In my lecture notes there is the following application of Borel-Cantelli's 2nd lemma: Let $(X_n)_{n\geq 1}$ be a sequence of independent $\mathcal{N}(0,\sigma^2)$-distributed random variables, with $\sigma > 0$. From Borel-Cantelli 2, it follows that $$\text{P-a.s}\ \limsup_nX_n = \infty$$
I know that for events $(A_n)$, $\limsup_nA_n = \bigcap_{n\geq 1}\bigcup_{m\geq n} A_m$, but I can't figure out how can I apply the lemma to random variables instead events.