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

Then:
 * $(1): \quad \Z_{\paren p} = \Q \cap \Z_p$.
 * $(2): \quad \Z_{\paren p}$ 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:
 * $\Z_{\paren p} = \set {x \in \Q: \norm x_p \le 1}$

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 $\Z_{\paren p} = \Q \cap \Z_p$.

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 $\Z_{\paren p}$ is a subring of $\Z_p$.

This proves $(2)$ above.