Partition of a 6 or more element integer set. Combinatorial lemma > Let $S=\\{n_1, n_2, \ldots, n_k\\}$ be a set of distinct integers such that $k\geq 6$. Suppose that $A$ and $B$ form a partition of $S$. Prove that t...--prophetes.ai

Partition of a 6 or more element integer set. Combinatorial lemma > Let $S=\\{n_1, n_2, \ldots, n_k\\}$ be a set of distinct integers such that $k\geq 6$. Suppose that $A$ and $B$ form a partition of $S$. Prove that there is an $m\in A$ and an $n\in B$ such that $|m-n|\geq 3$. This is a combinatorial lemma I found myself using while proving the irreducibility of a family of integer polynomials. I can see that it's true intuitively, but I'm having difficulty formalizing the argument. I can most likely attack this by case-analysis and induction, but I'm hoping there's a sleeker approach. Any help is appreciated.

Assume without loss of generality that $n_1