Sphere is Set Difference of Closed Ball with 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 $\map {{B_\epsilon}^-} a$ denote the $\epsilon$-closed ball of $a$ in $\struct{R, \norm{\,\cdot\,}}$.

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

Let $\map {S_\epsilon} a$ denote the $\epsilon$-sphere of $a$ in $\struct{R, \norm{\,\cdot\,}}$.

Then:
 * $\map {S_\epsilon} a = \map {{B_\epsilon}^-} a \setminus \map {B_\epsilon} a$

Proof
The result follows directly from:
 * * Closed Ball in Normed Division Ring is Closed Ball in Induced Metric
 * * Open Ball in Normed Division Ring is Open Ball in Induced Metric
 * * Sphere in Normed Division Ring is Sphere in Induced Metric
 * * Leigh.Samphier/Sandbox/Sphere is Set Difference of Closed Ball with Open Ball