I am trying to understand a part of the following theorem:
Theorem. Assume that $f:[a,b]\to\mathbb{R}$ is bounded, and let $c\in(a,b)$. Then, $f$ is integrable on $[a,b]$ if and only if $f$ is integrable on $[a,c]$ and $[c,b]$. In this case, we have $\int_a^bf=\int_a^cf+\int_c^bf.$ Proof. If $f$ is integrable on $[a,b]$, then for every $\epsilon>0$ there exists a partition $P$ such that U(f,P)-L(f,P)<\epsilon. Because refining a partition can only potentially bring the upper and lower sums closer together, we can simply add $c$ to $P$ if it is not already there. Then, let $P_1=P\cap[a,c]$ be a partition of $[a,c]$, and $P_2=P\cap[c,b]$ be a partition of $[c,b]$. It follows that U(f,P_1)-L(f,P_1)<\epsilon\text{ and }U(f,P_2)-L(f,P_2)<\epsilon, implying that $f$ is integrable on $[a,c]$ and $[c,b]$.
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How does that last expression "follow?" Neither $P_1$ nor $P_2$ are refinements of $P$, but they are still somehow less than $\epsilon$; will that not make their difference larger? That is, $U(f,P_i)-L(f,P_i)\geqslant U(f,P)-L(f,P),$ for $i=1,2$? Thanks in advance!