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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?
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.
