# Closed Ball in Normed Division Ring is Closed Ball in Induced Metric

## Theorem

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

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

Let $a \in R$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the closed ball in the normed division ring $\struct {R, \norm {\,\cdot\,} }$.

Let $\map {{B_\epsilon}^-} {a; d }$ denote the closed ball in the metric space $\struct {R, d}$.

Then:

$\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ = $\map {{B_\epsilon}^-} {a; d }$

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {x - a}$ $\le$ $\displaystyle \epsilon$ Definition of Closed Ball of Normed Division Ring $\displaystyle \leadstoandfrom \ \$ $\displaystyle \map d {x, a}$ $\le$ $\displaystyle \epsilon$ Definition of Metric Induced by Norm on Division Ring $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle \map { {B_\epsilon}^-} {a; d }$ Definition of Closed Ball

The result follows from Equality of Sets.

$\blacksquare$