TITLE: Proving that limit points of $E$ being countable $\implies$ $E$ countable
QUESTION [5 upvotes]: Let $E'$ denote the set of the limit points of $E$.

Prove: If $E$ is a subset of $\mathbb{R}^n$, and $E'$ is countable, then $E$ is countable.

REPLY [3 votes]: Any point of $E$ that is not a limit point of $E$ is an isolated point of $E$.  If $x$ is an isolated point of $E$, there is a point with rational coordinates that is closer
to $x$ than to any other point of $E$.  This implies that there are at most countably
many isolated points of $E$.  The union of two countable sets is countable, so ...