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

Suppose that the sequence of functions converges pointwise to on an interval , that each is differentiable on and that the sequence converges uniformly to some on . Then, is differentiable on and .

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 and for any given and in the interval,

then for these numbers.

Now, for any , for some in (or ), by the Mean Value Theorem.

Since , it follows that for some in (or ).

This equation is true for all sufficiently small (in order that be in the interval).

//now comes the fishy step

Moreover, since as h approaches 0, the interval eventually can only contain the number .

Therefore, 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 might not be the same for all , 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.)