# Use fundamental theorem of calculus to prove ${f'(x)}=\lim_{n\to \infty}{f'_n(x)}$ for all $x\in{[a,b]}$

Suppose $(f_n)_{n=1}^\infty$ is a sequence of continuous functions on [a, b] and that ${f'_n}$ exists and is continuous on [a, b]. Suppose further that $\lim_{n\to \infty}{f_n}=f$ uniformly and $\lim_{n\to \infty}{f'_n}=g$ uniformly for some continuous functions f and g.