Results 1 to 5 of 5

Math Help - Converging Subsequence

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    225

    Converging Subsequence

    Let x_n be a bounded sequence and let x=sup {x_n: n is a natural number} and let x_n < x for all n. Prove there is a subsequence convergent to x.

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member abhishekkgp's Avatar
    Joined
    Jan 2011
    From
    India
    Posts
    495
    Thanks
    1

    Re: Converging Subsequence

    Quote Originally Posted by veronicak5678 View Post
    Let x_n be a bounded sequence and let x=sup {x_n: n is a natural number} and let x_n < x for all n. Prove there is a subsequence convergent to x.

    First shoe that \forall M \in \mathbb{Z}^+ \, \exists k \in (x_n) such that  k \in (x-\frac{1}{N},x). Denote this k as x_{n_M}.

    now show that the subsequence (x_{n_M}) converges to x.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2008
    Posts
    225

    Re: Converging Subsequence

    Thanks for the help. That makes sense now, but how do I come up with things like that? I never know where to start these proofs.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,444
    Thanks
    1863

    Re: Converging Subsequence

    You should have noticed that the key to the condition "x_n< x for all n" is that no member of the sequence is equal to x. For example, if {x_n}= {1/n} then x= sup {1/n}= 1 but {1/n} is a sequence converging to 0 so has no subsequence converging to 1. Now, the fact, that x is a sup but NOT a maximum should lead you to what abhishekkgp did.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Aug 2008
    Posts
    225

    Re: Converging Subsequence

    I see. I'm trying to think more like that. Thanks for the answers.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. subsequence converging to p...
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 27th 2011, 10:24 PM
  2. Subsequence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 5th 2010, 06:17 AM
  3. Converging Subsequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 9th 2010, 08:47 PM
  4. Show there exists a SUBSEQUENCE converging to L
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 23rd 2010, 11:38 AM
  5. subsequence
    Posted in the Calculus Forum
    Replies: 7
    Last Post: July 7th 2008, 09:45 AM

Search Tags


/mathhelpforum @mathhelpforum