Division Ring Norm is Continuous on Induced Metric Space

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

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

The mapping $\norm {\,\cdot\,} : \struct {R, d} \to \R$ is continuous.

Proof
Let $x_0 \in R$.

Let $\epsilon \in \R_{>0}$.

Let $x \in R: \norm {x - x_0} < \epsilon$.

Then:

By the definition of metric induced by a norm and the definition of a continuous mapping, $\norm {\,\cdot\,}$ is continuous.