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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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05f71249-0159-4369-afdd-b84b5bf21c00%2F313c9976-787d-47dd-b7a1-d9adf68f6156%2Fuw17tai_processed.png&w=3840&q=75)
Transcribed Image Text:**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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