id	sid	tid	token	lemma	pos
ejde-508	1	1	electronic	electronic	ADJ
ejde-508	1	2	journal	journal	NOUN
ejde-508	1	3	of	of	ADP
ejde-508	1	4	differential	differential	ADJ
ejde-508	1	5	equations	equation	NOUN
ejde-508	1	6	,	,	PUNCT
ejde-508	1	7	vol	vol	NOUN
ejde-508	1	8	.	.	PUNCT
ejde-508	1	9	2020	2020	NUM
ejde-508	1	10	(	(	PUNCT
ejde-508	1	11	2020	2020	NUM
ejde-508	1	12	)	)	PUNCT
ejde-508	1	13	,	,	PUNCT
ejde-508	1	14	no	no	INTJ
ejde-508	1	15	.	.	PROPN
ejde-508	1	16	87	87	NUM
ejde-508	1	17	,	,	PUNCT
ejde-508	1	18	pp	pp	PROPN
ejde-508	1	19	.	.	PUNCT
ejde-508	2	1	1–14	1–14	PROPN
ejde-508	2	2	.	.	PUNCT
ejde-508	3	1	issn	issn	PROPN
ejde-508	3	2	:	:	PUNCT
ejde-508	3	3	1072	1072	NUM
ejde-508	3	4	-	-	SYM
ejde-508	3	5	6691	6691	NUM
ejde-508	3	6	.	.	PUNCT
ejde-508	4	1	url	url	PROPN
ejde-508	4	2	:	:	PUNCT
ejde-508	4	3	http://ejde.math.txstate.edu	http://ejde.math.txstate.edu	PROPN
ejde-508	4	4	or	or	CCONJ
ejde-508	4	5	http://ejde.math.unt.edu	http://ejde.math.unt.edu	VERB
ejde-508	4	6	oscillatory	oscillatory	ADJ
ejde-508	4	7	behavior	behavior	NOUN
ejde-508	4	8	for	for	ADP
ejde-508	4	9	nonlinear	nonlinear	ADJ
ejde-508	4	10	homogeneous	homogeneous	ADJ
ejde-508	4	11	neutral	neutral	ADJ
ejde-508	4	12	difference	difference	NOUN
ejde-508	4	13	equations	equation	NOUN
ejde-508	4	14	of	of	ADP
ejde-508	4	15	second	second	ADJ
ejde-508	4	16	order	order	NOUN
ejde-508	4	17	with	with	ADP
ejde-508	4	18	coefficient	coefficient	NOUN
ejde-508	4	19	changing	change	VERB
ejde-508	4	20	sign	sign	NOUN
ejde-508	4	21	ajit	ajit	PROPN
ejde-508	4	22	kumar	kumar	PROPN
ejde-508	4	23	bhuyan	bhuyan	PROPN
ejde-508	4	24	,	,	PUNCT
ejde-508	4	25	laxmi	laxmi	PROPN
ejde-508	4	26	narayan	narayan	PROPN
ejde-508	4	27	padhy	padhy	PROPN
ejde-508	4	28	,	,	PUNCT
ejde-508	4	29	radhanath	radhanath	PROPN
ejde-508	4	30	rath	rath	PROPN
ejde-508	4	31	abstract	abstract	NOUN
ejde-508	4	32	.	.	PUNCT
ejde-508	5	1	in	in	ADP
ejde-508	5	2	this	this	DET
ejde-508	5	3	article	article	NOUN
ejde-508	5	4	,	,	PUNCT
ejde-508	5	5	we	we	PRON
ejde-508	5	6	obtain	obtain	VERB
ejde-508	5	7	sufficient	sufficient	ADJ
ejde-508	5	8	conditions	condition	NOUN
ejde-508	5	9	so	so	SCONJ
ejde-508	5	10	that	that	SCONJ
ejde-508	5	11	all	all	DET
ejde-508	5	12	solutions	solution	NOUN
ejde-508	5	13	of	of	ADP
ejde-508	5	14	the	the	DET
ejde-508	5	15	neutral	neutral	ADJ
ejde-508	5	16	difference	difference	NOUN
ejde-508	5	17	equation	equation	NOUN
ejde-508	5	18	∆2	∆2	PROPN
ejde-508	5	19	(	(	PUNCT
ejde-508	5	20	yn	yn	PROPN
ejde-508	5	21	−	−	PROPN
ejde-508	5	22	pnl(yn−s	pnl(yn−s	NOUN
ejde-508	5	23	)	)	PUNCT
ejde-508	5	24	)	)	PUNCT
ejde-508	6	1	+	+	CCONJ
ejde-508	6	2	qng(yn−k	qng(yn−k	X
ejde-508	6	3	)	)	PUNCT
ejde-508	6	4	=	=	SYM
ejde-508	6	5	0	0	NUM
ejde-508	6	6	,	,	PUNCT
ejde-508	6	7	and	and	CCONJ
ejde-508	6	8	all	all	DET
ejde-508	6	9	unbounded	unbounded	ADJ
ejde-508	6	10	solutions	solution	NOUN
ejde-508	6	11	of	of	ADP
ejde-508	6	12	the	the	DET
ejde-508	6	13	neutral	neutral	ADJ
ejde-508	6	14	difference	difference	NOUN
ejde-508	6	15	equation	equation	NOUN
ejde-508	7	1	∆2	∆2	PROPN
ejde-508	7	2	(	(	PUNCT
ejde-508	7	3	yn	yn	PROPN
ejde-508	7	4	−	−	PROPN
ejde-508	7	5	pnl(yn−s	pnl(yn−s	NOUN
ejde-508	7	6	)	)	PUNCT
ejde-508	7	7	)	)	PUNCT
ejde-508	8	1	+	+	CCONJ
ejde-508	8	2	qng(yn−k)−	qng(yn−k)−	VERB
ejde-508	8	3	unh(yα(n	unh(yα(n	NOUN
ejde-508	8	4	)	)	PUNCT
ejde-508	8	5	)	)	PUNCT
ejde-508	9	1	=	=	SYM
ejde-508	9	2	0	0	NUM
ejde-508	9	3	are	be	AUX
ejde-508	9	4	oscillatory	oscillatory	ADJ
ejde-508	9	5	,	,	PUNCT
ejde-508	9	6	where	where	SCONJ
ejde-508	9	7	∆yn	∆yn	NOUN
ejde-508	9	8	=	=	SYM
ejde-508	9	9	yn+1	yn+1	PROPN
ejde-508	9	10	−	−	PROPN
ejde-508	9	11	yn	yn	PROPN
ejde-508	9	12	,	,	PUNCT
ejde-508	9	13	∆2yn	∆2yn	ADJ
ejde-508	9	14	=	=	SYM
ejde-508	9	15	∆(∆yn	∆(∆yn	PROPN
ejde-508	9	16	)	)	PUNCT
ejde-508	9	17	.	.	PUNCT
ejde-508	10	1	different	different	ADJ
ejde-508	10	2	types	type	NOUN
ejde-508	10	3	of	of	ADP
ejde-508	10	4	super	super	ADJ
ejde-508	10	5	linear	linear	ADJ
ejde-508	10	6	and	and	CCONJ
ejde-508	10	7	sub	sub	VERB
ejde-508	10	8	linear	linear	ADJ
ejde-508	10	9	conditions	condition	NOUN
ejde-508	10	10	are	be	AUX
ejde-508	10	11	imposed	impose	VERB
ejde-508	10	12	on	on	ADP
ejde-508	10	13	g	g	PROPN
ejde-508	10	14	to	to	PART
ejde-508	10	15	prevent	prevent	VERB
ejde-508	10	16	the	the	DET
ejde-508	10	17	solution	solution	NOUN
ejde-508	10	18	approaching	approach	VERB
ejde-508	10	19	zero	zero	NUM
ejde-508	10	20	or	or	CCONJ
ejde-508	10	21	±∞.	±∞.	NUM
ejde-508	10	22	1	1	NUM
ejde-508	10	23	.	.	PUNCT
ejde-508	11	1	introduction	introduction	NOUN
ejde-508	11	2	in	in	ADP
ejde-508	11	3	this	this	DET
ejde-508	11	4	article	article	NOUN
ejde-508	11	5	,	,	PUNCT
ejde-508	11	6	we	we	PRON
ejde-508	11	7	obtain	obtain	VERB
ejde-508	11	8	sufficient	sufficient	ADJ
ejde-508	11	9	conditions	condition	NOUN
ejde-508	11	10	so	so	SCONJ
ejde-508	11	11	that	that	SCONJ
ejde-508	11	12	all	all	DET
ejde-508	11	13	solutions	solution	NOUN
ejde-508	11	14	of	of	ADP
ejde-508	11	15	the	the	DET
ejde-508	11	16	neutral	neutral	ADJ
ejde-508	11	17	difference	difference	NOUN
ejde-508	11	18	equation	equation	NOUN
ejde-508	11	19	∆2	∆2	PROPN
ejde-508	11	20	(	(	PUNCT
ejde-508	11	21	yn	yn	PROPN
ejde-508	11	22	−	−	PROPN
ejde-508	11	23	pnl(yn−s	pnl(yn−s	NOUN
ejde-508	11	24	)	)	PUNCT
ejde-508	11	25	)	)	PUNCT
ejde-508	12	1	+	+	CCONJ
ejde-508	12	2	qng(yn−k	qng(yn−k	X
ejde-508	12	3	)	)	PUNCT
ejde-508	12	4	=	=	SYM
ejde-508	12	5	0	0	NUM
ejde-508	12	6	,	,	PUNCT
ejde-508	12	7	n	n	PRON
ejde-508	12	8	≥	≥	NOUN
ejde-508	12	9	n0	n0	NUM
ejde-508	12	10	,	,	PUNCT
ejde-508	12	11	(	(	PUNCT
ejde-508	12	12	1.1	1.1	NUM
ejde-508	12	13	)	)	PUNCT
ejde-508	12	14	and	and	CCONJ
ejde-508	12	15	all	all	DET
ejde-508	12	16	unbounded	unbounded	ADJ
ejde-508	12	17	solutions	solution	NOUN
ejde-508	12	18	of	of	ADP
ejde-508	12	19	the	the	DET
ejde-508	12	20	neutral	neutral	ADJ
ejde-508	12	21	difference	difference	NOUN
ejde-508	12	22	equation	equation	NOUN
ejde-508	12	23	∆2	∆2	PROPN
ejde-508	12	24	(	(	PUNCT
ejde-508	12	25	yn	yn	PROPN
ejde-508	12	26	−	−	PROPN
ejde-508	12	27	pnl(yn−s	pnl(yn−s	NOUN
ejde-508	12	28	)	)	PUNCT
ejde-508	12	29	)	)	PUNCT
ejde-508	13	1	+	+	CCONJ
ejde-508	13	2	qng(yn−k)−	qng(yn−k)−	VERB
ejde-508	13	3	unh(yα(n	unh(yα(n	NOUN
ejde-508	13	4	)	)	PUNCT
ejde-508	13	5	)	)	PUNCT
ejde-508	14	1	=	=	SYM
ejde-508	14	2	0	0	NUM
ejde-508	14	3	,	,	PUNCT
ejde-508	14	4	n	n	PRON
ejde-508	14	5	≥	≥	NOUN
ejde-508	14	6	n0	n0	NUM
ejde-508	14	7	(	(	PUNCT
ejde-508	14	8	1.2	1.2	NUM
ejde-508	14	9	)	)	PUNCT
ejde-508	14	10	are	be	AUX
ejde-508	14	11	oscillatory	oscillatory	ADJ
ejde-508	14	12	,	,	PUNCT
ejde-508	14	13	where	where	SCONJ
ejde-508	14	14	∆	∆	PROPN
ejde-508	14	15	is	be	AUX
ejde-508	14	16	the	the	DET
ejde-508	14	17	forward	forward	ADJ
ejde-508	14	18	difference	difference	NOUN
ejde-508	14	19	operator	operator	NOUN
ejde-508	14	20	∆yn	∆yn	NOUN
ejde-508	15	1	=	=	SYM
ejde-508	16	1	yn+1−yn	yn+1−yn	PROPN
ejde-508	16	2	,	,	PUNCT
ejde-508	16	3	∆2yn	∆2yn	ADJ
ejde-508	16	4	=	=	SYM
ejde-508	16	5	∆(∆yn	∆(∆yn	PROPN
ejde-508	16	6	)	)	PUNCT
ejde-508	16	7	,	,	PUNCT
ejde-508	16	8	{	{	PUNCT
ejde-508	16	9	qn	qn	NOUN
ejde-508	16	10	}	}	PUNCT
ejde-508	16	11	and	and	CCONJ
ejde-508	16	12	{	{	PUNCT
ejde-508	16	13	un	un	PROPN
ejde-508	16	14	}	}	PUNCT
ejde-508	16	15	are	be	AUX
ejde-508	16	16	sequences	sequence	NOUN
ejde-508	16	17	of	of	ADP
ejde-508	16	18	real	real	ADJ
ejde-508	16	19	numbers	number	NOUN
ejde-508	16	20	with	with	ADP
ejde-508	16	21	qn	qn	INTJ
ejde-508	16	22	>	>	X
ejde-508	16	23	0	0	PROPN
ejde-508	16	24	,	,	PUNCT
ejde-508	16	25	un	un	PROPN
ejde-508	16	26	≥	≥	PROPN
ejde-508	16	27	0	0	NUM
ejde-508	16	28	,	,	PUNCT
ejde-508	16	29	and	and	CCONJ
ejde-508	16	30	g	g	NOUN
ejde-508	16	31	,	,	PUNCT
ejde-508	16	32	h	h	NOUN
ejde-508	16	33	,	,	PUNCT
ejde-508	16	34	l	l	PROPN
ejde-508	16	35	∈	∈	PROPN
ejde-508	16	36	c(r	c(r	NOUN
ejde-508	16	37	,	,	PUNCT
ejde-508	16	38	r	r	NOUN
ejde-508	16	39	)	)	PUNCT
ejde-508	16	40	.	.	PUNCT
ejde-508	17	1	we	we	PRON
ejde-508	17	2	assume	assume	VERB
ejde-508	17	3	that	that	SCONJ
ejde-508	17	4	α(n	α(n	NOUN
ejde-508	17	5	)	)	PUNCT
ejde-508	17	6	<	<	X
ejde-508	17	7	n−	n−	NOUN
ejde-508	17	8	1	1	NUM
ejde-508	17	9	and	and	CCONJ
ejde-508	17	10	it	it	PRON
ejde-508	17	11	approaches	approach	VERB
ejde-508	17	12	∞	∞	PROPN
ejde-508	17	13	as	as	ADP
ejde-508	17	14	n→∞	n→∞	NUM
ejde-508	17	15	,	,	PUNCT
ejde-508	17	16	and	and	CCONJ
ejde-508	17	17	s	s	X
ejde-508	17	18	,	,	PUNCT
ejde-508	17	19	k	k	X
ejde-508	17	20	are	be	AUX
ejde-508	17	21	positive	positive	ADJ
ejde-508	17	22	integers	integer	NOUN
ejde-508	17	23	.	.	PUNCT
ejde-508	18	1	further	far	ADV
ejde-508	18	2	,	,	PUNCT
ejde-508	18	3	we	we	PRON
ejde-508	18	4	assume	assume	VERB
ejde-508	18	5	that	that	SCONJ
ejde-508	18	6	g(−x	g(−x	NOUN
ejde-508	18	7	)	)	PUNCT
ejde-508	18	8	=	=	SYM
ejde-508	18	9	−g(x	−g(x	NUM
ejde-508	18	10	)	)	PUNCT
ejde-508	18	11	,	,	PUNCT
ejde-508	18	12	h(−x	h(−x	NOUN
ejde-508	18	13	)	)	PUNCT
ejde-508	18	14	=	=	SYM
ejde-508	18	15	−h(x	−h(x	PROPN
ejde-508	18	16	)	)	PUNCT
ejde-508	18	17	,	,	PUNCT
ejde-508	18	18	l(−x	l(−x	NOUN
ejde-508	18	19	)	)	PUNCT
ejde-508	18	20	=	=	SYM
ejde-508	19	1	−l(x	−l(x	PROPN
ejde-508	19	2	)	)	PUNCT
ejde-508	19	3	,	,	PUNCT
ejde-508	20	1	∀x	∀x	VERB
ejde-508	20	2	∈	∈	NOUN
ejde-508	20	3	r	r	NOUN
ejde-508	20	4	xg(x	xg(x	PUNCT
ejde-508	20	5	)	)	PUNCT
ejde-508	20	6	>	>	X
ejde-508	20	7	0	0	NUM
ejde-508	20	8	,	,	PUNCT
ejde-508	20	9	xh(x	xh(x	PUNCT
ejde-508	20	10	)	)	PUNCT
ejde-508	20	11	>	>	X
ejde-508	20	12	0	0	NUM
ejde-508	20	13	,	,	PUNCT
ejde-508	20	14	xl(x	xl(x	NUM
ejde-508	20	15	)	)	PUNCT
ejde-508	20	16	>	>	X
ejde-508	20	17	0	0	PUNCT
ejde-508	21	1	∀x	∀x	X
ejde-508	21	2	>	>	X
ejde-508	21	3	0	0	NUM
ejde-508	21	4	.	.	PUNCT
ejde-508	22	1	(	(	PUNCT
ejde-508	22	2	1.3	1.3	NUM
ejde-508	22	3	)	)	PUNCT
ejde-508	22	4	some	some	PRON
ejde-508	22	5	of	of	ADP
ejde-508	22	6	the	the	DET
ejde-508	22	7	following	follow	VERB
ejde-508	22	8	assumptions	assumption	NOUN
ejde-508	22	9	are	be	AUX
ejde-508	22	10	used	use	VERB
ejde-508	22	11	later	later	ADV
ejde-508	22	12	in	in	ADP
ejde-508	22	13	this	this	DET
ejde-508	22	14	article	article	NOUN
ejde-508	22	15	.	.	PUNCT
ejde-508	23	1	(	(	PUNCT
ejde-508	23	2	a1	a1	NOUN
ejde-508	23	3	)	)	PUNCT
ejde-508	23	4	there	there	ADV
ejde-508	23	5	exists	exist	VERB
ejde-508	23	6	δ	δ	PROPN
ejde-508	23	7	>	>	X
ejde-508	23	8	0	0	NUM
ejde-508	24	1	such	such	ADJ
ejde-508	24	2	that	that	PRON
ejde-508	24	3	for	for	ADP
ejde-508	24	4	each	each	DET
ejde-508	24	5	x	x	SYM
ejde-508	24	6	>	>	X
ejde-508	24	7	0	0	NUM
ejde-508	24	8	,	,	PUNCT
ejde-508	24	9	l(x	l(x	PROPN
ejde-508	24	10	)	)	PUNCT
ejde-508	24	11	≤	≤	NUM
ejde-508	24	12	δx	δx	VERB
ejde-508	24	13	;	;	PUNCT
ejde-508	24	14	(	(	PUNCT
ejde-508	24	15	a2	a2	PROPN
ejde-508	24	16	)	)	PUNCT
ejde-508	24	17	qn	qn	NOUN
ejde-508	24	18	>	>	X
ejde-508	24	19	0	0	PUNCT
ejde-508	25	1	and	and	CCONJ
ejde-508	25	2	∑∞	∑∞	NOUN
ejde-508	25	3	n	n	CCONJ
ejde-508	25	4	=	=	SYM
ejde-508	25	5	n0	n0	NUM
ejde-508	25	6	qn	qn	NOUN
ejde-508	25	7	=	=	PROPN
ejde-508	25	8	∞	∞	PROPN
ejde-508	25	9	;	;	PUNCT
ejde-508	25	10	(	(	PUNCT
ejde-508	25	11	a3	a3	NOUN
ejde-508	25	12	)	)	PUNCT
ejde-508	25	13	∑∞	∑∞	NOUN
ejde-508	25	14	n	n	CCONJ
ejde-508	25	15	=	=	SYM
ejde-508	25	16	n1	n1	NOUN
ejde-508	25	17	q∗n	q∗n	PUNCT
ejde-508	25	18	=	=	NOUN
ejde-508	25	19	∞	∞	PROPN
ejde-508	25	20	,	,	PUNCT
ejde-508	25	21	where	where	SCONJ
ejde-508	25	22	q∗	q∗	NOUN
ejde-508	25	23	=	=	SYM
ejde-508	25	24	min{qn	min{qn	PROPN
ejde-508	25	25	,	,	PUNCT
ejde-508	25	26	qn−s	qn−s	PROPN
ejde-508	25	27	}	}	PUNCT
ejde-508	25	28	;	;	PUNCT
ejde-508	25	29	(	(	PUNCT
ejde-508	25	30	a4	a4	X
ejde-508	25	31	)	)	PUNCT
ejde-508	25	32	lim	lim	PROPN
ejde-508	25	33	infn→∞	infn→∞	PROPN
ejde-508	26	1	qn	qn	INTJ
ejde-508	26	2	>	>	X
ejde-508	26	3	0	0	NUM
ejde-508	26	4	;	;	PUNCT
ejde-508	26	5	(	(	PUNCT
ejde-508	26	6	a5	a5	NUM
ejde-508	26	7	)	)	PUNCT
ejde-508	26	8	g	g	PROPN
ejde-508	26	9	is	be	AUX
ejde-508	26	10	non	non	NOUN
ejde-508	26	11	decreasing	decrease	VERB
ejde-508	26	12	;	;	PUNCT
ejde-508	26	13	2010	2010	NUM
ejde-508	26	14	mathematics	mathematic	NOUN
ejde-508	26	15	subject	subject	NOUN
ejde-508	26	16	classification	classification	NOUN
ejde-508	26	17	.	.	PUNCT
ejde-508	27	1	39a10	39a10	NUM
ejde-508	27	2	,	,	PUNCT
ejde-508	27	3	39a12	39a12	NUM
ejde-508	27	4	.	.	PUNCT
ejde-508	28	1	key	key	ADJ
ejde-508	28	2	words	word	NOUN
ejde-508	28	3	and	and	CCONJ
ejde-508	28	4	phrases	phrase	NOUN
ejde-508	28	5	.	.	PUNCT
ejde-508	29	1	oscillatory	oscillatory	ADJ
ejde-508	29	2	solution	solution	NOUN
ejde-508	29	3	;	;	PUNCT
ejde-508	29	4	nonoscillatory	nonoscillatory	ADJ
ejde-508	29	5	solution	solution	NOUN
ejde-508	29	6	;	;	PUNCT
ejde-508	29	7	asymptotic	asymptotic	ADJ
ejde-508	29	8	behavior	behavior	NOUN
ejde-508	29	9	;	;	PUNCT
ejde-508	29	10	difference	difference	NOUN
ejde-508	29	11	equation	equation	NOUN
ejde-508	29	12	.	.	PUNCT
ejde-508	30	1	c	c	X
ejde-508	30	2	©	©	PROPN
ejde-508	30	3	2020	2020	NUM
ejde-508	30	4	texas	texas	PROPN
ejde-508	30	5	state	state	PROPN
ejde-508	30	6	university	university	PROPN
ejde-508	30	7	.	.	PUNCT
ejde-508	31	1	submitted	submit	VERB
ejde-508	31	2	june	june	PROPN
ejde-508	31	3	3	3	NUM
ejde-508	31	4	,	,	PUNCT
ejde-508	31	5	2020	2020	NUM
ejde-508	31	6	.	.	PUNCT
ejde-508	32	1	published	publish	VERB
ejde-508	32	2	august	august	PROPN
ejde-508	32	3	12	12	NUM
ejde-508	32	4	,	,	PUNCT
ejde-508	32	5	2020	2020	NUM
ejde-508	32	6	.	.	PUNCT
ejde-508	33	1	1	1	NUM
ejde-508	33	2	2	2	NUM
ejde-508	33	3	a.	a.	NOUN
ejde-508	33	4	k.	k.	PROPN
ejde-508	33	5	bhuyan	bhuyan	PROPN
ejde-508	33	6	,	,	PUNCT
ejde-508	33	7	l.	l.	PROPN
ejde-508	33	8	n.	n.	PROPN
ejde-508	33	9	padhy	padhy	PROPN
ejde-508	33	10	,	,	PUNCT
ejde-508	33	11	r.	r.	PROPN
ejde-508	33	12	n.	n.	PROPN
ejde-508	33	13	rath	rath	PROPN
ejde-508	33	14	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	33	15	(	(	PUNCT
ejde-508	33	16	a6	a6	NOUN
ejde-508	33	17	)	)	PUNCT
ejde-508	33	18	∑∞	∑∞	NOUN
ejde-508	33	19	n	n	CCONJ
ejde-508	33	20	=	=	SYM
ejde-508	33	21	n0	n0	X
ejde-508	33	22	nun	nun	NOUN
ejde-508	33	23	<	<	PROPN
ejde-508	33	24	∞	∞	PROPN
ejde-508	33	25	;	;	PUNCT
ejde-508	33	26	(	(	PUNCT
ejde-508	33	27	a7	a7	ADJ
ejde-508	33	28	)	)	PUNCT
ejde-508	33	29	h	h	NOUN
ejde-508	33	30	is	be	AUX
ejde-508	33	31	bounded	bound	VERB
ejde-508	33	32	.	.	PUNCT
ejde-508	34	1	for	for	ADP
ejde-508	34	2	the	the	DET
ejde-508	34	3	sequence	sequence	NOUN
ejde-508	34	4	{	{	PUNCT
ejde-508	34	5	pn	pn	NOUN
ejde-508	34	6	}	}	PUNCT
ejde-508	34	7	we	we	PRON
ejde-508	34	8	state	state	VERB
ejde-508	34	9	the	the	DET
ejde-508	34	10	following	follow	VERB
ejde-508	34	11	conditions	condition	NOUN
ejde-508	34	12	:	:	PUNCT
ejde-508	34	13	0	0	NUM
ejde-508	34	14	≤	≤	NUM
ejde-508	34	15	pn	pn	NOUN
ejde-508	34	16	≤	≤	PROPN
ejde-508	34	17	p	p	X
ejde-508	34	18	,	,	PUNCT
ejde-508	34	19	(	(	PUNCT
ejde-508	34	20	1.4	1.4	NUM
ejde-508	34	21	)	)	PUNCT
ejde-508	34	22	0	0	NUM
ejde-508	35	1	≤	≤	NUM
ejde-508	35	2	pn	pn	X
ejde-508	35	3	≤	≤	ADV
ejde-508	35	4	1	1	NUM
ejde-508	35	5	,	,	PUNCT
ejde-508	35	6	(	(	PUNCT
ejde-508	35	7	1.5	1.5	NUM
ejde-508	35	8	)	)	PUNCT
ejde-508	35	9	−p	−p	ADJ
ejde-508	35	10	≤	≤	NUM
ejde-508	35	11	pn	pn	X
ejde-508	35	12	<	<	X
ejde-508	35	13	0	0	PROPN
ejde-508	35	14	,	,	PUNCT
ejde-508	35	15	(	(	PUNCT
ejde-508	35	16	1.6	1.6	NUM
ejde-508	35	17	)	)	PUNCT
ejde-508	35	18	pn	pn	PROPN
ejde-508	35	19	changes	change	NOUN
ejde-508	35	20	sign	sign	VERB
ejde-508	35	21	and	and	CCONJ
ejde-508	35	22	−	−	PROPN
ejde-508	35	23	p	p	PROPN
ejde-508	35	24	≤	≤	NUM
ejde-508	35	25	pn	pn	NOUN
ejde-508	35	26	≤	≤	PROPN
ejde-508	35	27	p	p	X
ejde-508	35	28	,	,	PUNCT
ejde-508	35	29	(	(	PUNCT
ejde-508	35	30	1.7	1.7	NUM
ejde-508	35	31	)	)	PUNCT
ejde-508	35	32	1	1	NUM
ejde-508	35	33	≤	≤	NUM
ejde-508	35	34	pn	pn	NOUN
ejde-508	35	35	≤	≤	PROPN
ejde-508	35	36	p	p	X
ejde-508	35	37	,	,	PUNCT
ejde-508	35	38	(	(	PUNCT
ejde-508	35	39	1.8	1.8	NUM
ejde-508	35	40	)	)	PUNCT
ejde-508	35	41	−1	−1	NOUN
ejde-508	35	42	<	<	X
ejde-508	35	43	−b	−b	ADJ
ejde-508	35	44	≤	≤	X
ejde-508	35	45	pn	pn	X
ejde-508	35	46	≤	≤	ADV
ejde-508	35	47	0	0	NUM
ejde-508	35	48	,	,	PUNCT
ejde-508	35	49	(	(	PUNCT
ejde-508	35	50	1.9	1.9	NUM
ejde-508	35	51	)	)	PUNCT
ejde-508	35	52	where	where	SCONJ
ejde-508	35	53	p	p	NOUN
ejde-508	35	54	and	and	CCONJ
ejde-508	35	55	b	b	NOUN
ejde-508	35	56	are	be	AUX
ejde-508	35	57	positive	positive	ADJ
ejde-508	35	58	constants	constant	NOUN
ejde-508	35	59	.	.	PUNCT
ejde-508	36	1	as	as	ADP
ejde-508	36	2	of	of	ADP
ejde-508	36	3	now	now	ADV
ejde-508	36	4	,	,	PUNCT
ejde-508	36	5	many	many	ADJ
ejde-508	36	6	researchers	researcher	NOUN
ejde-508	36	7	all	all	ADV
ejde-508	36	8	over	over	ADP
ejde-508	36	9	the	the	DET
ejde-508	36	10	world	world	NOUN
ejde-508	36	11	are	be	AUX
ejde-508	36	12	engaged	engage	VERB
ejde-508	36	13	to	to	PART
ejde-508	36	14	find	find	VERB
ejde-508	36	15	necessary	necessary	ADJ
ejde-508	36	16	or	or	CCONJ
ejde-508	36	17	sufficient	sufficient	ADJ
ejde-508	36	18	conditions	condition	NOUN
ejde-508	36	19	for	for	ADP
ejde-508	36	20	oscillation	oscillation	NOUN
ejde-508	36	21	or	or	CCONJ
ejde-508	36	22	non	non	ADJ
ejde-508	36	23	oscillation	oscillation	NOUN
ejde-508	36	24	for	for	ADP
ejde-508	36	25	neutral	neutral	ADJ
ejde-508	36	26	difference	difference	NOUN
ejde-508	36	27	equations	equation	NOUN
ejde-508	36	28	,	,	PUNCT
ejde-508	36	29	because	because	SCONJ
ejde-508	36	30	of	of	ADP
ejde-508	36	31	its	its	PRON
ejde-508	36	32	important	important	ADJ
ejde-508	36	33	applications	application	NOUN
ejde-508	36	34	in	in	ADP
ejde-508	36	35	different	different	ADJ
ejde-508	36	36	fields	field	NOUN
ejde-508	36	37	of	of	ADP
ejde-508	36	38	science	science	NOUN
ejde-508	36	39	and	and	CCONJ
ejde-508	36	40	technology	technology	NOUN
ejde-508	36	41	.	.	PUNCT
ejde-508	37	1	for	for	ADP
ejde-508	37	2	the	the	DET
ejde-508	37	3	fundamentals	fundamental	NOUN
ejde-508	37	4	and	and	CCONJ
ejde-508	37	5	some	some	DET
ejde-508	37	6	recent	recent	ADJ
ejde-508	37	7	results	result	NOUN
ejde-508	37	8	on	on	ADP
ejde-508	37	9	the	the	DET
ejde-508	37	10	subject	subject	NOUN
ejde-508	37	11	,	,	PUNCT
ejde-508	37	12	one	one	PRON
ejde-508	37	13	may	may	AUX
ejde-508	37	14	go	go	VERB
ejde-508	37	15	through	through	ADP
ejde-508	37	16	the	the	DET
ejde-508	37	17	monograph	monograph	NOUN
ejde-508	38	1	[	[	X
ejde-508	38	2	1	1	NUM
ejde-508	38	3	,	,	PUNCT
ejde-508	38	4	5	5	NUM
ejde-508	38	5	]	]	PUNCT
ejde-508	38	6	and	and	CCONJ
ejde-508	38	7	the	the	DET
ejde-508	38	8	research	research	NOUN
ejde-508	38	9	articles	article	NOUN
ejde-508	38	10	[	[	X
ejde-508	38	11	2	2	NUM
ejde-508	38	12	,	,	PUNCT
ejde-508	38	13	4	4	NUM
ejde-508	38	14	,	,	PUNCT
ejde-508	38	15	12	12	NUM
ejde-508	38	16	,	,	PUNCT
ejde-508	38	17	14	14	NUM
ejde-508	38	18	]	]	PUNCT
ejde-508	38	19	and	and	CCONJ
ejde-508	38	20	the	the	DET
ejde-508	38	21	references	reference	NOUN
ejde-508	38	22	cited	cite	VERB
ejde-508	38	23	there	there	ADV
ejde-508	38	24	in	in	ADP
ejde-508	38	25	.	.	PUNCT
ejde-508	39	1	sufficient	sufficient	ADJ
ejde-508	39	2	conditions	condition	NOUN
ejde-508	39	3	are	be	AUX
ejde-508	39	4	found	find	VERB
ejde-508	39	5	,	,	PUNCT
ejde-508	39	6	in	in	ADP
ejde-508	39	7	[	[	PUNCT
ejde-508	39	8	3	3	NUM
ejde-508	39	9	,	,	PUNCT
ejde-508	39	10	4	4	NUM
ejde-508	39	11	,	,	PUNCT
ejde-508	39	12	7	7	NUM
ejde-508	39	13	,	,	PUNCT
ejde-508	39	14	12	12	NUM
ejde-508	39	15	,	,	PUNCT
ejde-508	39	16	13	13	NUM
ejde-508	39	17	,	,	PUNCT
ejde-508	39	18	14	14	NUM
ejde-508	39	19	,	,	PUNCT
ejde-508	39	20	15	15	NUM
ejde-508	39	21	,	,	PUNCT
ejde-508	39	22	16	16	NUM
ejde-508	39	23	]	]	PUNCT
ejde-508	39	24	,	,	PUNCT
ejde-508	39	25	and	and	CCONJ
ejde-508	39	26	more	more	ADV
ejde-508	39	27	recently	recently	ADV
ejde-508	39	28	in	in	ADP
ejde-508	39	29	[	[	X
ejde-508	39	30	2	2	NUM
ejde-508	39	31	,	,	PUNCT
ejde-508	39	32	3	3	NUM
ejde-508	39	33	]	]	PUNCT
ejde-508	39	34	,	,	PUNCT
ejde-508	39	35	so	so	SCONJ
ejde-508	39	36	that	that	SCONJ
ejde-508	39	37	every	every	DET
ejde-508	39	38	solutions	solution	NOUN
ejde-508	39	39	of	of	ADP
ejde-508	39	40	the	the	DET
ejde-508	39	41	non	non	ADJ
ejde-508	39	42	linear	linear	ADJ
ejde-508	39	43	neutral	neutral	ADJ
ejde-508	39	44	difference	difference	NOUN
ejde-508	39	45	equation	equation	NOUN
ejde-508	39	46	∆2	∆2	PROPN
ejde-508	39	47	(	(	PUNCT
ejde-508	39	48	yn	yn	PROPN
ejde-508	39	49	−	−	PROPN
ejde-508	39	50	pnyn−s	pnyn−s	INTJ
ejde-508	39	51	)	)	PUNCT
ejde-508	40	1	+	+	CCONJ
ejde-508	40	2	qng(yn−k)−	qng(yn−k)−	NOUN
ejde-508	40	3	unh(yn−r	unh(yn−r	NOUN
ejde-508	40	4	)	)	PUNCT
ejde-508	40	5	=	=	SYM
ejde-508	40	6	fn	fn	NOUN
ejde-508	40	7	,	,	PUNCT
ejde-508	40	8	n	n	PRON
ejde-508	40	9	≥	≥	NOUN
ejde-508	40	10	n0	n0	NUM
ejde-508	40	11	,	,	PUNCT
ejde-508	40	12	(	(	PUNCT
ejde-508	40	13	1.10	1.10	NUM
ejde-508	40	14	)	)	PUNCT
ejde-508	40	15	(	(	PUNCT
ejde-508	40	16	or	or	CCONJ
ejde-508	40	17	of	of	ADP
ejde-508	40	18	its	its	PRON
ejde-508	40	19	particular	particular	ADJ
ejde-508	40	20	case	case	NOUN
ejde-508	40	21	un	un	PROPN
ejde-508	40	22	≡	≡	PROPN
ejde-508	40	23	0	0	NUM
ejde-508	40	24	,	,	PUNCT
ejde-508	40	25	fn	fn	PROPN
ejde-508	40	26	≡	≡	PROPN
ejde-508	40	27	0	0	NUM
ejde-508	40	28	)	)	PUNCT
ejde-508	40	29	oscillates	oscillate	NOUN
ejde-508	40	30	or	or	CCONJ
ejde-508	40	31	tends	tend	VERB
ejde-508	40	32	to	to	ADP
ejde-508	40	33	zero	zero	NUM
ejde-508	40	34	or	or	CCONJ
ejde-508	40	35	to	to	ADP
ejde-508	40	36	±∞	±∞	PROPN
ejde-508	40	37	at	at	ADP
ejde-508	40	38	∞.	∞.	PROPN
ejde-508	40	39	the	the	DET
ejde-508	40	40	asymptotic	asymptotic	ADJ
ejde-508	40	41	behavior	behavior	NOUN
ejde-508	40	42	of	of	ADP
ejde-508	40	43	the	the	DET
ejde-508	40	44	solution	solution	NOUN
ejde-508	40	45	is	be	AUX
ejde-508	40	46	probably	probably	ADV
ejde-508	40	47	due	due	ADJ
ejde-508	40	48	to	to	ADP
ejde-508	40	49	the	the	DET
ejde-508	40	50	presence	presence	NOUN
ejde-508	40	51	of	of	ADP
ejde-508	40	52	the	the	DET
ejde-508	40	53	forcing	force	VERB
ejde-508	40	54	term	term	NOUN
ejde-508	40	55	fn	fn	NOUN
ejde-508	40	56	in	in	ADP
ejde-508	40	57	(	(	PUNCT
ejde-508	40	58	1.10	1.10	NUM
ejde-508	40	59	)	)	PUNCT
ejde-508	40	60	.	.	PUNCT
ejde-508	41	1	the	the	DET
ejde-508	41	2	objective	objective	NOUN
ejde-508	41	3	of	of	ADP
ejde-508	41	4	this	this	DET
ejde-508	41	5	work	work	NOUN
ejde-508	41	6	is	be	AUX
ejde-508	41	7	to	to	PART
ejde-508	41	8	find	find	VERB
ejde-508	41	9	sufficient	sufficient	ADJ
ejde-508	41	10	conditions	condition	NOUN
ejde-508	41	11	so	so	SCONJ
ejde-508	41	12	that	that	SCONJ
ejde-508	41	13	all	all	DET
ejde-508	41	14	solutions	solution	NOUN
ejde-508	41	15	of	of	ADP
ejde-508	41	16	(	(	PUNCT
ejde-508	41	17	1.2	1.2	NUM
ejde-508	41	18	)	)	PUNCT
ejde-508	41	19	are	be	AUX
ejde-508	41	20	oscillatory	oscillatory	ADJ
ejde-508	41	21	under	under	ADP
ejde-508	41	22	different	different	ADJ
ejde-508	41	23	cases	case	NOUN
ejde-508	41	24	of	of	ADP
ejde-508	41	25	pn	pn	PROPN
ejde-508	41	26	>	>	X
ejde-508	41	27	0	0	PROPN
ejde-508	41	28	,	,	PUNCT
ejde-508	41	29	pn	pn	X
ejde-508	41	30	<	<	X
ejde-508	41	31	0	0	PUNCT
ejde-508	41	32	or	or	CCONJ
ejde-508	41	33	pn	pn	X
ejde-508	41	34	changing	change	VERB
ejde-508	41	35	sign	sign	NOUN
ejde-508	41	36	.	.	PUNCT
ejde-508	42	1	for	for	ADP
ejde-508	42	2	that	that	PRON
ejde-508	42	3	,	,	PUNCT
ejde-508	42	4	we	we	PRON
ejde-508	42	5	had	have	VERB
ejde-508	42	6	to	to	PART
ejde-508	42	7	prevent	prevent	VERB
ejde-508	42	8	the	the	DET
ejde-508	42	9	bounded	bounded	ADJ
ejde-508	42	10	solutions	solution	NOUN
ejde-508	42	11	of	of	ADP
ejde-508	42	12	(	(	PUNCT
ejde-508	42	13	1.2	1.2	NUM
ejde-508	42	14	)	)	PUNCT
ejde-508	42	15	from	from	ADP
ejde-508	42	16	approaching	approach	VERB
ejde-508	42	17	zero	zero	NUM
ejde-508	42	18	by	by	ADP
ejde-508	42	19	imposing	impose	VERB
ejde-508	42	20	a	a	DET
ejde-508	42	21	sub	sub	NOUN
ejde-508	42	22	linear	linear	ADJ
ejde-508	42	23	condition	condition	NOUN
ejde-508	42	24	(	(	PUNCT
ejde-508	42	25	4.4	4.4	NUM
ejde-508	42	26	)	)	PUNCT
ejde-508	42	27	or	or	CCONJ
ejde-508	42	28	(	(	PUNCT
ejde-508	42	29	4.1	4.1	NUM
ejde-508	42	30	)	)	PUNCT
ejde-508	42	31	on	on	ADP
ejde-508	42	32	g	g	PROPN
ejde-508	42	33	as	as	ADV
ejde-508	42	34	well	well	ADV
ejde-508	42	35	as	as	ADP
ejde-508	42	36	stop	stop	VERB
ejde-508	42	37	the	the	DET
ejde-508	42	38	unbounded	unbounded	ADJ
ejde-508	42	39	solution	solution	NOUN
ejde-508	42	40	of	of	ADP
ejde-508	42	41	(	(	PUNCT
ejde-508	42	42	1.2	1.2	NUM
ejde-508	42	43	)	)	PUNCT
ejde-508	42	44	from	from	ADP
ejde-508	42	45	approaching	approach	VERB
ejde-508	42	46	±∞	±∞	PROPN
ejde-508	42	47	by	by	ADP
ejde-508	42	48	imposing	impose	VERB
ejde-508	42	49	a	a	DET
ejde-508	42	50	super	super	ADJ
ejde-508	42	51	linear	linear	ADJ
ejde-508	42	52	condition	condition	NOUN
ejde-508	42	53	(	(	PUNCT
ejde-508	42	54	3.5	3.5	NUM
ejde-508	42	55	)	)	PUNCT
ejde-508	42	56	or	or	CCONJ
ejde-508	42	57	(	(	PUNCT
ejde-508	42	58	3.2	3.2	NUM
ejde-508	42	59	)	)	PUNCT
ejde-508	42	60	on	on	ADP
ejde-508	42	61	g.	g.	PROPN
ejde-508	43	1	then	then	ADV
ejde-508	43	2	the	the	DET
ejde-508	43	3	results	result	NOUN
ejde-508	43	4	for	for	ADP
ejde-508	43	5	(	(	PUNCT
ejde-508	43	6	1.2	1.2	NUM
ejde-508	43	7	)	)	PUNCT
ejde-508	43	8	are	be	AUX
ejde-508	43	9	applied	apply	VERB
ejde-508	43	10	to	to	PART
ejde-508	43	11	study	study	VERB
ejde-508	43	12	the	the	DET
ejde-508	43	13	oscillatory	oscillatory	ADJ
ejde-508	43	14	behavior	behavior	NOUN
ejde-508	43	15	of	of	ADP
ejde-508	43	16	the	the	DET
ejde-508	43	17	unbounded	unbounded	ADJ
ejde-508	43	18	solutions	solution	NOUN
ejde-508	43	19	of	of	ADP
ejde-508	43	20	neutral	neutral	ADJ
ejde-508	43	21	difference	difference	NOUN
ejde-508	43	22	equation	equation	NOUN
ejde-508	43	23	∆2	∆2	PROPN
ejde-508	43	24	(	(	PUNCT
ejde-508	43	25	yn	yn	PROPN
ejde-508	43	26	−	−	PROPN
ejde-508	43	27	pnl(yn−s	pnl(yn−s	NOUN
ejde-508	43	28	)	)	PUNCT
ejde-508	43	29	)	)	PUNCT
ejde-508	44	1	+	+	CCONJ
ejde-508	44	2	vng(yn−k	vng(yn−k	X
ejde-508	44	3	)	)	PUNCT
ejde-508	44	4	=	=	SYM
ejde-508	44	5	0	0	NUM
ejde-508	44	6	,	,	PUNCT
ejde-508	44	7	n	n	PRON
ejde-508	44	8	≥	≥	NOUN
ejde-508	44	9	n0	n0	NUM
ejde-508	44	10	,	,	PUNCT
ejde-508	44	11	(	(	PUNCT
ejde-508	44	12	1.11	1.11	NUM
ejde-508	44	13	)	)	PUNCT
ejde-508	44	14	where	where	SCONJ
ejde-508	44	15	vn	vn	PROPN
ejde-508	44	16	changes	change	NOUN
ejde-508	44	17	sign	sign	VERB
ejde-508	44	18	.	.	PUNCT
ejde-508	45	1	our	our	PRON
ejde-508	45	2	results	result	NOUN
ejde-508	45	3	generalize	generalize	VERB
ejde-508	45	4	and	and	CCONJ
ejde-508	45	5	extend	extend	VERB
ejde-508	45	6	some	some	DET
ejde-508	45	7	results	result	NOUN
ejde-508	45	8	in	in	ADP
ejde-508	45	9	[	[	X
ejde-508	45	10	2	2	NUM
ejde-508	45	11	,	,	PUNCT
ejde-508	45	12	11	11	NUM
ejde-508	45	13	]	]	PUNCT
ejde-508	45	14	.	.	PUNCT
ejde-508	46	1	let	let	VERB
ejde-508	46	2	n0	n0	X
ejde-508	46	3	be	be	AUX
ejde-508	46	4	a	a	DET
ejde-508	46	5	fixed	fix	VERB
ejde-508	46	6	nonnegative	nonnegative	ADJ
ejde-508	46	7	integer	integer	NOUN
ejde-508	46	8	.	.	PUNCT
ejde-508	47	1	let	let	VERB
ejde-508	47	2	ρ	ρ	PROPN
ejde-508	47	3	=	=	SYM
ejde-508	47	4	min	min	PROPN
ejde-508	47	5	{	{	PUNCT
ejde-508	47	6	n0−s	n0−s	PROPN
ejde-508	47	7	,	,	PUNCT
ejde-508	47	8	n0−k	n0−k	PROPN
ejde-508	47	9	,	,	PUNCT
ejde-508	47	10	infn≥n0	infn≥n0	X
ejde-508	47	11	{	{	PUNCT
ejde-508	47	12	α(n	α(n	NOUN
ejde-508	47	13	)	)	PUNCT
ejde-508	47	14	}	}	PUNCT
ejde-508	47	15	}	}	PUNCT
ejde-508	47	16	.	.	PUNCT
ejde-508	48	1	by	by	ADP
ejde-508	48	2	a	a	DET
ejde-508	48	3	solution	solution	NOUN
ejde-508	48	4	of	of	ADP
ejde-508	48	5	(	(	PUNCT
ejde-508	48	6	1.2	1.2	NUM
ejde-508	48	7	)	)	PUNCT
ejde-508	48	8	we	we	PRON
ejde-508	48	9	mean	mean	VERB
ejde-508	48	10	a	a	DET
ejde-508	48	11	real	real	ADJ
ejde-508	48	12	sequence	sequence	NOUN
ejde-508	48	13	{	{	PUNCT
ejde-508	48	14	yn	yn	NOUN
ejde-508	48	15	}	}	PUNCT
ejde-508	48	16	which	which	PRON
ejde-508	48	17	is	be	AUX
ejde-508	48	18	defined	define	VERB
ejde-508	48	19	for	for	ADP
ejde-508	48	20	all	all	DET
ejde-508	48	21	integers	integer	NOUN
ejde-508	48	22	n	n	PRON
ejde-508	48	23	≥	≥	NOUN
ejde-508	48	24	ρ	ρ	NOUN
ejde-508	48	25	and	and	CCONJ
ejde-508	48	26	satisfies	satisfie	NOUN
ejde-508	48	27	(	(	PUNCT
ejde-508	48	28	1.2	1.2	NUM
ejde-508	48	29	)	)	PUNCT
ejde-508	48	30	for	for	ADP
ejde-508	48	31	n	n	PRON
ejde-508	48	32	≥	≥	NOUN
ejde-508	48	33	n0	n0	NUM
ejde-508	48	34	.	.	PUNCT
ejde-508	49	1	clearly	clearly	ADV
ejde-508	49	2	if	if	SCONJ
ejde-508	49	3	the	the	DET
ejde-508	49	4	initial	initial	ADJ
ejde-508	49	5	condition	condition	NOUN
ejde-508	49	6	yn	yn	NOUN
ejde-508	49	7	=	=	PUNCT
ejde-508	49	8	an	an	PRON
ejde-508	49	9	for	for	ADP
ejde-508	49	10	ρ	ρ	PROPN
ejde-508	49	11	≤	≤	NUM
ejde-508	49	12	n	n	PRON
ejde-508	49	13	≤	≤	PROPN
ejde-508	49	14	n0	n0	NOUN
ejde-508	49	15	+	+	CCONJ
ejde-508	49	16	1	1	NUM
ejde-508	49	17	,	,	PUNCT
ejde-508	49	18	(	(	PUNCT
ejde-508	49	19	1.12	1.12	NUM
ejde-508	49	20	)	)	PUNCT
ejde-508	49	21	is	be	AUX
ejde-508	49	22	given	give	VERB
ejde-508	49	23	then	then	ADV
ejde-508	49	24	equation	equation	NOUN
ejde-508	49	25	(	(	PUNCT
ejde-508	49	26	1.2	1.2	NUM
ejde-508	49	27	)	)	PUNCT
ejde-508	49	28	has	have	VERB
ejde-508	49	29	a	a	DET
ejde-508	49	30	unique	unique	ADJ
ejde-508	49	31	solution	solution	NOUN
ejde-508	49	32	satisfying	satisfy	VERB
ejde-508	49	33	(	(	PUNCT
ejde-508	49	34	1.12	1.12	NUM
ejde-508	49	35	)	)	PUNCT
ejde-508	49	36	.	.	PUNCT
ejde-508	50	1	a	a	DET
ejde-508	50	2	non	non	ADJ
ejde-508	50	3	trivial	trivial	ADJ
ejde-508	50	4	solution	solution	NOUN
ejde-508	50	5	{	{	PUNCT
ejde-508	50	6	yn	yn	NOUN
ejde-508	50	7	}	}	PUNCT
ejde-508	50	8	of	of	ADP
ejde-508	50	9	(	(	PUNCT
ejde-508	50	10	1.2	1.2	NUM
ejde-508	50	11	)	)	PUNCT
ejde-508	50	12	is	be	AUX
ejde-508	50	13	said	say	VERB
ejde-508	50	14	to	to	PART
ejde-508	50	15	be	be	AUX
ejde-508	50	16	oscillatory	oscillatory	ADJ
ejde-508	50	17	if	if	SCONJ
ejde-508	50	18	for	for	ADP
ejde-508	50	19	every	every	DET
ejde-508	50	20	positive	positive	ADJ
ejde-508	50	21	integer	integer	NOUN
ejde-508	50	22	n0	n0	X
ejde-508	50	23	>	>	X
ejde-508	50	24	0	0	PROPN
ejde-508	50	25	,	,	PUNCT
ejde-508	50	26	there	there	PRON
ejde-508	50	27	exists	exist	VERB
ejde-508	50	28	n	n	PRON
ejde-508	50	29	≥	≥	NUM
ejde-508	50	30	n0	n0	NUM
