AUSTRIAN-POLISH 1980 Problem 6

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality $$|a_{k+m}-a_k-a_m| \leq 1 \text{for all} \ k,m \in \mathbb{Z}_{>0}.$$ Show that the following inequality holds for all positive integers $k,m$ $$\left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}.$$