Normed Division Ring Operations are Continuous/Negation

Theorem
Let $\struct {R, +, *, \norm {\,\cdot\,} }$ be a normed division ring.

Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.

Then the mapping:
 * $\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$

is continuous.

Proof
Let $x_0 \in R$.

Let $\epsilon > 0$ be given.

Let $x \in R$ such that:
 * $\map d {x, x_0} < \epsilon$

Then:

Since $x_0$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
 * $\eta: \struct {R, d} \to \struct {R, d} : \map \eta x = -x$

is continuous.