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Partitioning $\mathbb{N}$

Source: own. For APMC 2016 #3

December 26, 2016
combinatorics

Problem Statement

Let a1,a2,⋯a_1,a_2,\cdots be a strictly increasing sequence on positive integers.
Is it always possible to partition the set of natural numbers N\mathbb{N} into infinitely many subsets with infinite cardinality A1,A2,⋯A_1,A_2,\cdots, so that for every subset AiA_i, if we denote b1<b2<⋯b_1<b_2<\cdots be the elements of AiA_i, then for every k∈Nk\in \mathbb{N} and for every 1≤i≤ak1\le i\le a_k, it satisfies bi+1−bi≤kb_{i+1}-b_{i}\le k?
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