just write |f(f(x))-f(x)| < k|f(x)-x|, |f(f(f(x)))-f(f(x))| < k^2 |f(x)-x| and so on...
Let be a function defined on all of , and assume there is a constant such that and for all .
a) Show is continuous on .
Let be a real number and . Then by definition. So, let . It follows that implies .
b) Pick some point and construct the sequence
.
In general, if , show that the resulting sequence is a Cauchy sequence. Hence, we may let .
I am not sure how to approach this one. Some help getting started would be appreciated. It seems almost trivial, but I am not sure how formalize it.