Question 80
Let $(a_{n})$ be a strictly increasing sequence of positive integers such that $a_{1}=1$ and, for all $n \geqslant 1$, $a_{n+1} \leqslant 2n$. Prove that for every integer $n \geqslant 1$, there exist two indices $p$ and $q$ such that $a_{p}-a_{q}=n$.