In studying Rudin's Mathematical Analysis, it seems that he uses the following statement:(adsbygoogle = window.adsbygoogle || []).push({});

If every infinite subset of a set C, has a limit point in C, then C is compact.

I can prove the converse of this statement, and its converse is also proven in the text, but I am unsure on how to prove the above statement, since I must provide a collection of finitely many open sets, whose union contains C, but the only thing I know is that there is a point in C complement s.t. every neighborhood of that point contains at least one point of C. The neighborhood of such a point is indeed open, but I don't know how to construct a finite collection of open sets whose union contains C. So, I was wondering if the above statement is actually true, and if so, what is the general reasoning behind it.

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# Compactness and every infinite subset has a limit point

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