Thai Mathematical Olympiad

Let $f:\mathbb{R}\to \mathbb{R}$ be a function satisfying $$|f(x+y)-f(x)-f(y)|<1\text{ for all }x,y\in \mathbb{R}.$$ Prove that $\left|f\left(\frac{x}{2008}\right)-\frac{f(x)}{2008}\right|<1$ for all $x\in \mathbb{R}$.