KöMaL 2012 A552

Prove that for an arbitrary sequence $a_1,a_2,\ldots$ of nonnegative real numbers and $\varepsilon$>0 there exist infinitely many positive integers $n$ for which $n^2 \big(4a_n(1-a_{n-1})-1 \big) \le \frac14+\varepsilon$.