Mathematics 250 Fall 2008 Proof Practice # 3 For a continuous real-valued function f deﬁned on an interval [ a,b ] in R , and a partition P of [ a,b ], let R P ( f ) denote the lower Riemann sum for f and P , obtained by evaluating f at its minimum point in each subinterval deﬁned by P . Similarly, let R P ( f ) denote the upper Riemann sum for f and P . Let Q be another partition, and suppose that Q is a reﬁnement of P . Show that R P ( f ) ≤ R Q ( f ) ≤ R Q ( f ) ≤ R Q ( f ) . ( Mon ) Response: A main issue with writing a reasonable argument for this statement is managing the situation so that the notation doesn’t become overwhelming. One strategy for doing this is to think of a special case that is simple but representative. In this situation, such a special case can be found. We recall that a partition Q reﬁnes a partition P if every subdivision point of P also belongs to Q . In other words, Q can be obtained from P by adding more points. This suggests the idea of adding the points
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Riemann integral, Subintervals, Riemann sum, Bernhard Riemann