Results 1 to 2 of 2

Math Help - prove uniform convergence to 0 on I

  1. #1
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    prove uniform convergence to 0 on I

    Suppose that \displaystyle \sum u_n(x) is uniformly convergent on an interval I. Prove that the sequence { u_n(x)} is uniformly convergent to zero on I
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by wopashui View Post
    Suppose that \displaystyle \sum u_n(x) is uniformly convergent on an interval I. Prove that the sequence { u_n(x)} is uniformly convergent to zero on I
    Let \displaystyle \|\cdot\|_\infty denote the infinity norm on I, then by assumption for every \displaystyle \varepsilon>0 there exists N\in\mathbb{N} such that n\geqslant m\geqslant N implies \displaystyle \left\|\sum_{k=1}^{n}u_k(x)-\sum_{k=1}^{m}u_k(x)\right\|_\infty<\varepsilon. Take n=m+1 and conclude.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uniform convergence vs pointwise convergence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 15th 2012, 11:03 PM
  2. Replies: 1
    Last Post: October 31st 2010, 07:09 PM
  3. how to prove the Abel's test for uniform convergence?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 27th 2010, 11:58 AM
  4. Pointwise Convergence vs. Uniform Convergence
    Posted in the Calculus Forum
    Replies: 8
    Last Post: October 31st 2007, 05:47 PM
  5. Uniform Continuous and Uniform Convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 28th 2007, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum