Results 1 to 5 of 5

Math Help - a_n has no convergent subsequences => lima_n = infinity

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    9

    a_n has no convergent subsequences => lima_n = infinity

    Prove that if a_n is a sequence that contains no convergent subsequences, the lim abs(a_n) = infinity
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,922
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by Jazz10 View Post
    Prove that if a_n is a sequence that contains no convergent subsequences, the lim abs(a_n) = infinity
    Can you prove that each sequence contains a monotone subsequence?
    But any bounded monotone sequence converges.
    So if you prove the first then you have solved this problem.

    Here is a start. Define S = \left\{ {k:\left( {\forall n > k} \right)\left[ {a_n  > a_k } \right]} \right\}
    Two cases: S is infinite or S is finite.
    Case one gives a increasing subsequence.
    Case two gives a decreasing subsequence.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    9
    I appreciate the help and see where you are going with it, however, can you clarify what you mean by each seqeuence? as in each subsequence has a monotone subsequence itself? not really sure how that ties in to your hint at the end
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,922
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by Jazz10 View Post
    I appreciate the help and see where you are going with it, however, can you clarify what you mean by each seqeuence? as in each subsequence has a monotone subsequence itself? not really sure how that ties in to your hint at the end
    Frankly, after I posted that I realized a simpler proof.
    Define \left( {\forall N \in \mathbb{Z}^ +  } \right)\left[ {S_N  = \left\{ {a_k :\left| {a_k } \right| \leqslant N} \right\}} \right].

    What if some S_J were infinite?
    Would we have a convergent subsequence? WHY?

    One real problems with help-sites such as this is simply that helpers have no idea what theorems the poster knows and therefore can use.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2009
    Posts
    9
    ooo, nice, so using your setup, if we had an set Sj that were infinite, then that infinite set would be bounded and would therefor have an accumulation point, and thus we could find subsequence that was convergent. Since there are no convergent subsequences, there cannot exist a j that bounds any set, and therefore the sequence must diverge to inifity or negative infinity.

    is that where you were going?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Two convergent subsequences, divergent sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 3rd 2011, 07:59 PM
  2. Replies: 5
    Last Post: January 26th 2010, 05:56 AM
  3. Convergent Subsequences....
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 19th 2009, 10:23 AM
  4. Convergent subsequences
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 20th 2009, 05:04 PM
  5. Replies: 3
    Last Post: April 6th 2009, 11:03 PM

Search Tags


/mathhelpforum @mathhelpforum