P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 3

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 such that:
 * $(1): \quad \forall n \in \N: \alpha_n \in \Z$ and $0 \le \alpha_n \le p^n - 1$
 * $(2): \quad \forall n \in \N: \alpha_n \equiv \alpha_{n - 1} \pmod {p^{n - 1} }$
 * $(3): \quad \displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$

$\sequence {\alpha_n}$ is a unique sequence satisfying properties $(1)$, $(2)$ and $(3)$ above.

Proof
Suppose that there exists a sequence $\sequence {\alpha'_n}$ with:
 * $(1'): \quad \forall n \in \N: \alpha'_n \in \Z$ and $0 \le \alpha'_n \le p^n - 1$
 * $(2'): \quad \forall n \in \N: \alpha'_n \equiv \alpha'_{n - 1} \pmod {p^{n - 1} }$
 * $(3'): \quad \displaystyle \lim_{n \mathop \to \infty} \alpha'_n = x$


 * $\alpha'_N \ne \alpha_N$ for some $N \in \N$
 * $\alpha'_N \ne \alpha_N$ for some $N \in \N$

By Initial Segment of Natural Numbers forms Complete Residue System:
 * $\alpha'_N \not \equiv \alpha_N \pmod {p^N}$

Then for all $n > N$:

That is, for all $n > N$:
 * $\alpha'_n \not \equiv \alpha_n \pmod {p^N}$

Hence for all $n > N$:
 * $\norm {\alpha'_n - \alpha_n}_p > p^{-N}$

By $(3)$ the limit of $\sequence{\alpha_n}$ is $x$:
 * $\exists N_1 \in \N: \forall n \ge N_1: \norm {x - \alpha_n}_p \le p^{-N}$

Similarly for $\sequence{\alpha'_n}$:
 * $\exists N_2 \in \N: \forall n \ge N_2: \norm {x - \alpha'_n}_p \le p^{-N}$

Let $M = \max \set {N+1, N_1, N_2}$.

Then:
 * $\norm {\alpha'_M - \alpha_M}_p > p^{-N}$
 * $\norm {x - \alpha_M} _p\le p^{-N}$
 * $\norm {x - \alpha'_M}_p \le p^{-N}$

But:

This contradicts the previous assertion that:
 * $\norm {\alpha'_M - \alpha_M}_p > p^{-N}$

Hence:
 * $\sequence{\alpha'_n} = \sequence{\alpha_n}$.

The result follows.