A metric defined in terms of the minimum distance to a subset of metric space For a set $S$ in a metric space $(M,d)$ and $x \in M$, let

$$
d(x,S):=\inf_{s \in S}d(x,s)
$$

Show that the distance for points in the closure of a set is zero:

$$
d(x,S)=0 \Leftrightarrow x \in \bar{S}
$$
Hint: $x \in \bar{S}$ if and only if $\forall \epsilon &gt; 0$, $\exists s \in S$ such that $d(x,s)&lt;\epsilon$.
Hint: $x \in \bar{S}$ if and only if $\forall \epsilon &gt; 0$, $\exists s \in S$ such that $d(x,s)&lt;\epsilon$.