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.