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!

Convergence of a sequence.

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume [itex] a_n [/itex] is a bounded sequence with the property that every convergent sub sequence of [itex] a_n [/itex] converges to the same limit a. Show that
    [itex] a_n [/itex] must converge to a.
    3. The attempt at a solution
    Could I do a proof by contradiction. And assume that [itex] a_n [/itex] does not converge
    to a. but then this would imply that there would be a sub sequence that did not converge
    to a and this is a contradiction because I could pick a sub sequence that converged to the same thing that [itex] a_n [/itex] did
     
  2. jcsd
  3. Feb 26, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, that will work. Just fill in the details- why does the fact that [itex]a_n[/itex] does not converge to a imply that there exist a subsequence that does not converge to a? You will need to look at several cases- the sequence does not converge or it converges to some number other than a.
     
  4. Feb 26, 2012 #3
    Could I say that eventually a sub sequence will have the same end behavior as
    [itex] a_n [/itex] Or I could take 2 sub sequences that when put together would equal
    [itex] a_n [/itex] Sub sequences aren't like subsets in the sense that a sub set could equal the set itself.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Convergence of a sequence.
  1. Sequence Convergence (Replies: 6)

  2. Sequence Convergence (Replies: 2)

Loading...