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.)

Answered: 1.62 Use the following steps to prove…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

1.62 please

1.61 Suppose In →∞. Prove that every subsequence Ink →∞ as k →∞ as
well. (Hint: The sequence xn is divergent, so it is not enough to quote Theorem
1.5.1.)
1.62 Use the following steps to prove that the sequence n has no convergent
subsequences 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 that
In 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.)

Expert Solution

Step 1

In this solution we will discuss and prove that a sequence $x_{n}$ has no convergent subsequences if and only if $|x_{n}| \to \infty$ as $n \to \infty$ .

We know that a sequence $y_{n} \to \infty$ as $n \to \infty$ , if for any real number $M > 0$ , there exists a natural number $n_{0}$ such that $y_{n} > M$ for all $n \geq n_{0}$ .