Uniform convergence and integration

Uniform convergence and integration (Riemann): Let $f_n:[a,b]\to\mathbb{R}$ be Riemann integrable for each $n$, and suppose $f_n\to f$ uniformly on $[a,b]$. Then:

• $f$ is Riemann integrable on $[a,b]$, and
• the integrals converge to the integral of the limit: $\lim_{n\to\infty}\int_a^b f_n(x)\,dx=\int_a^b f(x)\,dx.$

This theorem justifies passing limits through integrals when convergence is uniform, and it is a standard tool in approximation and series-of-functions arguments.

Proof sketch: Uniform convergence gives $\|f-f_n\|_\infty\to 0$. For the integral identity, use $\left|\int_a^b f-\int_a^b f_n\right| \le \int_a^b |f-f_n| \le (b-a)\|f-f_n\|_\infty \to 0.$ To show $f$ is integrable, combine integrability of some $f_n$ (choose $n$ large) with the estimate that upper /lower sums for $f$ differ from those of $f_n$ by at most $(b-a)\|f-f_n\|_\infty$.