Infimum of Upper Sums Never Smaller than Lower Sum

Theorem

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

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

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

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

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

Then:

$\inf_P \map U P \ge \map L S$

Proof

From Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions, $\map L S$ is a lower bound for the real set:

$T = \leftset {\map U P: P}$ is a finite subdivision of $\rightset {\closedint a b}$

Since $\inf_P \map U P$ is the infumum of $T$:

$\inf_P \map U P \ge \map L S$

Hence the result.

$\blacksquare$