RO-NMO 2004 Problem 101

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$ , for all $x,y \in \mathbb{R}$ .

Prove that if for any real $x$ , the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$ , for all $x \in \mathbb R$ .