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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

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Question 5.
Given a metric spaceX,p>
(a) If the sequence (2n)neN C X 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?
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 (Tn)neN, (yn)neN C X. Suppose that they converge to the same
limit a E X. Show that the metric distance p(xn, Yn) → 0 as no? Is it true that
if p(xn, Yn) → 0 as n→∞, then the two sequences have the same limit? Justify your
answer.

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