Almost sure convergence of the maximum of iid exponential radom variables

Question

Let $X_1, \ldots, X_n$ be independent exponential random variables with density $Exp(1)$. Let $$Y_n = \max_{1 \leq i \leq n} X_i$$ Show that $$\frac{Y_n}{\log n} \rightarrow 1 \;\; \text{almost surely}$$

I can show that $\frac{Y_n}{\log n} \rightarrow 1 \;\; \text{in probability}$ but have no idea how to show the almost sure convergence.

Thank you!