Exercise :
Let $X_n, n=1,2,3,\dots$ be a sequence of independent continuous random variables. It's also given that all of $X_n$ have the same probability density function $f(x)$, for which it is $f(x) > 0 \space \forall \space x \in \mathbb R$. For a given real number $b$, consider the following events : $$A=\lim \sup_n \{X_n >b\}, \space B=\lim \inf_n \{X_n>b\}$$ $$G = \lim \sup_n \{X_n \leq b\}, \space D = \lim \inf_n\{X_n\leq b\}$$ Explain with simple words what each of the events above means and calculate their probabilities of happening.
Discussion :
It seems something that may have to do like Borrel - Cantelli Theorems, but I'm not even close to sure on how to proceed and make conclusions. Any help will be greatly appreciated as this is a past exam problem.