Valuation Ring of P-adic Norm is Subring of P-adic Integers

Theorem
Let $p$ be a prime number.

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

Let $\Z_p$ be the $p$-adic integers.

Let $\Z_{(p)}$ be the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$.

Then:
 * $\quad \Z_{(p)} = \Q \cap \Z_p$.
 * $\quad \Z_{(p)}$ is a subring of $\Z_p$.

Proof
The $p$-adic integers is defined as:
 * $\Z_p = \set {x \in \Q_p: \norm {x}_p \le 1}$

The induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$ is defined as:
 * $\Z_{(p)} = \set {x \in \Q: \norm {x}_p \le 1}$

By definition, the $p$-adic norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is an extension of the $p$-adic norm $\norm {\,\cdot\,}_p$ on $\Q$.

It follows that $\Z_{(p)} = \Q \cap \Z_p$.

This proves 1. above.

By Valuation Ring of Non-Archimedean Division Ring is Subring then $\Z_p$ is a subring of $\Q_p$.

By definition of $p$-adic integers then $\Q$ is a subring of $\Q_p$.

By Intersection of Subrings is Largest Subring Contained in all Subrings then $\Z_{(p)}$ is a subring of $\Q_p$ that is contained in $\Z_p$.

By Subset is Subring of Subring Iff it is Subring then $\Z_{(p)}$ is a subring of $\Z_p$.

This proves 2. above.