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Absolute Valued Sequence Inequality

Source: 1980 Austrian-Polish Math Competition

August 24, 2016
inequalitiesabsolute valueSequenceSequencesalgebra

Problem Statement

Let a1,a2,a3,…a_1,a_2,a_3,\dots be a sequence of real numbers satisfying the inequality |a_{k+m}-a_k-a_m| \leq 1   \text{for all} \ k,m \in \mathbb{Z}_{>0}. Show that the following inequality holds for all positive integers k,mk,m ∣akk−amm∣<1k+1m. \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}.
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