ejde-508	50	31	such	such	ADJ
ejde-508	50	32	that	that	PRON
ejde-508	50	33	ynyn+1	ynyn+1	PROPN
ejde-508	50	34	≤	≤	ADV
ejde-508	50	35	0	0	NUM
ejde-508	50	36	,	,	PUNCT
ejde-508	50	37	otherwise	otherwise	ADV
ejde-508	50	38	{	{	PUNCT
ejde-508	50	39	yn	yn	NOUN
ejde-508	50	40	}	}	PUNCT
ejde-508	50	41	is	be	AUX
ejde-508	50	42	said	say	VERB
ejde-508	50	43	to	to	PART
ejde-508	50	44	be	be	AUX
ejde-508	50	45	nonoscillatory	nonoscillatory	ADJ
ejde-508	50	46	.	.	PUNCT
ejde-508	51	1	2	2	X
ejde-508	51	2	.	.	X
ejde-508	51	3	some	some	PRON
ejde-508	51	4	lemmas	lemma	NOUN
ejde-508	51	5	in	in	ADP
ejde-508	51	6	this	this	DET
ejde-508	51	7	section	section	NOUN
ejde-508	51	8	,	,	PUNCT
ejde-508	51	9	we	we	PRON
ejde-508	51	10	present	present	VERB
ejde-508	51	11	some	some	DET
ejde-508	51	12	lemmas	lemma	NOUN
ejde-508	51	13	to	to	PART
ejde-508	51	14	be	be	AUX
ejde-508	51	15	applied	apply	VERB
ejde-508	51	16	in	in	ADP
ejde-508	51	17	next	next	ADJ
ejde-508	51	18	section	section	NOUN
ejde-508	51	19	.	.	PUNCT
ejde-508	52	1	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	52	2	oscillation	oscillation	NOUN
ejde-508	52	3	for	for	ADP
ejde-508	52	4	second	second	ADJ
ejde-508	52	5	order	order	NOUN
ejde-508	52	6	neutral	neutral	ADJ
ejde-508	52	7	equations	equation	NOUN
ejde-508	52	8	3	3	NUM
ejde-508	52	9	lemma	lemma	PROPN
ejde-508	52	10	2.1	2.1	NUM
ejde-508	52	11	.	.	PUNCT
ejde-508	53	1	[	[	X
ejde-508	53	2	5	5	NUM
ejde-508	53	3	,	,	PUNCT
ejde-508	53	4	theorem	theorem	VERB
ejde-508	53	5	7.6.1	7.6.1	NUM
ejde-508	53	6	,	,	PUNCT
ejde-508	53	7	page	page	NOUN
ejde-508	53	8	184	184	NUM
ejde-508	53	9	]	]	PUNCT
ejde-508	53	10	let	let	AUX
ejde-508	53	11	{	{	PUNCT
ejde-508	53	12	rn	rn	VERB
ejde-508	53	13	}	}	PUNCT
ejde-508	53	14	be	be	AUX
ejde-508	53	15	a	a	DET
ejde-508	53	16	non	non	ADJ
ejde-508	53	17	negative	negative	ADJ
ejde-508	53	18	sequence	sequence	NOUN
ejde-508	53	19	of	of	ADP
ejde-508	53	20	real	real	ADJ
ejde-508	53	21	numbers	number	NOUN
ejde-508	53	22	,	,	PUNCT
ejde-508	53	23	k	k	PROPN
ejde-508	53	24	a	a	DET
ejde-508	53	25	positive	positive	ADJ
ejde-508	53	26	integer	integer	NOUN
ejde-508	53	27	and	and	CCONJ
ejde-508	53	28	lim	lim	PROPN
ejde-508	53	29	inf	inf	PROPN
ejde-508	53	30	n→∞	n→∞	X
ejde-508	53	31	n−1∑	n−1∑	PROPN
ejde-508	53	32	i	i	PRON
ejde-508	53	33	=	=	NOUN
ejde-508	53	34	n−k	n−k	NOUN
ejde-508	53	35	ri	ri	INTJ
ejde-508	53	36	>	>	X
ejde-508	54	1	(	(	PUNCT
ejde-508	54	2	k	k	PROPN
ejde-508	54	3	k	k	PROPN
ejde-508	54	4	+	+	PROPN
ejde-508	54	5	1	1	X
ejde-508	54	6	)	)	PUNCT
ejde-508	54	7	k+1	k+1	X
ejde-508	54	8	.	.	PUNCT
ejde-508	55	1	(	(	PUNCT
ejde-508	55	2	2.1	2.1	NUM
ejde-508	55	3	)	)	PUNCT
ejde-508	55	4	then	then	ADV
ejde-508	55	5	the	the	DET
ejde-508	55	6	following	follow	VERB
ejde-508	55	7	statements	statement	NOUN
ejde-508	55	8	are	be	AUX
ejde-508	55	9	true	true	ADJ
ejde-508	55	10	.	.	PUNCT
ejde-508	56	1	(	(	PUNCT
ejde-508	56	2	a	a	X
ejde-508	56	3	)	)	PUNCT
ejde-508	56	4	∆xn+rnxn−k	∆xn+rnxn−k	NOUN
ejde-508	56	5	≤	≤	NOUN
ejde-508	56	6	0	0	NUM
ejde-508	56	7	has	have	VERB
ejde-508	56	8	no	no	DET
ejde-508	56	9	eventually	eventually	ADV
ejde-508	56	10	positive	positive	ADJ
ejde-508	56	11	solutions	solution	NOUN
ejde-508	56	12	,	,	PUNCT
ejde-508	56	13	which	which	PRON
ejde-508	56	14	implies	imply	VERB
ejde-508	56	15	∆xn+	∆xn+	PROPN
ejde-508	56	16	rnxn−k	rnxn−k	VERB
ejde-508	56	17	≥	≥	NOUN
ejde-508	56	18	0	0	NUM
ejde-508	56	19	has	have	VERB
ejde-508	56	20	no	no	DET
ejde-508	56	21	eventually	eventually	ADV
ejde-508	56	22	negative	negative	ADJ
ejde-508	56	23	solutions	solution	NOUN
ejde-508	56	24	.	.	PUNCT
ejde-508	57	1	(	(	PUNCT
ejde-508	57	2	b	b	X
ejde-508	57	3	)	)	PUNCT
ejde-508	57	4	∆xn−rnxn+k	∆xn−rnxn+k	PROPN
ejde-508	57	5	≥	≥	X
ejde-508	57	6	0	0	PUNCT
ejde-508	57	7	has	have	VERB
ejde-508	57	8	no	no	DET
ejde-508	57	9	eventually	eventually	ADV
ejde-508	57	10	positive	positive	ADJ
ejde-508	57	11	solutions	solution	NOUN
ejde-508	57	12	,	,	PUNCT
ejde-508	57	13	which	which	PRON
ejde-508	57	14	implies	imply	VERB
ejde-508	57	15	∆xn−	∆xn−	PROPN
ejde-508	57	16	rnxn+k	rnxn+k	ADV
ejde-508	57	17	≤	≤	NUM
ejde-508	57	18	0	0	PUNCT
ejde-508	57	19	has	have	VERB
ejde-508	57	20	no	no	DET
ejde-508	57	21	eventually	eventually	ADV
ejde-508	57	22	negative	negative	ADJ
ejde-508	57	23	solutions	solution	NOUN
ejde-508	57	24	.	.	PUNCT
ejde-508	58	1	lemma	lemma	PROPN
ejde-508	58	2	2.2	2.2	NUM
ejde-508	58	3	.	.	PUNCT
ejde-508	58	4	suppose	suppose	VERB
ejde-508	58	5	that	that	SCONJ
ejde-508	58	6	(	(	PUNCT
ejde-508	58	7	a6	a6	NOUN
ejde-508	58	8	)	)	PUNCT
ejde-508	58	9	and	and	CCONJ
ejde-508	58	10	(	(	PUNCT
ejde-508	58	11	a7	a7	PROPN
ejde-508	58	12	)	)	PUNCT
ejde-508	58	13	hold	hold	NOUN
ejde-508	58	14	,	,	PUNCT
ejde-508	58	15	and	and	CCONJ
ejde-508	58	16	yn	yn	PROPN
ejde-508	58	17	is	be	AUX
ejde-508	58	18	an	an	DET
ejde-508	58	19	eventually	eventually	ADV
ejde-508	58	20	positive	positive	ADJ
ejde-508	58	21	solution	solution	NOUN
ejde-508	58	22	of	of	ADP
ejde-508	58	23	(	(	PUNCT
ejde-508	58	24	1.2	1.2	NUM
ejde-508	58	25	)	)	PUNCT
ejde-508	58	26	.	.	PUNCT
ejde-508	59	1	then	then	ADV
ejde-508	59	2	the	the	DET
ejde-508	59	3	sequence	sequence	NOUN
ejde-508	59	4	cn	cn	NOUN
ejde-508	59	5	=	=	PUNCT
ejde-508	59	6	−	−	PROPN
ejde-508	59	7	∞∑	∞∑	NUM
ejde-508	59	8	i	i	PRON
ejde-508	59	9	=	=	NOUN
ejde-508	59	10	n	n	PRON
ejde-508	59	11	(	(	PUNCT
ejde-508	59	12	i−	i−	PROPN
ejde-508	59	13	n+	n+	PUNCT
ejde-508	59	14	1)uih(yα(i	1)uih(yα(i	NUM
ejde-508	59	15	)	)	PUNCT
ejde-508	59	16	)	)	PUNCT
ejde-508	59	17	(	(	PUNCT
ejde-508	59	18	2.2	2.2	NUM
ejde-508	59	19	)	)	PUNCT
ejde-508	59	20	satisfies	satisfie	NOUN
ejde-508	59	21	lim	lim	PROPN
ejde-508	59	22	n→∞	n→∞	X
ejde-508	59	23	cn	cn	PROPN
ejde-508	59	24	=	=	SYM
ejde-508	59	25	0	0	PROPN
ejde-508	59	26	,	,	PUNCT
ejde-508	59	27	cn	cn	PROPN
ejde-508	59	28	≤	≤	ADV
ejde-508	59	29	0	0	NUM
ejde-508	59	30	,	,	PUNCT
ejde-508	59	31	∆cn	∆cn	VERB
ejde-508	59	32	≥	≥	NOUN
ejde-508	59	33	0	0	NUM
ejde-508	59	34	,	,	PUNCT
ejde-508	59	35	(	(	PUNCT
ejde-508	59	36	2.3	2.3	NUM
ejde-508	59	37	)	)	PUNCT
ejde-508	59	38	for	for	ADP
ejde-508	59	39	n	n	CCONJ
ejde-508	59	40	large	large	ADJ
ejde-508	59	41	enough	enough	ADV
ejde-508	59	42	,	,	PUNCT
ejde-508	59	43	and	and	CCONJ
ejde-508	59	44	∆2cn	∆2cn	ADJ
ejde-508	59	45	=	=	SYM
ejde-508	59	46	−unh(yα(n	−unh(yα(n	NOUN
ejde-508	59	47	)	)	PUNCT
ejde-508	59	48	)	)	PUNCT
ejde-508	59	49	.	.	PUNCT
ejde-508	60	1	(	(	PUNCT
ejde-508	60	2	2.4	2.4	NUM
ejde-508	60	3	)	)	PUNCT
ejde-508	60	4	proof	proof	NOUN
ejde-508	60	5	.	.	PUNCT
ejde-508	61	1	clearly	clearly	ADV
ejde-508	61	2	,	,	PUNCT
ejde-508	61	3	applying	apply	VERB
ejde-508	61	4	∆2	∆2	PROPN
ejde-508	61	5	to	to	ADP
ejde-508	61	6	(	(	PUNCT
ejde-508	61	7	2.2	2.2	NUM
ejde-508	61	8	)	)	PUNCT
ejde-508	61	9	,	,	PUNCT
ejde-508	61	10	we	we	PRON
ejde-508	61	11	obtain	obtain	VERB
ejde-508	61	12	∆2cn	∆2cn	ADJ
ejde-508	61	13	=	=	ADJ
ejde-508	61	14	−unh(yα(n	−unh(yα(n	NOUN
ejde-508	61	15	)	)	PUNCT
ejde-508	61	16	)	)	PUNCT
ejde-508	61	17	.	.	PUNCT
ejde-508	62	1	by	by	ADP
ejde-508	62	2	(	(	PUNCT
ejde-508	62	3	a6	a6	NOUN
ejde-508	62	4	)	)	PUNCT
ejde-508	62	5	and	and	CCONJ
ejde-508	62	6	(	(	PUNCT
ejde-508	62	7	a7	a7	PROPN
ejde-508	62	8	)	)	PUNCT
ejde-508	62	9	,	,	PUNCT
ejde-508	62	10	∑∞	∑∞	NOUN
ejde-508	62	11	i	i	PROPN
ejde-508	62	12	=	=	NOUN
ejde-508	62	13	n	n	X
ejde-508	62	14	iuih(yα(i	iuih(yα(i	PUNCT
ejde-508	62	15	)	)	PUNCT
ejde-508	62	16	)	)	PUNCT
ejde-508	63	1	<	<	X
ejde-508	63	2	∞.	∞.	PROPN
ejde-508	63	3	comparing	compare	VERB
ejde-508	63	4	this	this	DET
ejde-508	63	5	infinite	infinite	ADJ
ejde-508	63	6	series	series	NOUN
ejde-508	63	7	with	with	ADP
ejde-508	63	8	(	(	PUNCT
ejde-508	63	9	2.2	2.2	NUM
ejde-508	63	10	)	)	PUNCT
ejde-508	63	11	,	,	PUNCT
ejde-508	63	12	we	we	PRON
ejde-508	63	13	show	show	VERB
ejde-508	63	14	that	that	SCONJ
ejde-508	63	15	{	{	PUNCT
ejde-508	63	16	cn	cn	ADJ
ejde-508	63	17	}	}	PUNCT
ejde-508	63	18	converges	converge	VERB
ejde-508	63	19	absolutely	absolutely	ADV
ejde-508	63	20	to	to	ADP
ejde-508	63	21	zero	zero	NUM
ejde-508	63	22	.	.	PUNCT
ejde-508	64	1	the	the	DET
ejde-508	64	2	other	other	ADJ
ejde-508	64	3	statements	statement	NOUN
ejde-508	64	4	follow	follow	VERB
ejde-508	64	5	easily	easily	ADV
ejde-508	64	6	.	.	PUNCT
ejde-508	65	1	�	�	PROPN
ejde-508	65	2	note	note	VERB
ejde-508	65	3	that	that	SCONJ
ejde-508	65	4	if	if	SCONJ
ejde-508	65	5	yn	yn	PRON
ejde-508	65	6	is	be	AUX
ejde-508	65	7	eventually	eventually	ADV
ejde-508	65	8	negative	negative	ADJ
ejde-508	65	9	,	,	PUNCT
ejde-508	65	10	then	then	ADV
ejde-508	65	11	cn	cn	PROPN
ejde-508	65	12	≥	≥	PROPN
ejde-508	65	13	0	0	NUM
ejde-508	65	14	and	and	CCONJ
ejde-508	65	15	∆cn	∆cn	VERB
ejde-508	65	16	≤	≤	NOUN
ejde-508	65	17	0	0	X
ejde-508	65	18	.	.	PUNCT
ejde-508	66	1	next	next	ADJ
ejde-508	66	2	,	,	PUNCT
ejde-508	66	3	we	we	PRON
ejde-508	66	4	prove	prove	VERB
ejde-508	66	5	an	an	DET
ejde-508	66	6	important	important	ADJ
ejde-508	66	7	lemma	lemma	PROPN
ejde-508	66	8	to	to	PART
ejde-508	66	9	be	be	AUX
ejde-508	66	10	used	use	VERB
ejde-508	66	11	later	later	ADV
ejde-508	66	12	.	.	PUNCT
ejde-508	67	1	lemma	lemma	PROPN
ejde-508	67	2	2.3	2.3	NUM
ejde-508	67	3	.	.	PUNCT
ejde-508	68	1	let	let	VERB
ejde-508	68	2	(	(	PUNCT
ejde-508	68	3	a1	a1	NOUN
ejde-508	68	4	)	)	PUNCT
ejde-508	68	5	,	,	PUNCT
ejde-508	68	6	(	(	PUNCT
ejde-508	68	7	a6	a6	NOUN
ejde-508	68	8	)	)	PUNCT
ejde-508	68	9	,	,	PUNCT
ejde-508	68	10	(	(	PUNCT
ejde-508	68	11	a7	a7	ADJ
ejde-508	68	12	)	)	PUNCT
ejde-508	68	13	hold	hold	NOUN
ejde-508	68	14	,	,	PUNCT
ejde-508	68	15	yn	yn	PRON
ejde-508	68	16	be	be	VERB
ejde-508	68	17	an	an	DET
ejde-508	68	18	eventually	eventually	ADV
ejde-508	68	19	positive	positive	ADJ
ejde-508	68	20	solution	solution	NOUN
ejde-508	68	21	of	of	ADP
ejde-508	68	22	(	(	PUNCT
ejde-508	68	23	1.2	1.2	NUM
ejde-508	68	24	)	)	PUNCT
ejde-508	68	25	,	,	PUNCT
ejde-508	68	26	and	and	CCONJ
ejde-508	68	27	cn	cn	PROPN
ejde-508	68	28	be	be	AUX
ejde-508	68	29	defined	define	VERB
ejde-508	68	30	by	by	ADP
ejde-508	68	31	(	(	PUNCT
ejde-508	68	32	2.2	2.2	NUM
ejde-508	68	33	)	)	PUNCT
ejde-508	68	34	.	.	PUNCT
ejde-508	69	1	then	then	ADV
ejde-508	69	2	for	for	ADP
ejde-508	69	3	the	the	DET
ejde-508	69	4	sequences	sequence	NOUN
ejde-508	69	5	zn	zn	PROPN
ejde-508	69	6	=	=	SYM
ejde-508	69	7	yn	yn	PROPN
ejde-508	69	8	−	−	PROPN
ejde-508	69	9	pnl(yn−s	pnl(yn−s	NOUN
ejde-508	69	10	)	)	PUNCT
ejde-508	69	11	,	,	PUNCT
ejde-508	69	12	(	(	PUNCT
ejde-508	69	13	2.5	2.5	NUM
ejde-508	69	14	)	)	PUNCT
ejde-508	69	15	wn	wn	NOUN
ejde-508	69	16	=	=	SYM
ejde-508	69	17	zn	zn	PROPN
ejde-508	69	18	+	+	CCONJ
ejde-508	69	19	cn	cn	PROPN
ejde-508	69	20	(	(	PUNCT
ejde-508	69	21	2.6	2.6	NUM
ejde-508	69	22	)	)	PUNCT
ejde-508	69	23	we	we	PRON
ejde-508	69	24	have	have	VERB
ejde-508	69	25	the	the	DET
ejde-508	69	26	following	following	ADJ
ejde-508	69	27	statements	statement	NOUN
ejde-508	69	28	:	:	PUNCT
ejde-508	69	29	(	(	PUNCT
ejde-508	69	30	a	a	X
ejde-508	69	31	)	)	PUNCT
ejde-508	69	32	if	if	SCONJ
ejde-508	69	33	(	(	PUNCT
ejde-508	69	34	a2	a2	PROPN
ejde-508	69	35	)	)	PUNCT
ejde-508	69	36	and	and	CCONJ
ejde-508	69	37	(	(	PUNCT
ejde-508	69	38	a5	a5	PROPN
ejde-508	69	39	)	)	PUNCT
ejde-508	69	40	hold	hold	VERB
ejde-508	69	41	and	and	CCONJ
ejde-508	69	42	pn	pn	NOUN
ejde-508	69	43	satisfy	satisfy	VERB
ejde-508	69	44	(	(	PUNCT
ejde-508	69	45	1.4	1.4	NUM
ejde-508	69	46	)	)	PUNCT
ejde-508	69	47	,	,	PUNCT
ejde-508	69	48	then	then	ADV
ejde-508	69	49	either	either	CCONJ
ejde-508	69	50	∆wn	∆wn	PROPN
ejde-508	69	51	<	<	X
ejde-508	69	52	0	0	PROPN
ejde-508	69	53	for	for	ADP
ejde-508	69	54	large	large	ADJ
ejde-508	69	55	n	n	NOUN
ejde-508	69	56	which	which	PRON
ejde-508	69	57	implies	imply	VERB
ejde-508	69	58	lim	lim	PROPN
ejde-508	69	59	n→∞	n→∞	NUM
ejde-508	69	60	wn	wn	PROPN
ejde-508	69	61	=	=	PROPN
ejde-508	69	62	−∞	−∞	PROPN
ejde-508	69	63	,	,	PUNCT
ejde-508	69	64	(	(	PUNCT
ejde-508	69	65	2.7	2.7	NUM
ejde-508	69	66	)	)	PUNCT
ejde-508	69	67	or	or	CCONJ
ejde-508	69	68	∆wn	∆wn	X
ejde-508	69	69	>	>	X
ejde-508	69	70	0	0	PUNCT
ejde-508	70	1	for	for	ADP
ejde-508	70	2	large	large	ADJ
ejde-508	70	3	n	n	NOUN
ejde-508	70	4	which	which	PRON
ejde-508	70	5	implies	imply	VERB
ejde-508	70	6	lim	lim	PROPN
ejde-508	70	7	n→∞	n→∞	NUM
ejde-508	70	8	wn	wn	PROPN
ejde-508	70	9	=	=	PROPN
ejde-508	70	10	0	0	PROPN
ejde-508	70	11	,	,	PUNCT
ejde-508	70	12	(	(	PUNCT
ejde-508	70	13	2.8	2.8	NUM
ejde-508	70	14	)	)	PUNCT
ejde-508	70	15	wn	wn	NOUN
ejde-508	70	16	<	<	X
ejde-508	70	17	0	0	PROPN
ejde-508	70	18	,	,	PUNCT
ejde-508	70	19	lim	lim	PROPN
ejde-508	70	20	n→∞	n→∞	X
ejde-508	70	21	∆wn	∆wn	X
ejde-508	70	22	=	=	SYM
ejde-508	71	1	0	0	X
ejde-508	71	2	.	.	PUNCT
ejde-508	72	1	(	(	PUNCT
ejde-508	72	2	2.9	2.9	NUM
ejde-508	72	3	)	)	PUNCT
ejde-508	72	4	(	(	PUNCT
ejde-508	72	5	b	b	X
ejde-508	72	6	)	)	PUNCT
ejde-508	72	7	if	if	SCONJ
ejde-508	72	8	in	in	ADP
ejde-508	72	9	addition	addition	NOUN
ejde-508	72	10	pδ	pδ	ADP
ejde-508	72	11	≤	≤	NUM
ejde-508	72	12	1	1	NUM
ejde-508	72	13	,	,	PUNCT
ejde-508	72	14	then	then	ADV
ejde-508	72	15	only	only	ADV
ejde-508	72	16	(	(	PUNCT
ejde-508	72	17	2.8	2.8	NUM
ejde-508	72	18	)	)	PUNCT
ejde-508	72	19	and	and	CCONJ
ejde-508	72	20	(	(	PUNCT
ejde-508	72	21	2.9	2.9	NUM
ejde-508	72	22	)	)	PUNCT
ejde-508	72	23	hold	hold	VERB
ejde-508	72	24	.	.	PUNCT
ejde-508	73	1	proof	proof	NOUN
ejde-508	73	2	.	.	PUNCT
ejde-508	74	1	suppose	suppose	VERB
ejde-508	74	2	that	that	SCONJ
ejde-508	74	3	yn	yn	PROPN
ejde-508	74	4	is	be	AUX
ejde-508	74	5	an	an	DET
ejde-508	74	6	eventually	eventually	ADV
ejde-508	74	7	positive	positive	ADJ
ejde-508	74	8	solution	solution	NOUN
ejde-508	74	9	of	of	ADP
ejde-508	74	10	(	(	PUNCT
ejde-508	74	11	1.2	1.2	NUM
ejde-508	74	12	)	)	PUNCT
ejde-508	74	13	.	.	PUNCT
ejde-508	75	1	then	then	ADV
ejde-508	75	2	there	there	PRON
ejde-508	75	3	exits	exit	VERB
ejde-508	75	4	an	an	DET
ejde-508	75	5	integer	integer	NOUN
ejde-508	75	6	n1	n1	PROPN
ejde-508	75	7	≥	≥	NOUN
ejde-508	75	8	n0	n0	NUM
ejde-508	75	9	such	such	ADJ
ejde-508	75	10	that	that	SCONJ
ejde-508	75	11	yn	yn	PROPN
ejde-508	75	12	>	>	X
ejde-508	75	13	0	0	PROPN
ejde-508	75	14	,	,	PUNCT
ejde-508	75	15	yn−s	yn−s	NOUN
ejde-508	75	16	>	>	X
ejde-508	75	17	0	0	NUM
ejde-508	75	18	,	,	PUNCT
ejde-508	75	19	yn−k	yn−k	NOUN
ejde-508	75	20	and	and	CCONJ
ejde-508	75	21	yα(n	yα(n	NOUN
ejde-508	75	22	)	)	PUNCT
ejde-508	75	23	>	>	X
ejde-508	75	24	0	0	PUNCT
ejde-508	75	25	for	for	ADP
ejde-508	75	26	n	n	PRON
ejde-508	75	27	≥	≥	NOUN
ejde-508	75	28	n1	n1	PROPN
ejde-508	75	29	.	.	PUNCT
ejde-508	76	1	then	then	ADV
ejde-508	76	2	setting	set	VERB
ejde-508	76	3	cn	cn	PROPN
ejde-508	76	4	,	,	PUNCT
ejde-508	76	5	zn	zn	PROPN
ejde-508	76	6	and	and	CCONJ
ejde-508	76	7	wn	wn	PROPN
ejde-508	76	8	as	as	ADP
ejde-508	76	9	in	in	ADP
ejde-508	76	10	(	(	PUNCT
ejde-508	76	11	2.2	2.2	NUM
ejde-508	76	12	)	)	PUNCT
ejde-508	76	13	,	,	PUNCT
ejde-508	76	14	(	(	PUNCT
ejde-508	76	15	2.5	2.5	NUM
ejde-508	76	16	)	)	PUNCT
ejde-508	76	17	,	,	PUNCT
ejde-508	76	18	(	(	PUNCT
ejde-508	76	19	2.6	2.6	NUM
ejde-508	76	20	)	)	PUNCT
ejde-508	76	21	,	,	PUNCT
ejde-508	76	22	and	and	CCONJ
ejde-508	76	23	using	use	VERB
ejde-508	76	24	(	(	PUNCT
ejde-508	76	25	1.2	1.2	NUM
ejde-508	76	26	)	)	PUNCT
ejde-508	76	27	,	,	PUNCT
ejde-508	76	28	(	(	PUNCT
ejde-508	76	29	2.5	2.5	NUM
ejde-508	76	30	)	)	PUNCT
ejde-508	76	31	,	,	PUNCT
ejde-508	76	32	(	(	PUNCT
ejde-508	76	33	2.6	2.6	NUM
ejde-508	76	34	)	)	PUNCT
ejde-508	76	35	,	,	PUNCT
ejde-508	76	36	and	and	CCONJ
ejde-508	76	37	lemma	lemma	PROPN
ejde-508	76	38	2.2	2.2	NUM
ejde-508	76	39	,	,	PUNCT
ejde-508	76	40	we	we	PRON
ejde-508	76	41	obtain	obtain	VERB
ejde-508	76	42	∆2wn	∆2wn	ADJ
ejde-508	76	43	=	=	SYM
ejde-508	76	44	−qng(yn−k	−qng(yn−k	PROPN
ejde-508	76	45	)	)	PUNCT
ejde-508	76	46	≤	≤	NOUN
ejde-508	76	47	0	0	NUM
ejde-508	76	48	for	for	ADP
ejde-508	76	49	n	n	PROPN
ejde-508	76	50	>	>	X
ejde-508	76	51	n1	n1	PROPN
ejde-508	76	52	.	.	PUNCT
ejde-508	77	1	(	(	PUNCT
ejde-508	77	2	2.10	2.10	NUM
ejde-508	77	3	)	)	PUNCT
ejde-508	77	4	4	4	NUM
ejde-508	77	5	a.	a.	NOUN
ejde-508	77	6	k.	k.	PROPN
ejde-508	77	7	bhuyan	bhuyan	PROPN
ejde-508	77	8	,	,	PUNCT
ejde-508	77	9	l.	l.	PROPN
ejde-508	77	10	n.	n.	PROPN
ejde-508	77	11	padhy	padhy	PROPN
ejde-508	77	12	,	,	PUNCT
ejde-508	77	13	r.	r.	PROPN
ejde-508	77	14	n.	n.	PROPN
ejde-508	77	15	rath	rath	PROPN
ejde-508	77	16	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	77	17	then	then	ADV
ejde-508	77	18	∆wn	∆wn	PROPN
ejde-508	77	19	is	be	AUX
ejde-508	77	20	decreasing	decrease	VERB
ejde-508	77	21	.	.	PUNCT
ejde-508	78	1	hence	hence	ADV
ejde-508	78	2	∆wn	∆wn	PROPN
ejde-508	78	3	is	be	AUX
ejde-508	78	4	monotonic	monotonic	ADJ
ejde-508	78	5	and	and	CCONJ
ejde-508	78	6	of	of	ADP
ejde-508	78	7	single	single	ADJ
ejde-508	78	8	sign	sign	NOUN
ejde-508	78	9	for	for	ADP
ejde-508	78	10	n	n	X
ejde-508	78	11	large	large	ADJ
ejde-508	78	12	enough	enough	ADV
ejde-508	78	13	.	.	PUNCT
ejde-508	79	1	it	it	PRON
ejde-508	79	2	follows	follow	VERB
ejde-508	79	3	that	that	SCONJ
ejde-508	80	1	either	either	CCONJ
ejde-508	80	2	∆wn	∆wn	PROPN
ejde-508	80	3	<	<	X
ejde-508	80	4	0	0	PROPN
ejde-508	80	5	or	or	CCONJ
ejde-508	80	6	∆wn	∆wn	X
ejde-508	80	7	>	>	X
ejde-508	80	8	0	0	X
ejde-508	80	9	.	.	PUNCT
ejde-508	81	1	if	if	SCONJ
ejde-508	81	2	∆wn	∆wn	PROPN
ejde-508	81	3	<	<	X
ejde-508	81	4	0	0	PROPN
ejde-508	81	5	,	,	PUNCT
ejde-508	81	6	then	then	ADV
ejde-508	81	7	wn	wn	PROPN
ejde-508	81	8	is	be	AUX
ejde-508	81	9	decreasing	decrease	VERB
ejde-508	81	10	,	,	PUNCT
ejde-508	81	11	and	and	CCONJ
ejde-508	81	12	using	use	VERB
ejde-508	81	13	that	that	SCONJ
ejde-508	81	14	∆wn	∆wn	PROPN
ejde-508	81	15	is	be	AUX
ejde-508	81	16	decreasing	decrease	VERB
ejde-508	81	17	,	,	PUNCT
ejde-508	81	18	we	we	PRON
ejde-508	81	19	have	have	VERB
ejde-508	81	20	lim	lim	PROPN
ejde-508	81	21	n→∞	n→∞	PRON
ejde-508	81	22	∆wn	∆wn	X
ejde-508	81	23	=	=	SYM
ejde-508	81	24	−∞.	−∞.	PROPN
ejde-508	81	25	(	(	PUNCT
ejde-508	81	26	2.11	2.11	NUM
ejde-508	81	27	)	)	PUNCT
ejde-508	81	28	if	if	SCONJ
ejde-508	81	29	∆wn	∆wn	PROPN
ejde-508	81	30	>	>	X
ejde-508	81	31	0	0	PROPN
ejde-508	81	32	,	,	PUNCT
ejde-508	81	33	then	then	ADV
ejde-508	81	34	wn	wn	PROPN
ejde-508	81	35	is	be	AUX
ejde-508	81	36	increasing	increase	VERB
ejde-508	81	37	,	,	PUNCT
ejde-508	81	38	and	and	CCONJ
ejde-508	81	39	using	use	VERB
ejde-508	81	40	that	that	SCONJ
ejde-508	81	41	∆wn	∆wn	PROPN
ejde-508	81	42	is	be	AUX
ejde-508	81	43	decreasing	decrease	VERB
ejde-508	81	44	,	,	PUNCT
ejde-508	81	45	we	we	PRON
ejde-508	81	46	have	have	VERB
ejde-508	81	47	lim	lim	PROPN
ejde-508	81	48	n→∞	n→∞	PRON
ejde-508	82	1	∆wn	∆wn	X
ejde-508	82	2	=	=	SYM
ejde-508	82	3	ζ	ζ	PROPN
ejde-508	82	4	(	(	PUNCT
ejde-508	82	5	a	a	DET
ejde-508	82	6	finite	finite	ADJ
ejde-508	82	7	number	number	NOUN
ejde-508	82	8	)	)	PUNCT
ejde-508	82	9	.	.	PUNCT
ejde-508	83	1	(	(	PUNCT
ejde-508	83	2	2.12	2.12	NUM
ejde-508	83	3	)	)	PUNCT
ejde-508	83	4	let	let	VERB
ejde-508	83	5	us	we	PRON
ejde-508	83	6	prove	prove	VERB
ejde-508	83	7	part	part	NOUN
ejde-508	83	8	(	(	PUNCT
ejde-508	83	9	a	a	NOUN
ejde-508	83	10	)	)	PUNCT
ejde-508	83	11	.	.	PUNCT
ejde-508	84	1	if	if	SCONJ
ejde-508	84	2	(	(	PUNCT
ejde-508	84	3	2.11	2.11	NUM
ejde-508	84	4	)	)	PUNCT
ejde-508	84	5	holds	hold	VERB
ejde-508	84	6	then	then	ADV
ejde-508	84	7	clearly	clearly	ADV
ejde-508	84	8	(	(	PUNCT
ejde-508	84	9	2.7	2.7	NUM
ejde-508	84	10	)	)	PUNCT
ejde-508	84	11	follows	follow	VERB
ejde-508	84	12	.	.	PUNCT
ejde-508	85	1	if	if	SCONJ
ejde-508	85	2	(	(	PUNCT
ejde-508	85	3	2.12	2.12	NUM
ejde-508	85	4	)	)	PUNCT
ejde-508	85	5	holds	hold	VERB
ejde-508	85	6	then	then	ADV
ejde-508	85	7	,	,	PUNCT
ejde-508	85	8	summing	sum	VERB
ejde-508	85	9	(	(	PUNCT
ejde-508	85	10	2.10	2.10	NUM
ejde-508	85	11	)	)	PUNCT
ejde-508	85	12	from	from	ADP
ejde-508	85	13	n2	n2	PROPN
ejde-508	85	14	>	>	X
ejde-508	85	15	n1	n1	PROPN
ejde-508	85	16	to	to	ADP
ejde-508	85	17	∞	∞	PROPN
ejde-508	85	18	we	we	PRON
ejde-508	85	19	obtain	obtain	VERB
ejde-508	85	20	∞∑	∞∑	NUM
ejde-508	85	21	n	n	CCONJ
ejde-508	85	22	=	=	ADJ
ejde-508	85	23	n2	n2	PROPN
ejde-508	85	24	qng(yn−k	qng(yn−k	PROPN
ejde-508	85	25	)	)	PUNCT
ejde-508	85	26	<	<	X
ejde-508	85	27	∞	∞	PROPN
ejde-508	85	28	,	,	PUNCT
ejde-508	85	29	(	(	PUNCT
ejde-508	85	30	2.13	2.13	NUM
ejde-508	85	31	)	)	PUNCT
ejde-508	85	32	which	which	PRON
ejde-508	85	33	by	by	ADP
ejde-508	85	34	using	use	VERB
ejde-508	85	35	(	(	PUNCT
ejde-508	85	36	a2	a2	NOUN
ejde-508	85	37	)	)	PUNCT
ejde-508	85	38	yields	yield	NOUN
ejde-508	85	39	lim	lim	PROPN
ejde-508	85	40	inf	inf	PROPN
ejde-508	85	41	n→∞	n→∞	X
ejde-508	85	42	yn	yn	X
ejde-508	85	43	=	=	NOUN
ejde-508	85	44	0	0	PROPN
ejde-508	85	45	.	.	PUNCT
ejde-508	86	1	(	(	PUNCT
ejde-508	86	2	2.14	2.14	NUM
ejde-508	86	3	)	)	PUNCT
ejde-508	86	4	then	then	ADV
ejde-508	86	5	we	we	PRON
ejde-508	86	6	find	find	VERB
ejde-508	86	7	a	a	DET
ejde-508	86	8	subsequence	subsequence	NOUN
ejde-508	86	9	{	{	PUNCT
ejde-508	86	10	ynk	ynk	NOUN
ejde-508	86	11	}	}	PUNCT
ejde-508	86	12	such	such	ADJ
ejde-508	86	13	that	that	DET
ejde-508	86	14	ynk	ynk	NOUN
ejde-508	86	15	→	→	SYM
ejde-508	86	16	0	0	PUNCT
ejde-508	86	17	as	as	SCONJ
ejde-508	86	18	k	k	PROPN
ejde-508	86	19	→∞.	→∞.	X
ejde-508	86	20	now	now	ADV
ejde-508	86	21	using	use	VERB
ejde-508	86	22	(	(	PUNCT
ejde-508	86	23	1.4	1.4	NUM
ejde-508	86	24	)	)	PUNCT
ejde-508	86	25	,	,	PUNCT
ejde-508	86	26	(	(	PUNCT
ejde-508	86	27	a1	a1	NOUN
ejde-508	86	28	)	)	PUNCT
ejde-508	86	29	and	and	CCONJ
ejde-508	86	30	lemma	lemma	PROPN
ejde-508	86	31	2.2	2.2	NUM
ejde-508	86	32	we	we	PRON
ejde-508	86	33	obtain	obtain	VERB
ejde-508	86	34	wnk	wnk	PROPN
ejde-508	86	35	<	<	X
ejde-508	86	36	ynk	ynk	PROPN
ejde-508	86	37	+	+	CCONJ
ejde-508	86	38	cnk	cnk	PROPN
ejde-508	86	39	→	→	SYM
ejde-508	86	40	0	0	PUNCT
ejde-508	86	41	as	as	ADP
ejde-508	86	42	k	k	PROPN
ejde-508	86	43	→∞	→∞	PROPN
ejde-508	86	44	(	(	PUNCT
ejde-508	86	45	2.15	2.15	NUM
ejde-508	86	46	)	)	PUNCT
ejde-508	86	47	and	and	CCONJ
ejde-508	86	48	wnk+s	wnk+s	NUM
ejde-508	86	49	>	>	PUNCT
ejde-508	86	50	−pδynk	−pδynk	NOUN
ejde-508	87	1	+	+	PUNCT
ejde-508	87	2	cnk+s	cnk+s	PUNCT
ejde-508	87	3	→	→	SYM
ejde-508	87	4	0	0	PUNCT
ejde-508	87	5	as	as	ADP
ejde-508	87	6	k	k	PROPN
ejde-508	87	7	→∞.	→∞.	PROPN
ejde-508	87	8	(	(	PUNCT
ejde-508	87	9	2.16	2.16	NUM
ejde-508	87	10	)	)	PUNCT
ejde-508	87	11	since	since	SCONJ
ejde-508	87	12	wn	wn	PROPN
ejde-508	87	13	is	be	AUX
ejde-508	87	14	monotonic	monotonic	ADJ
ejde-508	87	15	,	,	PUNCT
ejde-508	87	16	it	it	PRON
ejde-508	87	17	follows	follow	VERB
ejde-508	87	18	that	that	SCONJ
ejde-508	87	19	limn→∞	limn→∞	PROPN
ejde-508	87	20	wn	wn	NOUN
ejde-508	87	21	=	=	SYM
ejde-508	87	22	0	0	PROPN
ejde-508	87	23	,	,	PUNCT
ejde-508	87	24	which	which	PRON
ejde-508	87	25	is	be	AUX
ejde-508	87	26	(	(	PUNCT
ejde-508	87	27	2.8	2.8	NUM
ejde-508	87	28	)	)	PUNCT
ejde-508	87	29	.	.	PUNCT
ejde-508	88	1	then	then	ADV
ejde-508	88	2	(	(	PUNCT
ejde-508	88	3	2.9	2.9	NUM
ejde-508	88	4	)	)	PUNCT
ejde-508	88	5	follows	follow	VERB
ejde-508	88	6	from	from	ADP
ejde-508	88	7	(	(	PUNCT
ejde-508	88	8	2.8	2.8	NUM
ejde-508	88	9	)	)	PUNCT
ejde-508	88	10	.	.	PUNCT
ejde-508	89	1	the	the	DET
ejde-508	89	2	proof	proof	NOUN
ejde-508	89	3	of	of	ADP
ejde-508	89	4	part	part	NOUN
ejde-508	89	5	(	(	PUNCT
ejde-508	89	6	a	a	NOUN
ejde-508	89	7	)	)	PUNCT
ejde-508	89	8	is	be	AUX
ejde-508	89	9	complete	complete	ADJ
ejde-508	89	10	.	.	PUNCT
ejde-508	90	1	to	to	PART
ejde-508	90	2	prove	prove	VERB
ejde-508	90	3	part	part	NOUN
ejde-508	90	4	(	(	PUNCT
ejde-508	90	5	b	b	NOUN
ejde-508	90	6	)	)	PUNCT
ejde-508	90	7	of	of	ADP
ejde-508	90	8	the	the	DET
ejde-508	90	9	lemma	lemma	PROPN
ejde-508	90	10	,	,	PUNCT
ejde-508	90	11	we	we	PRON
ejde-508	90	12	show	show	VERB
ejde-508	90	13	that	that	SCONJ
ejde-508	90	14	(	(	PUNCT
ejde-508	90	15	2.7	2.7	NUM
ejde-508	90	16	)	)	PUNCT
ejde-508	90	17	can	can	AUX
ejde-508	90	18	not	not	PART
ejde-508	90	19	happen	happen	VERB
ejde-508	90	20	;	;	PUNCT
ejde-508	90	21	therefore	therefore	ADV
ejde-508	90	22	(	(	PUNCT
ejde-508	90	23	2.8	2.8	NUM
ejde-508	90	24	)	)	PUNCT
ejde-508	90	25	and	and	CCONJ
ejde-508	90	26	(	(	PUNCT
ejde-508	90	27	2.9	2.9	NUM
ejde-508	90	28	)	)	PUNCT
ejde-508	90	29	must	must	AUX
ejde-508	90	30	occur	occur	VERB
ejde-508	90	31	.	.	PUNCT
ejde-508	91	1	to	to	PART
ejde-508	91	2	obtain	obtain	VERB
ejde-508	91	3	a	a	DET
ejde-508	91	4	contradiction	contradiction	NOUN
ejde-508	91	5	,	,	PUNCT
ejde-508	91	6	let	let	VERB
ejde-508	91	7	us	we	PRON
ejde-508	91	8	assume	assume	VERB
ejde-508	91	9	that	that	SCONJ
ejde-508	91	10	limn→∞	limn→∞	PROPN
ejde-508	91	11	wn	wn	PROPN
ejde-508	91	12	=	=	PUNCT
ejde-508	91	13	−∞.	−∞.	PROPN
ejde-508	91	14	note	note	VERB
ejde-508	91	15	that	that	SCONJ
ejde-508	91	16	from	from	ADP
ejde-508	91	17	(	(	PUNCT
ejde-508	91	18	2.6	2.6	NUM
ejde-508	91	19	)	)	PUNCT
ejde-508	91	20	and	and	CCONJ
ejde-508	91	21	lemma	lemma	PROPN
ejde-508	91	22	2.2	2.2	NUM
ejde-508	91	23	we	we	PRON
ejde-508	91	24	have	have	VERB
ejde-508	91	25	lim	lim	PROPN
ejde-508	91	26	n→∞	n→∞	NUM
ejde-508	92	1	wn	wn	PROPN
ejde-508	92	2	=	=	PROPN
ejde-508	92	3	lim	lim	PROPN
ejde-508	92	4	n→∞	n→∞	X
ejde-508	92	5	zn	zn	PROPN
ejde-508	92	6	;	;	PUNCT
ejde-508	92	7	(	(	PUNCT
ejde-508	92	8	2.17	2.17	NUM
ejde-508	92	9	)	)	PUNCT
ejde-508	92	10	thus	thus	ADV
ejde-508	92	11	limn→∞	limn→∞	X
ejde-508	92	12	zn	zn	X
ejde-508	92	13	=	=	SYM
ejde-508	92	14	−∞.	−∞.	PROPN
ejde-508	93	1	this	this	PRON
ejde-508	93	2	implies	imply	VERB
ejde-508	93	3	that	that	SCONJ
ejde-508	93	4	for	for	ADP
ejde-508	93	5	large	large	ADJ
ejde-508	93	6	n	n	CCONJ
ejde-508	93	7	,	,	PUNCT
ejde-508	93	8	there	there	PRON
ejde-508	93	9	exists	exist	VERB
ejde-508	93	10	η	η	PROPN
ejde-508	93	11	>	>	X
ejde-508	93	12	0	0	PROPN
ejde-508	93	13	,	,	PUNCT
ejde-508	93	14	however	however	ADV
ejde-508	93	15	large	large	ADJ
ejde-508	93	16	,	,	PUNCT
ejde-508	93	17	such	such	ADJ
ejde-508	93	18	that	that	PRON
ejde-508	93	19	for	for	ADP
ejde-508	93	20	n	n	PRON
ejde-508	93	21	≥	≥	NOUN
ejde-508	93	22	n3	n3	NOUN
ejde-508	93	23	implies	imply	VERB
ejde-508	93	24	zn	zn	PROPN
ejde-508	93	25	<	<	X
ejde-508	93	26	−η	−η	NOUN
ejde-508	93	27	which	which	PRON
ejde-508	93	28	implies	imply	VERB
ejde-508	93	29	by	by	ADP
ejde-508	93	30	(	(	PUNCT
ejde-508	93	31	a1	a1	PROPN
ejde-508	93	32	)	)	PUNCT
ejde-508	93	33	that	that	PRON
ejde-508	93	34	yn	yn	PRON
ejde-508	93	35	<	<	X
ejde-508	93	36	−η	−η	NOUN
ejde-508	93	37	+	+	CCONJ
ejde-508	93	38	pδyn−s	pδyn−	VERB
ejde-508	93	39	<	<	X
ejde-508	93	40	yn−s	yn−s	NOUN
ejde-508	93	41	.	.	PUNCT
ejde-508	94	1	then	then	ADV
ejde-508	94	2	yn	yn	PROPN
ejde-508	94	3	is	be	AUX
ejde-508	94	4	bounded	bound	VERB
ejde-508	94	5	.	.	PUNCT
ejde-508	95	1	consequently	consequently	ADV
ejde-508	95	2	zn	zn	PROPN
ejde-508	95	3	and	and	CCONJ
ejde-508	95	4	wn	wn	PROPN
ejde-508	95	5	are	be	AUX
ejde-508	95	6	bounded	bound	VERB
ejde-508	95	7	,	,	PUNCT
ejde-508	95	8	which	which	PRON
ejde-508	95	9	contradicts	contradict	VERB
ejde-508	95	10	(	(	PUNCT
ejde-508	95	11	2.7	2.7	NUM
ejde-508	95	12	)	)	PUNCT
ejde-508	95	13	.	.	PUNCT
ejde-508	96	1	as	as	ADP
ejde-508	96	2	a	a	DET
ejde-508	96	3	result	result	NOUN
ejde-508	96	4	,	,	PUNCT
ejde-508	96	5	(	(	PUNCT
ejde-508	96	6	2.7	2.7	NUM
ejde-508	96	7	)	)	PUNCT
ejde-508	96	8	can	can	AUX
ejde-508	96	9	not	not	PART
ejde-508	96	10	hold	hold	VERB
ejde-508	96	11	and	and	CCONJ
ejde-508	96	12	so	so	ADV
ejde-508	96	13	,	,	PUNCT
ejde-508	96	14	(	(	PUNCT
ejde-508	96	15	2.8	2.8	NUM
ejde-508	96	16	)	)	PUNCT
ejde-508	96	17	holds	hold	NOUN
ejde-508	96	18	,	,	PUNCT
ejde-508	96	19	which	which	PRON
ejde-508	96	20	implies	imply	VERB
ejde-508	96	21	(	(	PUNCT
ejde-508	96	22	2.9	2.9	NUM
ejde-508	96	23	)	)	PUNCT
ejde-508	96	24	.	.	PUNCT
ejde-508	97	1	the	the	DET
ejde-508	97	2	proof	proof	NOUN
ejde-508	97	3	is	be	AUX
ejde-508	97	4	complete	complete	ADJ
ejde-508	97	5	.	.	PUNCT
ejde-508	98	1	�	�	PROPN
ejde-508	98	2	remark	remark	VERB
ejde-508	98	3	2.4	2.4	NUM
ejde-508	98	4	.	.	PUNCT
ejde-508	99	1	if	if	SCONJ
ejde-508	99	2	yn	yn	PROPN
ejde-508	99	3	is	be	AUX
ejde-508	99	4	an	an	DET
ejde-508	99	5	eventually	eventually	ADV
ejde-508	99	6	negative	negative	ADJ
ejde-508	99	7	solution	solution	NOUN
ejde-508	99	8	of	of	ADP
ejde-508	99	9	(	(	PUNCT
ejde-508	99	10	1.2	1.2	NUM
ejde-508	99	11	)	)	PUNCT
ejde-508	99	12	,	,	PUNCT
ejde-508	99	13	then	then	ADV
ejde-508	99	14	using	use	VERB
ejde-508	99	15	(	(	PUNCT
ejde-508	99	16	1.3	1.3	NUM
ejde-508	99	17	)	)	PUNCT
ejde-508	99	18	,	,	PUNCT
ejde-508	99	19	we	we	PRON
ejde-508	99	20	observe	observe	VERB
ejde-508	99	21	that	that	SCONJ
ejde-508	99	22	xn	xn	PUNCT
ejde-508	100	1	=	=	PRON
ejde-508	100	2	−yn	−yn	NOUN
ejde-508	100	3	is	be	AUX
ejde-508	100	4	a	a	DET
ejde-508	100	5	positive	positive	ADJ
ejde-508	100	6	solution	solution	NOUN
ejde-508	100	7	of	of	ADP
ejde-508	100	8	(	(	PUNCT
ejde-508	100	9	1.2	1.2	NUM
ejde-508	100	10	)	)	PUNCT
ejde-508	100	11	.	.	PUNCT
ejde-508	101	1	so	so	ADV
ejde-508	101	2	that	that	SCONJ
ejde-508	101	3	all	all	DET
ejde-508	101	4	the	the	DET
ejde-508	101	5	oscillation	oscillation	NOUN
ejde-508	101	6	results	result	VERB
ejde-508	101	7	for	for	ADP
ejde-508	101	8	the	the	DET
ejde-508	101	9	positive	positive	ADJ
ejde-508	101	10	solutions	solution	NOUN
ejde-508	101	11	also	also	ADV
ejde-508	101	12	apply	apply	VERB
ejde-508	101	13	to	to	ADP
ejde-508	101	14	negative	negative	ADJ
ejde-508	101	15	solutions	solution	NOUN
ejde-508	101	16	.	.	PUNCT
ejde-508	102	1	lemma	lemma	PROPN
ejde-508	102	2	2.5	2.5	NUM
ejde-508	102	3	.	.	PUNCT
ejde-508	103	1	let	let	VERB
ejde-508	103	2	yn	yn	PRON
ejde-508	103	3	be	be	AUX
ejde-508	103	4	an	an	DET
ejde-508	103	5	eventually	eventually	ADV
ejde-508	103	6	positive	positive	ADJ
ejde-508	103	7	solution	solution	NOUN
ejde-508	103	8	of	of	ADP
ejde-508	103	9	(	(	PUNCT
ejde-508	103	10	1.2	1.2	NUM
ejde-508	103	11	)	)	PUNCT
ejde-508	103	12	,	,	PUNCT
ejde-508	103	13	with	with	ADP
ejde-508	103	14	wn	wn	PROPN
ejde-508	103	15	as	as	ADP
ejde-508	103	16	in	in	ADP
ejde-508	103	17	(	(	PUNCT
ejde-508	103	18	2.6	2.6	NUM
ejde-508	103	19	)	)	PUNCT
ejde-508	103	20	.	.	PUNCT
ejde-508	104	1	then	then	ADV
ejde-508	104	2	the	the	DET
ejde-508	104	3	following	follow	VERB
