Condition for Point being in Closure/Metric Space

Theorem
Let $M = \struct {S, d}$ be a metric space.

Let $H \subseteq S$.

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

Let $x \in S$.

Then $x \in H^-$ :
 * $\forall \epsilon \in \R_{>0}: \map {B_\epsilon} x \cap H \ne \O$

where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$.