How to show that $\{E_n\}$ is a convergent sequence if and only if there is no point $x\in X$ such that $x\in E_n$, $x\in X - E_m$ hold for infinitely many $n$ and infinitely many $m$.
Convergent sequence of sets
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0Your phrasing was poor: "$x\in E_n, x\in X-E_n$ for infinitely many $n$" is always false, because read literally, you are asking for **both** conditions to hold for the same $n$, infinitely many times. Also: perhaps you can look at how your posts are getting $\LaTeX$-ified and try to do it yourself in the future? – 2012-02-11
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0Perhaps you can add what you mean by "convergent sequence of sets"? I assume it means $\liminf\{E_n\} = \limsup\{E_n\}$, but it would be good to be sure. – 2012-02-11
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0@ArturoMagidin surprisingly that is just how it is in this eastern economy edition book. I felt uncomfortable changing it to post here as i was not confident in myself that i would n't unknowingly change the meaning of the question. Maybe I should read a different book on this topic perhaps. – 2012-02-11
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0@ArturoMagidin I am trying to learn Latex, but the issue i have been encountering is that i would like a table where i can look up what to type for each symbol, unfortunately i have n't come across a thorough one. Please share if you know of one. – 2012-02-11
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1An incomplete but very useful list is [here](http://amath.colorado.edu/documentation/LaTeX/Symbols.pdf). A comprehensive guide can be found [here](ftp://tug.ctan.org/pub/tex-archive/info/symbols/comprehensive/symbols-letter.pdf). You can also learn by right-clicking on $\LaTeX$ expressions and selecting ‘Show Math As’ and ‘TeX Commands’. – 2012-02-11
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0@Hardy: Googling quickly gave [this](http://omega.albany.edu:8008/Symbols.html), [this](http://web.ift.uib.no/Teori/KURS/WRK/TeX/symALL.html), [this](http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols), and [this](ftp://tug.ctan.org/pub/tex-archive/info/symbols/comprehensive/symbols-letter.pdf), among many, many, many others. There's also [DeTeXify](http://detexify.kirelabs.org/classify.html) (write the symbol, get the code) and [Web equation](http://webdemo.visionobjects.com/equation.html) (similar, but for entire expressions). How hard did you look? – 2012-02-11
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0Thanks guys i 'll try using these tools for my future posts. – 2012-02-12
2 Answers
Use the ideas at the beginning of Henno’s answer to this earlier question of yours. If $x\in E_n$ for infinitely many $n$, then $x\in\limsup_nE_n$. If $x\in X\setminus E_n$ for infinitely many $n$, then it is not true that $x\in E_n$ for all but finitely many $n$, so $x\notin \liminf_nE_n$. Thus, if $x\in E_n$ for infinitely many $n$ and $x\in X\setminus E_n$ for infinitely many $n$, then $$x\in\limsup_nE_n\setminus\liminf_nE_n\;,$$ so $\liminf_nE_n\ne\limsup_nE_n$, and $\lim_nE_n$ does not exist.
Now you just need to check that all of the implications reverse.
I'm going to assume that you define convergence of an arbitrary family of sets in terms of $\limsup$ and $\liminf$; recall that $$\begin{align*} \liminf_{n\to\infty}A_n &= \bigcup_{m=1}^{\infty}\bigcap_{n=m}^{\infty}A_n,\\ \limsup_{n\to\infty}A_n &= \bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}A_n. \end{align*}$$ Now, what you want to show is that:
Proposition. Let $\{A_n\}$ be a family of sets. Then:
- $x\in\liminf\limits_{n\to\infty} A_n$ if and only if there exists $N\gt 0$ such that $x\in A_n$ for all $n\geq N$; if and only if $x$ is, eventually, in all $A_n$; if and only if $x$ is in $X-A_n$ for only finitely many $n$ (where $X=\cup A_n$).
- $x\in\limsup\limits_{n\to\infty} A_n$ if and only if $x$ is in infinitely many $A_n$.
Now, $\liminf E_n$ is always contained in $\limsup E_n$. So in order for equality to hold, you need every element of $\limsup E_n$ to be in $\liminf E_n$. Verify that this holds precisely when your condition holds.