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I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions), and came across the exercise:

Prove that a norm $\|.\|$ on a field $F$ is non-Archimedean if and only if $\{x\in F : \|x\| < 1 \} \cap \{x\in F : \|x-1\| < 1 \} = \emptyset.$

In one direction, the proof was trivial, and in the other direction somewhat harder, but my question is really about where this question comes from. If I were trying to think up exercises on this topic, I don't think I would have thought of this one in a million years.

I am (gradually) getting used to the "eccentricities" of non-Archimedean metrics, but if someone could give me some idea of the intuition that lies behind this particular property, I would be grateful.

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    Perhaps the down-voter would be good enough to give me a clue as to what is wrong with my question?2013-12-05

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The defining property of ultrametrics is that in every triangle two longer sides are equal: more precisely, if ABC is a triangle (=triple of points) and $|AB|\ge |BC|\ge |AC|$ then $|AB|=|BC|$. Now, the exercise asks about the existence of a triangle in which one side has length 1 while the other two are strictly shorter: designed to be a contradiction to the definition.

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    @OldJohn It also matters that the distance between them is 1.2012-07-17