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:

\((\text N 1)\)	$:$		Positive Definiteness:	\(\ds \forall x \in \Q_p:\)		\(\ds \norm x_p = 0 \)	\(\ds \iff \)	\(\ds x = 0 \)
\((\text N 2)\)	$:$		Multiplicativity:	\(\ds \forall x, y \in \Q_p:\)		\(\ds \norm {x \cdot y}_p \)	\(\ds = \)	\(\ds \norm x_p \times \norm y_p \)
\((\text N 4)\)	$:$		Ultrametric Inequality:	\(\ds \forall x, y \in Q_p:\)		\(\ds \norm {x + y}_p \)	\(\ds \le \)	\(\ds \max \set {\norm x_p, \norm y_p} \)

Proof

From P-adic Numbers form Non-Archimedean Valued Field:

$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a non-Archimedean norm

$\blacksquare$