ILL 1988 Problem 74

Let $\{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: $$a_k - 2 a_{k + 1} + a_{k + 2} \geq 0$$

and $\sum^k_{j = 1} a_j \leq 1$ for all $k = 1,2, \ldots$. Prove that: $$0 \leq a_{k} - a_{k + 1} < \frac {2}{k^2}$$

for all $k = 1,2, \ldots$.