I recently was asked to prove the following in my real analysis course:
Suppose $f:[a.b]\rightarrow \mathbb{R}$ is a bounded function and $c\in(a,b)$. Prove that $f$ is Riemann integrable on $[a,b]$ if and only if $f$ is Riemann integrable on $[a,c]$ and $f$ is Riemann integrable on $[c,b]$. Furthermore, prove that if these conditions hold, then $$\int_{a}^{b} f=\int_{a}^c f+\int_{c}^{b} f.$$
My instructor told me that I put together a good proof, but I now realize that I don't totally understand some of the steps I made. One step in particular has me scratching my head. I will include my proof up until this confusing step for clear context:
Proof. If $f$ is Riemann integrable on $[a,b],$ then there exists, by definition, some partition $P$ of $[a,b]$ such that for any $\epsilon>0$ we have $$U(f,P)-L(f,P)<\epsilon.$$ Let $P=\{a=x_0,x_1,x_2,\ldots, x_{n-1},x_n=b\}$ be such a partition. Then, construct the partition $P'$ to be $P'=P\cup\{c\}$, and note that this is a refinement of $P$. Now consider the two partitions formed by $$Q_1=P'\cap [a,c]$$ and $$Q_2=P'\cap [c,b].$$ Observe that $Q_1$ is simply the partition with endpoints $a$ and $c$, i.e. $Q_1=\{a=x_0,x_1,x_2,\ldots,c=x_m\}$ for some $m\in\{1,2,\ldots,n-1\}$. A symmetrical statement can be made for $Q_2$ with $a$ and $c$ replaced by $c$ and $b$, respectively. Since $P'=Q_1\cup Q_2$, we have \begin{align*} U(f,P')&=U(f,Q_1)+U(f,Q_2),\\ L(f,P')&=L(f,Q_1)+L(f,Q_2). \end{align*} We can then write $U(f,Q_1)=U(f,P')-U(f,Q_2)$ and $L(f,Q_1)=L(f,P')-L(f,Q_2)$. Subtracting these two equations yields \begin{align*} U(f,Q_1)-L(f,Q_1)&=U(f,P')-L(f,P')-(U(f,Q_2)-L(f,Q_2))\\ &\leq U(f,P)-L(f,P)<\epsilon. \end{align*}
My question is actually the very last line here.
In particular, why was I allowed to conclude that $$U(f,P')-L(f,P')-(U(f,Q_2)-L(f,Q_2))\leq U(f,P)-L(f,P)?$$
I think the main linking concept is that upper sums get smaller over refinement, and lower sums get bigger over refinement. See Theorem 6-3 in Kirkwood, 2nd Ed., p. 134, for example.
You could insert a step or two to make it more explicit: \begin{align*} U(f, Q_1)-L(f, Q_1)&=U(f, P')-L(f, P')-(U(f, Q_2)-L(f, Q_2)) \\ &\le U(f, P')-L(f, P') \\ &\le U(f, P)-L(f, P). \end{align*}