Triangle Inequality on Distance from Point to Subset

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

Let $H \subseteq A$.

Then:
 * $\forall x, y \in A: \map d {x, H} \le \map d {x, y} + \map d {y, H}$

where $\map d {x, H}$ denotes the distance between $x$ and $H$.

Proof
Hence the result.