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

Hello,

I have a question concerning convergence of the non-monotonic sequences which takes place when the Cauchy criterion is satisfied.

I understand that [tex]|a_n - a_m| <ε[/tex] for all n,m[tex]N\ni[/tex]

2. Relevant equations

What I don't see is how [tex](a_{n+1} - a_n) →0[/tex]is not equivalent/enough. Doesn't the fact that the difference tends to zero mean that it can eventually be smaller than any ε?

3. The attempt at a solution

I know the examples when this doesn't work, like [tex](\sqrt{n})[/tex] but I don't understand the difference between these two criteria.

Could anyone explain why they aren't equivalent?

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# Homework Help: Cauchy sequnce and convergence of a non-monotonic sequence.

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