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

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$.

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

Let $d$ be non-Archimedean.

Then:

$\norm {\, \cdot \,}$ is a non-Archimedean norm.


Proof

Let $x, y \in R$.

\(\ds \norm {x + y}\) \(=\) \(\ds \norm {x - \paren {-y} }\)
\(\ds \) \(=\) \(\ds \map d {x, - y}\) Definition of Metric Induced by $\norm {\, \cdot \,}$
\(\ds \) \(\le\) \(\ds \max \set {\map d {x, 0}, \map d {0, -y} }\) Definition of Non-Archimedean Metric
\(\ds \) \(=\) \(\ds \max \set {\norm {x - 0 }, \norm {0 - \paren {-y} } }\) Definition of Metric Induced by $\norm {\, \cdot \,}$
\(\ds \) \(=\) \(\ds \max \set {\norm x, \norm y}\)


Hence $\norm {\,\cdot\,}$ is non-Archimedean.

$\blacksquare$


Sources