Anyone have a proof for this? Or a starting point?
Prove that for every uniform contraction f there exists a unique x* in R such that f(x*)=x*.
A contraction is a function such that for all and some .
For uniqueness, assume (for a contradiction) and . What can you say about ?
As for existence, define a sequence such that . Try to find a formula in terms of and for and use this to show that is Cauchy (and therefore converges to some point ). Then use the continuity of to show that .
What you are proving is known as the Contraction Mapping Principle or alternatively, the Banach Fixed Point Theorem.