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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