P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2

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:
 * $\norm {x -\alpha_n}_p \le p^{-\paren{n + 1}}$

Then:
 * $\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$

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

From Multiple Rule for Real Sequences:
 * $\displaystyle \lim_{n \mathop \to \infty} p^{-\paren{n + 1}} = 0$

By the Squeeze Theorem for Real Sequences :
 * $\displaystyle \lim_{n \mathop \to \infty} \norm{x - \alpha_n}_p = 0$.

Hence the limit of $\sequence {\alpha_n}$ is by definition:
 * $\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$