CIIM 2010 Problem 4

Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$ , and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\lim_{n\to+\infty} d(A_n) = 0$ , where $d(A)$ is the diameter of the set $A$ .

Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$