7.4. A Criterion for Integrability. THEOREM 7.11. Let ƒ : [a, b] → R be bounded. Then f is integrable if and only if for all € > 0 there exists a partition P of [a, b] such that U(f, Pe) — L(f, Pc) < €. COROLLARY 7.12. f: [a, b] → R is integrable if and only if there exists a sequence P₁ of partitions of [a, b] such that lim [U(f, Pn) L(f, Pn)] = 0. n→∞

7.4. A Criterion for Integrability. THEOREM 7.11. Let ƒ : [a, b] → R be bounded. Then f is integrable if and only if for all € > 0 there exists a partition P of [a, b] such that U(f, Pe) — L(f, Pc) < €. COROLLARY 7.12. f: [a, b] → R is integrable if and only if there exists a sequence P₁ of partitions of [a, b] such that lim [U(f, Pn) L(f, Pn)] = 0. n→∞

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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