If $(X_n)_n$ is an independent non-negative sequence, or r.v's with common mean 1. Then, I am trying to prove that
$$ P(\sup_n\prod_{I=1}^n X_i\ge \varepsilon)\le\frac{1}{\varepsilon}$$ for all $\varepsilon>0$. Since we have a mean of 1, we know that $Y_n=\prod_i X_i$ is a martingale and also has mean 1. however I am unsure of where to go from here.