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

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

Let $\map F X \in \Z_p \sqbrk X$ be a polynomial.

Let $\alpha_0 \in \Z_p$ be a $p$-adic integer:
 * $\map F {\alpha_0} \equiv 0 \pmod {p\Z_p}$

Let there exist a $p$-adic expansion $\ds \sum_{n = 0}^\infty d_n p^n$:
 * $\forall k : a_k = \ds \sum_{n = 0}^k d_n p_n$ satisfies:
 * $(1) \quad \map F {a_k} \equiv 0 \pmod {p^{k+1}\Z_p}$
 * $(2) \quad a_k \equiv \alpha_0 \pmod {p\Z_p}$

Let:
 * $\alpha = \ds \sum_{n = 0}^\infty d_n p^n$

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