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
By Valuation Ring of P-adic Norm on Rationals contains Integers then:
 * $\Z$ Is a subring of $\Z_{(p)}$

where $\Z_{(p)}$ is the valuation ring induced by $\norm {\,\cdot\,}_p$ on $\Q$.

By P-adic Integers contains Valuation Ring of P-adic Norm then:
 * $\Z_{(p)} = \Q \cap \Z_p$

The result follows.