To integrability Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assumed: If is a sequence of Riemann integrable functions defined on a compact interval which uniformly converge with limit , then is Riemann integrable and its integral can be computed as the limit of the integrals of the : In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for n sufficiently large, the graph of is within of the graph of f, and so the upper sum and lower sum of are each within of the value of the upper and lower sums of , respectively.
To integrability Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assu,med: If is a sequence of Riemann integrable functions defined on a compact interval which uniformly converge with limit , then is Riemann integrable and its integral can be computed as the limit of the integrals of the : In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for n sufficiently large, the graph of is within of the graph of f and so the upper sum and lower sum of are each within of the value of the upper and lower sums of , respectively.