MathDB

Assume that the set of all positive integers is decomposed into

r

(disjoint) subsets A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \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

A_i

with 0 < a_{j \plus{} 1} \minus{} a_j \leq m, (1 \leq j \leq k \minus{} 1).

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