Normed Division Ring Operations are Continuous

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 $R \times R$.

Let $R^* = R \setminus \set{0}$

Let $d^*$ be the restriction of $d$ to $R^*$.

Then the following results hold:

Addition is Continuous
The mapping:

Negation is Continuous
The mapping:

Multiplication is Continuous
The mapping:

Inversion is Continuous
The mapping:

Corollary
.