Bounded comparison forces equality of multiplicative absolute values ($c|x{|}_1\le |x{|}_2\le C|x{|}_1⟹|x{|}_1=|x{|}_2$)

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.