Recursive sequence

Let $x\in (0,1)$. Define sequence $a_1=x$ and $$a_{n+1}=a_n-a_n^2, \ \ \forall n\ge 1$$.
Prove that $$\lim_{n\to\infty} n a_n=1$$

A solution could be done along the lines of Cesaro-Stolz theorem.

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