# A Problem on the Limit of a Sequence

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.