ejde-508	104	4	statements	statement	NOUN
ejde-508	104	5	hold	hold	VERB
ejde-508	104	6	.	.	PUNCT
ejde-508	105	1	(	(	PUNCT
ejde-508	105	2	a	a	X
ejde-508	105	3	)	)	PUNCT
ejde-508	105	4	if	if	SCONJ
ejde-508	105	5	(	(	PUNCT
ejde-508	105	6	2.7	2.7	NUM
ejde-508	105	7	)	)	PUNCT
ejde-508	105	8	holds	hold	VERB
ejde-508	105	9	,	,	PUNCT
ejde-508	105	10	then	then	ADV
ejde-508	105	11	(	(	PUNCT
ejde-508	105	12	2.10	2.10	NUM
ejde-508	105	13	)	)	PUNCT
ejde-508	105	14	implies	imply	VERB
ejde-508	105	15	∆wn+1	∆wn+1	X
ejde-508	105	16	+	+	NUM
ejde-508	106	1	qng(yn−k	qng(yn−k	X
ejde-508	106	2	)	)	PUNCT
ejde-508	106	3	≤	≤	NOUN
ejde-508	106	4	0	0	NUM
ejde-508	106	5	,	,	PUNCT
ejde-508	106	6	(	(	PUNCT
ejde-508	106	7	2.18	2.18	NUM
ejde-508	106	8	)	)	PUNCT
ejde-508	106	9	which	which	PRON
ejde-508	106	10	further	far	ADV
ejde-508	106	11	implies	imply	VERB
ejde-508	106	12	∆zn+1	∆zn+1	NOUN
ejde-508	106	13	+	+	CCONJ
ejde-508	106	14	qng(yn−k	qng(yn−k	X
ejde-508	106	15	)	)	PUNCT
ejde-508	106	16	≤	≤	NOUN
ejde-508	106	17	0	0	NUM
ejde-508	106	18	.	.	PUNCT
ejde-508	107	1	(	(	PUNCT
ejde-508	107	2	2.19	2.19	NUM
ejde-508	107	3	)	)	PUNCT
ejde-508	107	4	(	(	PUNCT
ejde-508	107	5	b	b	X
ejde-508	107	6	)	)	PUNCT
ejde-508	107	7	if	if	SCONJ
ejde-508	107	8	(	(	PUNCT
ejde-508	107	9	2.8	2.8	NUM
ejde-508	107	10	)	)	PUNCT
ejde-508	107	11	holds	hold	VERB
ejde-508	107	12	,	,	PUNCT
ejde-508	107	13	then	then	ADV
ejde-508	107	14	(	(	PUNCT
ejde-508	107	15	2.10	2.10	NUM
ejde-508	107	16	)	)	PUNCT
ejde-508	107	17	implies	imply	VERB
ejde-508	107	18	∆wn	∆wn	PROPN
ejde-508	107	19	−	−	PROPN
ejde-508	107	20	qng(yn−k	qng(yn−k	PROPN
ejde-508	107	21	)	)	PUNCT
ejde-508	107	22	≥	≥	NOUN
ejde-508	107	23	0	0	NUM
ejde-508	107	24	.	.	PUNCT
ejde-508	108	1	(	(	PUNCT
ejde-508	108	2	2.20	2.20	NUM
ejde-508	108	3	)	)	PUNCT
ejde-508	108	4	ejde-2020/87	ejde-2020/87	VERB
ejde-508	108	5	oscillation	oscillation	NOUN
ejde-508	108	6	for	for	ADP
ejde-508	108	7	second	second	ADJ
ejde-508	108	8	order	order	NOUN
ejde-508	108	9	neutral	neutral	ADJ
ejde-508	108	10	equations	equation	NOUN
ejde-508	108	11	5	5	NUM
ejde-508	108	12	proof	proof	NOUN
ejde-508	108	13	.	.	PUNCT
ejde-508	109	1	if	if	SCONJ
ejde-508	109	2	(	(	PUNCT
ejde-508	109	3	2.7	2.7	NUM
ejde-508	109	4	)	)	PUNCT
ejde-508	109	5	holds	hold	VERB
ejde-508	109	6	then	then	ADV
ejde-508	109	7	∆wn	∆wn	ADV
ejde-508	109	8	<	<	X
ejde-508	109	9	0	0	PUNCT
ejde-508	109	10	and	and	CCONJ
ejde-508	109	11	∆w2	∆w2	VERB
ejde-508	109	12	n	n	CCONJ
ejde-508	109	13	<	<	X
ejde-508	109	14	0	0	NUM
ejde-508	109	15	.	.	PUNCT
ejde-508	110	1	we	we	PRON
ejde-508	110	2	write	write	VERB
ejde-508	110	3	(	(	PUNCT
ejde-508	110	4	2.10	2.10	NUM
ejde-508	110	5	)	)	PUNCT
ejde-508	110	6	as	as	ADP
ejde-508	110	7	∆wn+1	∆wn+1	X
ejde-508	110	8	+	+	CCONJ
ejde-508	110	9	qng(yn−k	qng(yn−k	X
ejde-508	110	10	)	)	PUNCT
ejde-508	110	11	=	=	PRON
ejde-508	110	12	∆wn	∆wn	VERB
ejde-508	110	13	≤	≤	ADV
ejde-508	110	14	0	0	NUM
ejde-508	110	15	.	.	PUNCT
ejde-508	111	1	thus	thus	ADV
ejde-508	111	2	,	,	PUNCT
ejde-508	111	3	(	(	PUNCT
ejde-508	111	4	2.18	2.18	NUM
ejde-508	111	5	)	)	PUNCT
ejde-508	111	6	holds	hold	VERB
ejde-508	111	7	.	.	PUNCT
ejde-508	112	1	from	from	ADP
ejde-508	112	2	(	(	PUNCT
ejde-508	112	3	2.6	2.6	NUM
ejde-508	112	4	)	)	PUNCT
ejde-508	112	5	,	,	PUNCT
ejde-508	112	6	it	it	PRON
ejde-508	112	7	follows	follow	VERB
ejde-508	112	8	that	that	PRON
ejde-508	112	9	∆wn+1	∆wn+1	X
ejde-508	112	10	=	=	SYM
ejde-508	112	11	∆zn+1+∆cn+1	∆zn+1+∆cn+1	NOUN
ejde-508	112	12	.	.	PUNCT
ejde-508	113	1	therefore	therefore	ADV
ejde-508	113	2	(	(	PUNCT
ejde-508	113	3	2.18	2.18	NUM
ejde-508	113	4	)	)	PUNCT
ejde-508	113	5	implies	imply	VERB
ejde-508	113	6	∆zn+1	∆zn+1	NOUN
ejde-508	113	7	+	+	CCONJ
ejde-508	113	8	qng(yn−k	qng(yn−k	X
ejde-508	113	9	)	)	PUNCT
ejde-508	113	10	=	=	PRON
ejde-508	113	11	−∆cn+1	−∆cn+1	VERB
ejde-508	113	12	≤	≤	NOUN
ejde-508	113	13	0	0	NUM
ejde-508	113	14	by	by	ADP
ejde-508	113	15	lemma	lemma	PROPN
ejde-508	113	16	2.2	2.2	NUM
ejde-508	113	17	.	.	PUNCT
ejde-508	114	1	hence	hence	ADV
ejde-508	114	2	(	(	PUNCT
ejde-508	114	3	a	a	X
ejde-508	114	4	)	)	PUNCT
ejde-508	114	5	is	be	AUX
ejde-508	114	6	proved	prove	VERB
ejde-508	114	7	.	.	PUNCT
ejde-508	115	1	let	let	VERB
ejde-508	115	2	us	we	PRON
ejde-508	115	3	prove	prove	VERB
ejde-508	115	4	(	(	PUNCT
ejde-508	115	5	b	b	NOUN
ejde-508	115	6	)	)	PUNCT
ejde-508	115	7	.	.	PUNCT
ejde-508	116	1	if	if	SCONJ
ejde-508	116	2	(	(	PUNCT
ejde-508	116	3	2.8	2.8	NUM
ejde-508	116	4	)	)	PUNCT
ejde-508	116	5	holds	hold	VERB
ejde-508	116	6	then	then	ADV
ejde-508	116	7	(	(	PUNCT
ejde-508	116	8	2.9	2.9	NUM
ejde-508	116	9	)	)	PUNCT
ejde-508	116	10	follows	follow	VERB
ejde-508	116	11	as	as	ADP
ejde-508	116	12	a	a	DET
ejde-508	116	13	consequence	consequence	NOUN
ejde-508	116	14	,	,	PUNCT
ejde-508	116	15	which	which	PRON
ejde-508	116	16	implies	imply	VERB
ejde-508	116	17	wn	wn	PROPN
ejde-508	116	18	<	<	X
ejde-508	116	19	0	0	PROPN
ejde-508	116	20	and	and	CCONJ
ejde-508	116	21	∆wn	∆wn	X
ejde-508	116	22	>	>	X
ejde-508	117	1	0	0	X
ejde-508	117	2	.	.	PUNCT
ejde-508	118	1	using	use	VERB
ejde-508	118	2	(	(	PUNCT
ejde-508	118	3	2.9	2.9	NUM
ejde-508	118	4	)	)	PUNCT
ejde-508	118	5	,	,	PUNCT
ejde-508	118	6	we	we	PRON
ejde-508	118	7	write	write	VERB
ejde-508	118	8	(	(	PUNCT
ejde-508	118	9	2.10	2.10	NUM
ejde-508	118	10	)	)	PUNCT
ejde-508	118	11	,	,	PUNCT
ejde-508	118	12	as	as	ADP
ejde-508	118	13	−∆wn	−∆wn	PROPN
ejde-508	118	14	+	+	CCONJ
ejde-508	118	15	qng(yn−k	qng(yn−k	X
ejde-508	118	16	)	)	PUNCT
ejde-508	118	17	=	=	PUNCT
ejde-508	118	18	−∆wn+1	−∆wn+1	ADJ
ejde-508	118	19	≤	≤	NOUN
ejde-508	118	20	0	0	NUM
ejde-508	118	21	,	,	PUNCT
ejde-508	118	22	which	which	PRON
ejde-508	118	23	implies	imply	VERB
ejde-508	118	24	∆wn	∆wn	PROPN
ejde-508	118	25	−	−	PROPN
ejde-508	118	26	qng(yn−k	qng(yn−k	PROPN
ejde-508	118	27	)	)	PUNCT
ejde-508	118	28	=	=	SYM
ejde-508	118	29	∆wn+1	∆wn+1	X
ejde-508	118	30	≥	≥	NOUN
ejde-508	118	31	0	0	NUM
ejde-508	118	32	.	.	PUNCT
ejde-508	119	1	this	this	PRON
ejde-508	119	2	proves	prove	VERB
ejde-508	119	3	of	of	ADP
ejde-508	119	4	(	(	PUNCT
ejde-508	119	5	b	b	NOUN
ejde-508	119	6	)	)	PUNCT
ejde-508	119	7	,	,	PUNCT
ejde-508	119	8	and	and	CCONJ
ejde-508	119	9	completes	complete	VERB
ejde-508	119	10	the	the	DET
ejde-508	119	11	proof	proof	NOUN
ejde-508	119	12	.	.	PUNCT
ejde-508	120	1	�	�	PROPN
ejde-508	120	2	lemma	lemma	PROPN
ejde-508	120	3	2.6	2.6	NUM
ejde-508	120	4	.	.	PUNCT
ejde-508	121	1	let	let	VERB
ejde-508	121	2	(	(	PUNCT
ejde-508	121	3	a1	a1	NOUN
ejde-508	121	4	)	)	PUNCT
ejde-508	121	5	,	,	PUNCT
ejde-508	121	6	(	(	PUNCT
ejde-508	121	7	a3	a3	NOUN
ejde-508	121	8	)	)	PUNCT
ejde-508	121	9	,	,	PUNCT
ejde-508	121	10	(	(	PUNCT
ejde-508	121	11	a6	a6	NOUN
ejde-508	121	12	)	)	PUNCT
ejde-508	121	13	,	,	PUNCT
ejde-508	121	14	(	(	PUNCT
ejde-508	121	15	a7	a7	ADJ
ejde-508	121	16	)	)	PUNCT
ejde-508	121	17	hold	hold	VERB
ejde-508	121	18	.	.	PUNCT
ejde-508	122	1	assume	assume	VERB
ejde-508	122	2	that	that	SCONJ
ejde-508	122	3	there	there	PRON
ejde-508	122	4	exists	exist	VERB
ejde-508	122	5	λ	λ	PROPN
ejde-508	122	6	>	>	X
ejde-508	122	7	0	0	NUM
ejde-508	122	8	such	such	ADJ
ejde-508	122	9	that	that	PRON
ejde-508	122	10	for	for	ADP
ejde-508	122	11	all	all	DET
ejde-508	122	12	x	x	NOUN
ejde-508	122	13	,	,	PUNCT
ejde-508	122	14	y	y	PROPN
ejde-508	122	15	∈	∈	PROPN
ejde-508	122	16	r	r	NOUN
ejde-508	122	17	with	with	ADP
ejde-508	122	18	x+	x+	PROPN
ejde-508	122	19	y	y	PROPN
ejde-508	122	20	>	>	X
ejde-508	122	21	0	0	PROPN
ejde-508	122	22	,	,	PUNCT
ejde-508	122	23	we	we	PRON
ejde-508	122	24	have	have	VERB
ejde-508	122	25	g(x	g(x	NOUN
ejde-508	122	26	)	)	PUNCT
ejde-508	123	1	+	+	NOUN
ejde-508	123	2	g(y	g(y	X
ejde-508	123	3	)	)	PUNCT
ejde-508	123	4	≥	≥	NOUN
ejde-508	123	5	λg(x+	λg(x+	ADV
ejde-508	123	6	y	y	PROPN
ejde-508	123	7	)	)	PUNCT
ejde-508	123	8	.	.	PUNCT
ejde-508	124	1	(	(	PUNCT
ejde-508	124	2	2.21	2.21	NUM
ejde-508	124	3	)	)	PUNCT
ejde-508	124	4	further	far	ADV
ejde-508	124	5	,	,	PUNCT
ejde-508	124	6	we	we	PRON
ejde-508	124	7	assume	assume	VERB
ejde-508	124	8	that	that	SCONJ
ejde-508	124	9	g(x)g(y	g(x)g(y	PROPN
ejde-508	124	10	)	)	PUNCT
ejde-508	124	11	≥	≥	PROPN
ejde-508	124	12	g(xy	g(xy	PROPN
ejde-508	124	13	)	)	PUNCT
ejde-508	124	14	for	for	ADP
ejde-508	124	15	all	all	DET
ejde-508	124	16	x	x	NOUN
ejde-508	124	17	,	,	PUNCT
ejde-508	124	18	y	y	PROPN
ejde-508	124	19	>	>	X
ejde-508	124	20	0	0	PROPN
ejde-508	124	21	.	.	PUNCT
ejde-508	125	1	(	(	PUNCT
ejde-508	125	2	2.22	2.22	NUM
ejde-508	125	3	)	)	PUNCT
ejde-508	125	4	let	let	VERB
ejde-508	125	5	yn	yn	PRON
ejde-508	125	6	be	be	AUX
ejde-508	125	7	an	an	DET
ejde-508	125	8	eventually	eventually	ADV
ejde-508	125	9	positive	positive	ADJ
ejde-508	125	10	solution	solution	NOUN
ejde-508	125	11	of	of	ADP
ejde-508	125	12	(	(	PUNCT
ejde-508	125	13	1.2	1.2	NUM
ejde-508	125	14	)	)	PUNCT
ejde-508	125	15	.	.	PUNCT
ejde-508	126	1	define	define	VERB
ejde-508	126	2	cn	cn	PROPN
ejde-508	126	3	,	,	PUNCT
ejde-508	126	4	zn	zn	PROPN
ejde-508	126	5	and	and	CCONJ
ejde-508	126	6	wn	wn	PROPN
ejde-508	126	7	as	as	ADP
ejde-508	126	8	in	in	ADP
ejde-508	126	9	(	(	PUNCT
ejde-508	126	10	2.2	2.2	NUM
ejde-508	126	11	)	)	PUNCT
ejde-508	126	12	,	,	PUNCT
ejde-508	126	13	(	(	PUNCT
ejde-508	126	14	2.5	2.5	NUM
ejde-508	126	15	)	)	PUNCT
ejde-508	126	16	and	and	CCONJ
ejde-508	126	17	(	(	PUNCT
ejde-508	126	18	2.6	2.6	NUM
ejde-508	126	19	)	)	PUNCT
ejde-508	126	20	respectively	respectively	ADV
ejde-508	126	21	.	.	PUNCT
ejde-508	127	1	if	if	SCONJ
ejde-508	127	2	pn	pn	PROPN
ejde-508	127	3	satisfies	satisfie	NOUN
ejde-508	127	4	(	(	PUNCT
ejde-508	127	5	1.6	1.6	NUM
ejde-508	127	6	)	)	PUNCT
ejde-508	127	7	or	or	CCONJ
ejde-508	127	8	(	(	PUNCT
ejde-508	127	9	1.7	1.7	NUM
ejde-508	127	10	)	)	PUNCT
ejde-508	127	11	,	,	PUNCT
ejde-508	127	12	then	then	ADV
ejde-508	127	13	limn→∞	limn→∞	ADJ
ejde-508	127	14	wn	wn	PROPN
ejde-508	127	15	=	=	PROPN
ejde-508	127	16	0	0	PROPN
ejde-508	127	17	.	.	PUNCT
ejde-508	128	1	consequently	consequently	ADV
ejde-508	128	2	,	,	PUNCT
ejde-508	128	3	(	(	PUNCT
ejde-508	128	4	2.9	2.9	NUM
ejde-508	128	5	)	)	PUNCT
ejde-508	128	6	holds	hold	VERB
ejde-508	128	7	.	.	PUNCT
ejde-508	129	1	proof	proof	NOUN
ejde-508	129	2	.	.	PUNCT
ejde-508	130	1	suppose	suppose	VERB
ejde-508	130	2	yn	yn	PRON
ejde-508	130	3	is	be	AUX
ejde-508	130	4	an	an	DET
ejde-508	130	5	eventually	eventually	ADV
ejde-508	130	6	positive	positive	ADJ
ejde-508	130	7	or	or	CCONJ
ejde-508	130	8	eventually	eventually	ADV
ejde-508	130	9	negative	negative	ADJ
ejde-508	130	10	solution	solution	NOUN
ejde-508	130	11	of	of	ADP
ejde-508	130	12	(	(	PUNCT
ejde-508	130	13	1.2	1.2	NUM
ejde-508	130	14	)	)	PUNCT
ejde-508	130	15	and	and	CCONJ
ejde-508	130	16	pn	pn	X
ejde-508	130	17	satisfies	satisfie	NOUN
ejde-508	130	18	(	(	PUNCT
ejde-508	130	19	1.6	1.6	NUM
ejde-508	130	20	)	)	PUNCT
ejde-508	130	21	.	.	PUNCT
ejde-508	131	1	from	from	ADP
ejde-508	131	2	(	(	PUNCT
ejde-508	131	3	1.2	1.2	NUM
ejde-508	131	4	)	)	PUNCT
ejde-508	131	5	,	,	PUNCT
ejde-508	131	6	using	use	VERB
ejde-508	131	7	(	(	PUNCT
ejde-508	131	8	2.6	2.6	NUM
ejde-508	131	9	)	)	PUNCT
ejde-508	131	10	,	,	PUNCT
ejde-508	131	11	(	(	PUNCT
ejde-508	131	12	2.5	2.5	NUM
ejde-508	131	13	)	)	PUNCT
ejde-508	131	14	,	,	PUNCT
ejde-508	131	15	(	(	PUNCT
ejde-508	131	16	2.2	2.2	NUM
ejde-508	131	17	)	)	PUNCT
ejde-508	131	18	and	and	CCONJ
ejde-508	131	19	lemma	lemma	PROPN
ejde-508	131	20	2.2	2.2	NUM
ejde-508	131	21	,	,	PUNCT
ejde-508	131	22	we	we	PRON
ejde-508	131	23	obtain	obtain	VERB
ejde-508	131	24	(	(	PUNCT
ejde-508	131	25	2.10	2.10	NUM
ejde-508	131	26	)	)	PUNCT
ejde-508	131	27	.	.	PUNCT
ejde-508	132	1	this	this	PRON
ejde-508	132	2	implies	imply	VERB
ejde-508	132	3	wn	wn	PROPN
ejde-508	132	4	and	and	CCONJ
ejde-508	132	5	∆wn	∆wn	PROPN
ejde-508	132	6	are	be	AUX
ejde-508	132	7	monotonic	monotonic	ADJ
ejde-508	132	8	and	and	CCONJ
ejde-508	132	9	single	single	ADJ
ejde-508	132	10	sign	sign	NOUN
ejde-508	132	11	.	.	PUNCT
ejde-508	133	1	hence	hence	ADV
ejde-508	133	2	,	,	PUNCT
ejde-508	133	3	it	it	PRON
ejde-508	133	4	follows	follow	VERB
ejde-508	133	5	that	that	SCONJ
ejde-508	133	6	(	(	PUNCT
ejde-508	133	7	2.17	2.17	NUM
ejde-508	133	8	)	)	PUNCT
ejde-508	133	9	holds	hold	VERB
ejde-508	133	10	and	and	CCONJ
ejde-508	133	11	let	let	VERB
ejde-508	133	12	limn→∞	limn→∞	PROPN
ejde-508	133	13	zn	zn	X
ejde-508	133	14	=	=	SYM
ejde-508	133	15	β	β	X
ejde-508	133	16	.	.	PUNCT
ejde-508	134	1	clearly	clearly	ADV
ejde-508	134	2	,	,	PUNCT
ejde-508	134	3	zn	zn	PROPN
ejde-508	134	4	>	>	X
ejde-508	134	5	0	0	PUNCT
ejde-508	134	6	by	by	ADP
ejde-508	134	7	(	(	PUNCT
ejde-508	134	8	1.6	1.6	NUM
ejde-508	134	9	)	)	PUNCT
ejde-508	134	10	.	.	PUNCT
ejde-508	135	1	this	this	PRON
ejde-508	135	2	implies	imply	VERB
ejde-508	135	3	,	,	PUNCT
ejde-508	135	4	β	β	X
ejde-508	135	5	in	in	ADP
ejde-508	135	6	(	(	PUNCT
ejde-508	135	7	2.17	2.17	NUM
ejde-508	135	8	)	)	PUNCT
ejde-508	135	9	,	,	PUNCT
ejde-508	135	10	can	can	AUX
ejde-508	135	11	not	not	PART
ejde-508	135	12	be	be	AUX
ejde-508	135	13	in	in	ADP
ejde-508	135	14	negative	negative	ADJ
ejde-508	135	15	.	.	PUNCT
ejde-508	136	1	if	if	SCONJ
ejde-508	136	2	β	β	X
ejde-508	136	3	>	>	X
ejde-508	136	4	0	0	NUM
ejde-508	136	5	,	,	PUNCT
ejde-508	136	6	then	then	ADV
ejde-508	136	7	then	then	ADV
ejde-508	136	8	there	there	PRON
ejde-508	136	9	exists	exist	VERB
ejde-508	136	10	a	a	DET
ejde-508	136	11	positive	positive	ADJ
ejde-508	136	12	scalar	scalar	NOUN
ejde-508	136	13	χ	χ	ADP
ejde-508	136	14	such	such	ADJ
ejde-508	136	15	that	that	SCONJ
ejde-508	136	16	zn	zn	PROPN
ejde-508	136	17	>	>	X
ejde-508	137	1	χ	χ	X
ejde-508	137	2	>	>	X
ejde-508	137	3	0	0	PUNCT
ejde-508	138	1	for	for	ADP
ejde-508	138	2	large	large	ADJ
ejde-508	138	3	n.	n.	NOUN
ejde-508	138	4	clearly	clearly	ADV
ejde-508	138	5	,	,	PUNCT
ejde-508	138	6	∆wn	∆wn	AUX
ejde-508	138	7	>	>	X
ejde-508	138	8	0	0	PROPN
ejde-508	138	9	,	,	PUNCT
ejde-508	138	10	otherwise	otherwise	ADV
ejde-508	138	11	,	,	PUNCT
ejde-508	138	12	β	β	X
ejde-508	138	13	=	=	SYM
ejde-508	138	14	−∞	−∞	PROPN
ejde-508	138	15	,	,	PUNCT
ejde-508	138	16	a	a	DET
ejde-508	138	17	contradiction	contradiction	NOUN
ejde-508	138	18	.	.	PUNCT
ejde-508	139	1	since	since	SCONJ
ejde-508	139	2	∆wn	∆wn	PROPN
ejde-508	139	3	is	be	AUX
ejde-508	139	4	decreasing	decrease	VERB
ejde-508	139	5	,	,	PUNCT
ejde-508	139	6	limn→∞∆wn	limn→∞∆wn	PROPN
ejde-508	139	7	exists	exist	VERB
ejde-508	139	8	.	.	PUNCT
ejde-508	140	1	if	if	SCONJ
ejde-508	140	2	x	x	PROPN
ejde-508	140	3	>	>	X
ejde-508	140	4	y	y	PROPN
ejde-508	140	5	then	then	ADV
ejde-508	140	6	using	use	VERB
ejde-508	140	7	(	(	PUNCT
ejde-508	140	8	1.3	1.3	NUM
ejde-508	140	9	)	)	PUNCT
ejde-508	140	10	and	and	CCONJ
ejde-508	140	11	(	(	PUNCT
ejde-508	140	12	2.21	2.21	NUM
ejde-508	140	13	)	)	PUNCT
ejde-508	140	14	,	,	PUNCT
ejde-508	140	15	we	we	PRON
ejde-508	140	16	note	note	VERB
ejde-508	140	17	that	that	SCONJ
ejde-508	140	18	0	0	PUNCT
ejde-508	140	19	<	<	X
ejde-508	140	20	λg(x−y	λg(x−y	NOUN
ejde-508	140	21	)	)	PUNCT
ejde-508	140	22	≤	≤	NUM
ejde-508	140	23	g(x)+g(−y	g(x)+g(−y	NOUN
ejde-508	140	24	)	)	PUNCT
ejde-508	140	25	=	=	PUNCT
ejde-508	141	1	g(x)−g(y	g(x)−g(y	NOUN
ejde-508	141	2	)	)	PUNCT
ejde-508	141	3	.	.	PUNCT
ejde-508	142	1	thus	thus	ADV
ejde-508	142	2	,	,	PUNCT
ejde-508	142	3	(	(	PUNCT
ejde-508	142	4	a5	a5	NOUN
ejde-508	142	5	)	)	PUNCT
ejde-508	142	6	holds	hold	NOUN
ejde-508	142	7	,	,	PUNCT
ejde-508	142	8	i.e	i.e	X
ejde-508	142	9	;	;	PUNCT
ejde-508	142	10	g	g	PROPN
ejde-508	142	11	is	be	AUX
ejde-508	142	12	non	non	NOUN
ejde-508	142	13	decreasing	decrease	VERB
ejde-508	142	14	.	.	PUNCT
ejde-508	143	1	then	then	ADV
ejde-508	143	2	using	use	VERB
ejde-508	143	3	(	(	PUNCT
ejde-508	143	4	a5	a5	PROPN
ejde-508	143	5	)	)	PUNCT
ejde-508	143	6	,	,	PUNCT
ejde-508	143	7	(	(	PUNCT
ejde-508	143	8	a1	a1	NOUN
ejde-508	143	9	)	)	PUNCT
ejde-508	143	10	and	and	CCONJ
ejde-508	143	11	(	(	PUNCT
ejde-508	143	12	1.6	1.6	NUM
ejde-508	143	13	)	)	PUNCT
ejde-508	143	14	in	in	ADP
ejde-508	143	15	(	(	PUNCT
ejde-508	143	16	2.5	2.5	NUM
ejde-508	143	17	)	)	PUNCT
ejde-508	143	18	,	,	PUNCT
ejde-508	143	19	we	we	PRON
ejde-508	143	20	have	have	VERB
ejde-508	143	21	zn	zn	NOUN
ejde-508	143	22	≤	≤	PROPN
ejde-508	143	23	yn	yn	PROPN
ejde-508	144	1	+	+	PROPN
ejde-508	145	1	pδyn−s	pδyn−s	PROPN
ejde-508	145	2	.	.	PUNCT
ejde-508	146	1	(	(	PUNCT
ejde-508	146	2	2.23	2.23	NUM
ejde-508	146	3	)	)	PUNCT
ejde-508	146	4	from	from	ADP
ejde-508	146	5	(	(	PUNCT
ejde-508	146	6	2.10	2.10	NUM
ejde-508	146	7	)	)	PUNCT
ejde-508	146	8	,	,	PUNCT
ejde-508	146	9	by	by	ADP
ejde-508	146	10	using	use	VERB
ejde-508	146	11	(	(	PUNCT
ejde-508	146	12	a3	a3	NOUN
ejde-508	146	13	)	)	PUNCT
ejde-508	146	14	,	,	PUNCT
ejde-508	146	15	(	(	PUNCT
ejde-508	146	16	2.21	2.21	NUM
ejde-508	146	17	)	)	PUNCT
ejde-508	146	18	,	,	PUNCT
ejde-508	146	19	(	(	PUNCT
ejde-508	146	20	2.22	2.22	NUM
ejde-508	146	21	)	)	PUNCT
ejde-508	146	22	and	and	CCONJ
ejde-508	146	23	(	(	PUNCT
ejde-508	146	24	2.23	2.23	NUM
ejde-508	146	25	)	)	PUNCT
ejde-508	146	26	it	it	PRON
ejde-508	146	27	follows	follow	VERB
ejde-508	146	28	that	that	SCONJ
ejde-508	146	29	0	0	NUM
ejde-508	146	30	≥	≥	NOUN
ejde-508	146	31	∆2wn	∆2wn	VERB
ejde-508	146	32	+	+	CCONJ
ejde-508	146	33	qng(yn−k	qng(yn−k	X
ejde-508	146	34	)	)	PUNCT
ejde-508	147	1	+	+	NOUN
ejde-508	147	2	g(pδ)[∆2wn−s	g(pδ)[∆2wn−s	ADJ
ejde-508	147	3	+	+	CCONJ
ejde-508	147	4	qng(yn−s−k	qng(yn−s−k	NOUN
ejde-508	147	5	)	)	PUNCT
ejde-508	147	6	]	]	PUNCT
ejde-508	147	7	≥	≥	NOUN
ejde-508	147	8	∆2wn	∆2wn	VERB
ejde-508	148	1	+	+	ADP
ejde-508	148	2	g(pδ)∆2wn−s	g(pδ)∆2wn−s	PROPN
ejde-508	148	3	+	+	NUM
ejde-508	148	4	q∗n	q∗n	X
ejde-508	148	5	(	(	PUNCT
ejde-508	148	6	g(yn−k	g(yn−k	INTJ
ejde-508	148	7	)	)	PUNCT
ejde-508	148	8	+	+	NOUN
ejde-508	148	9	g(pδ)g(yn−s−k	g(pδ)g(yn−s−k	NOUN
ejde-508	148	10	)	)	PUNCT
ejde-508	148	11	)	)	PUNCT
ejde-508	148	12	≥	≥	NOUN
ejde-508	148	13	∆2wn	∆2wn	VERB
ejde-508	149	1	+	+	ADP
ejde-508	150	1	g(pδ)∆2wn−s	g(pδ)∆2wn−s	PROPN
ejde-508	150	2	+	+	CCONJ
ejde-508	150	3	λq∗n	λq∗n	X
ejde-508	150	4	(	(	PUNCT
ejde-508	150	5	g(zn−k	g(zn−k	NOUN
ejde-508	150	6	)	)	PUNCT
ejde-508	150	7	)	)	PUNCT
ejde-508	151	1	≥	≥	NOUN
ejde-508	151	2	∆2wn	∆2wn	VERB
ejde-508	152	1	+	+	ADP
ejde-508	153	1	g(pδ)∆2wn−s	g(pδ)∆2wn−s	PROPN
ejde-508	153	2	+	+	NUM
ejde-508	153	3	λg(χ)q∗n	λg(χ)q∗n	X
ejde-508	153	4	(	(	PUNCT
ejde-508	153	5	2.24	2.24	NUM
ejde-508	153	6	)	)	PUNCT
ejde-508	153	7	for	for	ADP
ejde-508	153	8	n	n	PRON
ejde-508	153	9	≥	≥	NOUN
ejde-508	153	10	n2	n2	PROPN
ejde-508	153	11	>	>	X
ejde-508	153	12	n1	n1	PROPN
ejde-508	153	13	.	.	PUNCT
ejde-508	154	1	then	then	ADV
ejde-508	154	2	taking	take	VERB
ejde-508	154	3	summation	summation	NOUN
ejde-508	154	4	in	in	ADP
ejde-508	154	5	(	(	PUNCT
ejde-508	154	6	2.24	2.24	NUM
ejde-508	154	7	)	)	PUNCT
ejde-508	154	8	from	from	ADP
ejde-508	154	9	n2	n2	ADJ
ejde-508	154	10	to	to	ADP
ejde-508	154	11	l	l	NOUN
ejde-508	154	12	−	−	PROPN
ejde-508	154	13	1	1	NUM
ejde-508	154	14	and	and	CCONJ
ejde-508	154	15	then	then	ADV
ejde-508	154	16	letting	let	VERB
ejde-508	154	17	l	l	NOUN
ejde-508	154	18	→	→	SYM
ejde-508	154	19	∞	∞	PROPN
ejde-508	154	20	,	,	PUNCT
ejde-508	154	21	we	we	PRON
ejde-508	154	22	obtain	obtain	VERB
ejde-508	154	23	a	a	DET
ejde-508	154	24	contradiction	contradiction	NOUN
ejde-508	154	25	to	to	ADP
ejde-508	154	26	(	(	PUNCT
ejde-508	154	27	a3	a3	NOUN
ejde-508	154	28	)	)	PUNCT
ejde-508	154	29	.	.	PUNCT
ejde-508	155	1	thus	thus	ADV
ejde-508	155	2	β	β	X
ejde-508	155	3	=	=	SYM
ejde-508	155	4	limn→∞	limn→∞	PROPN
ejde-508	155	5	wn	wn	X
ejde-508	155	6	=	=	SYM
ejde-508	155	7	0	0	PROPN
ejde-508	155	8	,	,	PUNCT
ejde-508	155	9	which	which	PRON
ejde-508	155	10	implies	imply	VERB
ejde-508	155	11	(	(	PUNCT
ejde-508	155	12	2.9	2.9	NUM
ejde-508	155	13	)	)	PUNCT
ejde-508	155	14	.	.	PUNCT
ejde-508	156	1	suppose	suppose	VERB
ejde-508	156	2	pn	pn	PROPN
ejde-508	156	3	satisfies	satisfie	NOUN
ejde-508	156	4	(	(	PUNCT
ejde-508	156	5	1.7	1.7	NUM
ejde-508	156	6	)	)	PUNCT
ejde-508	156	7	.	.	PUNCT
ejde-508	157	1	if	if	SCONJ
ejde-508	157	2	β	β	X
ejde-508	157	3	>	>	X
ejde-508	157	4	0	0	PUNCT
ejde-508	157	5	then	then	ADV
ejde-508	157	6	proceeding	proceed	VERB
ejde-508	157	7	as	as	ADP
ejde-508	157	8	above	above	ADV
ejde-508	157	9	,	,	PUNCT
ejde-508	157	10	we	we	PRON
ejde-508	157	11	obtain	obtain	VERB
ejde-508	157	12	a	a	DET
ejde-508	157	13	similar	similar	ADJ
ejde-508	157	14	contradiction	contradiction	NOUN
ejde-508	157	15	.	.	PUNCT
ejde-508	158	1	if	if	SCONJ
ejde-508	158	2	β	β	X
ejde-508	158	3	<	<	X
ejde-508	158	4	0	0	PUNCT
ejde-508	158	5	then	then	ADV
ejde-508	158	6	using	use	VERB
ejde-508	158	7	(	(	PUNCT
ejde-508	158	8	1.7	1.7	NUM
ejde-508	158	9	)	)	PUNCT
ejde-508	158	10	,	,	PUNCT
ejde-508	158	11	we	we	PRON
ejde-508	158	12	have	have	VERB
ejde-508	158	13	wn	wn	PROPN
ejde-508	158	14	≥	≥	NOUN
ejde-508	158	15	−pδyn−s	−pδyn−s	ADJ
ejde-508	158	16	+	+	X
ejde-508	159	1	cn	cn	VERB
ejde-508	159	2	.	.	PUNCT
ejde-508	160	1	this	this	PRON
ejde-508	160	2	implies	imply	VERB
ejde-508	160	3	yn	yn	PROPN
ejde-508	160	4	≥	≥	PRON
ejde-508	160	5	cn+s	cn+	VERB
ejde-508	160	6	pδ	pδ	ADP
ejde-508	160	7	−	−	PROPN
ejde-508	160	8	wn+s	wn+s	PROPN
ejde-508	160	9	pδ	pδ	PROPN
ejde-508	160	10	.	.	PUNCT
ejde-508	161	1	then	then	ADV
ejde-508	161	2	taking	take	VERB
ejde-508	161	3	limit	limit	NOUN
ejde-508	161	4	inferior	inferior	ADJ
ejde-508	161	5	on	on	ADP
ejde-508	161	6	both	both	DET
ejde-508	161	7	sides	side	NOUN
ejde-508	161	8	of	of	ADP
ejde-508	161	9	this	this	DET
ejde-508	161	10	inequality	inequality	NOUN
ejde-508	161	11	,	,	PUNCT
ejde-508	161	12	we	we	PRON
ejde-508	161	13	obtain	obtain	VERB
ejde-508	161	14	lim	lim	PROPN
ejde-508	161	15	inf	inf	PROPN
ejde-508	161	16	n→∞	n→∞	X
ejde-508	161	17	yn	yn	PROPN
ejde-508	161	18	≥	≥	PROPN
ejde-508	161	19	lim	lim	PROPN
ejde-508	161	20	inf	inf	PROPN
ejde-508	161	21	n→∞	n→∞	X
ejde-508	161	22	cn+s	cn+s	CCONJ
ejde-508	161	23	pδ	pδ	NOUN
ejde-508	162	1	+	+	CCONJ
ejde-508	162	2	lim	lim	PROPN
ejde-508	162	3	inf	inf	PROPN
ejde-508	162	4	n→∞	n→∞	X
ejde-508	162	5	−wn+s	−wn+s	PROPN
ejde-508	162	6	pδ	pδ	PROPN
ejde-508	162	7	≥	≥	NOUN
ejde-508	162	8	−β	−β	PROPN
ejde-508	162	9	/	/	SYM
ejde-508	162	10	pδ	pδ	NOUN
ejde-508	162	11	>	>	X
ejde-508	162	12	0	0	X
ejde-508	162	13	.	.	PROPN
ejde-508	162	14	6	6	NUM
ejde-508	162	15	a.	a.	PROPN
ejde-508	162	16	k.	k.	PROPN
ejde-508	162	17	bhuyan	bhuyan	PROPN
ejde-508	162	18	,	,	PUNCT
ejde-508	162	19	l.	l.	PROPN
ejde-508	162	20	n.	n.	PROPN
ejde-508	162	21	padhy	padhy	PROPN
ejde-508	162	22	,	,	PUNCT
ejde-508	162	23	r.	r.	PROPN
ejde-508	162	24	n.	n.	PROPN
ejde-508	162	25	rath	rath	PROPN
ejde-508	162	26	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	162	27	in	in	ADP
ejde-508	162	28	the	the	DET
ejde-508	162	29	above	above	ADV
ejde-508	162	30	we	we	PRON
ejde-508	162	31	used	use	VERB
ejde-508	162	32	limn→∞	limn→∞	PROPN
ejde-508	162	33	cn	cn	PROPN
ejde-508	162	34	=	=	SYM
ejde-508	162	35	0	0	PROPN
ejde-508	162	36	and	and	CCONJ
ejde-508	162	37	limn→∞	limn→∞	PROPN
ejde-508	162	38	wn	wn	NOUN
ejde-508	162	39	=	=	PUNCT
ejde-508	162	40	β	β	X
ejde-508	162	41	<	<	X
ejde-508	162	42	0	0	NUM
ejde-508	162	43	.	.	PUNCT
ejde-508	162	44	for	for	ADP
ejde-508	162	45	−β/(3pδ	−β/(3pδ	PROPN
ejde-508	162	46	)	)	PUNCT
ejde-508	162	47	=	=	PUNCT
ejde-508	162	48	ε	ε	PROPN
ejde-508	162	49	>	>	X
ejde-508	162	50	0	0	PROPN
ejde-508	162	51	,	,	PUNCT
ejde-508	162	52	we	we	PRON
ejde-508	162	53	find	find	VERB
ejde-508	162	54	n3	n3	ADJ
ejde-508	162	55	≥	≥	PROPN
ejde-508	162	56	n2	n2	PROPN
ejde-508	162	57	such	such	ADJ
ejde-508	162	58	that	that	SCONJ
ejde-508	162	59	n	n	PROPN
ejde-508	162	60	>	>	X
ejde-508	162	61	n3	n3	PROPN
ejde-508	162	62	implies	imply	VERB
ejde-508	162	63	yn	yn	PROPN
ejde-508	162	64	>	>	X
ejde-508	162	65	2ε	2ε	PROPN
ejde-508	162	66	.	.	PUNCT
ejde-508	163	1	as	as	ADP
ejde-508	163	2	cn	cn	PROPN
ejde-508	163	3	→	→	SYM
ejde-508	163	4	0	0	PROPN
ejde-508	163	5	,	,	PUNCT
ejde-508	163	6	from	from	ADP
ejde-508	163	7	(	(	PUNCT
ejde-508	163	8	2.6	2.6	NUM
ejde-508	163	9	)	)	PUNCT
ejde-508	163	10	it	it	PRON
ejde-508	163	11	follows	follow	VERB
ejde-508	163	12	that	that	PRON
ejde-508	163	13	pnl(yn−s	pnl(yn−s	PROPN
ejde-508	163	14	)	)	PUNCT
ejde-508	163	15	>	>	PUNCT
ejde-508	164	1	yn	yn	PROPN
ejde-508	165	1	+	+	CCONJ
ejde-508	165	2	cn	cn	ADJ
ejde-508	165	3	>	>	X
ejde-508	165	4	ε	ε	PROPN
ejde-508	165	5	>	>	X
ejde-508	165	6	0	0	PROPN
ejde-508	165	7	.	.	PUNCT
ejde-508	166	1	this	this	PRON
ejde-508	166	2	further	far	ADV
ejde-508	166	3	implies	imply	VERB
ejde-508	166	4	pn+s	pn+s	PROPN
ejde-508	166	5	>	>	SYM
ejde-508	166	6	ε	ε	PROPN
ejde-508	166	7	l(yn	l(yn	PROPN
ejde-508	166	8	)	)	PUNCT
ejde-508	166	9	≥	≥	NOUN
ejde-508	166	10	ε	ε	PROPN
ejde-508	166	11	δyn	δyn	PROPN
ejde-508	166	12	>	>	X
ejde-508	166	13	0	0	NUM
ejde-508	166	14	,	,	PUNCT
ejde-508	166	15	for	for	ADP
ejde-508	166	16	n	n	PRON
ejde-508	166	17	≥	≥	NOUN
ejde-508	166	18	n3	n3	NOUN
ejde-508	166	19	,	,	PUNCT
ejde-508	166	20	which	which	PRON
ejde-508	166	21	contradicts	contradict	VERB
ejde-508	166	22	that	that	SCONJ
ejde-508	166	23	pn	pn	PROPN
ejde-508	166	24	changes	change	NOUN
ejde-508	166	25	sign	sign	NOUN
ejde-508	166	26	.	.	PUNCT
ejde-508	167	1	thus	thus	ADV
ejde-508	167	2	β	β	X
ejde-508	167	3	can	can	AUX
ejde-508	167	4	not	not	PART
ejde-508	167	5	be	be	AUX
ejde-508	167	6	in	in	ADP
ejde-508	167	7	negative	negative	ADJ
ejde-508	167	8	,	,	PUNCT
ejde-508	167	9	hence	hence	ADV
ejde-508	167	10	limn→∞	limn→∞	ADJ
ejde-508	167	11	wn	wn	NOUN
ejde-508	167	12	=	=	PUNCT
ejde-508	167	13	β	β	X
ejde-508	167	14	=	=	SYM
ejde-508	167	15	0	0	X
ejde-508	167	16	.	.	PUNCT
ejde-508	168	1	consequently	consequently	ADV
ejde-508	168	2	(	(	PUNCT
ejde-508	168	3	2.9	2.9	NUM
ejde-508	168	4	)	)	PUNCT
ejde-508	168	5	holds	hold	VERB
ejde-508	168	6	.	.	PUNCT
ejde-508	169	1	similarly	similarly	ADV
ejde-508	169	2	,	,	PUNCT
ejde-508	169	3	if	if	SCONJ
ejde-508	169	4	yn	yn	PRON
ejde-508	169	5	be	be	VERB
ejde-508	169	6	an	an	DET
ejde-508	169	7	eventually	eventually	ADV
ejde-508	169	8	negative	negative	ADJ
ejde-508	169	9	solution	solution	NOUN
ejde-508	169	10	of	of	ADP
ejde-508	169	11	(	(	PUNCT
ejde-508	169	12	1.2	1.2	NUM
ejde-508	169	13	)	)	PUNCT
ejde-508	169	14	then	then	ADV
ejde-508	169	15	proceeding	proceed	VERB
ejde-508	169	16	with	with	ADP
ejde-508	169	17	substitution	substitution	NOUN
ejde-508	169	18	xn	xn	PUNCT
ejde-508	170	1	=	=	SYM
ejde-508	170	2	−yn	−yn	NOUN
ejde-508	170	3	and	and	CCONJ
ejde-508	170	4	taking	take	VERB
ejde-508	170	5	note	note	NOUN
ejde-508	170	6	of	of	ADP
ejde-508	170	7	remark	remark	NOUN
ejde-508	170	8	2.4	2.4	NUM
ejde-508	170	9	,	,	PUNCT
ejde-508	170	10	it	it	PRON
ejde-508	170	11	could	could	AUX
ejde-508	170	12	be	be	AUX
ejde-508	170	13	shown	show	VERB
ejde-508	170	14	β	β	X
ejde-508	170	15	=	=	PUNCT
ejde-508	170	16	limn→∞	limn→∞	X
ejde-508	170	17	wn	wn	NOUN
ejde-508	170	18	=	=	SYM
ejde-508	170	19	0	0	PROPN
ejde-508	170	20	and	and	CCONJ
ejde-508	170	21	the	the	DET
ejde-508	170	22	proof	proof	NOUN
ejde-508	170	23	is	be	AUX
ejde-508	170	24	complete	complete	ADJ
ejde-508	170	25	.	.	PUNCT
ejde-508	171	1	�	�	PROPN
ejde-508	171	2	next	next	ADV
ejde-508	171	3	we	we	PRON
ejde-508	171	4	have	have	VERB
ejde-508	171	5	the	the	DET
ejde-508	171	6	following	following	ADJ
ejde-508	171	7	remark	remark	NOUN
ejde-508	171	8	,	,	PUNCT
ejde-508	171	9	which	which	PRON
ejde-508	171	10	would	would	AUX
ejde-508	171	11	be	be	AUX
ejde-508	171	12	helpful	helpful	ADJ
ejde-508	171	13	in	in	ADP
ejde-508	171	14	proving	prove	VERB
ejde-508	171	15	results	result	NOUN
ejde-508	171	16	concerned	concern	VERB
ejde-508	171	17	with	with	ADP
ejde-508	171	18	neutral	neutral	ADJ
ejde-508	171	19	equation	equation	NOUN
ejde-508	171	20	(	(	PUNCT
ejde-508	171	21	1.1	1.1	NUM
ejde-508	171	22	)	)	PUNCT
ejde-508	171	23	.	.	PUNCT
ejde-508	172	1	remark	remark	PROPN
ejde-508	172	2	2.7	2.7	NUM
ejde-508	172	3	.	.	PUNCT
ejde-508	173	1	lemmas	lemmas	PROPN
ejde-508	173	2	2.3	2.3	NUM
ejde-508	173	3	,	,	PUNCT
ejde-508	173	4	2.5	2.5	NUM
ejde-508	173	5	and	and	CCONJ
ejde-508	173	6	2.6	2.6	NUM
ejde-508	173	7	hold	hold	NOUN
ejde-508	173	8	for	for	ADP
ejde-508	173	9	un	un	PROPN
ejde-508	173	10	≡	≡	PROPN
ejde-508	173	11	0	0	PROPN
ejde-508	173	12	.	.	PUNCT
ejde-508	174	1	in	in	ADP
ejde-508	174	2	that	that	DET
ejde-508	174	3	case	case	NOUN
ejde-508	174	4	cn	cn	X
ejde-508	174	5	=	=	NOUN
ejde-508	174	6	0	0	PROPN
ejde-508	174	7	and	and	CCONJ
ejde-508	174	8	wn	wn	PROPN
ejde-508	174	9	=	=	NOUN
ejde-508	174	10	zn	zn	PROPN
ejde-508	174	11	.	.	PUNCT
ejde-508	175	1	the	the	DET
ejde-508	175	2	following	follow	VERB
ejde-508	175	3	lemmas	lemmas	PROPN
ejde-508	175	4	follow	follow	VERB
ejde-508	175	5	from	from	ADP
ejde-508	175	6	lemmas	lemmas	PROPN
ejde-508	175	7	2.3	2.3	NUM
ejde-508	175	8	,	,	PUNCT
ejde-508	175	9	2.5	2.5	NUM
ejde-508	175	10	,	,	PUNCT
ejde-508	175	11	and	and	CCONJ
ejde-508	175	12	2.6	2.6	NUM
ejde-508	175	13	as	as	ADP
ejde-508	175	14	a	a	DET
ejde-508	175	15	consequence	consequence	NOUN
ejde-508	175	16	of	of	ADP
ejde-508	175	17	the	the	DET
ejde-508	175	18	above	above	ADJ
ejde-508	175	19	remark	remark	NOUN
ejde-508	175	20	.	.	PUNCT
ejde-508	176	1	lemma	lemma	PROPN
ejde-508	176	2	2.8	2.8	NUM
ejde-508	176	3	.	.	PUNCT
ejde-508	177	1	assume	assume	VERB
ejde-508	177	2	(	(	PUNCT
ejde-508	177	3	a1	a1	NOUN
ejde-508	177	4	)	)	PUNCT
ejde-508	177	5	holds	hold	VERB
ejde-508	177	6	.	.	PUNCT
ejde-508	178	1	let	let	VERB
ejde-508	178	2	yn	yn	PRON
ejde-508	178	3	be	be	AUX
ejde-508	178	4	an	an	DET
ejde-508	178	5	eventually	eventually	ADV
ejde-508	178	6	positive	positive	ADJ
ejde-508	178	7	solution	solution	NOUN
ejde-508	178	8	of	of	ADP
ejde-508	178	9	(	(	PUNCT
ejde-508	178	10	1.1	1.1	NUM
ejde-508	178	11	)	)	PUNCT
ejde-508	178	12	,	,	PUNCT
ejde-508	178	13	and	and	CCONJ
ejde-508	178	14	zn	zn	PROPN
ejde-508	178	15	be	be	AUX
ejde-508	178	16	defined	define	VERB
ejde-508	178	17	as	as	ADP
ejde-508	178	18	in	in	ADP
ejde-508	178	19	(	(	PUNCT
ejde-508	178	20	2.5	2.5	NUM
ejde-508	178	21	)	)	PUNCT
ejde-508	178	22	.	.	PUNCT
ejde-508	179	1	then	then	ADV
ejde-508	179	2	∆2zn	∆2zn	PROPN
ejde-508	179	3	=	=	SYM
ejde-508	179	4	−qng(yn−k	−qng(yn−k	PROPN
ejde-508	179	5	)	)	PUNCT
