TITLE: Help in proof of Compactness implies limit point compactness.
QUESTION [9 upvotes]: Munkres' Topology says

Theorem 28.1.  Compactness implies limit point compactness, but not conversely.
Proof.  Let $X$ be a compact space.  Given a subset $A$ of $X$, we wish to prove that if $A$ is infinite, then $A$ has a limit point.  We prove the contrapositive - if $A$ has no limit point, then $A$ must be finite.
      Suppose $A$ has no limit point.  Then $A$ contains all its limit points, so that $A$ is closed. ....

That's confusing.  Do they mean vacuously $A$ must be closed since it has no limit points even to contain?

REPLY [10 votes]: If a set has no limit points, then it is closed, as it automatically contains all its limit points. By the same reasoning, each subset of $A$ is closed, as a limit point of the subset would also be a limit point of $A$. Therefore, it is discrete. Conversely, being closed and discrete implies that a set has no limit points.
As a closed subset of $X$, $A$ is compact. But a discrete compact set must be finite.