AUSTRIAN-POLISH 1981 Problem 9

For a function $f : [0,1] \to [0,1]$ we define $f^1 = f$ and $f^{n+1} (x) = f (f^n(x))$ for $0 \le x \le 1$ and $n \in N$ . Given that there is a $n$ such that $|f^n(x) - f^n(y)| < |x - y|$ for all distinct $x, y \in [0,1]$ , prove that there is a unique $x_0 \in [0,1]$ such that $f (x_0) = x_0$ .