Question 81
Prove that for any integer $n \geq 2$, there exists a unique finite sequence $x_{0}, x_{1}, \ldots, x_{n}$ of real numbers which satisfies $x_{0}=x_{n}=0$ and $x_{i+1}-8 x_{i}^{3}- 4 x_{i}+3 x_{i-1}+1=0$ for all $i=1,2, \ldots, n-1$. Prove moreover that $\left|x_{i}\right| \leq \frac{1}{2}$ for all $i=1,2, \ldots, n-1$.