IMC 2002 · Problem 4

Let $f : [a,b] \to [a,b]$ be a continuous function and let $p \in [a,b]$. Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2, \dots$ Suppose that the set $T_p = \{p_n : n = 0, 1, 2, \dots\}$ is closed, i.e., if $x \notin T_p$ then there is a $\delta > 0$ such that for all $x' \in T_p$ we have $|x' - x| \ge \delta$. Show that $T_p$ has finitely many elements.