Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once, show that there exist integers $\ell $ and $m,\ 1\lt\ell \lt m$, such that $a_1+a_m=2a_{\ell}$.

Let $\ell $ be the smallest index for which $a_\ell > a_1$. Then choose $ m > \ell $ so that $a_m = 2a_\ell - a_1.$ Such an $m$ exists since $ 2a_\ell - a_1$ occurs somewhere in the sequence, and $2a_\ell - a_1 > a_\ell,$ so it can't occur before $\ell$.

Let $\ell $ be the smallest index for which $a_\ell > a_1$. Then choose $ m > \ell $ so that $a_m = 2a_\ell - a_1.$ Such an $m$ exists since $ 2a_\ell - a_1$ occurs somewhere in the sequence, and $2a_\ell - a_1 > a_\ell,$ so it can't occur before $\ell$.