Interchange limit and integral under uniform convergence

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

Corollary:

• The limit function $f$ is Riemann integrable, and
• one may interchange limit and integral: $\lim_{n\to\infty}\int_a^b f_n(x)\,dx=\int_a^b \lim_{n\to\infty} f_n(x)\,dx.$

Connection to parent theorem: This is the “uniform convergence and integration ” theorem (often stated in exactly this corollary form). Uniform boundedness is automatic under uniform convergence and boundedness of some $f_N$.