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.

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- Nov 15th 2009, 11:18 AMthaopandaShow it converges uniformly
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.

- Nov 15th 2009, 11:45 AMredsoxfan325
- Nov 15th 2009, 12:21 PMthaopanda
I don't think I've learned that yet...

- Nov 15th 2009, 12:59 PMredsoxfan325
(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.