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Math Help - I got to figure this out tonight

  1. #1
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    I got to figure this out tonight

    please help.

    I need to find a uniformly convergent sequence of functions on (0,1) whose derivatives do not converge.

    then i need to find a convergent sequence of integrable functions whose limit is not integrable

    im thinking fn = 1/nsinx for the first one and for the second I don't really have a clue.
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  2. #2
    Senior Member euclid2's Avatar
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    Quote Originally Posted by Smancer View Post
    please help.

    I need to find a uniformly convergent sequence of functions on (0,1) whose derivatives do not converge.

    then i need to find a convergent sequence of integrable functions whose limit is not integrable

    im thinking fn = 1/nsinx for the first one and for the second I don't really have a clue.
    For the second, let rn be the countable rational numbers in the interval [0,1] and define the functions.

    gn(x)= {1,0 if x= r1,r2,r2,... rn and g(x) = 1 if x = irrational and g(x)=0 if x=rational

    then g(n) converges point wise to g
    the limit function is not integrable

    Does this help or are you looking for something else?
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  3. #3
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    I think I get it... thanks I was trying to build small intervals around the rational numbers and have them shrink as n approached infinity. I couldn't quite think of a way to do that though.

    Any ideas on the first?
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