Triangle Inequality/Examples/4 Points

Theorem

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

Let $x, y, z, t \in A$.

Then:

$\map d {x, z} + \map d {y, t} \ge \size {\map d {x, y} - \map d {z, t} }$

Proof

We have that $\map d {x, z}$, $\map d {y, t}$, $\map d {x, y}$, $\map d {z, t}$ are themselves all real numbers.

Hence the Euclidean metric on the real number line can be applied:

\(\ds \size {\map d {x, y} - \map d {z, t} }\)

\(\le\)

\(\ds \size {\map d {x, y} - \map d {y, z} } + \size {\map d {y, z} - \map d {z, t} }\)

\(\ds \)

\(\le\)

\(\ds \map d {x, z} + \map d {y, t}\)

Reverse Triangle Inequality twice

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 3$