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


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