ejde-508	179	6	≤	≤	NOUN
ejde-508	179	7	0	0	NUM
ejde-508	179	8	,	,	PUNCT
ejde-508	179	9	(	(	PUNCT
ejde-508	179	10	2.25	2.25	NUM
ejde-508	179	11	)	)	PUNCT
ejde-508	179	12	and	and	CCONJ
ejde-508	179	13	the	the	DET
ejde-508	179	14	following	following	ADJ
ejde-508	179	15	statements	statement	NOUN
ejde-508	179	16	hold	hold	VERB
ejde-508	179	17	.	.	PUNCT
ejde-508	180	1	(	(	PUNCT
ejde-508	180	2	a	a	X
ejde-508	180	3	)	)	PUNCT
ejde-508	180	4	if	if	SCONJ
ejde-508	180	5	(	(	PUNCT
ejde-508	180	6	a2	a2	PROPN
ejde-508	180	7	)	)	PUNCT
ejde-508	180	8	,	,	PUNCT
ejde-508	180	9	(	(	PUNCT
ejde-508	180	10	a5	a5	NOUN
ejde-508	180	11	)	)	PUNCT
ejde-508	180	12	hold	hold	VERB
ejde-508	180	13	and	and	CCONJ
ejde-508	180	14	pn	pn	NOUN
ejde-508	180	15	satisfies	satisfie	NOUN
ejde-508	180	16	(	(	PUNCT
ejde-508	180	17	1.4	1.4	NUM
ejde-508	180	18	)	)	PUNCT
ejde-508	180	19	,	,	PUNCT
ejde-508	180	20	then	then	ADV
ejde-508	180	21	either	either	CCONJ
ejde-508	180	22	∆wn	∆wn	PROPN
ejde-508	180	23	<	<	X
ejde-508	180	24	0	0	PROPN
ejde-508	180	25	for	for	ADP
ejde-508	180	26	large	large	ADJ
ejde-508	180	27	n	n	NOUN
ejde-508	180	28	which	which	PRON
ejde-508	180	29	implies	imply	VERB
ejde-508	180	30	lim	lim	PROPN
ejde-508	180	31	n→∞	n→∞	X
ejde-508	180	32	zn	zn	PROPN
ejde-508	180	33	=	=	SYM
ejde-508	180	34	−∞	−∞	PROPN
ejde-508	180	35	,	,	PUNCT
ejde-508	180	36	(	(	PUNCT
ejde-508	180	37	2.26	2.26	NUM
ejde-508	180	38	)	)	PUNCT
ejde-508	180	39	or	or	CCONJ
ejde-508	180	40	∆wn	∆wn	X
ejde-508	180	41	>	>	X
ejde-508	180	42	0	0	PUNCT
ejde-508	181	1	for	for	ADP
ejde-508	181	2	large	large	ADJ
ejde-508	181	3	n	n	NOUN
ejde-508	181	4	which	which	PRON
ejde-508	181	5	implies	imply	VERB
ejde-508	181	6	lim	lim	PROPN
ejde-508	181	7	n→∞	n→∞	X
ejde-508	181	8	zn	zn	PROPN
ejde-508	181	9	=	=	SYM
ejde-508	181	10	0	0	PROPN
ejde-508	181	11	,	,	PUNCT
ejde-508	181	12	(	(	PUNCT
ejde-508	181	13	2.27	2.27	NUM
ejde-508	181	14	)	)	PUNCT
ejde-508	181	15	zn	zn	NOUN
ejde-508	181	16	<	<	X
ejde-508	181	17	0	0	PROPN
ejde-508	181	18	,	,	PUNCT
ejde-508	181	19	∆zn	∆zn	PROPN
ejde-508	181	20	>	>	X
ejde-508	181	21	0	0	PROPN
ejde-508	181	22	,	,	PUNCT
ejde-508	181	23	lim	lim	PROPN
ejde-508	181	24	n→∞	n→∞	NUM
ejde-508	181	25	∆zn	∆zn	PROPN
ejde-508	182	1	=	=	NOUN
ejde-508	182	2	0	0	X
ejde-508	182	3	.	.	PUNCT
ejde-508	183	1	(	(	PUNCT
ejde-508	183	2	2.28	2.28	NUM
ejde-508	183	3	)	)	PUNCT
ejde-508	183	4	(	(	PUNCT
ejde-508	183	5	b	b	X
ejde-508	183	6	)	)	PUNCT
ejde-508	183	7	if	if	SCONJ
ejde-508	183	8	in	in	ADP
ejde-508	183	9	addition	addition	NOUN
ejde-508	183	10	δ	δ	NOUN
ejde-508	183	11	≤	≤	ADV
ejde-508	183	12	1	1	NUM
ejde-508	184	1	and	and	CCONJ
ejde-508	184	2	if	if	SCONJ
ejde-508	184	3	pn	pn	PROPN
ejde-508	184	4	satisfy	satisfy	VERB
ejde-508	184	5	(	(	PUNCT
ejde-508	184	6	1.5	1.5	NUM
ejde-508	184	7	)	)	PUNCT
ejde-508	184	8	,	,	PUNCT
ejde-508	184	9	then	then	ADV
ejde-508	184	10	only	only	ADV
ejde-508	184	11	(	(	PUNCT
ejde-508	184	12	2.27	2.27	NUM
ejde-508	184	13	)	)	PUNCT
ejde-508	184	14	and	and	CCONJ
ejde-508	184	15	(	(	PUNCT
ejde-508	184	16	2.28	2.28	NUM
ejde-508	184	17	)	)	PUNCT
ejde-508	184	18	hold	hold	NOUN
ejde-508	184	19	.	.	PUNCT
ejde-508	185	1	lemma	lemma	PROPN
ejde-508	185	2	2.9	2.9	NUM
ejde-508	185	3	.	.	PUNCT
ejde-508	186	1	if	if	SCONJ
ejde-508	186	2	yn	yn	PRON
ejde-508	186	3	is	be	AUX
ejde-508	186	4	any	any	DET
ejde-508	186	5	eventually	eventually	ADV
ejde-508	186	6	positive	positive	ADJ
ejde-508	186	7	solution	solution	NOUN
ejde-508	186	8	of	of	ADP
ejde-508	186	9	(	(	PUNCT
ejde-508	186	10	1.1	1.1	NUM
ejde-508	186	11	)	)	PUNCT
ejde-508	186	12	,	,	PUNCT
ejde-508	186	13	with	with	ADP
ejde-508	186	14	zn	zn	PROPN
ejde-508	186	15	as	as	ADP
ejde-508	186	16	in	in	ADP
ejde-508	186	17	(	(	PUNCT
ejde-508	186	18	2.5	2.5	NUM
ejde-508	186	19	)	)	PUNCT
ejde-508	186	20	,	,	PUNCT
ejde-508	186	21	then	then	ADV
ejde-508	186	22	the	the	DET
ejde-508	186	23	following	following	ADJ
ejde-508	186	24	statements	statement	NOUN
ejde-508	186	25	hold	hold	VERB
ejde-508	186	26	.	.	PUNCT
ejde-508	187	1	(	(	PUNCT
ejde-508	187	2	a	a	X
ejde-508	187	3	)	)	PUNCT
ejde-508	187	4	if	if	SCONJ
ejde-508	187	5	(	(	PUNCT
ejde-508	187	6	2.26	2.26	NUM
ejde-508	187	7	)	)	PUNCT
ejde-508	187	8	holds	hold	VERB
ejde-508	187	9	then	then	ADV
ejde-508	187	10	,	,	PUNCT
ejde-508	187	11	(	(	PUNCT
ejde-508	187	12	2.25	2.25	NUM
ejde-508	187	13	)	)	PUNCT
ejde-508	188	1	implies	imply	VERB
ejde-508	188	2	(	(	PUNCT
ejde-508	188	3	2.19	2.19	NUM
ejde-508	188	4	)	)	PUNCT
ejde-508	188	5	,	,	PUNCT
ejde-508	188	6	i.e	i.e	PROPN
ejde-508	188	7	;	;	PUNCT
ejde-508	188	8	∆zn+1	∆zn+1	X
ejde-508	188	9	+	+	CCONJ
ejde-508	188	10	qng(yn−k	qng(yn−k	X
ejde-508	188	11	)	)	PUNCT
ejde-508	188	12	≤	≤	NOUN
ejde-508	188	13	0	0	NUM
ejde-508	188	14	.	.	PUNCT
ejde-508	189	1	(	(	PUNCT
ejde-508	189	2	b	b	X
ejde-508	189	3	)	)	PUNCT
ejde-508	189	4	if	if	SCONJ
ejde-508	189	5	(	(	PUNCT
ejde-508	189	6	2.27	2.27	NUM
ejde-508	189	7	)	)	PUNCT
ejde-508	189	8	holds	hold	VERB
ejde-508	189	9	,	,	PUNCT
ejde-508	189	10	then	then	ADV
ejde-508	189	11	(	(	PUNCT
ejde-508	189	12	2.25	2.25	NUM
ejde-508	189	13	)	)	PUNCT
ejde-508	189	14	implies	imply	VERB
ejde-508	189	15	∆zn	∆zn	PROPN
ejde-508	189	16	−	−	PROPN
ejde-508	190	1	qng(yn−k	qng(yn−k	PROPN
ejde-508	190	2	)	)	PUNCT
ejde-508	190	3	≥	≥	NOUN
ejde-508	190	4	0	0	NUM
ejde-508	190	5	.	.	PUNCT
ejde-508	191	1	(	(	PUNCT
ejde-508	191	2	2.29	2.29	NUM
ejde-508	191	3	)	)	PUNCT
ejde-508	191	4	lemma	lemma	PROPN
ejde-508	191	5	2.10	2.10	NUM
ejde-508	191	6	.	.	PUNCT
ejde-508	192	1	let	let	VERB
ejde-508	192	2	(	(	PUNCT
ejde-508	192	3	a1	a1	NOUN
ejde-508	192	4	)	)	PUNCT
ejde-508	192	5	,	,	PUNCT
ejde-508	192	6	(	(	PUNCT
ejde-508	192	7	a3	a3	NOUN
ejde-508	192	8	)	)	PUNCT
ejde-508	192	9	,	,	PUNCT
ejde-508	192	10	(	(	PUNCT
ejde-508	192	11	2.21	2.21	NUM
ejde-508	192	12	)	)	PUNCT
ejde-508	192	13	,	,	PUNCT
ejde-508	192	14	and	and	CCONJ
ejde-508	192	15	(	(	PUNCT
ejde-508	192	16	2.22	2.22	NUM
ejde-508	192	17	)	)	PUNCT
ejde-508	192	18	hold	hold	NOUN
ejde-508	192	19	,	,	PUNCT
ejde-508	192	20	let	let	VERB
ejde-508	192	21	yn	yn	PRON
ejde-508	192	22	be	be	AUX
ejde-508	192	23	an	an	DET
ejde-508	192	24	eventually	eventually	ADV
ejde-508	192	25	positive	positive	ADJ
ejde-508	192	26	or	or	CCONJ
ejde-508	192	27	eventually	eventually	ADV
ejde-508	192	28	negative	negative	ADJ
ejde-508	192	29	solution	solution	NOUN
ejde-508	192	30	of	of	ADP
ejde-508	192	31	(	(	PUNCT
ejde-508	192	32	1.1	1.1	NUM
ejde-508	192	33	)	)	PUNCT
ejde-508	192	34	,	,	PUNCT
ejde-508	192	35	and	and	CCONJ
ejde-508	192	36	let	let	VERB
ejde-508	192	37	zn	zn	PRON
ejde-508	192	38	be	be	AUX
ejde-508	192	39	as	as	ADP
ejde-508	192	40	in	in	ADP
ejde-508	192	41	(	(	PUNCT
ejde-508	192	42	2.5	2.5	NUM
ejde-508	192	43	)	)	PUNCT
ejde-508	192	44	.	.	PUNCT
ejde-508	193	1	if	if	SCONJ
ejde-508	193	2	pn	pn	PROPN
ejde-508	193	3	satisfies	satisfie	NOUN
ejde-508	193	4	(	(	PUNCT
ejde-508	193	5	1.6	1.6	NUM
ejde-508	193	6	)	)	PUNCT
ejde-508	193	7	or	or	CCONJ
ejde-508	193	8	(	(	PUNCT
ejde-508	193	9	1.7	1.7	NUM
ejde-508	193	10	)	)	PUNCT
ejde-508	193	11	then	then	ADV
ejde-508	193	12	limn→∞	limn→∞	VERB
ejde-508	193	13	zn	zn	PROPN
ejde-508	193	14	=	=	SYM
ejde-508	193	15	0	0	PROPN
ejde-508	193	16	.	.	PUNCT
ejde-508	194	1	consequently	consequently	ADV
ejde-508	194	2	,	,	PUNCT
ejde-508	194	3	zn	zn	PROPN
ejde-508	194	4	<	<	X
ejde-508	194	5	0	0	PROPN
ejde-508	194	6	,	,	PUNCT
ejde-508	194	7	∆zn	∆zn	NOUN
ejde-508	194	8	>	>	X
ejde-508	194	9	0	0	PUNCT
ejde-508	194	10	for	for	ADP
ejde-508	194	11	yn	yn	PRON
ejde-508	194	12	>	>	X
ejde-508	194	13	0	0	PROPN
ejde-508	194	14	and	and	CCONJ
ejde-508	194	15	zn	zn	X
ejde-508	194	16	>	>	X
ejde-508	194	17	0	0	PROPN
ejde-508	194	18	,	,	PUNCT
ejde-508	194	19	∆zn	∆zn	NOUN
ejde-508	194	20	<	<	X
ejde-508	194	21	0	0	PROPN
ejde-508	194	22	for	for	ADP
ejde-508	194	23	yn	yn	PRON
ejde-508	194	24	<	<	X
ejde-508	194	25	0	0	NUM
ejde-508	194	26	.	.	PUNCT
ejde-508	195	1	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	195	2	oscillation	oscillation	NOUN
ejde-508	195	3	for	for	ADP
ejde-508	195	4	second	second	ADJ
ejde-508	195	5	order	order	NOUN
ejde-508	195	6	neutral	neutral	ADJ
ejde-508	195	7	equations	equation	NOUN
ejde-508	195	8	7	7	NUM
ejde-508	195	9	3	3	NUM
ejde-508	195	10	.	.	PUNCT
ejde-508	195	11	main	main	ADJ
ejde-508	195	12	results	result	NOUN
ejde-508	195	13	part	part	NOUN
ejde-508	195	14	i	i	PRON
ejde-508	195	15	in	in	ADP
ejde-508	195	16	this	this	DET
ejde-508	195	17	section	section	NOUN
ejde-508	195	18	,	,	PUNCT
ejde-508	195	19	we	we	PRON
ejde-508	195	20	find	find	VERB
ejde-508	195	21	sufficient	sufficient	ADJ
ejde-508	195	22	conditions	condition	NOUN
ejde-508	195	23	,	,	PUNCT
ejde-508	195	24	so	so	SCONJ
ejde-508	195	25	that	that	SCONJ
ejde-508	195	26	,	,	PUNCT
ejde-508	195	27	all	all	DET
ejde-508	195	28	unbounded	unbounded	ADJ
ejde-508	195	29	solutions	solution	NOUN
ejde-508	195	30	of	of	ADP
ejde-508	195	31	(	(	PUNCT
ejde-508	195	32	1.2	1.2	NUM
ejde-508	195	33	)	)	PUNCT
ejde-508	195	34	oscillate	oscillate	NOUN
ejde-508	195	35	.	.	PUNCT
ejde-508	196	1	remark	remark	PROPN
ejde-508	196	2	3.1	3.1	NUM
ejde-508	196	3	(	(	PUNCT
ejde-508	196	4	[	[	X
ejde-508	196	5	6	6	NUM
ejde-508	196	6	,	,	PUNCT
ejde-508	196	7	remark	remark	NOUN
ejde-508	196	8	4.8	4.8	NUM
ejde-508	196	9	]	]	PUNCT
ejde-508	196	10	)	)	PUNCT
ejde-508	196	11	.	.	PUNCT
ejde-508	197	1	assumption	assumption	NOUN
ejde-508	197	2	(	(	PUNCT
ejde-508	197	3	a4	a4	NOUN
ejde-508	197	4	)	)	PUNCT
ejde-508	197	5	and	and	CCONJ
ejde-508	197	6	the	the	DET
ejde-508	197	7	condition	condition	NOUN
ejde-508	197	8	∞∑	∞∑	NUM
ejde-508	197	9	j=1	j=1	NOUN
ejde-508	197	10	qnj	qnj	VERB
ejde-508	197	11	=	=	NOUN
ejde-508	197	12	∞	∞	PROPN
ejde-508	197	13	,	,	PUNCT
ejde-508	197	14	where	where	SCONJ
ejde-508	197	15	qnj	qnj	NOUN
ejde-508	197	16	is	be	AUX
ejde-508	197	17	any	any	DET
ejde-508	197	18	subsequence	subsequence	NOUN
ejde-508	197	19	of	of	ADP
ejde-508	197	20	qn	qn	NOUN
ejde-508	197	21	(	(	PUNCT
ejde-508	197	22	3.1	3.1	NUM
ejde-508	197	23	)	)	PUNCT
ejde-508	197	24	are	be	AUX
ejde-508	197	25	equivalent	equivalent	ADJ
ejde-508	197	26	.	.	PUNCT
ejde-508	198	1	theorem	theorem	ADJ
ejde-508	198	2	3.2	3.2	NUM
ejde-508	198	3	.	.	PUNCT
ejde-508	199	1	let	let	VERB
ejde-508	199	2	(	(	PUNCT
ejde-508	199	3	a1	a1	NOUN
ejde-508	199	4	)	)	PUNCT
ejde-508	199	5	,	,	PUNCT
ejde-508	199	6	(	(	PUNCT
ejde-508	199	7	a4)–(a7	a4)–(a7	ADV
ejde-508	199	8	)	)	PUNCT
ejde-508	199	9	hold	hold	NOUN
ejde-508	199	10	,	,	PUNCT
ejde-508	199	11	and	and	CCONJ
ejde-508	199	12	s	s	VERB
ejde-508	199	13	>	>	X
ejde-508	199	14	k	k	PROPN
ejde-508	200	1	+	+	PROPN
ejde-508	200	2	1	1	NUM
ejde-508	200	3	,	,	PUNCT
ejde-508	200	4	(	(	PUNCT
ejde-508	200	5	1.4	1.4	NUM
ejde-508	200	6	)	)	PUNCT
ejde-508	200	7	be	be	AUX
ejde-508	200	8	satisfied	satisfied	ADJ
ejde-508	200	9	.	.	PUNCT
ejde-508	201	1	if∣∣	if∣∣	PROPN
ejde-508	201	2	∫	∫	PROPN
ejde-508	201	3	∞	∞	PROPN
ejde-508	201	4	a	a	DET
ejde-508	201	5	du	du	NOUN
ejde-508	201	6	g(u	g(u	X
ejde-508	201	7	)	)	PUNCT
ejde-508	201	8	∣∣	∣∣	X
ejde-508	201	9	<	<	X
ejde-508	201	10	∞	∞	PROPN
ejde-508	201	11	,	,	PUNCT
ejde-508	201	12	∀a	∀a	NOUN
ejde-508	201	13	∈	∈	NOUN
ejde-508	201	14	r	r	NOUN
ejde-508	201	15	,	,	PUNCT
ejde-508	201	16	(	(	PUNCT
ejde-508	201	17	3.2	3.2	NUM
ejde-508	201	18	)	)	PUNCT
ejde-508	201	19	then	then	ADV
ejde-508	201	20	every	every	DET
ejde-508	201	21	unbounded	unbounded	ADJ
ejde-508	201	22	solution	solution	NOUN
ejde-508	201	23	of	of	ADP
ejde-508	201	24	(	(	PUNCT
ejde-508	201	25	1.2	1.2	NUM
ejde-508	201	26	)	)	PUNCT
ejde-508	201	27	oscillates	oscillate	NOUN
ejde-508	201	28	.	.	PUNCT
ejde-508	202	1	proof	proof	NOUN
ejde-508	202	2	.	.	PUNCT
ejde-508	203	1	to	to	PART
ejde-508	203	2	obtain	obtain	VERB
ejde-508	203	3	a	a	DET
ejde-508	203	4	contradiction	contradiction	NOUN
ejde-508	203	5	,	,	PUNCT
ejde-508	203	6	let	let	VERB
ejde-508	203	7	yn	yn	PRON
ejde-508	203	8	be	be	AUX
ejde-508	203	9	an	an	DET
ejde-508	203	10	eventually	eventually	ADV
ejde-508	203	11	positive	positive	ADJ
ejde-508	203	12	solution	solution	NOUN
ejde-508	203	13	of	of	ADP
ejde-508	203	14	(	(	PUNCT
ejde-508	203	15	1.2	1.2	NUM
ejde-508	203	16	)	)	PUNCT
ejde-508	203	17	.	.	PUNCT
ejde-508	204	1	setting	set	VERB
ejde-508	204	2	zn	zn	PROPN
ejde-508	204	3	,	,	PUNCT
ejde-508	204	4	wn	wn	PROPN
ejde-508	204	5	and	and	CCONJ
ejde-508	204	6	cn	cn	PROPN
ejde-508	204	7	as	as	ADP
ejde-508	204	8	in	in	ADP
ejde-508	204	9	(	(	PUNCT
ejde-508	204	10	2.5	2.5	NUM
ejde-508	204	11	)	)	PUNCT
ejde-508	204	12	,	,	PUNCT
ejde-508	204	13	(	(	PUNCT
ejde-508	204	14	2.6	2.6	NUM
ejde-508	204	15	)	)	PUNCT
ejde-508	204	16	and	and	CCONJ
ejde-508	204	17	(	(	PUNCT
ejde-508	204	18	2.2	2.2	NUM
ejde-508	204	19	)	)	PUNCT
ejde-508	204	20	respectively	respectively	ADV
ejde-508	204	21	,	,	PUNCT
ejde-508	204	22	we	we	PRON
ejde-508	204	23	obtain	obtain	VERB
ejde-508	204	24	(	(	PUNCT
ejde-508	204	25	2.10	2.10	NUM
ejde-508	204	26	)	)	PUNCT
ejde-508	204	27	.	.	PUNCT
ejde-508	205	1	note	note	VERB
ejde-508	205	2	that	that	SCONJ
ejde-508	205	3	(	(	PUNCT
ejde-508	205	4	a4	a4	NOUN
ejde-508	205	5	)	)	PUNCT
ejde-508	205	6	implies	imply	VERB
ejde-508	205	7	(	(	PUNCT
ejde-508	205	8	a2	a2	PROPN
ejde-508	205	9	)	)	PUNCT
ejde-508	205	10	.	.	PUNCT
ejde-508	206	1	hence	hence	ADV
ejde-508	206	2	,	,	PUNCT
ejde-508	206	3	by	by	ADP
ejde-508	206	4	lemma	lemma	PROPN
ejde-508	206	5	2.3(a	2.3(a	NUM
ejde-508	206	6	)	)	PUNCT
ejde-508	206	7	,	,	PUNCT
ejde-508	206	8	we	we	PRON
ejde-508	206	9	observe	observe	VERB
ejde-508	206	10	that	that	SCONJ
ejde-508	206	11	either	either	CCONJ
ejde-508	206	12	(	(	PUNCT
ejde-508	206	13	2.7	2.7	NUM
ejde-508	206	14	)	)	PUNCT
ejde-508	206	15	or	or	CCONJ
ejde-508	206	16	(	(	PUNCT
ejde-508	206	17	2.8	2.8	NUM
ejde-508	206	18	)	)	PUNCT
ejde-508	206	19	holds	hold	VERB
ejde-508	206	20	.	.	PUNCT
ejde-508	207	1	first	first	ADV
ejde-508	207	2	we	we	PRON
ejde-508	207	3	consider	consider	VERB
ejde-508	207	4	the	the	DET
ejde-508	207	5	case	case	NOUN
ejde-508	207	6	when	when	SCONJ
ejde-508	207	7	(	(	PUNCT
ejde-508	207	8	2.7	2.7	NUM
ejde-508	207	9	)	)	PUNCT
ejde-508	207	10	holds	hold	VERB
ejde-508	207	11	.	.	PUNCT
ejde-508	208	1	using	use	VERB
ejde-508	208	2	lemma	lemma	PROPN
ejde-508	208	3	2.5(a	2.5(a	NUM
ejde-508	208	4	)	)	PUNCT
ejde-508	208	5	,	,	PUNCT
ejde-508	208	6	we	we	PRON
ejde-508	208	7	show	show	VERB
ejde-508	208	8	that	that	SCONJ
ejde-508	208	9	(	(	PUNCT
ejde-508	208	10	2.10	2.10	NUM
ejde-508	208	11	)	)	PUNCT
ejde-508	208	12	implies	imply	VERB
ejde-508	208	13	(	(	PUNCT
ejde-508	208	14	2.19	2.19	NUM
ejde-508	208	15	)	)	PUNCT
ejde-508	208	16	.	.	PUNCT
ejde-508	209	1	from	from	ADP
ejde-508	209	2	(	(	PUNCT
ejde-508	209	3	2.7	2.7	NUM
ejde-508	209	4	)	)	PUNCT
ejde-508	209	5	,	,	PUNCT
ejde-508	209	6	(	(	PUNCT
ejde-508	209	7	2.17	2.17	NUM
ejde-508	209	8	)	)	PUNCT
ejde-508	209	9	and	and	CCONJ
ejde-508	209	10	lemma	lemma	PROPN
ejde-508	209	11	2.2	2.2	NUM
ejde-508	209	12	,	,	PUNCT
ejde-508	209	13	it	it	PRON
ejde-508	209	14	follows	follow	VERB
ejde-508	209	15	that	that	SCONJ
ejde-508	209	16	limn→∞	limn→∞	PROPN
ejde-508	209	17	zn	zn	X
ejde-508	209	18	=	=	SYM
ejde-508	209	19	−∞	−∞	PROPN
ejde-508	209	20	,	,	PUNCT
ejde-508	209	21	which	which	PRON
ejde-508	209	22	implies	imply	VERB
ejde-508	209	23	∆zn	∆zn	PROPN
ejde-508	209	24	<	<	X
ejde-508	209	25	0	0	PUNCT
ejde-508	209	26	and	and	CCONJ
ejde-508	209	27	zn	zn	X
ejde-508	209	28	<	<	X
ejde-508	209	29	0	0	NUM
ejde-508	209	30	for	for	ADP
ejde-508	209	31	large	large	ADJ
ejde-508	209	32	n.	n.	NOUN
ejde-508	209	33	if	if	SCONJ
ejde-508	209	34	pn	pn	PROPN
ejde-508	209	35	=	=	NOUN
ejde-508	209	36	0	0	PROPN
ejde-508	210	1	then	then	ADV
ejde-508	210	2	zn	zn	PROPN
ejde-508	210	3	=	=	SYM
ejde-508	210	4	yn	yn	PROPN
ejde-508	210	5	<	<	X
ejde-508	210	6	0	0	PROPN
ejde-508	210	7	,	,	PUNCT
ejde-508	210	8	a	a	DET
ejde-508	210	9	contradiction	contradiction	NOUN
ejde-508	210	10	.	.	PUNCT
ejde-508	211	1	hence	hence	ADV
ejde-508	211	2	pn	pn	VERB
ejde-508	211	3	>	>	X
ejde-508	211	4	0	0	PROPN
ejde-508	211	5	.	.	PUNCT
ejde-508	212	1	from	from	ADP
ejde-508	212	2	(	(	PUNCT
ejde-508	212	3	2.5	2.5	NUM
ejde-508	212	4	)	)	PUNCT
ejde-508	212	5	,	,	PUNCT
ejde-508	212	6	we	we	PRON
ejde-508	212	7	find	find	VERB
ejde-508	212	8	yn−k	yn−k	PROPN
ejde-508	212	9	≥	≥	PROPN
ejde-508	212	10	−zn+s−k/(pδ	−zn+s−k/(pδ	PROPN
ejde-508	212	11	)	)	PUNCT
ejde-508	212	12	.	.	PUNCT
ejde-508	213	1	using	use	VERB
ejde-508	213	2	this	this	PRON
ejde-508	213	3	in	in	ADP
ejde-508	213	4	(	(	PUNCT
ejde-508	213	5	2.19	2.19	NUM
ejde-508	213	6	)	)	PUNCT
ejde-508	213	7	,	,	PUNCT
ejde-508	213	8	we	we	PRON
ejde-508	213	9	obtain	obtain	VERB
ejde-508	213	10	∆zn+1	∆zn+1	X
ejde-508	213	11	+	+	CCONJ
ejde-508	213	12	qng	qng	PROPN
ejde-508	213	13	(	(	PUNCT
ejde-508	213	14	−zn+s−k	−zn+s−k	NOUN
ejde-508	213	15	pδ	pδ	NOUN
ejde-508	213	16	)	)	PUNCT
ejde-508	213	17	≤	≤	NOUN
ejde-508	213	18	0	0	NUM
ejde-508	213	19	.	.	PUNCT
ejde-508	214	1	(	(	PUNCT
ejde-508	214	2	3.3	3.3	NUM
ejde-508	214	3	)	)	PUNCT
ejde-508	214	4	note	note	VERB
ejde-508	214	5	that	that	SCONJ
ejde-508	214	6	−zn/(pδ	−zn/(pδ	PROPN
ejde-508	214	7	)	)	PUNCT
ejde-508	214	8	=	=	NOUN
ejde-508	214	9	vn	vn	PROPN
ejde-508	214	10	implies	imply	VERB
ejde-508	214	11	∆zn	∆zn	PROPN
ejde-508	214	12	=	=	PUNCT
ejde-508	215	1	−pδ∆vn	−pδ∆vn	NOUN
ejde-508	215	2	.	.	PUNCT
ejde-508	216	1	then	then	ADV
ejde-508	216	2	,	,	PUNCT
ejde-508	216	3	substituting	substitute	VERB
ejde-508	216	4	this	this	DET
ejde-508	216	5	expression	expression	NOUN
ejde-508	216	6	in	in	ADP
ejde-508	216	7	the	the	DET
ejde-508	216	8	above	above	NOUN
ejde-508	216	9	,	,	PUNCT
ejde-508	216	10	we	we	PRON
ejde-508	216	11	obtain	obtain	VERB
ejde-508	216	12	pδ∆vn+1	pδ∆vn+1	NOUN
ejde-508	216	13	−	−	NOUN
ejde-508	216	14	qng(vn+s−k	qng(vn+s−k	NOUN
ejde-508	216	15	)	)	PUNCT
ejde-508	216	16	≥	≥	NOUN
ejde-508	216	17	0	0	NUM
ejde-508	216	18	.	.	PUNCT
ejde-508	217	1	note	note	VERB
ejde-508	217	2	that	that	SCONJ
ejde-508	217	3	vn	vn	PROPN
ejde-508	217	4	>	>	X
ejde-508	217	5	0	0	PROPN
ejde-508	217	6	,	,	PUNCT
ejde-508	217	7	limn→∞	limn→∞	PROPN
ejde-508	217	8	vn	vn	X
ejde-508	217	9	=	=	SYM
ejde-508	217	10	∞	∞	PROPN
ejde-508	217	11	and	and	CCONJ
ejde-508	217	12	vn	vn	PROPN
ejde-508	217	13	is	be	AUX
ejde-508	217	14	increasing	increase	VERB
ejde-508	217	15	.	.	PUNCT
ejde-508	218	1	dividing	divide	VERB
ejde-508	218	2	both	both	DET
ejde-508	218	3	sides	side	NOUN
ejde-508	218	4	by	by	ADP
ejde-508	218	5	g(vn+s−k	g(vn+s−k	NOUN
ejde-508	218	6	)	)	PUNCT
ejde-508	218	7	,	,	PUNCT
ejde-508	218	8	we	we	PRON
ejde-508	218	9	obtain	obtain	VERB
ejde-508	218	10	pδ	pδ	PRON
ejde-508	218	11	∆vn+1	∆vn+1	NUM
ejde-508	218	12	g(vn+s−k	g(vn+s−k	NOUN
ejde-508	218	13	)	)	PUNCT
ejde-508	218	14	≥	≥	PROPN
ejde-508	218	15	qn	qn	NOUN
ejde-508	218	16	.	.	PROPN
ejde-508	219	1	(	(	PUNCT
ejde-508	219	2	3.4	3.4	NUM
ejde-508	219	3	)	)	PUNCT
ejde-508	219	4	then	then	ADV
ejde-508	219	5	writing	write	VERB
ejde-508	219	6	∆vn+1	∆vn+1	PUNCT
ejde-508	219	7	=	=	SYM
ejde-508	219	8	∫	∫	PROPN
ejde-508	220	1	vn+2	vn+2	PROPN
ejde-508	220	2	vn+1	vn+1	PROPN
ejde-508	220	3	dx	dx	PROPN
ejde-508	220	4	,	,	PUNCT
ejde-508	220	5	where	where	SCONJ
ejde-508	220	6	vn+1	vn+1	PROPN
ejde-508	220	7	≤	≤	NUM
ejde-508	220	8	x	x	PUNCT
ejde-508	220	9	≤	≤	NOUN
ejde-508	220	10	vn+2	vn+2	NUM
ejde-508	220	11	,	,	PUNCT
ejde-508	220	12	and	and	CCONJ
ejde-508	220	13	using	use	VERB
ejde-508	220	14	s−	s−	PROPN
ejde-508	220	15	k	k	PROPN
ejde-508	220	16	≥	≥	NUM
ejde-508	220	17	2	2	NUM
ejde-508	220	18	,	,	PUNCT
ejde-508	220	19	we	we	PRON
ejde-508	220	20	obtain	obtain	VERB
ejde-508	220	21	qn	qn	NOUN
ejde-508	220	22	≤	≤	NOUN
ejde-508	220	23	pδ	pδ	ADP
ejde-508	220	24	∫	∫	PROPN
ejde-508	220	25	vn+2	vn+2	PROPN
ejde-508	220	26	vn+1	vn+1	PROPN
ejde-508	220	27	dx	dx	PROPN
ejde-508	220	28	g(x	g(x	PROPN
ejde-508	220	29	)	)	PUNCT
ejde-508	220	30	.	.	PUNCT
ejde-508	221	1	summing	sum	VERB
ejde-508	221	2	n2	n2	NOUN
ejde-508	221	3	to	to	ADP
ejde-508	221	4	l	l	NOUN
ejde-508	221	5	−	−	PROPN
ejde-508	221	6	1	1	NUM
ejde-508	221	7	,	,	PUNCT
ejde-508	221	8	and	and	CCONJ
ejde-508	221	9	then	then	ADV
ejde-508	221	10	taking	take	VERB
ejde-508	221	11	limit	limit	NOUN
ejde-508	221	12	l→∞	l→∞	NOUN
ejde-508	221	13	,	,	PUNCT
ejde-508	221	14	we	we	PRON
ejde-508	221	15	obtain	obtain	VERB
ejde-508	221	16	∞∑	∞∑	NUM
ejde-508	221	17	n	n	CCONJ
ejde-508	221	18	=	=	NOUN
ejde-508	221	19	n2	n2	NOUN
ejde-508	221	20	qn	qn	NOUN
ejde-508	221	21	≤	≤	PROPN
ejde-508	221	22	pδ	pδ	ADP
ejde-508	221	23	∫	∫	PROPN
ejde-508	221	24	∞	∞	NUM
ejde-508	221	25	vn2	vn2	PROPN
ejde-508	222	1	+	+	PROPN
ejde-508	222	2	1	1	NUM
ejde-508	222	3	dx	dx	NOUN
ejde-508	222	4	g(x	g(x	PROPN
ejde-508	222	5	)	)	PUNCT
ejde-508	223	1	<	<	X
ejde-508	223	2	∞	∞	NUM
ejde-508	223	3	,	,	PUNCT
ejde-508	223	4	by	by	ADP
ejde-508	223	5	(	(	PUNCT
ejde-508	223	6	3.2	3.2	NUM
ejde-508	223	7	)	)	PUNCT
ejde-508	223	8	,	,	PUNCT
ejde-508	223	9	which	which	PRON
ejde-508	223	10	contradicts	contradict	VERB
ejde-508	223	11	(	(	PUNCT
ejde-508	223	12	a2	a2	NOUN
ejde-508	223	13	)	)	PUNCT
ejde-508	223	14	.	.	PUNCT
ejde-508	224	1	now	now	ADV
ejde-508	224	2	we	we	PRON
ejde-508	224	3	consider	consider	VERB
ejde-508	224	4	the	the	DET
ejde-508	224	5	case	case	NOUN
ejde-508	224	6	when	when	SCONJ
ejde-508	224	7	(	(	PUNCT
ejde-508	224	8	2.8	2.8	NUM
ejde-508	224	9	)	)	PUNCT
ejde-508	224	10	holds	hold	VERB
ejde-508	224	11	.	.	PUNCT
ejde-508	225	1	consequently	consequently	ADV
ejde-508	225	2	,	,	PUNCT
ejde-508	225	3	we	we	PRON
ejde-508	225	4	obtain	obtain	VERB
ejde-508	225	5	(	(	PUNCT
ejde-508	225	6	2.9	2.9	NUM
ejde-508	225	7	)	)	PUNCT
ejde-508	225	8	.	.	PUNCT
ejde-508	226	1	then	then	ADV
ejde-508	226	2	taking	take	VERB
ejde-508	226	3	summation	summation	NOUN
ejde-508	226	4	in	in	ADP
ejde-508	226	5	(	(	PUNCT
ejde-508	226	6	2.10	2.10	NUM
ejde-508	226	7	)	)	PUNCT
ejde-508	226	8	from	from	ADP
ejde-508	226	9	n2	n2	PROPN
ejde-508	226	10	to∞	to∞	PROPN
ejde-508	226	11	we	we	PRON
ejde-508	226	12	find	find	VERB
ejde-508	226	13	(	(	PUNCT
ejde-508	226	14	2.13	2.13	NUM
ejde-508	226	15	)	)	PUNCT
ejde-508	226	16	.	.	PUNCT
ejde-508	227	1	as	as	SCONJ
ejde-508	227	2	yn	yn	PROPN
ejde-508	227	3	is	be	AUX
ejde-508	227	4	unbounded	unbounde	VERB
ejde-508	227	5	,	,	PUNCT
ejde-508	227	6	we	we	PRON
ejde-508	227	7	can	can	AUX
ejde-508	227	8	find	find	VERB
ejde-508	227	9	a	a	DET
ejde-508	227	10	subsequence	subsequence	NOUN
ejde-508	227	11	{	{	PUNCT
ejde-508	227	12	ynj	ynj	PROPN
ejde-508	227	13	}	}	PUNCT
ejde-508	227	14	of	of	ADP
ejde-508	227	15	{	{	PUNCT
ejde-508	227	16	yn	yn	PROPN
ejde-508	227	17	}	}	PUNCT
ejde-508	227	18	which	which	PRON
ejde-508	227	19	approaches∞	approaches∞	VERB
ejde-508	227	20	as	as	ADP
ejde-508	227	21	j	j	PROPN
ejde-508	227	22	→∞.	→∞.	PROPN
ejde-508	227	23	then	then	ADV
ejde-508	227	24	there	there	PRON
ejde-508	227	25	exists	exist	VERB
ejde-508	227	26	η	η	PROPN
ejde-508	227	27	>	>	X
ejde-508	227	28	0	0	NUM
ejde-508	228	1	such	such	ADJ
ejde-508	228	2	that	that	SCONJ
ejde-508	228	3	ynj	ynj	PROPN
ejde-508	228	4	>	>	X
ejde-508	228	5	η	η	PROPN
ejde-508	228	6	for	for	ADP
ejde-508	228	7	large	large	ADJ
ejde-508	228	8	j.	j.	PROPN
ejde-508	228	9	then	then	ADV
ejde-508	228	10	∑∞	∑∞	X
ejde-508	228	11	j	j	PROPN
ejde-508	228	12	=	=	PROPN
ejde-508	228	13	n3	n3	PROPN
ejde-508	228	14	qnj	qnj	PROPN
ejde-508	228	15	g(ynj	g(ynj	PROPN
ejde-508	228	16	)	)	PUNCT
ejde-508	228	17	>	>	X
ejde-508	228	18	g(η	g(η	PROPN
ejde-508	228	19	)	)	PUNCT
ejde-508	228	20	∑∞	∑∞	NOUN
ejde-508	228	21	j	j	PROPN
ejde-508	228	22	=	=	PROPN
ejde-508	228	23	n3	n3	NOUN
ejde-508	228	24	qnj	qnj	NOUN
ejde-508	228	25	→	→	SYM
ejde-508	228	26	8	8	NUM
ejde-508	228	27	a.	a.	PROPN
ejde-508	228	28	k.	k.	PROPN
ejde-508	228	29	bhuyan	bhuyan	PROPN
ejde-508	228	30	,	,	PUNCT
ejde-508	228	31	l.	l.	PROPN
ejde-508	228	32	n.	n.	PROPN
ejde-508	228	33	padhy	padhy	PROPN
ejde-508	228	34	,	,	PUNCT
ejde-508	228	35	r.	r.	PROPN
ejde-508	228	36	n.	n.	PROPN
ejde-508	228	37	rath	rath	PROPN
ejde-508	228	38	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	229	1	+	+	NOUN
ejde-508	229	2	∞	∞	PROPN
ejde-508	229	3	by	by	ADP
ejde-508	229	4	(	(	PUNCT
ejde-508	229	5	a4	a4	NOUN
ejde-508	229	6	)	)	PUNCT
ejde-508	229	7	.	.	PUNCT
ejde-508	230	1	this	this	PRON
ejde-508	230	2	contradicts	contradict	VERB
ejde-508	230	3	(	(	PUNCT
ejde-508	230	4	2.13	2.13	NUM
ejde-508	230	5	)	)	PUNCT
ejde-508	230	6	which	which	PRON
ejde-508	230	7	follows	follow	VERB
ejde-508	230	8	from	from	ADP
ejde-508	230	9	(	(	PUNCT
ejde-508	230	10	2.8	2.8	NUM
ejde-508	230	11	)	)	PUNCT
ejde-508	230	12	.	.	PUNCT
ejde-508	231	1	the	the	DET
ejde-508	231	2	proof	proof	NOUN
ejde-508	231	3	for	for	ADP
ejde-508	231	4	the	the	DET
ejde-508	231	5	case	case	NOUN
ejde-508	231	6	yn	yn	X
ejde-508	231	7	<	<	X
ejde-508	231	8	0	0	NUM
ejde-508	231	9	,	,	PUNCT
ejde-508	231	10	and	and	CCONJ
ejde-508	231	11	unbounded	unbounded	ADJ
ejde-508	231	12	is	be	AUX
ejde-508	231	13	similar	similar	ADJ
ejde-508	231	14	.	.	PUNCT
ejde-508	232	1	thus	thus	ADV
ejde-508	232	2	,	,	PUNCT
ejde-508	232	3	the	the	DET
ejde-508	232	4	proof	proof	NOUN
ejde-508	232	5	is	be	AUX
ejde-508	232	6	complete	complete	ADJ
ejde-508	232	7	.	.	PUNCT
ejde-508	233	1	�	�	PROPN
ejde-508	233	2	theorem	theorem	VERB
ejde-508	233	3	3.3	3.3	NUM
ejde-508	233	4	.	.	PUNCT
ejde-508	234	1	let	let	AUX
ejde-508	234	2	(	(	PUNCT
ejde-508	234	3	a1	a1	NOUN
ejde-508	234	4	)	)	PUNCT
ejde-508	234	5	,	,	PUNCT
ejde-508	234	6	(	(	PUNCT
ejde-508	234	7	a4)–(a7	a4)–(a7	NOUN
ejde-508	234	8	)	)	PUNCT
ejde-508	234	9	,	,	PUNCT
ejde-508	234	10	and	and	CCONJ
ejde-508	234	11	(	(	PUNCT
ejde-508	234	12	3.2	3.2	NUM
ejde-508	234	13	)	)	PUNCT
ejde-508	234	14	,	,	PUNCT
ejde-508	234	15	s	s	PART
ejde-508	234	16	>	>	X
ejde-508	234	17	k+	k+	NOUN
ejde-508	234	18	1	1	NUM
ejde-508	234	19	,	,	PUNCT
ejde-508	234	20	(	(	PUNCT
ejde-508	234	21	1.8	1.8	NUM
ejde-508	234	22	)	)	PUNCT
ejde-508	234	23	be	be	AUX
ejde-508	234	24	satisfied	satisfied	ADJ
ejde-508	234	25	.	.	PUNCT
ejde-508	235	1	then	then	ADV
ejde-508	235	2	every	every	DET
ejde-508	235	3	unbounded	unbounded	ADJ
ejde-508	235	4	solution	solution	NOUN
ejde-508	235	5	of	of	ADP
ejde-508	235	6	(	(	PUNCT
ejde-508	235	7	1.2	1.2	NUM
ejde-508	235	8	)	)	PUNCT
ejde-508	235	9	oscillates	oscillate	NOUN
ejde-508	235	10	.	.	PUNCT
ejde-508	236	1	the	the	DET
ejde-508	236	2	proof	proof	NOUN
ejde-508	236	3	of	of	ADP
ejde-508	236	4	the	the	DET
ejde-508	236	5	above	above	ADJ
ejde-508	236	6	theorem	theorem	NOUN
ejde-508	236	7	is	be	AUX
ejde-508	236	8	similar	similar	ADJ
ejde-508	236	9	to	to	ADP
ejde-508	236	10	that	that	PRON
ejde-508	236	11	of	of	ADP
ejde-508	236	12	theorem	theorem	NOUN
ejde-508	236	13	3.2	3.2	NUM
ejde-508	236	14	;	;	PUNCT
ejde-508	236	15	we	we	PRON
ejde-508	236	16	omit	omit	VERB
ejde-508	236	17	it	it	PRON
ejde-508	236	18	.	.	PUNCT
ejde-508	237	1	theorem	theorem	VERB
ejde-508	237	2	3.4	3.4	NUM
ejde-508	237	3	.	.	PUNCT
ejde-508	238	1	let	let	VERB
ejde-508	238	2	(	(	PUNCT
ejde-508	238	3	a1	a1	NOUN
ejde-508	238	4	)	)	PUNCT
ejde-508	238	5	,	,	PUNCT
ejde-508	238	6	(	(	PUNCT
ejde-508	238	7	a4)–(a7	a4)–(a7	ADV
ejde-508	238	8	)	)	PUNCT
ejde-508	238	9	hold	hold	NOUN
ejde-508	238	10	.	.	PUNCT
ejde-508	239	1	suppose	suppose	VERB
ejde-508	239	2	pn	pn	PROPN
ejde-508	239	3	satisfies	satisfie	NOUN
ejde-508	239	4	(	(	PUNCT
ejde-508	239	5	1.8	1.8	NUM
ejde-508	239	6	)	)	PUNCT
ejde-508	239	7	,	,	PUNCT
ejde-508	239	8	s	s	VERB
ejde-508	239	9	>	>	X
ejde-508	239	10	k	k	PROPN
ejde-508	240	1	+	+	PROPN
ejde-508	240	2	1	1	NUM
ejde-508	240	3	,	,	PUNCT
ejde-508	240	4	and	and	CCONJ
ejde-508	240	5	lim	lim	PROPN
ejde-508	240	6	inf	inf	PROPN
ejde-508	240	7	|x|→∞	|x|→∞	PROPN
ejde-508	240	8	g(x	g(x	NOUN
ejde-508	240	9	)	)	PUNCT
ejde-508	240	10	x	x	X
ejde-508	240	11	>	>	X
ejde-508	240	12	γ	γ	X
ejde-508	240	13	>	>	X
ejde-508	240	14	0	0	PROPN
ejde-508	240	15	.	.	PUNCT
ejde-508	241	1	(	(	PUNCT
ejde-508	241	2	3.5	3.5	NUM
ejde-508	241	3	)	)	PUNCT
ejde-508	241	4	suppose	suppose	VERB
ejde-508	241	5	that	that	SCONJ
ejde-508	241	6	lim	lim	PROPN
ejde-508	241	7	inf	inf	PROPN
ejde-508	241	8	n→∞	n→∞	X
ejde-508	241	9	n−1∑	n−1∑	PROPN
ejde-508	241	10	i	i	PRON
ejde-508	241	11	=	=	PRON
ejde-508	241	12	n−s+k+1	n−s+k+1	VERB
ejde-508	241	13	qi	qi	PROPN
ejde-508	241	14	>	>	X
ejde-508	241	15	pδ	pδ	ADP
ejde-508	241	16	γ	γ	X
ejde-508	241	17	(	(	PUNCT
ejde-508	241	18	s−	s−	PROPN
ejde-508	241	19	k	k	PROPN
ejde-508	242	1	−	−	PROPN
ejde-508	242	2	1	1	NUM
ejde-508	242	3	s−	s−	PROPN
ejde-508	242	4	k	k	PROPN
ejde-508	242	5	)	)	PUNCT
ejde-508	242	6	s−k	s−k	NOUN
ejde-508	242	7	(	(	PUNCT
ejde-508	242	8	3.6	3.6	NUM
ejde-508	242	9	)	)	PUNCT
ejde-508	242	10	then	then	ADV
ejde-508	242	11	every	every	DET
ejde-508	242	12	unbounded	unbounded	ADJ
ejde-508	242	13	solution	solution	NOUN
ejde-508	242	14	of	of	ADP
ejde-508	242	15	(	(	PUNCT
ejde-508	242	16	1.2	1.2	NUM
ejde-508	242	17	)	)	PUNCT
ejde-508	242	18	oscillates	oscillate	NOUN
ejde-508	242	19	.	.	PUNCT
ejde-508	243	1	proof	proof	NOUN
ejde-508	243	2	.	.	PUNCT
ejde-508	244	1	to	to	PART
ejde-508	244	2	obtain	obtain	VERB
ejde-508	244	3	a	a	DET
ejde-508	244	4	contradiction	contradiction	NOUN
ejde-508	244	5	,	,	PUNCT
ejde-508	244	6	let	let	VERB
ejde-508	244	7	yn	yn	PRON
ejde-508	244	8	be	be	AUX
ejde-508	244	9	an	an	DET
ejde-508	244	10	eventually	eventually	ADV
ejde-508	244	11	positive	positive	ADJ
ejde-508	244	12	solution	solution	NOUN
ejde-508	244	13	of	of	ADP
ejde-508	244	14	(	(	PUNCT
ejde-508	244	15	1.2	1.2	NUM
ejde-508	244	16	)	)	PUNCT
ejde-508	244	17	.	.	PUNCT
ejde-508	245	1	proceeding	proceed	VERB
ejde-508	245	2	as	as	ADP
ejde-508	245	3	in	in	ADP
ejde-508	245	4	the	the	DET
ejde-508	245	5	proof	proof	NOUN
ejde-508	245	6	of	of	ADP
ejde-508	245	7	theorem	theorem	ADJ
ejde-508	245	8	3.2	3.2	NUM
ejde-508	245	9	,	,	PUNCT
ejde-508	245	10	we	we	PRON
ejde-508	245	11	show	show	VERB
ejde-508	245	12	that	that	SCONJ
ejde-508	245	13	if	if	SCONJ
ejde-508	245	14	(	(	PUNCT
ejde-508	245	15	2.7	2.7	NUM
ejde-508	245	16	)	)	PUNCT
ejde-508	245	17	holds	hold	VERB
ejde-508	245	18	then	then	ADV
ejde-508	245	19	∆zn+1	∆zn+1	NUM
ejde-508	245	20	+	+	PUNCT
ejde-508	245	21	qng(−zn+s−k	qng(−zn+s−k	NOUN
ejde-508	245	22	pδ	pδ	NOUN
ejde-508	245	23	)	)	PUNCT
