P-adic Norm satisfies Non-Archimedean Norm Axioms

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \to \R_{\ge 0}$.

Then $\norm {\,\cdot\,}_p$ satisfies the non-Archimedean norm axioms:

Proof
By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$.

By definition of a non-Archimedean valued field, $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

By definition of a non-Archimedean norm, $\norm {\,\cdot\,}_p$ satisfies the axioms: $\text N 1$, $\text N 2$ and $\text N 4$.