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:

\(\ds \size {\norm x - \norm {x_0} }\)

\(\le\)

\(\ds \norm {x - x_0}\)

Reverse Triangle Inequality on Normed Division Ring

\(\ds \)

\(<\)

\(\ds \epsilon\)

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

$\blacksquare$