Normed Division Ring Operations are Continuous/Multiplication

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

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

Let $p \in \R_{\ge 1} \cup \set \infty$.

Let $d_p$ be the $p$-Product Metric on $R \times R$.

Then the mapping:
 * $* : \struct {R \times R, d_p} \to \struct {R, d}$

is continuous.

Proof
By $p$-Product Metric Induces Product Topology and Continuous Mapping is Continuous on Induced Topological Spaces, it suffices to consider the case $p = \infty$.

Let $\tuple {x_0, y_0} \in R \times R$.

Let $\epsilon > 0$ be given.

Let $\delta = \min \set {\dfrac \epsilon {1 + \norm {y_0} + \norm {x_0} }, 1}$

Since $1 + \norm {y_0} + \norm {x_0} > 0$ then $\delta > 0$

Let $\tuple {x,y} \in R \times R$ such that:
 * $\map {d_\infty} {\tuple {x, y}, \tuple{x_0, y_0} } < \delta$

By the definition of the $p$-product metric $d_\infty$:
 * $\max \set {\map d {x, x_0}, \map d {y, y_0}} < \delta$

or equivalently:
 * $\map d {x, x_0} < \delta$
 * $\map d {y, y_0} < \delta$

Then:

Hence:

We have that $\tuple {x_0, y_0}$ and $\epsilon$ are arbitrary.

Hence, by the definition of continuity, the mapping:
 * $* : \struct {R \times R, d_\infty} \to \struct {R, d}$

is continuous.