Lower Sum of Refinement

Theorem
Let $\closedint a b$ be a closed interval.

Let $P$ be a finite subdivision of $\closedint a b$.

Let $Q$ be a refinement of $P$.

Then:


 * $\map L {f, P} \le \map L {f, Q}$

where $\map L {f, P}$ denotes the lower sum of $f$ with respect to $P$.