Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1

Theorem
Let $p$ be a prime number.

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

The set of integers $\Z$ is a subring of $\Z_p$.

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

By Integers form Subring of Valuation Ring of P-adic Norm on Rationals then:
 * $\Z$ is a subring of $\Z_{\paren p}$

By Valuation Ring of P-adic Norm is Subring of P-adic Integers then:
 * $\Z_{\paren p}$ is a subring of $\Z_p$

The result follows.