Center is Element of Closed 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 $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

Then:

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

Proof

By definition, $\map { {B_\epsilon}^-} a$ is the closed $\epsilon$-ball of $a$ in the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

From Center is Element of Closed Ball in Normed Division Ring: $a \in \map { {B_\epsilon}^-} a$

$\blacksquare$