Division Ring Norm is Continuous on Induced Metric Space

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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 $\epsilon \in \R_{\gt 0}$.

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

Then:

\(\displaystyle \size {\norm {x} - \norm {x_0} }\) \(\le\) \(\displaystyle \norm {x - x_0}\) Reverse triangle inequality
\(\displaystyle \) \(\lt\) \(\displaystyle \epsilon\)

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

$\blacksquare$