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
The result follows from Equality of Sets.

Also see

 * Open Ball in Normed Division Ring is Open Ball in Induced Metric
 * Sphere in Normed Division Ring is Sphere in Induced Metric