Let $Y_1,Y_2,\dots$ be non-negative i.i.d random variable with $EY_m=1$ and $P(Y_m=1)<1 $. $X_n=\Pi_{1\leq m\leq n}Y_m$ is martingale. Use the strong law of large numbers to conclude $(1/n)\log X_n \rightarrow c <0$.
To use strong law of large number, I think $E|\log Y_i|<\infty$ is needed. However , if $P(Y_i=0)>0$, how could this possible?