Borel Cantelli Lemma for Lim inf

Question

Let ($\mathcal{A}$$_{n}$)$_{n \in \mathbb{N}}$ be a sequence of events in a probability space.

a) Show that lim inf$_{n \to \infty}$ $\mathcal{A}$$_{n}$ = $\overline{lim sup_{n \to \infty} \overline{\mathcal{A}_{n}}}$.

b) State the Borel-Cantelli Lemma for Lim inf$_{n \to \infty}$ $\mathcal{A}_{n}$.

My attempts:

Part a) is clear to me with the use of De Morgan's laws. However I am not sure to what extent I can negate or transform the Borel-Cantelli Lemma in part b), to have P(Lim inf$_{n \to \infty}$ $\mathcal{A}_{n}$) = 0.

Any help is appreciated

Answer 1

I think the "goal" is to proof $\sum_{n \in \mathbb{N}}P(A_n^c) < \infty \implies P(liminf_{n \rightarrow \infty} A_n) = 1.$

This follows immediately from part a) + using the first part of Borel-Cantelli Lemma + using $P(B^c) = 1-P(B)$.

Best P