Assume that the set of all positive integers is decomposed into
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
(disjoint) subsets
![A_1 \cup A_2 \cup \ldots \cup A_r = \mathbb{N}.](/media/m/1/8/e/18e58632c3aa2626c823bbfa10b42277.png)
Prove that one of them, say
![A_i,](/media/m/8/9/0/8909da203075f2091aaacfa1eb6e9f1e.png)
has the following property: There exists a positive
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
such that for any
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
one can find numbers
![a_1, a_2, \ldots, a_k](/media/m/9/c/2/9c2cad8f80a2395453d76f398ab7b5f1.png)
in
![A_i](/media/m/5/f/0/5f0935569a883b13bb70b83ea33eee14.png)
with
![(1 \leq j \leq k - 1)](/media/m/1/6/4/16437f622c686b1bffefab1cd71f2ffb.png)
.
%V0
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 $0 < a_{j + 1} - a_j \leq m,$ $(1 \leq j \leq k - 1)$.