I'm studying p-adic numbers in "A course in Arithmetic" by Serre. I need some help to understand a proposition about solutions of polynomials with p-adic coefficients. I've seen that there is already a question about this proposition here:
II. p-adic equations [2.1. Solutions] (J.-P. Serre)
I'm interested in the first proposition after the lemma:
Proposition: Let $f^{(i)} \in \mathbb{Z}_p[X1,…,Xm]$ be polynomials with p-adic integer coefficients. The following are equivalent:
i) The $f^{(i)}$ have a common zero in $(\mathbb{Z}_p)^m$.
ii) For all $n>1$ the polynomials $f^{(i)}_n$ have a common zero in $(A_n)^m$.
Here $A_n=\mathbb{Z}/{p^n\mathbb{Z}}$ and $\mathbb{Z}_p=\lim A_n$.
$f_n$ is the polynomial $f$ reduced mod $p^n$.
In the proof Serre says that, if $D$ is the set of the common zeros in $(\mathbb{Z}_p)^m$ and $D_n$ of common zeros in $(A_n)^m$, then $D=\lim D_n$. Then he applies a lemma to say that $D$ is non empty if and only if the $D_n$ are non empty. I have understand the lemma but i don't understand why $D=\lim D_n$ and how to prove it. I'm not sure about reducing mod $p^n$ a polynomial. Is it equivalent to take only the $n$ components of coefficients?
Thanks to anyone who will help me.