Let $P = \{a=x_0,x_1, \ldots, x_n=b\}$ be a partition on the interval $[a,b]$. Then $\lVert P\rVert$ denotes the norm of partition $P$, defined to be the length of the greatest subinterval of the form $[x_{i-1},x_i]$.
Let $f:[a,b] \to \mathbb{R}$ be a bounded function. Define $M_i = \operatorname{sup} f([x_{i-1},x_i])$ for all $i = 1, \ldots,n$ and the upper sum of $f$ corresponding to $P$ is $U(P,f)= \sum_{i=1}^{n}{M_i\Delta x_i}$ (where $\Delta x_i = x_i - x_{i-1}$). Here is the real question:
Suppose $P$, $Q$ are two partitions on $[a,b]$ such that $\lVert P\rVert \leq \lVert Q\rVert$. Then is it true that $U(P,f) \leq U(Q,f)$?
This seems to be true intuitively, but I can't find a solid argument to claim so. I tried to look for counter-examples too, but without any success. I would appreciate any help regarding this problem. Thanks and regards.