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 $\tau$ be the topology induced by the metric $d$.

Let $\tau \times \tau$ be the product topology on $R \times R$.

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
.