I have been going through Resnick's 'A Probability Path', and at one point he is trying to prove a version of Fatou's lemma: $$P(\liminf_{n\rightarrow\infty}A_n)\le\liminf_{n\rightarrow\infty}P(A_n)$$ In the first line of the proof he writes: $$P(\liminf_{n\rightarrow\infty}A_n)=P\left(\lim_{n\rightarrow\infty}\uparrow\left(\bigcap_{k\ge n}A_k\right)\right)$$ $$=\lim_{n\rightarrow\infty}\uparrow P\left(\bigcap_{k\ge n}A_k\right)$$ As justification he writes 'from the monotone continuity property'.
I understand that $P$ is continuous in that if $A_n\uparrow A$ then $P(A_n)\uparrow P(A)$, but what I don't understand is how you can just go from a $\liminf$, which is always defined, to a $\lim$, which is not always defined, especially since in this case there are no requirements on $A_n$, i.e. they are just unrelated sets from our $\sigma$-field. Any help in understanding this would be greatly appreciated.
For every $n$, let $B_n=\bigcap\limits_{k\geqslant n}A_k$. For every sequence $(A_n)$, the sequence $(B_n)$ is nondecreasing hence has a limit, called $\liminf\limits_{n\rightarrow\infty}A_n$.
Using the sets $C_n=\bigcup\limits_{k\geqslant n}A_k$, a similar result holds for $\limsup\limits_{n\rightarrow\infty}A_n$.