Let $Y$, $Y_1$, $Y_2$, $\dots$, be nonnegative i.i.d random variables with mean $1$. Let $$X_n = \prod_{1\le m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim\limits_{n\to\infty}X_n = 0$ almost surely.
I feel like this question has something to do with the idea that $(X_n)$ is a martingale (which I can prove easily) but I am not sure if I am overthinking it or not. I was trying to use Doob's upcrossing inequalities in a clever way but there might be an easier approach to the problem.
$\dfrac{\log X_n}{n} = \dfrac{\sum_{m \le n} \log Y_m}{n} \to E\log Y_1$ almost surely by the strong law of large number.
And by Jensen's inequality, $E\log Y_1 \leq \log EY_1 =0$ since $EY_1 = 1$.
Since $P(Y = 1) < 1$, $E\log Y_1 < \log EY_1 =0$.
So we get that $\dfrac{\log X_n}{n}$ converges to a strictly negative number almost surely, thus $\log X_n \to -\infty$, i.e. $X_n \to 0$