IMO 1988 - Shortlisted problems Question 24

(SWE 2) Let $\{a_k\}_1^\infty$ be a sequence of nonnegative real numbers such that $a_k-2a_{k+1}+a_{k+2}\ge0$ and $\sum_{j=1}^k a_j\le1$ for all $k=1,2,\ldots$. Prove that $0\le(a_k-a_{k+1})\lt \frac2{k^2}$ for all $k=1,2,\ldots$.