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Sequence satisfying the condition a_(m+n) < a_n + a_m

Source: Chinese Mathematical Olympiad 1997 Problem 6

August 26, 2013
inductioninequalities

Problem Statement

Let (an)(a_n) be a sequence of non-negative real numbers satisfying an+m≤an+ama_{n+m}\le a_n+a_m for all non-negative integers m,nm,n. Prove that if n≥mn\ge m then an≤ma1+(nm−1)ama_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m holds.
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