Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions

Theorem

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

Let $f$ be a bounded real function defined on $\closedint a b$.

Let $P$ and $Q$ be finite subdivisions of $\closedint a b$.

Let $\map L P$ be the lower Darboux sum of $f$ on $\closedint a b$ with respect to $P$.

Let $\map U Q$ be the upper Darboux sum of $f$ on $\closedint a b$ with respect to $Q$.

Then $\map L P \le \map U Q$.

Proof

Let $P' = P \cup Q$.

We observe:

$P'$ is either equal to $P$ or finer than $P$

$P'$ is either equal to $Q$ or finer than $Q$

We find:

$\map L P \le \map L {P'}$ by Lower Sum of Refinement

$\map L {P'} \le \map U {P'}$ by Upper Darboux Sum Never Smaller than Lower Darboux Sum

$\map U {P'} \le \map U Q$ by Upper Sum of Refinement

By combining these inequalities, we conclude:

$\map L P \le \map U Q$

$\blacksquare$

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $2.5$: The Riemann Integral