Valuation Ring of Non-Archimedean Division Ring is Clopen/Corollary 1

Theorem

Let $p$ be a prime number.

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

Then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric.

Proof

The $p$-adic integers $\Z_p$ is the valuation ring induced by $\norm {\,\cdot\,}_p$ by definition.

By Valuation Ring of Non-Archimedean Division Ring is Clopen then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric.

$\blacksquare$