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Math Help - intersection of union

  1. #1
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    intersection of union

    i do not understand how this notation works. \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m
    help me please.
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  2. #2
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    Hi, i don't know if the following visualisation is helpful:
    \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m = \bigcap_{n=1}^\infty (A_n \cup A_{n+1}\cup A_{n+2}\cup A_{n+3}\ldots) =

    = (A_1 \cup A_{2}\cup A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap (A_{2}\cup A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap (A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap\ldots

    but maybe this little proposition will make things clear:

    x \in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m \mbox{ if and only if } x \in A_k \mbox{ for infinitely many } k \in \mathbb{N}.

    Proof: "=>" assume x\in A_k for only finitely many k\in\mathbb{N}, let l be the biggest such  k, then for all m\ge l+1 we have x\not \in A_m. This means x \not \in \bigcup_{m\ge l+1}A_m, so x\not\in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m.

    "<=" assume x\not\in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m, this means there's some n\in\mathbb{N} such that x \not \in \bigcup_{m\ge n}A_m, that is, x\not\in A_m for all m\ge n, so x belongs to only finitely many A_k.
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  3. #3
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    So if we define lim_{n \rightarrow \infty}sup  A_m= \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m, what is sup <br />
A_5?
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  4. #4
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    Quote Originally Posted by Kat-M View Post
    So if we define lim_{n \rightarrow \infty}sup  A_m= \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m, what is sup <br />
A_5?
    this doesn't make sense.
    it is "limes superior of a sequence of sets \{A_n:\, n\in \mathbb{N}\}" , by definition:
    \limsup_{n\rightarrow \infty}A_n =\bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m
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