KMO 1999 Problem 2

Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$ , $n$ . Show that for all positive integers $k$ : $$\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}$$.