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 $\displaystyle f$ such that $\displaystyle d(f(x),f(y))\leq kd(x,y)$ for all $\displaystyle x,y$ and some $\displaystyle k<1$.
For uniqueness, assume (for a contradiction) $\displaystyle f(x^*)=x^*$ and $\displaystyle f(y^*)=y^*$. What can you say about $\displaystyle d(x^*,y^*)$?
As for existence, define a sequence $\displaystyle \{x_n\}$ such that $\displaystyle x_{n+1}=f(x_n)$. Try to find a formula in terms of $\displaystyle k$ and $\displaystyle d(x_1,x_0)$ for $\displaystyle d(x_n,x_m)$ and use this to show that $\displaystyle \{x_n\}$ is Cauchy (and therefore converges to some point $\displaystyle x^*$). Then use the continuity of $\displaystyle f$ to show that $\displaystyle f(x^*)=x^*$.
What you are proving is known as the Contraction Mapping Principle or alternatively, the Banach Fixed Point Theorem.