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

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## 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

\(\displaystyle x \in \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }\) | \(\leadstoandfrom\) | \(\displaystyle \norm {x - a} \le \epsilon\) | Definition of closed ball in $\struct {R, \norm {\,\cdot\,} }$ | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle \map d {x, a} \le \epsilon\) | Definition of metric induced by $\norm {\,\cdot\,}$ | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle x \in \map { {B_\epsilon}^-} {a; d }\) | Definition of closed ball in $\struct {R, d}$ |

The result follows from Equality of Sets.

$\blacksquare$