Refinement lemma for upper and lower sums

Refinement lemma: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be bounded. If $P'$ is a refinement of a partition $P$, then

where $U(f,P)$ and $L(f,P)$ denote the upper sum and lower sum of $f$ with respect to $P$.

This monotonicity under refinement underlies the definition of the Riemann integral as the common value of the infimum of upper sums and the supremum of lower sums.