Let $\{x_{n}\}_{n=0}^{\infty}$ be decreasing sequence of non-negative reals. Suppose that there exist constants $a, s>0$ and $b>1$ such that $$x_{n+1}\leq ab^{n}x_{n}^{1+s}$$ and $$x_{0}\leq a^{-1/s}b^{-1/s^{2}}.$$ Is it true that then $$\lim_{n\to \infty}x_{n}=0?$$

Any help is appreciated!