Full disclosure: a philosophical logic person trying to learn math. Here is the problem from Gallant(1997), a measure-theoretic intro to econometrics:
Let $F_i$ where $i = 1, 2, …$ be an infinite sequence of events from
the sample space $\Omega$. Let $F$ be the set of points that are in
all but a finite number of the events $F_i$. Prove that
$F=\bigcup_{k=1}^\infty \bigcap_{i=k}^\infty F_i$.
Per the suggestion of the problem, I'm starting by trying to prove $\omega \in F \rightarrow \omega \in \bigcup_{k=1}^\infty \bigcap_{i=k}^\infty F_i$
Importantly, the notion of "all but a finite number" isn't formalized in the text (beyond what we're trying to prove). I'm starting with something like $\exists m \in \mathbb N$ s.t. $\omega \in \bigcap_{i=n}^\infty F_i$ and $m \le n$ but also $\omega \notin \bigcap_{i=1}^\infty F_i.$ Not sure if this gets me to where I want to go, however, and not sure of the next steps. Any proof I can think of that doesn't rely on an explicit formalism of the notion basically just looks like I'm describing how the $\bigcup_{k=1}^\infty \bigcap_{i=k}^\infty F_i$ structure works, which doesn't feel like a proof to me.
This is a problem early in the text, so it should be provable by fairly primitive notions. I think mentioning the philosophical logic part is relevant because maybe the somewhat more informal proof writing rules of math are throwing me for a loop. (For instance, earlier in the text they "proved" DeMorgan's laws for pairwise union and intersection by assuming DeMorgan's laws for 'or' and 'and,' which seemed like a cop-out, not that proving DM's laws for those notions are challenging).
Any help on this proof?
Best Answer
If $\omega\in F$, then $\omega$ is in all but a finite number of the events $F_i$. Therefore, there is a natural number $N$ such that $\omega$ is in every $F_i$ when $i\geqslant N$. So, $\omega$ belongs to each$$\bigcap_{i=k}^\infty F_i$$as long as $k\geqslant N$ and therefore$$\omega\in\bigcup_{k=1}^\infty\bigcap_{i=k}^\infty F_i.$$