P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 5

Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $f \paren{X} \in \Z [X]$ be the polynomial:
 * $X^2 - a$

for some $a \in \Z$

Let $\sequence {x_n}$ be a sequence of integers such that
 * $\forall n: \map f {x_n} \equiv 0 \pmod {p^n}$

Then:
 * $\displaystyle \lim_{n \to \infty} x_n^2 = a$

Proof
By assumption:
 * $\forall n \in \N: p^n \divides \paren{x_n^2 - a}$

By the definition of the $p$-adic norm:
 * $\forall n \in \N: \norm{x_n^2 - a}_p \le \dfrac 1 {p^n}$

By Sequence of Powers of Number less than One then:
 * $\displaystyle \lim_{n \to \infty} \dfrac 1 {p^n} = 0$

By Inequality Rule for Real Sequences then:
 * $\displaystyle \lim_{n \to \infty} \norm{x_n^2 - a}_p = 0$.

By the definition of convergence in $\struct {\Q, \norm{\,\cdot\,}_p}$ then:
 * $\displaystyle \lim_{n \to \infty} x_n^2 = a$