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 $\{a_n\}_{n=1}^{\infty}$ be any sequence and $L$ any real number. Prove that if $\{a_n\}_{n=1}^{\infty}$ does not converge to $L$, then there exists an $\varepsilon > 0$ and a subsequence $\{a_{n_k}\}_{k=1}^{\infty}$ such that $|a_{n_k} - L| > \varepsilon$ for all $k$.*Tip:* Start by writing down what it means that $\{a_n\}_{n=1}^{\infty}$ does not converge to $L$, i.e., write down the formal negation of the statement $\{a_n\}_{n=1}^{\infty}$ converges to $L$. Use this formal negation and induction on $k$ to construct an increasing sequence of natural numbers $\{n_k\}_{k=1}^{\infty}$, such that $|a_{n_k} - L| > \varepsilon$ for all $k$.

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 $\{a_n\}_{n=1}^{\infty}$ be any sequence and $L$ any real number. Prove that if $\{a_n\}_{n=1}^{\infty}$ does not converge to $L$, then there exists an $\varepsilon > 0$ and a subsequence $\{a_{n_k}\}_{k=1}^{\infty}$ such that $|a_{n_k} - L| > \varepsilon$ for all $k$.

*Tip:* Start by writing down what it means that $\{a_n\}_{n=1}^{\infty}$ does not converge to $L$, i.e., write down the formal negation of the statement $\{a_n\}_{n=1}^{\infty}$ converges to $L$. Use this formal negation and induction on $k$ to construct an increasing sequence of natural numbers $\{n_k\}_{k=1}^{\infty}$, such that $|a_{n_k} - L| > \varepsilon$ for all $k$.

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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