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There's a theorem that says if $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.

Why can it not be done with $K$ only being closed? If $K$ is closed then $\overline{E} \subset K$, so if $x$ is a limit point of $E$, then $x \in \overline{E} \implies x \in K$.

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As a counterexample: $\Bbb R$ is closed, and $\Bbb Z$ is an infinite subset of $\Bbb R$. However, $\Bbb Z$ has no limit points.

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    $\begingroup$ @JohnMa Sorry! I think you beat me to the punch, too $\endgroup$ – Omnomnomnom Oct 5 '15 at 5:14
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Your conclusion fails because you assumed that $x$, a limit point of $E$, exists.

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