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