Characterization of Polynomial has Root in P-adic Integers/Sufficient Condition

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 over $\Z_p$.

Let $a \in \Z_p$.

Let there exist a sequence $\sequence{a_n}$ of integers:
 * $(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
 * $(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$

where $\map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$ denotes congruence modulo the ideal $p^{n + 1} \Z_p$.

Then:
 * $\map F a = 0$

Proof
Let there exist a sequence $\sequence{a_n}$ of integers:
 * $(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$
 * $(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$

We have: