Closure of Subset of Closed Set of Metric Space is Subset

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Let $F$ be a closed set of $M$.

Let $H \subseteq F$ be a subset of $F$.

Let $H^-$ denote the closure of $H$.

Then $H^- \subseteq F$.

Proof
Let $M = \left({A, d}\right)$ be a metric space.

Let $F$ be a closed set of $M$.

Let $H \subseteq F$ be a subset of $F$.

Then: