Question #35 please 35. Suppose x is an accumulation point of {a,: n E J). Show that there is a subsequence ofthat converges to x.(a)*36. Let (an) be a bounded sequence of real numbers. Prove that (al has a convergentsubsequence. (Hint: You may want to use the Bolzano-Weierstrass Theorem.)*37. Prove that if (a) decreasing and bounded, then (a,) converges.38. Prove that if c> 1, then (Vc), converges to 1.

35. Let x is an accumulation point of A=an:n∈J.Then choose xn∈x-1n,x+1n∩A.

Answered: Suppose x is an accumulation point of…

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Suppose x is an accumulation point of (a,: n E J). Show that there is a subsequence of (a) that converges to x.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

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Question

Question #35 please

35. Suppose x is an accumulation point of {a,: n E J). Show that there is a subsequence of
that converges to x.
(a)
*36. Let (an) be a bounded sequence of real numbers. Prove that (al has a convergent
subsequence. (Hint: You may want to use the Bolzano-Weierstrass Theorem.)
*37. Prove that if (a) decreasing and bounded, then (a,) converges.
38. Prove that if c> 1, then (Vc), converges to 1.

Expert Solution

Step 1

35.

$L e t x i s a n a c c u m u l a t i o n p o i n t o f A = \{a_{n} : n \in J\} . T h e n c h o o s e x_{n} \in (x - \frac{1}{n}, x + \frac{1}{n}) \cap A .$

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