Normed Division Ring Operations are Continuous/Addition

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:
 * $\phi : \struct {R \times R, d_p} \to \struct{R,d} : \phi \tuple {x,y} = x + y$

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 $\tuple {x,y} \in R \times R$ such that:
 * $d_\infty \tuple {\tuple {x,y},\tuple{x_0,y_0}} \lt \dfrac \epsilon 2$

By the definition of the product metric $d_\infty$ then:
 * $\max \set { d \tuple {x, x_0}, d \tuple {y, y_0}} \lt \dfrac \epsilon 2$

or equivalently:
 * $d \tuple {x, x_0} \lt \dfrac \epsilon 2$
 * $d \tuple {y, y_0} \lt \dfrac \epsilon 2$

Then:

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

is continuous.