# Limit superior and limit inferior of sets

I have searched for the answer in wikipedia and math stackexchange. However, I do not have any background in real analysis and all the answers seem very complicated to me to understand. I am wondering if anyone can provide an intuitive answer or a graphical answer to visualize the concept easily.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
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Commented Apr 21, 2022 at 9:56
• See these questions. One characterization for a sequence $A_1,$ $A_2,$ $A_3,\;\ldots\;A_n,\;\ldots$ of sets: $\limsup A_n$ is the set consisting of all elements that belong to infinitely many of the sets in this sequence and $\liminf A_n$ is the set consisting of all elements that belong to at most finitely many of the sets in this sequence. FYI, some or all of the sets in such a sequence may be the same, and "infinitely many" and "finitely many" refer to the subscripts $1,\,2,\,3,\,\ldots$ Commented Apr 21, 2022 at 13:11

If $$(\mathcal P,\leq)$$ is a lattice (i.e. a poset s.t. all pair of element $$(a,b)$$ has an infimum and a supremum), then for a sequence $$(a_n)\in \mathcal P^{\mathbb N}$$, $$\limsup_{n\to \infty }a_n:=\inf_{n\in\mathbb N}\sup_{k\geq n}a_ n\quad \text{and}\quad \liminf_{n\to \infty }a_n:=\sup_{n\in\mathbb N}\inf_{k\geq n}a_k.$$
Now, consider $$(\mathcal P(\mathbb R),\subset )$$ is a lattice and $$\inf(A,B):=A\cap B\quad \text{and}\quad \sup(A,B):=A\cup B,$$ for all $$A,B\in \mathcal P(\mathbb R)$$.
So, for a sequence $$(A_n)\in \mathcal P(\mathbb R)^{\mathbb N}$$, $$\inf_{k\geq n}A_k=\bigcap_{k\geq n}A_k,$$ and thus $$\liminf_{n\to \infty }A_n:=\sup_{n\in\mathbb N}\inf_{k\geq n}A_k=\bigcup_{n\in\mathbb N}\bigcap_{k\geq n}A_k.$$ And similarly, $$\limsup_{n\to \infty }A_n=\bigcap_{n\in\mathbb N}\bigcup_{k\geq n}A_k.$$
• Given the OP's assertion "I do not have any background in real analysis and all the answers seem very complicated to me to understand", my guess is that your answer should include (probably at the beginning) a less symbolic description. For example, the OP probably doesn't know what $\mathcal P^{\mathbb N}$ means. Commented Apr 21, 2022 at 13:20