUN
ejpam-677	109	41	2n	2n	NUM
ejpam-677	109	42	f	f	X
ejpam-677	109	43	(	(	PUNCT
ejpam-677	109	44	2−n	2−n	NUM
ejpam-677	109	45	x	x	NOUN
ejpam-677	109	46	)	)	PUNCT
ejpam-677	109	47	,	,	PUNCT
ejpam-677	109	48	α>1	α>1	NOUN
ejpam-677	109	49	for	for	ADP
ejpam-677	109	50	all	all	DET
ejpam-677	109	51	x	x	SYM
ejpam-677	109	52	∈	∈	PROPN
ejpam-677	109	53	x	x	X
ejpam-677	109	54	and	and	CCONJ
ejpam-677	109	55	n	n	CCONJ
ejpam-677	109	56	∈	∈	PROPN
ejpam-677	109	57	n	n	NOUN
ejpam-677	109	58	=	=	PUNCT
ejpam-677	109	59	{	{	PUNCT
ejpam-677	109	60	0,1,2	0,1,2	NOUN
ejpam-677	109	61	,	,	PUNCT
ejpam-677	109	62	.	.	PUNCT
ejpam-677	109	63	.	.	PUNCT
ejpam-677	110	1	.	.	PUNCT
ejpam-677	111	1	}	}	PUNCT
ejpam-677	111	2	,	,	PUNCT
ejpam-677	111	3	which	which	PRON
ejpam-677	111	4	is	be	AUX
ejpam-677	111	5	the	the	DET
ejpam-677	111	6	set	set	NOUN
ejpam-677	111	7	of	of	ADP
ejpam-677	111	8	natural	natural	ADJ
ejpam-677	111	9	numbers	number	NOUN
ejpam-677	111	10	and	and	CCONJ
ejpam-677	111	11	||	||	NUM
ejpam-677	111	12	f	f	PROPN
ejpam-677	112	1	(	(	PUNCT
ejpam-677	112	2	x)−	x)−	PROPN
ejpam-677	112	3	a(x)||	a(x)||	NOUN
ejpam-677	112	4	≤	≤	NOUN
ejpam-677	112	5	3ǫ	3ǫ	VERB
ejpam-677	112	6	|2−	|2−	PROPN
ejpam-677	112	7	2α|	2α|	NUM
ejpam-677	112	8	||x	||x	NOUN
ejpam-677	112	9	||	||	NOUN
ejpam-677	113	1	α	α	PROPN
ejpam-677	113	2	for	for	ADP
ejpam-677	113	3	some	some	DET
ejpam-677	113	4	fixed	fixed	ADJ
ejpam-677	113	5	ǫ	ǫ	PRON
ejpam-677	113	6	>	>	X
ejpam-677	113	7	0	0	NUM
ejpam-677	113	8	,	,	PUNCT
ejpam-677	113	9	α	α	PROPN
ejpam-677	113	10	6=	6=	ADP
ejpam-677	113	11	1	1	NUM
ejpam-677	113	12	and	and	CCONJ
ejpam-677	113	13	all	all	DET
ejpam-677	113	14	x	x	SYM
ejpam-677	113	15	∈	∈	ADJ
ejpam-677	113	16	x	x	X
ejpam-677	113	17	.	.	PUNCT
ejpam-677	114	1	if	if	SCONJ
ejpam-677	114	2	,	,	PUNCT
ejpam-677	114	3	moreover	moreover	ADV
ejpam-677	114	4	,	,	PUNCT
ejpam-677	114	5	f	f	PROPN
ejpam-677	114	6	(	(	PUNCT
ejpam-677	114	7	t	t	NOUN
ejpam-677	114	8	x	x	VERB
ejpam-677	114	9	)	)	PUNCT
ejpam-677	114	10	is	be	AUX
ejpam-677	114	11	continuous	continuous	ADJ
ejpam-677	114	12	in	in	ADP
ejpam-677	114	13	t	t	PROPN
ejpam-677	114	14	∈	∈	NOUN
ejpam-677	114	15	r	r	NOUN
ejpam-677	114	16	for	for	ADP
ejpam-677	114	17	each	each	DET
ejpam-677	114	18	fixed	fix	VERB
ejpam-677	114	19	x	x	SYM
ejpam-677	114	20	∈	∈	PROPN
ejpam-677	114	21	x	x	X
ejpam-677	114	22	,	,	PUNCT
ejpam-677	114	23	then	then	ADV
ejpam-677	114	24	a(t	a(t	VERB
ejpam-677	114	25	x	x	PRON
ejpam-677	114	26	)	)	PUNCT
ejpam-677	114	27	=	=	SYM
ejpam-677	114	28	ta(x	ta(x	NOUN
ejpam-677	114	29	)	)	PUNCT
ejpam-677	114	30	for	for	ADP
ejpam-677	114	31	all	all	DET
ejpam-677	114	32	t	t	NOUN
ejpam-677	114	33	∈	∈	NOUN
ejpam-677	114	34	r	r	NOUN
ejpam-677	114	35	and	and	CCONJ
ejpam-677	114	36	x	x	NOUN
ejpam-677	114	37	∈	∈	PROPN
ejpam-677	114	38	x	x	X
ejpam-677	114	39	.	.	PUNCT
ejpam-677	115	1	a	a	PRON
ejpam-677	115	2	:	:	PUNCT
ejpam-677	115	3	x	x	X
ejpam-677	115	4	→	→	SYM
ejpam-677	115	5	y	y	PROPN
ejpam-677	115	6	is	be	AUX
ejpam-677	115	7	a	a	DET
ejpam-677	115	8	unique	unique	ADJ
ejpam-677	115	9	linear	linear	NOUN
ejpam-677	115	10	cauchy	cauchy	NOUN
ejpam-677	115	11	type	type	NOUN
ejpam-677	115	12	additive	additive	ADJ
ejpam-677	115	13	mapping	mapping	NOUN
ejpam-677	115	14	satisfying	satisfy	VERB
ejpam-677	115	15	equation	equation	NOUN
ejpam-677	115	16	a(x	a(x	NOUN
ejpam-677	115	17	+	+	PROPN
ejpam-677	115	18	y	y	NOUN
ejpam-677	115	19	)	)	PUNCT
ejpam-677	116	1	+	+	CCONJ
ejpam-677	116	2	a(x	a(x	PROPN
ejpam-677	116	3	−	−	PROPN
ejpam-677	116	4	y	y	NOUN
ejpam-677	116	5	)	)	PUNCT
ejpam-677	117	1	+	+	CCONJ
ejpam-677	117	2	a(y	a(y	PROPN
ejpam-677	117	3	−	−	NOUN
ejpam-677	117	4	x	x	NOUN
ejpam-677	117	5	)	)	PUNCT
ejpam-677	117	6	=	=	SYM
ejpam-677	117	7	a(x)+	a(x)+	ADP
ejpam-677	117	8	a(y	a(y	PROPN
ejpam-677	117	9	)	)	PUNCT
ejpam-677	117	10	.	.	PUNCT
ejpam-677	118	1	(	(	PUNCT
ejpam-677	118	2	11	11	NUM
ejpam-677	118	3	)	)	PUNCT
ejpam-677	118	4	proof	proof	NOUN
ejpam-677	118	5	.	.	PUNCT
ejpam-677	119	1	we	we	PRON
ejpam-677	119	2	start	start	VERB
ejpam-677	119	3	our	our	PRON
ejpam-677	119	4	proof	proof	NOUN
ejpam-677	119	5	considering	consider	VERB
ejpam-677	119	6	:	:	PUNCT
ejpam-677	119	7	−∞	−∞	X
ejpam-677	119	8	<	<	X
ejpam-677	119	9	α	α	X
ejpam-677	119	10	<	<	X
ejpam-677	119	11	1	1	NUM
ejpam-677	119	12	.	.	PUNCT
ejpam-677	119	13	step	step	NOUN
ejpam-677	119	14	1	1	NUM
ejpam-677	119	15	by	by	ADP
ejpam-677	119	16	substituting	substitute	VERB
ejpam-677	119	17	x	x	PUNCT
ejpam-677	119	18	=	=	PUNCT
ejpam-677	119	19	y	y	PROPN
ejpam-677	119	20	=	=	SYM
ejpam-677	119	21	0	0	PROPN
ejpam-677	119	22	and	and	CCONJ
ejpam-677	119	23	x	x	X
ejpam-677	119	24	=	=	SYM
ejpam-677	119	25	y	y	PROPN
ejpam-677	119	26	in	in	ADP
ejpam-677	119	27	(	(	PUNCT
ejpam-677	119	28	6	6	NUM
ejpam-677	119	29	)	)	PUNCT
ejpam-677	119	30	,	,	PUNCT
ejpam-677	119	31	respectively	respectively	ADV
ejpam-677	119	32	,	,	PUNCT
ejpam-677	119	33	we	we	PRON
ejpam-677	119	34	can	can	AUX
ejpam-677	119	35	observe	observe	VERB
ejpam-677	120	1	that	that	SCONJ
ejpam-677	120	2	f	f	PROPN
ejpam-677	120	3	(	(	PUNCT
ejpam-677	120	4	0	0	NUM
ejpam-677	120	5	)	)	PUNCT
ejpam-677	120	6	=	=	SYM
ejpam-677	120	7	0	0	NUM
ejpam-677	120	8	and	and	CCONJ
ejpam-677	120	9	||	||	NUM
ejpam-677	120	10	f	f	X
ejpam-677	120	11	(	(	PUNCT
ejpam-677	120	12	x)−	x)−	PROPN
ejpam-677	121	1	2−1	2−1	NUM
ejpam-677	121	2	f	f	X
ejpam-677	121	3	(	(	PUNCT
ejpam-677	121	4	2x)||	2x)||	NUM
ejpam-677	121	5	≤	≤	NUM
ejpam-677	121	6	3	3	NUM
ejpam-677	121	7	2	2	NUM
ejpam-677	121	8	ǫ||x	ǫ||x	PROPN
ejpam-677	121	9	||α	||α	NOUN
ejpam-677	121	10	.	.	PUNCT
ejpam-677	122	1	hence	hence	ADV
ejpam-677	122	2	,	,	PUNCT
ejpam-677	122	3	for	for	ADP
ejpam-677	122	4	n	n	PRON
ejpam-677	122	5	∈	∈	PROPN
ejpam-677	122	6	n	n	PRON
ejpam-677	122	7	−	−	PROPN
ejpam-677	122	8	{	{	PUNCT
ejpam-677	122	9	0	0	NUM
ejpam-677	122	10	}	}	PUNCT
ejpam-677	122	11	||	||	NUM
ejpam-677	123	1	f	f	X
ejpam-677	123	2	(	(	PUNCT
ejpam-677	123	3	x)−	x)−	PROPN
ejpam-677	123	4	2−n	2−n	NUM
ejpam-677	123	5	f	f	PROPN
ejpam-677	123	6	(	(	PUNCT
ejpam-677	123	7	2n	2n	NUM
ejpam-677	123	8	x)||	x)||	NOUN
ejpam-677	123	9	≤	≤	NUM
ejpam-677	123	10	||	||	PUNCT
ejpam-677	124	1	f	f	X
ejpam-677	124	2	(	(	PUNCT
ejpam-677	124	3	x)−	x)−	PROPN
ejpam-677	124	4	2−1	2−1	NUM
ejpam-677	124	5	f	f	X
ejpam-677	124	6	(	(	PUNCT
ejpam-677	124	7	2x)||+	2x)||+	NUM
ejpam-677	124	8	||2−1	||2−1	NOUN
ejpam-677	124	9	f	f	X
ejpam-677	124	10	(	(	PUNCT
ejpam-677	124	11	2x)−	2x)−	NUM
ejpam-677	124	12	2−2	2−2	NUM
ejpam-677	124	13	f	f	NOUN
ejpam-677	124	14	(	(	PUNCT
ejpam-677	124	15	22	22	NUM
ejpam-677	124	16	x)||+	x)||+	PROPN
ejpam-677	124	17	.	.	PUNCT
ejpam-677	124	18	.	.	PUNCT
ejpam-677	124	19	.	.	PUNCT
ejpam-677	125	1	+	+	CCONJ
ejpam-677	125	2	||2−(n−1	||2−(n−1	X
ejpam-677	125	3	)	)	PUNCT
ejpam-677	125	4	f	f	NOUN
ejpam-677	125	5	(	(	PUNCT
ejpam-677	125	6	2n−1	2n−1	NUM
ejpam-677	125	7	x)−	x)−	PROPN
ejpam-677	125	8	2−n	2−n	NUM
ejpam-677	125	9	f	f	NOUN
ejpam-677	125	10	(	(	PUNCT
ejpam-677	125	11	2nx)||	2nx)||	NUM
ejpam-677	125	12	≤	≤	NUM
ejpam-677	125	13	3	3	NUM
ejpam-677	125	14	2	2	NUM
ejpam-677	125	15	(	(	PUNCT
ejpam-677	125	16	1	1	NUM
ejpam-677	125	17	+	+	NUM
ejpam-677	125	18	2α−1	2α−1	NUM
ejpam-677	125	19	+	+	NUM
