How to remember $\limsup$ and $\liminf$ of sets

Question

I have an understanding of what $\limsup$ and $\liminf$ are as set theoretic limits. I've also found a lot of explanations of how to think about them. I'm having trouble reproducing the arguments quickly though. Besides memorizing, do you have a quick way of remebering which order the intersection and union should be in?

Answer 1

$\limsup$ is a set which is as small as possible given certain restraints (it is a "least upper bound", after all). That means it should be an intersection (of unions).

Similarly, $\liminf$ is a set which is as large as possible, given certain restraints. So it should be a union (of intersections).

Answer 2

Well, I remember this by translating to logic (using $\bigvee$ for $\exists$ and $\bigwedge$ for $\forall$) and back. $\limsup A_n$ is the set of points in infinitely many of the $A_n$, and $\liminf A_n$ the set of points that are eventually in all of them (so clearly $\liminf A_n\subset \limsup A_n$).

If $x\in \limsup A_n$, then for infinitely many $n$, $x\in A_n$. That means for every $n$ ($\bigvee\limits_n$) there exists a $k\ge n$ ($\bigwedge\limits_{k\ge n}$) with $x\in A_k$. Translating from logic to set theory, $$\limsup A_n=\bigcap_n \bigcup_{k\ge n} A_k.$$

Similarly, if $x\in \liminf A_n$, then there exists $n$ ($\bigvee\limits_{n}$) such that for all $k\ge n$ ($\bigwedge\limits_{k\ge n}$), $x\in A_k$. So $$\liminf A_n=\bigcup_n \bigcap_{k\ge n} A_k.$$

Answer 3

When we are talking about sequence of real numbers I always think in limsup as the largest acumulation point and liminf as the lowest acumulation of a sequence if they exists, so if I forgot the order I ask to my self "how to produce the largest (respec. lowest) acumulation point?" and then the order of sup and inf came for me naturaly...

$$ \limsup x_n = \inf_{n} \sup_{n\geq k}x_k~~\text{and}~~\liminf x_n = \sup_{n} \inf_{n\geq k}x_k $$

In the sequence of set contexts I like to think in the same path where we have sup I put the union, and where we have inf I put intersection.

Answer 4

In order theory notation $\bigvee A$ stands for $\sup A$ and $\bigwedge A$ for $\inf A$.

Then: $$\limsup\cdots=\inf\sup\cdots=\bigwedge\bigvee\cdots\approx\bigcap\bigcup\cdots$$ and:$$\lim\inf\cdots=\sup\inf\cdots=\bigvee\bigwedge\cdots\approx\bigcup\bigcap\cdots$$

This could serve as mnemonic.