Refinement lemma for upper and lower sums

Refinement lemma: Let $f:[a,b]\to\mathbb{R}$ be bounded . If $P$ and $Q$ are partitions of $[a,b]$ and $Q$ is a refinement of $P$ (i.e., every partition point of $P$ is also a partition point of $Q$), then $L(f,P)\le L(f,Q)\le U(f,Q)\le U(f,P),$ where $L(f,P)$ and $U(f,P)$ are the lower and upper Riemann sums of $f$ with respect to $P$.

An analogous statement holds for Riemann–Stieltjes upper/lower sums when the integrator $\alpha$ is increasing .

This lemma formalizes the idea that making the partition finer can only improve the approximation: lower sums go up and upper sums go down.

Proof sketch: It suffices to check the effect of inserting a single new point $t$ into one subinterval $[x_{i-1},x_i]$ of $P$. On the refined partition, the infimum over $[x_{i-1},x_i]$ is $\le$ the infimum over each smaller subinterval, so the weighted sum of infima cannot decrease; similarly the supremum over the large interval is $\ge$ each smaller-interval supremum, so the weighted sum of suprema cannot increase. Iterate over all inserted points.