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:

$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$

Let $x_0 \in R$.

Let $\epsilon > 0$ be given.

Let $x \in R$ such that:

$\map d {x, x_0} < \epsilon$

Then:

$\ds \map d {-x, -x_0}$

$=$

$\ds \norm {-x - \paren {-x_0} }$

$\ds $

$=$

$\ds \norm {-x + x_0}$

$\ds $

$=$

$\ds \norm {x_0 - x}$

$\ds $

$=$

$\ds \map d {x_0, x}$

$\ds $

$=$

$\ds \map d {x, x_0}$

$\ds $

$<$

$\ds \epsilon$

Since $x_0$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:

$\eta: \struct {R, d} \to \struct {R, d} : \map \eta x = -x$

$\blacksquare$