P-adic Integers is Valuation Ring Induced by P-adic Norm

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

Then:
 * the $p$-adic integers, $\Z_p$, is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

Proof
By the definition of the $p$-adic integers:
 * $\Z_p = \set {x \in \Q_p : \norm x_p \le 1}$

From P-adic Numbers form Non-Archimedean Valued Field:
 * $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a non-Archimedean valued field.

By definition of the valuation ring induced by a non-Archimedean norm:
 * $\Z_p$ is the valuation ring induced by $\norm {\,\cdot\,}_p$