Three Points in Ultrametric Space have Two Equal Distances/Corollary 3

Theorem

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

Let $x, y \in R$ and $\norm x \lt \norm y$.

Then:

$\norm {x + y} = \norm {x - y} = \norm {y - x} = \norm y$

Proof

By Corollary 2 then:

$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} = \norm y$

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed Fields: Remark $1.16$