Bounded-integer criterion for a non-Archimedean absolute value
= Bounded-integer criterion for a non-Archimedean absolute value
An absolute value on a field is non-Archimedean exactly when its values on the image of $\mathbb Z$ are bounded. The reverse implication follows by applying the ordinary triangle inequality to the binomial expansion of $(x+y)^n$ and taking $n$th roots as $n\to\infty$.