6.(a) A sequence {an}neN is said to be bounded if there exists M > 0 such that|an| ≤ M for any n E N(i) Let {an}nen and {bn}nen be two sequences such that {an}nen is bounded0. Prove that lim anbn = 0=n→∞Give a counterexample to (a)(i) when {an}nen is not bounded. That is,{an}neN and {bn}nen are two sequences such that lim bn = 0, but lim anbn ‡0n→∞n→∞(ii)and lim bn*8个&(b) By part (a)(i), evaluate limn→∞(sin n) (2n-1)n² + 1

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Answered: 6. (a) A sequence {anneN is said to be…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

6.
(a) A sequence {an}neN is said to be bounded if there exists M > 0 such that
|an| ≤ M for any n E N
(i) Let {an}nen and {bn}nen be two sequences such that {an}nen is bounded
0. Prove that lim anbn = 0
=
n→∞
Give a counterexample to (a)(i) when {an}nen is not bounded. That is,
{an}neN and {bn}nen are two sequences such that lim bn = 0, but lim anbn ‡
0
n→∞
n→∞
(ii)
and lim bn
*
8个&
(b) By part (a)(i), evaluate lim
n→∞
(sin n) (2n-1)
n² + 1

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