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Math Help - A proof for the union of a family of sets..

  1. #1
    Junior Member krtica's Avatar
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    A proof for the union of a family of sets..


    • What is  \bigcup_{n \in N} A_n where  A_n=\{n, n+1, n+2,..., 2n\}<br />
?
      I want to say that the union is \mathbb{N}, could I prove this by induction? What would be the simplest approach?


    Last edited by krtica; October 20th 2010 at 05:15 PM. Reason: Fix LaTex
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  2. #2
    MHF Contributor
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    The simplest approach is to note that for every n\in\mathbb{N}, n\in A_n, so n\in\bigcup_{n\in\mathbb{N}}A_n. (And, obviously, \bigcup_{n\in\mathbb{N}}A_n\subseteq\mathbb{N} because each A_n\subseteq\mathbb{N}.)
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  3. #3
    Behold, the power of SARDINES!
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    In General if you want to show that two sets A \text { and }  B are equal show that

    A \subset B and B \subset A

    Notice that for all x \in \mathbb{N} that
     \displaystyle x \in A_x \implies x \in \cup_{n \in \mathbb{N}}^{\infty}A_n

    This should get you started
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  4. #4
    Junior Member krtica's Avatar
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    Got it. Thank you both for responding.
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