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I want to prove this from the theorem from Characterization of Compact metric spaces. I don't know how to go about with it. Should I prove that if every infinite set has a cluster point then every sequence is bounded? Then I'll be able to prove it using Bolzano Weierstrass theorem. I'm not sure about how to proceed with it. Is there any other way?

Dianne Su
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1 Answers1

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So you want to prove that limit point compactness implies sequential compactness, which is true for metrizable spaces but not true for a general topological space.

If the space $X$ is limit point compact, given a sequence $\langle x_n\rangle\in X$, consider the set $\{x_n\}$. If the set $\{x_n\}$ is finite, then there is some $x_0$ which occurs infinitely many times and we are done. If $\{x_n\}$ is infinite, then $\{x_n\}$ has a limit point $x_0$, and for every $k\in\mathbb N_+$, we choose some $x_{n_k}\in B(x_0,\frac1k)$. Then the subsequence $\langle x_{n_k}\rangle$ converges to $x_0$.

Ricky
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