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.