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.