Intuition on Limit Sup and Inf for sequences of sets So if $(x_n)_{n=1}^{\infty}$ is a sequence in $\mathbb{R}$, then we can define
$$a_n = \sup\{x_k, k \geq n\}$$
Then $\limsup_{n \to \infty}x_n = \lim_{n\to\infty}a_n$, similarly we can do the same thing for infimum.
So now suppose $(X_n)_{n=1}^{\infty}$ is a sequence of sets. Since sets are partially ordered, I think a reasonable way to define a supremum is the smallest set that contains them all.
So something like $A_n = \bigcup_{k \geq n}X_k$.
Based on this, that I think it is reasonable to define $\limsup_{n \to\infty}X_n = \lim_{n\to\infty}A_n$.
But the actually definition is this
$$\limsup_{n \to\infty}X_n = \bigcap_{n \geq 1}A_n.$$
So what's the intuition behind the actual definition? I mean I understand that both definition would lead to the same thing. But the "official" definition just doesn't look like the limit of anything...

For a sequence $(x_n)_{n=1}^\infty$, we have
$$
\limsup_{n\to\infty}x_n = \lim_{n\to\infty}\sup_{k\geq n}x_k = \inf_{n}\sup_{k\geq n}x_k
$$
because $(\sup_{k\ge n}x_k)_n$ is a decreasing sequence. Replacing $\sup$ by union and $\inf$ by intersection, we obtain a natural definition
$$
\limsup_{n\to\infty}X_n := \bigcap_n\bigcup_{k\ge n} X_k.
$$
This definition is usually presented, rather than $\limsup_{n\to\infty}X_n:=\lim_{n\to\infty}\cup_{k\ge n}X_k$, because for this to make sense one first has to define what it means to take a limit of sets.
As an aside, the way one "thinks" about the limit supremum and infimum for sets is as follows:
\begin{align*}
\limsup_{n\to\infty}X_n &= \{x \mid x\in X_n\ \text{for infinitely many $n$}\} \\
\liminf_{n\to\infty}X_n &= \{x \mid x\in X_n\ \text{for all but finitely many $n$}\} 
\end{align*}
