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Math Help - Limit set question

  1. #1
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    Limit set question

    Suppose you have an infinite tower of subsets of the integers: A_1 \subsetneq A_2 \subsetneq \cdots \subset \mathbb{Z} such that for any x \in \mathbb{Z}, there exists k \in A_n such that x \equiv k \pmod{2^n}. Can I determine whether or not x \in \bigcup_{n\ge 1}A_n?
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  2. #2
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    Re: Limit set question

    I figured out the answer is no. For example, if A_n = [2n] = \{1,\ldots, 2n\}, then this satisfies the claim that for any x \in \mathbb{Z}, there exists k \in A_n with x \equiv k \pmod {2^n}. However, \bigcup_{n\ge 1} A_n = \mathbb{N}, so it would not contain any non-positive integers.
    Last edited by SlipEternal; April 24th 2014 at 05:47 AM.
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