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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 $m$.

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    Your 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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    Perhaps 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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    @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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    @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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    An 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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    @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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    Thanks guys i 'll try using these tools for my future posts.2012-02-12

2 Answers 2

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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.

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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:

  1. $x\in\liminf\limits_{n\to\infty} A_n$ 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 $X=\cup A_n$).
  2. $x\in\limsup\limits_{n\to\infty} A_n$ 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.