I would like your opinion about the following question:
Let $\{f_n\}$ be a sequence of integrable functions in $[a,b]$which uniformly converges to integrable function $f$, then let $F(x)=\int_a^xf(t)dt $, and $F_n(x)=\int_a^xf_n(t)dt$, I want to prove that $F_n$ uniformly converges to $F$ on $[a,b]$.
Can I just use the continuous of the integral and say that $\lim_{n \to \infty} F_n(x)=\lim_{n \to \infty}\int_{a}^{x}f_n(t)=\int_{a}^{x}\lim _{n \to \infty}f_n(t)dt=\int_{a}^{x}f(t)dt=F(x)?$
Thanks a lot!