P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1

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 $x \in \Z_p$.

Let $\sequence{\alpha_n}$ be a sequence of integers:
 * $(1): \quad 0 \le \alpha_n \le p^n - 1$
 * $(2): \quad \norm {x -\alpha_n}_p \le p^{-n}$

Then:
 * $\forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^n}$

Proof
For any $n \in \N$ then:

Hence:
 * $p^n \divides \paren {\alpha_{n + 1} - \alpha_n} $

or equivalently:
 * $\alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1} }$

The result follows.