Meaning of liminf of sets I've been struggling for a while to understand the meaning of liminf of a sequence of sets. 
I know that the definition is 
$\liminf_{n\to\infty}A_n:=\bigcup_{n\in\mathbb{N}}\bigcap_{m\geq n} A_n$.
When I break it up into pieces, I get $(A_1 \cap A_2 \cap A_3 \cap A_4 \cap \ldots) \cup (A_2 \cap A_3 \cap A_4 \ldots) \cup (A_3 \cap A_4 \cap \dots) \cup (A_4 \cap \ldots) \cup \ldots$.
Here, $A_n$ occurs infinitely many times, doesn't it? Because it is in each set of parentheses?
Or do the unions mean OR, so that $A_n$ might or might not be in any of the sets? 
However, this is wrong! Why? I'm confused.
 A: Let's break it up.  Define:
$$
B_n=A_n\cap A_{n+1}\cap A_{n+2}\cap\ldots
$$
Then you've got $B_1\cup B_2\cup\ldots$  Now, $B_n$ is all the elements that live in all $A_i$ for $i\geq n$.  By taking a union you can maybe convince yourself that the limit infimum is the set of all elements that are in all but finitely many $A_i$.  
A: You get the definition slightly wrong: It should be $$\liminf_{n\to\infty}A_n:=\bigcup_{n\in\mathbb{N}}\bigcap_{m\geq n} A_m$$
(the difference being the index at the end).
Now any given $A_k$ occurs in only finitely many of the infinite unions $\bigcap_{m\geq n} A_m$, namely for $n\le k$.
The reason this is called the liminf is that you may think of $B_n=\bigcap_{m\geq n} A_m$ as being the infimum of the sets $A_n$, $A_{n+1}$, … as it is the largest set contained in each of them. And now the sets $B_1$, $B_2$, … form an increasing sequence of sets, so the “limit” of this sequence will be their union.
And yes, union means “or”. An infinite union becomes an infinitely big “or”, so being a member of the union means being a member of at least one of the sets. You will find that $x$ belongs to $\bigcup_{n\in\mathbb{N}}\bigcap_{m\geq n} A_m$ if and only if it is in $\bigcap_{m\geq n} A_m$ for at least some $n\in\mathbb{N}$, which means it is in every $A_m$ for $m$ larger than some $n$. (In short, “for every sufficiently large $m$”.)
