Prove the following, unofficially known as the Awesome Lemma or Awesome Theorem: Suppose {an} is a sequence that converges to a real number a. (a) If a < b, then there exists an NEN such that, if n > N, then an < b. (b) If a > b, then there exists an NEN such that, if n > N, then an > b. (You may omit the proof of part (b) and simply state that it follows from part (a) “by symmetric argument.")

Prove the following, unofficially known as the Awesome Lemma or Awesome Theorem: Suppose {an} is a sequence that converges to a real number a. (a) If a < b, then there exists an NEN such that, if n > N, then an < b. (b) If a > b, then there exists an NEN such that, if n > N, then an > b. (You may omit the proof of part (b) and simply state that it follows from part (a) “by symmetric argument.")

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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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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Prove the following, unofficially known as the Awesome Lemma or Awesome Theorem:
Suppose {a,} is a sequence that converges to a real number a.
(a) If a < b, then there exists an NEN such that, if n > N, then an < b.
(b) If a > b, then there exists an NEN such that, if n > N, then an > b.
(You may omit the proof of part (b) and simply state that it follows from part (a)
“by symmetric argument.")

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