In the context of understanding Borel-Cantelli lemmas, I have come across the expression for a sequence of events $\{E_n\}$:
$$\bigcap_{n=1}^\infty \bigcup_{k\geq n}^\infty E_k$$
following Wikipedia notation. This is verbalized as:
- Limit supremum of the sequence of events when $n\rightarrow\infty$: $\limsup\limits_{\small n\rightarrow\infty}E_n$.
- The event that occurs "infinitely often": $\{E_n \,\text{i.o.}\}$ or the set that belongs to "infinitely many" $E_n$'s.
The explanation is that $\displaystyle\bigcup_{k\geq n}^\infty E_k$ is the event that at least one of the $E_n,E_{n+1},E_{n+2,\cdots}$ events occurs.
The question is about the meaning of $\displaystyle\bigcap_{n=1}^\infty $ in front.
The way I look at it is that it would play the role of excluding every $\{E_n\}$ imaginable with $n\in\mathbb N$. For instance, when $n=1$, the union part of the expression would render: $\{E_1,E_2,E_3,\cdots\}$; and when $n=2$, this same union operation would yield: $\{E_2,E_3,\cdots\}$. So the intersection in front of the expression would eliminate $\{E_1\}$ and leave the rest. Next would come the intersection with $\{E_3,E_4,\cdots\}$, excluding $\{E_2\}$. Doing this exercise at infinity would manage to exclude every single $\{E_n\}$.
Is this what the intersection/union operation is supposed to do: "clip" out every single set? My understanding is that this is not the case, and that we want to end up with a "tail" of events after point $k$.
What is the correct interpretation, and what is the mistake I am making?
NOTE to self: It is not about the "clipped out" sets at the beginning, but rather about the tail events. The union part indicates that any of the events after event $n$ occurs, defining the "$n$-th tail event." The intersection is defines the event that all the $n$-th tail events occur.