ejpam-677	125	20	...	...	PUNCT
ejpam-677	126	1	+	+	CCONJ
ejpam-677	126	2	2(n−1)(α−1))ǫ||x	2(n−1)(α−1))ǫ||x	ADJ
ejpam-677	126	3	||α	||α	NOUN
ejpam-677	126	4	=	=	SYM
ejpam-677	126	5	3	3	NUM
ejpam-677	126	6	2−	2−	NUM
ejpam-677	126	7	2α	2α	NOUN
ejpam-677	126	8	(	(	PUNCT
ejpam-677	126	9	1−	1−	NUM
ejpam-677	126	10	2n(α−1))ǫ||x	2n(α−1))ǫ||x	NUM
ejpam-677	126	11	||α	||α	NOUN
ejpam-677	126	12	.	.	PUNCT
ejpam-677	127	1	m.	m.	NOUN
ejpam-677	127	2	rassias	rassias	PROPN
ejpam-677	127	3	/	/	SYM
ejpam-677	127	4	eur	eur	PROPN
ejpam-677	127	5	.	.	PUNCT
ejpam-677	128	1	j.	j.	PROPN
ejpam-677	128	2	pure	pure	PROPN
ejpam-677	128	3	appl	appl	PROPN
ejpam-677	128	4	.	.	PROPN
ejpam-677	128	5	math	math	PROPN
ejpam-677	128	6	,	,	PUNCT
ejpam-677	128	7	4	4	NUM
ejpam-677	128	8	(	(	PUNCT
ejpam-677	128	9	2011	2011	NUM
ejpam-677	128	10	)	)	PUNCT
ejpam-677	128	11	,	,	PUNCT
ejpam-677	128	12	50	50	NUM
ejpam-677	128	13	-	-	SYM
ejpam-677	128	14	58	58	NUM
ejpam-677	128	15	55	55	NUM
ejpam-677	128	16	thus	thus	ADV
ejpam-677	128	17	,	,	PUNCT
ejpam-677	128	18	||	||	PROPN
ejpam-677	129	1	f	f	X
ejpam-677	129	2	(	(	PUNCT
ejpam-677	129	3	x)−	x)−	PROPN
ejpam-677	129	4	2−n	2−n	NUM
ejpam-677	129	5	f	f	NOUN
ejpam-677	129	6	(	(	PUNCT
ejpam-677	129	7	2nx)||	2nx)||	NUM
ejpam-677	129	8	≤	≤	NOUN
ejpam-677	129	9	3	3	NUM
ejpam-677	129	10	2−	2−	NUM
ejpam-677	129	11	2α	2α	NOUN
ejpam-677	129	12	(	(	PUNCT
ejpam-677	129	13	1−	1−	NUM
ejpam-677	129	14	2n(α−1))ǫ||x	2n(α−1))ǫ||x	NUM
ejpam-677	129	15	||α	||α	NOUN
ejpam-677	129	16	,	,	PUNCT
ejpam-677	129	17	for	for	ADP
ejpam-677	129	18	n	n	PRON
ejpam-677	129	19	∈	∈	PROPN
ejpam-677	129	20	n	n	PRON
ejpam-677	129	21	−	−	PROPN
ejpam-677	129	22	{	{	PUNCT
ejpam-677	129	23	0	0	NUM
ejpam-677	129	24	}	}	PUNCT
ejpam-677	129	25	and	and	CCONJ
ejpam-677	129	26	−∞	−∞	ADP
ejpam-677	129	27	<	<	X
ejpam-677	129	28	α	α	X
ejpam-677	129	29	<	<	X
ejpam-677	129	30	1	1	NUM
ejpam-677	129	31	.	.	PUNCT
ejpam-677	129	32	step	step	NOUN
ejpam-677	129	33	2	2	NUM
ejpam-677	129	34	following	follow	VERB
ejpam-677	129	35	,	,	PUNCT
ejpam-677	129	36	we	we	PRON
ejpam-677	129	37	need	need	VERB
ejpam-677	129	38	to	to	PART
ejpam-677	129	39	show	show	VERB
ejpam-677	129	40	that	that	SCONJ
ejpam-677	129	41	if	if	SCONJ
ejpam-677	129	42	there	there	PRON
ejpam-677	129	43	is	be	VERB
ejpam-677	129	44	a	a	DET
ejpam-677	129	45	sequence	sequence	NOUN
ejpam-677	129	46	{	{	PUNCT
ejpam-677	129	47	fn	fn	NOUN
ejpam-677	129	48	}	}	PUNCT
ejpam-677	129	49	:	:	PUNCT
ejpam-677	129	50	fn(x	fn(x	X
ejpam-677	129	51	)	)	PUNCT
ejpam-677	129	52	=	=	SYM
ejpam-677	129	53	2−n	2−n	NUM
ejpam-677	129	54	f	f	NOUN
ejpam-677	129	55	(	(	PUNCT
ejpam-677	129	56	2nx	2nx	NOUN
ejpam-677	129	57	)	)	PUNCT
ejpam-677	129	58	,	,	PUNCT
ejpam-677	129	59	then	then	ADV
ejpam-677	129	60	{	{	PUNCT
ejpam-677	129	61	fn	fn	NOUN
ejpam-677	129	62	}	}	PUNCT
ejpam-677	129	63	converges	converge	NOUN
ejpam-677	129	64	.	.	PUNCT
ejpam-677	130	1	for	for	ADP
ejpam-677	130	2	every	every	DET
ejpam-677	130	3	n	n	CCONJ
ejpam-677	130	4	>	>	X
ejpam-677	130	5	m	m	PROPN
ejpam-677	130	6	>	>	X
ejpam-677	130	7	0	0	NUM
ejpam-677	130	8	,	,	PUNCT
ejpam-677	130	9	we	we	PRON
ejpam-677	130	10	can	can	AUX
ejpam-677	130	11	obtain	obtain	VERB
ejpam-677	130	12	||	||	NOUN
ejpam-677	131	1	fn(x)−	fn(x)−	NOUN
ejpam-677	132	1	fm(x)||	fm(x)||	PROPN
ejpam-677	132	2	=	=	PUNCT
ejpam-677	132	3	||2−n	||2−n	NUM
ejpam-677	132	4	f	f	PROPN
ejpam-677	132	5	(	(	PUNCT
ejpam-677	132	6	2nx)−	2nx)−	NUM
ejpam-677	132	7	2−m	2−m	NUM
ejpam-677	132	8	f	f	X
ejpam-677	132	9	(	(	PUNCT
ejpam-677	132	10	2mx)||	2mx)||	NUM
ejpam-677	132	11	=	=	SYM
ejpam-677	132	12	2−m||	2−m||	NUM
ejpam-677	132	13	f	f	NOUN
ejpam-677	132	14	(	(	PUNCT
ejpam-677	132	15	2mx)−	2mx)−	PROPN
ejpam-677	132	16	2−(n−m	2−(n−m	NUM
ejpam-677	132	17	)	)	PUNCT
ejpam-677	132	18	f	f	PROPN
ejpam-677	132	19	(	(	PUNCT
ejpam-677	132	20	2(n−m)2mx)||	2(n−m)2mx)||	NUM
ejpam-677	132	21	≤	≤	NOUN
ejpam-677	132	22	2−m	2−m	NUM
ejpam-677	132	23	3ǫ	3ǫ	NOUN
ejpam-677	132	24	2−	2−	NUM
ejpam-677	132	25	2α	2α	NOUN
ejpam-677	132	26	(	(	PUNCT
ejpam-677	132	27	1−	1−	NUM
ejpam-677	132	28	2(n−m)(α−1))||x	2(n−m)(α−1))||x	NOUN
ejpam-677	132	29	||α	||α	NOUN
ejpam-677	132	30	<	<	X
ejpam-677	132	31	2−m	2−m	NUM
ejpam-677	132	32	3ǫ	3ǫ	NUM
ejpam-677	132	33	2−	2−	NUM
ejpam-677	132	34	2α	2α	NOUN
ejpam-677	132	35	||x	||x	NOUN
ejpam-677	132	36	||α→	||α→	X
ejpam-677	132	37	0	0	NUM
ejpam-677	132	38	,	,	PUNCT
ejpam-677	132	39	for	for	ADP
ejpam-677	132	40	m	m	PROPN
ejpam-677	132	41	→	→	SYM
ejpam-677	132	42	∞	∞	PROPN
ejpam-677	132	43	,	,	PUNCT
ejpam-677	132	44	as	as	ADP
ejpam-677	132	45	α	α	X
ejpam-677	132	46	<	<	X
ejpam-677	132	47	1	1	NUM
ejpam-677	132	48	.	.	PUNCT
ejpam-677	133	1	therefore	therefore	ADV
ejpam-677	133	2	,	,	PUNCT
ejpam-677	133	3	{	{	PUNCT
ejpam-677	133	4	fn	fn	NOUN
ejpam-677	133	5	}	}	PUNCT
ejpam-677	133	6	is	be	AUX
ejpam-677	133	7	a	a	DET
ejpam-677	133	8	cauchy	cauchy	ADJ
ejpam-677	133	9	sequence	sequence	NOUN
ejpam-677	133	10	.	.	PUNCT
ejpam-677	134	1	since	since	SCONJ
ejpam-677	134	2	y	y	PROPN
ejpam-677	134	3	is	be	AUX
ejpam-677	134	4	complete	complete	ADJ
ejpam-677	134	5	we	we	PRON
ejpam-677	134	6	can	can	AUX
ejpam-677	134	7	conclude	conclude	VERB
ejpam-677	134	8	that	that	PRON
ejpam-677	134	9	{	{	PUNCT
ejpam-677	134	10	fn	fn	NOUN
ejpam-677	134	11	}	}	PUNCT
ejpam-677	134	12	is	be	AUX
ejpam-677	134	13	convergent	convergent	ADJ
ejpam-677	134	14	.	.	PUNCT
ejpam-677	135	1	thus	thus	ADV
ejpam-677	135	2	,	,	PUNCT
ejpam-677	135	3	there	there	PRON
ejpam-677	135	4	is	be	VERB
ejpam-677	135	5	a	a	DET
ejpam-677	135	6	well	well	ADV
ejpam-677	135	7	-	-	PUNCT
ejpam-677	135	8	defined	define	VERB
ejpam-677	135	9	a	a	DET
ejpam-677	135	10	:	:	PUNCT
ejpam-677	135	11	x	x	SYM
ejpam-677	135	12	→	→	SYM
ejpam-677	135	13	y	y	NUM
ejpam-677	135	14	such	such	ADJ
ejpam-677	135	15	that	that	SCONJ
ejpam-677	135	16	a(x	a(x	NOUN
ejpam-677	135	17	)	)	PUNCT
ejpam-677	135	18	=	=	PUNCT
ejpam-677	135	19	limn→∞	limn→∞	PROPN
ejpam-677	135	20	2−n	2−n	NUM
ejpam-677	135	21	f	f	NOUN
ejpam-677	135	22	(	(	PUNCT
ejpam-677	135	23	2nx	2nx	NOUN
ejpam-677	135	24	)	)	PUNCT
ejpam-677	135	25	,	,	PUNCT
ejpam-677	135	26	for	for	ADP
ejpam-677	135	27	−∞	−∞	X
ejpam-677	135	28	<	<	X
ejpam-677	135	29	α	α	X
ejpam-677	135	30	<	<	X
ejpam-677	135	31	1	1	NUM
ejpam-677	135	32	.	.	PUNCT
ejpam-677	135	33	step	step	NOUN
ejpam-677	135	34	3	3	NUM
ejpam-677	135	35	observe	observe	VERB
ejpam-677	135	36	that	that	PRON
ejpam-677	135	37	||	||	NOUN
ejpam-677	136	1	f	f	X
ejpam-677	136	2	(	(	PUNCT
ejpam-677	136	3	x)−	x)−	PROPN
ejpam-677	136	4	fn(x)||=	fn(x)||=	PROPN
ejpam-677	136	5	||	||	PUNCT
ejpam-677	137	1	f	f	X
ejpam-677	137	2	(	(	PUNCT
ejpam-677	137	3	x)−	x)−	PROPN
ejpam-677	137	4	2−n	2−n	NUM
ejpam-677	137	5	f	f	NOUN
ejpam-677	137	6	(	(	PUNCT
ejpam-677	137	7	2nx)||	2nx)||	NUM
ejpam-677	137	8	≤	≤	NOUN
ejpam-677	137	9	3ǫ	3ǫ	NOUN
ejpam-677	137	10	2−	2−	NUM
ejpam-677	137	11	2α	2α	NOUN
ejpam-677	137	12	(	(	PUNCT
ejpam-677	137	13	1−	1−	NUM
ejpam-677	137	14	2n(α−1))||x	2n(α−1))||x	NUM
ejpam-677	137	15	||α	||α	NOUN
ejpam-677	137	16	,	,	PUNCT
ejpam-677	137	17	from	from	ADP
ejpam-677	137	18	which	which	PRON
ejpam-677	137	19	by	by	ADP
ejpam-677	137	20	letting	let	VERB
ejpam-677	137	21	n→∞	n→∞	PRON
ejpam-677	137	22	we	we	PRON
