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Math Help - Proving the intersection and union of the indexed...

  1. #1
    Senior Member Danneedshelp's Avatar
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    Proving the intersection and union of the indexed...

    For every n∈N, let An=[0,1/n], closed interval of the real line. Find the union and the intersection of the indexed family {An}n∈N. Justify your answer.

    Well, I got the union:

    ∪F=[0,1] (where F=the above indexed family)

    and the intersection:

    ∩F={0}

    I am having troubles proving this.

    For the union I choose an arbitrary x∈∪F. x is an element of ∪F ⇔ ∃n∈N(x∈An). Clearly, if x∈An for some n∈N ⇒ x∈[0,1] for some n∈N. So, 0<x<1/n≤1.
    Now, for the second inclusion we choose some x∈[0,1]....?

    I don't really know how to prove either. My teacher says the archimedean principle should be in the proof. I have my final tomorrow and wouldl appreciate some help with this. It's one of the concepts I never really got in the class.

    Thanks.
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  2. #2
    MHF Contributor

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    There is nothing to prove about union.
    A_1=[0,1] so the union is [0,1].

    If 0<x<1 then \left( {\exists n} \right)\left[ {\frac{1}<br />
{n} < x} \right]\; \Rightarrow \;x \notin A_n \; \Rightarrow \;x\notin\bigcap\limits_n {A_n }
    But \left( {\forall n} \right)\left[ {0 \in A_n } \right]\; \Rightarrow \;0 \in \bigcap\limits_n {A_n }
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