How $\Pi_{i=0}^n\xi_i$ converges a.s. to $0$ provided $\xi_n>0 $, iid and $E(\xi_n)=1$

Question

suppose $\{\xi_n,n \ge 0\}$ are iid and positive random variables. $E(\xi_0)=1$. show $\Pi_{i=0}^n\xi_i$ is a positive martingale converging to $0$ provided $P[\xi_0=1]\not=1$

It's easy to prove it's a martingale by definition. About the convergence, any idea? Thank you!

Answer 1

Hint: Rewrite $\prod\limits_{k=1}^n\xi_k$ as $\exp\left(\sum\limits_{k=1}^n\eta_k\right)$ with $\eta_k=\log\xi_k$ and show that $E[\eta_k]\lt0$.