ejpam-677	137	23	obtain	obtain	VERB
ejpam-677	137	24	||	||	PUNCT
ejpam-677	138	1	f	f	X
ejpam-677	138	2	(	(	PUNCT
ejpam-677	138	3	x)−	x)−	PROPN
ejpam-677	138	4	a(x)||	a(x)||	NOUN
ejpam-677	138	5	≤	≤	PROPN
ejpam-677	138	6	3ǫ	3ǫ	PROPN
ejpam-677	138	7	2−	2−	NUM
ejpam-677	138	8	2α	2α	NOUN
ejpam-677	138	9	||x	||x	NOUN
ejpam-677	138	10	||α	||α	NOUN
ejpam-677	138	11	.	.	PUNCT
ejpam-677	139	1	(	(	PUNCT
ejpam-677	139	2	12	12	NUM
ejpam-677	139	3	)	)	PUNCT
ejpam-677	139	4	step	step	NOUN
ejpam-677	139	5	4	4	NUM
ejpam-677	139	6	claim	claim	VERB
ejpam-677	139	7	that	that	SCONJ
ejpam-677	139	8	mapping	map	VERB
ejpam-677	139	9	a	a	X
ejpam-677	139	10	:	:	PUNCT
ejpam-677	139	11	x	x	X
ejpam-677	139	12	→	→	SYM
ejpam-677	139	13	y	y	NUM
ejpam-677	139	14	satisfies	satisfie	NOUN
ejpam-677	139	15	(	(	PUNCT
ejpam-677	139	16	11	11	NUM
ejpam-677	139	17	)	)	PUNCT
ejpam-677	139	18	.	.	PUNCT
ejpam-677	140	1	in	in	ADP
ejpam-677	140	2	fact	fact	NOUN
ejpam-677	140	3	,	,	PUNCT
ejpam-677	140	4	by	by	ADP
ejpam-677	140	5	letting	let	VERB
ejpam-677	140	6	x	x	PRON
ejpam-677	140	7	→	→	SYM
ejpam-677	140	8	2nx	2nx	NOUN
ejpam-677	140	9	and	and	CCONJ
ejpam-677	140	10	y	y	PROPN
ejpam-677	140	11	→	→	SYM
ejpam-677	140	12	2n	2n	NUM
ejpam-677	140	13	y	y	PROPN
ejpam-677	140	14	,	,	PUNCT
ejpam-677	140	15	from	from	ADP
ejpam-677	140	16	(	(	PUNCT
ejpam-677	140	17	6	6	NUM
ejpam-677	140	18	)	)	PUNCT
ejpam-677	140	19	,	,	PUNCT
ejpam-677	140	20	we	we	PRON
ejpam-677	140	21	have	have	VERB
ejpam-677	140	22	:	:	PUNCT
ejpam-677	140	23	||	||	NUM
ejpam-677	140	24	f	f	PROPN
ejpam-677	140	25	�	�	PROPN
ejpam-677	140	26	2n(x	2n(x	PROPN
ejpam-677	140	27	+	+	CCONJ
ejpam-677	140	28	y	y	X
ejpam-677	140	29	)	)	PUNCT
ejpam-677	140	30	�	�	PROPN
ejpam-677	141	1	+	+	CCONJ
ejpam-677	141	2	f	f	PROPN
ejpam-677	141	3	�	�	PROPN
ejpam-677	141	4	2n(x	2n(x	NUM
ejpam-677	141	5	−	−	PROPN
ejpam-677	141	6	y	y	X
ejpam-677	141	7	)	)	PUNCT
ejpam-677	141	8	�	�	PROPN
ejpam-677	141	9	+	+	CCONJ
ejpam-677	141	10	f	f	PROPN
ejpam-677	141	11	�	�	PROPN
ejpam-677	141	12	2n(y	2n(y	NUM
ejpam-677	141	13	−	−	PROPN
ejpam-677	141	14	x	x	SYM
ejpam-677	141	15	)	)	PUNCT
ejpam-677	141	16	�	�	PROPN
ejpam-677	141	17	−	−	PROPN
ejpam-677	141	18	f	f	PROPN
ejpam-677	141	19	�	�	PROPN
ejpam-677	141	20	2n	2n	NUM
ejpam-677	141	21	x	x	SYM
ejpam-677	141	22	�	�	PROPN
ejpam-677	141	23	−	−	PROPN
ejpam-677	141	24	f	f	PROPN
ejpam-677	141	25	�	�	PROPN
ejpam-677	141	26	2n	2n	NUM
ejpam-677	141	27	y	y	PROPN
ejpam-677	141	28	�	�	PROPN
ejpam-677	141	29	||	||	PROPN
ejpam-677	141	30	≤	≤	NUM
ejpam-677	142	1	ǫ	ǫ	X
ejpam-677	142	2	�	�	NOUN
ejpam-677	142	3	||2nx	||2nx	PROPN
ejpam-677	142	4	||	||	PROPN
ejpam-677	142	5	α2	α2	PROPN
ejpam-677	142	6	||2n	||2n	PROPN
ejpam-677	142	7	y||	y||	PROPN
ejpam-677	142	8	α2	α2	PROPN
ejpam-677	142	9	+	+	CCONJ
ejpam-677	142	10	||2nx	||2nx	PROPN
ejpam-677	142	11	||α+	||α+	PROPN
ejpam-677	142	12	||2n	||2n	PROPN
ejpam-677	142	13	y||α	y||α	PROPN
ejpam-677	142	14	�	�	PROPN
ejpam-677	142	15	.	.	PUNCT
ejpam-677	143	1	next	next	ADV
ejpam-677	143	2	,	,	PUNCT
ejpam-677	143	3	by	by	ADP
ejpam-677	143	4	multiplying	multiply	VERB
ejpam-677	143	5	with	with	ADP
ejpam-677	143	6	2−n	2−n	NUM
ejpam-677	143	7	we	we	PRON
ejpam-677	143	8	obtain	obtain	VERB
ejpam-677	143	9	0	0	NUM
ejpam-677	143	10	≤	≤	NOUN
ejpam-677	143	11	||2−n	||2−n	NUM
ejpam-677	143	12	f	f	PROPN
ejpam-677	143	13	�	�	PROPN
ejpam-677	143	14	2n(x	2n(x	NUM
ejpam-677	143	15	+	+	CCONJ
ejpam-677	143	16	y	y	X
ejpam-677	143	17	)	)	PUNCT
ejpam-677	143	18	�	�	PROPN
ejpam-677	144	1	+	+	CCONJ
ejpam-677	144	2	2−n	2−n	NUM
ejpam-677	144	3	f	f	PROPN
ejpam-677	144	4	�	�	PROPN
ejpam-677	144	5	2n(x	2n(x	NUM
ejpam-677	144	6	−	−	PROPN
ejpam-677	144	7	y	y	X
ejpam-677	144	8	)	)	PUNCT
ejpam-677	144	9	�	�	PROPN
ejpam-677	145	1	+	+	CCONJ
ejpam-677	145	2	2−n	2−n	NUM
ejpam-677	145	3	f	f	PROPN
ejpam-677	145	4	�	�	PROPN
ejpam-677	145	5	2n(y	2n(y	NUM
ejpam-677	145	6	−	−	PROPN
ejpam-677	145	7	x	x	SYM
ejpam-677	145	8	)	)	PUNCT
ejpam-677	145	9	�	�	PROPN
ejpam-677	145	10	−	−	NOUN
ejpam-677	145	11	2−n	2−n	NUM
ejpam-677	145	12	f	f	PROPN
ejpam-677	145	13	�	�	PROPN
ejpam-677	145	14	2n	2n	NUM
ejpam-677	145	15	x	x	SYM
ejpam-677	145	16	�	�	PROPN
ejpam-677	145	17	−	−	NOUN
ejpam-677	145	18	2−n	2−n	NUM
ejpam-677	145	19	f	f	PROPN
ejpam-677	145	20	�	�	PROPN
ejpam-677	145	21	2n	2n	NUM
ejpam-677	145	22	y	y	PROPN
ejpam-677	145	23	�	�	PROPN
ejpam-677	145	24	||	||	PROPN
ejpam-677	145	25	≤	≤	NUM
ejpam-677	145	26	2n(α−1)ǫ	2n(α−1)ǫ	NUM
ejpam-677	145	27	�	�	NOUN
ejpam-677	145	28	||x	||x	PROPN
ejpam-677	145	29	||	||	NOUN
ejpam-677	145	30	α2	α2	PROPN
ejpam-677	145	31	||y||	||y||	PROPN
ejpam-677	145	32	α2	α2	PROPN
ejpam-677	145	33	+	+	CCONJ
ejpam-677	145	34	||x	||x	NOUN
ejpam-677	145	35	||α+	||α+	PROPN
ejpam-677	145	36	||y||α	||y||α	PROPN
ejpam-677	145	37	�	�	PROPN
ejpam-677	145	38	and	and	CCONJ
ejpam-677	145	39	by	by	ADP
ejpam-677	145	40	letting	let	VERB
ejpam-677	145	41	n→∞	n→∞	NUM
ejpam-677	145	42	,	,	PUNCT
ejpam-677	145	43	for	for	ADP
ejpam-677	145	44	−∞	−∞	X
ejpam-677	145	45	<	<	X
ejpam-677	145	46	α	α	X
ejpam-677	145	47	<	<	X
ejpam-677	145	48	1	1	NUM
ejpam-677	145	49	we	we	PRON
ejpam-677	145	50	can	can	AUX
ejpam-677	145	51	conclude	conclude	VERB
ejpam-677	145	52	that	that	SCONJ
ejpam-677	145	53	an	an	DET
ejpam-677	145	54	a	a	PRON
ejpam-677	145	55	:	:	PUNCT
ejpam-677	145	56	x	x	SYM
ejpam-677	145	57	→	→	SYM
ejpam-677	145	58	y	y	PRON
ejpam-677	145	59	truly	truly	ADV
ejpam-677	145	60	exists	exist	VERB
ejpam-677	145	61	such	such	ADJ
ejpam-677	145	62	that	that	SCONJ
ejpam-677	145	63	:	:	PUNCT
ejpam-677	145	64	a(x	a(x	NOUN
ejpam-677	145	65	)	)	PUNCT
ejpam-677	145	66	=	=	PUNCT
ejpam-677	145	67	limn→∞	limn→∞	PROPN
ejpam-677	145	68	2−n	2−n	NUM
ejpam-677	145	69	f	f	NOUN
ejpam-677	145	70	(	(	PUNCT
ejpam-677	145	71	2nx	2nx	ADJ
ejpam-677	145	72	)	)	PUNCT
ejpam-677	145	73	satisfies	satisfy	VERB
ejpam-677	145	74	the	the	DET
ejpam-677	145	75	cauchy	cauchy	NOUN
ejpam-677	145	76	-	-	PUNCT
ejpam-677	145	77	type	type	NOUN
ejpam-677	145	78	additivity	additivity	NOUN
ejpam-677	145	79	property	property	NOUN
ejpam-677	145	80	a(x	a(x	NOUN
ejpam-677	145	81	+	+	PROPN
ejpam-677	145	82	y	y	NOUN
ejpam-677	145	83	)	)	PUNCT
ejpam-677	146	1	+	+	VERB
ejpam-677	146	2	a(x	a(x	NOUN
ejpam-677	146	3	−	−	PROPN
ejpam-677	146	4	y	y	NOUN
ejpam-677	146	5	)	)	PUNCT
ejpam-677	146	6	+	+	CCONJ
ejpam-677	146	7	a(y	a(y	PROPN
ejpam-677	146	8	−	−	NOUN
ejpam-677	146	9	x	x	NOUN
ejpam-677	146	10	)	)	PUNCT
ejpam-677	146	11	=	=	SYM
ejpam-677	146	12	a(x)+	a(x)+	ADP
ejpam-677	146	13	a(y	a(y	PROPN
ejpam-677	146	14	)	)	PUNCT
ejpam-677	146	15	.	.	PUNCT
ejpam-677	147	1	(	(	PUNCT
ejpam-677	147	2	13	13	NUM
ejpam-677	147	3	)	)	PUNCT
ejpam-677	147	4	therefore	therefore	ADV
ejpam-677	147	5	,	,	PUNCT
ejpam-677	147	6	existence	existence	NOUN
ejpam-677	147	7	of	of	ADP
ejpam-677	147	8	theorem	theorem	ADJ
ejpam-677	147	9	holds	hold	NOUN
ejpam-677	147	10	.	.	PUNCT
ejpam-677	148	1	m.	m.	NOUN
ejpam-677	148	2	rassias	rassias	PROPN
ejpam-677	148	3	/	/	SYM
ejpam-677	148	4	eur	eur	PROPN
ejpam-677	148	5	.	.	PUNCT
ejpam-677	149	1	j.	j.	PROPN
ejpam-677	149	2	pure	pure	PROPN
ejpam-677	149	3	appl	appl	PROPN
ejpam-677	149	4	.	.	PROPN
ejpam-677	149	5	math	math	PROPN
ejpam-677	149	6	,	,	PUNCT
