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Math Help - Convergence and Continuity of function sequences

  1. #1
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    Convergence and Continuity of function sequences

    Hi,

    I would like to know how to prove the following:

    If fn converge pointwise to F on D and each fn is monotone increasing on D, then F is monotone increasing on D (and also for monotone decreasing).

    Can someone please help me on this? Thank you!
    ( Do we have to make like a gn(x) = fn(x) - F(x) ...)
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  2. #2
    Senior Member Tinyboss's Avatar
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    Assume F isn't increasing...then you have x<y with F(x)>F(y). Pick an epsilon smaller than half of |F(x)-F(y)|, and find an n such that f_n is within epsilon of F at both x and y.
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  3. #3
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    But I think that we have to somehow use the fact that fn converge pointwise to F on D . . .
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  4. #4
    Senior Member Tinyboss's Avatar
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    Quote Originally Posted by zxcv View Post
    But I think that we have to somehow use the fact that fn converge pointwise to F on D . . .
    That's why n exists.
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  5. #5
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    Can you please further elaborate as it is not too clear to me.
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  6. #6
    Senior Member Tinyboss's Avatar
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    Quote Originally Posted by zxcv View Post
    Can you please further elaborate as it is not too clear to me.
    Since \{f_n\} converges pointwise on D, if we have x,y\in D then there's some N_x such that |f_n(x)-F(x)|<\varepsilon for every n>N_x, and likewise there exists an N_y for the point y. Take the maximum of those two, and continue the proof.
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