Undergrad Upper and Lower Darboux Sum Inequality

[Magnetons]

TL;DR

L(f,P) $\leq$ L(f,Q) $\leq$ U(f,Q) $\leq$ U(f,P)

Lemma
Let f be a bounded function on [a,b]. If P & Q are partitions of [a,b] and P $\subseteq$ Q , then

L(f,P) $\leq$ L(f,Q) $\leq$ U(f,Q) $\leq$ U(f,P) .

---

Question is "How can P have bigger upper darboux sum than Q while it is a subset of Q"

 

[FactChecker]

The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller $\Delta x$s.

 

[Magnetons]

The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller $\Delta x$s.