## Refinements

Theorem. Let $[a,b] \subseteq K \subseteq \mathbb{R}$, and let $f: K \to \mathbb{R}$ be bounded on $[a,b]$. If $P$ is a partition of $[a,b]$ and $Q$ is a refinement of $P$, then $\mathcal{L}(f,P) \leq \mathcal{L}(f,Q) \leq \mathcal{U}(f,Q) \leq \mathcal{U}(f,P)$.

Proof. By definition, $\mathcal{U}(f,P) = \sum_{i=1}^{n} M_{i}(x_{i}-x_{i-1})$ and $\mathcal{L}(f,P) = \sum_{i=1}^{n} m_{i}(x_{i}-x_{i-1})$. In the refinement $Q$, we are "throwing" extra points in $P$. If we sample exactly the same points from each partition, there will be equality. However, if we sample different points, $\mathcal{L}(f,Q)$ and $\mathcal{U}(f,Q)$ will have larger values for $x_i-x_{i-1}$. $\diamond$

Is this correct?