Results 1 to 3 of 3

Thread: unbounded sequence

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    172

    unbounded sequence

    let $\displaystyle x_n $ be an unbounded sequence . prove that there is a subsequence $\displaystyle x_{n_k} $ of $\displaystyle x_n $ such that $\displaystyle \frac{1}{x_{n_k}} \to 0 $
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Nov 2009
    Posts
    12
    If epsilon > 0 then 1/x_n - 0 < epsilon => x_n > 1/epsilon. If you choose a strictly increasing subsequence of {x_n} (which is possible since it is unbounded), then all but a finite number of x_n > 1/epsilon, so 1/x_n converges to 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,742
    Thanks
    2814
    Awards
    1
    Quote Originally Posted by flower3 View Post
    let $\displaystyle x_n $ be an unbounded sequence . prove that there is a subsequence $\displaystyle x_{n_k} $ of $\displaystyle x_n $ such that $\displaystyle \frac{1}{x_{n_k}} \to 0 $
    Suppose that $\displaystyle \left(x_n\right)$ is not bounded above.
    Then $\displaystyle \left( {\exists x_{N_1 } } \right)\left[ {1 < x_{N_1 } } \right]$.
    Let $\displaystyle B_1 = \max \left\{ {2,N_1 } \right\}$ then $\displaystyle \left( {\exists x_{N_2 } } \right)\left[ {B_1 < x_{N_2 } } \right]$.
    Now you can see how to construct the required subsequence.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] to prove that the sequence is unbounded.
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: May 6th 2011, 04:15 AM
  2. Bounded sequence whose set of acc points is unbounded.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 13th 2009, 03:59 PM
  3. Unbounded Sequence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 4th 2009, 12:30 AM
  4. Unbounded sequence?
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: Oct 4th 2008, 04:35 AM
  5. Unbounded sequence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Sep 6th 2008, 07:59 AM

Search Tags


/mathhelpforum @mathhelpforum