4. (30 points) In lecture, we've mentioned how the lower and upper Darboux sums 'bound' possible...

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4. (30 points) In lecture, we've mentioned how the lower and upper Darboux sums 'bound' possible Riemann sums. In this problem, let's make that statement more rigorous. (a) Let f : [a,b] R be a bounded function, and let P = {xo,, n} be a partition of [a, b]. We say a set of points 7 := {C1,, Cn} is a tagging of P if xi-1 Ci xi for all i=1,..., n. Given any partition P and tagging 7 of P, show that n L(P, f) f(ci)\xi U(P, ) i=1 (b) Suppose f [a, b] R is Riemann integrable. Show that for all > 0, there exists a partition P such that for any tagging 7, rb L a n (c)^r; 4. (30 points) In lecture, we've mentioned how the lower and upper Darboux sums 'bound' possible Riemann sums. In this problem, let's make that statement more rigorous. (a) Let f : [a,b] R be a bounded function, and let P = {xo,, n} be a partition of [a, b]. We say a set of points 7 := {C1,, Cn} is a tagging of P if xi-1 Ci xi for all i=1,..., n. Given any partition P and tagging 7 of P, show that n L(P, f) f(ci)\xi U(P, ) i=1 (b) Suppose f [a, b] R is Riemann integrable. Show that for all > 0, there exists a partition P such that for any tagging 7, rb L a n (c)^r;