# 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

$\blacksquare$