Applying the Mean Value Theorem to sequences of function

1. Jan 18, 2012

jdinatale

As usual, I typed up the problem and my attempt in LaTeX:

Maybe I'm not applying the MVT correctly, but my result does not seem to help me solve the problem in anyway. What are your thoughts?

2. Jan 18, 2012

micromass

Staff Emeritus
Maybe you shouldn't apply the mean value theorem on [a,b] but rather on [x,x0]??

3. Jan 19, 2012

jdinatale

Thanks, that would help. Ok I tried your suggestion and here is the result. The problem is now making use of the $f'_n(c) - f'_m(c)$ since they do not appear in the triangle inequality. And also, I am not using the assumption that $f'_n$ converges uniformly.

4. Jan 19, 2012

micromass

Staff Emeritus
In your triangle inequality, you have a term

$$|(f_n(x)-f_m(x))-(f_n(x_0)-f_m(x_0))|$$

Isn't this the same as the numerator of what you got after applying the mean value theorem??

5. Jan 20, 2012

jdinatale

Well, no, it's slightly different:

$$|f_n(x) + f_m(x) - f_n(x_0) - f_m(x_0)|$$

We need it to be $$f_n(x) - f_m(x)$$ instead of +

Oh btw, I noticed in your profile that it says you are 15. Is that true? Because that's incredible that you know so much math at that age, really astonishing.

6. Jan 20, 2012

micromass

Staff Emeritus
Oh, how did you get that + in the mean value theorem? That isn't correct. The mean value theorem says that

$$(f_n-f_m)^\prime(c)(x-x_0)=(f_n-f_m)(x)-(f_n-f_m)(x_0)$$

You seemed to have applied the mean-value theorem to $f_n+f_m$...

7. Jan 22, 2012

jdinatale

Micromass, I believe I have found the solution! Here is my completed proof, what do you think?

8. Jan 22, 2012

micromass

Staff Emeritus
Hmm, you know that that c is depend on n and m right?? You act like there is only one c, but there are multiple c's.

And to be sure you really get it: where exactly do you need uniform convergence of $(f^\prime_n)_n$??

9. Jan 22, 2012

jdinatale

Oh, well it was my understanding that the c was only dependent on the $$[x, x_0]$$. I guess I didn't understand that c could be different for different m's and n's.

We need uniform convergence of $(f^\prime_n)_n$ because we have to use the Cauchy Criterion for Uniform Convergence.

Well, since my proof is incorrect then, could you point me in the right direction so that I can try to complete this?

10. Jan 22, 2012

micromass

Staff Emeritus
The mean value theorem on a function actually implies:

$$|g(a)-g(b)|\leq |a-b| \sup_{c\in [a,b]}|g^\prime(c)|$$

I suggest you use this. It will become apparent where the uniform convergence is used.

11. Jan 22, 2012

jdinatale

I guess I'm not quite sure I understand. My book has no mention of this variation to the MVT. Are you certain that it is necessary to solve this problem?

12. Jan 22, 2012

micromass

Staff Emeritus
It's quite easy to prove from the MVT.

I think it's the easiest and cleanest way...