Uniform limit of integrable functions

Uniform limit of integrable functions: Let $f_n:[a,b]\to\mathbb{R}$ be Riemann integrable for every $n$. If $f_n\to f$ uniformly on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

With the additional conclusion of interchanging limit and integral, one also gets convergence of the corresponding Riemann integrals.