Darboux sum refinement monotonicity
= Darboux sum refinement monotonicity
{c}
For a bounded real function, inserting partition points increases its <lower Darboux sum> and decreases its <upper Darboux sum>. This follows interval by interval because infima increase on smaller sets while suprema decrease, and the new widths sum to the original width. A common <partition refinement> consequently proves $s(f,D_1)\leq S(f,D_2)$ for arbitrary two partitions.