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

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

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

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

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

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

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

//now comes the fishy step

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

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

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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 $\displaystyle f_n'(y) $ might not be the same for all $\displaystyle 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.)