Triangle Inequality on Distance from Point to Subset

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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.

$\blacksquare$


Sources