I've been given some independent and identically distributed random variables $X_i$ such $E(X_i)=1$, $P(X_i <1)>0$ and $\log X_i$ is integrable. We define $M_n=X_1 \dots X_n$. How can I prove that $M_n$ converges almost surely to 0? I have tried directly with the definition of almost surely convergence and I wonder if I could take the limit outside and somehow split the probabilities, since the $X_i$ are independent.
Thank you so much in advance.
Note that $M_n \to 0 \iff \ln(M_n) \to -\infty$. As mentioned in the comment, the main point is to apply the Strong Law of Large Numbers to $\ln(X_n)$. Further, you need to show that $\mathbb{E}\ln(X_1)<1$ which can be done by noting the inequality $ln(x)\leq x-1$, with equality only if $x=1$. In particular, since we know by the assumption $\mathbb{P}(X_1<1)>0$ that $X_1$ is not a.s. equal to 1, we have the strict inequality $\mathbb{E}(\ln(X_1))<\mathbb{E}X_1-1=0$. From here, set $\mu=\mathbb{E}\ln(X_1)<0$, and then by the SLLN $$\frac{1}{n}\sum_{k=1}^n\ln(X_k) \to \mu$$ almost surely, and so $\sum_{k=1}^n\ln(X_k) \to -\infty$. But this expression is just $\ln(M_n)$ so we are done.