# 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

 $\ds \forall z \in H: \,$ $\ds \map d {y, z}$ $\ge$ $\ds \map d {x, z} - \map d {x, y}$ Metric Space Axiom $\text M 2$ $\ds \leadsto \ \$ $\ds \forall z \in H: \,$ $\ds \map d {y, z}$ $\ge$ $\ds \map d {x, H} - \map d {x, y}$ Definition of Distance between Element and Subset of Metric Space $\ds \leadsto \ \$ $\ds \map d {y, H}$ $\ge$ $\ds \map d {x, H} - \map d {x, y}$ Definition of Distance between Element and Subset of Metric Space

Hence the result.

$\blacksquare$