Normed Division Ring Operations are Continuous/Inversion

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

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

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

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

Then the mapping:
 * $\xi : \struct {R^* ,d^*} \to \struct{R,d} : \xi \tuple {x} = x^{-1}$

is continuous.

Proof
Let $x_0, y_0 \in R$.

Let $\epsilon \gt 0$ be given.

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

Then:

By the definition of continuity then the mapping:
 * $\xi : \struct {R ,d} \to \struct{R,d} : \xi \tuple {x} = x^{-1}$

is continuous.