= Contraction property under equivalent metrics
The same self-map can be a <contraction mapping> for one metric but not for a <bilipschitz equivalent> metric. On three points with distances $d'(x,y)=d'(x,z)=2$, $d'(y,z)=1$, the map $f(x)=y$, $f(y)=f(z)=z$ has Lipschitz constant $1/2$. For the <discrete metric>, its Lipschitz constant is one. The metrics satisfy $d\le d'\le2d$, so they preserve topology and completeness but not the strict numerical contraction condition. A <contraction mapping theorem> fixed-point proof therefore needs the specified metric and its contraction constant.
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