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Uniform convergence

  1. Dec 14, 2009 #1

    hth

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    1. The problem statement, all variables and given/known data

    Let fn(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly.

    2. Relevant equations

    3. The attempt at a solution

    I know why it is not uniformly convergent. Even though fn(x) goes to zero monotonically on the interval (0,1), it's not continuous on a compact interval. How would go about showing this formally/by example?
     
  2. jcsd
  3. Dec 15, 2009 #2
    A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?
     
  4. Dec 15, 2009 #3

    hth

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    Isn't that the solution to it if f_n(x) = x^(n)? How does that apply here?
     
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