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How I can show the truth of this relationship?

$$\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$$

The definition of $\limsup x_n$ is $$\limsup x_n = \inf_n\left[\sup_{k\ge n}x_k\right]$$

TRiG
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meysam
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    What is the definition of $\limsup$ that you're starting from? – Ben Grossmann Feb 20 '14 at 13:32
  • I added it to question. @Omnomnomnom – meysam Feb 20 '14 at 13:34
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    So what do the $\bigvee$ and $\bigwedge$ mean? Do they not mean inf / sup? – Najib Idrissi Feb 20 '14 at 13:41
  • I don't know!, in finite case we have $f\vee g = max{f,g}$ but in this case, i can't find any definition for that. @nik – meysam Feb 20 '14 at 13:52
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    @meysam: I don't have context so I can say for sure, but it's very probably they also mean sup and inf. So the two things are equal, they're just different notations. – Najib Idrissi Feb 20 '14 at 14:08
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    Somewhat related: http://math.stackexchange.com/questions/164136/what-is-the-meaning-of-bigvee-bigvee-operator – Martin Sleziak Feb 20 '14 at 14:24
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    Perhaps you could say where you have seen this. (Providing name of the textbook, link to the course notes or something similar could help the potential answerers.) But $\bigvee$ and $\bigwedge$ as a different notation for $\sup$ and $\inf$ seems to be the most probable explanation. – Martin Sleziak Feb 20 '14 at 14:26
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    @nik: I'm thinking that you are right, i re-note to context and it's seem that writer define $\sup$ as infinite $\vee$, but without definition notation! You can see text in http://books.google.com/books?id=m40ivUwAonUC&lpg=PR3&dq=aliprantis&pg=PA24#v=onepage&q&f=false – meysam Feb 20 '14 at 14:39
  • @MartinSleziak: Thanks martin, "Wedges and vees" (∧,∨) are usually used to denote "meets and joins" (respectively) in lattice theory. Roughly speaking "meet" means "greatest lower bound" and "join" means "least upper bound" clear definition. – meysam Feb 20 '14 at 14:41
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    @meysam: I quote from the link which you have provided: If we write $\sup_{k\ge n} x_k=\bigvee_{k=n}^\infty x_k$ ... then the preceding formulas can be rewritten as $\limsup x_n=\bigwedge\limits_{n=1}^\infty\left[\bigvee\limits_{k=n}^\infty x_k\right]$. I would understand this sentence as definition of $\bigvee$ and $\bigwedge$. – Martin Sleziak Feb 20 '14 at 14:53
  • @MartinSleziak: Maybe! But on mathematical definition, it's a tradition that on definition of an expression, we put the object that we want define on the left hand side of equation and use $:=$ notation for definition. – meysam Feb 20 '14 at 15:17
  • I think Meynam does not know the traditions of mathematics very well. – GEdgar Mar 06 '14 at 20:26
  • Maybe! I cant improve my skill if you guide me to some good sources! :) @GEdgar – meysam Mar 07 '14 at 11:13

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