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

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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 $\struct {R, d}$.


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


\(\displaystyle x\) \(\in\) \(\displaystyle \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \norm {x - a}\) \(\le\) \(\displaystyle \epsilon\) 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.


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