= Bounded comparison forces equality of multiplicative absolute values
{title2=$c|x|_1\le|x|_2\le C|x|_1\Longrightarrow |x|_1=|x|_2$}
If two multiplicative <absolute values on a field> satisfy a comparison with fixed positive constants $c,C$, apply it to $x^n$ and take $n$th roots. Letting $n\to\infty$ eliminates both constants. This stronger bounded comparison differs from equivalence merely as topologies: replacing an absolute value by its square generally preserves topology but not equality.
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