ejpam-677	149	7	4	4	NUM
ejpam-677	149	8	(	(	PUNCT
ejpam-677	149	9	2011	2011	NUM
ejpam-677	149	10	)	)	PUNCT
ejpam-677	149	11	,	,	PUNCT
ejpam-677	149	12	50	50	NUM
ejpam-677	149	13	-	-	SYM
ejpam-677	149	14	58	58	NUM
ejpam-677	149	15	56	56	NUM
ejpam-677	149	16	step	step	NOUN
ejpam-677	149	17	5	5	NUM
ejpam-677	149	18	we	we	PRON
ejpam-677	149	19	need	need	VERB
ejpam-677	149	20	to	to	PART
ejpam-677	149	21	prove	prove	VERB
ejpam-677	149	22	that	that	SCONJ
ejpam-677	149	23	a	a	PRON
ejpam-677	149	24	is	be	AUX
ejpam-677	149	25	unique	unique	ADJ
ejpam-677	149	26	.	.	PUNCT
ejpam-677	150	1	observe	observe	VERB
ejpam-677	150	2	,	,	PUNCT
ejpam-677	150	3	from	from	ADP
ejpam-677	150	4	(	(	PUNCT
ejpam-677	150	5	13	13	NUM
ejpam-677	150	6	)	)	PUNCT
ejpam-677	150	7	,	,	PUNCT
ejpam-677	150	8	that	that	PRON
ejpam-677	150	9	a(0	a(0	PROPN
ejpam-677	150	10	)	)	PUNCT
ejpam-677	150	11	=	=	SYM
ejpam-677	150	12	0	0	NUM
ejpam-677	150	13	and	and	CCONJ
ejpam-677	150	14	a(2x	a(2x	NUM
ejpam-677	150	15	)	)	PUNCT
ejpam-677	150	16	=	=	SYM
ejpam-677	150	17	2a(x	2a(x	NOUN
ejpam-677	150	18	)	)	PUNCT
ejpam-677	150	19	.	.	PUNCT
ejpam-677	151	1	therefore	therefore	ADV
ejpam-677	151	2	,	,	PUNCT
ejpam-677	151	3	by	by	ADP
ejpam-677	151	4	induction	induction	NOUN
ejpam-677	151	5	we	we	PRON
ejpam-677	151	6	can	can	AUX
ejpam-677	151	7	show	show	VERB
ejpam-677	151	8	that	that	SCONJ
ejpam-677	151	9	a(2nx	a(2nx	NOUN
ejpam-677	151	10	)	)	PUNCT
ejpam-677	151	11	=	=	SYM
ejpam-677	152	1	2a(2n−1x	2a(2n−1x	X
ejpam-677	152	2	)	)	PUNCT
ejpam-677	152	3	=	=	SYM
ejpam-677	152	4	2na(x	2na(x	NUM
ejpam-677	152	5	)	)	PUNCT
ejpam-677	152	6	or	or	CCONJ
ejpam-677	152	7	equivalently	equivalently	ADV
ejpam-677	152	8	a(x	a(x	PROPN
ejpam-677	152	9	)	)	PUNCT
ejpam-677	152	10	=	=	PUNCT
ejpam-677	152	11	2−na(2nx	2−na(2nx	NUM
ejpam-677	152	12	)	)	PUNCT
ejpam-677	152	13	.	.	PUNCT
ejpam-677	153	1	(	(	PUNCT
ejpam-677	153	2	14	14	NUM
ejpam-677	153	3	)	)	PUNCT
ejpam-677	153	4	assume	assume	VERB
ejpam-677	153	5	,	,	PUNCT
ejpam-677	153	6	now	now	ADV
ejpam-677	153	7	,	,	PUNCT
ejpam-677	153	8	the	the	DET
ejpam-677	153	9	existence	existence	NOUN
ejpam-677	153	10	of	of	ADP
ejpam-677	153	11	another	another	PRON
ejpam-677	153	12	a′	a′	NOUN
ejpam-677	153	13	:	:	PUNCT
ejpam-677	153	14	x	x	X
ejpam-677	153	15	→	→	SYM
ejpam-677	153	16	y	y	PROPN
ejpam-677	153	17	,	,	PUNCT
ejpam-677	153	18	such	such	ADJ
ejpam-677	153	19	that	that	PRON
ejpam-677	153	20	a′(x	a′(x	NOUN
ejpam-677	153	21	)	)	PUNCT
ejpam-677	153	22	=	=	PUNCT
ejpam-677	154	1	2−na′(2n	2−na′(2n	NUM
ejpam-677	154	2	x	x	NOUN
ejpam-677	154	3	)	)	PUNCT
ejpam-677	154	4	.	.	PUNCT
ejpam-677	155	1	with	with	ADP
ejpam-677	155	2	the	the	DET
ejpam-677	155	3	aid	aid	NOUN
ejpam-677	155	4	of	of	ADP
ejpam-677	155	5	the	the	DET
ejpam-677	155	6	(	(	PUNCT
ejpam-677	155	7	12)-(14	12)-(14	NUM
ejpam-677	155	8	)	)	PUNCT
ejpam-677	155	9	and	and	CCONJ
ejpam-677	155	10	the	the	DET
ejpam-677	155	11	triangular	triangular	NOUN
ejpam-677	155	12	inequality	inequality	NOUN
ejpam-677	155	13	,	,	PUNCT
ejpam-677	155	14	one	one	PRON
ejpam-677	155	15	gets	get	VERB
ejpam-677	155	16	0≤	0≤	ADJ
ejpam-677	156	1	||a(x)−	||a(x)−	PROPN
ejpam-677	156	2	a′(x)||	a′(x)||	NOUN
ejpam-677	156	3	=	=	NOUN
ejpam-677	156	4	||2−na(2nx)−	||2−na(2nx)−	PROPN
ejpam-677	156	5	2−na′(2nx)||	2−na′(2nx)||	PROPN
ejpam-677	156	6	≤	≤	ADJ
ejpam-677	156	7	||2−na(2nx)−	||2−na(2nx)−	PROPN
ejpam-677	156	8	2−n	2−n	NUM
ejpam-677	156	9	f	f	PROPN
ejpam-677	156	10	(	(	PUNCT
ejpam-677	156	11	2n	2n	NUM
ejpam-677	156	12	x)||+	x)||+	PROPN
ejpam-677	156	13	||2−n	||2−n	NUM
ejpam-677	156	14	f	f	PROPN
ejpam-677	156	15	(	(	PUNCT
ejpam-677	156	16	2nx)−	2nx)−	PROPN
ejpam-677	156	17	2−na′(2nx)||	2−na′(2nx)||	NUM
ejpam-677	156	18	≤	≤	NUM
ejpam-677	156	19	2n(α−1	2n(α−1	NUM
ejpam-677	156	20	)	)	PUNCT
ejpam-677	156	21	3ǫ	3ǫ	NOUN
ejpam-677	156	22	2−	2−	NUM
ejpam-677	156	23	2α	2α	NOUN
ejpam-677	156	24	||x	||x	NOUN
ejpam-677	156	25	||α	||α	NOUN
ejpam-677	156	26	→	→	SYM
ejpam-677	156	27	0	0	NUM
ejpam-677	156	28	,	,	PUNCT
ejpam-677	156	29	as	as	ADP
ejpam-677	156	30	n	n	X
ejpam-677	156	31	→	→	SYM
ejpam-677	156	32	∞	∞	PROPN
ejpam-677	156	33	,	,	PUNCT
ejpam-677	156	34	(	(	PUNCT
ejpam-677	156	35	−∞	−∞	X
ejpam-677	156	36	<	<	X
ejpam-677	156	37	α	α	X
ejpam-677	156	38	<	<	X
ejpam-677	156	39	1	1	NUM
ejpam-677	156	40	)	)	PUNCT
ejpam-677	156	41	.	.	PUNCT
ejpam-677	157	1	thus	thus	ADV
ejpam-677	157	2	,	,	PUNCT
ejpam-677	157	3	the	the	DET
ejpam-677	157	4	uniqueness	uniqueness	NOUN
ejpam-677	157	5	of	of	ADP
ejpam-677	157	6	a	a	PRON
ejpam-677	157	7	is	be	AUX
ejpam-677	157	8	proved	prove	VERB
ejpam-677	157	9	and	and	CCONJ
ejpam-677	157	10	the	the	DET
ejpam-677	157	11	stability	stability	NOUN
ejpam-677	157	12	of	of	ADP
ejpam-677	157	13	cauchy	cauchy	NOUN
ejpam-677	157	14	-	-	PUNCT
ejpam-677	157	15	type	type	NOUN
ejpam-677	157	16	additive	additive	NOUN
ejpam-677	157	17	mapping	mapping	NOUN
ejpam-677	157	18	a	a	PRON
ejpam-677	157	19	:	:	PUNCT
ejpam-677	157	20	x	x	SYM
ejpam-677	157	21	→	→	SYM
ejpam-677	157	22	y	y	PROPN
ejpam-677	157	23	is	be	AUX
ejpam-677	157	24	established	establish	VERB
ejpam-677	157	25	.	.	PUNCT
ejpam-677	158	1	step	step	NOUN
ejpam-677	158	2	6	6	NUM
ejpam-677	158	3	to	to	PART
ejpam-677	158	4	complete	complete	VERB
ejpam-677	158	5	the	the	DET
ejpam-677	158	6	proof	proof	NOUN
ejpam-677	158	7	of	of	ADP
ejpam-677	158	8	theorem	theorem	NOUN
ejpam-677	158	9	6	6	NUM
ejpam-677	158	10	,	,	PUNCT
ejpam-677	158	11	we	we	PRON
ejpam-677	158	12	only	only	ADV
ejpam-677	158	13	need	need	VERB
ejpam-677	158	14	to	to	PART
ejpam-677	158	15	examine	examine	VERB
ejpam-677	158	16	whether	whether	SCONJ
ejpam-677	158	17	a	a	PRON
ejpam-677	158	18	:	:	PUNCT
ejpam-677	158	19	x	x	SYM
ejpam-677	158	20	→	→	SYM
ejpam-677	158	21	y	y	PROPN
ejpam-677	158	22	is	be	AUX
ejpam-677	158	23	a	a	DET
ejpam-677	158	24	linear	linear	ADJ
ejpam-677	158	25	cauchy	cauchy	NOUN
ejpam-677	158	26	-	-	PUNCT
ejpam-677	158	27	type	type	NOUN
ejpam-677	158	28	mapping	mapping	NOUN
ejpam-677	158	29	.	.	PUNCT
ejpam-677	159	1	to	to	PART
ejpam-677	159	2	be	be	AUX
ejpam-677	159	3	more	more	ADV
ejpam-677	159	4	precise	precise	ADJ
ejpam-677	159	5	,	,	PUNCT
ejpam-677	159	6	we	we	PRON
ejpam-677	159	7	need	need	VERB
ejpam-677	159	8	to	to	PART
ejpam-677	159	9	show	show	VERB
ejpam-677	159	10	that	that	SCONJ
ejpam-677	159	11	:	:	PUNCT
ejpam-677	159	12	(	(	PUNCT
ejpam-677	159	13	1	1	X
ejpam-677	159	14	)	)	PUNCT
ejpam-677	159	15	a(x	a(x	PROPN
ejpam-677	159	16	+	+	NUM
ejpam-677	159	17	y	y	NOUN
ejpam-677	159	18	)	)	PUNCT
ejpam-677	159	19	+	+	CCONJ
ejpam-677	159	20	a(x	a(x	PROPN
ejpam-677	159	21	−	−	PROPN
ejpam-677	159	22	y	y	NOUN
ejpam-677	159	23	)	)	PUNCT
ejpam-677	160	1	+	+	CCONJ
ejpam-677	160	2	a(y	a(y	PROPN
ejpam-677	160	3	−	−	NOUN
ejpam-677	160	4	x	x	NOUN
ejpam-677	160	5	)	)	PUNCT
ejpam-677	160	6	=	=	SYM
ejpam-677	160	7	a(x)+	a(x)+	ADP
ejpam-677	160	8	a(y	a(y	PROPN
ejpam-677	160	9	)	)	PUNCT
ejpam-677	160	10	,	,	PUNCT
ejpam-677	160	11	and	and	CCONJ
ejpam-677	160	12	(	(	PUNCT
ejpam-677	160	13	2	2	NUM
ejpam-677	160	14	)	)	PUNCT
ejpam-677	160	15	a(r	a(r	NOUN
ejpam-677	160	16	x	x	SYM
ejpam-677	160	17	)	)	PUNCT
ejpam-677	160	18	=	=	SYM
ejpam-677	160	19	ra(x	ra(x	NOUN
ejpam-677	160	20	)	)	PUNCT
ejpam-677	160	21	,	,	PUNCT
