Suppose that g : R R is such that there exists r (0,1) with the property that |g'(x)| r for every x R. Let : R R be a bounded function and define recursively := , for each n 1. Show that { } converges uniformly on R.
(You might want to prove this using induction.)
Let . For , by the triangle inequality,
(You might also want to prove this by induction.)
Since (and is bounded), , such that implies . So for all , implies
so converges uniformly by the Cauchy criterion.
Note: The above proof I gave is basically the proof of the CMP.