Center is Element of Open Ball/Normed Division Ring

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

Let $a \in R$.

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

Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$.

Then:
 * $a \in \map {B_\epsilon} a$

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

From Open Ball in Normed Division Ring is Open Ball in Induced Metric, $\map {B_\epsilon} a$ is the open $\epsilon$-ball of $a$ in the metric space $\struct{R,d}$.

From Center is Element of Open Ball:
 * $a \in \map {B_\epsilon} a$