$l(f,P) \leq u(f,Q)$

Question

Let $P$ and $Q$ be two partitions of $[a,b]$ and $f$ continuous on $[a,b]$. Prove that $l(f,P) \leq u(f,Q)$, where $l(f,P)$ and $u(f,Q)$ denote the lower Darboux sum and the upper Darboux sum respectively.

Answer 1

Observe that if we pass to the common refinement $P\cup Q$,we have

$l(f,P) \le l(f,P\cup Q)$

$l(f,P\cup Q)\le u(f,P \cup Q)$ and

$u(f,P \cup Q)\le u(f,Q).$

The first and third inequality are true by direct calculation. (First add one point to $P$, and relate $\min f$ and $\max f$ on the two new subintervals,to the original one. The result follows by induction). The second inequality is a triviality.

Combining these, we have

$l(f,P) \le l(f,P\cup Q) \le u(f,P \cup Q) \le u(f,Q).$

Answer 2

Here is a hint: show that if $P$ and $Q$ are two partitions and $P \subset Q$ (interpreting a partition as a finite subset of $[a,b]$) then $l(f,P) \le l(f,Q)$ and $u(f,Q) \le u(f,P)$.

Then $$l(f,P) \le l(f,P\cup Q) \le u(f,P \cup Q) \le u(f,Q).$$