A convergence of the sequence 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?
 A: This sequence is called Logistic map & it is an example of a discrete dynamical system with chaos. You can consider it as a dynamical system with initial condition:
$$\begin{cases}x_{i+1} = f(x_i), \\ x_0 = c \in (0,1)\end{cases}$$ This system for $\lambda \in (0,4)$ has two fixed points, that is $x^{*}: f(x^{*}) = x^*$. One of these points is, obviously, 0, and the other one can be found:
$$
x^* = f(x^*) = \lambda x^* (1-x^*) \Rightarrow x^* = 1 - \frac{1}{\lambda}
$$
Zero is fixed point for all $\lambda$, whereas $x^* = 1 - \frac{1}{\lambda}$ is in the range (0,1) only if $\lambda \ge 1$.
These fixed points for the range of $\lambda$ mentioned before are the only attractors of the dynamical system, so they are only candidates for being limits of your sequence. Now, for $x^* = 1 - \frac{1}{\lambda}$ you can express $\lambda$ in terms of $x^* \equiv L$ and it will be $\lambda = \frac{1}{1-L}$.
Deep analysis of this map and explaining what happens when $\lambda > 4$ you can find here: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536
