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 subspace metric on $R^*$.

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

is continuous.

Proof
Let $x_0 \in R^*$.

Let $\epsilon \gt 0$ be given.

Let $\delta = \min \set {\dfrac {\norm {x_0}} 2, \dfrac {\norm {x_0}^2 \epsilon} 2 }$

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

By the definition of the subspace metric on $R^*$ and the definition of the metric induced by the norm on $R$ then:
 * $d^* \tuple {x,x_0} = d \tuple {x,x_0} = \norm {x - x_0} \lt \delta$

Then:

Hence:

Since $x_0$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
 * $\xi : \struct {R^* ,d^*} \to \struct{R,d} : \xi \tuple {x} = x^{-1}$

is continuous.