# True of False: Every sequence contains a Cauchy subsequence?

For my practice midterm exam this is a question: State whether the following are true or false AND briefly explain why (or give counterexample):

Statement: Every sequence contains a Cauchy subsequence.

I said this is false because I used the example that the sequence of{x_n} = n is divergent. Therefore a convergent subsequence cannot exist, therefor a Cauchy subsequence cannot exist since all Cauchy sequences are convergent.

Am I right or am I missing/getting mixed up some proper information?

• This is correct. – Joppy Sep 30 '18 at 8:44

## 1 Answer

Your example is fine, but not your explanation. The sequence $$\bigl((-1)^n\bigr)_{n\in\mathbb N}$$ is also divergent, but it has Cauchy subsequences.

However, the distance between any two distinct terms of the sequence $$(n)_{n\in\mathbb N}$$ is at least $$1$$. That proves that it has no Cauchy subsequence.