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:
 * $\psi : \struct {R \times R, d_p} \to \struct{R,d} : \map \psi {x,y} = xy$

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 $\epsilon \gt 0$ be given.

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

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

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

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

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

Then:

Hence:

Since $\tuple {x_0, y_0}$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
 * $\psi : \struct {R \times R, d_\infty} \to \struct{R,d} : \map \psi {x,y} = xy$

is continuous.