Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the limit.
I believe the limit is $0$. I tried to show that for any $\epsilon>0$ , $\sum_{i=0}^\infty P(|\dfrac{\log X_n}{n}|>\epsilon)<\infty$. However, I found that if $F_n$ is allowed to be the distribution function of $X_n$ then $$F_n(x)=x\sum_{i=0}^n\dfrac{(-\log x)^i}{i!}$$for $x\in(0,1)$. Now $F_n(e^{-n\epsilon})\to1$ (using Central Limit Theorem with $\mathrm{Poisson}(n\epsilon)$) so the series $\sum_{i=0}^\infty P(|\dfrac{\log X_n}{n}|>\epsilon)$ does not converge.
How can one show almost sure convergence?