Interesting convergence of sequence

[grusini]

Homework Statement

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence such that $a_0\in(0,1)$ and $a_{n+1}=a_n-a_n^2$
Does $\lim_{n\rightarrow\infty}na_n$ exist? If yes, calculate it.

Homework Equations

The Attempt at a Solution

I have a solution but I'd like to see other solutions..

 

[Dick]

Homework Statement

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence such that $a_0\in(0,1)$ and $a_{n+1}=a_n-a_n^2$
Does $\lim_{n\rightarrow\infty}na_n$ exist? If yes, calculate it.

Homework Equations

The Attempt at a Solution

I have a solution but I'd like to see other solutions..

Asking for a solution when you haven't shown your own goes against the general spirit of the homework help forum. Try another forum if you don't want to show your own proof.