Closure of Subset of Closed Set of Metric Space is Subset/Proof 1

Proof
From Metric Induces Topology, the topology $\tau$ induced by the metric $d$ is a topology on $M$.

From Metric Closure and Topological Closure of Subset are Equivalent, it is sufficient to show that the topological closure of $H$ is contained in $F$.

From Set is Closed in Metric Space iff Closed in Induced Topological Space:
 * $F$ is closed in the topological space $\struct{A, \tau}$ induced by the metric $d$.

We have: