P-adic Numbers form Completion of Rational Numbers with P-adic Norm

Theorem
Let $p$ be a prime number.

Let $\norm {\,\cdot\,}^\Q_p$ be the p-adic norm on the rationals $\Q$.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Then $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$

Proof

 * Completion of Normed Division Ring for a proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.


 * Completion Theorem for a proof that the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$ exists and is unique up to isometric isomorphism.