Proof of Reverse Borel Cantelli Lemma

Question

I read a proof on The converse Borel Cantelli lemma unfortunately I can't upload a photo from my app I don't know why

In the proof they use $P(lim inf {A_n}^c) = {lim}_{n \rightarrow \infty} P(\bigcap_{k≥n} {A_k}^c)$

Can anyone explain why they can interchange limit and probability? Thanks

Answer 1

This statement follows from the continuity of the probability measure

$$P\left(\liminf_{n} A_{n}\right) = P\left(\bigcup_{n \geq 1} \bigcap_{k \geq n} A_k\right) = \lim_{n \to \infty}P \left( \bigcap_{k \geq n} A_k \right)$$