I want to prove this:
If $P^{*}$ is a finer partition than $P$, then show that $L(f,P, \alpha) \leq L(f,P^{*}, \alpha)$ and $U(f,P^{*}, \alpha) \leq U(f,P, \alpha)$.
If you have a set $S = \{1.2.3 \}$ then adding an element can change the infimum. If $S' = \{\frac{1}{2},1.2.3 \}$ then $\inf S' \leq \inf S$. I don't get how the above holds then.