Chebyshev Distance on Real Vector Space is Metric/Proof 1

Theorem
The maximum metric on $\R^n$:


 * $\displaystyle \forall x, y \in \R^n: d_\infty \left({x, y}\right):= \max_{i \mathop = 1}^n {\left\vert{x_i - y_i}\right\vert}$

is a metric.

Proof
This is an instance of the maximum metric on the general cartesian product of metric spaces $A_1, A_2, \ldots, A_3$.

This is proved in Maximum Metric is Metric.