Let $(X,d)$ be a metric space and $\emptyset\neq Y\subseteq X$. For $x\in X$ define $d(x,Y)=\inf\{d(x,y)|y\in Y\}$
Show: $d(x,Y)=0$ iff $x\in\overline{Y}$
Proof:
We have,
$d(x,Y)=0\Leftrightarrow \inf\{d(x,y)|y\in Y\}=0\Leftrightarrow\forall\varepsilon >0 \exists y\in Y: d(x,y)<\varepsilon$
$\Leftrightarrow \forall\varepsilon>0:B_\varepsilon(x)\cap Y\neq\emptyset\Leftrightarrow x\in\overline{Y}$
Completing the proof.