P-adic Numbers form Non-Archimedean Valued Field

Theorem
Let $p$ be a prime number.

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

Then:
 * $\struct{\Q_p, \norm {\,\cdot\,}_p}$ is a valued field
 * $\norm {\,\cdot\,}_p$ is a non-Archimedean norm

That is, the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a valued field with a non-Archimedean norm.

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

From P-adic Norm is Non-Archimedean Norm:
 * $\struct{Q, \norm {\,\cdot\,}^\Q_p}$ is a valued field with non-Archimedean norm $\norm {\,\cdot\,}_p$

By definition of the $p$-adic numbers:
 * $\Q_p$ is the quotient ring $\CC \, \big / \NN$

where:
 * $\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

and
 * $\NN$ is the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

From Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring:
 * $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field.

From Completion of Normed Division Ring:
 * $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a normed division ring completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$

From Non-Archimedean Division Ring iff Non-Archimedean Completion:
 * $\norm {\, \cdot \,}_p$ on $\Q_p$ is a non-Archimedean norm.