Let $X_1, \dots$ be a sequence of independent random variables with $P(X_i = \frac{1}{2})=\frac{1}{2}$ and $P(X_i = \frac{3}{2})=\frac{1}{2}$. Let $S_n := \prod_{i=1}^n X_i$.
I've proven that $S_n$ is a martingale and using the convergence theorem for martingales there exists an $S_\infty$ such that $S_n \to S_\infty$ a.s. Now I'd like to compute $S_\infty$ and this is my attempt so far:
Using the strong law of large numbers it is clear that:
$\frac{1}{n}\log(S_n) = \frac{1}{n}\sum_{i=1}^n \log(X_i) \to E(\log(X_1)) = \frac{1}{2}\log(\frac{3}{2})-\frac{\log(2)}{2} < 0 $ a.s.
Therefore $\log(S_n) \to -\infty$ a.s. and so $S_n \to 0$ a.s which means $S_\infty = 0$.
Is the proof correct or is there something lacking?
I agree that the proof is correct.