Let $I=[a_1,b_1]\times...\times[a_n,b_n]$ an interval in $\mathbb{R^n}$ and $P=\{P_1,...,P_n\}$ a partition of $I$. Let $f:I \rightarrow \mathbb{R}$ be a bounded function in $I$. For each $J \in P$ we define $m_J(f)=inf\{f(x):x \in J\}$, $M_J(f)=sup\{f(x):x \in J\}$.
We define the inferior sum of $f$ with respect of $P$ as $L(f,P)= \sum_{J\in P} m_J(f)V(J)$.
I'm trying to prove this statement:
If $P$ is finer than $Q$ then $L(f,Q) \leq L(f,P)$.
Here is my proof:
Let $J\in Q$ and $J' \in P$ such that $J' \subset J$. Because $J$ contains more elements than $J'$, then $m_J(f) \leq m_{J'}(f) \Rightarrow m_J(f)V(J') \leq m_{J'}(f) V(J') \Rightarrow \sum_{J'\in J} m_J(f)V(J') \leq \sum_{J'\in J} m_{J'}(f)V(J') \Rightarrow m_J(f)V(J) \leq \sum_{J'\in J} m_{J'}(f)V(J')$
$L(f,Q)=\sum_{J\in Q} m_J(f)V(J) \leq \sum_{J\in Q} (\sum_{J'\in J} m_{J'}(f)V(J'))= \sum_{J'\in Q} m_{J'}(f)V(J') = L(f,P) $
Is this OK?