Question: Let {an} be a sequence and L is any real number. Prove that {an} =1 converges to L if and only if every monotone subsequence of {an} converges to L . Tip: prove the contrapositive by using a) in image, and the Bolzano-Weierstrass Theorem. (a) Let {an} be any sequence and L any real number. Prove that if {a,} doesnot converge to L, then there exists an e> 0 and a subsequence {a} suchthat an - L > e for all k.Tip: Start by writing down what it means that {am} does not converge to L,i.e. write down the formal negation of the statement {a,} converges to L. Usethis formal negation and induction on k to construct an increasing sequence ofnatural numbers {ng} such that |an - L| > e for all k.3D1

Name: Question: Let {an} be a sequence and L is any real number. Prove that
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Description: Question: Let {an} be a sequence and L is any real number. Prove that {an} =1 converges to L if and only if every monotone subsequence of {an} converges to L . Tip: prove the contrapositive by using a) in image, and the Bolzano-Weierstrass Theorem

Answered: Question: Let {an} be a sequence and L…

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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Question: Let {an} be a sequence and L is any real number. Prove that {an} =1 converges to L if and only if every monotone subsequence of {an} converges to L .

Tip: prove the contrapositive by using a) in image, and the Bolzano-Weierstrass Theorem.

(a) Let {an} be any sequence and L any real number. Prove that if {a,} does
not converge to L, then there exists an e> 0 and a subsequence {a} such
that an - L > e for all k.
Tip: Start by writing down what it means that {am} does not converge to L,
i.e. write down the formal negation of the statement {a,} converges to L. Use
this formal negation and induction on k to construct an increasing sequence of
natural numbers {ng} such that |an - L| > e for all k.
3D1

Expert Solution

Step 1

Consider a sequence, $\{a_{n}\}$ and $L$ be any real number.

To prove that, $\{a_{n}\}$ converges to $L$ if and only if every monotone sub-sequence of $\{a_{n}\}$ converges to $L$ .

Suppose that $\{a_{n}\}$ converges to $L$ .

Since every convergent sequence is bounded, $\{a_{n}\}$ is bounded.

Then by Bolzano-Weierstrass Theorem,

A bounded sequence of real numbers has a convergent sub-sequence.

Let $\{a_{n_{k}}\}$ is a monotone sub-sequence of $\{a_{n}\}$

Since, $\{a_{n}\}$ converges to $L$ , by definition,

For every $ε > 0,$ there exists a positive integer $m$ such that,

$|a_{n} - L| < ε \forall n \geq m$

In particular,

$|a_{n_{k}} - L| < ε \forall n \geq m$

Therefore, $\{a_{n_{k}}\}$ converges to $L$ .

So, if $\{a_{n}\}$ converges to $L$ , then every monotone sub-sequence of $\{a_{n}\}$ converges to $L$ .

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