(a) Given two partitions P and P', is it always true that if P has more points than P', then $U_P - L_P \leq U_{P'} - L_{P'}$? My line of though was that if the distance between points is massive then the approximations are not necessarily better. Meaning suppose a function is defined on domain [0,1], then let $P =\{0, 0.01, 0.9, 0.99 , 1\}$ and let $P' = \{0, 1/3, 2/3, 1\}$. Then certainly its not true that $U_P - L_P \leq U_{P'} - L_{P'}$.
(b) Given two partitions $P$ and $P'$, is $U_{P \cup P'} - L_{P \cup P'} \leq U_P - L_P$ always true? how would you go about proving this? I know how to prove this is it was the case that $P \subseteq P'$ since it would mean that $P \cup P'$ has more points than P and would give a smaller difference between upper and lower sums
For (a) you are correct that it might not be true.
In general for partitions $P$ and $Q$ where $P \subseteq Q$, we have $U_Q - L_Q \le U_P - L_P$. One way to prove this is to show that $L_P \le L_Q \le U_Q \le U_P$ by noting that the intervals created by $Q$ are sub-intervals of the intervals created by $P$.
Applying this result with $Q = P \cup P'$ yields (b).