P-adic Closed Ball is Instance of Closed Ball of a Norm

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 $B \subseteq \Q_p$.

Then:
 * $B$ is a closed ball in $p$-adic numbers with radius $\epsilon$ and centre $a$


 * $B$ is a closed ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$
 * $B$ is a closed ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$

That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a closed ball in a normed division ring.

Proof
By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

The definition of a closed ball in $p$-adic numbers is identical to the definition of a closed ball of a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$.