# Center is Element of Closed Ball/Normed Division Ring

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## 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 closed $\epsilon$-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 Closed Ball in Normed Division Ring is Closed Ball in Induced Metric, $\map { {B_\epsilon}^-} a$ is the closed $\epsilon$-ball of $a$ in the metric space $\struct{R,d}$.

From Center is Element of Closed Ball:

- $a \in \map { {B_\epsilon}^-} a$

$\blacksquare$