Prove that for every uniform contraction function $f$ there exists a unique real $z$ such that $f(z)=z$. A function $f:\mathbb R\to\mathbb R$ is called a uniform contraction if there exists an $a$ in $(0,1)$ such that for all real $x$ and $y$, $|f(x)-f(y)|\leq a|x-y|$.
I proved $f$ is continuous. I let a sequence $\{x_n\}$ be defined as $x_{n+1}=f(x_n)$. I'm having trouble proving $\{x_n\}$ converges to $z$. From there I know how to complete the proof. Any help will be greatly appreciated.