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Let $M$ be an infinite compact metric space. Given any $e>0$, does there always exist a positive integer $n$ such that any subset of $M$ which contains at least $n$ points, must contain two distinct points whose distance apart is not greater than $e$?

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There will be finitely many $e$-neighborhoods which will cover $M$ (since such neighborhoods form an open cover and therefore have a finite subcover). If we take more points than there are neighborhoods in this subcover, the pigeonhole principle dictates that two points will belong to the same neighborhood, thus having distance $\le e$.

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    Many thanks. I tried several approaches to prove this but completely forgot about that basic covering lemma. – 2017-01-25
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    @GarabedGulbenkian My pleasure! I was also rather surprised to find such a neat answer.... – 2017-01-25