1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Science Advisor

    you might want to show that if x > 0, there exists k in N with 1/k < x, first.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook