Let $\displaystyle [a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.
Prove that if $\displaystyle P_1$ and $\displaystyle P_2$ are partitions of $\displaystyle [a,b]$ then:
$\displaystyle L(P_1,f) \leq{U(P_2,f)}$
Thanks a lot
To do this you need to understand a refinement of a partition.
Prove that if $\displaystyle Q' $ is a refinement of the partition $\displaystyle Q$ then
$\displaystyle L(Q,f) \leqslant L(Q',f) \leqslant U(Q',f) \leqslant U(Q,f) $.
Then to do your problem find a refinement of $\displaystyle P_1\cup P_2$.