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.
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.