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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

    http://en.wikipedia.org/wiki/Uniform_convergence

    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

    pasmith

    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
    [tex]
    M_n = \sup_{x \in I} |f(x) - f_n(x)|
    [/tex]
    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.
     
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