Why is this differential limit theorem showing uniform convergence

Question

link to the capture 1

This is so called a variant of differentiable limit theorem

I have no trouble to understand the first capture, but I am stuck on the second capture theorem.
I am pretty sure they have a subtle difference, the first one has $f_n$ uniform convergence as given, whereas the second is to show $f_n$ uniform convergence.

Especially the quota if $f_n$ converge at some $x_0\in[a,b]$, they converge (uniformly) to a continuously differentiable function and….

1. what exactly $f_n$ converge at some $x_0\in[a,b]$ means, is it just $lim_{x\to x_0}f_n(x)=f(x_0)$ and this fixing $x_0$ convergence behavior seems pointwise converges to me.
   pointwise convergence definition

2. what's the point to say something like *some conditons..blablabla…if $f_n$ pointwise converges to $f$ then it is uniform converges….. *?

3. In the second capture proof, I have no idea what that preceding theorem is.

please enlighten me on those three questions.

Answer 1

1. It means there is at least one $x_0\in[a,b]$ for which the limit $\lim_{n\to\infty}f_n(x_0)$ exists. As of yet, there is no concept of "$f$".
2. They are not saying: "if $f_n$ converges pointwise to $f$, then $f_n$ converges uniformly to $f$". Though, that would still be an interesting theorem since we will have shown the extra conditions are strong enough to convert the a priori weaker statement of "pointwise" convergence into an a priori stronger statement of "uniform" convergence. What they actually are saying is this: If $f_n$ converges pointwise at at least one point, then $f_n$ converges uniformly everywhere. This is a huuuuge strengthening, and there is a real point in saying this. We have promoted "pointwise" to "uniform" and we have promoted "somewhere" to "everywhere"
3. The preceding theorem is the "theorem that has come before". Namely, the one at the top of the page. It is applied to the sequence $(f_n')\rightrightarrows g$ :