It is shown that the following martingale \begin{equation} X_n = \prod _{i=1}^n e^{Y_i - \frac{1}{2}} \end{equation} where $Y_i$ has a standard normal distribution, converges almost surely.
To this end, notice that $$ E(|X_n|) = E(X_n) = e^{-n / 2}E(e^{Y_i})^n = 1. $$ so $sup_n E(|X_n|) = 1$ and almost sure convergence follows.
How can we conclude almost sure convergence by noting that $sup_n E(|X_n|) = 1$?