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Assume that the set of all positive integers is decomposed into r (disjoint) subsets A_1 \cup A_2 \cup \ldots \cup 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).

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