Results 1 to 2 of 2

Math Help - why does this sequence of functions converge uniformly on different sets?

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    why does this sequence of functions converge uniformly on different sets?

    f_{n}(x)=x^{n} on [0,1] and

    (0,1)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466
    Quote Originally Posted by szpengchao View Post
    f_{n}(x)=x^{n} on [0,1] and

    (0,1)
    What exactly is your question? A sequence may converge uniformly on many sets. In fact, one of the first things you should have learned about "uniform convergence" is "if a sequence converges uniformly on a set, then it converges uniformly on any subset of that set". Another is "if a sequence converges on a compact set, then it converges uniformly on that set".

    A standard way of proving that a sequence of functions converges uniformly on (0, 1) is to look at the closure of that set, [0, 1]. Since that is both closed and bounded it is compact so you only have to prove the sequence converges on it to know that it converges uniformly. Then, since (0,1) is a subset the sequence also converges uniformly on (0,1).
    Last edited by HallsofIvy; October 18th 2008 at 02:00 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 13th 2010, 12:44 AM
  2. Replies: 4
    Last Post: June 26th 2010, 07:04 AM
  3. Uniformly convergent sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 16th 2009, 08:50 PM
  4. show this series converge uniformly to a continuous function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 4th 2009, 12:07 AM
  5. Replies: 1
    Last Post: October 18th 2008, 03:50 PM

Search Tags


/mathhelpforum @mathhelpforum