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

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.

$\blacksquare$