**Problem 5.**Prove the following three statements:(a) Any convergent sequence $(x_n)$ is bounded.(b) Let $(x_n)$ and $(y_n)$ be two convergent sequences, and suppose that their limits are $x_n \rightarrow L$ and $y_n \rightarrow M$. Show that the sequence $(x_n + y_n)$, obtained by summing them termwise, is a convergent sequence, and in fact \[x_n + y_n \rightarrow (L + M).\](c) There exist bounded sequences which are not convergent.

Name: **Problem 5.**Prove the following three statements:(a) Any convergent
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Description: **Problem 5.**Prove the following three statements:(a) Any convergent sequence $(x_n)$ is bounded.(b) Let $(x_n)$ and $(y_n)$ be two convergent sequences, and suppose that their limits are $x_n \rightarrow L$ and $y_n \rightarrow M$. Show

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UIIree Statements. (a) Any convergent sequence (xn) is bounded. (b) Let (xn) and (yn) be two convergent sequences, and suppose that their limits are xn → L and yn → M. Show that the sequence (xn + Yn), obtained by summing them termwise, is a convergent sequence, and in fact Xn + Yn → (L+ M). (c) There exist bounded sequences which are not convergent.

UIIree Statements. (a) Any convergent sequence (xn) is bounded. (b) Let (xn) and (yn) be two convergent sequences, and suppose that their limits are xn → L and yn → M. Show that the sequence (xn + Yn), obtained by summing them termwise, is a convergent sequence, and in fact Xn + Yn → (L+ M). (c) There exist bounded sequences which are not convergent.

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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Concept explainers

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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**Problem 5.**

Prove the following three statements:

(a) Any convergent sequence $(x_n)$ is bounded.

(b) Let $(x_n)$ and $(y_n)$ be two convergent sequences, and suppose that their limits are $x_n \rightarrow L$ and $y_n \rightarrow M$. Show that the sequence $(x_n + y_n)$, obtained by summing them termwise, is a convergent sequence, and in fact

\[x_n + y_n \rightarrow (L + M).\]

(c) There exist bounded sequences which are not convergent.

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