I am trying to prove that a metric space in which every infinite subset has a limit point is compact. I am trying to prove it with Heine-Borel theorem for general metric spaces, since I have not studied countable bases and separability.
My proof is as follows:
- Suppose you have a metric space where every infinite subset has a limit point.
- If the metric space is unbounded (thus infinite), one can form an infinite subset with points at distance between any two points greater than a certain minimum, which will be without a limit point since the space is unbounded.
- If the metric space is bounded and finite then it won't be a problem as there will be no infinite set.
- But if the metric space is bounded and infinite, then it has to be complete because if not, then there is a point not belonging to the metric space, to which a Cauchy sequence converges (a limit point not in the metric space), but in the given metric space every infinite subset has a limit point. So the metric space has to be bounded and complete, meaning complete, by the Heine-Borel theorem for general metric spaces.
If I missed something or did something wrong, please help me complete it.