1990 IMO 第 4 题

(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 $0<a_{j+1}-a_{j} \leq m$ $(1 \leq j \leq k-1)$.

(CZS 2) 假设所有正整数的集合被分解为 $r$ (不相交)子集 $A_{1} \cup A_{2} \cup \cdots A_{r}=\mathbb{N}$。证明其中一个，比如 $A_{i}$，具有以下性质：存在一个正 $m$，使得对于任何 $k$ 都可以在 $A_{i}$ 中找到数字 $a_{1}, a_{2}, \ldots, a_{k}$ 且 $0<a_{j+1}-a_{j} \leq m$ $(1 \leq j \leq k-1)$。