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.

Printable View

- November 15th 2009, 12:18 PMthaopandaShow 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.

- November 15th 2009, 12:45 PMredsoxfan325
- November 15th 2009, 01:21 PMthaopanda
I don't think I've learned that yet...

- November 15th 2009, 01:59 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.