Chebyshev Distance on Real Vector Space is Metric

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.