Triangle Inequality/Examples/4 Points

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

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


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


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