cuts. If is a refinement of then and If P1, P2 are two partitions of the same interval (one need not be a refinement of the other), then and it follows that Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if and together make a tagged partition (as in the definition of the Riemann integral), and if the Riemann sum of is equal to R corresponding to P and T, then From the previous fact, Riemann integrals are at least as strong as Darboux integrals: if the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. There is (see below) a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well. {| class="toccolours collapsible collapsed" width="90%" style="text-align:left" !Details of finding the tags |- | For this proof, we shall use superscripts to index and variables related to it. Let be a sequence of arbitrary partitions of such that , whose tags are to be determined. By the definition of infimum, for any , we can always find a such that Thus, Let , we have Taking limits of both sides, Similarly, (with a different sequences of tags) Thus, we have which means that the Darboux integral exist and equals . |} See also Regulated integral Lebesgue integration Minimum bounding rectangle Notes References Darboux integral at Encyclopaedia of Mathematics Definitions of mathematical integration