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Prove intersection is empty

  1. Feb 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that [itex] \bigcap_{n=0}^{\inf} (0,\frac{1}{n})=\emptyset [/itex]
    3. The attempt at a solution
    since 0 is not included in our interval. eventually I will get to
    (0,0) because I could pick a real as close to zero as I wanted and there would be a natural such that [itex] \frac{1}{n}<y [/itex] therefore this intersection is empty.
    but if my orginal intersection was [0,1/n] then this would not be empty.
     
  2. jcsd
  3. Feb 2, 2012 #2

    jbunniii

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    You have the right idea, and you're correct that if (0,1/n) was replaced by [0,1/n], the intersection would not be empty. It would contain exactly one point: 0.

    You could word your proof a bit precisely as follows:

    Suppose the intersection is non-empty. Then there exists some x in the intersection:

    [tex]x \in \bigcap_{n=1}^{\infty} (0, 1/n)[/tex]

    Since x is in the intersection, it means that x must be in every interval of the form (0, 1/n), and this is impossible because...
     
  4. Feb 2, 2012 #3
    ok thanks for the input
     
  5. Feb 3, 2012 #4

    Deveno

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    you might want to show that if x > 0, there exists k in N with 1/k < x, first.
     
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