ejpam-677	160	22	∀r	∀r	PROPN
ejpam-677	160	23	∈	∈	PROPN
ejpam-677	160	24	r.	r.	PROPN
ejpam-677	160	25	recall	recall	NOUN
ejpam-677	160	26	that	that	PRON
ejpam-677	160	27	we	we	PRON
ejpam-677	160	28	have	have	AUX
ejpam-677	160	29	shown	show	VERB
ejpam-677	160	30	already	already	ADV
ejpam-677	160	31	that	that	SCONJ
ejpam-677	160	32	(	(	PUNCT
ejpam-677	160	33	1	1	X
ejpam-677	160	34	)	)	PUNCT
ejpam-677	160	35	holds	hold	VERB
ejpam-677	160	36	.	.	PUNCT
ejpam-677	161	1	therefore	therefore	ADV
ejpam-677	161	2	,	,	PUNCT
ejpam-677	161	3	we	we	PRON
ejpam-677	161	4	only	only	ADV
ejpam-677	161	5	need	need	VERB
ejpam-677	161	6	to	to	PART
ejpam-677	161	7	show	show	VERB
ejpam-677	161	8	that	that	SCONJ
ejpam-677	161	9	(	(	PUNCT
ejpam-677	161	10	2	2	X
ejpam-677	161	11	)	)	PUNCT
ejpam-677	161	12	is	be	AUX
ejpam-677	161	13	valid	valid	ADJ
ejpam-677	161	14	∀r	∀r	X
ejpam-677	161	15	∈	∈	PROPN
ejpam-677	161	16	r.	r.	NOUN
ejpam-677	161	17	for	for	ADP
ejpam-677	161	18	that	that	SCONJ
ejpam-677	161	19	we	we	PRON
ejpam-677	161	20	will	will	AUX
ejpam-677	161	21	study	study	VERB
ejpam-677	161	22	four	four	NUM
ejpam-677	161	23	cases	case	NOUN
ejpam-677	161	24	.	.	PUNCT
ejpam-677	162	1	case	case	NOUN
ejpam-677	162	2	1	1	NUM
ejpam-677	162	3	:	:	PUNCT
ejpam-677	162	4	let	let	VERB
ejpam-677	162	5	r	r	NOUN
ejpam-677	162	6	=	=	PUNCT
ejpam-677	162	7	k	k	PROPN
ejpam-677	162	8	∈	∈	PROPN
ejpam-677	162	9	n	n	NOUN
ejpam-677	162	10	=	=	PUNCT
ejpam-677	162	11	{	{	PUNCT
ejpam-677	162	12	0,1,2	0,1,2	NOUN
ejpam-677	162	13	,	,	PUNCT
ejpam-677	162	14	.	.	PUNCT
ejpam-677	162	15	.	.	PUNCT
ejpam-677	162	16	.	.	PUNCT
ejpam-677	162	17	}	}	PUNCT
ejpam-677	162	18	.	.	PUNCT
ejpam-677	163	1	for	for	ADP
ejpam-677	163	2	k	k	PROPN
ejpam-677	163	3	=	=	SYM
ejpam-677	163	4	0	0	PROPN
ejpam-677	163	5	,	,	PUNCT
ejpam-677	163	6	from	from	ADP
ejpam-677	163	7	(	(	PUNCT
ejpam-677	163	8	2	2	NUM
ejpam-677	163	9	)	)	PUNCT
ejpam-677	163	10	,	,	PUNCT
ejpam-677	163	11	we	we	PRON
ejpam-677	163	12	have	have	AUX
ejpam-677	163	13	a(0	a(0	PROPN
ejpam-677	163	14	)	)	PUNCT
ejpam-677	164	1	=	=	SYM
ejpam-677	164	2	0	0	X
ejpam-677	164	3	.	.	PUNCT
ejpam-677	165	1	this	this	PRON
ejpam-677	165	2	is	be	AUX
ejpam-677	165	3	verified	verify	VERB
ejpam-677	165	4	if	if	SCONJ
ejpam-677	165	5	we	we	PRON
ejpam-677	165	6	substitute	substitute	VERB
ejpam-677	165	7	x	x	PUNCT
ejpam-677	165	8	=	=	PUNCT
ejpam-677	165	9	y	y	PROPN
ejpam-677	165	10	=	=	SYM
ejpam-677	165	11	0	0	NUM
ejpam-677	165	12	in	in	ADP
ejpam-677	165	13	(	(	PUNCT
ejpam-677	165	14	13	13	NUM
ejpam-677	165	15	)	)	PUNCT
ejpam-677	165	16	.	.	PUNCT
ejpam-677	166	1	assume	assume	VERB
ejpam-677	166	2	,	,	PUNCT
ejpam-677	166	3	that	that	SCONJ
ejpam-677	166	4	a	a	DET
ejpam-677	166	5	�	�	PROPN
ejpam-677	166	6	(	(	PUNCT
ejpam-677	166	7	k−	k−	PROPN
ejpam-677	166	8	1)x	1)x	NUM
ejpam-677	166	9	�	�	PROPN
ejpam-677	166	10	=	=	SYM
ejpam-677	166	11	(	(	PUNCT
ejpam-677	166	12	k−	k−	PROPN
ejpam-677	166	13	1)a(x	1)a(x	PROPN
ejpam-677	166	14	)	)	PUNCT
ejpam-677	166	15	is	be	AUX
ejpam-677	166	16	true	true	ADJ
ejpam-677	166	17	∀k	∀k	NOUN
ejpam-677	166	18	.	.	PUNCT
ejpam-677	167	1	then	then	ADV
ejpam-677	167	2	,	,	PUNCT
ejpam-677	167	3	we	we	PRON
ejpam-677	167	4	need	need	VERB
ejpam-677	167	5	to	to	PART
ejpam-677	167	6	prove	prove	VERB
ejpam-677	167	7	that	that	PRON
ejpam-677	167	8	a(kx	a(kx	NOUN
ejpam-677	167	9	)	)	PUNCT
ejpam-677	167	10	=	=	SYM
ejpam-677	167	11	ka(x	ka(x	NOUN
ejpam-677	167	12	)	)	PUNCT
ejpam-677	167	13	.	.	PUNCT
ejpam-677	168	1	note	note	VERB
ejpam-677	168	2	that	that	SCONJ
ejpam-677	168	3	for	for	ADP
ejpam-677	168	4	x	x	SYM
ejpam-677	168	5	=	=	SYM
ejpam-677	168	6	x	x	X
ejpam-677	168	7	,	,	PUNCT
ejpam-677	168	8	and	and	CCONJ
ejpam-677	168	9	y	y	PROPN
ejpam-677	168	10	=	=	NOUN
ejpam-677	168	11	0	0	NUM
ejpam-677	168	12	from	from	ADP
ejpam-677	168	13	(	(	PUNCT
ejpam-677	168	14	13	13	NUM
ejpam-677	168	15	)	)	PUNCT
ejpam-677	168	16	,	,	PUNCT
ejpam-677	168	17	we	we	PRON
ejpam-677	168	18	can	can	AUX
ejpam-677	168	19	easily	easily	ADV
ejpam-677	168	20	obtain	obtain	VERB
ejpam-677	168	21	a(−x	a(−x	NOUN
ejpam-677	168	22	)	)	PUNCT
ejpam-677	168	23	=	=	SYM
ejpam-677	168	24	(	(	PUNCT
ejpam-677	168	25	−1)a(x	−1)a(x	NOUN
ejpam-677	168	26	)	)	PUNCT
ejpam-677	168	27	.	.	PUNCT
ejpam-677	169	1	let	let	VERB
ejpam-677	169	2	x	x	PUNCT
ejpam-677	169	3	=	=	PUNCT
ejpam-677	169	4	x	x	X
ejpam-677	169	5	and	and	CCONJ
ejpam-677	169	6	y	y	PROPN
ejpam-677	169	7	=	=	SYM
ejpam-677	169	8	(	(	PUNCT
ejpam-677	169	9	k−	k−	PROPN
ejpam-677	169	10	1)x	1)x	NUM
ejpam-677	169	11	in	in	ADP
ejpam-677	169	12	(	(	PUNCT
ejpam-677	169	13	13	13	NUM
ejpam-677	169	14	)	)	PUNCT
ejpam-677	169	15	.	.	PUNCT
ejpam-677	170	1	then	then	ADV
ejpam-677	170	2	,	,	PUNCT
ejpam-677	170	3	a(kx)+	a(kx)+	PROPN
ejpam-677	170	4	a	a	DET
ejpam-677	170	5	�	�	NOUN
ejpam-677	170	6	−	−	PROPN
ejpam-677	170	7	(	(	PUNCT
ejpam-677	170	8	k−	k−	PROPN
ejpam-677	170	9	2)x	2)x	NUM
ejpam-677	170	10	�	�	NOUN
ejpam-677	170	11	+	+	CCONJ
ejpam-677	170	12	a	a	DET
ejpam-677	170	13	�	�	PROPN
ejpam-677	170	14	(	(	PUNCT
ejpam-677	170	15	k−	k−	PROPN
ejpam-677	170	16	2)x	2)x	NUM
ejpam-677	170	17	�	�	NOUN
ejpam-677	170	18	=	=	PUNCT
ejpam-677	170	19	a(x)+	a(x)+	ADP
ejpam-677	170	20	a	a	DET
ejpam-677	170	21	�	�	PROPN
ejpam-677	170	22	(	(	PUNCT
ejpam-677	170	23	k−	k−	PROPN
ejpam-677	170	24	1)x	1)x	NUM
ejpam-677	170	25	�	�	PROPN
ejpam-677	170	26	,	,	PUNCT
ejpam-677	170	27	or	or	CCONJ
ejpam-677	170	28	a(kx	a(kx	NUM
ejpam-677	170	29	)	)	PUNCT
ejpam-677	170	30	=	=	SYM
ejpam-677	170	31	ka(x	ka(x	NOUN
ejpam-677	170	32	)	)	PUNCT
ejpam-677	170	33	,	,	PUNCT
ejpam-677	170	34	∀k	∀k	X
ejpam-677	170	35	∈	∈	PROPN
ejpam-677	170	36	n	n	NOUN
ejpam-677	170	37	=	=	PUNCT
ejpam-677	170	38	{	{	PUNCT
ejpam-677	170	39	0,1,2	0,1,2	NOUN
ejpam-677	170	40	,	,	PUNCT
ejpam-677	170	41	.	.	PUNCT
ejpam-677	170	42	.	.	PUNCT
ejpam-677	170	43	.	.	PUNCT
ejpam-677	170	44	}	}	PUNCT
ejpam-677	170	45	.	.	PUNCT
ejpam-677	171	1	m.	m.	NOUN
ejpam-677	171	2	rassias	rassias	PROPN
ejpam-677	171	3	/	/	SYM
ejpam-677	171	4	eur	eur	PROPN
ejpam-677	171	5	.	.	PUNCT
ejpam-677	172	1	j.	j.	PROPN
ejpam-677	172	2	pure	pure	PROPN
ejpam-677	172	3	appl	appl	PROPN
ejpam-677	172	4	.	.	PROPN
ejpam-677	172	5	math	math	PROPN
ejpam-677	172	6	,	,	PUNCT
ejpam-677	172	7	4	4	NUM
ejpam-677	172	8	(	(	PUNCT
ejpam-677	172	9	2011	2011	NUM
ejpam-677	172	10	)	)	PUNCT
ejpam-677	172	11	,	,	PUNCT
ejpam-677	172	12	50	50	NUM
ejpam-677	172	13	-	-	SYM
ejpam-677	172	14	58	58	NUM
ejpam-677	172	15	57	57	NUM
ejpam-677	172	16	case	case	NOUN
ejpam-677	172	17	2	2	NUM
ejpam-677	172	18	:	:	PUNCT
ejpam-677	172	19	let	let	VERB
ejpam-677	172	20	r	r	NOUN
ejpam-677	172	21	=	=	PUNCT
ejpam-677	172	22	k	k	PROPN
ejpam-677	172	23	∈	∈	PROPN
ejpam-677	172	24	z	z	NOUN
ejpam-677	172	25	.	.	PUNCT
ejpam-677	173	1	we	we	PRON
ejpam-677	173	2	only	only	ADV
ejpam-677	173	3	need	need	VERB
ejpam-677	173	4	to	to	PART
ejpam-677	173	5	observe	observe	VERB
ejpam-677	173	6	that	that	SCONJ
ejpam-677	173	7	a	a	PRON
ejpam-677	173	8	is	be	AUX
ejpam-677	173	9	odd	odd	ADJ
ejpam-677	173	10	.	.	PUNCT
ejpam-677	174	1	since	since	ADV
ejpam-677	174	2	,	,	PUNCT
ejpam-677	174	3	we	we	PRON
ejpam-677	174	4	have	have	AUX
ejpam-677	174	5	already	already	ADV
