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Math Help - Is this proof correct? (about the derivative of the limit of a sequence of functions)

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    Is this proof correct? (about the derivative of the limit of a sequence of functions)

    Hi, I am trying to prove the following statement (part of a bigger proof):

    Suppose that the sequence of functions  \left\{f_n \right\} converges pointwise to f on an interval  \left\[a,b \right\] , that each f_n is differentiable on  \left\[a,b \right\] and that the sequence  \{f_n'\} converges uniformly to some  g on  \left\[a,b\right\] . Then,  f  is differentiable on  \left\(a,b\right\) and  f'=g .

    My attempt at a proof (which may or may not be correct, but one step seems a little fishy to me):

    ************************************************** *********
    Since  \lim_{n\rightarrow \infty}{f_n(x)}} = f(x) and  \lim_{n\rightarrow \infty}{f_n(x+h)}} = f(x+h) for any given  x and  x+h in the interval,

    then  \lim_{n\rightarrow \infty}{\frac{f_n(x+h)-f_n(x)}{h}}} = \frac{f(x+h)-f(x)}{h} for these numbers.


    Now, for any  n ,  \frac{f_n(x+h)-f_n(x)}{h} = f_n'(y) for some  y in  \left\[x,x+h \right\] (or  \left\[x+h,x\right\] ), by the Mean Value Theorem.

    Since  \lim_{n\rightarrow \infty}{f_n'(y)}} = g(y) , it follows that \frac{f(x+h)-f(x)}{h} = g(y) for some  y in  \left\[x,x+h \right\] (or  \left\[x+h,x\right\] ).

    This equation is true for all sufficiently small |h|>0 (in order that  x+h be in the interval).

    //now comes the fishy step

    Moreover,  \lim_{h\rightarrow\0}{g(y)}} = g(x) since as h approaches 0, the interval  \left\[x,x+h\right] eventually can only contain the number  y = x .

    Therefore,  \lim_{n\rightarrow\infty}{\frac{f(x+h)-f(x)}{h}} = g(x) which proves the theorem.
    **********************************************

    This seems simple, but I'm worried about how rigorous the last step is. I tried using a more formal "epsilon-delta" argument but I could not construct the inequalities necessary (one reason is that h depends on n, and at the same time n depends on h). Also, I am worried that I did not even use uniform convergence of the {fn'} sequence...
    Is this flawed, or is my proof actually valid?

    Thank you!

    (EDIT: I think I found another flaw. The y that works for  f_n'(y) might not be the same for all  n , which makes taking the limit as I did incorrect. I am really stumped because it doesn't seem like such a hard statement to prove, but I just can't get my head around it... A few days earlier I had thought of a proof using an "epsilon over 3" argument but I forgot what it was (I lost the paper) and I am not even sure it was correct either.)
    Last edited by boorglar; June 25th 2012 at 10:02 PM.
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