I'm having some trouble with an exercise. Suppose $(X_n)_n$ is a sequence of independent r.v with $\mathbb{P}(X_n > x) = e^{-x}$ for positive x and one otherwise. The exercise first asks for what values of $\alpha>0$ is it almost surely the case the $X_n > \alpha \log \ n$ i.o. which I have managed.
Then it asks if $L= \limsup_n \frac{X_n}{\log \ n}$ show that $\mathbb{P}(L=1)=1$. So far I can show $\mathbb{P}(L \geq 1)=1$ and now want to show $\mathbb{P}(L > 1)=0$. But this part is giving me some problems.