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

A sequence (X_{n}) is Cauchy if and only if, for every ε>0, there exists an open interval length ε that contains all except for finitely many terms of (X_{n}).

2. Relevant equations

The Cauchy Definition is:

A sequence X = (x_{n}) of real numbers is said to be a Cauchy sequence if for every ε>0 there exists a natural number H(ε) such that for all natural numbers n,m≥H(ε) satisfy |x_{n}- x_{m}| < ε.

I want to check my contrapositive for the left-to-right implication:

There exists ε>0, such ALL open intervals ε contain all except for finitely many terms of (x_{n}) => (x_{n}) is not Cauchy.

3. The attempt at a solution

[STRIKE]I want to say this is false. As the right side holds the criterion that there is an interval around some point p such that the interval bounds are (p-ε/2 , p+ε/2) while the first only requires that the terms of x_{n}only be between (x_{m}- 1, x_{m}+1). [/STRIKE] I'm certain now that this is true

[STRIKE]However, I cannot come up with a counter example as when I try it seems that both sides imply each other.[/STRIKE] So then I assumed it was true and to prove it. Now, I believe that the right side does indeed imply Cauchy. So for the left to right implication, I decided to prove it with the contrapositive:

If the contrapositive is the correct form, then it is easy to see that the hypothesis is false as now finite interval that can be any where on the number line can contain all the terms of a sequence.

Does this make sense? If not, please let me know what I could clarify. Thank you!

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# Homework Help: (dis)prove an if and only if statement of a Cauchy sequence and and interval.

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