ISL 2002 A2

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$ , such that $$\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \text{for all }i,\ j \text{ with } i \neq j.$$ Prove that $c\geq1$ .

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