Solution:

The essential point is that if we plot the values $a_r$ against $r$, then the curve formed by joining the points is cup shaped. Its two endpoints are on the axis, so the other points cannot be above it. There are many ways of turning this insight into a formal proof. Barry Paul's was neater than mine: $a_{r + 1} - a_r \geq a_r - a_{r - 1}$. Hence (easy induction) if $a_s - a_{s - 1} > 0$, then $a_n > a_s$. Take $a_s$ to be the first positive, then certainly $a_s > a_{s - 1}$, so $a_n > 0$. Contradiction.