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Infinite intersection of indexed sets

  1. Jul 16, 2011 #1
    Consider the set [itex] A_n=\{0.9, 0.99, 0.999,...\} [/itex], where the greatest element of [itex] A_n [/itex] has [itex] n [/itex] 9s in its decimal expansion. Then [itex]0.999\ldots=1\in\bigcap_{n=1}^\infty{A_n}[/itex]. Is this possible even though [itex]\not\exists{n}(1\in{A_n})[/itex]?

    Edit: I see that [itex]0.999\ldots=1\not\in\bigcap_{n=1}^\infty{A_n}[/itex]. Sorry :(.
     
    Last edited: Jul 16, 2011
  2. jcsd
  3. Jul 17, 2011 #2

    HallsofIvy

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    It is quite common that the limit of a sequence has some property that no member of the sequence has. Nothing at all strange about that.
     
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