P-adic Metric on P-adic Numbers is Non-Archimedean Metric/Corollary 1

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Theorem

Let $p$ be a prime number.

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


Then:

$\forall x, y, z \in R: \norm{x - y}_p \le \max \set{\norm{x - z}_p, \norm{y - z}_p}$

Proof

Let $d_p$ be the $p$-adic metric on $\Q_p$:

$\forall x, y \in \Q_p: \map {d_p} {x, y} = \norm{x - y}_p$

From P-adic Metric on P-adic Numbers is Non-Archimedean Metric, $d_p$ is a non-Archimedean norm.

By definition of a non-Archimedean norm:

$\forall x, y, z \in R: \norm{x - y}_p \le \max \set{\norm{x - z}_p, \norm{y - z}_p}$

$\blacksquare$