Results 1 to 3 of 3

Math Help - analysis

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    172

    analysis

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,615
    Thanks
    1578
    Awards
    1
    Quote Originally Posted by flower3 View Post
    Here is an outline of a proof.
    We may as well prove this statement: Every bounded sequence contains a convergent subsequence.

    Lemma 1, Every sequence, \left(x_n\right), contains a monotone subsequence.
    Easy proof: Let S=\{n:~\forall j>n,~x_j>x_n\}. Two cases: S is infinite or finite.
    In the first case the subsequence is increasing.
    In the second case the subsequence is non-increasing.

    Lemma 2, Every bounded monotone sequence converges.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2006
    From
    Florida
    Posts
    228
    I assume that you mean a sequence in \mathbb{R}, since this need not be true in other metric spaces.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with Analysis please :)
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 16th 2009, 04:38 AM
  2. analysis help
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 8th 2009, 03:11 PM
  3. analysis help
    Posted in the Calculus Forum
    Replies: 0
    Last Post: January 19th 2009, 02:38 PM
  4. analysis
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 15th 2008, 03:07 PM
  5. Analysis Help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 8th 2008, 05:59 PM

Search Tags


/mathhelpforum @mathhelpforum