1. ## Help with proof about uniform contraction

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*.

2. Originally Posted by paupsers
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 $f$ such that $d(f(x),f(y))\leq kd(x,y)$ for all $x,y$ and some $k<1$.

For uniqueness, assume (for a contradiction) $f(x^*)=x^*$ and $f(y^*)=y^*$. What can you say about $d(x^*,y^*)$?

As for existence, define a sequence $\{x_n\}$ such that $x_{n+1}=f(x_n)$. Try to find a formula in terms of $k$ and $d(x_1,x_0)$ for $d(x_n,x_m)$ and use this to show that $\{x_n\}$ is Cauchy (and therefore converges to some point $x^*$). Then use the continuity of $f$ to show that $f(x^*)=x^*$.

What you are proving is known as the Contraction Mapping Principle or alternatively, the Banach Fixed Point Theorem.