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If xn → 0, then for every e > 0 there exists N e N such that n > N implies xn < e. If for every e > 0 there exists N € N such that n > N implies xn < €, then xn + 0. If xn → x and xn → y, then x = = y.

If xn → 0, then for every e > 0 there exists N e N such that n > N implies xn < e. If for every e > 0 there exists N € N such that n > N implies xn < €, then xn + 0. If xn → x and xn → y, then x = = y.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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If xn → 0, then for every e > 0 there exists N E N such that n > N implies xm < €. If for every e > 0 there exists N E N such that n > N implies xn < €, then xn → 0. If xn → x and xn → y, then x = y. If xn → x and xn > 0 for all n, then x > 0. If a monotone sequence is bounded, then it converges. If a convergent sequence is monotone, then it is bounded. If a convergent sequence is bounded, then it is monotone. (Xn)1 conveges to x iff every subsequence of (xn) converges to x. Let (xn) be a bounded sequence and let m = exists N E N such that n > N implies Xn > m – E. lim sup xn. Then for every e > 0 there If (xn) is unbounded above, then +∞ is a subsequential limit point of (xn).
Transcribed Image Text:If xn → 0, then for every e > 0 there exists N E N such that n > N implies xm < €. If for every e > 0 there exists N E N such that n > N implies xn < €, then xn → 0. If xn → x and xn → y, then x = y. If xn → x and xn > 0 for all n, then x > 0. If a monotone sequence is bounded, then it converges. If a convergent sequence is monotone, then it is bounded. If a convergent sequence is bounded, then it is monotone. (Xn)1 conveges to x iff every subsequence of (xn) converges to x. Let (xn) be a bounded sequence and let m = exists N E N such that n > N implies Xn > m – E. lim sup xn. Then for every e > 0 there If (xn) is unbounded above, then +∞ is a subsequential limit point of (xn).
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