Let $(X_n)_n$ be a sequence of nonnegative independent random variables, such that for all $n \in \mathbb{N},E[X_n]=1.$ Prove that $Y_n=\prod_{k=1}^nX_k$ converges a.s.
Is it possible to prove this without using martingale convergence theorem?
(Since we don't know the limit, one to do it is to prove that for every $\epsilon>0, P(\sup_{k}|Y_{n+k}-Y_n|>\epsilon)\to_{n}0$ to conclude with the completeness of $\mathbb{R}$, another way is if there exist a sequence of nonnegative real numbers $(\epsilon_n)_n$ such that $\sum_{n}\epsilon_n,\sum_nP(|Y_{n+1}-Y_n|>\epsilon_n)$ converge).