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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!

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1 Answer 1

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By induction on $n$, we check that $$x_n\le a^{-1/s}b^{-1/s^2-n/s}$$ for all integers $n\ge0$.

Now the desired result immediately follows.


The condition that $x_n$ is decreasing in $n$ was not needed or used.

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  • $\begingroup$ Thank you very much! Could the induction basis be something else here or is it related to the above condition? $\endgroup$
    – Shaq155
    May 3, 2022 at 6:35
  • $\begingroup$ @Shaq155 : The initial condition, on $x_0$, is very important here. I think the upper bound on $x_0$ cannot be improved, but I have not checked that. I am wondering where this problem came from. $\endgroup$ May 3, 2022 at 13:19
  • $\begingroup$ Do you remember what that paper was? $\endgroup$ May 3, 2022 at 15:54

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