Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring.

Let $d$ be the metric induced by $\norm {\,\cdot\,}$.


Then $d$ is a non-Archimedean metric.


Proof

By Metric Induced by Norm on Normed Division Ring is Metric then $d$ satisfies the metric space axioms $(\text M 1)$ to $(\text M 4)$.

To complete the proof, all that remains is to show that $d$ is non-Archimedean.


Let $x, y, z \in R$.

\(\ds \map d {x, y}\) \(=\) \(\ds \norm {x - y}\) Definition of Metric Induced by $\norm {\,\cdot\,}$
\(\ds \) \(=\) \(\ds \norm {\paren {x - z} + \paren {z - y} }\)
\(\ds \) \(\le\) \(\ds \max \set {\norm {x - z}, \norm {z - y} }\) Definition of Non-Archimedean Division Ring Norm
\(\ds \) \(=\) \(\ds \max \set {\map d {x, z}, \map d {z, y} }\) Definition of Metric Induced by $\norm {\,\cdot\,}$


$\blacksquare$


Sources