While I investigate the property of positive random variables, I encountered the following question, not easy to solve.

Let $\left\{ a_n \right\}_{n=1}^{\infty}$ be a sequence that satisfies $a_n > 1$ and $\displaystyle\lim_{n\rightarrow\infty}\left(a_1 a_2 \cdots a_n \right)=\infty$. Then define sequences $U_n$ and $T_n$ inductively by: $$ U_0 = 1,\quad U_{n+1}= \frac{2U_n}{a_{n+1}}+1,\quad n=0,1,2,.... $$ $$ T_0 = 1,\quad T_{n+1}= \frac{2T_n}{{a_{n+1}}^2}+1,\quad n=0,1,2,.... $$ Assuming $\displaystyle\lim_{n \rightarrow \infty} U_n = \infty$, prove or disprove that $\displaystyle\lim_{n \rightarrow \infty} \frac{U_n}{T_n} = \infty$.

I tried in various way but could not succeed. I would be very happy if someone could answer this question.


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