So this thought popped into my head while attempting to solve a grad-level real analysis assignment. Though this question is probably basic, I am finding it hard to justify a claim I am making (I am not a math major).
Firstly, the question I am trying to tackle basically states the following: $\{f_n, n = 1, ...\}$ are real valued and twice differentiable. They also converge uniformly to an arbitrary function, $f$, and the second derivatives, $f_n''$, are uniformly bounded. I essentially have to show that the first derivative, $f_n'$, converges uniformly to $f'$. I suppose this is a standard problem (precisely, this is the Lipschitz condition).
I was thinking in the lines of proof by contradiction. At first, I assumed $f_n'$ doesn't converge to $f'$. So, roughly speaking, there will be some $x^*$ where the difference between $f_n(x)$ and $f_m(x)$ is greater than some $\epsilon > 0$ $\forall n,m > n_0$ for some $n_0 \in \mathbb{N}$. Then, I wished to consider a $\delta$-neighborhood of $x^*$. Then, for all $x$ in this neighborhood, without loss of generality, if we assume that $f_n'$ converges to $f'$ pointwise, then I intuitively think that the second derivative should become arbitrarily large at that point. Because, for all $x$ in this neighborhood, $| f_n'(x) - f_m'(x)| < \epsilon$ (for sufficiently large $n, m$) while but at $x^*$ it is greater than the same $\epsilon$ (where I pick $\epsilon$ as in the definition of $x^*$).
I am pretty sure that to make the above argument rigorous, I need to use the fact that the second derivative is uniformly bounded. I also have a hunch that if I can argue that since the second derivative is uniformly bounded, then for any chord (that is line segment joining two points of a curve of a function in $\mathbb{R}$, just my "term" for this) then I can pick one of the two points to be $x^*$ and the other point to be in the $\delta$-neighborhood and then somehow use this property to write in mathematical terms the argument I presented above.
I want to know if my argument is sound and if I can indeed claim the above property via uniform boundedness and if indeed the above technique can prove my argument mathematically.
A clarification of the original question: The question needs me to show that $f_n' \rightarrow f'$ uniformly and that $\exists C > 0$ such that $|f'(x) - f'(y)| \leq C|x - y| \hspace{1 pc}\forall x, y \in [a, b]$.