use the method in second image example 20.2 Prove that both of the following sequences are Cauchy sequences, using just the definition (not any Theorems.)(a) {"}°n=18.(b) {까1Sn=1n+1 lam – an| =-Example 20.2. I claim the sequence {÷} is a Cauchy sequence.Proof. Pick e > 0. (Scratch work: we need to figure out how big mand n need to be in order that -< e. Note that if both and 1are between 0 and ɛ, then the distance between them will be at mostɛ, and this occurs so long as n, m >!. So we will set N = !. Back tothe proof.) Let N = !. Let m, n eN be such that m > N and n > N.Then 0 < < = e and 0 < 습 <★m%3D= €. It follows thatm11lam – anand this proves the sequence is Cauchy.

(a) Consider the sequence, an=-1nnn=1∞ Let ε>0, For, n,m∈ℕ, if n≥m then 1n≤1m Then,…

Answered: Prove that both of the following…

Prove that both of the following sequences are Cauchy sequences, using just the definition (not any Theorems.) (a) {4"} (b) {„}, (-1)" | n-1 n+1

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

use the method in second image example 20.2

Prove that both of the following sequences are Cauchy sequences, using just the definition (not any Theorems.)
(a) {"}°
n=1
8.
(b) {까1Sn=1
n+1

lam – an| =-
Example 20.2. I claim the sequence {÷} is a Cauchy sequence.
Proof. Pick e > 0. (Scratch work: we need to figure out how big m
and n need to be in order that -< e. Note that if both and 1
are between 0 and ɛ, then the distance between them will be at most
ɛ, and this occurs so long as n, m >!. So we will set N = !. Back to
the proof.) Let N = !. Let m, n eN be such that m > N and n > N.
Then 0 < < = e and 0 < 습 <★
m
%3D
= €. It follows that
m
1
1
lam – an
and this proves the sequence is Cauchy.

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