Results 1 to 2 of 2

Math Help - Rearrangement of a Sequence

  1. #1
    Jes
    Jes is offline
    Newbie
    Joined
    Oct 2008
    Posts
    12

    Rearrangement of a Sequence

    Hi, I want to prove that a converging sequence will still converge to the same limit even if all the terms are permuted. I figured the problematic terms are the first N-1 terms since they can be located anywhere in the permuted sequence. To get rid of them, I want to find the largest permuted index of these finite terms and define that as the new starting point for the tail sequence. Here's what I got:

    Suppose \{a_n\} converges to L. Let \phi : \mathbb{N} \rightarrow \mathbb{N} be a bijection. Define a new sequence by \{ a_{\phi(n)} \}. We want to show this new sequence converges to L .

    Given any \epsilon > 0, there is some N' \in \mathbb{N} such that if n \geq N', then d(a_n, L) < \epsilon. Define N = \max \{\phi(1), \phi(2), \ldots, \phi(N'), N' \} and suppose \phi(n) \geq N. Then d(a_{\phi(n)}, L) \leq \epsilon.

    It seems like that should do it but I'm always skeptical when it seems easy. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    That looks like a good proof to me.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Algebraic Rearrangement
    Posted in the Algebra Forum
    Replies: 5
    Last Post: August 10th 2011, 09:12 AM
  2. Algebra rearrangement help...
    Posted in the Algebra Forum
    Replies: 2
    Last Post: July 27th 2011, 01:53 PM
  3. Help with an algebraic rearrangement
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 11th 2011, 07:38 AM
  4. Rearrangement of Terms
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 11th 2011, 07:23 AM
  5. simple rearrangement which i cant do!
    Posted in the Algebra Forum
    Replies: 5
    Last Post: December 2nd 2010, 12:17 PM

Search Tags


/mathhelpforum @mathhelpforum