When does convergence of function imply convergence of its derivative?
Solution 1:
You need to add the assumption that $F_n'$ converges uniformly on a closed interval $[a,b]$. In fact:
Theorem: Suppose $\{f_n\}$ is a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point $x_0$ on $[a,b]$. If $\{f_n'\}$ converges uniformly on $[a,b]$, then $\{f_n\}$ converges uniformly on $[a,b]$, to a function $f$, and $$f'(x)=\lim_{n\to\infty}f_n'(x),\quad(a\leq x\leq b).$$
Source: Rudin, Principles of Mathematical Analysis, Theorem 7.17.