Given $x_{n+1}=x_n(1-x_n)$, find $\lim_{n \to \infty}n.x_{n}$

Question

Let the sequence $\{x_n\}$ be such that $0<x_{1}<1$ and $x_{n+1}=x_{n}\left(1-x_{n}\right)$. Find $\lim_{n \rightarrow \infty}n.x_{n}$

So far I have only been able to prove that the sequence is monotonically decreasing.

I tried finding $x_2,x_3,..$ in terms of $x_1$ but they didnt seem to form a pattern(i.e. I couldnt find $x_n$ explicitly in terms of $x_1$)

Any hint in the right direction is appreciated.