I was reading the proof of Ostrowski's theorem (which BTW is a beauty) and I got stumped here:
[O]ne has $|nr|_∗ \leq n|r|_∗$. [C]hoosing $r=n^{-1}$ shows that for all positive integer $n$, it holds $n^{-1}\leq |n|_{*}$ [...].
How exactly does setting $r$ to $n^{-1}$ leads to $n^{-1} \leq |n|_*$? If we can't obtain the result this way, how can we?
Note that $|\cdot|_*$ is not necessarily the usual real absolute value ($|x| = x$ if $x \geq 0$, else $-x$), but any absolute value on $\mathbb{Q}$.
In fact, the statement is not true. For non-archimedean valuations $| \cdot |$, the function $| \cdot |^\lambda$ is still a valuation, for any $\lambda >0$. That is, e.g. $|x|_\ast := |x|_p^2$ is a counterexample with $|p|_\ast = p^{-2} < p^{-1}$.
Accordingly, the definition of equivalence of valuations in that WP article is imprecise (for starters, it obviously does not define an equivalence relation: One has to allow $0<\lambda<\infty$.)
For archimedean values one has to be careful as soon as $\lambda >1$ (in general, too big $\lambda$ will make $|\cdot |^\lambda$ not be a valuation anymore), but in that case, the inequality in question is not needed, but rather comes out of the proof automatically. (All archimedan valuations on $\mathbb Q$ are of the form $|\cdot|_\infty^\lambda$ for some $0<\lambda\le 1$, and all equivalent to each other.)
UPDATE: I have corrected the Wikipedia article, in particular the definition of equivalence in there, and deleted the erroneous claim as well as its supposed usage in the proof. Note that the redundant wrong claim had only been introduced into the WP article less than a month ago, in this edit.