ejde-508	245	24	≤	≤	NOUN
ejde-508	245	25	0	0	NUM
ejde-508	245	26	.	.	PUNCT
ejde-508	246	1	applying	apply	VERB
ejde-508	246	2	(	(	PUNCT
ejde-508	246	3	3.5	3.5	NUM
ejde-508	246	4	)	)	PUNCT
ejde-508	246	5	to	to	ADP
ejde-508	246	6	the	the	DET
ejde-508	246	7	above	above	ADJ
ejde-508	246	8	inequality	inequality	NOUN
ejde-508	246	9	,	,	PUNCT
ejde-508	246	10	we	we	PRON
ejde-508	246	11	obtain	obtain	VERB
ejde-508	246	12	∆zn+1	∆zn+1	ADJ
ejde-508	246	13	−	−	PROPN
ejde-508	246	14	γqn	γqn	PROPN
ejde-508	246	15	(	(	PUNCT
ejde-508	246	16	zn+s−k	zn+s−k	NOUN
ejde-508	246	17	pδ	pδ	PROPN
ejde-508	246	18	)	)	PUNCT
ejde-508	246	19	≤	≤	NOUN
ejde-508	246	20	0	0	NUM
ejde-508	246	21	.	.	PUNCT
ejde-508	247	1	note	note	VERB
ejde-508	247	2	that	that	SCONJ
ejde-508	247	3	zn	zn	PROPN
ejde-508	247	4	<	<	X
ejde-508	247	5	0	0	NUM
ejde-508	247	6	for	for	ADP
ejde-508	247	7	large	large	ADJ
ejde-508	247	8	n.	n.	NOUN
ejde-508	247	9	substituting	substituting	NOUN
ejde-508	247	10	(	(	PUNCT
ejde-508	247	11	zn+1/(pδ	zn+1/(pδ	NOUN
ejde-508	247	12	)	)	PUNCT
ejde-508	247	13	)	)	PUNCT
ejde-508	248	1	=	=	SYM
ejde-508	248	2	vn	vn	PROPN
ejde-508	248	3	and	and	CCONJ
ejde-508	248	4	∆zn+1	∆zn+1	PROPN
ejde-508	248	5	=	=	SYM
ejde-508	248	6	pδ∆vn	pδ∆vn	PROPN
ejde-508	248	7	,	,	PUNCT
ejde-508	248	8	in	in	ADP
ejde-508	248	9	the	the	DET
ejde-508	248	10	above	above	ADV
ejde-508	248	11	we	we	PRON
ejde-508	248	12	obtain	obtain	VERB
ejde-508	248	13	∆vn	∆vn	PROPN
ejde-508	248	14	−	−	ADP
ejde-508	248	15	γ	γ	PROPN
ejde-508	248	16	pδ	pδ	ADP
ejde-508	248	17	qnvn+s−k−1	qnvn+s−k−1	PROPN
ejde-508	248	18	≤	≤	NUM
ejde-508	248	19	0	0	NUM
ejde-508	248	20	.	.	PUNCT
ejde-508	249	1	since	since	SCONJ
ejde-508	249	2	s−k−1	s−k−1	PROPN
ejde-508	249	3	>	>	SYM
ejde-508	249	4	0	0	NUM
ejde-508	250	1	this	this	PRON
ejde-508	250	2	is	be	AUX
ejde-508	250	3	an	an	DET
ejde-508	250	4	advanced	advanced	ADJ
ejde-508	250	5	difference	difference	NOUN
ejde-508	250	6	inequality	inequality	NOUN
ejde-508	250	7	with	with	ADP
ejde-508	250	8	a	a	DET
ejde-508	250	9	negative	negative	ADJ
ejde-508	250	10	solution	solution	NOUN
ejde-508	250	11	vn	vn	PROPN
ejde-508	250	12	,	,	PUNCT
ejde-508	250	13	which	which	PRON
ejde-508	250	14	contradicts	contradict	VERB
ejde-508	250	15	lemma	lemma	PROPN
ejde-508	250	16	2.1(b	2.1(b	NUM
ejde-508	250	17	)	)	PUNCT
ejde-508	250	18	.	.	PUNCT
ejde-508	251	1	next	next	ADV
ejde-508	251	2	consider	consider	VERB
ejde-508	251	3	the	the	DET
ejde-508	251	4	case	case	NOUN
ejde-508	251	5	that	that	SCONJ
ejde-508	251	6	(	(	PUNCT
ejde-508	251	7	2.8	2.8	NUM
ejde-508	251	8	)	)	PUNCT
ejde-508	251	9	holds	hold	VERB
ejde-508	251	10	.	.	PUNCT
ejde-508	252	1	proceeding	proceed	VERB
ejde-508	252	2	as	as	ADP
ejde-508	252	3	in	in	ADP
ejde-508	252	4	the	the	DET
ejde-508	252	5	proof	proof	NOUN
ejde-508	252	6	of	of	ADP
ejde-508	252	7	theorem	theorem	ADJ
ejde-508	252	8	3.2	3.2	NUM
ejde-508	252	9	we	we	PRON
ejde-508	252	10	obtain	obtain	VERB
ejde-508	252	11	a	a	DET
ejde-508	252	12	contradiction	contradiction	NOUN
ejde-508	252	13	.	.	PUNCT
ejde-508	253	1	the	the	DET
ejde-508	253	2	proof	proof	NOUN
ejde-508	253	3	for	for	ADP
ejde-508	253	4	the	the	DET
ejde-508	253	5	case	case	NOUN
ejde-508	253	6	yn	yn	X
ejde-508	253	7	<	<	X
ejde-508	253	8	0	0	NUM
ejde-508	253	9	,	,	PUNCT
ejde-508	253	10	and	and	CCONJ
ejde-508	253	11	unbounded	unbounded	ADJ
ejde-508	253	12	is	be	AUX
ejde-508	253	13	similar	similar	ADJ
ejde-508	253	14	.	.	PUNCT
ejde-508	254	1	thus	thus	ADV
ejde-508	254	2	,	,	PUNCT
ejde-508	254	3	the	the	DET
ejde-508	254	4	proof	proof	NOUN
ejde-508	254	5	is	be	AUX
ejde-508	254	6	complete	complete	ADJ
ejde-508	254	7	.	.	PUNCT
ejde-508	255	1	�	�	PROPN
ejde-508	255	2	remark	remark	VERB
ejde-508	255	3	3.5	3.5	NUM
ejde-508	255	4	.	.	PUNCT
ejde-508	256	1	condition	condition	NOUN
ejde-508	256	2	(	(	PUNCT
ejde-508	256	3	3.6	3.6	NUM
ejde-508	256	4	)	)	PUNCT
ejde-508	256	5	implies	imply	VERB
ejde-508	256	6	(	(	PUNCT
ejde-508	256	7	a2	a2	PROPN
ejde-508	256	8	)	)	PUNCT
ejde-508	256	9	.	.	PUNCT
ejde-508	257	1	if	if	SCONJ
ejde-508	257	2	(	(	PUNCT
ejde-508	257	3	3.6	3.6	NUM
ejde-508	257	4	)	)	PUNCT
ejde-508	257	5	holds	hold	VERB
ejde-508	257	6	and	and	CCONJ
ejde-508	257	7	(	(	PUNCT
ejde-508	257	8	a2	a2	PROPN
ejde-508	257	9	)	)	PUNCT
ejde-508	257	10	fails	fail	VERB
ejde-508	257	11	,	,	PUNCT
ejde-508	257	12	we	we	PRON
ejde-508	257	13	have∑∞	have∑∞	PUNCT
ejde-508	257	14	n	n	CCONJ
ejde-508	257	15	=	=	SYM
ejde-508	257	16	n1	n1	NOUN
ejde-508	257	17	qn	qn	NOUN
ejde-508	257	18	<	<	X
ejde-508	257	19	∞	∞	NUM
ejde-508	257	20	which	which	PRON
ejde-508	257	21	implies	imply	VERB
ejde-508	257	22	pδ	pδ	PROPN
ejde-508	257	23	γ	γ	X
ejde-508	257	24	(	(	PUNCT
ejde-508	257	25	s−	s−	PROPN
ejde-508	257	26	k	k	PROPN
ejde-508	258	1	−	−	PROPN
ejde-508	258	2	1	1	NUM
ejde-508	258	3	s−	s−	PROPN
ejde-508	258	4	k	k	PROPN
ejde-508	258	5	)	)	PUNCT
ejde-508	258	6	s−k	s−k	NOUN
ejde-508	258	7	<	<	X
ejde-508	258	8	lim	lim	PROPN
ejde-508	258	9	inf	inf	PROPN
ejde-508	258	10	n→∞	n→∞	X
ejde-508	259	1	n−1∑	n−1∑	PROPN
ejde-508	259	2	i	i	NOUN
ejde-508	259	3	=	=	PROPN
ejde-508	259	4	n−s+k	n−s+k	X
ejde-508	259	5	qi	qi	PROPN
ejde-508	259	6	≤	≤	PROPN
ejde-508	259	7	lim	lim	PROPN
ejde-508	259	8	sup	sup	PROPN
ejde-508	259	9	n→∞	n→∞	NUM
ejde-508	260	1	(	(	PUNCT
ejde-508	260	2	n−1∑	n−1∑	PROPN
ejde-508	260	3	i	i	PROPN
ejde-508	260	4	=	=	NOUN
ejde-508	260	5	n1	n1	PROPN
ejde-508	260	6	qi	qi	NOUN
ejde-508	260	7	−	−	PROPN
ejde-508	260	8	n−s+k−1∑	n−s+k−1∑	VERB
ejde-508	260	9	i	i	PROPN
ejde-508	260	10	=	=	NOUN
ejde-508	260	11	n1	n1	PROPN
ejde-508	260	12	qi	qi	NOUN
ejde-508	260	13	)	)	PUNCT
ejde-508	261	1	=	=	SYM
ejde-508	261	2	0	0	NUM
ejde-508	261	3	,	,	PUNCT
ejde-508	261	4	a	a	DET
ejde-508	261	5	contradiction	contradiction	NOUN
ejde-508	261	6	.	.	PUNCT
ejde-508	262	1	theorem	theorem	VERB
ejde-508	262	2	3.6	3.6	NUM
ejde-508	262	3	.	.	PUNCT
ejde-508	263	1	suppose	suppose	VERB
ejde-508	263	2	(	(	PUNCT
ejde-508	263	3	a1	a1	NOUN
ejde-508	263	4	)	)	PUNCT
ejde-508	263	5	,	,	PUNCT
ejde-508	263	6	(	(	PUNCT
ejde-508	263	7	a4)–(a7	a4)–(a7	ADV
ejde-508	263	8	)	)	PUNCT
ejde-508	263	9	hold	hold	NOUN
ejde-508	263	10	,	,	PUNCT
ejde-508	263	11	and	and	CCONJ
ejde-508	263	12	(	(	PUNCT
ejde-508	263	13	1.5	1.5	NUM
ejde-508	263	14	)	)	PUNCT
ejde-508	263	15	and	and	CCONJ
ejde-508	263	16	δ	δ	PROPN
ejde-508	263	17	≤	≤	NOUN
ejde-508	263	18	1	1	NUM
ejde-508	263	19	are	be	AUX
ejde-508	263	20	satisfied	satisfied	ADJ
ejde-508	263	21	.	.	PUNCT
ejde-508	264	1	then	then	ADV
ejde-508	264	2	every	every	DET
ejde-508	264	3	unbounded	unbounded	ADJ
ejde-508	264	4	solution	solution	NOUN
ejde-508	264	5	of	of	ADP
ejde-508	264	6	(	(	PUNCT
ejde-508	264	7	1.2	1.2	NUM
ejde-508	264	8	)	)	PUNCT
ejde-508	264	9	oscillates	oscillate	NOUN
ejde-508	264	10	.	.	PUNCT
ejde-508	265	1	proof	proof	NOUN
ejde-508	265	2	.	.	PUNCT
ejde-508	266	1	let	let	VERB
ejde-508	266	2	yn	yn	PRON
ejde-508	266	3	be	be	AUX
ejde-508	266	4	an	an	DET
ejde-508	266	5	unbounded	unbounded	ADJ
ejde-508	266	6	and	and	CCONJ
ejde-508	266	7	eventually	eventually	ADV
ejde-508	266	8	positive	positive	ADJ
ejde-508	266	9	solution	solution	NOUN
ejde-508	266	10	of	of	ADP
ejde-508	266	11	(	(	PUNCT
ejde-508	266	12	1.2	1.2	NUM
ejde-508	266	13	)	)	PUNCT
ejde-508	266	14	.	.	PUNCT
ejde-508	267	1	setting	set	VERB
ejde-508	267	2	cn	cn	PROPN
ejde-508	267	3	,	,	PUNCT
ejde-508	267	4	zn	zn	PROPN
ejde-508	267	5	and	and	CCONJ
ejde-508	267	6	wn	wn	PROPN
ejde-508	267	7	as	as	ADP
ejde-508	267	8	in	in	ADP
ejde-508	267	9	(	(	PUNCT
ejde-508	267	10	2.2	2.2	NUM
ejde-508	267	11	)	)	PUNCT
ejde-508	267	12	,	,	PUNCT
ejde-508	267	13	(	(	PUNCT
ejde-508	267	14	2.5	2.5	NUM
ejde-508	267	15	)	)	PUNCT
ejde-508	267	16	and	and	CCONJ
ejde-508	267	17	(	(	PUNCT
ejde-508	267	18	2.6	2.6	NUM
ejde-508	267	19	)	)	PUNCT
ejde-508	267	20	respectively	respectively	ADV
ejde-508	267	21	,	,	PUNCT
ejde-508	267	22	we	we	PRON
ejde-508	267	23	obtain	obtain	VERB
ejde-508	267	24	(	(	PUNCT
ejde-508	267	25	2.10	2.10	NUM
ejde-508	267	26	)	)	PUNCT
ejde-508	267	27	.	.	PUNCT
ejde-508	268	1	by	by	ADP
ejde-508	268	2	lemma	lemma	PROPN
ejde-508	268	3	2.3(b	2.3(b	NUM
ejde-508	268	4	)	)	PUNCT
ejde-508	268	5	,	,	PUNCT
ejde-508	268	6	we	we	PRON
ejde-508	268	7	have	have	VERB
ejde-508	268	8	limn→∞	limn→∞	PROPN
ejde-508	268	9	wn	wn	PROPN
ejde-508	268	10	=	=	SYM
ejde-508	268	11	0	0	PROPN
ejde-508	268	12	.	.	PUNCT
ejde-508	269	1	using	use	VERB
ejde-508	269	2	this	this	PRON
ejde-508	269	3	,	,	PUNCT
ejde-508	269	4	unboundedness	unboundedness	NOUN
ejde-508	269	5	of	of	ADP
ejde-508	269	6	yn	yn	PROPN
ejde-508	269	7	and	and	CCONJ
ejde-508	269	8	(	(	PUNCT
ejde-508	269	9	a4	a4	NOUN
ejde-508	269	10	)	)	PUNCT
ejde-508	269	11	,	,	PUNCT
ejde-508	269	12	and	and	CCONJ
ejde-508	269	13	proceeding	proceed	VERB
ejde-508	269	14	as	as	ADP
ejde-508	269	15	in	in	ADP
ejde-508	269	16	the	the	DET
ejde-508	269	17	last	last	ADJ
ejde-508	269	18	part	part	NOUN
ejde-508	269	19	of	of	ADP
ejde-508	269	20	the	the	DET
ejde-508	269	21	proof	proof	NOUN
ejde-508	269	22	of	of	ADP
ejde-508	269	23	theorem	theorem	ADJ
ejde-508	269	24	3.2	3.2	NUM
ejde-508	269	25	we	we	PRON
ejde-508	269	26	obtain	obtain	VERB
ejde-508	269	27	a	a	DET
ejde-508	269	28	contradiction	contradiction	NOUN
ejde-508	269	29	.	.	PUNCT
ejde-508	270	1	a	a	DET
ejde-508	270	2	similar	similar	ADJ
ejde-508	270	3	contradiction	contradiction	NOUN
ejde-508	270	4	could	could	AUX
ejde-508	270	5	be	be	AUX
ejde-508	270	6	obtained	obtain	VERB
ejde-508	270	7	if	if	SCONJ
ejde-508	270	8	yn	yn	PRON
ejde-508	270	9	be	be	VERB
ejde-508	270	10	an	an	DET
ejde-508	270	11	eventually	eventually	ADV
ejde-508	270	12	negative	negative	ADJ
ejde-508	270	13	and	and	CCONJ
ejde-508	270	14	unbounded	unbounded	ADJ
ejde-508	270	15	solution	solution	NOUN
ejde-508	270	16	of	of	ADP
ejde-508	270	17	(	(	PUNCT
ejde-508	270	18	1.2	1.2	NUM
ejde-508	270	19	)	)	PUNCT
ejde-508	270	20	.	.	PUNCT
ejde-508	271	1	this	this	PRON
ejde-508	271	2	completes	complete	VERB
ejde-508	271	3	the	the	DET
ejde-508	271	4	proof	proof	NOUN
ejde-508	271	5	.	.	PUNCT
ejde-508	272	1	�	�	PROPN
ejde-508	272	2	ejde-2020/87	ejde-2020/87	VERB
ejde-508	272	3	oscillation	oscillation	NOUN
ejde-508	272	4	for	for	ADP
ejde-508	272	5	second	second	ADJ
ejde-508	272	6	order	order	NOUN
ejde-508	272	7	neutral	neutral	ADJ
ejde-508	272	8	equations	equation	NOUN
ejde-508	272	9	9	9	NUM
ejde-508	272	10	theorem	theorem	VERB
ejde-508	272	11	3.7	3.7	NUM
ejde-508	272	12	.	.	PUNCT
ejde-508	273	1	suppose	suppose	VERB
ejde-508	273	2	(	(	PUNCT
ejde-508	273	3	a1	a1	NOUN
ejde-508	273	4	)	)	PUNCT
ejde-508	273	5	,	,	PUNCT
ejde-508	273	6	(	(	PUNCT
ejde-508	273	7	a3	a3	NOUN
ejde-508	273	8	)	)	PUNCT
ejde-508	273	9	,	,	PUNCT
ejde-508	273	10	(	(	PUNCT
ejde-508	273	11	a4	a4	NOUN
ejde-508	273	12	)	)	PUNCT
ejde-508	273	13	,	,	PUNCT
ejde-508	273	14	(	(	PUNCT
ejde-508	273	15	a6	a6	NOUN
ejde-508	273	16	)	)	PUNCT
ejde-508	273	17	,	,	PUNCT
ejde-508	273	18	(	(	PUNCT
ejde-508	273	19	a7	a7	ADJ
ejde-508	273	20	)	)	PUNCT
ejde-508	273	21	hold	hold	VERB
ejde-508	273	22	,	,	PUNCT
ejde-508	273	23	(	(	PUNCT
ejde-508	273	24	1.6	1.6	NUM
ejde-508	273	25	)	)	PUNCT
ejde-508	273	26	or	or	CCONJ
ejde-508	273	27	(	(	PUNCT
ejde-508	273	28	1.7	1.7	NUM
ejde-508	273	29	)	)	PUNCT
ejde-508	273	30	,	,	PUNCT
ejde-508	273	31	and	and	CCONJ
ejde-508	273	32	(	(	PUNCT
ejde-508	273	33	2.21	2.21	NUM
ejde-508	273	34	)	)	PUNCT
ejde-508	273	35	and	and	CCONJ
ejde-508	273	36	(	(	PUNCT
ejde-508	273	37	2.22	2.22	NUM
ejde-508	273	38	)	)	PUNCT
ejde-508	273	39	be	be	AUX
ejde-508	273	40	satisfied	satisfied	ADJ
ejde-508	273	41	.	.	PUNCT
ejde-508	274	1	then	then	ADV
ejde-508	274	2	every	every	DET
ejde-508	274	3	unbounded	unbounded	ADJ
ejde-508	274	4	solution	solution	NOUN
ejde-508	274	5	of	of	ADP
ejde-508	274	6	(	(	PUNCT
ejde-508	274	7	1.2	1.2	NUM
ejde-508	274	8	)	)	PUNCT
ejde-508	274	9	oscillates	oscillate	NOUN
ejde-508	274	10	.	.	PUNCT
ejde-508	275	1	proof	proof	NOUN
ejde-508	275	2	.	.	PUNCT
ejde-508	276	1	on	on	ADP
ejde-508	276	2	the	the	DET
ejde-508	276	3	contrary	contrary	NOUN
ejde-508	276	4	suppose	suppose	VERB
ejde-508	276	5	yn	yn	PRON
ejde-508	276	6	be	be	AUX
ejde-508	276	7	an	an	DET
ejde-508	276	8	eventually	eventually	ADV
ejde-508	276	9	positive	positive	ADJ
ejde-508	276	10	and	and	CCONJ
ejde-508	276	11	unbounded	unbounded	ADJ
ejde-508	276	12	solution	solution	NOUN
ejde-508	276	13	of	of	ADP
ejde-508	276	14	(	(	PUNCT
ejde-508	276	15	1.2	1.2	NUM
ejde-508	276	16	)	)	PUNCT
ejde-508	276	17	.	.	PUNCT
ejde-508	277	1	setting	set	VERB
ejde-508	277	2	zn	zn	PROPN
ejde-508	277	3	and	and	CCONJ
ejde-508	277	4	wn	wn	PROPN
ejde-508	277	5	as	as	ADP
ejde-508	277	6	in	in	ADP
ejde-508	277	7	(	(	PUNCT
ejde-508	277	8	2.5	2.5	NUM
ejde-508	277	9	)	)	PUNCT
ejde-508	277	10	and	and	CCONJ
ejde-508	277	11	(	(	PUNCT
ejde-508	277	12	2.6	2.6	NUM
ejde-508	277	13	)	)	PUNCT
ejde-508	277	14	,	,	PUNCT
ejde-508	277	15	we	we	PRON
ejde-508	277	16	obtain	obtain	VERB
ejde-508	277	17	(	(	PUNCT
ejde-508	277	18	2.10	2.10	NUM
ejde-508	277	19	)	)	PUNCT
ejde-508	277	20	.	.	PUNCT
ejde-508	278	1	application	application	NOUN
ejde-508	278	2	of	of	ADP
ejde-508	278	3	lemma	lemma	PROPN
ejde-508	278	4	2.6	2.6	NUM
ejde-508	278	5	yields	yield	NOUN
ejde-508	278	6	β	β	X
ejde-508	278	7	=	=	SYM
ejde-508	278	8	limn→∞	limn→∞	X
ejde-508	278	9	wn	wn	X
ejde-508	278	10	=	=	SYM
ejde-508	278	11	0	0	PROPN
ejde-508	278	12	,	,	PUNCT
ejde-508	278	13	wn	wn	X
ejde-508	278	14	<	<	X
ejde-508	278	15	0	0	PROPN
ejde-508	278	16	and	and	CCONJ
ejde-508	278	17	∆wn	∆wn	X
ejde-508	278	18	>	>	X
ejde-508	278	19	0	0	X
ejde-508	278	20	.	.	PUNCT
ejde-508	279	1	then	then	ADV
ejde-508	279	2	using	use	VERB
ejde-508	279	3	this	this	PRON
ejde-508	279	4	,	,	PUNCT
ejde-508	279	5	unboundedness	unboundedness	NOUN
ejde-508	279	6	of	of	ADP
ejde-508	279	7	yn	yn	PROPN
ejde-508	279	8	,	,	PUNCT
ejde-508	279	9	(	(	PUNCT
ejde-508	279	10	a4	a4	NOUN
ejde-508	279	11	)	)	PUNCT
ejde-508	279	12	and	and	CCONJ
ejde-508	279	13	proceeding	proceed	VERB
ejde-508	279	14	as	as	ADP
ejde-508	279	15	in	in	ADP
ejde-508	279	16	the	the	DET
ejde-508	279	17	last	last	ADJ
ejde-508	279	18	part	part	NOUN
ejde-508	279	19	of	of	ADP
ejde-508	279	20	the	the	DET
ejde-508	279	21	proof	proof	NOUN
ejde-508	279	22	of	of	ADP
ejde-508	279	23	theorem	theorem	ADJ
ejde-508	279	24	3.2	3.2	NUM
ejde-508	279	25	we	we	PRON
ejde-508	279	26	obtain	obtain	VERB
ejde-508	279	27	a	a	DET
ejde-508	279	28	contradiction	contradiction	NOUN
ejde-508	279	29	.	.	PUNCT
ejde-508	280	1	a	a	DET
ejde-508	280	2	similar	similar	ADJ
ejde-508	280	3	contradiction	contradiction	NOUN
ejde-508	280	4	could	could	AUX
ejde-508	280	5	be	be	AUX
ejde-508	280	6	obtain	obtain	ADJ
ejde-508	280	7	if	if	SCONJ
ejde-508	280	8	yn	yn	PRON
ejde-508	280	9	be	be	VERB
ejde-508	280	10	an	an	DET
ejde-508	280	11	eventually	eventually	ADV
ejde-508	280	12	negative	negative	ADJ
ejde-508	280	13	and	and	CCONJ
ejde-508	280	14	unbounded	unbounded	ADJ
ejde-508	280	15	solution	solution	NOUN
ejde-508	280	16	of	of	ADP
ejde-508	280	17	(	(	PUNCT
ejde-508	280	18	1.2	1.2	NUM
ejde-508	280	19	)	)	PUNCT
ejde-508	280	20	.	.	PUNCT
ejde-508	281	1	this	this	PRON
ejde-508	281	2	completes	complete	VERB
ejde-508	281	3	the	the	DET
ejde-508	281	4	proof	proof	NOUN
ejde-508	281	5	.	.	PUNCT
ejde-508	282	1	�	�	PROPN
ejde-508	282	2	note	note	VERB
ejde-508	282	3	that	that	SCONJ
ejde-508	282	4	the	the	DET
ejde-508	282	5	condition	condition	NOUN
ejde-508	282	6	lim	lim	PROPN
ejde-508	282	7	inf	inf	PROPN
ejde-508	282	8	n→∞	n→∞	X
ejde-508	282	9	|xn|	|xn|	X
ejde-508	282	10	>	>	X
ejde-508	282	11	0	0	NUM
ejde-508	282	12	implies	imply	VERB
ejde-508	282	13	lim	lim	PROPN
ejde-508	282	14	inf	inf	PROPN
ejde-508	282	15	n→∞	n→∞	X
ejde-508	282	16	|g(xn)|	|g(xn)|	NOUN
ejde-508	282	17	>	>	X
ejde-508	282	18	0	0	X
ejde-508	282	19	.	.	PUNCT
ejde-508	283	1	(	(	PUNCT
ejde-508	283	2	3.7	3.7	NUM
ejde-508	283	3	)	)	PUNCT
ejde-508	283	4	is	be	AUX
ejde-508	283	5	equivalent	equivalent	ADJ
ejde-508	283	6	to	to	ADP
ejde-508	283	7	lim	lim	PROPN
ejde-508	283	8	inf	inf	PROPN
ejde-508	283	9	u→±∞	u→±∞	PROPN
ejde-508	283	10	g(u	g(u	PROPN
ejde-508	283	11	)	)	PUNCT
ejde-508	283	12	6=	6=	ADP
ejde-508	283	13	0	0	NUM
ejde-508	283	14	(	(	PUNCT
ejde-508	283	15	3.8	3.8	NUM
ejde-508	283	16	)	)	PUNCT
ejde-508	283	17	and	and	CCONJ
ejde-508	283	18	note	note	VERB
ejde-508	283	19	that	that	SCONJ
ejde-508	283	20	(	(	PUNCT
ejde-508	283	21	a5	a5	PROPN
ejde-508	283	22	)	)	PUNCT
ejde-508	283	23	implies	imply	VERB
ejde-508	283	24	(	(	PUNCT
ejde-508	283	25	3.8	3.8	NUM
ejde-508	283	26	)	)	PUNCT
ejde-508	283	27	.	.	PUNCT
ejde-508	284	1	consequently	consequently	ADV
ejde-508	284	2	,	,	PUNCT
ejde-508	284	3	we	we	PRON
ejde-508	284	4	quote	quote	VERB
ejde-508	284	5	a	a	DET
ejde-508	284	6	particular	particular	ADJ
ejde-508	284	7	case	case	NOUN
ejde-508	284	8	of	of	ADP
ejde-508	284	9	[	[	X
ejde-508	284	10	13	13	NUM
ejde-508	284	11	,	,	PUNCT
ejde-508	284	12	theorem	theorem	VERB
ejde-508	284	13	2.5	2.5	NUM
ejde-508	284	14	,	,	PUNCT
ejde-508	284	15	p.236	p.236	X
ejde-508	284	16	]	]	PUNCT
ejde-508	284	17	for	for	ADP
ejde-508	284	18	fn	fn	PROPN
ejde-508	284	19	≡	≡	PROPN
ejde-508	284	20	0	0	PUNCT
ejde-508	285	1	as	as	ADP
ejde-508	285	2	our	our	PRON
ejde-508	285	3	next	next	ADJ
ejde-508	285	4	result	result	NOUN
ejde-508	285	5	.	.	PUNCT
ejde-508	286	1	theorem	theorem	VERB
ejde-508	286	2	3.8	3.8	NUM
ejde-508	286	3	.	.	PUNCT
ejde-508	287	1	suppose	suppose	VERB
ejde-508	287	2	(	(	PUNCT
ejde-508	287	3	a2	a2	PROPN
ejde-508	287	4	)	)	PUNCT
ejde-508	287	5	,	,	PUNCT
ejde-508	287	6	(	(	PUNCT
ejde-508	287	7	a6	a6	NOUN
ejde-508	287	8	)	)	PUNCT
ejde-508	287	9	,	,	PUNCT
ejde-508	287	10	(	(	PUNCT
ejde-508	287	11	a7	a7	ADJ
ejde-508	287	12	)	)	PUNCT
ejde-508	287	13	hold	hold	VERB
ejde-508	287	14	,	,	PUNCT
ejde-508	287	15	and	and	CCONJ
ejde-508	287	16	(	(	PUNCT
ejde-508	287	17	1.9	1.9	NUM
ejde-508	287	18	)	)	PUNCT
ejde-508	287	19	and	and	CCONJ
ejde-508	287	20	(	(	PUNCT
ejde-508	287	21	3.8	3.8	NUM
ejde-508	287	22	)	)	PUNCT
ejde-508	287	23	are	be	AUX
ejde-508	287	24	satisfied	satisfied	ADJ
ejde-508	287	25	.	.	PUNCT
ejde-508	288	1	if	if	SCONJ
ejde-508	288	2	l(x	l(x	PROPN
ejde-508	288	3	)	)	PUNCT
ejde-508	288	4	=	=	SYM
ejde-508	289	1	x	x	NOUN
ejde-508	289	2	,	,	PUNCT
ejde-508	289	3	then	then	ADV
ejde-508	289	4	every	every	DET
ejde-508	289	5	non	non	ADJ
ejde-508	289	6	-	-	ADJ
ejde-508	289	7	oscillatory	oscillatory	ADJ
ejde-508	289	8	solution	solution	NOUN
ejde-508	289	9	of	of	ADP
ejde-508	289	10	(	(	PUNCT
ejde-508	289	11	1.2	1.2	NUM
ejde-508	289	12	)	)	PUNCT
ejde-508	289	13	is	be	AUX
ejde-508	289	14	bounded	bound	VERB
ejde-508	289	15	.	.	PUNCT
ejde-508	290	1	or	or	CCONJ
ejde-508	290	2	equivalently	equivalently	ADV
ejde-508	290	3	every	every	DET
ejde-508	290	4	unbounded	unbounded	ADJ
ejde-508	290	5	solution	solution	NOUN
ejde-508	290	6	of	of	ADP
ejde-508	290	7	(	(	PUNCT
ejde-508	290	8	1.2	1.2	NUM
ejde-508	290	9	)	)	PUNCT
ejde-508	290	10	oscillates	oscillate	NOUN
ejde-508	290	11	.	.	PUNCT
ejde-508	291	1	4	4	X
ejde-508	291	2	.	.	X
ejde-508	291	3	main	main	ADJ
ejde-508	291	4	results	result	NOUN
ejde-508	291	5	part	part	NOUN
ejde-508	291	6	ii	ii	NOUN
ejde-508	291	7	in	in	ADP
ejde-508	291	8	this	this	DET
ejde-508	291	9	section	section	NOUN
ejde-508	291	10	,	,	PUNCT
ejde-508	291	11	we	we	PRON
ejde-508	291	12	find	find	VERB
ejde-508	291	13	sufficient	sufficient	ADJ
ejde-508	291	14	conditions	condition	NOUN
ejde-508	291	15	so	so	SCONJ
ejde-508	291	16	that	that	SCONJ
ejde-508	291	17	all	all	DET
ejde-508	291	18	solutions	solution	NOUN
ejde-508	291	19	of	of	ADP
ejde-508	291	20	(	(	PUNCT
ejde-508	291	21	1.1	1.1	NUM
ejde-508	291	22	)	)	PUNCT
ejde-508	291	23	oscillate	oscillate	NOUN
ejde-508	291	24	under	under	ADP
ejde-508	291	25	condition	condition	NOUN
ejde-508	291	26	(	(	PUNCT
ejde-508	291	27	a2	a2	PROPN
ejde-508	291	28	)	)	PUNCT
ejde-508	291	29	,	,	PUNCT
ejde-508	291	30	which	which	PRON
ejde-508	291	31	is	be	AUX
ejde-508	291	32	less	less	ADV
ejde-508	291	33	restrictive	restrictive	ADJ
ejde-508	291	34	than	than	ADP
ejde-508	291	35	(	(	PUNCT
ejde-508	291	36	a4	a4	NOUN
ejde-508	291	37	)	)	PUNCT
ejde-508	291	38	.	.	PUNCT
ejde-508	292	1	theorem	theorem	VERB
ejde-508	292	2	4.1	4.1	NUM
ejde-508	292	3	.	.	PUNCT
ejde-508	293	1	suppose	suppose	VERB
ejde-508	293	2	(	(	PUNCT
ejde-508	293	3	a1	a1	NOUN
ejde-508	293	4	)	)	PUNCT
ejde-508	293	5	,	,	PUNCT
ejde-508	293	6	(	(	PUNCT
ejde-508	293	7	a2	a2	PROPN
ejde-508	293	8	)	)	PUNCT
ejde-508	293	9	,	,	PUNCT
ejde-508	293	10	(	(	PUNCT
ejde-508	293	11	a5	a5	NOUN
ejde-508	293	12	)	)	PUNCT
ejde-508	293	13	hold	hold	NOUN
ejde-508	293	14	,	,	PUNCT
ejde-508	293	15	and	and	CCONJ
ejde-508	293	16	(	(	PUNCT
ejde-508	293	17	1.4	1.4	NUM
ejde-508	293	18	)	)	PUNCT
ejde-508	293	19	and	and	CCONJ
ejde-508	293	20	s	s	VERB
ejde-508	293	21	<	<	X
ejde-508	293	22	k	k	X
ejde-508	293	23	are	be	AUX
ejde-508	293	24	satisfied	satisfied	ADJ
ejde-508	293	25	.	.	PUNCT
ejde-508	294	1	if	if	SCONJ
ejde-508	294	2	∣∣	∣∣	NUM
ejde-508	294	3	∫	∫	PROPN
ejde-508	294	4	±c	±c	PROPN
ejde-508	294	5	0	0	NUM
ejde-508	294	6	du	du	PROPN
ejde-508	294	7	g(u	g(u	PROPN
ejde-508	294	8	)	)	PUNCT
ejde-508	294	9	∣∣	∣∣	X
ejde-508	294	10	<	<	X
ejde-508	294	11	∞	∞	PROPN
ejde-508	294	12	,	,	PUNCT
ejde-508	294	13	for	for	ADP
ejde-508	294	14	any	any	DET
ejde-508	294	15	finite	finite	NOUN
ejde-508	294	16	positive	positive	ADJ
ejde-508	294	17	c	c	NOUN
ejde-508	294	18	∈	∈	PROPN
ejde-508	294	19	r	r	NOUN
ejde-508	294	20	,	,	PUNCT
ejde-508	294	21	(	(	PUNCT
ejde-508	294	22	4.1	4.1	NUM
ejde-508	294	23	)	)	PUNCT
ejde-508	294	24	then	then	ADV
ejde-508	294	25	every	every	DET
ejde-508	294	26	bounded	bounded	ADJ
ejde-508	294	27	solution	solution	NOUN
ejde-508	294	28	of	of	ADP
ejde-508	294	29	(	(	PUNCT
ejde-508	294	30	1.1	1.1	NUM
ejde-508	294	31	)	)	PUNCT
ejde-508	294	32	oscillates	oscillate	NOUN
ejde-508	294	33	.	.	PUNCT
ejde-508	295	1	proof	proof	NOUN
ejde-508	295	2	.	.	PUNCT
ejde-508	296	1	on	on	ADP
ejde-508	296	2	the	the	DET
ejde-508	296	3	contrary	contrary	NOUN
ejde-508	296	4	let	let	VERB
ejde-508	296	5	yn	yn	PRON
ejde-508	296	6	be	be	AUX
ejde-508	296	7	a	a	DET
ejde-508	296	8	bounded	bound	VERB
ejde-508	296	9	eventually	eventually	ADV
ejde-508	296	10	positive	positive	ADJ
ejde-508	296	11	solution	solution	NOUN
ejde-508	296	12	of	of	ADP
ejde-508	296	13	(	(	PUNCT
ejde-508	296	14	1.1	1.1	NUM
ejde-508	296	15	)	)	PUNCT
ejde-508	296	16	.	.	PUNCT
ejde-508	297	1	setting	set	VERB
ejde-508	297	2	zn	zn	NOUN
ejde-508	297	3	as	as	ADP
ejde-508	297	4	in	in	ADP
ejde-508	297	5	(	(	PUNCT
ejde-508	297	6	2.5	2.5	NUM
ejde-508	297	7	)	)	PUNCT
ejde-508	297	8	we	we	PRON
ejde-508	297	9	obtain	obtain	VERB
ejde-508	297	10	(	(	PUNCT
ejde-508	297	11	2.25	2.25	NUM
ejde-508	297	12	)	)	PUNCT
ejde-508	297	13	.	.	PUNCT
ejde-508	298	1	then	then	ADV
ejde-508	298	2	zn	zn	PROPN
ejde-508	298	3	is	be	AUX
ejde-508	298	4	bounded	bound	VERB
ejde-508	298	5	and	and	CCONJ
ejde-508	298	6	by	by	ADP
ejde-508	298	7	lemma	lemma	PROPN
ejde-508	298	8	2.8(a	2.8(a	NUM
ejde-508	298	9	)	)	PUNCT
ejde-508	298	10	,	,	PUNCT
ejde-508	298	11	we	we	PRON
ejde-508	298	12	find	find	VERB
ejde-508	298	13	that	that	SCONJ
ejde-508	298	14	(	(	PUNCT
ejde-508	298	15	2.26	2.26	NUM
ejde-508	298	16	)	)	PUNCT
ejde-508	298	17	can	can	AUX
ejde-508	298	18	not	not	PART
ejde-508	298	19	hold	hold	VERB
ejde-508	298	20	because	because	SCONJ
ejde-508	298	21	boundedness	boundedness	NOUN
ejde-508	298	22	of	of	ADP
ejde-508	298	23	zn	zn	PROPN
ejde-508	298	24	,	,	PUNCT
ejde-508	298	25	as	as	ADP
ejde-508	298	26	a	a	DET
ejde-508	298	27	result	result	NOUN
ejde-508	298	28	,	,	PUNCT
ejde-508	298	29	(	(	PUNCT
ejde-508	298	30	2.27	2.27	NUM
ejde-508	298	31	)	)	PUNCT
ejde-508	298	32	holds	hold	VERB
ejde-508	298	33	.	.	PUNCT
ejde-508	299	1	then	then	ADV
ejde-508	299	2	(	(	PUNCT
ejde-508	299	3	2.28	2.28	NUM
ejde-508	299	4	)	)	PUNCT
ejde-508	299	5	follows	follow	VERB
ejde-508	299	6	as	as	ADP
ejde-508	299	7	a	a	DET
ejde-508	299	8	consequence	consequence	NOUN
ejde-508	299	9	which	which	PRON
ejde-508	299	10	implies	imply	VERB
ejde-508	299	11	that	that	SCONJ
ejde-508	299	12	zn	zn	PROPN
ejde-508	299	13	<	<	X
ejde-508	299	14	0	0	PUNCT
ejde-508	299	15	and	and	CCONJ
ejde-508	299	16	increasing	increase	VERB
ejde-508	299	17	.	.	PUNCT
ejde-508	300	1	if	if	SCONJ
ejde-508	300	2	pn	pn	PROPN
ejde-508	300	3	=	=	SYM
ejde-508	300	4	0	0	PROPN
ejde-508	300	5	,	,	PUNCT
ejde-508	300	6	then	then	ADV
ejde-508	300	7	zn	zn	PROPN
ejde-508	300	8	=	=	SYM
ejde-508	300	9	yn	yn	PROPN
ejde-508	300	10	<	<	X
ejde-508	300	11	0	0	NUM
ejde-508	300	12	,	,	PUNCT
ejde-508	300	13	is	be	AUX
ejde-508	300	14	a	a	DET
ejde-508	300	15	contradiction	contradiction	NOUN
ejde-508	300	16	.	.	PUNCT
ejde-508	301	1	hence	hence	ADV
ejde-508	301	2	pn	pn	VERB
ejde-508	301	3	>	>	X
ejde-508	301	4	0	0	PROPN
ejde-508	301	5	.	.	PUNCT
ejde-508	302	1	from	from	ADP
ejde-508	302	2	(	(	PUNCT
ejde-508	302	3	a1	a1	NOUN
ejde-508	302	4	)	)	PUNCT
ejde-508	302	5	and	and	CCONJ
ejde-508	302	6	(	(	PUNCT
ejde-508	302	7	1.4	1.4	NUM
ejde-508	302	8	)	)	PUNCT
ejde-508	302	9	it	it	PRON
ejde-508	302	10	follows	follow	VERB
ejde-508	302	11	that	that	SCONJ
ejde-508	302	12	yn−k	yn−k	PROPN
ejde-508	302	13	≥	≥	PRON
ejde-508	302	14	zn+s−k	zn+s−k	PROPN
ejde-508	302	15	−pδ	−pδ	NOUN
ejde-508	302	16	.	.	PUNCT
ejde-508	303	1	hence	hence	ADV
ejde-508	303	2	,	,	PUNCT
ejde-508	303	3	(	(	PUNCT
ejde-508	303	4	2.25	2.25	NUM
ejde-508	303	5	)	)	PUNCT
ejde-508	303	6	with	with	ADP
ejde-508	303	7	lemma	lemma	PROPN
ejde-508	303	8	2.9	2.9	NUM
ejde-508	303	9	(	(	PUNCT
ejde-508	303	10	b	b	NOUN
ejde-508	303	11	)	)	PUNCT
ejde-508	303	12	yields	yield	NOUN
ejde-508	303	13	∆zn	∆zn	NOUN
ejde-508	303	14	−	−	PROPN
ejde-508	303	15	qng(zn+s−k/(−pδ	qng(zn+s−k/(−pδ	NOUN
ejde-508	303	16	)	)	PUNCT
ejde-508	303	17	)	)	PUNCT
ejde-508	303	18	≥	≥	NOUN
ejde-508	303	19	0	0	NUM
ejde-508	303	20	.	.	PUNCT
ejde-508	304	1	substituting	substitute	VERB
ejde-508	304	2	vn	vn	PROPN
ejde-508	304	3	=	=	SYM
ejde-508	304	4	zn/(−pδ	zn/(−pδ	NOUN
ejde-508	304	5	)	)	PUNCT
ejde-508	304	6	,	,	PUNCT
ejde-508	304	7	which	which	PRON
ejde-508	304	8	implies	imply	VERB
ejde-508	304	9	−pδ∆vn	−pδ∆vn	X
ejde-508	304	10	=	=	SYM
ejde-508	304	11	∆zn	∆zn	PROPN
ejde-508	304	12	,	,	PUNCT
ejde-508	304	13	we	we	PRON
ejde-508	304	14	find	find	VERB
ejde-508	304	15	that	that	SCONJ
ejde-508	304	16	pδ∆vn	pδ∆vn	PROPN
ejde-508	304	17	+	+	CCONJ
ejde-508	304	18	qng(vn+s−k	qng(vn+s−k	SYM
ejde-508	304	19	)	)	PUNCT
ejde-508	304	20	≤	≤	NOUN
ejde-508	304	21	0	0	NUM
ejde-508	304	22	,	,	PUNCT
ejde-508	304	23	(	(	PUNCT
ejde-508	304	24	4.2	4.2	NUM
ejde-508	304	25	)	)	PUNCT
ejde-508	304	26	which	which	PRON
ejde-508	304	27	together	together	ADV
ejde-508	304	28	with	with	ADP
ejde-508	304	29	s	s	X
ejde-508	304	30	<	<	X
ejde-508	304	31	k	k	PROPN
ejde-508	304	32	and	and	CCONJ
ejde-508	304	33	vn	vn	PROPN
ejde-508	304	34	is	be	AUX
ejde-508	304	35	positive	positive	ADJ
ejde-508	304	36	and	and	CCONJ
ejde-508	304	37	decreasing	decrease	VERB
ejde-508	304	38	,	,	PUNCT
ejde-508	304	39	implies	imply	VERB
ejde-508	304	40	pδ∆vn	pδ∆vn	NOUN
ejde-508	304	41	+	+	NUM
ejde-508	304	42	qng(vn	qng(vn	NOUN
ejde-508	304	43	)	)	PUNCT
ejde-508	304	44	≤	≤	NOUN
ejde-508	304	45	0	0	NUM
ejde-508	304	46	,	,	PUNCT
ejde-508	304	47	then	then	ADV
ejde-508	304	48	dividing	divide	VERB
ejde-508	304	49	both	both	DET
ejde-508	304	50	sides	side	NOUN
ejde-508	304	51	of	of	ADP
ejde-508	304	52	the	the	DET
ejde-508	304	53	above	above	ADJ
ejde-508	304	54	by	by	ADP
ejde-508	304	55	g(vn	g(vn	NOUN
ejde-508	304	56	)	)	PUNCT
ejde-508	304	57	we	we	PRON
ejde-508	304	58	obtain	obtain	VERB
ejde-508	304	59	pδ∆vn	pδ∆vn	NUM
ejde-508	304	60	g(vn	g(vn	NUM
ejde-508	304	61	)	)	PUNCT
ejde-508	305	1	+	+	CCONJ
ejde-508	305	2	qn	qn	NOUN
ejde-508	305	3	≤	≤	NUM
ejde-508	305	4	0	0	NUM
ejde-508	305	5	.	.	PUNCT
ejde-508	306	1	10	10	NUM
ejde-508	306	2	a.	a.	PROPN
ejde-508	306	3	k.	k.	PROPN
ejde-508	306	4	bhuyan	bhuyan	PROPN
ejde-508	306	5	,	,	PUNCT
ejde-508	306	6	l.	l.	PROPN
ejde-508	306	7	n.	n.	PROPN
ejde-508	306	8	padhy	padhy	PROPN
ejde-508	306	9	,	,	PUNCT
ejde-508	306	10	r.	r.	PROPN
ejde-508	306	11	n.	n.	PROPN
ejde-508	306	12	rath	rath	PROPN
ejde-508	306	13	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	306	14	then	then	ADV
ejde-508	306	15	using	use	VERB
ejde-508	306	16	∆vn	∆vn	PROPN
ejde-508	306	17	=	=	SYM
ejde-508	306	18	∫	∫	PROPN
ejde-508	306	19	vn+1	vn+1	PROPN
ejde-508	306	20	vn	vn	PROPN
ejde-508	306	21	dx	dx	PROPN
ejde-508	306	22	and	and	CCONJ
ejde-508	306	23	taking	take	VERB
ejde-508	306	24	vn+1	vn+1	PROPN
ejde-508	306	25	≤	≤	NUM
ejde-508	306	26	x	x	PUNCT
ejde-508	306	27	≤	≤	NUM
ejde-508	306	28	vn	vn	NOUN
ejde-508	306	29	we	we	PRON
ejde-508	306	30	have	have	VERB
ejde-508	306	31	pδ	pδ	PROPN
ejde-508	306	32	∫	∫	PROPN
ejde-508	306	33	vn+1	vn+1	PROPN
ejde-508	307	1	vn	vn	PROPN
ejde-508	307	2	dx	dx	PROPN
ejde-508	307	3	g(vn	g(vn	PROPN
ejde-508	307	4	)	)	PUNCT
ejde-508	308	1	+	+	CCONJ
ejde-508	308	2	qn	qn	X
ejde-508	308	3	≤	≤	NUM
ejde-508	308	4	0	0	NUM
ejde-508	308	5	,	,	PUNCT
ejde-508	308	6	which	which	PRON
ejde-508	308	7	implies	imply	VERB
ejde-508	308	8	,	,	PUNCT
ejde-508	308	9	because	because	SCONJ
ejde-508	308	10	of	of	ADP
ejde-508	308	11	the	the	DET
ejde-508	308	12	nondecreasing	nondecreasing	ADJ
ejde-508	308	13	character	character	NOUN
ejde-508	308	14	of	of	ADP
ejde-508	308	15	g	g	PROPN
ejde-508	308	16	that	that	PRON
ejde-508	308	17	pδ	pδ	PROPN
ejde-508	308	18	∫	∫	PROPN
ejde-508	308	19	vn+1	vn+1	PROPN
ejde-508	308	20	vn	vn	PROPN
ejde-508	308	21	dx	dx	PROPN
ejde-508	308	22	g(x	g(x	PROPN
ejde-508	308	23	)	)	PUNCT
ejde-508	309	1	+	+	CCONJ
ejde-508	309	2	qn	qn	X
ejde-508	309	3	≤	≤	NUM
ejde-508	309	4	0	0	NUM
ejde-508	309	5	.	.	PUNCT
ejde-508	309	6	summing	sum	VERB
ejde-508	309	7	from	from	ADP
ejde-508	309	8	n	n	NOUN
ejde-508	309	9	=	=	SYM
ejde-508	309	10	n1	n1	NOUN
ejde-508	309	11	to	to	PART
ejde-508	309	12	l	l	NOUN
ejde-508	309	13	−	−	NUM
ejde-508	309	14	1	1	NUM
ejde-508	309	15	we	we	PRON
ejde-508	309	16	obtain	obtain	VERB
ejde-508	309	17	pδ	pδ	PRON
ejde-508	309	18	∫	∫	PROPN
ejde-508	309	19	vl	vl	PROPN
ejde-508	309	20	vn1	vn1	PROPN
ejde-508	309	21	dx	dx	PROPN
