Integers are Arbitrarily Close to 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 $x \in \Z_p$.

Then for $n \in \N$ there exists unique $\alpha \in \Z$:
 * $\quad0 \le \alpha \le p^n - 1$
 * $\quad\norm { x -\alpha}_p \le p^{-n}$

Proof
Let $n \in \N$.

By definition of the $p$-adic numbers, the rational numbers are dense in $\Q_p$.

So there exists $\dfrac a b \in \Q: \norm {x - \dfrac a b}_p \le p^{-n} \lt 1$

we can assume that $\dfrac a b$ is in canonical form.

Lemma 2
By Integer is Congruent to Integer less than Modulus, then there exists $\alpha \in \Z$:
 * $\alpha \equiv am \pmod {p^n}$.
 * $0 \le \alpha \le p^n - 1$

Then $\norm {am - \alpha}_p \le p^{-n}$

Hence:

Lemma 3
The result follows.