I wish to prove:
Suppose $f:[a, b] \rightarrow \mathbf{R}$ is a bounded function and $P, P^{\prime}$ are partitions of $[a, b]$ such that the list defining $P$ is a sublist of the list defining $P^{\prime}$. Then $$ L(f, P,[a, b]) \leq L\left(f, P^{\prime},[a, b]\right) $$
Is the following correct?
Let $P:=\{b_1, b_2,...,b_n\}$ and $P':=\{a_1, a_2,...,a_{N}\}$. Wlog, suppose $N=n+1$ (the process outlined below may be applied $k$ times for the general $N=n+k$ case).
Let $a^*$ be the element in $P'$ that is not in $P$, and suppose $a^*\in (b_{w}, b_{w+1})$ for some $w \in \{1,....,n\}$. So then the list of intervals corresponding to $P'$ is of the form $[b_1,b_2], [b_2,b_3],...,[b_w, a^*],[a^*, b_{w+1}],...,[b_{n-1}, b_n]$. So: $$L\left(f, P^{\prime},[a, b]\right)=\sum_{i=1}^{w-1}(b_{i}-b_{i-1})\inf_{[b_{i-1},b_i]}f +(a^*-b_{w-1})\inf_{[b_{w-1},a^*]}f + (b_{w}-a^*)\inf_{[a^*,b_{w}]}f + \sum_{w+1}^{n}(b_{i}-b_{i-1})\inf_{[b_{i}-b_{i-1}]}f \\ \geq \sum_{i=1}^{w-1}(b_{i}-b_{i-1})\inf_{[b_{i-1},b_i]}f \hspace{1mm}+(a^*-b_{w-1})\inf_{[b_{w-1},b_{w}]}f + (b_{w}-a^*)\inf_{[b_{w-1},b_{w}]}f+\sum_{w+1}^{n}(b_{i}-b_{i-1})\inf_{[b_{i}-b_{i-1}]}f \\ =\sum_{i=1}^{w-1}(b_{i}-b_{i-1})\inf_{[b_{i-1},b_i]}f + (b_{w}-b_{w-1})\inf_{[b_{w-1},b_{w}]}f+\sum_{w+1}^{n}(b_{i}-b_{i-1})\inf_{[b_{i}-b_{i-1}]}f=L(f, P,[a, b])$$
where the inequality follows from the fact that $[b_{w-1},a^*], [a^{*}, b_{w}] \subset [b_{w-1}, b_{w}]$, and the properties of $\inf$.