Closure of Subset of Closed Set of Metric Space is Subset/Proof 2

Proof
Let $x \in H^-$.

From Point in Closure of Subset of Metric Space iff Limit of Sequence
 * there exists a sequence $\sequence {a_n}$ of points of $H$ which converges to the limit $x$.

By assumption:
 * $\sequence {a_n}$ is also a sequence of points of $F$

From Subset of Metric Space contains Limits of Sequences iff Closed:
 * $x \in F$

Thus it has been shown:
 * $H^- \subseteq F$