1. Not finding help here? Sign up for a free 30min 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!

Cauchy sequence in Q not converging to zero.

  1. Mar 7, 2012 #1
    I have the following exercise:

    Let s_n be a cauchy sequence in Q(rationals) not converging to 0. Show that there exists an e(epsilon) >0 and a natural number N such that either for all n>N, s_n > e or for all n>N, -s_n >e.

    I know that since Q is not complete, we cannot assume that there exists a point (say s) such that s_n coverges to it since this s could well be in the real numbers.

    I am hessitant with the the answer I came up with since i didnt use the fact that s_n is cauchy. What i did is the following:

    From the fact that s_n does not converge to zero, then we can deduce that: there's an e>0 and N such that for all n>N, d(s_n, 0) is not less than e. so this implies that d(s_n, 0) >= e. So absolute value of (s_n - 0) > e in implies that s_n>e or -s_n>e. Now i need to show that s_n=e is not possible, but as of right now i havent been able to do so.

    Any advice and guidence will be greatly appreciated.
    Thank you very much.
     
  2. jcsd
  3. Mar 7, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Here's why you need Cauchy. Define s_n to be 1 if n is even and 1/n if n is odd. I.e. {1,1,1/3,1,1/5,1,1/7,1...}. s_n does not converge to zero, but it doesn't avoid any neighborhood of zero either. And it's not Cauchy.
     
  4. Mar 7, 2012 #3
    I see that makes sense. Thank you Dick
    Now, as far as my semi-complete proof goes, is it correct? and how could I implement the fact that s_n is cauchy in the proof?
    Thank you again
     
  5. Mar 7, 2012 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Start by getting the correct statement that corresponds to "s_n does not converge to 0". Write down the definition of s_n converges to 0 and negate it. Carefully. You didn't get it right the first time.
     
  6. Mar 7, 2012 #5
    Thank you for your patience Dick.

    So I start with this for "s_n converges to zero":
    For all ε>0, there exists a natural number N such that n≥N implies that abs. value of (s_n - 0) < ε.

    Now I tried negating every part but it doesn't seem to be right.
    What seems somewhat correct is that the negation will read:
    There exists an ε>0, for all natural numbers N such that n≥N implies that abs. value of (s_n - 0) ≥ ε

    If this is correct, will s_N=ε? so that way we take n=N out of the final statement.
     
  7. Mar 7, 2012 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Almost correct. Change it to this:
    There exists an ε>0 such that for all natural numbers N THERE EXISTS an n≥N such that abs. value of (s_n - 0) ≥ ε

    In other words, there is an infinite subsequence s_k of s_n that satisfies |s_k|> ε. That's makes sense, doesn't it?
     
  8. Mar 7, 2012 #7
    ok, so now if I replace ≤ by < then I'm able to include all n greater then N and abs. value (s_n) thus becomes only > ε. And this is it?
     
  9. Mar 7, 2012 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Well, no! You want to say that ALL of the points with n greater than some N are farther from 0 than some ε, so far you only have an infinite sequence of them. That's where being Cauchy comes in.
     
  10. Mar 7, 2012 #9
    Yes, now it's clear.
    Thank you so much Dick, I think I got it!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cauchy sequence in Q not converging to zero.
Loading...