VMO 2021 Problem 1

Let $(x_n)$ define by $x_1\in \left(0;\dfrac{1}{2}\right)$ and $x_{n+1}=3x_n^2-2nx_n^3$ for all $n\ge 1$ . a) Prove that $(x_n)$ convergence to $0$ .

b) For each $n\ge 1$ , let $y_n=x_1+2x_2+\cdots+n x_n$ . Prove that $(y_n)$ has a limit.