I'm stuck in the middle of a proof on page 19/20 of Local Fields, by Cassels.
Background: Definition: A valuation is a real-valued function $|.|$ on a field $k$ such that (1)$|b|\geq 0$ with equality only for $b=0$. (2)$|bc|=|b||c|$ for all $b,c\in k$. (3)There is some $C \in \mathbb{R}$ such that $|b|\leq 1 \Rightarrow |1+b| \leq C$.
Theorem: Suppose that $|.|_1$ and $|.|_2$ are two valuations on a field $k$. Suppose that $|.|_1$ is non-trivial and that $ |a|_1<1 \Rightarrow |a|_2 <1. $ I want to prove that these valuations are equivalent.
WHERE I GET STUCK We know that for b,c be non-zero elements of $k$ and $m,n \in \mathbb{Z}$ $ m \log |b|_1+n\log |c|_1 >0 \Rightarrow m \log |b|_2+n\log |c|_2>0$
$m \log |b|_1+n\log |c|_1 =0 \Rightarrow m \log |b|_2+n\log |c|_2=0$
$ m \log |b|_1+n\log |c|_1 <0 \Rightarrow m \log |b|_2+n\log |c|_2<0$
Then apparently given $|c|_1 \not=1$, "it readily follows that
$\log |b|_1=\lambda \log |b|_2$ where
$\lambda=\frac{\log|c|_1}{\log |c|_2}$".
I am not finding this result forthcoming.