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 sum of $f$ on $\closedint a b$ with respect to $P$.

Let $\map U Q$ be the upper 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 the definition of lower sum and $P'$ refining $P$
$\map L {P'} \le \map U {P'}$ by Upper Sum Never Smaller than Lower Sum
$\map U {P'} \le \map U Q$ by the definition of upper sum and $P'$ refining $Q$

By combining these inequalities, we conclude:

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

$\blacksquare$