Generalization of improper integrals under a closed interval to the regular integral

Question

Let $f$ be integrable in the regular sense. I want to show that $\int_a^b f(x) dx = lim_{r\rightarrow b^-} \int_a^r f(x) dx$

I thought about implementing each explicitly.

$\int_a^b f(x) dx =(F.T.C) F(b) - F(a)$

$lim_{r\rightarrow b^-} \int_a^r f(x) dx =(F.T.C) lim_{r\rightarrow b^-} F(r)-F(a) = F(b^-) -F(a)$.

Am I done?

Thanks in advance!

Answer 1

Sort of. You are using the wrong statement of the FTC. What you have written assumes the result. You want to use the fact that the integral is uniformly continuous for integrable functions.