ejpam-677	174	6	proved	prove	VERB
ejpam-677	174	7	that	that	SCONJ
ejpam-677	174	8	(	(	PUNCT
ejpam-677	174	9	2	2	X
ejpam-677	174	10	)	)	PUNCT
ejpam-677	174	11	is	be	AUX
ejpam-677	174	12	valid	valid	ADJ
ejpam-677	174	13	∀k	∀k	NOUN
ejpam-677	174	14	∈	∈	PROPN
ejpam-677	174	15	n	n	NOUN
ejpam-677	174	16	=	=	PUNCT
ejpam-677	174	17	{	{	PUNCT
ejpam-677	174	18	0,1,2	0,1,2	NOUN
ejpam-677	174	19	,	,	PUNCT
ejpam-677	174	20	.	.	PUNCT
ejpam-677	174	21	.	.	PUNCT
ejpam-677	175	1	.	.	PUNCT
ejpam-677	176	1	}	}	PUNCT
ejpam-677	177	1	we	we	PRON
ejpam-677	177	2	can	can	AUX
ejpam-677	177	3	then	then	ADV
ejpam-677	177	4	conclude	conclude	VERB
ejpam-677	177	5	that	that	PRON
ejpam-677	177	6	a(kx	a(kx	NOUN
ejpam-677	177	7	)	)	PUNCT
ejpam-677	177	8	=	=	SYM
ejpam-677	177	9	ka(x	ka(x	NOUN
ejpam-677	177	10	)	)	PUNCT
ejpam-677	177	11	,	,	PUNCT
ejpam-677	177	12	∀k	∀k	NOUN
ejpam-677	177	13	∈	∈	PROPN
ejpam-677	177	14	z	z	X
ejpam-677	177	15	.	.	PUNCT
ejpam-677	178	1	case	case	NOUN
ejpam-677	178	2	3	3	X
ejpam-677	178	3	:	:	PUNCT
ejpam-677	178	4	let	let	VERB
ejpam-677	178	5	r	r	NOUN
ejpam-677	178	6	=	=	PUNCT
ejpam-677	178	7	k	k	PROPN
ejpam-677	178	8	l	l	NOUN
ejpam-677	178	9	∈q	∈q	NOUN
ejpam-677	178	10	,	,	PUNCT
ejpam-677	178	11	for	for	ADP
ejpam-677	178	12	k	k	PROPN
ejpam-677	178	13	∈	∈	PROPN
ejpam-677	178	14	z	z	PROPN
ejpam-677	178	15	,	,	PUNCT
ejpam-677	178	16	l	l	PROPN
ejpam-677	178	17	∈	∈	PROPN
ejpam-677	178	18	z	z	NOUN
ejpam-677	178	19	−	−	PROPN
ejpam-677	178	20	{	{	PUNCT
ejpam-677	178	21	0	0	NUM
ejpam-677	178	22	}	}	PUNCT
ejpam-677	178	23	.	.	PUNCT
ejpam-677	179	1	then	then	ADV
ejpam-677	179	2	,	,	PUNCT
ejpam-677	179	3	a(x	a(x	PROPN
ejpam-677	179	4	)	)	PUNCT
ejpam-677	179	5	=	=	PUNCT
ejpam-677	179	6	a	a	DET
ejpam-677	179	7	�	�	PROPN
ejpam-677	179	8	l	l	NOUN
ejpam-677	179	9	1	1	NUM
ejpam-677	179	10	l	l	NOUN
ejpam-677	179	11	x	x	X
ejpam-677	179	12	�	�	PROPN
ejpam-677	179	13	=	=	SYM
ejpam-677	179	14	la	la	X
ejpam-677	179	15	�	�	PROPN
ejpam-677	179	16	1	1	NUM
ejpam-677	179	17	l	l	NOUN
ejpam-677	179	18	x	x	X
ejpam-677	179	19	�	�	PROPN
ejpam-677	179	20	,	,	PUNCT
ejpam-677	179	21	for	for	ADP
ejpam-677	179	22	l	l	PROPN
ejpam-677	179	23	∈	∈	PROPN
ejpam-677	179	24	z	z	PROPN
ejpam-677	179	25	−	−	PROPN
ejpam-677	179	26	{	{	PUNCT
ejpam-677	179	27	0	0	NUM
ejpam-677	179	28	}	}	PUNCT
ejpam-677	179	29	.	.	PUNCT
ejpam-677	180	1	hence	hence	ADV
ejpam-677	180	2	,	,	PUNCT
ejpam-677	180	3	a	a	DET
ejpam-677	180	4	�	�	NOUN
ejpam-677	180	5	1	1	NUM
ejpam-677	180	6	l	l	NOUN
ejpam-677	180	7	x	x	X
ejpam-677	180	8	�	�	PROPN
ejpam-677	180	9	=	=	SYM
ejpam-677	180	10	1	1	NUM
ejpam-677	180	11	l	l	NOUN
ejpam-677	180	12	a(x	a(x	NOUN
ejpam-677	180	13	)	)	PUNCT
ejpam-677	180	14	.	.	PUNCT
ejpam-677	181	1	besides	besides	SCONJ
ejpam-677	181	2	,	,	PUNCT
ejpam-677	181	3	for	for	ADP
ejpam-677	181	4	k	k	PROPN
ejpam-677	181	5	∈	∈	PROPN
ejpam-677	181	6	z	z	PROPN
ejpam-677	181	7	,	,	PUNCT
ejpam-677	181	8	a	a	DET
ejpam-677	181	9	�	�	PROPN
ejpam-677	181	10	k	k	NOUN
ejpam-677	181	11	l	l	PUNCT
ejpam-677	181	12	x	x	X
ejpam-677	181	13	�	�	PROPN
ejpam-677	181	14	=	=	PUNCT
ejpam-677	181	15	a	a	DET
ejpam-677	181	16	�	�	PROPN
ejpam-677	181	17	k	k	NOUN
ejpam-677	181	18	1	1	NUM
ejpam-677	181	19	l	l	NOUN
ejpam-677	181	20	x	x	X
ejpam-677	181	21	�	�	PROPN
ejpam-677	181	22	=	=	PUNCT
ejpam-677	181	23	ka(1	ka(1	NOUN
ejpam-677	181	24	l	l	NOUN
ejpam-677	181	25	x	x	NOUN
ejpam-677	181	26	)	)	PUNCT
ejpam-677	181	27	,	,	PUNCT
ejpam-677	181	28	from	from	ADP
ejpam-677	181	29	case	case	NOUN
ejpam-677	181	30	2	2	NUM
ejpam-677	181	31	.	.	PUNCT
ejpam-677	181	32	thus	thus	ADV
ejpam-677	181	33	,	,	PUNCT
ejpam-677	181	34	a	a	DET
ejpam-677	181	35	�	�	PROPN
ejpam-677	181	36	k	k	NOUN
ejpam-677	181	37	l	l	PUNCT
ejpam-677	181	38	x	x	X
ejpam-677	181	39	�	�	PROPN
ejpam-677	181	40	=	=	PUNCT
ejpam-677	181	41	k	k	PROPN
ejpam-677	181	42	l	l	NOUN
ejpam-677	181	43	a(x	a(x	PROPN
ejpam-677	181	44	)	)	PUNCT
ejpam-677	181	45	,	,	PUNCT
ejpam-677	181	46	or	or	CCONJ
ejpam-677	181	47	a(r	a(r	PROPN
ejpam-677	181	48	x	x	X
ejpam-677	181	49	)	)	PUNCT
ejpam-677	181	50	=	=	SYM
ejpam-677	181	51	ra(x	ra(x	NOUN
ejpam-677	181	52	)	)	PUNCT
ejpam-677	181	53	for	for	ADP
ejpam-677	181	54	r	r	NOUN
ejpam-677	181	55	∈q	∈q	PROPN
ejpam-677	181	56	.	.	PUNCT
ejpam-677	181	57	case	case	NOUN
ejpam-677	181	58	4	4	NUM
ejpam-677	181	59	:	:	PUNCT
ejpam-677	181	60	let	let	VERB
ejpam-677	181	61	r	r	NOUN
ejpam-677	181	62	∈	∈	NOUN
ejpam-677	181	63	r	r	NOUN
ejpam-677	181	64	,	,	PUNCT
ejpam-677	181	65	where	where	SCONJ
ejpam-677	181	66	r	r	NOUN
ejpam-677	181	67	=	=	PUNCT
ejpam-677	181	68	qn	qn	NOUN
ejpam-677	181	69	:	:	PUNCT
ejpam-677	181	70	rational	rational	ADJ
ejpam-677	181	71	numbers	number	NOUN
ejpam-677	181	72	.	.	PUNCT
ejpam-677	182	1	since	since	SCONJ
ejpam-677	182	2	r	r	NOUN
ejpam-677	182	3	is	be	AUX
ejpam-677	182	4	a	a	DET
ejpam-677	182	5	complete	complete	ADJ
ejpam-677	182	6	space	space	NOUN
ejpam-677	182	7	,	,	PUNCT
ejpam-677	182	8	every	every	DET
ejpam-677	182	9	sequence	sequence	NOUN
ejpam-677	182	10	{	{	PUNCT
ejpam-677	182	11	qn	qn	NOUN
ejpam-677	182	12	}	}	PUNCT
ejpam-677	182	13	converges	converge	NOUN
ejpam-677	182	14	in	in	ADP
ejpam-677	182	15	r	r	NOUN
ejpam-677	182	16	,	,	PUNCT
ejpam-677	182	17	i.e.	i.e.	X
ejpam-677	182	18	limn→∞	limn→∞	X
ejpam-677	182	19	qn	qn	NOUN
ejpam-677	182	20	=	=	X
ejpam-677	182	21	q	q	PROPN
ejpam-677	182	22	∈	∈	PROPN
ejpam-677	182	23	r.	r.	NOUN
ejpam-677	182	24	recall	recall	VERB
ejpam-677	182	25	that	that	SCONJ
ejpam-677	182	26	a(x	a(x	NOUN
ejpam-677	182	27	)	)	PUNCT
ejpam-677	182	28	=	=	PUNCT
ejpam-677	182	29	limn→∞	limn→∞	PROPN
ejpam-677	182	30	2−n	2−n	NUM
ejpam-677	182	31	f	f	X
ejpam-677	182	32	(	(	PUNCT
ejpam-677	182	33	2n	2n	NUM
ejpam-677	182	34	x	x	NOUN
ejpam-677	182	35	)	)	PUNCT
ejpam-677	182	36	and	and	CCONJ
ejpam-677	182	37	f	f	PROPN
ejpam-677	182	38	(	(	PUNCT
ejpam-677	182	39	t	t	NOUN
ejpam-677	182	40	x	x	VERB
ejpam-677	182	41	)	)	PUNCT
ejpam-677	182	42	is	be	AUX
ejpam-677	182	43	continuous	continuous	ADJ
ejpam-677	182	44	in	in	ADP
ejpam-677	182	45	t	t	PROPN
ejpam-677	182	46	for	for	ADP
ejpam-677	182	47	each	each	DET
ejpam-677	182	48	fixed	fix	VERB
ejpam-677	182	49	x	x	PUNCT
ejpam-677	182	50	in	in	ADP
ejpam-677	182	51	x	x	X
ejpam-677	182	52	.	.	PUNCT
ejpam-677	183	1	therefore	therefore	ADV
ejpam-677	183	2	,	,	PUNCT
ejpam-677	183	3	a(t	a(t	PROPN
ejpam-677	183	4	x	x	PRON
ejpam-677	183	5	)	)	PUNCT
ejpam-677	183	6	is	be	AUX
ejpam-677	183	7	continuous	continuous	ADJ
ejpam-677	183	8	in	in	ADP
ejpam-677	183	9	t	t	PROPN
ejpam-677	183	10	for	for	ADP
ejpam-677	183	11	each	each	DET
ejpam-677	183	12	fixed	fix	VERB
ejpam-677	183	13	x	x	PUNCT
ejpam-677	183	14	in	in	ADP
ejpam-677	183	15	x	x	X
ejpam-677	183	16	.	.	PUNCT
ejpam-677	184	1	besides	besides	ADV
ejpam-677	184	2	,	,	PUNCT
ejpam-677	184	3	lim	lim	PROPN
ejpam-677	184	4	n→∞a(qnx	n→∞a(qnx	PROPN
ejpam-677	184	5	)	)	PUNCT
ejpam-677	184	6	=	=	PUNCT
ejpam-677	184	7	a	a	DET
ejpam-677	184	8	�	�	PROPN
ejpam-677	184	9	lim	lim	PROPN
ejpam-677	184	10	n→∞qn	n→∞qn	VERB
ejpam-677	184	11	x	x	SYM
ejpam-677	184	12	�	�	PROPN
ejpam-677	184	13	=	=	SYM
ejpam-677	184	14	a(qx	a(qx	PROPN
ejpam-677	184	15	)	)	PUNCT
ejpam-677	184	16	(	(	PUNCT
ejpam-677	184	17	15	15	NUM
ejpam-677	184	18	)	)	PUNCT
