1.62 please 1.61 Suppose In →∞. Prove that every subsequence Ink →∞ as k →∞ aswell. (Hint: The sequence xn is divergent, so it is not enough to quote Theorem1.5.1.)1.62 Use the following steps to prove that the sequence n has no convergentsubsequences if and only if |xn| →∞ as n →∞.a) Suppose that the sequence on has no convergent subsequences. Let M >0. Prove that there exist at most finitely many values of n such thatIn E[-M, M]. Explain why this implies |xn|→∞ as n →∞.b) Suppose an →∞ as n → ∞. Show that n has no convergent subse-quence. (Hint: Exercise 1.61 may help.)

1.62 Use the following steps to prove that the sequence n has no convergent subsequences if and only if |xn| →∞as nx. a) Suppose that the sequence xn has no convergent subsequences. Let M > 0. Prove that there exist at most finitely many values of n such that In E[-M, M]. Explain why this implies |xn| →∞ as nx. b) Suppose an →∞ as n → ∞o. Show that xn has no convergent subse- quence. (Hint: Exercise 1.61 may help.)

1.62 Use the following steps to prove that the sequence n has no convergent subsequences if and only if |xn| →∞as nx. a) Suppose that the sequence xn has no convergent subsequences. Let M > 0. Prove that there exist at most finitely many values of n such that In E[-M, M]. Explain why this implies |xn| →∞ as nx. b) Suppose an →∞ as n → ∞o. Show that xn has no convergent subse- quence. (Hint: Exercise 1.61 may help.)

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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