Set is Closed iff Equals Topological Closure/Proof 1

Proof
Let $H'$ denote the derived set of $H$.

By Closed Set iff Contains all its Limit Points, $H$ is closed in $T$ $H' \subseteq H$.

By Union with Superset is Superset, $H' \subseteq H$ $H = H \cup H'$.

The result follows from the definition of closure.