Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma/Proof 1

Theorem

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

Let $c$ be a real number.

Let $a < c < b$.

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

Let $\map L S$ be the lower Darboux sum of $f$ on $\closedint a b$ where $S$ is a subdivision of $\closedint a b$.

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

Let:

$Q = P \cup \set c$.

Then:

$\map L Q \ge \map L P$

Proof

This is an instance of Lower Sum of Refinement.

$\blacksquare$