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A refinement of a finite interval partition inserts additional points without removing original points. The union of two partitions is a common refinement. Refinement makes each subinterval smaller, increasing its infimum and decreasing its supremum; this proves Darboux sum refinement monotonicity.

For a partition $D=\{a=x_0<x_1<\cdots <x_n=b\}$, put $\Delta x_k=x_k-x_{k-1}$ andBoundedness makes these finite. The lower Darboux sum and upper Darboux sum areSince $m_k\le M_k$ and the widths are positive, $s(f,D)\le S(f,D)$.

Splitting a partition interval into smaller intervals can only increase each infimum and decrease each supremum. The smaller widths sum to the original width. Its new lower contribution is therefore at least its old lower contribution, and its new upper contribution at most its old upper contribution. Repeating this for every inserted point proves Darboux sum refinement monotonicity:For arbitrary partitions, take their common partition refinement $D_{\ast }=D_1\cup D_2$. ThenNo compatibility of the original partitions is required.

Finally suppose $f\ge 0$ is Riemann integrable. Then $t_k=m_k\Delta x_k\ge 0$, so every product factor is nonnegative and the exponential inequality can be multiplied safely:A lower Darboux sum is no greater than the Riemann integral, since the integral equals the supremum of all lower sums. Monotonicity of the exponential therefore gives the exponential bound for a lower-sum productFor a degenerate interval $a=b$, the empty product and the exponential are both one.