RU-SPB 2008 Problem 107

In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$ . Prove that $0.99<x_{100}<0.991$ .

Fresh translation. This problem may be similar to one of the 9th grade problems.