Real Analysis I need help with this 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

Name: Real Analysis I need help with this 2. Show directly that sup(4) = 1 f
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Description: Real Analysis I need help with this 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

As we know that, 1>1-1n ∀ n∈ℕ Therefore 1 is the upper bound of the set 1-1n:n∈ℕ. And for each…

Answered: 2. Show directly that sup(4) = 1 for A…

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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Real Analysis I need help with this

2. Show directly that sup(4) = 1 for A = {1- : n eN}.

Expert Solution

Step 1

As we know that,

$1 > 1 - \frac{1}{n} \forall n \in ℕ$

Therefore 1 is the upper bound of the set $\{1 - \frac{1}{n} : n \in ℕ\}$ .

And for each $ε > 0$ , by Archimedean Property ,

$\exists N \in ℕ s.t. N \geq \frac{1}{ε} Then we get 1 - \frac{1}{N} > 1 - ε$

Therefore $1 - ε$ is not an upper bound for the set for each $ε > 0$ .

Therefore, 1 is the greatest upper bound and hence $s u p (A) .$

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