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

Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p \ge 3$.

Then:


 * $\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete normed division ring.

That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.

Proof
Hensel's Lemma is used to prove the existence of a Cauchy sequence that does not converge.

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

The formal derivative $f' \paren{X} \in \Z [X]$ is by definition:
 * $2X$

Let $x_0 = \dfrac {p + 1} 2$

Since $p$ is odd then $p + 1$ is even and $x_0 \in \N$

Then:

and

Hence:
 * $0 \lt x_0 \lt p$

By Corollary to Absolute Value of Integer is not less than Divisors then:
 * $p \nmid x_0$
 * $p \nmid 2$

By Euclid's Lemma for Prime Divisors then:
 * $p \nmid 2x_0 = \map {f'} {x_0}$