= Uniform contraction from a comparison function on a bounded set
{title2=$d(Tx,Ty)\le\psi(d(x,y))\le qd(x,y)$}
Suppose all distances lie in $[0,D]$, $\psi$ is <continuous>, $0<\psi(r)<r$ for positive $r$, and $\lim_{r\to0}\psi(r)/r<1$. The ratio is uniformly <bounded> below one near zero, and its maximum on the remaining compact <scalar> <interval> is also below one. Thus one constant $q<1$ works on the entire <bounded set>. If the <set> is complete and the map preserves it, <Banach fixed-point theorem>[Banach's theorem] gives a unique <fixed point>. <Compactness> of the <set> itself is not required.
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