Results 1 to 4 of 4

Math Help - Uniform convergence of a function sequence

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    21

    Uniform convergence of a function sequence

    Hey everyone
    Given function f(x) which is differentiable on R , and f'(x) is continuous on R, and a sequence (f_n(x)) such as : f_n(x)=n(f(x+\frac{1}{n}))-f(x))
    I need to prove that f_n(x) is uniformly converges on each interval [a,b].

    I have managed to show that (f_n(x)) approaches f'(x), but I couldn't get any further.
    I know nothing about the values of f(x), so I can't know if f_n(x) is monotonic or not , and therefore I can't use Dini's theorem. Plus , I can't say much about |f_n(x)-f'(x)| , so I can't figure out how do I show that for every epsilon > 0 , there is N such as for every n>N , |f_n(x)-f'(x)| < epsilon

    Any ideas how to get any further? Thanks people!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by Gok2 View Post
    Hey everyone
    Given function f(x) which is differentiable on R , and f'(x) is continuous on R, and a sequence (f_n(x)) such as : f_n(x)=n(f(x+\frac{1}{n}))-f(x))
    I need to prove that f_n(x) is uniformly converges on each interval [a,b].

    I have managed to show that (f_n(x)) approaches f'(x), but I couldn't get any further.
    I know nothing about the values of f(x), so I can't know if f_n(x) is monotonic or not , and therefore I can't use Dini's theorem. Plus , I can't say much about |f_n(x)-f'(x)| , so I can't figure out how do I show that for every epsilon > 0 , there is N such as for every n>N , |f_n(x)-f'(x)| < epsilon

    Any ideas how to get any further? Thanks people!
    http://www.math.uchicago.edu/~ershov/16300/uniform2.pdf
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2010
    Posts
    21
    Thank you , but I knew that.
    That's what I explained,
    I don't know nothing about |f_n(x)-f'(x)| , therefore I can't find a sequence c_n that |f_n(x)-f'(x)| <= c_n , and from the seam reason I can't find sup |f_n(x)-f(x)|, therefore I am kinda stuck...
    Any ideas people?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jan 2010
    Posts
    21
    Anyone?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: April 19th 2011, 06:39 PM
  2. Uniform Convergence of the Sequence of Compositions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 5th 2010, 08:33 PM
  3. Uniform convergence of a sequence of functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 17th 2010, 12:49 PM
  4. Uniform Convergence of a function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 10th 2010, 02:14 AM
  5. limit function / uniform convergence
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: June 28th 2009, 02:39 PM

Search Tags


/mathhelpforum @mathhelpforum