ejde-508	309	22	g(x	g(x	PROPN
ejde-508	309	23	)	)	PUNCT
ejde-508	310	1	+	+	CCONJ
ejde-508	310	2	l−1∑	l−1∑	ADJ
ejde-508	310	3	n1	n1	PROPN
ejde-508	310	4	qn	qn	NOUN
ejde-508	310	5	≤	≤	NOUN
ejde-508	310	6	0	0	NUM
ejde-508	310	7	.	.	PUNCT
ejde-508	311	1	as	as	ADP
ejde-508	311	2	l→∞	l→∞	NUM
ejde-508	311	3	,	,	PUNCT
ejde-508	311	4	vl	vl	X
ejde-508	311	5	→	→	X
ejde-508	311	6	0	0	NUM
ejde-508	311	7	,	,	PUNCT
ejde-508	311	8	and	and	CCONJ
ejde-508	311	9	so	so	ADV
ejde-508	311	10	in	in	ADP
ejde-508	311	11	the	the	DET
ejde-508	311	12	limiting	limit	VERB
ejde-508	311	13	case	case	NOUN
ejde-508	311	14	,	,	PUNCT
ejde-508	311	15	we	we	PRON
ejde-508	311	16	obtain	obtain	VERB
ejde-508	311	17	∞∑	∞∑	NUM
ejde-508	311	18	n1	n1	ADJ
ejde-508	311	19	qn	qn	NOUN
ejde-508	311	20	≤	≤	PROPN
ejde-508	311	21	pδ	pδ	ADP
ejde-508	311	22	∫	∫	PROPN
ejde-508	311	23	vn1	vn1	NOUN
ejde-508	311	24	0	0	NUM
ejde-508	311	25	dx	dx	PROPN
ejde-508	311	26	g(x	g(x	PROPN
ejde-508	311	27	)	)	PUNCT
ejde-508	312	1	<	<	X
ejde-508	312	2	∞	∞	NUM
ejde-508	312	3	by	by	ADP
ejde-508	312	4	(	(	PUNCT
ejde-508	312	5	4.1	4.1	NUM
ejde-508	312	6	)	)	PUNCT
ejde-508	312	7	,	,	PUNCT
ejde-508	312	8	which	which	PRON
ejde-508	312	9	contradicts	contradict	VERB
ejde-508	312	10	(	(	PUNCT
ejde-508	312	11	a2	a2	PROPN
ejde-508	312	12	)	)	PUNCT
ejde-508	312	13	.	.	PUNCT
ejde-508	313	1	the	the	DET
ejde-508	313	2	proof	proof	NOUN
ejde-508	313	3	for	for	ADP
ejde-508	313	4	the	the	DET
ejde-508	313	5	case	case	NOUN
ejde-508	313	6	yn	yn	PRON
ejde-508	313	7	being	be	AUX
ejde-508	313	8	eventually	eventually	ADV
ejde-508	313	9	negative	negative	ADJ
ejde-508	313	10	is	be	AUX
ejde-508	313	11	similar	similar	ADJ
ejde-508	313	12	and	and	CCONJ
ejde-508	313	13	this	this	PRON
ejde-508	313	14	completes	complete	VERB
ejde-508	313	15	the	the	DET
ejde-508	313	16	proof	proof	NOUN
ejde-508	313	17	.	.	PUNCT
ejde-508	314	1	�	�	PROPN
ejde-508	314	2	theorem	theorem	VERB
ejde-508	314	3	4.2	4.2	NUM
ejde-508	314	4	.	.	PUNCT
ejde-508	315	1	suppose	suppose	VERB
ejde-508	315	2	(	(	PUNCT
ejde-508	315	3	a1	a1	NOUN
ejde-508	315	4	)	)	PUNCT
ejde-508	315	5	,	,	PUNCT
ejde-508	315	6	(	(	PUNCT
ejde-508	315	7	a2	a2	PROPN
ejde-508	315	8	)	)	PUNCT
ejde-508	315	9	,	,	PUNCT
ejde-508	315	10	(	(	PUNCT
ejde-508	315	11	a5	a5	NOUN
ejde-508	315	12	)	)	PUNCT
ejde-508	315	13	hold	hold	NOUN
ejde-508	315	14	,	,	PUNCT
ejde-508	315	15	and	and	CCONJ
ejde-508	315	16	(	(	PUNCT
ejde-508	315	17	1.5	1.5	NUM
ejde-508	315	18	)	)	PUNCT
ejde-508	315	19	,	,	PUNCT
ejde-508	315	20	(	(	PUNCT
ejde-508	315	21	4.1	4.1	NUM
ejde-508	315	22	)	)	PUNCT
ejde-508	315	23	,	,	PUNCT
ejde-508	315	24	δ	δ	PROPN
ejde-508	315	25	≤	≤	ADV
ejde-508	315	26	1	1	NUM
ejde-508	315	27	and	and	CCONJ
ejde-508	315	28	s	s	X
ejde-508	315	29	<	<	X
ejde-508	315	30	k	k	X
ejde-508	315	31	are	be	AUX
ejde-508	315	32	satisfied	satisfied	ADJ
ejde-508	315	33	.	.	PUNCT
ejde-508	316	1	then	then	ADV
ejde-508	316	2	every	every	DET
ejde-508	316	3	solution	solution	NOUN
ejde-508	316	4	of	of	ADP
ejde-508	316	5	(	(	PUNCT
ejde-508	316	6	1.1	1.1	NUM
ejde-508	316	7	)	)	PUNCT
ejde-508	316	8	oscillates	oscillate	NOUN
ejde-508	316	9	.	.	PUNCT
ejde-508	317	1	proof	proof	NOUN
ejde-508	317	2	.	.	PUNCT
ejde-508	318	1	on	on	ADP
ejde-508	318	2	the	the	DET
ejde-508	318	3	contrary	contrary	NOUN
ejde-508	318	4	,	,	PUNCT
ejde-508	318	5	let	let	VERB
ejde-508	318	6	yn	yn	PRON
ejde-508	318	7	be	be	AUX
ejde-508	318	8	an	an	DET
ejde-508	318	9	eventually	eventually	ADV
ejde-508	318	10	positive	positive	ADJ
ejde-508	318	11	solution	solution	NOUN
ejde-508	318	12	of	of	ADP
ejde-508	318	13	(	(	PUNCT
ejde-508	318	14	1.1	1.1	NUM
ejde-508	318	15	)	)	PUNCT
ejde-508	318	16	.	.	PUNCT
ejde-508	319	1	setting	set	VERB
ejde-508	319	2	zn	zn	NOUN
ejde-508	319	3	as	as	ADP
ejde-508	319	4	in	in	ADP
ejde-508	319	5	(	(	PUNCT
ejde-508	319	6	2.5	2.5	NUM
ejde-508	319	7	)	)	PUNCT
ejde-508	319	8	,	,	PUNCT
ejde-508	319	9	we	we	PRON
ejde-508	319	10	obtain	obtain	VERB
ejde-508	319	11	(	(	PUNCT
ejde-508	319	12	2.25	2.25	NUM
ejde-508	319	13	)	)	PUNCT
ejde-508	319	14	.	.	PUNCT
ejde-508	320	1	then	then	ADV
ejde-508	320	2	by	by	ADP
ejde-508	320	3	lemma	lemma	PROPN
ejde-508	320	4	2.8(b	2.8(b	NUM
ejde-508	320	5	)	)	PUNCT
ejde-508	320	6	,	,	PUNCT
ejde-508	320	7	we	we	PRON
ejde-508	320	8	find	find	VERB
ejde-508	320	9	that	that	SCONJ
ejde-508	320	10	(	(	PUNCT
ejde-508	320	11	2.27	2.27	NUM
ejde-508	320	12	)	)	PUNCT
ejde-508	320	13	holds	hold	VERB
ejde-508	320	14	.	.	PUNCT
ejde-508	321	1	then	then	ADV
ejde-508	321	2	(	(	PUNCT
ejde-508	321	3	2.28	2.28	NUM
ejde-508	321	4	)	)	PUNCT
ejde-508	321	5	,	,	PUNCT
ejde-508	321	6	follows	follow	VERB
ejde-508	321	7	as	as	ADP
ejde-508	321	8	a	a	DET
ejde-508	321	9	consequence	consequence	NOUN
ejde-508	321	10	and	and	CCONJ
ejde-508	321	11	zn	zn	X
ejde-508	321	12	<	<	X
ejde-508	321	13	0	0	NUM
ejde-508	321	14	.	.	PUNCT
ejde-508	322	1	if	if	SCONJ
ejde-508	322	2	pn	pn	PROPN
ejde-508	322	3	=	=	NOUN
ejde-508	322	4	0	0	PROPN
ejde-508	323	1	then	then	ADV
ejde-508	323	2	zn	zn	PROPN
ejde-508	323	3	=	=	SYM
ejde-508	323	4	yn	yn	PROPN
ejde-508	323	5	<	<	X
ejde-508	323	6	0	0	NUM
ejde-508	323	7	which	which	PRON
ejde-508	323	8	is	be	AUX
ejde-508	323	9	a	a	DET
ejde-508	323	10	contradiction	contradiction	NOUN
ejde-508	323	11	.	.	PUNCT
ejde-508	324	1	hence	hence	ADV
ejde-508	324	2	from	from	ADP
ejde-508	324	3	(	(	PUNCT
ejde-508	324	4	2.25	2.25	NUM
ejde-508	324	5	)	)	PUNCT
ejde-508	324	6	,	,	PUNCT
ejde-508	324	7	and	and	CCONJ
ejde-508	324	8	lemma	lemma	PROPN
ejde-508	324	9	2.9(b	2.9(b	NUM
ejde-508	324	10	)	)	PUNCT
ejde-508	324	11	we	we	PRON
ejde-508	324	12	obtain	obtain	VERB
ejde-508	324	13	∆zn	∆zn	PROPN
ejde-508	324	14	−	−	PROPN
ejde-508	324	15	qng(yn−k	qng(yn−k	PROPN
ejde-508	324	16	)	)	PUNCT
ejde-508	324	17	≥	≥	NOUN
ejde-508	324	18	0	0	NUM
ejde-508	324	19	.	.	PUNCT
ejde-508	324	20	using	use	VERB
ejde-508	324	21	δ	δ	PROPN
ejde-508	324	22	≤	≤	ADV
ejde-508	324	23	1	1	NUM
ejde-508	324	24	and	and	CCONJ
ejde-508	324	25	(	(	PUNCT
ejde-508	324	26	1.5	1.5	NUM
ejde-508	324	27	)	)	PUNCT
ejde-508	324	28	,	,	PUNCT
ejde-508	324	29	we	we	PRON
ejde-508	324	30	find	find	VERB
ejde-508	324	31	yn−k	yn−k	NOUN
ejde-508	324	32	≥	≥	PRON
ejde-508	324	33	zn+s−k	zn+s−k	PROPN
ejde-508	324	34	−δ	−δ	ADJ
ejde-508	324	35	≥	≥	NOUN
ejde-508	324	36	−zn+s−k	−zn+s−k	NOUN
ejde-508	324	37	.	.	PUNCT
ejde-508	325	1	therefore	therefore	ADV
ejde-508	325	2	,	,	PUNCT
ejde-508	325	3	∆zn	∆zn	PROPN
ejde-508	325	4	−	−	PROPN
ejde-508	325	5	qng(−zn+s−k	qng(−zn+s−k	NOUN
ejde-508	325	6	)	)	PUNCT
ejde-508	325	7	≥	≥	NOUN
ejde-508	325	8	0	0	NUM
ejde-508	325	9	.	.	PUNCT
ejde-508	325	10	substituting	substitute	VERB
ejde-508	325	11	−zn	−zn	NOUN
ejde-508	325	12	=	=	SYM
ejde-508	325	13	vn	vn	PROPN
ejde-508	325	14	,	,	PUNCT
ejde-508	325	15	which	which	PRON
ejde-508	325	16	implies	imply	VERB
ejde-508	325	17	∆zn	∆zn	PROPN
ejde-508	325	18	=	=	SYM
ejde-508	325	19	−∆vn	−∆vn	PROPN
ejde-508	325	20	,	,	PUNCT
ejde-508	325	21	in	in	ADP
ejde-508	325	22	the	the	DET
ejde-508	325	23	above	above	NOUN
ejde-508	325	24	,	,	PUNCT
ejde-508	325	25	we	we	PRON
ejde-508	325	26	obtain	obtain	VERB
ejde-508	325	27	∆vn	∆vn	NOUN
ejde-508	325	28	+	+	CCONJ
ejde-508	325	29	qng(vn+s−k	qng(vn+s−k	SYM
ejde-508	325	30	)	)	PUNCT
ejde-508	325	31	≤	≤	NOUN
ejde-508	325	32	0	0	NUM
ejde-508	325	33	.	.	PUNCT
ejde-508	326	1	(	(	PUNCT
ejde-508	326	2	4.3	4.3	NUM
ejde-508	326	3	)	)	PUNCT
ejde-508	326	4	then	then	ADV
ejde-508	326	5	further	far	ADV
ejde-508	326	6	using	use	VERB
ejde-508	326	7	s	s	PRON
ejde-508	326	8	<	<	X
ejde-508	326	9	k	k	PROPN
ejde-508	326	10	and	and	CCONJ
ejde-508	326	11	vn	vn	X
ejde-508	326	12	>	>	X
ejde-508	326	13	0	0	PUNCT
ejde-508	327	1	and	and	CCONJ
ejde-508	327	2	decreasing	decrease	VERB
ejde-508	327	3	,	,	PUNCT
ejde-508	327	4	we	we	PRON
ejde-508	327	5	obtain	obtain	VERB
ejde-508	327	6	∆vn	∆vn	PROPN
ejde-508	327	7	+	+	CCONJ
ejde-508	327	8	qng(vn	qng(vn	X
ejde-508	327	9	)	)	PUNCT
ejde-508	327	10	≤	≤	NOUN
ejde-508	327	11	0	0	NUM
ejde-508	327	12	.	.	PUNCT
ejde-508	328	1	dividing	divide	VERB
ejde-508	328	2	both	both	DET
ejde-508	328	3	sides	side	NOUN
ejde-508	328	4	of	of	ADP
ejde-508	328	5	the	the	DET
ejde-508	328	6	above	above	ADJ
ejde-508	328	7	inequality	inequality	NOUN
ejde-508	328	8	,	,	PUNCT
ejde-508	328	9	by	by	ADP
ejde-508	328	10	g(vn	g(vn	PROPN
ejde-508	328	11	)	)	PUNCT
ejde-508	328	12	,	,	PUNCT
ejde-508	328	13	we	we	PRON
ejde-508	328	14	obtain	obtain	VERB
ejde-508	328	15	∆vn	∆vn	NUM
ejde-508	328	16	g(vn	g(vn	NUM
ejde-508	328	17	)	)	PUNCT
ejde-508	329	1	+	+	CCONJ
ejde-508	329	2	qn	qn	NOUN
ejde-508	329	3	≤	≤	NUM
ejde-508	329	4	0	0	NUM
ejde-508	329	5	.	.	PUNCT
ejde-508	330	1	taking	take	VERB
ejde-508	330	2	vn+1	vn+1	PROPN
ejde-508	330	3	≤	≤	NUM
ejde-508	330	4	v	v	PRON
ejde-508	330	5	≤	≤	NUM
ejde-508	330	6	vn	vn	NOUN
ejde-508	330	7	and	and	CCONJ
ejde-508	330	8	using	use	VERB
ejde-508	330	9	∆vn	∆vn	PROPN
ejde-508	330	10	=	=	SYM
ejde-508	330	11	∫	∫	PROPN
ejde-508	330	12	vn+1	vn+1	PROPN
ejde-508	330	13	vn	vn	PROPN
ejde-508	330	14	dv	dv	PROPN
ejde-508	330	15	,	,	PUNCT
ejde-508	330	16	we	we	PRON
ejde-508	330	17	proceed	proceed	VERB
ejde-508	330	18	as	as	ADP
ejde-508	330	19	in	in	ADP
ejde-508	330	20	the	the	DET
ejde-508	330	21	proof	proof	NOUN
ejde-508	330	22	of	of	ADP
ejde-508	330	23	theorem	theorem	ADJ
ejde-508	330	24	4.1	4.1	NUM
ejde-508	330	25	to	to	PART
ejde-508	330	26	obtain	obtain	VERB
ejde-508	330	27	∞∑	∞∑	NUM
ejde-508	330	28	n	n	CCONJ
ejde-508	330	29	=	=	SYM
ejde-508	330	30	n1	n1	NOUN
ejde-508	330	31	qn	qn	NOUN
ejde-508	330	32	≤	≤	NUM
ejde-508	330	33	∫	∫	PROPN
ejde-508	330	34	vn1	vn1	NOUN
ejde-508	330	35	0	0	NUM
ejde-508	331	1	dv	dv	PROPN
ejde-508	331	2	g(v	g(v	PROPN
ejde-508	331	3	)	)	PUNCT
ejde-508	332	1	<	<	X
ejde-508	332	2	∞	∞	NUM
ejde-508	332	3	by	by	ADP
ejde-508	332	4	(	(	PUNCT
ejde-508	332	5	4.1	4.1	NUM
ejde-508	332	6	)	)	PUNCT
ejde-508	332	7	,	,	PUNCT
ejde-508	332	8	which	which	PRON
ejde-508	332	9	contradicts	contradict	VERB
ejde-508	332	10	(	(	PUNCT
ejde-508	332	11	a2	a2	PROPN
ejde-508	332	12	)	)	PUNCT
ejde-508	332	13	.	.	PUNCT
ejde-508	333	1	the	the	DET
ejde-508	333	2	proof	proof	NOUN
ejde-508	333	3	for	for	ADP
ejde-508	333	4	the	the	DET
ejde-508	333	5	case	case	NOUN
ejde-508	333	6	when	when	SCONJ
ejde-508	333	7	yn	yn	PROPN
ejde-508	333	8	is	be	AUX
ejde-508	333	9	eventually	eventually	ADV
ejde-508	333	10	negative	negative	ADJ
ejde-508	333	11	is	be	AUX
ejde-508	333	12	similar	similar	ADJ
ejde-508	333	13	.	.	PUNCT
ejde-508	334	1	�	�	PROPN
ejde-508	334	2	ejde-2020/87	ejde-2020/87	VERB
ejde-508	334	3	oscillation	oscillation	NOUN
ejde-508	334	4	for	for	ADP
ejde-508	334	5	second	second	ADJ
ejde-508	334	6	order	order	NOUN
ejde-508	334	7	neutral	neutral	ADJ
ejde-508	334	8	equations	equation	NOUN
ejde-508	334	9	11	11	NUM
ejde-508	334	10	theorem	theorem	VERB
ejde-508	334	11	4.3	4.3	NUM
ejde-508	334	12	.	.	PUNCT
ejde-508	335	1	suppose	suppose	VERB
ejde-508	335	2	(	(	PUNCT
ejde-508	335	3	a1	a1	NOUN
ejde-508	335	4	)	)	PUNCT
ejde-508	335	5	,	,	PUNCT
ejde-508	335	6	(	(	PUNCT
ejde-508	335	7	a2	a2	PROPN
ejde-508	335	8	)	)	PUNCT
ejde-508	335	9	,	,	PUNCT
ejde-508	335	10	(	(	PUNCT
ejde-508	335	11	a5	a5	NOUN
ejde-508	335	12	)	)	PUNCT
ejde-508	335	13	hold	hold	NOUN
ejde-508	335	14	,	,	PUNCT
ejde-508	335	15	and	and	CCONJ
ejde-508	335	16	(	(	PUNCT
ejde-508	335	17	1.5	1.5	NUM
ejde-508	335	18	)	)	PUNCT
ejde-508	335	19	,	,	PUNCT
ejde-508	335	20	δ	δ	PROPN
ejde-508	335	21	≤	≤	ADV
ejde-508	335	22	1	1	NUM
ejde-508	335	23	and	and	CCONJ
ejde-508	335	24	s	s	X
ejde-508	335	25	<	<	X
ejde-508	335	26	k	k	X
ejde-508	335	27	are	be	AUX
ejde-508	335	28	satisfied	satisfied	ADJ
ejde-508	335	29	.	.	PUNCT
ejde-508	336	1	if	if	SCONJ
ejde-508	336	2	lim	lim	PROPN
ejde-508	336	3	inf	inf	PROPN
ejde-508	336	4	|x|→0	|x|→0	PROPN
ejde-508	336	5	g(x	g(x	PROPN
ejde-508	336	6	)	)	PUNCT
ejde-508	337	1	x	x	X
ejde-508	337	2	>	>	X
ejde-508	337	3	γ	γ	X
ejde-508	337	4	>	>	X
ejde-508	337	5	0	0	NUM
ejde-508	337	6	.	.	PUNCT
ejde-508	338	1	(	(	PUNCT
ejde-508	338	2	4.4	4.4	NUM
ejde-508	338	3	)	)	PUNCT
ejde-508	338	4	and	and	CCONJ
ejde-508	338	5	lim	lim	PROPN
ejde-508	338	6	inf	inf	PROPN
ejde-508	338	7	n→∞	n→∞	X
ejde-508	338	8	n−1∑	n−1∑	PROPN
ejde-508	338	9	i	i	NOUN
ejde-508	338	10	=	=	NOUN
ejde-508	338	11	n−k+s	n−k+s	X
ejde-508	338	12	qi	qi	NOUN
ejde-508	338	13	>	>	X
ejde-508	338	14	1	1	NUM
ejde-508	338	15	γ	γ	X
ejde-508	338	16	(	(	PUNCT
ejde-508	338	17	k	k	PROPN
ejde-508	338	18	−	−	PROPN
ejde-508	338	19	s	s	PART
ejde-508	338	20	k	k	X
ejde-508	338	21	−	−	NOUN
ejde-508	338	22	s+	s+	PUNCT
ejde-508	338	23	1	1	NUM
ejde-508	338	24	)	)	PUNCT
ejde-508	338	25	k−s+1	k−s+1	PROPN
ejde-508	338	26	,	,	PUNCT
ejde-508	338	27	(	(	PUNCT
ejde-508	338	28	4.5	4.5	NUM
ejde-508	338	29	)	)	PUNCT
ejde-508	338	30	then	then	ADV
ejde-508	338	31	every	every	DET
ejde-508	338	32	solution	solution	NOUN
ejde-508	338	33	of	of	ADP
ejde-508	338	34	(	(	PUNCT
ejde-508	338	35	1.1	1.1	NUM
ejde-508	338	36	)	)	PUNCT
ejde-508	338	37	oscillates	oscillate	NOUN
ejde-508	338	38	.	.	PUNCT
ejde-508	339	1	proof	proof	NOUN
ejde-508	339	2	.	.	PUNCT
ejde-508	340	1	as	as	SCONJ
ejde-508	340	2	s	s	X
ejde-508	340	3	<	<	X
ejde-508	340	4	k	k	NOUN
ejde-508	340	5	and	and	CCONJ
ejde-508	340	6	0	0	NUM
ejde-508	340	7	<	<	X
ejde-508	340	8	δ	δ	PROPN
ejde-508	340	9	≤	≤	ADV
ejde-508	340	10	1	1	NUM
ejde-508	340	11	proceeding	proceeding	NOUN
ejde-508	340	12	as	as	ADP
ejde-508	340	13	in	in	ADP
ejde-508	340	14	the	the	DET
ejde-508	340	15	proof	proof	NOUN
ejde-508	340	16	of	of	ADP
ejde-508	340	17	the	the	DET
ejde-508	340	18	theorem	theorem	ADJ
ejde-508	340	19	4.2	4.2	NUM
ejde-508	340	20	,	,	PUNCT
ejde-508	340	21	we	we	PRON
ejde-508	340	22	obtain	obtain	VERB
ejde-508	340	23	the	the	DET
ejde-508	340	24	first	first	ADJ
ejde-508	340	25	order	order	NOUN
ejde-508	340	26	delay	delay	NOUN
ejde-508	340	27	difference	difference	NOUN
ejde-508	340	28	inequality	inequality	NOUN
ejde-508	340	29	(	(	PUNCT
ejde-508	340	30	4.3	4.3	NUM
ejde-508	340	31	)	)	PUNCT
ejde-508	340	32	,	,	PUNCT
ejde-508	340	33	which	which	PRON
ejde-508	340	34	by	by	ADP
ejde-508	340	35	(	(	PUNCT
ejde-508	340	36	4.4	4.4	NUM
ejde-508	340	37	)	)	PUNCT
ejde-508	340	38	,	,	PUNCT
ejde-508	340	39	yields	yield	VERB
ejde-508	340	40	∆vn	∆vn	PROPN
ejde-508	340	41	+	+	CCONJ
ejde-508	340	42	γqnvn+s−k	γqnvn+s−k	ADP
ejde-508	340	43	≤	≤	NOUN
ejde-508	340	44	0	0	NUM
ejde-508	340	45	which	which	PRON
ejde-508	340	46	has	have	VERB
ejde-508	340	47	a	a	DET
ejde-508	340	48	positive	positive	ADJ
ejde-508	340	49	solution	solution	NOUN
ejde-508	340	50	.	.	PUNCT
ejde-508	341	1	this	this	PRON
ejde-508	341	2	,	,	PUNCT
ejde-508	341	3	contradicts	contradict	VERB
ejde-508	341	4	lemma	lemma	PROPN
ejde-508	341	5	2.1(a	2.1(a	NUM
ejde-508	341	6	)	)	PUNCT
ejde-508	341	7	.	.	PUNCT
ejde-508	342	1	the	the	DET
ejde-508	342	2	proof	proof	NOUN
ejde-508	342	3	for	for	ADP
ejde-508	342	4	the	the	DET
ejde-508	342	5	case	case	NOUN
ejde-508	342	6	when	when	SCONJ
ejde-508	342	7	yn	yn	PROPN
ejde-508	342	8	is	be	AUX
ejde-508	342	9	eventually	eventually	ADV
ejde-508	342	10	negative	negative	ADJ
ejde-508	342	11	is	be	AUX
ejde-508	342	12	similar	similar	ADJ
ejde-508	342	13	.	.	PUNCT
ejde-508	343	1	this	this	PRON
ejde-508	343	2	completes	complete	VERB
ejde-508	343	3	the	the	DET
ejde-508	343	4	proof	proof	NOUN
ejde-508	343	5	.	.	PUNCT
ejde-508	344	1	�	�	PROPN
ejde-508	344	2	theorem	theorem	VERB
ejde-508	344	3	4.4	4.4	NUM
ejde-508	344	4	.	.	PUNCT
ejde-508	345	1	suppose	suppose	VERB
ejde-508	345	2	(	(	PUNCT
ejde-508	345	3	a1	a1	NOUN
ejde-508	345	4	)	)	PUNCT
ejde-508	345	5	,	,	PUNCT
ejde-508	345	6	(	(	PUNCT
ejde-508	345	7	a3	a3	NOUN
ejde-508	345	8	)	)	PUNCT
ejde-508	345	9	hold	hold	VERB
ejde-508	345	10	,	,	PUNCT
ejde-508	345	11	and	and	CCONJ
ejde-508	345	12	(	(	PUNCT
ejde-508	345	13	1.6	1.6	NUM
ejde-508	345	14	)	)	PUNCT
ejde-508	345	15	,	,	PUNCT
ejde-508	345	16	(	(	PUNCT
ejde-508	345	17	2.21	2.21	NUM
ejde-508	345	18	)	)	PUNCT
ejde-508	345	19	and	and	CCONJ
ejde-508	345	20	(	(	PUNCT
ejde-508	345	21	2.22	2.22	NUM
ejde-508	345	22	)	)	PUNCT
ejde-508	345	23	are	be	AUX
ejde-508	345	24	satisfied	satisfied	ADJ
ejde-508	345	25	.	.	PUNCT
ejde-508	346	1	then	then	ADV
ejde-508	346	2	every	every	DET
ejde-508	346	3	solution	solution	NOUN
ejde-508	346	4	of	of	ADP
ejde-508	346	5	(	(	PUNCT
ejde-508	346	6	1.1	1.1	NUM
ejde-508	346	7	)	)	PUNCT
ejde-508	346	8	oscillates	oscillate	NOUN
ejde-508	346	9	.	.	PUNCT
ejde-508	347	1	proof	proof	NOUN
ejde-508	347	2	.	.	PUNCT
ejde-508	348	1	on	on	ADP
ejde-508	348	2	the	the	DET
ejde-508	348	3	contrary	contrary	NOUN
ejde-508	348	4	,	,	PUNCT
ejde-508	348	5	suppose	suppose	VERB
ejde-508	348	6	yn	yn	PRON
ejde-508	348	7	be	be	AUX
ejde-508	348	8	an	an	DET
ejde-508	348	9	eventually	eventually	ADV
ejde-508	348	10	positive	positive	ADJ
ejde-508	348	11	solution	solution	NOUN
ejde-508	348	12	of	of	ADP
ejde-508	348	13	(	(	PUNCT
ejde-508	348	14	1.1	1.1	NUM
ejde-508	348	15	)	)	PUNCT
ejde-508	348	16	.	.	PUNCT
ejde-508	349	1	setting	set	VERB
ejde-508	349	2	zn	zn	NOUN
ejde-508	349	3	as	as	ADP
ejde-508	349	4	in	in	ADP
ejde-508	349	5	(	(	PUNCT
ejde-508	349	6	2.5	2.5	NUM
ejde-508	349	7	)	)	PUNCT
ejde-508	349	8	,	,	PUNCT
ejde-508	349	9	we	we	PRON
ejde-508	349	10	obtain	obtain	VERB
ejde-508	349	11	(	(	PUNCT
ejde-508	349	12	2.25	2.25	NUM
ejde-508	349	13	)	)	PUNCT
ejde-508	349	14	.	.	PUNCT
ejde-508	350	1	then	then	ADV
ejde-508	350	2	applying	apply	VERB
ejde-508	350	3	lemma	lemma	PROPN
ejde-508	350	4	2.10	2.10	NUM
ejde-508	350	5	,	,	PUNCT
ejde-508	350	6	we	we	PRON
ejde-508	350	7	obtain	obtain	VERB
ejde-508	350	8	β	β	X
ejde-508	350	9	=	=	PUNCT
ejde-508	350	10	limn→∞	limn→∞	PROPN
ejde-508	350	11	zn	zn	PROPN
ejde-508	350	12	=	=	SYM
ejde-508	350	13	0	0	PROPN
ejde-508	350	14	,	,	PUNCT
ejde-508	350	15	which	which	PRON
ejde-508	350	16	implies	imply	VERB
ejde-508	350	17	zn	zn	X
ejde-508	350	18	<	<	X
ejde-508	350	19	0	0	PROPN
ejde-508	350	20	,	,	PUNCT
ejde-508	350	21	a	a	DET
ejde-508	350	22	contradiction	contradiction	NOUN
ejde-508	350	23	because	because	SCONJ
ejde-508	350	24	zn	zn	PROPN
ejde-508	350	25	≥	≥	VERB
ejde-508	350	26	0	0	NUM
ejde-508	350	27	by	by	ADP
ejde-508	350	28	(	(	PUNCT
ejde-508	350	29	1.6	1.6	NUM
ejde-508	350	30	)	)	PUNCT
ejde-508	350	31	.	.	PUNCT
ejde-508	351	1	the	the	DET
ejde-508	351	2	proof	proof	NOUN
ejde-508	351	3	for	for	ADP
ejde-508	351	4	the	the	DET
ejde-508	351	5	case	case	NOUN
ejde-508	351	6	when	when	SCONJ
ejde-508	351	7	yn	yn	PROPN
ejde-508	351	8	is	be	AUX
ejde-508	351	9	eventually	eventually	ADV
ejde-508	351	10	negative	negative	ADJ
ejde-508	351	11	,	,	PUNCT
ejde-508	351	12	is	be	AUX
ejde-508	351	13	similar	similar	ADJ
ejde-508	351	14	and	and	CCONJ
ejde-508	351	15	thus	thus	ADV
ejde-508	351	16	,	,	PUNCT
ejde-508	351	17	the	the	DET
ejde-508	351	18	proof	proof	NOUN
ejde-508	351	19	is	be	AUX
ejde-508	351	20	complete	complete	ADJ
ejde-508	351	21	.	.	PUNCT
ejde-508	352	1	�	�	PROPN
ejde-508	352	2	theorem	theorem	VERB
ejde-508	352	3	4.5	4.5	NUM
ejde-508	352	4	.	.	PUNCT
ejde-508	353	1	suppose	suppose	VERB
ejde-508	353	2	(	(	PUNCT
ejde-508	353	3	a1	a1	NOUN
ejde-508	353	4	)	)	PUNCT
ejde-508	353	5	,	,	PUNCT
ejde-508	353	6	(	(	PUNCT
ejde-508	353	7	a3	a3	NOUN
ejde-508	353	8	)	)	PUNCT
ejde-508	353	9	hold	hold	VERB
ejde-508	353	10	,	,	PUNCT
ejde-508	353	11	and	and	CCONJ
ejde-508	353	12	(	(	PUNCT
ejde-508	353	13	1.7	1.7	NUM
ejde-508	353	14	)	)	PUNCT
ejde-508	353	15	,	,	PUNCT
ejde-508	353	16	(	(	PUNCT
ejde-508	353	17	2.21	2.21	NUM
ejde-508	353	18	)	)	PUNCT
ejde-508	353	19	and	and	CCONJ
ejde-508	353	20	(	(	PUNCT
ejde-508	353	21	2.22	2.22	NUM
ejde-508	353	22	)	)	PUNCT
ejde-508	353	23	are	be	AUX
ejde-508	353	24	satisfied	satisfied	ADJ
ejde-508	353	25	.	.	PUNCT
ejde-508	354	1	then	then	ADV
ejde-508	354	2	every	every	DET
ejde-508	354	3	solution	solution	NOUN
ejde-508	354	4	of	of	ADP
ejde-508	354	5	(	(	PUNCT
ejde-508	354	6	1.1	1.1	NUM
ejde-508	354	7	)	)	PUNCT
ejde-508	354	8	oscillates	oscillate	NOUN
ejde-508	354	9	.	.	PUNCT
ejde-508	355	1	proof	proof	NOUN
ejde-508	355	2	.	.	PUNCT
ejde-508	356	1	on	on	ADP
ejde-508	356	2	the	the	DET
ejde-508	356	3	contrary	contrary	NOUN
ejde-508	356	4	,	,	PUNCT
ejde-508	356	5	assume	assume	VERB
ejde-508	356	6	yn	yn	PRON
ejde-508	356	7	be	be	AUX
ejde-508	356	8	an	an	DET
ejde-508	356	9	eventually	eventually	ADV
ejde-508	356	10	positive	positive	ADJ
ejde-508	356	11	solution	solution	NOUN
ejde-508	356	12	of	of	ADP
ejde-508	356	13	(	(	PUNCT
ejde-508	356	14	1.1	1.1	NUM
ejde-508	356	15	)	)	PUNCT
ejde-508	356	16	.	.	PUNCT
ejde-508	357	1	setting	set	VERB
ejde-508	357	2	zn	zn	NOUN
ejde-508	357	3	as	as	ADP
ejde-508	357	4	in	in	ADP
ejde-508	357	5	(	(	PUNCT
ejde-508	357	6	2.5	2.5	NUM
ejde-508	357	7	)	)	PUNCT
ejde-508	357	8	,	,	PUNCT
ejde-508	357	9	we	we	PRON
ejde-508	357	10	obtain	obtain	VERB
ejde-508	357	11	(	(	PUNCT
ejde-508	357	12	2.25	2.25	NUM
ejde-508	357	13	)	)	PUNCT
ejde-508	357	14	.	.	PUNCT
ejde-508	358	1	then	then	ADV
ejde-508	358	2	application	application	NOUN
ejde-508	358	3	of	of	ADP
ejde-508	358	4	lemma	lemma	PROPN
ejde-508	358	5	2.10	2.10	NUM
ejde-508	358	6	yields	yield	NOUN
ejde-508	358	7	β	β	X
ejde-508	358	8	=	=	PUNCT
ejde-508	358	9	limn→∞	limn→∞	PROPN
ejde-508	358	10	zn	zn	PROPN
ejde-508	358	11	=	=	SYM
ejde-508	358	12	0	0	PROPN
ejde-508	358	13	.	.	PUNCT
ejde-508	359	1	consequently	consequently	ADV
ejde-508	359	2	∆zn	∆zn	PROPN
ejde-508	359	3	>	>	X
ejde-508	359	4	0	0	PUNCT
ejde-508	360	1	and	and	CCONJ
ejde-508	360	2	zn	zn	X
ejde-508	360	3	<	<	X
ejde-508	360	4	0	0	X
ejde-508	360	5	.	.	PUNCT
ejde-508	360	6	again	again	ADV
ejde-508	360	7	,	,	PUNCT
ejde-508	360	8	this	this	PRON
ejde-508	360	9	would	would	AUX
ejde-508	360	10	lead	lead	VERB
ejde-508	360	11	to	to	ADP
ejde-508	360	12	pn	pn	PROPN
ejde-508	360	13	>	>	X
ejde-508	360	14	yn	yn	PROPN
ejde-508	360	15	l(yn−s	l(yn−s	PROPN
ejde-508	360	16	)	)	PUNCT
ejde-508	360	17	>	>	X
ejde-508	360	18	0	0	PUNCT
ejde-508	361	1	for	for	ADP
ejde-508	361	2	large	large	ADJ
ejde-508	361	3	n	n	CCONJ
ejde-508	361	4	,	,	PUNCT
ejde-508	361	5	which	which	PRON
ejde-508	361	6	is	be	AUX
ejde-508	361	7	a	a	DET
ejde-508	361	8	contradiction	contradiction	NOUN
ejde-508	361	9	,	,	PUNCT
ejde-508	361	10	because	because	SCONJ
ejde-508	361	11	pn	pn	PROPN
ejde-508	361	12	changes	change	NOUN
ejde-508	361	13	sign	sign	VERB
ejde-508	361	14	.	.	PUNCT
ejde-508	362	1	for	for	ADP
ejde-508	362	2	the	the	DET
ejde-508	362	3	proof	proof	NOUN
ejde-508	362	4	of	of	ADP
ejde-508	362	5	the	the	DET
ejde-508	362	6	case	case	NOUN
ejde-508	362	7	,	,	PUNCT
ejde-508	362	8	when	when	SCONJ
ejde-508	362	9	yn	yn	PROPN
ejde-508	362	10	is	be	AUX
ejde-508	362	11	eventually	eventually	ADV
ejde-508	362	12	negative	negative	ADJ
ejde-508	362	13	,	,	PUNCT
ejde-508	362	14	we	we	PRON
ejde-508	362	15	may	may	AUX
ejde-508	362	16	proceed	proceed	VERB
ejde-508	362	17	with	with	ADP
ejde-508	362	18	xn	xn	PROPN
ejde-508	362	19	=	=	SYM
ejde-508	362	20	−yn	−yn	NOUN
ejde-508	362	21	and	and	CCONJ
ejde-508	362	22	complete	complete	VERB
ejde-508	362	23	the	the	DET
ejde-508	362	24	proof	proof	NOUN
ejde-508	362	25	.	.	PUNCT
ejde-508	363	1	�	�	PROPN
ejde-508	363	2	theorem	theorem	VERB
ejde-508	363	3	4.6	4.6	NUM
ejde-508	363	4	.	.	PUNCT
ejde-508	364	1	suppose	suppose	VERB
ejde-508	364	2	(	(	PUNCT
ejde-508	364	3	a2	a2	PROPN
ejde-508	364	4	)	)	PUNCT
ejde-508	364	5	,	,	PUNCT
ejde-508	364	6	(	(	PUNCT
ejde-508	364	7	a5	a5	NOUN
ejde-508	364	8	)	)	PUNCT
ejde-508	364	9	hold	hold	NOUN
ejde-508	364	10	,	,	PUNCT
ejde-508	364	11	l(x	l(x	PROPN
ejde-508	364	12	)	)	PUNCT
ejde-508	364	13	=	=	SYM
ejde-508	365	1	x	x	NOUN
ejde-508	365	2	,	,	PUNCT
ejde-508	365	3	and	and	CCONJ
ejde-508	365	4	pn	pn	PROPN
ejde-508	365	5	satisfies	satisfie	NOUN
ejde-508	365	6	(	(	PUNCT
ejde-508	365	7	1.9	1.9	NUM
ejde-508	365	8	)	)	PUNCT
ejde-508	365	9	.	.	PUNCT
ejde-508	366	1	then	then	ADV
ejde-508	366	2	every	every	DET
ejde-508	366	3	solution	solution	NOUN
ejde-508	366	4	of	of	ADP
ejde-508	366	5	(	(	PUNCT
ejde-508	366	6	1.1	1.1	NUM
ejde-508	366	7	)	)	PUNCT
ejde-508	366	8	oscillates	oscillate	NOUN
ejde-508	366	9	.	.	PUNCT
ejde-508	367	1	proof	proof	NOUN
ejde-508	367	2	.	.	PUNCT
ejde-508	368	1	on	on	ADP
ejde-508	368	2	the	the	DET
ejde-508	368	3	contrary	contrary	ADJ
ejde-508	368	4	assume	assume	VERB
ejde-508	368	5	yn	yn	PRON
ejde-508	368	6	be	be	AUX
ejde-508	368	7	an	an	DET
ejde-508	368	8	eventually	eventually	ADV
ejde-508	368	9	positive	positive	ADJ
ejde-508	368	10	solution	solution	NOUN
ejde-508	368	11	of	of	ADP
ejde-508	368	12	(	(	PUNCT
ejde-508	368	13	1.1	1.1	NUM
ejde-508	368	14	)	)	PUNCT
ejde-508	368	15	.	.	PUNCT
ejde-508	369	1	setting	set	VERB
ejde-508	369	2	zn	zn	NOUN
ejde-508	369	3	as	as	ADP
ejde-508	369	4	in	in	ADP
ejde-508	369	5	(	(	PUNCT
ejde-508	369	6	2.5	2.5	NUM
ejde-508	369	7	)	)	PUNCT
ejde-508	369	8	,	,	PUNCT
ejde-508	369	9	we	we	PRON
ejde-508	369	10	obtain	obtain	VERB
ejde-508	369	11	(	(	PUNCT
ejde-508	369	12	2.25	2.25	NUM
ejde-508	369	13	)	)	PUNCT
ejde-508	369	14	.	.	PUNCT
ejde-508	370	1	note	note	VERB
ejde-508	370	2	that	that	SCONJ
ejde-508	370	3	(	(	PUNCT
ejde-508	370	4	a5	a5	PROPN
ejde-508	370	5	)	)	PUNCT
ejde-508	370	6	implies	imply	VERB
ejde-508	370	7	(	(	PUNCT
ejde-508	370	8	3.8	3.8	NUM
ejde-508	370	9	)	)	PUNCT
ejde-508	370	10	.	.	PUNCT
ejde-508	371	1	then	then	ADV
ejde-508	371	2	applying	apply	VERB
ejde-508	371	3	theorem	theorem	ADJ
ejde-508	371	4	3.8	3.8	NUM
ejde-508	371	5	for	for	ADP
ejde-508	371	6	un	un	PROPN
ejde-508	371	7	=	=	PROPN
ejde-508	371	8	0	0	PROPN
ejde-508	371	9	,	,	PUNCT
ejde-508	371	10	we	we	PRON
ejde-508	371	11	show	show	VERB
ejde-508	371	12	that	that	SCONJ
ejde-508	371	13	yn	yn	PROPN
ejde-508	371	14	is	be	AUX
ejde-508	371	15	bounded	bound	VERB
ejde-508	371	16	,	,	PUNCT
ejde-508	371	17	which	which	PRON
ejde-508	371	18	implies	imply	VERB
ejde-508	371	19	zn	zn	PROPN
ejde-508	371	20	is	be	AUX
ejde-508	371	21	bounded	bound	VERB
ejde-508	371	22	.	.	PUNCT
ejde-508	372	1	as	as	SCONJ
ejde-508	372	2	zn	zn	PROPN
ejde-508	372	3	is	be	AUX
ejde-508	372	4	monotonic	monotonic	ADJ
ejde-508	372	5	,	,	PUNCT
ejde-508	372	6	limn→∞	limn→∞	X
ejde-508	372	7	zn	zn	X
ejde-508	372	8	=	=	SYM
ejde-508	372	9	β	β	PROPN
ejde-508	372	10	∈	∈	PROPN
ejde-508	372	11	r.	r.	PROPN
ejde-508	372	12	summing	sum	VERB
ejde-508	372	13	(	(	PUNCT
ejde-508	372	14	2.25	2.25	NUM
ejde-508	372	15	)	)	PUNCT
ejde-508	372	16	from	from	ADP
ejde-508	372	17	n1	n1	PROPN
ejde-508	372	18	to	to	ADP
ejde-508	372	19	∞	∞	PROPN
ejde-508	372	20	,	,	PUNCT
ejde-508	372	21	we	we	PRON
ejde-508	372	22	obtain	obtain	VERB
ejde-508	372	23	(	(	PUNCT
ejde-508	372	24	2.13	2.13	NUM
ejde-508	372	25	)	)	PUNCT
ejde-508	372	26	,	,	PUNCT
ejde-508	372	27	which	which	PRON
ejde-508	372	28	implies	imply	VERB
ejde-508	372	29	lim	lim	PROPN
ejde-508	372	30	infn→∞	infn→∞	PROPN
ejde-508	373	1	yn	yn	PROPN
ejde-508	373	2	=	=	NOUN
ejde-508	373	3	0	0	PROPN
ejde-508	373	4	.	.	PUNCT
ejde-508	374	1	by	by	ADP
ejde-508	374	2	[	[	X
ejde-508	374	3	9	9	NUM
ejde-508	374	4	,	,	PUNCT
ejde-508	374	5	lemma	lemma	PROPN
ejde-508	374	6	2.1	2.1	NUM
ejde-508	374	7	]	]	PUNCT
ejde-508	374	8	,	,	PUNCT
ejde-508	374	9	we	we	PRON
ejde-508	374	10	have	have	VERB
ejde-508	374	11	limn→∞	limn→∞	PROPN
ejde-508	374	12	zn	zn	X
ejde-508	374	13	=	=	SYM
ejde-508	374	14	0	0	PROPN
ejde-508	374	15	.	.	PUNCT
ejde-508	375	1	as	as	ADP
ejde-508	375	2	a	a	DET
ejde-508	375	3	consequence	consequence	NOUN
ejde-508	375	4	(	(	PUNCT
ejde-508	375	5	2.28	2.28	NUM
ejde-508	375	6	)	)	PUNCT
ejde-508	375	7	holds	hold	NOUN
ejde-508	375	8	,	,	PUNCT
ejde-508	375	9	which	which	PRON
ejde-508	375	10	implies	imply	VERB
ejde-508	375	11	zn	zn	X
ejde-508	375	12	<	<	X
ejde-508	375	13	0	0	NUM
ejde-508	375	14	.	.	PUNCT
ejde-508	376	1	however	however	ADV
ejde-508	376	2	,	,	PUNCT
ejde-508	376	3	by	by	ADP
ejde-508	376	4	(	(	PUNCT
ejde-508	376	5	1.9	1.9	NUM
ejde-508	376	6	)	)	PUNCT
ejde-508	376	7	we	we	PRON
ejde-508	376	8	have	have	VERB
ejde-508	376	9	zn	zn	PROPN
ejde-508	376	10	>	>	X
ejde-508	376	11	0	0	PROPN
ejde-508	376	12	,	,	PUNCT
ejde-508	376	13	a	a	DET
ejde-508	376	14	contradiction	contradiction	NOUN
ejde-508	376	15	.	.	PUNCT
ejde-508	377	1	the	the	DET
ejde-508	377	2	proof	proof	NOUN
ejde-508	377	3	for	for	ADP
ejde-508	377	4	the	the	DET
ejde-508	377	5	case	case	NOUN
ejde-508	377	6	yn	yn	ADP
ejde-508	377	7	<	<	X
ejde-508	377	8	0	0	X
ejde-508	377	9	is	be	AUX
ejde-508	377	10	similar	similar	ADJ
ejde-508	377	11	.	.	PUNCT
ejde-508	378	1	thus	thus	ADV
ejde-508	378	2	proof	proof	NOUN
ejde-508	378	3	is	be	AUX
ejde-508	378	4	complete	complete	ADJ
ejde-508	378	5	.	.	PUNCT
ejde-508	379	1	�	�	PROPN
ejde-508	379	2	next	next	ADV
ejde-508	379	3	,	,	PUNCT
ejde-508	379	4	we	we	PRON
ejde-508	379	5	give	give	VERB
ejde-508	379	6	some	some	DET
ejde-508	379	7	examples	example	NOUN
ejde-508	379	8	to	to	PART
ejde-508	379	9	illustrate	illustrate	VERB
ejde-508	379	10	the	the	DET
ejde-508	379	11	results	result	NOUN
ejde-508	379	12	.	.	PUNCT
ejde-508	380	1	example	example	NOUN
ejde-508	380	2	4.7	4.7	NUM
ejde-508	380	3	.	.	PUNCT
ejde-508	381	1	consider	consider	VERB
ejde-508	381	2	the	the	DET
ejde-508	381	3	neutral	neutral	ADJ
ejde-508	381	4	difference	difference	NOUN
ejde-508	381	5	equation	equation	NOUN
ejde-508	381	6	∆2	∆2	PROPN
ejde-508	381	7	(	(	PUNCT
ejde-508	381	8	yn	yn	PROPN
ejde-508	381	9	−	−	PROPN
ejde-508	381	10	pyn−4	pyn−4	PROPN
ejde-508	381	11	)	)	PUNCT
ejde-508	382	1	+	+	CCONJ
ejde-508	382	2	18	18	NUM
ejde-508	382	3	(	(	PUNCT
ejde-508	382	4	1	1	NUM
ejde-508	382	5	22n	22n	NOUN
ejde-508	382	6	+	+	X
ejde-508	382	7	1−	1−	NUM
ejde-508	382	8	p	p	NOUN
ejde-508	382	9	16	16	NUM
ejde-508	382	10	)	)	PUNCT
ejde-508	382	11	yn−1	yn−1	ADV
ejde-508	382	12	−	−	PROPN
ejde-508	382	13	(	(	PUNCT
ejde-508	382	14	72	72	NUM
ejde-508	382	15	22n	22n	NOUN
ejde-508	382	16	+	+	CCONJ
ejde-508	382	17	9	9	NUM
ejde-508	382	18	2n	2n	NUM
ejde-508	382	19	)	)	PUNCT
ejde-508	382	20	h(yn−3	h(yn−3	PUNCT
ejde-508	382	21	)	)	PUNCT
ejde-508	382	22	=	=	SYM
ejde-508	382	23	0	0	NUM
ejde-508	382	24	(	(	PUNCT
ejde-508	382	25	4.6	4.6	NUM
ejde-508	382	26	)	)	PUNCT
ejde-508	382	27	12	12	NUM
ejde-508	382	28	a.	a.	NOUN
ejde-508	382	29	k.	k.	PROPN
ejde-508	382	30	bhuyan	bhuyan	PROPN
