Characterization of Polynomial has Root in P-adic Integers/Necessary 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 $\map F a = 0$.

Then:
 * there exists 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 $p^{n + 1} \Z_p$

Proof
Let $\map F a = 0$.

Let $a = \ds \sum_{j \mathop = 0}^\infty d_j p^j$ be the $p$-adic expansion of $a$.

For all $n \in \N_{>0}$, let:
 * $a_n = \ds \sum_{j \mathop = 0}^{n - 1} d_j p^j$

By definition of $p$-adic expansion:
 * $\ds \lim_{n \mathop \to \infty} {a_n} = a$

By definition of $p$-adic expansion of a $p$-adic integer:
 * $\forall n \in \N_{>0} : a_n \in \Z$

We have: