Suppose lim||P||0 S (P, f) = I E R. Show that if (Pn)neN is a sequence...

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Suppose lim||P||→0 S (P, f) = I E R. Show that if (Pn)neN is a sequence of partitions with ||Pn|| → 0, then limn→∞ S (Pn, f) = I. Let f € R[a, b]. i) If ƒ ≥ 0 on [a, b], show that √f € R[a, b]. ii) For n € N, show that fn = R[a, b] where f'(x) = (f(x))". iii) If there is a 6 > 0 such that f(x) ≥ 6 for all x = [a, b], then 1/f = R[a, b]. A similar result holds if f(x) < -6 for all x = [a, b]. Suppose lim||P||→0 S (P, f) = I E R. Show that if (Pn)neN is a sequence of partitions with ||Pn|| → 0, then limn→∞ S (Pn, f) = I. Let f € R[a, b]. i) If ƒ ≥ 0 on [a, b], show that √f € R[a, b]. ii) For n € N, show that fn = R[a, b] where f'(x) = (f(x))". iii) If there is a 6 > 0 such that f(x) ≥ 6 for all x = [a, b], then 1/f = R[a, b]. A similar result holds if f(x) < -6 for all x = [a, b].

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Suppose limPn f I R Show that if Pn is a sequence of partitions with Pn 0 then limn SPn f I Solution Let 0 be given Since limPn f I there exists a pos... View the full answer