P-adic Norm not Complete on Rational Numbers

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

Then:


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

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