ejde-508	382	31	,	,	PUNCT
ejde-508	382	32	l.	l.	PROPN
ejde-508	382	33	n.	n.	PROPN
ejde-508	382	34	padhy	padhy	PROPN
ejde-508	382	35	,	,	PUNCT
ejde-508	382	36	r.	r.	PROPN
ejde-508	382	37	n.	n.	PROPN
ejde-508	382	38	rath	rath	PROPN
ejde-508	382	39	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	382	40	where	where	SCONJ
ejde-508	382	41	|p|	|p|	PRON
ejde-508	382	42	<	<	X
ejde-508	382	43	16	16	NUM
ejde-508	382	44	.	.	PUNCT
ejde-508	383	1	suppose	suppose	VERB
ejde-508	383	2	p	p	X
ejde-508	383	3	=	=	NOUN
ejde-508	383	4	±2	±2	NOUN
ejde-508	383	5	or	or	CCONJ
ejde-508	383	6	p	p	NOUN
ejde-508	383	7	=	=	NOUN
ejde-508	383	8	±1/2	±1/2	NOUN
ejde-508	383	9	.	.	PUNCT
ejde-508	384	1	here	here	ADV
ejde-508	384	2	,	,	PUNCT
ejde-508	384	3	s	s	VERB
ejde-508	384	4	=	=	SYM
ejde-508	384	5	4	4	NUM
ejde-508	384	6	,	,	PUNCT
ejde-508	384	7	k	k	NOUN
ejde-508	384	8	=	=	SYM
ejde-508	384	9	1	1	NUM
ejde-508	384	10	,	,	PUNCT
ejde-508	384	11	qn	qn	NOUN
ejde-508	384	12	=	=	SYM
ejde-508	384	13	18	18	NUM
ejde-508	384	14	(	(	PUNCT
ejde-508	384	15	1	1	NUM
ejde-508	384	16	22n	22n	NOUN
ejde-508	384	17	+	+	CCONJ
ejde-508	384	18	1	1	NUM
ejde-508	384	19	−	−	NOUN
ejde-508	384	20	p	p	NOUN
ejde-508	384	21	16	16	NUM
ejde-508	384	22	)	)	PUNCT
ejde-508	384	23	,	,	PUNCT
ejde-508	384	24	un	un	PROPN
ejde-508	385	1	=	=	PRON
ejde-508	385	2	(	(	PUNCT
ejde-508	385	3	72	72	NUM
ejde-508	385	4	22n	22n	NOUN
ejde-508	385	5	+	+	CCONJ
ejde-508	385	6	9	9	NUM
ejde-508	385	7	2n	2n	NUM
ejde-508	385	8	)	)	PUNCT
ejde-508	385	9	and	and	CCONJ
ejde-508	385	10	h(u	h(u	PROPN
ejde-508	385	11	)	)	PUNCT
ejde-508	386	1	=	=	PUNCT
ejde-508	386	2	u/(1	u/(1	PROPN
ejde-508	386	3	+	+	CCONJ
ejde-508	386	4	|u|	|u|	NUM
ejde-508	386	5	)	)	PUNCT
ejde-508	386	6	.	.	PUNCT
ejde-508	387	1	clearly	clearly	ADV
ejde-508	387	2	,	,	PUNCT
ejde-508	387	3	the	the	DET
ejde-508	387	4	neutral	neutral	ADJ
ejde-508	387	5	difference	difference	NOUN
ejde-508	387	6	equation	equation	NOUN
ejde-508	387	7	(	(	PUNCT
ejde-508	387	8	4.6	4.6	NUM
ejde-508	387	9	)	)	PUNCT
ejde-508	387	10	satisfies	satisfie	NOUN
ejde-508	387	11	all	all	DET
ejde-508	387	12	the	the	DET
ejde-508	387	13	conditions	condition	NOUN
ejde-508	387	14	of	of	ADP
ejde-508	387	15	theorems	theorem	NOUN
ejde-508	387	16	3.4	3.4	NUM
ejde-508	387	17	,	,	PUNCT
ejde-508	387	18	3.6	3.6	NUM
ejde-508	387	19	,	,	PUNCT
ejde-508	387	20	3.7	3.7	NUM
ejde-508	387	21	and	and	CCONJ
ejde-508	387	22	3.8	3.8	NUM
ejde-508	387	23	.	.	PUNCT
ejde-508	388	1	as	as	ADP
ejde-508	388	2	a	a	DET
ejde-508	388	3	result	result	NOUN
ejde-508	388	4	,	,	PUNCT
ejde-508	388	5	it	it	PRON
ejde-508	388	6	has	have	VERB
ejde-508	388	7	an	an	DET
ejde-508	388	8	unbounded	unbounded	ADJ
ejde-508	388	9	solution	solution	NOUN
ejde-508	388	10	yn	yn	X
ejde-508	388	11	=	=	SYM
ejde-508	388	12	2n(−1)n	2n(−1)n	NUM
ejde-508	388	13	,	,	PUNCT
ejde-508	388	14	which	which	PRON
ejde-508	388	15	is	be	AUX
ejde-508	388	16	oscillatory	oscillatory	ADJ
ejde-508	388	17	.	.	PUNCT
ejde-508	388	18	example	example	NOUN
ejde-508	388	19	4.8	4.8	NUM
ejde-508	388	20	.	.	PUNCT
ejde-508	389	1	consider	consider	VERB
ejde-508	389	2	the	the	DET
ejde-508	389	3	neutral	neutral	ADJ
ejde-508	389	4	difference	difference	NOUN
ejde-508	389	5	equation	equation	NOUN
ejde-508	389	6	∆2	∆2	PROPN
ejde-508	389	7	(	(	PUNCT
ejde-508	389	8	yn	yn	PROPN
ejde-508	389	9	−	−	PROPN
ejde-508	389	10	byn−2	byn−2	PROPN
ejde-508	389	11	)	)	PUNCT
ejde-508	390	1	+	+	CCONJ
ejde-508	390	2	(	(	PUNCT
ejde-508	390	3	9(1−	9(1−	NUM
ejde-508	390	4	b/4)(2n	b/4)(2n	NOUN
ejde-508	390	5	+	+	CCONJ
ejde-508	390	6	128	128	NUM
ejde-508	390	7	)	)	PUNCT
ejde-508	391	1	+	+	CCONJ
ejde-508	391	2	2−2n	2−2n	NUM
ejde-508	391	3	)	)	PUNCT
ejde-508	392	1	g(yn−7)−	g(yn−7)−	PROPN
ejde-508	392	2	unh(yn−7	unh(yn−7	PROPN
ejde-508	392	3	)	)	PUNCT
ejde-508	392	4	=	=	SYM
ejde-508	392	5	0	0	NUM
ejde-508	392	6	(	(	PUNCT
ejde-508	392	7	4.7	4.7	NUM
ejde-508	392	8	)	)	PUNCT
ejde-508	392	9	where	where	SCONJ
ejde-508	392	10	|b|	|b|	PROPN
ejde-508	392	11	<	<	X
ejde-508	392	12	4	4	NUM
ejde-508	392	13	is	be	AUX
ejde-508	392	14	suitably	suitably	ADV
ejde-508	392	15	selected	select	VERB
ejde-508	392	16	constant	constant	ADJ
ejde-508	392	17	.	.	PUNCT
ejde-508	393	1	suppose	suppose	VERB
ejde-508	393	2	b	b	NOUN
ejde-508	393	3	=	=	NOUN
ejde-508	393	4	±1/2	±1/2	PROPN
ejde-508	393	5	.	.	PUNCT
ejde-508	394	1	here	here	ADV
ejde-508	394	2	,	,	PUNCT
ejde-508	394	3	s	s	VERB
ejde-508	394	4	=	=	SYM
ejde-508	394	5	2	2	NUM
ejde-508	394	6	,	,	PUNCT
ejde-508	394	7	k	k	NOUN
ejde-508	394	8	=	=	SYM
ejde-508	394	9	7	7	NUM
ejde-508	394	10	,	,	PUNCT
ejde-508	394	11	un	un	PROPN
ejde-508	394	12	=	=	PROPN
ejde-508	394	13	2	2	NUM
ejde-508	394	14	+	+	NOUN
ejde-508	394	15	2n−7	2n−7	NOUN
ejde-508	394	16	22n(1	22n(1	NOUN
ejde-508	394	17	+	+	NOUN
ejde-508	394	18	2n−7	2n−7	NOUN
ejde-508	394	19	)	)	PUNCT
ejde-508	394	20	,	,	PUNCT
ejde-508	394	21	qn	qn	NOUN
ejde-508	394	22	=	=	X
ejde-508	394	23	(	(	PUNCT
ejde-508	394	24	9(1	9(1	NUM
ejde-508	394	25	−	−	NOUN
ejde-508	394	26	b/4)(2n	b/4)(2n	NOUN
ejde-508	395	1	+	+	CCONJ
ejde-508	395	2	128	128	NUM
ejde-508	395	3	)	)	PUNCT
ejde-508	396	1	+	+	CCONJ
ejde-508	396	2	2−2n	2−2n	NUM
ejde-508	396	3	)	)	PUNCT
ejde-508	396	4	,	,	PUNCT
ejde-508	396	5	g(u	g(u	PROPN
ejde-508	396	6	)	)	PUNCT
ejde-508	396	7	=	=	SYM
ejde-508	396	8	u/(1	u/(1	PROPN
ejde-508	396	9	+	+	CCONJ
ejde-508	396	10	|u|	|u|	NUM
ejde-508	396	11	)	)	PUNCT
ejde-508	396	12	and	and	CCONJ
ejde-508	396	13	h(u	h(u	PROPN
ejde-508	396	14	)	)	PUNCT
ejde-508	397	1	=	=	SYM
ejde-508	397	2	u/(2	u/(2	PROPN
ejde-508	397	3	+	+	NUM
ejde-508	397	4	|u|	|u|	NUM
ejde-508	397	5	)	)	PUNCT
ejde-508	397	6	.	.	PUNCT
ejde-508	398	1	clearly	clearly	ADV
ejde-508	398	2	,	,	PUNCT
ejde-508	398	3	the	the	DET
ejde-508	398	4	neutral	neutral	ADJ
ejde-508	398	5	difference	difference	NOUN
ejde-508	398	6	equation	equation	NOUN
ejde-508	398	7	(	(	PUNCT
ejde-508	398	8	4.7	4.7	NUM
ejde-508	398	9	)	)	PUNCT
ejde-508	398	10	satisfies	satisfie	NOUN
ejde-508	398	11	all	all	DET
ejde-508	398	12	the	the	DET
ejde-508	398	13	conditions	condition	NOUN
ejde-508	398	14	of	of	ADP
ejde-508	398	15	theorems	theorem	NOUN
ejde-508	398	16	3.6	3.6	NUM
ejde-508	398	17	and	and	CCONJ
ejde-508	398	18	3.8	3.8	NUM
ejde-508	398	19	.	.	PUNCT
ejde-508	399	1	consequently	consequently	ADV
ejde-508	399	2	,	,	PUNCT
ejde-508	399	3	it	it	PRON
ejde-508	399	4	has	have	VERB
ejde-508	399	5	an	an	DET
ejde-508	399	6	unbounded	unbounded	ADJ
ejde-508	399	7	solution	solution	NOUN
ejde-508	399	8	yn	yn	X
ejde-508	399	9	=	=	SYM
ejde-508	399	10	2n(−1)n	2n(−1)n	NUM
ejde-508	399	11	,	,	PUNCT
ejde-508	399	12	which	which	PRON
ejde-508	399	13	is	be	AUX
ejde-508	399	14	oscillatory	oscillatory	ADJ
ejde-508	399	15	.	.	PUNCT
ejde-508	399	16	example	example	NOUN
ejde-508	399	17	4.9	4.9	NUM
ejde-508	399	18	.	.	PUNCT
ejde-508	400	1	consider	consider	VERB
ejde-508	400	2	the	the	DET
ejde-508	400	3	neutral	neutral	ADJ
ejde-508	400	4	difference	difference	NOUN
ejde-508	400	5	equation	equation	NOUN
ejde-508	400	6	∆2	∆2	PROPN
ejde-508	400	7	(	(	PUNCT
ejde-508	400	8	yn	yn	PROPN
ejde-508	400	9	−	−	PROPN
ejde-508	400	10	pl(yn−3	pl(yn−3	NUM
ejde-508	400	11	)	)	PUNCT
ejde-508	400	12	)	)	PUNCT
ejde-508	401	1	+	+	PUNCT
ejde-508	401	2	[	[	PUNCT
ejde-508	401	3	4(1	4(1	NUM
ejde-508	401	4	+	+	CCONJ
ejde-508	401	5	a+	a+	PUNCT
ejde-508	401	6	p	p	X
ejde-508	401	7	)	)	PUNCT
ejde-508	401	8	(	(	PUNCT
ejde-508	401	9	1	1	NUM
ejde-508	401	10	+	+	CCONJ
ejde-508	401	11	a)(1	a)(1	X
ejde-508	401	12	+	+	CCONJ
ejde-508	401	13	γ	γ	X
ejde-508	401	14	)	)	PUNCT
ejde-508	401	15	]	]	PUNCT
ejde-508	401	16	g(yn−5	g(yn−5	X
ejde-508	401	17	)	)	PUNCT
ejde-508	401	18	=	=	SYM
ejde-508	401	19	0	0	NUM
ejde-508	401	20	(	(	PUNCT
ejde-508	401	21	4.8	4.8	NUM
ejde-508	401	22	)	)	PUNCT
ejde-508	401	23	where	where	SCONJ
ejde-508	401	24	p	p	NOUN
ejde-508	401	25	>	>	X
ejde-508	401	26	0	0	NUM
ejde-508	401	27	is	be	AUX
ejde-508	401	28	any	any	DET
ejde-508	401	29	scalar	scalar	NOUN
ejde-508	401	30	.	.	PUNCT
ejde-508	402	1	here	here	ADV
ejde-508	402	2	l(x	l(x	PROPN
ejde-508	402	3	)	)	PUNCT
ejde-508	402	4	=	=	PUNCT
ejde-508	403	1	x/(a	x/(a	PROPN
ejde-508	403	2	+	+	CCONJ
ejde-508	403	3	|x|	|x|	PROPN
ejde-508	403	4	)	)	PUNCT
ejde-508	403	5	and	and	CCONJ
ejde-508	403	6	g(u	g(u	PROPN
ejde-508	403	7	)	)	PUNCT
ejde-508	404	1	=	=	SYM
ejde-508	404	2	u(γ	u(γ	PROPN
ejde-508	404	3	+	+	CCONJ
ejde-508	404	4	|u|	|u|	X
ejde-508	404	5	)	)	PUNCT
ejde-508	404	6	where	where	SCONJ
ejde-508	404	7	a	a	PRON
ejde-508	404	8	and	and	CCONJ
ejde-508	404	9	γ	γ	NOUN
ejde-508	404	10	are	be	AUX
ejde-508	404	11	positive	positive	ADJ
ejde-508	404	12	constants	constant	NOUN
ejde-508	404	13	.	.	PUNCT
ejde-508	405	1	this	this	DET
ejde-508	405	2	neutral	neutral	ADJ
ejde-508	405	3	equation	equation	NOUN
ejde-508	405	4	satisfies	satisfy	VERB
ejde-508	405	5	all	all	DET
ejde-508	405	6	the	the	DET
ejde-508	405	7	conditions	condition	NOUN
ejde-508	405	8	of	of	ADP
ejde-508	405	9	theorems	theorem	NOUN
ejde-508	405	10	4.3	4.3	NUM
ejde-508	405	11	.	.	PUNCT
ejde-508	406	1	as	as	ADP
ejde-508	406	2	such	such	ADJ
ejde-508	406	3	,	,	PUNCT
ejde-508	406	4	it	it	PRON
ejde-508	406	5	has	have	VERB
ejde-508	406	6	a	a	DET
ejde-508	406	7	solution	solution	NOUN
ejde-508	406	8	yn	yn	X
ejde-508	406	9	=	=	SYM
ejde-508	406	10	(	(	PUNCT
ejde-508	406	11	−1)n	−1)n	PROPN
ejde-508	406	12	,	,	PUNCT
ejde-508	406	13	which	which	PRON
ejde-508	406	14	is	be	AUX
ejde-508	406	15	oscillatory	oscillatory	ADJ
ejde-508	406	16	.	.	PUNCT
ejde-508	406	17	example	example	NOUN
ejde-508	407	1	4.10	4.10	NUM
ejde-508	407	2	.	.	PUNCT
ejde-508	408	1	consider	consider	VERB
ejde-508	408	2	the	the	DET
ejde-508	408	3	neutral	neutral	ADJ
ejde-508	408	4	difference	difference	NOUN
ejde-508	408	5	equation	equation	NOUN
ejde-508	408	6	∆2	∆2	PROPN
ejde-508	408	7	(	(	PUNCT
ejde-508	408	8	yn	yn	PROPN
ejde-508	408	9	+	+	CCONJ
ejde-508	408	10	p(−1)nl(yn−5	p(−1)nl(yn−5	PROPN
ejde-508	408	11	)	)	PUNCT
ejde-508	408	12	)	)	PUNCT
ejde-508	409	1	+	+	CCONJ
ejde-508	409	2	4y	4y	PROPN
ejde-508	409	3	1/3	1/3	NUM
ejde-508	409	4	n−1	n−1	PROPN
ejde-508	409	5	=	=	SYM
ejde-508	409	6	0	0	NUM
ejde-508	409	7	(	(	PUNCT
ejde-508	409	8	4.9	4.9	NUM
ejde-508	409	9	)	)	PUNCT
ejde-508	409	10	where	where	SCONJ
ejde-508	409	11	p	p	NOUN
ejde-508	409	12	>	>	X
ejde-508	409	13	0	0	NUM
ejde-508	409	14	is	be	AUX
ejde-508	409	15	any	any	DET
ejde-508	409	16	scalar	scalar	NOUN
ejde-508	409	17	.	.	PUNCT
ejde-508	410	1	here	here	ADV
ejde-508	410	2	pn	pn	PROPN
ejde-508	410	3	changes	change	NOUN
ejde-508	410	4	sign	sign	VERB
ejde-508	410	5	and	and	CCONJ
ejde-508	410	6	satisfies	satisfie	NOUN
ejde-508	410	7	(	(	PUNCT
ejde-508	410	8	1.7	1.7	NUM
ejde-508	410	9	)	)	PUNCT
ejde-508	410	10	.	.	PUNCT
ejde-508	411	1	further	far	ADV
ejde-508	411	2	,	,	PUNCT
ejde-508	411	3	l(x	l(x	PROPN
ejde-508	411	4	)	)	PUNCT
ejde-508	411	5	=	=	PUNCT
ejde-508	412	1	x/(a	x/(a	PROPN
ejde-508	412	2	+	+	CCONJ
ejde-508	412	3	|x|	|x|	PROPN
ejde-508	412	4	)	)	PUNCT
ejde-508	412	5	.	.	PUNCT
ejde-508	413	1	this	this	DET
ejde-508	413	2	neutral	neutral	ADJ
ejde-508	413	3	equation	equation	NOUN
ejde-508	413	4	satisfies	satisfy	VERB
ejde-508	413	5	all	all	DET
ejde-508	413	6	the	the	DET
ejde-508	413	7	conditions	condition	NOUN
ejde-508	413	8	of	of	ADP
ejde-508	413	9	theorem	theorem	NOUN
ejde-508	413	10	4.5	4.5	NUM
ejde-508	413	11	.	.	PUNCT
ejde-508	414	1	hence	hence	ADV
ejde-508	414	2	,	,	PUNCT
ejde-508	414	3	it	it	PRON
ejde-508	414	4	has	have	VERB
ejde-508	414	5	a	a	DET
ejde-508	414	6	solution	solution	NOUN
ejde-508	414	7	yn	yn	X
ejde-508	414	8	=	=	SYM
ejde-508	414	9	(	(	PUNCT
ejde-508	414	10	−1)3n	−1)3n	PROPN
ejde-508	414	11	,	,	PUNCT
ejde-508	414	12	which	which	PRON
ejde-508	414	13	is	be	AUX
ejde-508	414	14	oscillatory	oscillatory	ADJ
ejde-508	414	15	.	.	PUNCT
ejde-508	415	1	it	it	PRON
ejde-508	415	2	seems	seem	VERB
ejde-508	415	3	,	,	PUNCT
ejde-508	415	4	no	no	DET
ejde-508	415	5	result	result	NOUN
ejde-508	415	6	in	in	ADP
ejde-508	415	7	the	the	DET
ejde-508	415	8	literature	literature	NOUN
ejde-508	415	9	,	,	PUNCT
ejde-508	415	10	could	could	AUX
ejde-508	415	11	be	be	AUX
ejde-508	415	12	applied	apply	VERB
ejde-508	415	13	to	to	ADP
ejde-508	415	14	the	the	DET
ejde-508	415	15	neutral	neutral	ADJ
ejde-508	415	16	equations	equation	NOUN
ejde-508	415	17	(	(	PUNCT
ejde-508	415	18	4.8)–(4.9	4.8)–(4.9	NOUN
ejde-508	415	19	)	)	PUNCT
ejde-508	415	20	given	give	VERB
ejde-508	415	21	in	in	ADP
ejde-508	415	22	the	the	DET
ejde-508	415	23	examples	example	NOUN
ejde-508	415	24	above	above	ADV
ejde-508	415	25	,	,	PUNCT
ejde-508	415	26	because	because	SCONJ
ejde-508	415	27	of	of	ADP
ejde-508	415	28	the	the	DET
ejde-508	415	29	non	non	ADJ
ejde-508	415	30	linear	linear	ADJ
ejde-508	415	31	term	term	NOUN
ejde-508	415	32	inside	inside	ADP
ejde-508	415	33	∆2	∆2	PROPN
ejde-508	415	34	,	,	PUNCT
ejde-508	415	35	5	5	NUM
ejde-508	415	36	.	.	X
ejde-508	415	37	application	application	NOUN
ejde-508	415	38	to	to	ADP
ejde-508	415	39	neutral	neutral	ADJ
ejde-508	415	40	difference	difference	NOUN
ejde-508	415	41	equations	equation	NOUN
ejde-508	415	42	with	with	ADP
ejde-508	415	43	oscillating	oscillate	VERB
ejde-508	415	44	coefficients	coefficient	NOUN
ejde-508	415	45	in	in	ADP
ejde-508	415	46	this	this	DET
ejde-508	415	47	section	section	NOUN
ejde-508	415	48	,	,	PUNCT
ejde-508	415	49	we	we	PRON
ejde-508	415	50	find	find	VERB
ejde-508	415	51	sufficient	sufficient	ADJ
ejde-508	415	52	conditions	condition	NOUN
ejde-508	415	53	so	so	SCONJ
ejde-508	415	54	that	that	SCONJ
ejde-508	415	55	every	every	DET
ejde-508	415	56	unbounded	unbounded	ADJ
ejde-508	415	57	solution	solution	NOUN
ejde-508	415	58	of	of	ADP
ejde-508	415	59	the	the	DET
ejde-508	415	60	second	second	ADJ
ejde-508	415	61	order	order	NOUN
ejde-508	415	62	neutral	neutral	ADJ
ejde-508	415	63	difference	difference	NOUN
ejde-508	415	64	equation	equation	NOUN
ejde-508	415	65	(	(	PUNCT
ejde-508	415	66	1.11	1.11	NUM
ejde-508	415	67	)	)	PUNCT
ejde-508	415	68	oscillates	oscillate	NOUN
ejde-508	415	69	,	,	PUNCT
ejde-508	415	70	where	where	SCONJ
ejde-508	415	71	vn	vn	PROPN
ejde-508	415	72	is	be	AUX
ejde-508	415	73	allowed	allow	VERB
ejde-508	415	74	to	to	PART
ejde-508	415	75	change	change	VERB
ejde-508	415	76	sign	sign	NOUN
ejde-508	415	77	.	.	PUNCT
ejde-508	416	1	let	let	VERB
ejde-508	416	2	v+n	v+n	NUM
ejde-508	416	3	=	=	SYM
ejde-508	416	4	max{vn	max{vn	X
ejde-508	416	5	,	,	PUNCT
ejde-508	416	6	0	0	NUM
ejde-508	416	7	}	}	PUNCT
ejde-508	416	8	and	and	CCONJ
ejde-508	416	9	v−n	v−n	NOUN
ejde-508	416	10	=	=	SYM
ejde-508	416	11	max{−vn	max{−vn	PROPN
ejde-508	416	12	,	,	PUNCT
ejde-508	416	13	0	0	NUM
ejde-508	416	14	}	}	PUNCT
ejde-508	416	15	.	.	PUNCT
ejde-508	417	1	then	then	ADV
ejde-508	417	2	vn	vn	PROPN
ejde-508	417	3	=	=	SYM
ejde-508	417	4	v+n	v+n	PROPN
ejde-508	417	5	−	−	ADP
ejde-508	417	6	v−n	v−n	NOUN
ejde-508	417	7	and	and	CCONJ
ejde-508	417	8	the	the	DET
ejde-508	417	9	equation	equation	NOUN
ejde-508	417	10	(	(	PUNCT
ejde-508	417	11	1.11	1.11	NUM
ejde-508	417	12	)	)	PUNCT
ejde-508	417	13	can	can	AUX
ejde-508	417	14	be	be	AUX
ejde-508	417	15	written	write	VERB
ejde-508	417	16	as	as	ADP
ejde-508	417	17	∆2	∆2	X
ejde-508	417	18	[	[	PUNCT
ejde-508	417	19	yn	yn	PROPN
ejde-508	417	20	−	−	PROPN
ejde-508	417	21	pnl(yn−s	pnl(yn−s	PROPN
ejde-508	417	22	)	)	PUNCT
ejde-508	417	23	]	]	PUNCT
ejde-508	418	1	+	+	CCONJ
ejde-508	418	2	v+ng(yn−k)−	v+ng(yn−k)−	NOUN
ejde-508	418	3	v−ng(yn−k	v−ng(yn−k	NOUN
ejde-508	418	4	)	)	PUNCT
ejde-508	419	1	=	=	SYM
ejde-508	419	2	0	0	X
ejde-508	419	3	.	.	PUNCT
ejde-508	420	1	(	(	PUNCT
ejde-508	420	2	5.1	5.1	NUM
ejde-508	420	3	)	)	PUNCT
ejde-508	420	4	now	now	ADV
ejde-508	420	5	we	we	PRON
ejde-508	420	6	proceed	proceed	VERB
ejde-508	420	7	as	as	ADP
ejde-508	420	8	in	in	ADP
ejde-508	420	9	the	the	DET
ejde-508	420	10	previous	previous	ADJ
ejde-508	420	11	section	section	NOUN
ejde-508	420	12	by	by	ADP
ejde-508	420	13	setting	set	VERB
ejde-508	420	14	qn	qn	NOUN
ejde-508	420	15	=	=	PROPN
ejde-508	420	16	v+n	v+n	NUM
ejde-508	420	17	,	,	PUNCT
ejde-508	420	18	un	un	PROPN
ejde-508	420	19	=	=	PROPN
ejde-508	420	20	v−n	v−n	NOUN
ejde-508	420	21	and	and	CCONJ
ejde-508	420	22	h(x	h(x	PROPN
ejde-508	420	23	)	)	PUNCT
ejde-508	421	1	=	=	PUNCT
ejde-508	421	2	g(x	g(x	NOUN
ejde-508	421	3	)	)	PUNCT
ejde-508	421	4	.	.	PUNCT
ejde-508	422	1	assumptions	assumption	NOUN
ejde-508	422	2	(	(	PUNCT
ejde-508	422	3	a4	a4	NOUN
ejde-508	422	4	)	)	PUNCT
ejde-508	422	5	,	,	PUNCT
ejde-508	422	6	(	(	PUNCT
ejde-508	422	7	a3	a3	NOUN
ejde-508	422	8	)	)	PUNCT
ejde-508	422	9	and	and	CCONJ
ejde-508	422	10	(	(	PUNCT
ejde-508	422	11	a6	a6	NOUN
ejde-508	422	12	)	)	PUNCT
ejde-508	422	13	become	become	VERB
ejde-508	423	1	∞∑	∞∑	NUM
ejde-508	423	2	n	n	CCONJ
ejde-508	423	3	=	=	SYM
ejde-508	423	4	n0	n0	NUM
ejde-508	423	5	v+n	v+n	NUM
ejde-508	423	6	=	=	SYM
ejde-508	423	7	∞.	∞.	PROPN
ejde-508	423	8	(	(	PUNCT
ejde-508	423	9	5.2	5.2	NUM
ejde-508	423	10	)	)	PUNCT
ejde-508	423	11	lim	lim	PROPN
ejde-508	423	12	inf	inf	PROPN
ejde-508	423	13	n→∞	n→∞	X
ejde-508	423	14	v+n	v+n	NUM
ejde-508	423	15	>	>	X
ejde-508	423	16	0	0	X
ejde-508	423	17	.	.	PUNCT
ejde-508	423	18	(	(	PUNCT
ejde-508	423	19	5.3	5.3	NUM
ejde-508	423	20	)	)	PUNCT
ejde-508	423	21	∞∑	∞∑	NUM
ejde-508	423	22	n	n	CCONJ
ejde-508	423	23	=	=	SYM
ejde-508	423	24	n0	n0	NUM
ejde-508	423	25	v	v	NOUN
ejde-508	423	26	+	+	CCONJ
ejde-508	423	27	n	n	CCONJ
ejde-508	423	28	=	=	NOUN
ejde-508	423	29	∞	∞	NOUN
ejde-508	423	30	where	where	SCONJ
ejde-508	423	31	v	v	NOUN
ejde-508	423	32	+	+	CCONJ
ejde-508	423	33	n	n	CCONJ
ejde-508	423	34	=	=	SYM
ejde-508	423	35	min{v+n	min{v+n	PROPN
ejde-508	423	36	,	,	PUNCT
ejde-508	423	37	v+n−s	v+n−s	PROPN
ejde-508	423	38	}	}	PUNCT
ejde-508	423	39	.	.	PUNCT
ejde-508	424	1	(	(	PUNCT
ejde-508	424	2	5.4	5.4	NUM
ejde-508	424	3	)	)	PUNCT
ejde-508	424	4	ejde-2020/87	ejde-2020/87	VERB
ejde-508	424	5	oscillation	oscillation	NOUN
ejde-508	424	6	for	for	ADP
ejde-508	424	7	second	second	ADJ
ejde-508	424	8	order	order	NOUN
ejde-508	424	9	neutral	neutral	ADJ
ejde-508	424	10	equations	equation	NOUN
ejde-508	424	11	13	13	NUM
ejde-508	424	12	∞∑	∞∑	NUM
ejde-508	424	13	n	n	CCONJ
ejde-508	424	14	=	=	SYM
ejde-508	424	15	n0	n0	ADJ
ejde-508	424	16	nv−n	nv−n	PROPN
ejde-508	424	17	<	<	X
ejde-508	424	18	∞.	∞.	PROPN
ejde-508	424	19	(	(	PUNCT
ejde-508	424	20	5.5	5.5	NUM
ejde-508	424	21	)	)	PUNCT
ejde-508	424	22	respectively	respectively	ADV
ejde-508	424	23	,	,	PUNCT
ejde-508	424	24	which	which	PRON
ejde-508	424	25	are	be	AUX
ejde-508	424	26	feasible	feasible	ADJ
ejde-508	424	27	conditions	condition	NOUN
ejde-508	424	28	.	.	PUNCT
ejde-508	425	1	therefore	therefore	ADV
ejde-508	425	2	,	,	PUNCT
ejde-508	425	3	the	the	DET
ejde-508	425	4	study	study	NOUN
ejde-508	425	5	of	of	ADP
ejde-508	425	6	(	(	PUNCT
ejde-508	425	7	1.11	1.11	NUM
ejde-508	425	8	)	)	PUNCT
ejde-508	425	9	reduces	reduce	VERB
ejde-508	425	10	to	to	ADP
ejde-508	425	11	the	the	DET
ejde-508	425	12	study	study	NOUN
ejde-508	425	13	of	of	ADP
ejde-508	425	14	(	(	PUNCT
ejde-508	425	15	5.1	5.1	NUM
ejde-508	425	16	)	)	PUNCT
ejde-508	425	17	,	,	PUNCT
ejde-508	425	18	which	which	PRON
ejde-508	425	19	could	could	AUX
ejde-508	425	20	be	be	AUX
ejde-508	425	21	achieved	achieve	VERB
ejde-508	425	22	,	,	PUNCT
ejde-508	425	23	by	by	ADP
ejde-508	425	24	following	follow	VERB
ejde-508	425	25	the	the	DET
ejde-508	425	26	study	study	NOUN
ejde-508	425	27	of	of	ADP
ejde-508	425	28	(	(	PUNCT
ejde-508	425	29	1.2	1.2	NUM
ejde-508	425	30	)	)	PUNCT
ejde-508	425	31	for	for	ADP
ejde-508	425	32	different	different	ADJ
ejde-508	425	33	results	result	NOUN
ejde-508	425	34	in	in	ADP
ejde-508	425	35	section	section	NOUN
ejde-508	425	36	3	3	NUM
ejde-508	425	37	.	.	PUNCT
ejde-508	426	1	the	the	DET
ejde-508	426	2	following	follow	VERB
ejde-508	426	3	results	result	NOUN
ejde-508	426	4	for	for	ADP
ejde-508	426	5	(	(	PUNCT
ejde-508	426	6	1.11	1.11	NUM
ejde-508	426	7	)	)	PUNCT
ejde-508	426	8	(	(	PUNCT
ejde-508	426	9	with	with	ADP
ejde-508	426	10	vn	vn	NUM
ejde-508	426	11	changing	change	VERB
ejde-508	426	12	sign	sign	NOUN
ejde-508	426	13	)	)	PUNCT
ejde-508	426	14	follow	follow	VERB
ejde-508	426	15	from	from	ADP
ejde-508	426	16	theorems	theorem	NOUN
ejde-508	426	17	3.6	3.6	NUM
ejde-508	426	18	,	,	PUNCT
ejde-508	426	19	3.7	3.7	NUM
ejde-508	426	20	and	and	CCONJ
ejde-508	426	21	3.8	3.8	NUM
ejde-508	426	22	,	,	PUNCT
ejde-508	426	23	by	by	ADP
ejde-508	426	24	replacing	replace	VERB
ejde-508	426	25	qn	qn	NOUN
ejde-508	426	26	by	by	ADP
ejde-508	426	27	v+n	v+n	NUM
ejde-508	426	28	,	,	PUNCT
ejde-508	426	29	un	un	PROPN
ejde-508	426	30	by	by	ADP
ejde-508	426	31	v−n	v−n	NOUN
ejde-508	426	32	and	and	CCONJ
ejde-508	426	33	h	h	NOUN
ejde-508	426	34	by	by	ADP
ejde-508	426	35	g.	g.	PROPN
ejde-508	426	36	theorem	theorem	VERB
ejde-508	426	37	5.1	5.1	NUM
ejde-508	426	38	.	.	PUNCT
ejde-508	427	1	suppose	suppose	VERB
ejde-508	427	2	that	that	SCONJ
ejde-508	427	3	(	(	PUNCT
ejde-508	427	4	a1	a1	NOUN
ejde-508	427	5	)	)	PUNCT
ejde-508	427	6	holds	hold	VERB
ejde-508	427	7	with	with	ADP
ejde-508	427	8	δ	δ	PROPN
ejde-508	427	9	≤	≤	ADV
ejde-508	427	10	1	1	NUM
ejde-508	427	11	,	,	PUNCT
ejde-508	427	12	(	(	PUNCT
ejde-508	427	13	a5	a5	NOUN
ejde-508	427	14	)	)	PUNCT
ejde-508	427	15	holds	hold	VERB
ejde-508	427	16	,	,	PUNCT
ejde-508	427	17	pn	pn	PROPN
ejde-508	427	18	satisfies	satisfie	NOUN
ejde-508	427	19	(	(	PUNCT
ejde-508	427	20	1.5	1.5	NUM
ejde-508	427	21	)	)	PUNCT
ejde-508	427	22	,	,	PUNCT
ejde-508	427	23	g	g	PROPN
ejde-508	427	24	is	be	AUX
ejde-508	427	25	bounded	bound	VERB
ejde-508	427	26	,	,	PUNCT
ejde-508	427	27	and	and	CCONJ
ejde-508	427	28	(	(	PUNCT
ejde-508	427	29	5.3	5.3	NUM
ejde-508	427	30	)	)	PUNCT
ejde-508	427	31	and	and	CCONJ
ejde-508	427	32	(	(	PUNCT
ejde-508	427	33	5.5	5.5	NUM
ejde-508	427	34	)	)	PUNCT
ejde-508	427	35	are	be	AUX
ejde-508	427	36	satisfied	satisfied	ADJ
ejde-508	427	37	.	.	PUNCT
ejde-508	428	1	then	then	ADV
ejde-508	428	2	every	every	DET
ejde-508	428	3	unbounded	unbounded	ADJ
ejde-508	428	4	solution	solution	NOUN
ejde-508	428	5	of	of	ADP
ejde-508	428	6	(	(	PUNCT
ejde-508	428	7	1.11	1.11	NUM
ejde-508	428	8	)	)	PUNCT
ejde-508	428	9	(	(	PUNCT
ejde-508	428	10	with	with	ADP
ejde-508	428	11	vn	vn	NUM
ejde-508	428	12	changing	change	VERB
ejde-508	428	13	sign	sign	NOUN
ejde-508	428	14	)	)	PUNCT
ejde-508	428	15	oscillates	oscillate	NOUN
ejde-508	428	16	.	.	PUNCT
ejde-508	429	1	theorem	theorem	VERB
ejde-508	429	2	5.2	5.2	NUM
ejde-508	429	3	.	.	PUNCT
ejde-508	430	1	suppose	suppose	VERB
ejde-508	430	2	pn	pn	PROPN
ejde-508	430	3	satisfies	satisfie	NOUN
ejde-508	430	4	(	(	PUNCT
ejde-508	430	5	1.6	1.6	NUM
ejde-508	430	6	)	)	PUNCT
ejde-508	430	7	or	or	CCONJ
ejde-508	430	8	(	(	PUNCT
ejde-508	430	9	1.7	1.7	NUM
ejde-508	430	10	)	)	PUNCT
ejde-508	430	11	,	,	PUNCT
ejde-508	430	12	g	g	PROPN
ejde-508	430	13	is	be	AUX
ejde-508	430	14	bounded	bound	VERB
ejde-508	430	15	,	,	PUNCT
ejde-508	430	16	(	(	PUNCT
ejde-508	430	17	a1	a1	NOUN
ejde-508	430	18	)	)	PUNCT
ejde-508	430	19	holds	hold	NOUN
ejde-508	430	20	,	,	PUNCT
ejde-508	430	21	and	and	CCONJ
ejde-508	430	22	(	(	PUNCT
ejde-508	430	23	2.21	2.21	NUM
ejde-508	430	24	)	)	PUNCT
ejde-508	430	25	,	,	PUNCT
ejde-508	430	26	(	(	PUNCT
ejde-508	430	27	2.22	2.22	NUM
ejde-508	430	28	)	)	PUNCT
ejde-508	430	29	,	,	PUNCT
ejde-508	430	30	(	(	PUNCT
ejde-508	430	31	5.3	5.3	NUM
ejde-508	430	32	)	)	PUNCT
ejde-508	430	33	,	,	PUNCT
ejde-508	430	34	(	(	PUNCT
ejde-508	430	35	5.4	5.4	NUM
ejde-508	430	36	)	)	PUNCT
ejde-508	430	37	,	,	PUNCT
ejde-508	430	38	and	and	CCONJ
ejde-508	430	39	(	(	PUNCT
ejde-508	430	40	5.5	5.5	NUM
ejde-508	430	41	)	)	PUNCT
ejde-508	430	42	are	be	AUX
ejde-508	430	43	satisfied	satisfied	ADJ
ejde-508	430	44	.	.	PUNCT
ejde-508	431	1	then	then	ADV
ejde-508	431	2	every	every	DET
ejde-508	431	3	unbounded	unbounded	ADJ
ejde-508	431	4	solution	solution	NOUN
ejde-508	431	5	of	of	ADP
ejde-508	431	6	(	(	PUNCT
ejde-508	431	7	1.11	1.11	NUM
ejde-508	431	8	)	)	PUNCT
ejde-508	431	9	(	(	PUNCT
ejde-508	431	10	with	with	ADP
ejde-508	431	11	vn	vn	NUM
ejde-508	431	12	changing	change	VERB
ejde-508	431	13	sign	sign	NOUN
ejde-508	431	14	)	)	PUNCT
ejde-508	431	15	oscillates	oscillate	NOUN
ejde-508	431	16	.	.	PUNCT
ejde-508	432	1	theorem	theorem	VERB
ejde-508	432	2	5.3	5.3	NUM
ejde-508	432	3	.	.	PUNCT
ejde-508	433	1	suppose	suppose	VERB
ejde-508	433	2	pn	pn	PROPN
ejde-508	433	3	satisfy	satisfy	VERB
ejde-508	433	4	(	(	PUNCT
ejde-508	433	5	1.9	1.9	NUM
ejde-508	433	6	)	)	PUNCT
ejde-508	433	7	,	,	PUNCT
ejde-508	433	8	g	g	PROPN
ejde-508	433	9	is	be	AUX
ejde-508	433	10	bounded	bound	VERB
ejde-508	433	11	,	,	PUNCT
ejde-508	433	12	(	(	PUNCT
ejde-508	433	13	3.8	3.8	NUM
ejde-508	433	14	)	)	PUNCT
ejde-508	433	15	,	,	PUNCT
ejde-508	433	16	(	(	PUNCT
ejde-508	433	17	5.2	5.2	NUM
ejde-508	433	18	)	)	PUNCT
ejde-508	433	19	and	and	CCONJ
ejde-508	433	20	(	(	PUNCT
ejde-508	433	21	5.5	5.5	NUM
ejde-508	433	22	)	)	PUNCT
ejde-508	433	23	are	be	AUX
ejde-508	433	24	satisfied	satisfied	ADJ
ejde-508	433	25	.	.	PUNCT
ejde-508	434	1	if	if	SCONJ
ejde-508	434	2	l(x	l(x	PROPN
ejde-508	434	3	)	)	PUNCT
ejde-508	434	4	=	=	SYM
ejde-508	435	1	x	x	NOUN
ejde-508	435	2	,	,	PUNCT
ejde-508	435	3	then	then	ADV
ejde-508	435	4	every	every	DET
ejde-508	435	5	unbounded	unbounded	ADJ
ejde-508	435	6	solution	solution	NOUN
ejde-508	435	7	of	of	ADP
ejde-508	435	8	(	(	PUNCT
ejde-508	435	9	1.11	1.11	NUM
ejde-508	435	10	)	)	PUNCT
ejde-508	435	11	oscillates	oscillate	NOUN
ejde-508	435	12	.	.	PUNCT
ejde-508	436	1	6	6	X
ejde-508	436	2	.	.	X
ejde-508	436	3	final	final	ADJ
ejde-508	436	4	comments	comment	NOUN
ejde-508	436	5	before	before	SCONJ
ejde-508	436	6	we	we	PRON
ejde-508	436	7	close	close	VERB
ejde-508	436	8	this	this	DET
ejde-508	436	9	article	article	NOUN
ejde-508	436	10	,	,	PUNCT
ejde-508	436	11	we	we	PRON
ejde-508	436	12	would	would	AUX
ejde-508	436	13	like	like	VERB
ejde-508	436	14	to	to	PART
ejde-508	436	15	give	give	VERB
ejde-508	436	16	our	our	PRON
ejde-508	436	17	concluding	concluding	NOUN
ejde-508	436	18	remarks	remark	NOUN
ejde-508	436	19	,	,	PUNCT
ejde-508	436	20	which	which	PRON
ejde-508	436	21	may	may	AUX
ejde-508	436	22	be	be	AUX
ejde-508	436	23	helpful	helpful	ADJ
ejde-508	436	24	for	for	ADP
ejde-508	436	25	further	further	ADJ
ejde-508	436	26	research	research	NOUN
ejde-508	436	27	.	.	PUNCT
ejde-508	437	1	in	in	ADP
ejde-508	437	2	this	this	DET
ejde-508	437	3	paper	paper	NOUN
ejde-508	437	4	,	,	PUNCT
ejde-508	437	5	some	some	DET
ejde-508	437	6	oscillatory	oscillatory	ADJ
ejde-508	437	7	results	result	NOUN
ejde-508	437	8	are	be	AUX
ejde-508	437	9	obtained	obtain	VERB
ejde-508	437	10	for	for	ADP
ejde-508	437	11	the	the	DET
ejde-508	437	12	neutral	neutral	ADJ
ejde-508	437	13	difference	difference	NOUN
ejde-508	437	14	equation	equation	NOUN
ejde-508	437	15	(	(	PUNCT
ejde-508	437	16	1.2	1.2	NUM
ejde-508	437	17	)	)	PUNCT
ejde-508	437	18	and	and	CCONJ
ejde-508	437	19	(	(	PUNCT
ejde-508	437	20	1.1	1.1	NUM
ejde-508	437	21	)	)	PUNCT
ejde-508	437	22	by	by	ADP
ejde-508	437	23	imposing	impose	VERB
ejde-508	437	24	different	different	ADJ
ejde-508	437	25	super	super	ADJ
ejde-508	437	26	linear	linear	ADJ
ejde-508	437	27	conditions	condition	NOUN
ejde-508	437	28	like	like	VERB
ejde-508	437	29	(	(	PUNCT
ejde-508	437	30	3.5	3.5	NUM
ejde-508	437	31	)	)	PUNCT
ejde-508	437	32	or	or	CCONJ
ejde-508	437	33	(	(	PUNCT
ejde-508	437	34	3.2	3.2	NUM
ejde-508	437	35	)	)	PUNCT
ejde-508	437	36	,	,	PUNCT
ejde-508	437	37	and	and	CCONJ
ejde-508	437	38	sublinear	sublinear	VERB
ejde-508	437	39	conditions	condition	NOUN
ejde-508	437	40	like	like	ADP
ejde-508	437	41	(	(	PUNCT
ejde-508	437	42	4.4	4.4	NUM
ejde-508	437	43	)	)	PUNCT
ejde-508	437	44	or	or	CCONJ
ejde-508	437	45	(	(	PUNCT
ejde-508	437	46	4.1	4.1	NUM
ejde-508	437	47	)	)	PUNCT
ejde-508	437	48	on	on	ADP
ejde-508	437	49	g.	g.	PROPN
ejde-508	437	50	note	note	VERB
ejde-508	437	51	that	that	SCONJ
ejde-508	437	52	the	the	DET
ejde-508	437	53	super	super	ADJ
ejde-508	437	54	linear	linear	ADJ
ejde-508	437	55	condition	condition	NOUN
ejde-508	437	56	(	(	PUNCT
ejde-508	437	57	3.5	3.5	NUM
ejde-508	437	58	)	)	PUNCT
ejde-508	437	59	and	and	CCONJ
ejde-508	437	60	the	the	DET
ejde-508	437	61	sub	sub	NOUN
ejde-508	437	62	linear	linear	PROPN
ejde-508	437	63	condition	condition	NOUN
ejde-508	437	64	(	(	PUNCT
ejde-508	437	65	4.4	4.4	NUM
ejde-508	437	66	)	)	PUNCT
ejde-508	437	67	on	on	ADP
ejde-508	437	68	g	g	PROPN
ejde-508	437	69	include	include	VERB
ejde-508	437	70	their	their	PRON
ejde-508	437	71	corresponding	corresponding	ADJ
ejde-508	437	72	linear	linear	ADJ
ejde-508	437	73	case	case	NOUN
ejde-508	437	74	g(x	g(x	NOUN
ejde-508	437	75	)	)	PUNCT
ejde-508	438	1	=	=	SYM
ejde-508	438	2	x.	x.	NOUN
ejde-508	438	3	authors	author	NOUN
ejde-508	438	4	while	while	SCONJ
ejde-508	438	5	studying	study	VERB
ejde-508	438	6	the	the	DET
ejde-508	438	7	oscillatory	oscillatory	ADJ
ejde-508	438	8	and	and	CCONJ
ejde-508	438	9	asymptotic	asymptotic	ADJ
ejde-508	438	10	behavior	behavior	NOUN
ejde-508	438	11	of	of	ADP
ejde-508	438	12	(	(	PUNCT
ejde-508	438	13	1.2	1.2	NUM
ejde-508	438	14	)	)	PUNCT
ejde-508	438	15	or	or	CCONJ
ejde-508	438	16	(	(	PUNCT
ejde-508	438	17	1.1	1.1	NUM
ejde-508	438	18	)	)	PUNCT
ejde-508	438	19	,	,	PUNCT
ejde-508	438	20	very	very	ADV
ejde-508	438	21	often	often	ADV
ejde-508	438	22	find	find	VERB
ejde-508	438	23	difficulty	difficulty	NOUN
ejde-508	438	24	in	in	ADP
ejde-508	438	25	tackling	tackle	VERB
ejde-508	438	26	,	,	PUNCT
ejde-508	438	27	the	the	DET
ejde-508	438	28	case	case	NOUN
ejde-508	438	29	of	of	ADP
ejde-508	438	30	pn	pn	PROPN
ejde-508	438	31	≥	≥	PROPN
ejde-508	438	32	1	1	NUM
ejde-508	438	33	,	,	PUNCT
ejde-508	438	34	i.e	i.e	X
ejde-508	438	35	;	;	PUNCT
ejde-508	438	36	when	when	SCONJ
ejde-508	438	37	(	(	PUNCT
ejde-508	438	38	1.8	1.8	NUM
ejde-508	438	39	)	)	PUNCT
ejde-508	438	40	or	or	CCONJ
ejde-508	438	41	(	(	PUNCT
ejde-508	438	42	1.4	1.4	NUM
ejde-508	438	43	)	)	PUNCT
ejde-508	438	44	are	be	AUX
ejde-508	438	45	satisfied	satisfied	ADJ
ejde-508	438	46	.	.	PUNCT
ejde-508	439	1	that	that	PRON
ejde-508	439	2	is	be	AUX
ejde-508	439	3	why	why	SCONJ
ejde-508	439	4	,	,	PUNCT
