(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the following assertion: Suppose {xn} and {yn} are Cauchy sequences of real number. If {xn} is a cauchy sequence and for every η>0 there exists a pos. int. N such that for every n>N so that abs(xn-yn)<η then {yn} is a Cauchy sequence.

2. Relevant equations

None

3. The attempt at a solution

We will prove that {yn} is a cauchy sequence by showing that for every ε>0 there exists a pos. int. N so that both n and m >N so that abs(yn-ym)<ε.

Consider ε>0 arbitrary.

Since {xn} is a cauchy sequence and by hypothesis for every η>0 there exists an N so that every n>N abs(xn-yn)<η. Choose such an N.

Consider n,m>N arbitrary.

Then I know I need to get from abs(yn-ym) to ε but unsure how to use what I have to get there correctly and if I have the rest of the proof right.

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# Homework Help: Intro to Analysis (Cauchy)

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