Refinements

$Q$ is called a refinement of $P \iff$ $P$ and $Q$ are partitions of $[a,b]$ and $P\subseteq Q$.

In that case:

$$L(f;P) \le L(f;Q) \le U(f;Q) \le U(f;P)$$

If $P_1$ and $P_2$ are partitions of $[a,b]$, then $Q=P_1\cup P_2$ is a refinement of both $P_1$ and $P_2$.