BALKAN-SL 2008 A3

Let $(a_m)$ be a sequence satisfying $a_n \geq 0$ , $n=0,1,2,\ldots$ Suppose there exists $A >0$ , $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$ . Prove that there exists $B>0$ such that \begin{align*} a_n \le \frac{B}{n} \text{for }1 \le n \end{align*}