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Three sequences have a simultaneous relation

Source: 1961 All-Soviet Union Olympiad

August 4, 2015
algebraSequences

Problem Statement

Prove that for any three infinite sequences of natural numbers (an)n≥1(a_n)_{n\ge 1}, (bn)n≥1(b_n)_{n\ge 1}, (cn)n≥1(c_n)_{n\ge 1}, there exist numbers pp and qq such that ap≥aqa_p\ge a_q, bp≥bqb_p\ge b_q and cp≥cqc_p\ge c_q.
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