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Uniformly convergent on R

  1. Nov 30, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that [tex]f_{n} = \frac{x}{\sqrt{1+nx^2}}[/tex] is uniformly convergent to 0 on all real numbers


    2. Relevant equations
    {f_n} is said to converge uniformly on E if there is a function f:E->R such that for every epsilon >0, there is an N where n>=N implies that | f_n(x) - f(x) | < epsilon, for all x in E.


    3. The attempt at a solution
    Let f(x) = lim n-> infty f_n(x), and let epsilon > 0. Then it is obvious, that if n>1, that as n -> infty, the limit goes to 0, and thus we would need to show that [tex]\frac{x}{\sqrt{1+nx^2}} < epsilon[/tex]| , which happens as long as n > [tex]\frac{\frac{x}{epslion}-1}{x^2}[/tex]. So, I feel like I got it, except for the 'obvious' statement that f(x) = 0. Am I doing this right? Thanks ahead of time.
     
  2. jcsd
  3. Nov 30, 2008 #2

    Dick

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    Homework Helper

    i) you want to show lim n->infinity f_n(x)=0 for all x. That tells you the pointwise limit is 0. ii) your algebra is wrong, what happened to the sqrt? ii) There shouldn't be an x in your expression for choosing n. That's what 'uniform' convergence is all about.
     
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