Open and Closed Balls in P-adic Numbers are Compact Subspaces

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 $n \in \Z$.

Then the open ball $\map {B_{p^{-n}}} a$ and closed ball $\map {B^-_{p^{-n}}} a$ are compact.

Proof
We begin by proving the theorem for the closed ball $\map {B^-_{p^{-n}}} a$.

From Open Ball in P-adic Numbers is Closed Ball then the theorem will be proved.

From Leigh.Samphier/Sandbox/Open and Closed Balls in P-adic Numbers are Totally Bounded, the closed ball $\map {B^-_{p^{-n}}} a$ is totally bounded.

Recall that by definition, the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ are complete.

From Closed Ball in Normed Division Ring is Closed Ball in Induced Metric, the closed ball $\map {B^-_{p^{-n}}} a$ is closed.

From Subspace of Complete Metric Space is Closed iff Complete, the closed ball $\map {B^-_{p^{-n}}} a$ is complete.

From Complete and Totally Bounded Metric Space is Sequentially Compact, the closed ball $\map {B^-_{p^{-n}}} a$ is sequentially compact.

From Sequentially Compact Metric Space is Compact, the closed ball $\map {B^-_{p^{-n}}} a$ is compact.

The result follows.