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A simple question: uniform convergence of sequences

  1. Nov 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Find sequences {f_n} {g_n} which converge uniformly on some set E, but such that {f_n*g_n} does not converge uniformly on E.

    2. Relevant equations



    3. The attempt at a solution
    I looked at some sequences of functions known to be convergent but not uniformly convergent and tried to find {f_n} and {g_n} from that. However, I have not enough sequences at hand, I could not find a proper sequence.
    I guess it is not the right way to solve this question. But I've no idea how to construct such. Any hint? Thanks
     
  2. jcsd
  3. Nov 3, 2008 #2

    Dick

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    If you are looking at sequences that are not uniformly convergent, you are looking in the wrong place. Hint: is f_n(x)=x+1/n is uniformly convergent on R?
     
  4. Nov 4, 2008 #3
    Thanks!
    your f_n is uniformly convergent on R, with limit function f(x)=x.
    and f_n(x)*f_n(x) = g_n(x) = x^2+2x/n+1/(n^2) converges to h(x)=x^2, but not uniformly, since |g(n)-h(n)|>=2.
    Well, I do not understand how to get this function from nowhere. However, the behavior of this function is so simple that it could be memorized easily...
     
  5. Nov 4, 2008 #4

    Dick

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    To get it from nowhere, just think 'big number'*'small epsilon' isn't necessarily small if 'big number' can go to infinity.
     
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