ejde-508	439	5	the	the	DET
ejde-508	439	6	results	result	NOUN
ejde-508	439	7	[	[	X
ejde-508	439	8	10	10	NUM
ejde-508	439	9	,	,	PUNCT
ejde-508	439	10	theorems	theorem	VERB
ejde-508	439	11	2.6	2.6	NUM
ejde-508	439	12	and	and	CCONJ
ejde-508	439	13	2.7	2.7	NUM
ejde-508	439	14	]	]	PUNCT
ejde-508	439	15	appear	appear	VERB
ejde-508	439	16	to	to	PART
ejde-508	439	17	be	be	AUX
ejde-508	439	18	wrong	wrong	ADJ
ejde-508	439	19	,	,	PUNCT
ejde-508	439	20	as	as	ADP
ejde-508	439	21	the	the	DET
ejde-508	439	22	neutral	neutral	ADJ
ejde-508	439	23	equation	equation	NOUN
ejde-508	439	24	∆2(yn	∆2(yn	PROPN
ejde-508	439	25	−	−	PROPN
ejde-508	439	26	4yn−1	4yn−1	PROPN
ejde-508	439	27	)	)	PUNCT
ejde-508	440	1	+	+	CCONJ
ejde-508	440	2	4(n+1)/3y	4(n+1)/3y	NOUN
ejde-508	440	3	1/3	1/3	NUM
ejde-508	440	4	n−2	n−2	PROPN
ejde-508	440	5	=	=	SYM
ejde-508	440	6	0	0	NUM
ejde-508	440	7	satisfies	satisfy	VERB
ejde-508	440	8	all	all	DET
ejde-508	440	9	the	the	DET
ejde-508	440	10	conditions	condition	NOUN
ejde-508	440	11	of	of	ADP
ejde-508	440	12	the	the	DET
ejde-508	440	13	theorems	theorem	NOUN
ejde-508	440	14	,	,	PUNCT
ejde-508	440	15	but	but	CCONJ
ejde-508	440	16	,	,	PUNCT
ejde-508	440	17	it	it	PRON
ejde-508	440	18	admits	admit	VERB
ejde-508	440	19	a	a	DET
ejde-508	440	20	non	non	X
ejde-508	440	21	oscillatory	oscillatory	ADJ
ejde-508	440	22	solution	solution	NOUN
ejde-508	440	23	yn	yn	X
ejde-508	440	24	=	=	SYM
ejde-508	440	25	2n	2n	NUM
ejde-508	440	26	,	,	PUNCT
ejde-508	440	27	which	which	PRON
ejde-508	440	28	tends	tend	VERB
ejde-508	440	29	to	to	ADP
ejde-508	440	30	∞	∞	NUM
ejde-508	440	31	,	,	PUNCT
ejde-508	440	32	as	as	ADP
ejde-508	440	33	n	n	X
ejde-508	440	34	→	→	SYM
ejde-508	440	35	∞	∞	PROPN
ejde-508	440	36	,	,	PUNCT
ejde-508	440	37	contradicting	contradict	VERB
ejde-508	440	38	the	the	DET
ejde-508	440	39	theorems	theorem	NOUN
ejde-508	440	40	.	.	PUNCT
ejde-508	441	1	with	with	ADP
ejde-508	441	2	the	the	DET
ejde-508	441	3	super	super	ADJ
ejde-508	441	4	linear	linear	PROPN
ejde-508	441	5	g	g	NOUN
ejde-508	441	6	with	with	ADP
ejde-508	441	7	(	(	PUNCT
ejde-508	441	8	3.2	3.2	NUM
ejde-508	441	9	)	)	PUNCT
ejde-508	441	10	or	or	CCONJ
ejde-508	441	11	(	(	PUNCT
ejde-508	441	12	3.5	3.5	NUM
ejde-508	441	13	)	)	PUNCT
ejde-508	441	14	,	,	PUNCT
ejde-508	441	15	we	we	PRON
ejde-508	441	16	proved	prove	VERB
ejde-508	441	17	in	in	ADP
ejde-508	441	18	theorems	theorem	NOUN
ejde-508	441	19	3.2	3.2	NUM
ejde-508	441	20	and	and	CCONJ
ejde-508	441	21	3.4	3.4	NUM
ejde-508	441	22	that	that	PRON
ejde-508	441	23	(	(	PUNCT
ejde-508	441	24	a4	a4	NOUN
ejde-508	441	25	)	)	PUNCT
ejde-508	441	26	is	be	AUX
ejde-508	441	27	sufficient	sufficient	ADJ
ejde-508	441	28	for	for	SCONJ
ejde-508	441	29	all	all	DET
ejde-508	441	30	unbounded	unbounded	ADJ
ejde-508	441	31	solutions	solution	NOUN
ejde-508	441	32	of	of	ADP
ejde-508	441	33	(	(	PUNCT
ejde-508	441	34	3.2	3.2	NUM
ejde-508	441	35	)	)	PUNCT
ejde-508	441	36	to	to	PART
ejde-508	441	37	be	be	AUX
ejde-508	441	38	oscillatory	oscillatory	ADJ
ejde-508	441	39	which	which	PRON
ejde-508	441	40	is	be	AUX
ejde-508	441	41	more	more	ADV
ejde-508	441	42	restrictive	restrictive	ADJ
ejde-508	441	43	than	than	ADP
ejde-508	441	44	(	(	PUNCT
ejde-508	441	45	a2	a2	PROPN
ejde-508	441	46	)	)	PUNCT
ejde-508	441	47	.	.	PUNCT
ejde-508	442	1	hence	hence	ADV
ejde-508	442	2	,	,	PUNCT
ejde-508	442	3	one	one	PRON
ejde-508	442	4	may	may	AUX
ejde-508	442	5	extend	extend	VERB
ejde-508	442	6	this	this	DET
ejde-508	442	7	study	study	NOUN
ejde-508	442	8	to	to	PART
ejde-508	442	9	improve	improve	VERB
ejde-508	442	10	the	the	DET
ejde-508	442	11	results	result	NOUN
ejde-508	442	12	(	(	PUNCT
ejde-508	442	13	theorems	theorem	NOUN
ejde-508	442	14	3.2	3.2	NUM
ejde-508	442	15	and	and	CCONJ
ejde-508	442	16	3.4)by	3.4)by	NUM
ejde-508	442	17	attempting	attempt	VERB
ejde-508	442	18	to	to	PART
ejde-508	442	19	answer	answer	VERB
ejde-508	442	20	the	the	DET
ejde-508	442	21	following	following	ADJ
ejde-508	442	22	problem	problem	NOUN
ejde-508	442	23	.	.	PUNCT
ejde-508	443	1	problem	problem	NOUN
ejde-508	443	2	6.1	6.1	NUM
ejde-508	443	3	.	.	PUNCT
ejde-508	443	4	suppose	suppose	VERB
ejde-508	443	5	that	that	SCONJ
ejde-508	443	6	l(x	l(x	PROPN
ejde-508	443	7	)	)	PUNCT
ejde-508	443	8	=	=	SYM
ejde-508	443	9	x	x	NOUN
ejde-508	443	10	,	,	PUNCT
ejde-508	443	11	or	or	CCONJ
ejde-508	443	12	(	(	PUNCT
ejde-508	443	13	a1	a1	NOUN
ejde-508	443	14	)	)	PUNCT
ejde-508	443	15	holds	hold	NOUN
ejde-508	443	16	,	,	PUNCT
ejde-508	443	17	and	and	CCONJ
ejde-508	443	18	1	1	NUM
ejde-508	443	19	≤	≤	NUM
ejde-508	443	20	pn	pn	NOUN
ejde-508	443	21	≤	≤	PROPN
ejde-508	443	22	p.	p.	NOUN
ejde-508	443	23	assuming	assume	VERB
ejde-508	443	24	(	(	PUNCT
ejde-508	443	25	a2	a2	PROPN
ejde-508	443	26	)	)	PUNCT
ejde-508	443	27	,	,	PUNCT
ejde-508	443	28	(	(	PUNCT
ejde-508	443	29	a5	a5	PROPN
ejde-508	443	30	)	)	PUNCT
ejde-508	443	31	and	and	CCONJ
ejde-508	443	32	(	(	PUNCT
ejde-508	443	33	3.2	3.2	NUM
ejde-508	443	34	)	)	PUNCT
ejde-508	443	35	can	can	AUX
ejde-508	443	36	we	we	PRON
ejde-508	443	37	prove	prove	VERB
ejde-508	443	38	that	that	SCONJ
ejde-508	443	39	every	every	DET
ejde-508	443	40	unbounded	unbounded	ADJ
ejde-508	443	41	solution	solution	NOUN
ejde-508	443	42	of	of	ADP
ejde-508	443	43	(	(	PUNCT
ejde-508	443	44	1.1	1.1	NUM
ejde-508	443	45	)	)	PUNCT
ejde-508	443	46	oscillates	oscillate	NOUN
ejde-508	443	47	?	?	PUNCT
ejde-508	444	1	acknowledgment	acknowledgment	NOUN
ejde-508	444	2	.	.	PUNCT
ejde-508	445	1	the	the	DET
ejde-508	445	2	authors	author	NOUN
ejde-508	445	3	are	be	AUX
ejde-508	445	4	obliged	oblige	VERB
ejde-508	445	5	and	and	CCONJ
ejde-508	445	6	thankful	thankful	ADJ
ejde-508	445	7	to	to	ADP
ejde-508	445	8	the	the	DET
ejde-508	445	9	referee	referee	NOUN
ejde-508	445	10	and	and	CCONJ
ejde-508	445	11	the	the	DET
ejde-508	445	12	editor	editor	NOUN
ejde-508	445	13	for	for	ADP
ejde-508	445	14	their	their	PRON
ejde-508	445	15	various	various	ADJ
ejde-508	445	16	suggestions	suggestion	NOUN
ejde-508	445	17	to	to	PART
ejde-508	445	18	improve	improve	VERB
ejde-508	445	19	the	the	DET
ejde-508	445	20	presentation	presentation	NOUN
ejde-508	445	21	of	of	ADP
ejde-508	445	22	this	this	DET
ejde-508	445	23	article	article	NOUN
ejde-508	445	24	.	.	PUNCT
ejde-508	446	1	references	reference	NOUN
ejde-508	446	2	[	[	X
ejde-508	446	3	1	1	NUM
ejde-508	446	4	]	]	PUNCT
ejde-508	446	5	r.	r.	PROPN
ejde-508	446	6	p.	p.	PROPN
ejde-508	446	7	agarwal	agarwal	PROPN
ejde-508	446	8	;	;	PUNCT
ejde-508	446	9	difference	difference	NOUN
ejde-508	446	10	equations	equation	NOUN
ejde-508	446	11	and	and	CCONJ
ejde-508	446	12	inequalities	inequality	NOUN
ejde-508	446	13	,	,	PUNCT
ejde-508	446	14	marcel	marcel	PROPN
ejde-508	446	15	dekker	dekker	PROPN
ejde-508	446	16	,	,	PUNCT
ejde-508	446	17	newyork	newyork	PROPN
ejde-508	446	18	,	,	PUNCT
ejde-508	446	19	2000	2000	NUM
ejde-508	446	20	.	.	PUNCT
ejde-508	447	1	[	[	X
ejde-508	447	2	2	2	NUM
ejde-508	447	3	]	]	PUNCT
ejde-508	447	4	chittaranjan	chittaranjan	PROPN
ejde-508	447	5	behera	behera	PROPN
ejde-508	447	6	,	,	PUNCT
ejde-508	447	7	radhanath	radhanath	PROPN
ejde-508	447	8	rath	rath	PROPN
ejde-508	447	9	,	,	PUNCT
ejde-508	447	10	prayag	prayag	PROPN
ejde-508	447	11	prasad	prasad	PROPN
ejde-508	447	12	mishra	mishra	PROPN
ejde-508	447	13	;	;	PUNCT
ejde-508	447	14	oscillation	oscillation	NOUN
ejde-508	447	15	for	for	ADP
ejde-508	447	16	second	second	ADJ
ejde-508	447	17	order	order	NOUN
ejde-508	447	18	neutral	neutral	ADJ
ejde-508	447	19	difference	difference	NOUN
ejde-508	447	20	equations	equation	NOUN
ejde-508	447	21	with	with	ADP
ejde-508	447	22	variable	variable	ADJ
ejde-508	447	23	delays	delay	NOUN
ejde-508	447	24	,	,	PUNCT
ejde-508	447	25	international	international	PROPN
ejde-508	447	26	j.	j.	PROPN
ejde-508	447	27	of	of	ADP
ejde-508	447	28	mathematical	mathematical	PROPN
ejde-508	447	29	,	,	PUNCT
ejde-508	447	30	engineering	engineering	NOUN
ejde-508	447	31	and	and	CCONJ
ejde-508	447	32	management	management	NOUN
ejde-508	447	33	sciences	science	NOUN
ejde-508	447	34	,	,	PUNCT
ejde-508	447	35	5	5	NUM
ejde-508	447	36	,	,	PUNCT
ejde-508	447	37	(	(	PUNCT
ejde-508	447	38	2020	2020	NUM
ejde-508	447	39	)	)	PUNCT
ejde-508	447	40	,	,	PUNCT
ejde-508	447	41	no	no	INTJ
ejde-508	447	42	.	.	NOUN
ejde-508	447	43	4	4	NUM
ejde-508	447	44	,	,	PUNCT
ejde-508	447	45	663–681	663–681	NUM
ejde-508	447	46	.	.	PUNCT
ejde-508	448	1	[	[	X
ejde-508	448	2	3	3	NUM
ejde-508	448	3	]	]	PUNCT
ejde-508	448	4	chittaranjan	chittaranjan	PROPN
ejde-508	448	5	behera	behera	PROPN
ejde-508	448	6	,	,	PUNCT
ejde-508	448	7	radhanath	radhanath	PROPN
ejde-508	448	8	rath	rath	PROPN
ejde-508	448	9	;	;	PUNCT
ejde-508	448	10	oscillation	oscillation	NOUN
ejde-508	448	11	and	and	CCONJ
ejde-508	448	12	asymptotic	asymptotic	ADJ
ejde-508	448	13	behavior	behavior	NOUN
ejde-508	448	14	of	of	ADP
ejde-508	448	15	second	second	ADJ
ejde-508	448	16	-	-	PUNCT
ejde-508	448	17	order	order	NOUN
ejde-508	448	18	neutral	neutral	ADJ
ejde-508	448	19	delay	delay	NOUN
ejde-508	448	20	difference	difference	NOUN
ejde-508	448	21	equations	equation	NOUN
ejde-508	448	22	with	with	ADP
ejde-508	448	23	variable	variable	ADJ
ejde-508	448	24	delays	delay	NOUN
ejde-508	448	25	.	.	PUNCT
ejde-508	449	1	int	int	NOUN
ejde-508	449	2	.	.	PUNCT
ejde-508	450	1	j.	j.	PROPN
ejde-508	450	2	of	of	ADP
ejde-508	450	3	diff	diff	PROPN
ejde-508	450	4	.	.	PUNCT
ejde-508	451	1	equ	equ	PROPN
ejde-508	451	2	.	.	PROPN
ejde-508	451	3	15	15	NUM
ejde-508	451	4	(	(	PUNCT
ejde-508	451	5	2020	2020	NUM
ejde-508	451	6	)	)	PUNCT
ejde-508	451	7	,	,	PUNCT
ejde-508	451	8	no	no	INTJ
ejde-508	451	9	.	.	NOUN
ejde-508	451	10	1	1	NUM
ejde-508	451	11	,	,	PUNCT
ejde-508	451	12	107–126	107–126	NUM
ejde-508	451	13	.	.	PUNCT
ejde-508	451	14	14	14	NUM
ejde-508	451	15	a.	a.	PROPN
ejde-508	451	16	k.	k.	PROPN
ejde-508	451	17	bhuyan	bhuyan	PROPN
ejde-508	451	18	,	,	PUNCT
ejde-508	451	19	l.	l.	PROPN
ejde-508	451	20	n.	n.	PROPN
ejde-508	451	21	padhy	padhy	PROPN
ejde-508	451	22	,	,	PUNCT
ejde-508	451	23	r.	r.	PROPN
ejde-508	451	24	n.	n.	PROPN
ejde-508	451	25	rath	rath	PROPN
ejde-508	451	26	ejde-2020/87	ejde-2020/87	PROPN
ejde-508	452	1	[	[	X
ejde-508	452	2	4	4	NUM
ejde-508	452	3	]	]	PUNCT
ejde-508	452	4	chittaranjan	chittaranjan	PROPN
ejde-508	452	5	behera	behera	PROPN
ejde-508	452	6	,	,	PUNCT
ejde-508	452	7	radhanath	radhanath	PROPN
ejde-508	452	8	rath	rath	PROPN
ejde-508	452	9	,	,	PUNCT
ejde-508	452	10	prayag	prayag	PROPN
ejde-508	452	11	prasad	prasad	PROPN
ejde-508	452	12	mishra	mishra	PROPN
ejde-508	452	13	;	;	PUNCT
ejde-508	452	14	oscillation	oscillation	NOUN
ejde-508	452	15	and	and	CCONJ
ejde-508	452	16	asymptotic	asymptotic	ADJ
ejde-508	452	17	behavior	behavior	NOUN
ejde-508	452	18	of	of	ADP
ejde-508	452	19	a	a	DET
ejde-508	452	20	higher	high	ADJ
ejde-508	452	21	-	-	PUNCT
ejde-508	452	22	order	order	NOUN
ejde-508	452	23	neutral	neutral	ADJ
ejde-508	452	24	delay	delay	NOUN
ejde-508	452	25	difference	difference	NOUN
ejde-508	452	26	equation	equation	NOUN
ejde-508	452	27	with	with	ADP
ejde-508	452	28	variable	variable	ADJ
ejde-508	452	29	delays	delay	NOUN
ejde-508	452	30	under	under	ADP
ejde-508	452	31	∆m	∆m	PROPN
ejde-508	452	32	(	(	PUNCT
ejde-508	452	33	to	to	PART
ejde-508	452	34	appear	appear	VERB
ejde-508	452	35	in	in	ADP
ejde-508	452	36	math	math	NOUN
ejde-508	452	37	.	.	PUNCT
ejde-508	453	1	slovaca	slovaca	PROPN
ejde-508	453	2	,	,	PUNCT
ejde-508	453	3	70	70	NUM
ejde-508	453	4	(	(	PUNCT
ejde-508	453	5	dec	dec	PROPN
ejde-508	453	6	2020	2020	NUM
ejde-508	453	7	)	)	PUNCT
ejde-508	453	8	,	,	PUNCT
ejde-508	453	9	no	no	INTJ
ejde-508	453	10	.	.	NOUN
ejde-508	453	11	6	6	NUM
ejde-508	453	12	)	)	PUNCT
ejde-508	453	13	.	.	PUNCT
ejde-508	454	1	[	[	X
ejde-508	454	2	5	5	NUM
ejde-508	454	3	]	]	PUNCT
ejde-508	454	4	i.	i.	PROPN
ejde-508	454	5	gyori	gyori	PROPN
ejde-508	454	6	,	,	PUNCT
ejde-508	454	7	g.	g.	PROPN
ejde-508	454	8	ladas	ladas	PROPN
ejde-508	454	9	;	;	PUNCT
ejde-508	454	10	oscillation	oscillation	NOUN
ejde-508	454	11	theory	theory	NOUN
ejde-508	454	12	of	of	ADP
ejde-508	454	13	delay	delay	NOUN
ejde-508	454	14	-	-	PUNCT
ejde-508	454	15	differential	differential	NOUN
ejde-508	454	16	equations	equation	NOUN
ejde-508	454	17	with	with	ADP
ejde-508	454	18	applications	application	NOUN
ejde-508	454	19	,	,	PUNCT
ejde-508	454	20	clarendon	clarendon	PROPN
ejde-508	454	21	press	press	NOUN
ejde-508	454	22	,	,	PUNCT
ejde-508	454	23	oxford	oxford	NOUN
ejde-508	454	24	,	,	PUNCT
ejde-508	454	25	1991	1991	NUM
ejde-508	454	26	.	.	PUNCT
ejde-508	455	1	[	[	X
ejde-508	455	2	6	6	NUM
ejde-508	455	3	]	]	PUNCT
ejde-508	455	4	b.	b.	PROPN
ejde-508	455	5	karpuz	karpuz	PROPN
ejde-508	455	6	,	,	PUNCT
ejde-508	455	7	r.	r.	PROPN
ejde-508	455	8	n.	n.	PROPN
ejde-508	455	9	rath	rath	PROPN
ejde-508	455	10	,	,	PUNCT
ejde-508	455	11	s.	s.	PROPN
ejde-508	455	12	k.	k.	PROPN
ejde-508	455	13	rath	rath	PROPN
ejde-508	455	14	;	;	PUNCT
ejde-508	455	15	on	on	ADP
ejde-508	455	16	oscillation	oscillation	NOUN
ejde-508	455	17	and	and	CCONJ
ejde-508	455	18	asymptotic	asymptotic	ADJ
ejde-508	455	19	behaviour	behaviour	NOUN
ejde-508	455	20	of	of	ADP
ejde-508	455	21	a	a	DET
ejde-508	455	22	higher	high	ADJ
ejde-508	455	23	order	order	NOUN
ejde-508	455	24	functional	functional	ADJ
ejde-508	455	25	difference	difference	NOUN
ejde-508	455	26	equation	equation	NOUN
ejde-508	455	27	of	of	ADP
ejde-508	455	28	neutral	neutral	ADJ
ejde-508	455	29	type	type	NOUN
ejde-508	455	30	,	,	PUNCT
ejde-508	455	31	int	int	NOUN
ejde-508	455	32	.	.	PUNCT
ejde-508	456	1	j.	j.	PROPN
ejde-508	456	2	difference	difference	PROPN
ejde-508	456	3	equ	equ	PROPN
ejde-508	456	4	.	.	PROPN
ejde-508	457	1	4	4	NUM
ejde-508	457	2	(	(	PUNCT
ejde-508	457	3	2009	2009	NUM
ejde-508	457	4	)	)	PUNCT
ejde-508	457	5	,	,	PUNCT
ejde-508	457	6	no	no	INTJ
ejde-508	457	7	.	.	NOUN
ejde-508	457	8	1	1	NUM
ejde-508	457	9	,	,	PUNCT
ejde-508	457	10	69–96	69–96	NUM
ejde-508	457	11	.	.	PUNCT
ejde-508	458	1	[	[	X
ejde-508	458	2	7	7	X
ejde-508	458	3	]	]	X
ejde-508	458	4	b.	b.	NOUN
ejde-508	458	5	karpuz	karpuz	PROPN
ejde-508	458	6	,	,	PUNCT
ejde-508	458	7	o.	o.	PROPN
ejde-508	458	8	ocalan	ocalan	PROPN
ejde-508	458	9	,	,	PUNCT
ejde-508	458	10	m.	m.	PROPN
ejde-508	458	11	k.	k.	PROPN
ejde-508	458	12	yildiz	yildiz	PROPN
ejde-508	458	13	;	;	PUNCT
ejde-508	458	14	oscillation	oscillation	NOUN
ejde-508	458	15	of	of	ADP
ejde-508	458	16	a	a	DET
ejde-508	458	17	class	class	NOUN
ejde-508	458	18	of	of	ADP
ejde-508	458	19	difference	difference	NOUN
ejde-508	458	20	equations	equation	NOUN
ejde-508	458	21	of	of	ADP
ejde-508	458	22	second	second	ADJ
ejde-508	458	23	order	order	NOUN
ejde-508	458	24	,	,	PUNCT
ejde-508	458	25	math	math	NOUN
ejde-508	458	26	.	.	PUNCT
ejde-508	459	1	comput	comput	NOUN
ejde-508	459	2	.	.	PUNCT
ejde-508	460	1	modelling	modelling	NOUN
ejde-508	460	2	,	,	PUNCT
ejde-508	460	3	49	49	NUM
ejde-508	460	4	(	(	PUNCT
ejde-508	460	5	2009	2009	NUM
ejde-508	460	6	)	)	PUNCT
ejde-508	460	7	,	,	PUNCT
ejde-508	460	8	912	912	NUM
ejde-508	460	9	-	-	SYM
ejde-508	460	10	917	917	NUM
ejde-508	460	11	.	.	PUNCT
ejde-508	461	1	[	[	X
ejde-508	461	2	8	8	NUM
ejde-508	461	3	]	]	X
ejde-508	461	4	w.	w.	PROPN
ejde-508	461	5	g.	g.	PROPN
ejde-508	461	6	kelley	kelley	PROPN
ejde-508	461	7	,	,	PUNCT
ejde-508	461	8	a.	a.	PROPN
ejde-508	461	9	c.	c.	PROPN
ejde-508	461	10	peterson	peterson	PROPN
ejde-508	461	11	;	;	PUNCT
ejde-508	461	12	difference	difference	NOUN
ejde-508	461	13	equations	equation	NOUN
ejde-508	461	14	:	:	PUNCT
ejde-508	461	15	an	an	DET
ejde-508	461	16	introduction	introduction	NOUN
ejde-508	461	17	with	with	ADP
ejde-508	461	18	applications	application	NOUN
ejde-508	461	19	,	,	PUNCT
ejde-508	461	20	academic	academic	ADJ
ejde-508	461	21	press	press	NOUN
ejde-508	461	22	,	,	PUNCT
ejde-508	461	23	newyork,1991	newyork,1991	NOUN
ejde-508	461	24	.	.	PUNCT
ejde-508	462	1	[	[	X
ejde-508	462	2	9	9	NUM
ejde-508	462	3	]	]	X
ejde-508	462	4	n.	n.	NOUN
ejde-508	462	5	parhi	parhi	NOUN
ejde-508	462	6	,	,	PUNCT
ejde-508	462	7	a.	a.	PROPN
ejde-508	462	8	k.	k.	PROPN
ejde-508	462	9	tripathy	tripathy	PROPN
ejde-508	462	10	;	;	PUNCT
ejde-508	462	11	oscillation	oscillation	NOUN
ejde-508	462	12	of	of	ADP
ejde-508	462	13	forced	force	VERB
ejde-508	462	14	nonlinear	nonlinear	ADJ
ejde-508	462	15	neutral	neutral	ADJ
ejde-508	462	16	delay	delay	NOUN
ejde-508	462	17	difference	difference	NOUN
ejde-508	462	18	equations	equation	NOUN
ejde-508	462	19	of	of	ADP
ejde-508	462	20	first	first	ADJ
ejde-508	462	21	order	order	NOUN
ejde-508	462	22	,	,	PUNCT
ejde-508	462	23	czech	czech	PROPN
ejde-508	462	24	.	.	PUNCT
ejde-508	462	25	math	math	PROPN
ejde-508	462	26	.	.	PUNCT
ejde-508	463	1	j.	j.	PROPN
ejde-508	463	2	53	53	NUM
ejde-508	463	3	(	(	PUNCT
ejde-508	463	4	2003	2003	NUM
ejde-508	463	5	)	)	PUNCT
ejde-508	463	6	,	,	PUNCT
ejde-508	463	7	83	83	NUM
ejde-508	463	8	-	-	SYM
ejde-508	463	9	101	101	NUM
ejde-508	463	10	.	.	PUNCT
ejde-508	464	1	[	[	X
ejde-508	464	2	10	10	NUM
ejde-508	464	3	]	]	X
ejde-508	464	4	n.	n.	NOUN
ejde-508	464	5	parhi	parhi	NOUN
ejde-508	464	6	,	,	PUNCT
ejde-508	464	7	a.	a.	PROPN
ejde-508	464	8	k.	k.	PROPN
ejde-508	464	9	tripathy	tripathy	PROPN
ejde-508	464	10	;	;	PUNCT
ejde-508	464	11	oscillation	oscillation	NOUN
ejde-508	464	12	of	of	ADP
ejde-508	464	13	a	a	DET
ejde-508	464	14	class	class	NOUN
ejde-508	464	15	of	of	ADP
ejde-508	464	16	non	non	ADJ
ejde-508	464	17	-	-	ADJ
ejde-508	464	18	linear	linear	ADJ
ejde-508	464	19	neutral	neutral	ADJ
ejde-508	464	20	difference	difference	NOUN
ejde-508	464	21	equations	equation	NOUN
ejde-508	464	22	of	of	ADP
ejde-508	464	23	higher	high	ADJ
ejde-508	464	24	order	order	NOUN
ejde-508	464	25	,	,	PUNCT
ejde-508	464	26	j.	j.	PROPN
ejde-508	464	27	math	math	PROPN
ejde-508	464	28	.	.	PUNCT
ejde-508	465	1	anal	anal	PROPN
ejde-508	465	2	.	.	PUNCT
ejde-508	465	3	appl	appl	PROPN
ejde-508	465	4	.	.	PUNCT
ejde-508	466	1	284(2003	284(2003	NUM
ejde-508	466	2	)	)	PUNCT
ejde-508	466	3	,	,	PUNCT
ejde-508	466	4	756	756	NUM
ejde-508	466	5	-	-	SYM
ejde-508	466	6	774	774	NUM
ejde-508	466	7	.	.	PUNCT
ejde-508	467	1	[	[	X
ejde-508	467	2	11	11	NUM
ejde-508	467	3	]	]	X
ejde-508	467	4	n.	n.	NOUN
ejde-508	467	5	parhi	parhi	NOUN
ejde-508	467	6	,	,	PUNCT
ejde-508	467	7	a.	a.	PROPN
ejde-508	467	8	k.	k.	PROPN
ejde-508	467	9	tripathy	tripathy	PROPN
ejde-508	467	10	;	;	PUNCT
ejde-508	467	11	oscillation	oscillation	NOUN
ejde-508	467	12	of	of	ADP
ejde-508	467	13	a	a	DET
ejde-508	467	14	class	class	NOUN
ejde-508	467	15	of	of	ADP
ejde-508	467	16	neutral	neutral	ADJ
ejde-508	467	17	difference	difference	NOUN
ejde-508	467	18	equations	equation	NOUN
ejde-508	467	19	of	of	ADP
ejde-508	467	20	first	first	ADJ
ejde-508	467	21	order	order	NOUN
ejde-508	467	22	,	,	PUNCT
ejde-508	467	23	journal	journal	NOUN
ejde-508	467	24	of	of	ADP
ejde-508	467	25	difference	difference	NOUN
ejde-508	467	26	equations	equation	NOUN
ejde-508	467	27	and	and	CCONJ
ejde-508	467	28	applications	application	NOUN
ejde-508	467	29	9(2003	9(2003	NUM
ejde-508	467	30	)	)	PUNCT
ejde-508	467	31	no	no	NOUN
ejde-508	467	32	.	.	NOUN
ejde-508	467	33	10	10	NUM
ejde-508	467	34	,	,	PUNCT
ejde-508	467	35	933	933	NUM
ejde-508	467	36	-	-	SYM
ejde-508	467	37	946	946	NUM
ejde-508	467	38	.	.	PUNCT
ejde-508	468	1	[	[	X
ejde-508	468	2	12	12	NUM
ejde-508	468	3	]	]	PUNCT
ejde-508	468	4	r.	r.	PROPN
ejde-508	468	5	n.	n.	PROPN
ejde-508	468	6	rath	rath	PROPN
ejde-508	468	7	,	,	PUNCT
ejde-508	468	8	b.	b.	PROPN
ejde-508	468	9	l.	l.	PROPN
ejde-508	468	10	s.	s.	PROPN
ejde-508	468	11	barik	barik	PROPN
ejde-508	468	12	,	,	PUNCT
ejde-508	468	13	s.	s.	PROPN
ejde-508	468	14	k.	k.	PROPN
ejde-508	468	15	rath	rath	PROPN
ejde-508	468	16	;	;	PUNCT
ejde-508	468	17	oscillation	oscillation	NOUN
ejde-508	468	18	of	of	ADP
ejde-508	468	19	higher	high	ADJ
ejde-508	468	20	order	order	NOUN
ejde-508	468	21	neutral	neutral	ADJ
ejde-508	468	22	functional	functional	ADJ
ejde-508	468	23	difference	difference	NOUN
ejde-508	468	24	equations	equation	NOUN
ejde-508	468	25	with	with	ADP
ejde-508	468	26	positive	positive	ADJ
ejde-508	468	27	and	and	CCONJ
ejde-508	468	28	negative	negative	ADJ
ejde-508	468	29	coefficients	coefficient	NOUN
ejde-508	468	30	,	,	PUNCT
ejde-508	468	31	mathematica	mathematica	PROPN
ejde-508	468	32	slovaca	slovaca	PROPN
ejde-508	468	33	,	,	PUNCT
ejde-508	468	34	60	60	NUM
ejde-508	468	35	(	(	PUNCT
ejde-508	468	36	2010	2010	NUM
ejde-508	468	37	)	)	PUNCT
ejde-508	468	38	,	,	PUNCT
ejde-508	468	39	no	no	INTJ
ejde-508	468	40	.	.	NOUN
ejde-508	468	41	3	3	NUM
ejde-508	468	42	,	,	PUNCT
ejde-508	468	43	361–384	361–384	NUM
ejde-508	468	44	.	.	PUNCT
ejde-508	469	1	[	[	X
ejde-508	469	2	13	13	NUM
ejde-508	469	3	]	]	PUNCT
ejde-508	469	4	r.	r.	PROPN
ejde-508	469	5	n.	n.	PROPN
ejde-508	469	6	rath	rath	PROPN
ejde-508	469	7	,	,	PUNCT
ejde-508	469	8	c.	c.	PROPN
ejde-508	469	9	r.	r.	PROPN
ejde-508	469	10	behera	behera	PROPN
ejde-508	469	11	,	,	PUNCT
ejde-508	469	12	a.	a.	PROPN
ejde-508	469	13	k.	k.	PROPN
ejde-508	469	14	bhuyan	bhuyan	PROPN
ejde-508	469	15	;	;	PUNCT
ejde-508	469	16	oscillatory	oscillatory	ADJ
ejde-508	469	17	and	and	CCONJ
ejde-508	469	18	asymptotic	asymptotic	ADJ
ejde-508	469	19	behaviour	behaviour	NOUN
ejde-508	469	20	of	of	ADP
ejde-508	469	21	higher	high	ADJ
ejde-508	469	22	order	order	NOUN
ejde-508	469	23	neutral	neutral	ADJ
ejde-508	469	24	difference	difference	NOUN
ejde-508	469	25	equations	equation	NOUN
ejde-508	469	26	,	,	PUNCT
ejde-508	469	27	acta	acta	PROPN
ejde-508	469	28	mathematica	mathematica	PROPN
ejde-508	469	29	academiae	academiae	PROPN
ejde-508	469	30	paedagogicae	paedagogicae	VERB
ejde-508	469	31	nýıregyháziensis	nýıregyháziensis	NOUN
ejde-508	469	32	,	,	PUNCT
ejde-508	469	33	31(2015	31(2015	NUM
ejde-508	469	34	)	)	PUNCT
ejde-508	469	35	,	,	PUNCT
ejde-508	469	36	no	no	INTJ
ejde-508	469	37	.	.	NOUN
ejde-508	469	38	2	2	NUM
ejde-508	469	39	,	,	PUNCT
ejde-508	469	40	233	233	NUM
ejde-508	469	41	-	-	SYM
ejde-508	469	42	248	248	NUM
ejde-508	469	43	.	.	PUNCT
ejde-508	470	1	[	[	X
ejde-508	470	2	14	14	NUM
ejde-508	470	3	]	]	X
ejde-508	470	4	radhanath	radhanath	PROPN
ejde-508	470	5	rath	rath	PROPN
ejde-508	470	6	,	,	PUNCT
ejde-508	470	7	chittaranjan	chittaranjan	PROPN
ejde-508	470	8	behera	behera	PROPN
ejde-508	470	9	;	;	PUNCT
ejde-508	470	10	oscillatory	oscillatory	ADJ
ejde-508	470	11	and	and	CCONJ
ejde-508	470	12	asymptotic	asymptotic	ADJ
ejde-508	470	13	behaviour	behaviour	NOUN
ejde-508	470	14	of	of	ADP
ejde-508	470	15	a	a	DET
ejde-508	470	16	first	first	ADJ
ejde-508	470	17	order	order	NOUN
ejde-508	470	18	neutral	neutral	ADJ
ejde-508	470	19	equation	equation	NOUN
ejde-508	470	20	of	of	ADP
ejde-508	470	21	discrete	discrete	ADJ
ejde-508	470	22	type	type	NOUN
ejde-508	470	23	with	with	ADP
ejde-508	470	24	variable	variable	ADJ
ejde-508	470	25	several	several	ADJ
ejde-508	470	26	delay	delay	NOUN
ejde-508	470	27	under	under	ADP
ejde-508	470	28	∆	∆	PROPN
ejde-508	470	29	sign	sign	NOUN
ejde-508	470	30	.	.	PUNCT
ejde-508	471	1	international	international	ADJ
ejde-508	471	2	j.	j.	PROPN
ejde-508	471	3	of	of	ADP
ejde-508	471	4	math	math	NOUN
ejde-508	471	5	.	.	PUNCT
ejde-508	472	1	and	and	CCONJ
ejde-508	472	2	mathematical	mathematical	ADJ
ejde-508	472	3	sciences	science	NOUN
ejde-508	472	4	,	,	PUNCT
ejde-508	472	5	vol	vol	NOUN
ejde-508	472	6	2018	2018	NUM
ejde-508	472	7	,	,	PUNCT
ejde-508	472	8	article	article	NOUN
ejde-508	472	9	i	i	PROPN
ejde-508	472	10	d	d	PROPN
ejde-508	472	11	4586176	4586176	NUM
ejde-508	472	12	,	,	PUNCT
ejde-508	472	13	https//doi.org10.1155/20184586176	https//doi.org10.1155/20184586176	X
ejde-508	473	1	[	[	X
ejde-508	473	2	15	15	NUM
ejde-508	473	3	]	]	PUNCT
ejde-508	473	4	a.	a.	NOUN
ejde-508	473	5	k.	k.	PROPN
ejde-508	473	6	tripathy	tripathy	PROPN
ejde-508	473	7	,	,	PUNCT
ejde-508	473	8	s.	s.	PROPN
ejde-508	473	9	panigrahi	panigrahi	PROPN
ejde-508	473	10	;	;	PUNCT
ejde-508	473	11	oscillation	oscillation	NOUN
ejde-508	473	12	in	in	ADP
ejde-508	473	13	non	non	ADJ
ejde-508	473	14	linear	linear	ADJ
ejde-508	473	15	neutral	neutral	ADJ
ejde-508	473	16	difference	difference	NOUN
ejde-508	473	17	equations	equation	NOUN
ejde-508	473	18	with	with	ADP
ejde-508	473	19	positive	positive	ADJ
ejde-508	473	20	and	and	CCONJ
ejde-508	473	21	negative	negative	ADJ
ejde-508	473	22	coefficients	coefficient	NOUN
ejde-508	473	23	,	,	PUNCT
ejde-508	473	24	int	int	NOUN
ejde-508	473	25	.	.	PUNCT
ejde-508	474	1	j.	j.	PROPN
ejde-508	474	2	of	of	ADP
ejde-508	474	3	difference	difference	NOUN
ejde-508	474	4	equations	equation	NOUN
ejde-508	474	5	5(2010	5(2010	NUM
ejde-508	474	6	)	)	PUNCT
ejde-508	474	7	,	,	PUNCT
ejde-508	474	8	no	no	INTJ
ejde-508	474	9	.	.	NOUN
ejde-508	474	10	2	2	NUM
ejde-508	474	11	,	,	PUNCT
ejde-508	474	12	251–265	251–265	NUM
ejde-508	474	13	.	.	PUNCT
ejde-508	475	1	[	[	X
ejde-508	475	2	16	16	NUM
ejde-508	475	3	]	]	PUNCT
ejde-508	475	4	m.	m.	PROPN
ejde-508	475	5	k.	k.	PROPN
ejde-508	475	6	yildiz	yildiz	PROPN
ejde-508	475	7	;	;	PUNCT
ejde-508	475	8	oscillation	oscillation	NOUN
ejde-508	475	9	of	of	ADP
ejde-508	475	10	a	a	DET
ejde-508	475	11	class	class	NOUN
ejde-508	475	12	of	of	ADP
ejde-508	475	13	nonlinear	nonlinear	ADJ
ejde-508	475	14	difference	difference	NOUN
ejde-508	475	15	equations	equation	NOUN
ejde-508	475	16	of	of	ADP
ejde-508	475	17	second	second	ADJ
ejde-508	475	18	order	order	NOUN
ejde-508	475	19	with	with	ADP
ejde-508	475	20	oscillating	oscillate	VERB
ejde-508	475	21	coefficients	coefficient	NOUN
ejde-508	475	22	,	,	PUNCT
ejde-508	475	23	konuralp	konuralp	PROPN
ejde-508	475	24	j.	j.	PROPN
ejde-508	475	25	math	math	PROPN
ejde-508	475	26	.	.	PUNCT
ejde-508	475	27	,	,	PUNCT
ejde-508	475	28	3(2015	3(2015	NUM
ejde-508	475	29	)	)	PUNCT
ejde-508	475	30	,	,	PUNCT
ejde-508	475	31	no	no	INTJ
ejde-508	475	32	.	.	NOUN
ejde-508	475	33	2	2	NUM
ejde-508	475	34	,	,	PUNCT
ejde-508	475	35	211	211	NUM
ejde-508	475	36	-	-	SYM
ejde-508	475	37	218	218	NUM
ejde-508	475	38	.	.	PUNCT
ejde-508	476	1	ajit	ajit	PROPN
ejde-508	476	2	kumar	kumar	PROPN
ejde-508	476	3	bhuyan	bhuyan	PROPN
ejde-508	476	4	dept	dept	PROPN
ejde-508	476	5	.	.	PROPN
ejde-508	476	6	of	of	ADP
ejde-508	476	7	mathematics	mathematics	PROPN
ejde-508	476	8	,	,	PUNCT
ejde-508	476	9	sai	sai	PROPN
ejde-508	476	10	international	international	PROPN
ejde-508	476	11	school	school	PROPN
ejde-508	476	12	,	,	PUNCT
ejde-508	476	13	bhubaneswar	bhubaneswar	NOUN
ejde-508	476	14	,	,	PUNCT
ejde-508	476	15	odisha	odisha	PROPN
ejde-508	476	16	,	,	PUNCT
ejde-508	476	17	india	india	PROPN
ejde-508	476	18	email	email	NOUN
ejde-508	476	19	address	address	NOUN
ejde-508	476	20	:	:	PUNCT
ejde-508	476	21	ajitbhuyan13@gmail.com	ajitbhuyan13@gmail.com	X
ejde-508	476	22	laxmi	laxmi	PROPN
ejde-508	476	23	narayan	narayan	PROPN
ejde-508	476	24	padhy	padhy	PROPN
ejde-508	476	25	dept	dept	PROPN
ejde-508	476	26	.	.	PROPN
ejde-508	476	27	of	of	ADP
ejde-508	476	28	math	math	NOUN
ejde-508	476	29	and	and	CCONJ
ejde-508	476	30	computer	computer	NOUN
ejde-508	476	31	science	science	NOUN
ejde-508	476	32	,	,	PUNCT
ejde-508	476	33	konark	konark	PROPN
ejde-508	476	34	institute	institute	PROPN
ejde-508	476	35	of	of	ADP
ejde-508	476	36	science	science	NOUN
ejde-508	476	37	and	and	CCONJ
ejde-508	476	38	technology	technology	NOUN
ejde-508	476	39	,	,	PUNCT
ejde-508	476	40	bhubaneswar	bhubaneswar	NOUN
ejde-508	476	41	,	,	PUNCT
ejde-508	476	42	odisha	odisha	PROPN
ejde-508	476	43	,	,	PUNCT
ejde-508	476	44	india	india	PROPN
ejde-508	476	45	email	email	NOUN
ejde-508	476	46	address	address	NOUN
ejde-508	476	47	:	:	PUNCT
ejde-508	477	1	padhyln@gmail.com	padhyln@gmail.com	X
ejde-508	477	2	radhanath	radhanath	PROPN
ejde-508	477	3	rath	rath	PROPN
ejde-508	477	4	(	(	PUNCT
ejde-508	477	5	corresponding	correspond	VERB
ejde-508	477	6	author	author	NOUN
ejde-508	477	7	)	)	PUNCT
ejde-508	477	8	vssut	vssut	PROPN
ejde-508	477	9	burla	burla	PROPN
ejde-508	477	10	,	,	PUNCT
ejde-508	477	11	768018	768018	NUM
ejde-508	477	12	.	.	PUNCT
ejde-508	478	1	retired	retire	VERB
ejde-508	478	2	principalhallikote	principalhallikote	PROPN
ejde-508	478	3	autonomous	autonomous	ADJ
ejde-508	478	4	college	college	NOUN
ejde-508	478	5	,	,	PUNCT
ejde-508	478	6	berhampur	berhampur	NOUN
ejde-508	478	7	,	,	PUNCT
ejde-508	478	8	760001	760001	NUM
ejde-508	478	9	.	.	PUNCT
ejde-508	479	1	center	center	NOUN
ejde-508	479	2	point	point	NOUN
ejde-508	479	3	apartment	apartment	NOUN
ejde-508	479	4	,	,	PUNCT
ejde-508	479	5	flat	flat	ADJ
ejde-508	479	6	a-203	a-203	PROPN
ejde-508	479	7	,	,	PUNCT
ejde-508	479	8	shailashree	shailashree	PROPN
ejde-508	479	9	vihar	vihar	PROPN
ejde-508	479	10	ph-7	ph-7	PROPN
ejde-508	479	11	,	,	PUNCT
ejde-508	479	12	751024	751024	NUM
ejde-508	479	13	,	,	PUNCT
ejde-508	479	14	bhubaneswar	bhubaneswar	NOUN
ejde-508	479	15	,	,	PUNCT
ejde-508	479	16	odisha	odisha	PROPN
ejde-508	479	17	,	,	PUNCT
ejde-508	479	18	india	india	PROPN
ejde-508	479	19	email	email	NOUN
ejde-508	479	20	address	address	NOUN
ejde-508	479	21	:	:	PUNCT
ejde-508	479	22	radhanathmath@yahoo.co.in	radhanathmath@yahoo.co.in	PROPN
ejde-508	479	23	1	1	NUM
ejde-508	479	24	.	.	X
ejde-508	479	25	introduction	introduction	NOUN
ejde-508	479	26	2	2	NUM
ejde-508	479	27	.	.	PUNCT
ejde-508	480	1	some	some	DET
ejde-508	480	2	lemmas	lemmas	ADJ
ejde-508	480	3	3	3	NUM
ejde-508	480	4	.	.	NOUN
ejde-508	480	5	main	main	ADJ
ejde-508	480	6	results	result	NOUN
ejde-508	480	7	part	part	NOUN
ejde-508	480	8	i	i	PRON
ejde-508	480	9	4	4	NUM
ejde-508	480	10	.	.	PUNCT
ejde-508	480	11	main	main	ADJ
ejde-508	480	12	results	result	NOUN
ejde-508	480	13	part	part	NOUN
ejde-508	480	14	ii	ii	PROPN
ejde-508	480	15	5	5	NUM
ejde-508	480	16	.	.	PUNCT
ejde-508	481	1	application	application	NOUN
ejde-508	481	2	to	to	ADP
ejde-508	481	3	neutral	neutral	ADJ
ejde-508	481	4	difference	difference	NOUN
ejde-508	481	5	equations	equation	NOUN
ejde-508	481	6	with	with	ADP
ejde-508	481	7	oscillating	oscillate	VERB
ejde-508	481	8	coefficients	coefficient	NOUN
ejde-508	481	9	6	6	NUM
ejde-508	481	10	.	.	PUNCT
ejde-508	481	11	final	final	ADJ
ejde-508	481	12	comments	comment	NOUN
ejde-508	481	13	acknowledgment	acknowledgment	NOUN
ejde-508	481	14	references	reference	NOUN