Understanding the concept of infinitely often My book in stochastic processes has this section about i.o which I don’t really understand. Is there any example of any other to explain this? How can I understand this?
It states the following:
Let $A_{1}, A_{2}, \ldots$ be a sequence of subsets of $\Omega$. We define
$$
\left(A_{n} \text { i.o. }\right)=\bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} A_{m}
$$
The abbreviation i.o. stands for infinitely often.

Consider a point $x$. Then the point $x$ is in infinitely many of the sets $A_1, A_2, \ldots$ if and only if there exists no final $A_n$ containing $x$. This means for all $n$ there exists some $m \geq n$ such that $x \in A_m$. This is precisely the case when
$$ x \in \bigcap_{n=1}^{\infty} \bigcup_{m = n}^{\infty} A_m.$$
Thus $\bigcap_{n=1}^{\infty} \bigcup_{m = n}^{\infty} A_m$ is exactly the set of points which are in infinitely many of the $A_1, A_2, \ldots$
