Given the sequence: \begin{equation} (x_n): \begin{cases} x_0\in(0, 1),\\ x_{n+1}=x_n - x_n^2,&n\geq 0. \end{cases} \end{equation}
Find \begin{equation} \lim_{n \to +\infty }{\frac{n(1-nx_n)}{\ln{n}}}\tag{*} \end{equation}
My progress:
This is the second part of a problem, the first part is proving $\lim_{n \to +\infty }nx_n=1$ (**), which I already did.
And I'm not that naive to substitute (**) directly into (*), as it will return $0$, which is ridiculously simple.
And through trial and error, I found out the limit is $1$, but don't know how to prove it.
