Motivation to define $\limsup$ and $\liminf$ of sets Let $\{A_n\}$ a collection of set. We define $$\limsup_{n\to \infty }A_n:=\bigcap_{n\in\mathbb N}\bigcup_{m\geq n}A_m\quad \text{and}\quad \liminf_{n\to \infty }A_n:=\bigcup_{n\in\mathbb N}\bigcap_{m\geq n}A_m.$$
These definition looks a bit coming from nowhere.

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*Is there any reason to define the limsup and liminf of sets like that?

*Is there any connection with limsup and liminf defined for a real sequence for example ?


If $(x_n)$ is a sequence in a lattice $(P,\leq)$, $$\liminf_{n\to \infty }a_n:=\sup_{n\in\mathbb N} \inf_{k\geq n}x_k\quad \text{and}\quad \limsup_{n\to \infty }x_n=\inf_{n\in\mathbb N}\sup_{k\geq n}x_k.$$
If we use standard notation, we use$$a\wedge b:=\inf\{a,b\}\quad \text{and}\quad a\vee b:=\sup\{a,b\}.$$
So, $$\limsup_{n\to \infty }x_n:=\bigvee_{n\in\mathbb N}\bigwedge_{m\geq n}x_m\quad \text{and}\quad \liminf_{n\to \infty }x_n:=\bigwedge_{n\in\mathbb R}\bigvee_{m\geq n}x_m.$$
Take the lattice $(2^{\mathbb R},\subset )$ where the order is the usual inclusion. Then, $A\wedge B$ is the biggest set contain in $A$ and $B$ and $A\vee B$ is the smallest set where $A$ and $B$ are contained. Therefore,
$$A\wedge B:=A\cap B\quad \text{and}\quad A\vee B:=A\cup B.$$
So, $$\inf_{k\geq n}A_k:=\bigwedge_{k\geq n}A_k=\bigcap_{k\geq n}A_k,$$
and thus $$\sup_{n\in\mathbb N}\inf_{k\geq n}A_k=\bigvee_{n\in\mathbb N}\bigwedge_{k\geq n}A_k=\bigcup_{n\in\mathbb N}\bigcap_{k\geq n}A_k=:\liminf_{n\to \infty }A_k.$$
Same idea with the $\limsup$.
