IMC 2002 Problem 4

Let $f : [a, b] \rightarrow [a, b]$ be a continuous function and let $p \in [a, b]$ . Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2,...$ . Suppose that the set $T_p = \{p_n : n = 0, 1, 2,...\}$ is closed, i.e., if $x \not\in T_p$ then $\exists \delta > 0$ such that for all $x' \in T_p$ we have $|x'-x|\ge\delta$ .

Show that $T_p$ has finitely many elements.