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

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

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

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

For all $x,y \in \Z_p$, let:
 * $x \equiv y \pmod {p\Z_p}$

denote congruence modulo the ideal $p\Z_p$.

For all $k \in \N$, 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
From User:Leigh.Samphier/Sandbox/Ideals of P-adic Integers:
 * $p\Z_p$ is an ideal of the ring $\Z_p$

By the definition of congruence modulo an ideal:
 * $\forall k \in \N: a_k - \alpha_0 \in p\Z_p$

By definition of $p$-adic expansion:
 * $\alpha = \ds \lim_{k \mathop \to \infty} a_k$

From Sum Rule for Sequences in Normed Division Ring:
 * $\alpha - \alpha_0 = \ds \lim_{k \mathop \to \infty} a_k - \alpha_0$

From User:Leigh.Samphier/Sandbox/Closed Subgroups of P-adic Integers:
 * $p\Z_p$ is a closed set in the $p$-adic metric

From Subset of Metric Space contains Limits of Sequences iff Closed:
 * $\alpha - \alpha_0 \in p\Z_p$

By the definition of congruence modulo an ideal:
 * $\alpha \equiv \alpha_0 \pmod {p\Z_p}$