I have to prove two similar identities involving the limit superior/inferior of a sequence of events in the event space but I'm not sure how to proceed. Here are the identities:
\begin{align*}
\limsup_{n \rightarrow \infty }A_n &= \bigg\{ \omega \in \Omega: \big|\{n\in \mathbb{N}{:\omega \in A_n } \}\big| =\infty\bigg\} \\ \liminf_{n \rightarrow \infty }A_n &= \bigg\{ \omega \in \Omega: \big|\{n\in \mathbb{N}{:\omega \notin A_n } \}\big| <\infty\bigg\}.
\end{align*}
I know the relevant definitions as:
$$\limsup_{n \rightarrow \infty }A_n =\bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty}A_k,\qquad \liminf_{n \rightarrow \infty }A_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty}A_k $$
I have read that this identity basically states that the limit superior describes the event that infinitely many $A_i$'s occur and the limit inferior describes the event that all but a finite number of $A_i$'s occur but I haven't been able to construct something meaningful.
Best Answer
HINT
Another useful (and equivalent) way to rephrase the proposed definitions is given by: \begin{align*} \begin{cases} \displaystyle\liminf_{n\to\infty}A_{n} = \left\{\omega\in\Omega \mid \liminf_{n\to\infty}1_{A_{n}}(\omega) = 1\right\}\\\\ \displaystyle\limsup_{n\to\infty}A_{n} = \left\{\omega\in\Omega \mid \limsup_{n\to\infty}1_{A_{n}}(\omega) = 1\right\} \end{cases} \end{align*}
where $1_{A_{n}}$ is the indicator function of the set $A_{n}$.
Can you take it from here?