Denoting the Occurrence of Events Using Countable Unions and Intersections Let $A_1, A_2, ...$ be a sequence of events in a $\sigma$-algebra on a sample space $\Omega$. Can someone please explain to me why 
$\bigcap_{n \geq 1} \bigcup_{k \geq n} A_k$
denotes the occurrence of infinitely many $A_k$'s?
I understand that $\bigcup$ corresponds to saying "there exists" and $\bigcap$ means "for all", but is there another way of thinking about this?
Thanks so much :)
 A: You can think about $\bigcap_{n\geq 1}\bigcup_{k\geq n} A_k$ taking successively steps to see what is going on. Denote $B_n = \bigcup_{k\geq n} A_k$, for each $n$. Then we have $\bigcap_{n\geq 1}\bigcup_{k\geq n} A_k = \bigcap_{n \geq 1} B_n$.
Note that $$B_1 = A_1\cup A_2 \cup A_3 \cup \ldots $$
$$B_2 = A_2\cup A_3 \cup A_4 \cup \ldots $$
$$B_3 = A_3\cup A_4 \cup A_5 \cup \ldots $$
$$\vdots $$
Now note that $$B_1\cap B_2 = A_2 \cup A_3 \cup A_4 \ldots $$
$$B_1\cap B_2\cap B_3 = A_3\cup A_4 \cup A_5 \ldots $$
$$B_1\cap B_2\cap B_3 \cap B_4= A_4\cup A_5 \cup A_6 \ldots $$
$$ \vdots $$
With this, we can see $\bigcap_{n\geq 1}\bigcup_{k\geq n} A_k$ as $\displaystyle\lim_{n\to \infty} A_{n}\cup A_{n+1}\cup A_{n+2}\cup\ldots = \lim_{n\to\infty} \bigcup_{k\geq n} A_k$. It's important to realize this is totally informal. There is no definition of limit of sets, this is just to get the pattern. And the pattern is: we get the union of all $A_k$'s and start to "cut off" them one by one, starting from $A_1$ and going up. "In the end" we will have the event $\bigcap_{n\geq 1}\bigcup_{k\geq n} A_k$.
