MathDB

Assume that the set of all positive integers is decomposed into

r

disjoint subsets

A_{1}, A_{2}, \cdots, A_{r}

A_{1} \cup A_{2} \cup \cdots \cup A_{r}= \mathbb{N}

. Prove that one of them, say

A_{i}

, has the following property: There exist a positive integer

m

such that for any

k

one can find numbers

a_{1}, \cdots, a_{k}

A_{i}

with

0 < a_{j+1}-a_{j} \le m \; (1\le j \le k-1)

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