There are typical cases of various complexity where notions are defined or solvable by liminf or limsup. At the same time, from my perception, they are touched only superficially during the course of study up until (mostly facultative) measury theory.

How do you work with these limit points? How do you know when it is appropriate to try to apply these concepts? And how do you process a liminf-statement mentally?

As an example consider the root test for a series over a real sequence. Or the meausure-theoretic $$\limsup_{n \to \infty}A_n := \{x \in X | x \in A_n\text{ for infinitely many } n \in \mathbb{N}\} = \bigcap_{n \in \mathbb{N}} \bigcup_{k=n}A_k \equiv \inf_{n \in \mathbb{N}} \sup_{k \ge n} A_k$$ for an arbitrary sequence $(A_n) \in \mathcal{P}(X)^{\mathbb{N}}$ and a set $X$, where $ \mathcal{P}(X)$ denotes the power set of $X$.

I'd like to develop intuition for this concept.


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