Isolated Point of Closure of Subset is Isolated Point of Subset/Proof 1

Proof
Let $x \in \map \cl H$ be an isolated point of $\map \cl H$.

that $x$ is not an isolated point of $H$.

Then by the definition of a limit point, it follows that $x$ must be a limit point of $H$.

From Set is Subset of its Topological Closure:
 * $H \subseteq \map \cl H$

So by Limit Point of Subset is Limit Point of Set it follows that $x$ is a limit point of $\map \cl H$.

But then $x$ cannot be an isolated point of $\map \cl H$.

This is a contradiction.

Therefore every isolated point of $\map \cl H$ is also an isolated point of $H$.