Question 4
(CZS 2) Assume that the set of all positive integers is decomposed into $r$ (disjoint) subsets $A_1\cup A_2\cup\cdots A_r=\mathbb N$. Prove that one of them, say $A_i$, has the following property: There exists a positive $m$ such that for any $k$ one can find numbers $a_1,a_2,\ldots,a_k$ in $A_i$ with $0\lt a_{j+1}-a_j\le m$ $(1\le j\le k-1)$.