Normed Division Ring Operations are Continuous/Multiplication

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

is continuous.

Proof
Let $x_0, y_0 \in R$.

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 $x,y \in R$ such that:
 * $d \tuple {x,x_0} \lt \delta$
 * $d \tuple {y,y_0} \lt \delta$

Then:

Hence:

By the definition of continuity then the mapping:
 * $\psi : \struct {R \times R, d_\infty} \to \struct{R,d} : \psi \tuple {x,y} = xy$

is continuous.