1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proof that fn converges uniformly

  1. Dec 16, 2012 #1
    1. The problem statement, all variables and given/known data

    [itex]\frac{x}{1+n^2*x^2}[/itex] I must show if this converges uniformly or that it doesnt. So i must show that there is an N or that there isnt an N for which if n > N the inequality in the definition of uniform convergence holds for all x.

    2. Relevant equations


    3. The attempt at a solution
    Using point convergence one can easily see that the function converges to 0 for each x. So this can be used to see if there is an N for which |[itex]\frac{x}{1+n^2*x^2}[/itex]|<[itex]\epsilon[/itex]
  2. jcsd
  3. Dec 16, 2012 #2


    User Avatar
    Homework Helper

    It may be best to use the equivalent definition (given on the wikipedia page) that [itex]f_n(x) \to f(x)[/itex] uniformly on [itex]I[/itex] if and only if
    M_n = \sup_{x \in I} |f(x) - f_n(x)|
    is such that [itex]M_n \to 0[/itex].
  4. Dec 16, 2012 #3
    thank you, i solved it using that definition and by finding the maxima of fn by differentiation.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook