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:
 * $\Q_p$ 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

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


 * Corollary to Cauchy Sequences form Ring with Unity for a proof that $\CC$ is a commutative ring of Cauchy sequences.


 * Corollary to Null Sequences form Maximal Left and Right Ideal for a proof that $\NN$ is a maximal ideal of $\CC$.


 * Corollary to Quotient Ring of Cauchy Sequences is Division Ring for a proof that the quotient ring $\Q_p = \CC \big / \NN$ is a field.


 * Quotient Ring of Cauchy Sequences is Normed Division Ring for a proof that $\norm {\, \cdot \,}_p$ on $\Q_p$ is a norm.


 * Non-Archimedean Division Ring Iff Non-Archimedean Completion for a proof that $\norm {\, \cdot \,}_p$ on $\Q_p$ is a non-Archimedean norm.


 * Normed Division Ring is Field iff Completion is Field for a proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a valued field.


 * P-adic Norm satisfies Non-Archimedean Norm Axioms for as proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a valued field with a non-Archimedean norm.