Hensel's Lemma/P-adic Integers/Lemma 2

Theorem
Let $\Q_p$ be the $p$-adic numberss for some prime $p$.

Let $\alpha \in \Q_p$ be a $p$-adic number with $p$-adic expansion:
 * $\alpha = \ds \sum_{n = 0}^\infty d_n p^n$

Let $\Z_p$ be the $p$-adic integers.

Let $\alpha_0 \in \Z_p$ be a $p$-adic integer.

For all k, let the partial sum $a_k = \ds \sum_{n = 0}^k d_n p_n$ satisfy:
 * $a_k \equiv \alpha_0 \pmod {p\Z_p}$

Then:
 * $\alpha \equiv \alpha_0 \pmod {p\Z_p}$

Proof
Definition:Congruence Modulo an Ideal

P-adic Metric on P-adic Numbers is Non-Archimedean Metric

Subset of Metric Space contains Limits of Sequences iff Closed