Non-Archimedean Norm iff Non-Archimedean Metric

Theorem
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring with zero $0$.

Let $d$ be the metric induced by $\norm {\,\cdot\,}$.

Then:
 * $\norm {\,\cdot\,}$ is a non-Archimedean norm $d$ is a non-Archimedean metric.