ejpam-677	184	19	and	and	CCONJ
ejpam-677	184	20	lim	lim	PROPN
ejpam-677	184	21	n→∞a(qnx	n→∞a(qnx	PROPN
ejpam-677	184	22	)	)	PUNCT
ejpam-677	185	1	=	=	VERB
ejpam-677	185	2	lim	lim	PROPN
ejpam-677	185	3	n→∞qna(x	n→∞qna(x	NOUN
ejpam-677	185	4	)	)	PUNCT
ejpam-677	185	5	=	=	SYM
ejpam-677	185	6	qa(x	qa(x	NOUN
ejpam-677	185	7	)	)	PUNCT
ejpam-677	185	8	.	.	PUNCT
ejpam-677	186	1	(	(	PUNCT
ejpam-677	186	2	16	16	NUM
ejpam-677	186	3	)	)	PUNCT
ejpam-677	186	4	from	from	ADP
ejpam-677	186	5	(	(	PUNCT
ejpam-677	186	6	15	15	NUM
ejpam-677	186	7	)	)	PUNCT
ejpam-677	186	8	and	and	CCONJ
ejpam-677	186	9	(	(	PUNCT
ejpam-677	186	10	16	16	NUM
ejpam-677	186	11	)	)	PUNCT
ejpam-677	186	12	case	case	NOUN
ejpam-677	186	13	4	4	NUM
ejpam-677	186	14	.	.	PUNCT
ejpam-677	186	15	is	be	AUX
ejpam-677	186	16	now	now	ADV
ejpam-677	186	17	proved	prove	VERB
ejpam-677	186	18	,	,	PUNCT
ejpam-677	186	19	which	which	PRON
ejpam-677	186	20	completes	complete	VERB
ejpam-677	186	21	step	step	NOUN
ejpam-677	186	22	6	6	NUM
ejpam-677	186	23	.	.	PUNCT
ejpam-677	187	1	and	and	CCONJ
ejpam-677	187	2	thus	thus	ADV
ejpam-677	187	3	the	the	DET
ejpam-677	187	4	proof	proof	NOUN
ejpam-677	187	5	of	of	ADP
ejpam-677	187	6	our	our	PRON
ejpam-677	187	7	theorem	theorem	NOUN
ejpam-677	187	8	6	6	NUM
ejpam-677	187	9	for	for	ADP
ejpam-677	187	10	the	the	DET
ejpam-677	187	11	case	case	NOUN
ejpam-677	187	12	of	of	ADP
ejpam-677	187	13	−∞	−∞	X
ejpam-677	187	14	<	<	X
ejpam-677	187	15	α	α	X
ejpam-677	187	16	<	<	X
ejpam-677	187	17	1	1	NUM
ejpam-677	187	18	.	.	PUNCT
ejpam-677	188	1	the	the	DET
ejpam-677	188	2	proof	proof	NOUN
ejpam-677	188	3	for	for	ADP
ejpam-677	188	4	the	the	DET
ejpam-677	188	5	case	case	NOUN
ejpam-677	188	6	of	of	ADP
ejpam-677	188	7	α	α	PROPN
ejpam-677	188	8	>	>	X
ejpam-677	188	9	1	1	NUM
ejpam-677	188	10	is	be	AUX
ejpam-677	188	11	similar	similar	ADJ
ejpam-677	188	12	to	to	ADP
ejpam-677	188	13	the	the	DET
ejpam-677	188	14	proof	proof	NOUN
ejpam-677	188	15	for	for	ADP
ejpam-677	188	16	−∞	−∞	X
ejpam-677	188	17	<	<	X
ejpam-677	188	18	α	α	X
ejpam-677	188	19	<	<	X
ejpam-677	188	20	1	1	NUM
ejpam-677	188	21	.	.	PUNCT
ejpam-677	189	1	in	in	ADP
ejpam-677	189	2	fact	fact	NOUN
ejpam-677	189	3	,	,	PUNCT
ejpam-677	189	4	we	we	PRON
ejpam-677	189	5	can	can	AUX
ejpam-677	189	6	find	find	VERB
ejpam-677	189	7	the	the	DET
ejpam-677	189	8	general	general	ADJ
ejpam-677	189	9	inequality	inequality	NOUN
ejpam-677	189	10	||	||	PUNCT
ejpam-677	190	1	f	f	X
ejpam-677	190	2	(	(	PUNCT
ejpam-677	190	3	x)−	x)−	PROPN
ejpam-677	190	4	2n	2n	NUM
ejpam-677	190	5	f	f	X
ejpam-677	190	6	(	(	PUNCT
ejpam-677	190	7	2−nx)||	2−nx)||	NUM
ejpam-677	190	8	≤	≤	NOUN
ejpam-677	190	9	3ǫ	3ǫ	VERB
ejpam-677	190	10	2α−	2α−	NUM
ejpam-677	190	11	2	2	NUM
ejpam-677	190	12	(	(	PUNCT
ejpam-677	190	13	1−	1−	NUM
ejpam-677	190	14	2n(1−α))||x	2n(1−α))||x	NUM
ejpam-677	190	15	||α	||α	NOUN
ejpam-677	190	16	,	,	PUNCT
ejpam-677	190	17	(	(	PUNCT
ejpam-677	190	18	17	17	NUM
ejpam-677	190	19	)	)	PUNCT
ejpam-677	190	20	for	for	ADP
ejpam-677	190	21	all	all	DET
ejpam-677	190	22	n	n	PRON
ejpam-677	190	23	∈	∈	NOUN
ejpam-677	190	24	n	n	PRON
ejpam-677	190	25	−{0	−{0	NUM
ejpam-677	190	26	}	}	PUNCT
ejpam-677	190	27	.	.	PUNCT
ejpam-677	191	1	thus	thus	ADV
ejpam-677	191	2	from	from	ADP
ejpam-677	191	3	this	this	DET
ejpam-677	191	4	inequality	inequality	NOUN
ejpam-677	191	5	(	(	PUNCT
ejpam-677	191	6	17	17	NUM
ejpam-677	191	7	)	)	PUNCT
ejpam-677	191	8	and	and	CCONJ
ejpam-677	191	9	the	the	DET
ejpam-677	191	10	formula	formula	NOUN
ejpam-677	191	11	a(x	a(x	NOUN
ejpam-677	191	12	)	)	PUNCT
ejpam-677	191	13	=	=	SYM
ejpam-677	191	14	lim	lim	PROPN
ejpam-677	191	15	n→∞2n	n→∞2n	PUNCT
ejpam-677	191	16	f	f	PROPN
ejpam-677	191	17	(	(	PUNCT
ejpam-677	191	18	2−n	2−n	NUM
ejpam-677	191	19	x	x	NOUN
ejpam-677	191	20	)	)	PUNCT
ejpam-677	191	21	,	,	PUNCT
ejpam-677	191	22	for	for	ADP
ejpam-677	191	23	n→∞	n→∞	NUM
ejpam-677	191	24	,	,	PUNCT
ejpam-677	191	25	we	we	PRON
ejpam-677	191	26	get	get	VERB
ejpam-677	191	27	the	the	DET
ejpam-677	191	28	inequality	inequality	NOUN
ejpam-677	191	29	||	||	PUNCT
ejpam-677	192	1	f	f	PROPN
ejpam-677	192	2	(	(	PUNCT
ejpam-677	192	3	x)−a(x)||	x)−a(x)||	NOUN
ejpam-677	192	4	≤	≤	PROPN
ejpam-677	192	5	3ǫ	3ǫ	VERB
ejpam-677	192	6	2α−	2α−	NUM
ejpam-677	192	7	2	2	NUM
ejpam-677	192	8	||x	||x	NOUN
ejpam-677	192	9	||α	||α	NOUN
ejpam-677	192	10	,	,	PUNCT
ejpam-677	192	11	for	for	ADP
ejpam-677	192	12	α	α	PROPN
ejpam-677	192	13	>	>	X
ejpam-677	192	14	1	1	NUM
ejpam-677	192	15	.	.	PUNCT
ejpam-677	193	1	the	the	DET
ejpam-677	193	2	rest	rest	NOUN
ejpam-677	193	3	of	of	ADP
ejpam-677	193	4	the	the	DET
ejpam-677	193	5	proof	proof	NOUN
ejpam-677	193	6	for	for	ADP
ejpam-677	193	7	α	α	PROPN
ejpam-677	193	8	>	>	X
ejpam-677	193	9	1	1	NUM
ejpam-677	193	10	is	be	AUX
ejpam-677	193	11	omitted	omit	VERB
ejpam-677	193	12	as	as	ADP
ejpam-677	193	13	similar	similar	ADJ
ejpam-677	193	14	to	to	ADP
ejpam-677	193	15	the	the	DET
ejpam-677	193	16	above	above	ADJ
ejpam-677	193	17	mentioned	mention	VERB
ejpam-677	193	18	proof	proof	NOUN
ejpam-677	193	19	for	for	ADP
ejpam-677	193	20	−∞	−∞	X
ejpam-677	193	21	<	<	X
ejpam-677	193	22	α	α	X
ejpam-677	193	23	<	<	X
ejpam-677	193	24	1	1	NUM
ejpam-677	193	25	.	.	PUNCT
ejpam-677	193	26	references	reference	NOUN
ejpam-677	193	27	58	58	NUM
ejpam-677	193	28	references	reference	NOUN
ejpam-677	193	29	[	[	X
ejpam-677	193	30	1	1	NUM
ejpam-677	193	31	]	]	PUNCT
ejpam-677	193	32	t.	t.	PROPN
ejpam-677	193	33	aoki	aoki	PROPN
ejpam-677	193	34	.	.	PUNCT
ejpam-677	194	1	on	on	ADP
ejpam-677	194	2	the	the	DET
ejpam-677	194	3	stability	stability	NOUN
ejpam-677	194	4	of	of	ADP
ejpam-677	194	5	the	the	DET
ejpam-677	194	6	linear	linear	ADJ
ejpam-677	194	7	transformation	transformation	NOUN
ejpam-677	194	8	in	in	ADP
ejpam-677	194	9	banach	banach	NOUN
ejpam-677	194	10	spaces	space	NOUN
ejpam-677	194	11	.	.	PUNCT
ejpam-677	195	1	j.	j.	PROPN
ejpam-677	195	2	math	math	PROPN
ejpam-677	195	3	.	.	PUNCT
ejpam-677	196	1	soc	soc	PROPN
ejpam-677	196	2	.	.	PUNCT
ejpam-677	197	1	japan	japan	PROPN
ejpam-677	197	2	,	,	PUNCT
ejpam-677	197	3	2	2	NUM
ejpam-677	197	4	:	:	SYM
ejpam-677	197	5	64–66	64–66	NUM
ejpam-677	197	6	,	,	PUNCT
ejpam-677	197	7	1950	1950	NUM
ejpam-677	197	8	.	.	PUNCT
ejpam-677	198	1	[	[	X
ejpam-677	198	2	2	2	NUM
ejpam-677	198	3	]	]	X
ejpam-677	198	4	d.h	d.h	PROPN
ejpam-677	198	5	.	.	PROPN
ejpam-677	198	6	hyers	hyer	NOUN
ejpam-677	198	7	.	.	PUNCT
ejpam-677	199	1	on	on	ADP
ejpam-677	199	2	the	the	DET
ejpam-677	199	3	stability	stability	NOUN
ejpam-677	199	4	of	of	ADP
ejpam-677	199	5	the	the	DET
ejpam-677	199	6	linear	linear	ADJ
ejpam-677	199	7	functional	functional	ADJ
ejpam-677	199	8	equation	equation	NOUN
ejpam-677	199	9	.	.	PUNCT
ejpam-677	200	1	proc	proc	NOUN
ejpam-677	200	2	.	.	PUNCT
ejpam-677	201	1	nat	nat	PROPN
ejpam-677	201	2	.	.	PUNCT
ejpam-677	202	1	acad	acad	PROPN
ejpam-677	202	2	.	.	PUNCT
ejpam-677	203	1	sci	sci	PROPN
ejpam-677	203	2	.	.	PROPN
ejpam-677	203	3	,	,	PUNCT
ejpam-677	203	4	27	27	NUM
ejpam-677	203	5	:	:	PUNCT
ejpam-677	203	6	222–224	222–224	NUM
ejpam-677	203	7	,	,	PUNCT
ejpam-677	203	8	1941	1941	NUM
ejpam-677	203	9	.	.	PUNCT
ejpam-677	204	1	[	[	X
ejpam-677	204	2	3	3	X
ejpam-677	204	3	]	]	X
ejpam-677	204	4	j.m	j.m	PROPN
ejpam-677	204	5	.	.	PROPN
