Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers

Theorem
Let $p$ be a prime number.

Let $\struct{\Q_p,\norm{\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a \in \Q_p$.

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

Let $\map {{B_\epsilon}^-} a$ denote the $\epsilon$-closed ball of $a$ in $\Q_p$.

Let $\map {B_\epsilon} a$ denote the $\epsilon$-open ball of $a$ in $\Q_p$.

Let $\map {S_\epsilon} a$ denote the $\epsilon$-sphere of $a$ in $\Q_p$.

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

Proof
The result follows directly from:
 * P-adic Closed Ball is Instance of Closed Ball of a Norm
 * P-adic Open Ball is Instance of Open Ball of a Norm
 * P-adic Sphere is Instance of Sphere of a Norm
 * Sphere is Set Difference of Closed and Open Ball in Normed Division Ring