Integers are Dense in 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 $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$ restricted to the $p$-adic integers.

The integers $\Z$ are dense in the metric space $\struct{\Z_p, d_p}$.

Proof
By Open Ball Characterization of Denseness it is sufficient to show that every open ball of $\struct {\Z_p, d_p}$ contains an element of $\Z$.

Let $x \in \Z_p$ and $\epsilon \in \R_{\gt 0}$.

By definition the open ball $\map {B_\epsilon} x$ is:
 * $\map {B_\epsilon} x = \set {y \in \Z_p: \norm{y}_p \lt \epsilon }$

By Sequence of Powers of Number less than One then:
 * $\displaystyle \lim_{n \to \infty} p^{-n} = 0$

Hence there exists $N \in \N$:
 * $\forall n \ge N: p^{-n} \lt \epsilon$

Consider the open ball $\map {B_{p^{-N}}} x$.

Since $0 \lt p^{-n} \lt \epsilon$ then:
 * $\map {B_{p^{-N}}} x \subseteq \map {B_\epsilon} x$.

By Integers are Arbitrarily Close to P-adic Integers then:
 * $\exists \alpha \in \Z: \alpha \in \map {B_{p^{-N}}} x$

Hence $\alpha \in \map {B_\epsilon} x$.

Since $x$ and $\epsilon$ were arbitrary then every open ball of $\struct {\Z_p, d_p}$ contains an element of $\Z$.

By Open Ball Characterization of Denseness then $\Z$ is dense in the metric space $\struct{\Z_p, d_p}$.