ejpam-677	204	6	rassias	rassias	PROPN
ejpam-677	204	7	.	.	PUNCT
ejpam-677	205	1	on	on	ADP
ejpam-677	205	2	approximation	approximation	NOUN
ejpam-677	205	3	of	of	ADP
ejpam-677	205	4	approximately	approximately	ADV
ejpam-677	205	5	linear	linear	ADJ
ejpam-677	205	6	mappings	mapping	NOUN
ejpam-677	205	7	by	by	ADP
ejpam-677	205	8	linear	linear	ADJ
ejpam-677	205	9	mappings	mapping	NOUN
ejpam-677	205	10	.	.	PUNCT
ejpam-677	206	1	j.	j.	PROPN
ejpam-677	206	2	funct	funct	PROPN
ejpam-677	206	3	.	.	PUNCT
ejpam-677	207	1	anal	anal	PROPN
ejpam-677	207	2	.	.	PUNCT
ejpam-677	208	1	46	46	NUM
ejpam-677	208	2	:	:	PUNCT
ejpam-677	208	3	126–130	126–130	NUM
ejpam-677	208	4	,	,	PUNCT
ejpam-677	208	5	1982	1982	NUM
ejpam-677	208	6	.	.	PUNCT
ejpam-677	209	1	[	[	X
ejpam-677	209	2	4	4	NUM
ejpam-677	209	3	]	]	X
ejpam-677	209	4	j.m	j.m	PROPN
ejpam-677	209	5	.	.	PROPN
ejpam-677	209	6	rassias	rassias	PROPN
ejpam-677	209	7	.	.	PUNCT
ejpam-677	210	1	on	on	ADP
ejpam-677	210	2	approximation	approximation	NOUN
ejpam-677	210	3	of	of	ADP
ejpam-677	210	4	approximately	approximately	ADV
ejpam-677	210	5	linear	linear	ADJ
ejpam-677	210	6	mappings	mapping	NOUN
ejpam-677	210	7	by	by	ADP
ejpam-677	210	8	linear	linear	ADJ
ejpam-677	210	9	mappings	mapping	NOUN
ejpam-677	210	10	.	.	PUNCT
ejpam-677	211	1	bull	bull	NOUN
ejpam-677	211	2	.	.	PUNCT
ejpam-677	212	1	sci	sci	PROPN
ejpam-677	212	2	.	.	PUNCT
ejpam-677	212	3	math	math	PROPN
ejpam-677	212	4	.	.	PUNCT
ejpam-677	212	5	,	,	PUNCT
ejpam-677	213	1	108	108	NUM
ejpam-677	213	2	:	:	PUNCT
ejpam-677	213	3	445–446	445–446	NUM
ejpam-677	213	4	,	,	PUNCT
ejpam-677	213	5	1984	1984	NUM
ejpam-677	213	6	.	.	PUNCT
ejpam-677	214	1	[	[	X
ejpam-677	214	2	5	5	NUM
ejpam-677	214	3	]	]	X
ejpam-677	214	4	j.m	j.m	PROPN
ejpam-677	214	5	.	.	PROPN
ejpam-677	214	6	rassias	rassias	PROPN
ejpam-677	214	7	.	.	PUNCT
ejpam-677	215	1	solution	solution	NOUN
ejpam-677	215	2	of	of	ADP
ejpam-677	215	3	a	a	DET
ejpam-677	215	4	problem	problem	NOUN
ejpam-677	215	5	of	of	ADP
ejpam-677	215	6	ulam	ulam	PROPN
ejpam-677	215	7	.	.	PUNCT
ejpam-677	216	1	j.	j.	PROPN
ejpam-677	216	2	approx	approx	PROPN
ejpam-677	216	3	.	.	PUNCT
ejpam-677	217	1	theory	theory	NOUN
ejpam-677	217	2	,	,	PUNCT
ejpam-677	217	3	57	57	NUM
ejpam-677	217	4	:	:	PUNCT
ejpam-677	217	5	268–273	268–273	NUM
ejpam-677	217	6	,	,	PUNCT
ejpam-677	217	7	1989	1989	NUM
ejpam-677	217	8	.	.	PUNCT
ejpam-677	218	1	[	[	X
ejpam-677	218	2	6	6	NUM
ejpam-677	218	3	]	]	PUNCT
ejpam-677	218	4	th.m	th.m	PROPN
ejpam-677	218	5	.	.	PUNCT
ejpam-677	219	1	rassias	rassias	PROPN
ejpam-677	219	2	.	.	PUNCT
ejpam-677	220	1	on	on	ADP
ejpam-677	220	2	the	the	DET
ejpam-677	220	3	stability	stability	NOUN
ejpam-677	220	4	of	of	ADP
ejpam-677	220	5	the	the	DET
ejpam-677	220	6	linear	linear	ADJ
ejpam-677	220	7	mapping	mapping	NOUN
ejpam-677	220	8	in	in	ADP
ejpam-677	220	9	banach	banach	NOUN
ejpam-677	220	10	spaces	space	NOUN
ejpam-677	220	11	.	.	PUNCT
ejpam-677	221	1	proc	proc	NOUN
ejpam-677	221	2	.	.	PUNCT
ejpam-677	222	1	amer	amer	PROPN
ejpam-677	222	2	.	.	PUNCT
ejpam-677	222	3	math	math	PROPN
ejpam-677	222	4	.	.	PUNCT
ejpam-677	223	1	soc	soc	PROPN
ejpam-677	223	2	.	.	PUNCT
ejpam-677	223	3	,	,	PUNCT
ejpam-677	223	4	72	72	NUM
ejpam-677	223	5	:	:	PUNCT
ejpam-677	223	6	297–300	297–300	NUM
ejpam-677	223	7	,	,	PUNCT
ejpam-677	223	8	1978	1978	NUM
ejpam-677	223	9	.	.	PUNCT
ejpam-677	224	1	[	[	X
ejpam-677	224	2	7	7	X
ejpam-677	224	3	]	]	PUNCT
ejpam-677	224	4	k.	k.	PROPN
ejpam-677	224	5	ravi	ravi	PROPN
ejpam-677	224	6	,	,	PUNCT
ejpam-677	224	7	m.	m.	NOUN
ejpam-677	224	8	arunkumar	arunkumar	PROPN
ejpam-677	224	9	and	and	CCONJ
ejpam-677	224	10	j.m	j.m	PROPN
ejpam-677	224	11	.	.	PROPN
ejpam-677	224	12	rassias	rassias	PROPN
ejpam-677	224	13	.	.	PUNCT
ejpam-677	225	1	ulam	ulam	PROPN
ejpam-677	225	2	stability	stability	NOUN
ejpam-677	225	3	for	for	ADP
ejpam-677	225	4	the	the	DET
ejpam-677	225	5	orthogonally	orthogonally	ADV
ejpam-677	225	6	general	general	ADJ
ejpam-677	225	7	euler	euler	PROPN
ejpam-677	225	8	-	-	PUNCT
ejpam-677	225	9	lagrange	lagrange	NOUN
ejpam-677	225	10	type	type	NOUN
ejpam-677	225	11	functional	functional	ADJ
ejpam-677	225	12	equation	equation	NOUN
ejpam-677	225	13	.	.	PUNCT
ejpam-677	226	1	intern	intern	PROPN
ejpam-677	226	2	.	.	PUNCT
ejpam-677	227	1	j.	j.	PROPN
ejpam-677	227	2	math	math	PROPN
ejpam-677	227	3	.	.	PUNCT
ejpam-677	228	1	stat	stat	PROPN
ejpam-677	228	2	.	.	PUNCT
ejpam-677	228	3	,	,	PUNCT
ejpam-677	228	4	3(a08	3(a08	NUM
ejpam-677	228	5	):	):	PUNCT
ejpam-677	228	6	36–46	36–46	NUM
ejpam-677	228	7	,	,	PUNCT
ejpam-677	228	8	2008	2008	NUM
ejpam-677	228	9	.	.	PUNCT
ejpam-677	229	1	[	[	X
ejpam-677	229	2	8	8	NUM
ejpam-677	229	3	]	]	X
ejpam-677	229	4	m.b	m.b	PROPN
ejpam-677	229	5	.	.	PROPN
ejpam-677	229	6	savadkouhi	savadkouhi	PROPN
ejpam-677	229	7	,	,	PUNCT
ejpam-677	229	8	m.e	m.e	PROPN
ejpam-677	229	9	.	.	PROPN
ejpam-677	229	10	gordji	gordji	PROPN
ejpam-677	229	11	,	,	PUNCT
ejpam-677	229	12	j.m	j.m	PROPN
ejpam-677	229	13	.	.	PROPN
ejpam-677	229	14	rassias	rassias	PROPN
ejpam-677	229	15	and	and	CCONJ
ejpam-677	229	16	n.	n.	PROPN
ejpam-677	229	17	ghobadipour	ghobadipour	PROPN
ejpam-677	229	18	,	,	PUNCT
ejpam-677	229	19	approximate	approximate	ADJ
ejpam-677	229	20	ternary	ternary	ADJ
ejpam-677	229	21	jordan	jordan	PROPN
ejpam-677	229	22	derivations	derivation	NOUN
ejpam-677	229	23	on	on	ADP
ejpam-677	229	24	banach	banach	ADV
ejpam-677	229	25	ternary	ternary	ADJ
ejpam-677	229	26	algebras	algebra	NOUN
ejpam-677	229	27	.	.	PUNCT
ejpam-677	230	1	j.	j.	PROPN
ejpam-677	230	2	math	math	PROPN
ejpam-677	230	3	.	.	PUNCT
ejpam-677	231	1	phys	phy	NOUN
ejpam-677	231	2	.	.	PUNCT
ejpam-677	231	3	,	,	PUNCT
ejpam-677	231	4	50	50	NUM
ejpam-677	231	5	,	,	PUNCT
ejpam-677	231	6	042303	042303	NUM
ejpam-677	231	7	:	:	PUNCT
ejpam-677	231	8	1–9	1–9	NUM
ejpam-677	231	9	,	,	PUNCT
ejpam-677	231	10	2009	2009	NUM
ejpam-677	231	11	.	.	PUNCT
ejpam-677	232	1	[	[	X
ejpam-677	232	2	9	9	NUM
ejpam-677	232	3	]	]	SYM
ejpam-677	232	4	s.m	s.m	PROPN
ejpam-677	232	5	.	.	PROPN
ejpam-677	232	6	ulam	ulam	PROPN
ejpam-677	232	7	,	,	PUNCT
ejpam-677	232	8	a	a	DET
ejpam-677	232	9	collection	collection	NOUN
ejpam-677	232	10	of	of	ADP
ejpam-677	232	11	mathematical	mathematical	ADJ
ejpam-677	232	12	problems	problem	NOUN
ejpam-677	232	13	.	.	PUNCT
ejpam-677	233	1	interscience	interscience	NOUN
ejpam-677	233	2	publisher	publisher	PROPN
ejpam-677	233	3	,	,	PUNCT
ejpam-677	233	4	inc	inc	PROPN
ejpam-677	233	5	.	.	PROPN
ejpam-677	233	6	,	,	PUNCT
ejpam-677	233	7	no	no	INTJ
ejpam-677	233	8	.	.	NOUN
ejpam-677	233	9	8	8	NUM
ejpam-677	233	10	,	,	PUNCT
ejpam-677	233	11	new	new	PROPN
ejpam-677	233	12	york	york	PROPN
ejpam-677	233	13	;	;	PUNCT
ejpam-677	233	14	problems	problem	NOUN
ejpam-677	233	15	in	in	ADP
ejpam-677	233	16	modern	modern	ADJ
ejpam-677	233	17	mathematics	mathematic	NOUN
ejpam-677	233	18	.	.	PUNCT
ejpam-677	234	1	wiley	wiley	PROPN
ejpam-677	234	2	and	and	CCONJ
ejpam-677	234	3	sons	son	NOUN
ejpam-677	234	4	,	,	PUNCT
ejpam-677	234	5	new	new	PROPN
ejpam-677	234	6	york	york	PROPN
ejpam-677	234	7	,	,	PUNCT
ejpam-677	234	8	chapter	chapter	NOUN
ejpam-677	234	9	vi	vi	PROPN
ejpam-677	234	10	,	,	PUNCT
ejpam-677	234	11	1964	1964	NUM
ejpam-677	234	12	.	.	PUNCT