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Please help me prove this:

Let $A_1,A_2,\ldots$ be subsets of $\Omega$. Prove that $A_n\to A$ if and only if $I_{A_n}(\omega)\to I_A(\omega)$ for every $\omega\in\Omega$ (so that convergence of sets is the same as pointwise convergence of their indicator functions).

Note: $I_A(\omega)=1$ if $\omega\in A$, and $0$ if $\omega\notin A$. Use in the proof that $$\operatorname{lim\;inf}\limits_n\; x_n=\bigvee_{k=1}^\infty\bigwedge_{n=k}^\infty x_n\quad\text{ and }\quad\operatorname{lim\;sup}\limits_n\; x_n=\bigwedge_{k=1}^\infty\bigvee_{n=k}^\infty x_n.$$

Thank you very much!

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    What definition of convergence of sets are you using?2012-06-25
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    Since you are new, I want to let you know some things about the site. In order to get the best possible answers, it is helpful if you say what your thoughts on the problem are so far; this will prevent people from telling you things you already know, and help them write answers at an appropriate level. Also, people are much happier to help those who show they've tried the problem themselves first.2012-06-25
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    For Robert Israel, The sequence (An) converges to A if lim inf An = lim sup An = A. Thank you2012-06-25
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    Thanks, Zev Chonoles. We've only had two class meetings so far so I only know a few concepts about probability. I'm really having a hard time thinking where to start since I'm not even familiar with how a sequence of real numbers converge (which is required to be used in the proof).2012-06-25
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    It helps to think of the problem in the following way. The liminf of a sequence of sets is the set of points that occur in every single $A_n$ for large enough $n>N$, where $N$ depends on the point in question. Alternately the limsup is the set of points that occur infinitely often, in that they belong to some infinite sequence of $A_{n_k}$. Now rewrite these definitions in terms of indicator function convergence.2012-06-25

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