ILL 1974 Problem 25

Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ $$x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).$$ Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$