13. Let (an) be a Cauchy sequence, and let p : N → N be a one-to-one function. Show that the sequence (ap(n)) is a Cauchy sequence.n=1

Answered: 13. Let (an)=I be a Cauchy sequence,…

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

13. Let (an) be a Cauchy sequence, and let p : N → N be a one-to-one function. Show that the sequence (ap(n)) is a Cauchy sequence.
n=1

Expert Solution

Step 1

A sequence is called a Cauchy sequence if as the sequence progresses, the terms of the sequence become arbitrarily close to each other. For any given small positive distance, all but finite number of terms of the sequence are less than that of the small positive distance.

Let $(x_{1}, x_{2}, . . ., x_{m}, . . .) b e t h e s e q u e n c e$ . A sequence can be defined as a function from natural numbers ( that gives the positions of the elements of the sequence) to the elements at each position.

we get, $x_{n} \Rightarrow x i s t h e e l e m e n t a t t h e n^{t h} p o s i t i o n o f t h e s e q u e n c e$ .

This sequence is said to a Cauchy sequence if for every positive real number $\in > 0$ , there is a positive integer $p > 0$ such that for all natural numbers $m, n > p, |x_{n} - x_{m}| < \in$ , the condition is for requiring $|x_{n} - x_{m}|$ to be infinitesimal for every pair of infinite $m a n d n$ .