We assume that $Y_{n}$ are independent random variables and we let $Y_{n}$ have the values $\frac{3}{2}$or $\frac{1}{2}$ with probability $\frac{1}{2}$ each. We let $X_{n}=Y_{1}\cdot \cdot \cdot Y_{n}$. I showed that $\lbrace X_{n} \rbrace$ is a martingale. However, I am also suppose to show that $\lbrace X_{n} \rbrace$ converges to $X=0$, a.s, and that $E(\prod_{n=1}^{\infty}Y_{n}) \le \prod_{n=1}^{\infty}E(Y_{n})$.
This problem is from Patrick Billingsley's "Probability and Measure". The hint given was to let $S_{n}$ be the number of $k$ such that $1 \le k \le n$ and $Y_{k}=\frac{3}{2}$. Then $X_{n}=\frac{3^{S_{n}}}{2^{n}}$. Then we take logarithms and use the Strong Law of Large Numbers. I am having trouble completing this problem, even with the hint included.