Neighbourhood of Point Contains Point of Subset iff Distance is Zero

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

Let $A \subseteq X$ be a non-empty subset of $X$.

Let $x \in X$.

Then every neighborhood of $x$ contains a point of $A$ :
 * $d \left({x, A}\right) = 0$

where $d \left({x, A}\right)$ denotes the distance from $x$ to $A$.