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