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Thread: introduction to real analysis

  1. #1
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    introduction to real analysis

    I would like to see which steps are required in order to solve the problem.
    How to write N=N1 U N2 U N3.... U NnU.... as an infinite union of infinite subsets pairwise disjoint
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  2. #2
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    Re: introduction to real analysis

    You want to prove that it is possible? That's easy. There is the trivial case: $N_1 = N$, $N_k = \emptyset$ for all $k>1$.
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    Re: introduction to real analysis

    Thank you. I canīt believe I was complicating my life with this problem.
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  4. #4
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    Re: introduction to real analysis

    On the other hand, if you want to prove, not that there exist an intersection that can be written in that way but that every infinite union can be written as a union of [b]disjoint[b] sets, that's a bit harder..
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  5. #5
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    Re: introduction to real analysis

    True, given an infinite union of sets:

    $N = \bigcup_{k\ge 1}U_k$, and you want to find an infinite union of pairwise disjoint sets, define $A_n = \bigcup_{k=1}^n U_k$ Then define $N_1 = U_1$ and $N_{n+1} = U_{n+1} \setminus A_n$ for all $n>0$. Can you show that $N = \bigcup_{k\ge 1}N_k$? Can you also show that the $N_k$'s are pairwise disjoint?
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