How to understand limit sup and limit inf? Can anyone provide an intuitive explanation of $\limsup  A_{n}=\bigcap_{k=1}^{\infty }\bigcup_{i=k}^{\infty}A_{i}$ and $\liminf  A_{n}=\bigcup_{k=1}^{\infty }\bigcap_{i=k}^{\infty}A_{i}$ (maybe using representations of Venn diagrams)?
 A: First recall the intuitive meaning of $\limsup$ and $\liminf$ for sequences of numbers. Basically, given a sequence $(a_n)_{n\in\mathbb{N}}$ which has no limit, we can still try to look at what its limit should be "from above" and "from below." So, for example, in the sequence $${1\over 2}, {2\over 3},{1\over 3}, {3\over 4}, {1\over 4}, {4\over 5}, {1\over 5},...$$ we basically have two well-behaved "parts:" one going up $${1\over 2}, {2\over 3}, {3\over 4}, ...$$ and one going down $${1\over 2}, {1\over 3}, {1\over 4},...$$ which suggest different limit-y values, namely $1$ and $0$ respectively. These are the $\limsup$ and $\liminf$ of the sequence $\overline{a}$.

For sets, the picture is similar. Given a sequence of sets $\overline{A}=(A_i)_{i\in\mathbb{N}}$, it's natural to try to define the limit of the sequence $L$ by saying:


*

*$x\in L$ iff $x$ is "eventually always in" the $A_i$s.

*$x\not\in L$ iff $x$ is "eventually always out" of the $A_i$s.
But this only works if the sequence "settles" on every specific potential element: how are we to make sense of something like $$\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},...?$$
Well, there are two natural ideas here: 


*

*$0$ and $1$ each keep cropping up over and over, so the limit should be $\{0,1\}$.

*$0$ and $1$ each keep dropping out over and over, so the limit should be $\emptyset$.
These are the $\limsup$ and $\liminf$ of the sequence of sets, respectively. This isn't immediately obvious from the definitions, but it is a good exercise.

Incidentally, we can link the two notions of $\limsup/\liminf$ for sequences of numbers and sequences of sets as follows:
Suppose I have a sequence of sets $\overline{A}=(A_i)_{i\in\mathbb{N}}$. If $x$ is a possible element of the domain, we can consider the function $$In_x(i)=\begin{cases}1 \mbox{ if } x\in A_i,\\0 \mbox{ if } x\not\in A_i\end{cases}.$$
(If you're familiar with characteristic functions, you can think of this as a kind of dual: visualizing $\overline{A}$ as an array whose $n$th column is $A_n$, the characteristic functions of the $A_n$s focus on columns of the array while the $In_x$s look at rows. This may not be helpful, though, so ignore it if it's confusing.)
The point is that we then have the following: $$x\in\limsup(A_n)\quad\iff\quad\limsup(In_x(n))=1$$ (and similarly for the $\liminf$s).
