In my textbook is an example of an application of Borel-Cantelli's lemma which I don't understand.
Let $X_n$, $n\geq 1$, be a sequence of independent $N(0, \sigma^2)$- distributed random variables, with $\sigma > 0$. From the second lemma of Borel-Cantelli it follows that: P- almost surely (P-a.s.) $\limsup_n X_n = \infty$.
For sake of completeness: $N(0, \sigma^2$) denotes the normal distribution with mean $=0$ and variance $\sigma$. Our 2nd lemma of Borel Cantelli says:
Let $A_n$, $n \geq 1$, be a sequence of independent events on a probability space. Then: $$\sum_n P(A_n) = \infty \Rightarrow P\left[\limsup A_n\right]=1.$$
What I don't see is why the normal distribution of $X_n$ implies that $\sum_nP(\{X_n \leq x\}) = \infty$ (or does it not?) and how from this we then can apply Borel Cantelli's lemma.