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 $(\text M 2)$: Triangle Inequality

\(\ds \leadsto \ \ \)

\(\ds \forall z \in H: \, \)

\(\ds \map d {y, z}\)

\(\ge\)

\(\ds \map d {x, H} - \map d {x, y}\)

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

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Exercise $6$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: Exercise $4.3: 5 \ \text {(b)}$