Three Points in Ultrametric Space have Two Equal Distances/Corollary 4
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Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,
Let $x, y \in R$.
Then:
- $\norm {x + y} \lt \norm y \implies \norm x = \norm y$
- $\norm {x - y} \lt \norm y \implies \norm x = \norm y$
- $\norm {y - x} \lt \norm y \implies \norm x = \norm y$
Proof
The contrapositive statements are proved.
Let $\norm x \ne \norm y$
By Corollary 2 then:
- $\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} \ge \norm y$
The result follows.
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed Fields: Proposition $1.15$