CZ-SK-MO 1991 Problem 6

The set $N$ is partitioned into three subsets $A_1,A_2,A_3$ . Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$ .