(a) Let f [a, b] R be a bounded function, and let P = {xo,...,xn} be...

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(a) Let f [a, b] R be a bounded function, and let P = {xo,...,xn} be a partition of [a, b]. We say a set of points 7 := {C1, ..., Cn} is a tagging of P if xi-1 i xi for all i=1, ..., n. Given any partition P and tagging 7 of P, show that n L(P,f)f(c)Ax 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, h n f(c) Axi (a) Let f [a, b] R be a bounded function, and let P = {xo,...,xn} be a partition of [a, b]. We say a set of points 7 := {C1, ..., Cn} is a tagging of P if xi-1 i xi for all i=1, ..., n. Given any partition P and tagging 7 of P, show that n L(P,f)f(c)Ax 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, h n f(c) Axi