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?
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@Math_QED i know there are a zillion answers to this question -- I'm looking for something to put me on the right track. not full explanations -- which i can usually reproduce once I get started. the linked question still is in the realm of explanation – yoshi Sep 03 '18 at 16:46
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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 $$\liminf A_n=\bigcup_n \bigcap_{k\ge n} A_k.$$
Kusma
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$\limsup$ 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).
Arthur
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In this case, this is more of a mnemonic than an actual definition, but it is often enough the case that "the smallest set such that ..." is an intersection and "the largest set such that ..." is a union. They also often enough come in corresponding pairs such as this one. Another example of this would be the closures and interiors of subsets of topological spaces (closures being the intersection of all closed sets containing the given subset and interiors the union of all open sets contained in the given subset). – Arthur Sep 03 '18 at 16:47
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It would be nice if whoever down voted this could explain why. I would think you voted to signal that this answer could be better. Without concrete feedback there is no way I can know what to do better which means your down vote didn't achieve much at all. – Arthur Sep 03 '18 at 18:17
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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.
Eduardo
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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.
drhab
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