3. Prove that a sequence {a,} converges to the real number a if and only iflim sup a, = lim inf a, = a.(HINTS: For the direction, you'll need the two Lemmas given in Session 25. Forthe direction, the Squeeze Theorem is quite helpful.)Lemma: Every seguence{an} has aSubsequence {any}that converges to limsup an.Lemma: Every sequence žans has a subsequene {anpsthat converges to liming an.

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Description: 3. Prove that a sequence {a,} converges to the real number a if and only iflim sup a, = lim inf a, = a.(HINTS: For the direction, you'll need the two Lemmas given in Session 25. Forthe direction, the Squeeze Theorem is quite helpful.)Lemma: Every se

First suppose sequence an is converges to a Then we have to prove that limsupan=liminfan=a Fix an…

Answered: 3. Prove that a sequence {a,} converges…

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3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a lim inf an = a.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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3. Prove that a sequence {a,} converges to the real number a if and only if
lim sup a, = lim inf a, = a.
(HINTS: For the direction, you'll need the two Lemmas given in Session 25. For
the direction, the Squeeze Theorem is quite helpful.)
Lemma: Every seguence
{an} has a
Subsequence {any}
that converges to limsup an.
Lemma: Every sequence žans has a subsequene {anps
that converges to liming an.

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