Given a metric space.(a) If the sequence (n)neN CX is convergent, show that it is bounded.(b) If the sequence (Tn)neN C X is convergent, prove that it is Cauchy. Is the converse true?Justify your answer.(c) True or false? Justify your answer. If (n)neN is a bounded sequence in X, then it has aconvergent subsequence.(d) Given two sequences (zn)neN, (n)nEN C X. Suppose that they converge to the samelimit a € X. Show that the metric distance p(x, yn) → 0 as noo? Is it true thatif p(xn, Yn) → 0 as noo, then the two sequences have the same limit? Justify youranswer.

Thank you for the question. Since you have posted multiple questions we will solve the first…

Answered: (a) If the sequence ()neN CX is…

(a) If the sequence ()neN CX is convergent, show that it is bounded. (b) If the sequence (n)neN CX is convergent, prove that it is Cauchy. Is the converse true?

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

Given a metric space.
<X,p>
(a) If the sequence (n)neN CX is convergent, show that it is bounded.
(b) If the sequence (Tn)neN C X is convergent, prove that it is Cauchy. Is the converse true?
Justify your answer.
(c) True or false? Justify your answer. If (n)neN is a bounded sequence in X, then it has a
convergent subsequence.
(d) Given two sequences (zn)neN, (n)nEN C X. Suppose that they converge to the same
limit a € X. Show that the metric distance p(x, yn) → 0 as noo? Is it true that
if p(xn, Yn) → 0 as noo, then the two sequences have the same limit? Justify your
answer.

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