KöMaL 2010 A521

A. 521. There are given a positive integer $m$ and an infinite sequence $a_1$<$a_2$<... of positive integers such that $a_k \le mk$ holds for infinitely many indices $k$. Prove that there exist positive integers $b_1$,...,$b_m$ with the property that every integer can be written as $a_i$-$a_j$+$b_k$ where $i$,$j$ are positive integers and 1$\le k \le m$.