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 $\norm{\,\cdot\,}$.

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

Then the mapping:
 * $\theta : \struct {R ,\tau} \to \struct{R,\tau} : \theta \tuple {x} = -x$

is continuous.

Proof
Let $x_0 \in R$.

Let $\epsilon \gt 0$ be given.

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

Then:

By the definition of continuity then the function:
 * $\theta : \struct {R ,d} \to \struct{R,d} : \theta \tuple {x} = -x$

is continuous on metric spaces.

By Continuous Mapping is Continuous on Induced Topological Spaces then the function
 * $\theta : \struct {R ,\tau} \to \struct{R,\tau} : \theta \tuple {x} = -x$

is continuous on topological spaces.