I am currently taking first course in real analysis following Ross's Elementary Analysis textbook. When I was introduced to lim sup and lim inf, I found it hard to manage to play around or make meaningful conclusions from them because the terms are not in explicit forms but are in form of suprema and infima of a set.
For example, how to start tackling this problem:
if $\lim \sup |a_n| > 0$, prove that $\lim \sup |a_n|^\frac{1}{n} \geq 1$.
What might be the first ideas that I should try thinking about
Generally, how ones can approach lim sup and lim inf, starting intuitively and working technically?
Best Answer
My analysis teacher taught us about these concepts using the ideas of "eventually" and "frequently". A sequence is "eventually" in a set if there exists some $N$ such that for every $n>N$, we have $a_n$ in the set. A sequence is "frequently" in a set if, for every $N$, there exists an $n>N$ with $a_n$ in the set.
Step one is to meditate on that paragraph until it makes sense with your understanding of the words "eventually" and "frequently".
Once you have that, $\lim\sup$ works this way. To say $\lim\sup\{a_n\}=c$ means that the sequence $\{a_n\}$ is eventually less than $c+\epsilon$, and frequently greater than $c-\epsilon$, for any small $\epsilon>0$.
The first claim, that the sequence is eventually less than $c+\epsilon$, means that there is no accumulation point of the sequence strictly larger than $c$. Adding in the other fact, that it is frequently greater than $c-\epsilon$, we know that $c$ is an accumulation point itself.
For $\lim\inf$, just turn all that on its head. If $\lim\inf\{a_n\}=c$, then $\{a_n\}$ is eventually greater than $c-\epsilon$, and it is frequently less than $c+\epsilon$, again for any small $\epsilon>0$.
Once you have these definitions in your head, you can gain intuition by playing around with practice sequences that have various kinds of limit points. You should be able to find such exercises in any decent analysis textbook. If you google "lim sup lim inf practice problems", you can find many more.
A nice thing about these concepts, "eventually" and "frequently", is that they can be easily adapted to talk about other kinds of limits in mathematics, such as limits of sequences of sets, and possibly even direct limits of algebraic and topological structures, although I have less experience with those, so I'm not too sure.