BALKAN-SL 2007 A8

Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any $n$ $\in$ $\mathbb{N}$ . Prove, $a_1=0$