Let $(x_n)_{n=0}^{\infty}$ be a real sequence such that $x_0 \in (0,1)$ and for all $i \ge 1$ is $x_i = \lambda x_{i-1}(1-x_{i-1})$, where $\lambda$ is a real constant. Suppose it converges to the given limit $L$. Express $\lambda$ in terms of $L$.
I'd like to ask what's wrong with this "solution":
If $x_i \to L$, then $L=\lim_{i \to \infty} x_i = \lim_{i \to \infty}\lambda x_{i-1}(1-x_{i-1}) = \lambda L(1-L)$, solving this yields $\lambda = \frac{1}{1-L}$.
And also, how do I solve it correctly?