CMO 1995 Problem 5

$u$ is a real parameter such that $0<u<1$ . For $0\le x \le u$ , $f(x)=0$ . For $u\le x \le n$ , $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$ . The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$ . Show that there exists a positive integer $k$ for which $u_k=0$ .