P-adic Integer is Limit of Unique Coherent Sequence of 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 there exists a Cauchy sequence $\sequence {\alpha_n}$:
 * $\forall n \in \N: \alpha_n \in \Z$ and $0 \le \alpha_n \le p^n-1$
 * $\forall n \in \N: \alpha_n \equiv \alpha_{n-1} \pmod {p^{n-1}}$
 * $\displaystyle \lim_{n \to \infty} \alpha_n = x$