Clearly, A must be compact also, so I'm assuming this proof requires the use of all 3 factors - closed, discrete & compact - but I'm not quite sure how to actually use these in a proper proof. Any help/hints/explanation would be greatly appreciated!
2026-04-07 16:15:15.1775578515
Let A be a closed discrete set inside a compact set K. Show that A is finite.
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We know (or can prove) that a closed subset of a compact set is compact. Therefore, $A$ is compact. We are also given that $A$ is discrete, which means that, for each $x \in A$, we can find a neighborhood $N_x$ of $x$ such that $N_x \cap A = \{x\}$.
Notice that $\{N_x \}_{x \in X}$ is an open cover of $A$...