Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 9 / 1 / h / Solution

False. On the additive topological group $\mathbb{R}$, put $d(x,y)=|x-y|$ and $d^{\prime }(x,y)=\sqrt{|x-y|}$. Both are invariant under left and right translations and induce the usual topology. The triangle inequality for $d^{\prime }$ follows from $\sqrt{a+b}\le \sqrt{a}+\sqrt{b}$ for $a,b\ge 0$. Butcan be arbitrarily close to zero and arbitrarily large. Consequently no $K>0$ can satisfy both proposed inequalities. Compatibility of the two invariant metrics is topological equivalence, which need not be bilipschitz equivalence.