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I am wondering if anybody can help me with a problem regarding the definition of the limit superior of a sequence - or rather showing an alternate but equivalent defintion holds.

The question is: The limit superior of a numerical sequence $\{x_{k}\}$ (presumably this means the sequence is real valued) can be defined as the supremum of the set of limit points of the sequence. Show that this is the same thing as defining

$$ \limsup _{n \to \infty} ~ x_n = \bigwedge_{n=1}^{\infty} \bigvee_{k=n}^{\infty} x_k .$$

This question comes from Chapter 4 of "Probability and Measure" by Patrick Billingsley. The problem is that Billingsley assumes the reader knows what the symbols $\bigwedge$ and $\bigvee$ are - but I do not! The best I have come up with is that they are the "meet" and the "join" symbols used in a lattice? Could anybody shed some light on how this problem might be attacked?

Billingsley does give some hints to the problem. He says that the following are all equivalent: $x i.o. OR $x for some $k\geq n$, for all $n\geq 1$ OR $x<\bigvee_{k=n}^{\infty} x_k$ for all $n\geq 1$. Using this kind of logic he shows that

$$\bigwedge_{n=1}^{\infty}\bigvee_{k=n}^{\infty}x_{k} = \sup\{ x: x

Apparently the supremum of the set above can be seen to be the supremum of the limit points of the sequence - this would prove the result. I think I follow this derivation but I was taking the $\bigvee$ and $\bigwedge$ symbols to simply mean $\bigcup$ and $\bigcap$ for singleton sets $\{x_{k}\}$. I think this is the wrong assumption. I also cannot see the last assertion about the limit points of the sequence.

Any help would be much appreciated.

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    Maybe $\bigvee$ means sup and $\bigwedge$ means inf? What I have seen is that $\vee$ was used for max (between two numbers) and $\wedge$ for min; and in your case it would yield the right definition.2011-11-03
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    Many thanks Florian - the problem now makes more sense. I will see if I can complete it!2011-11-03
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    Probably it is written in this way because that way it readily generalizes to more general lattices. For example, it works [for sets](https://en.wikipedia.org/wiki/Set-theoretic_limit) with $\bigwedge\equiv\bigcap$ and $\bigvee\equiv\bigcup$.2017-02-06

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