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

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

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