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Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)

Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 17E: In each of the following parts, a relation R is defined on the power set (A) of the nonempty set A....
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Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)
Transcribed Image Text:Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)
Exercise 3 Suppose A and B are non-empty subsets of the real numbers such that for any x EA and y B, we have x ≤y. Prove that sup(A) ≤ inf(B). Exercise 5 Complete all the details of Proposition 7.9. Let f [a, b] R be a bounded function; say m≤ f(x) ≤ M for all x [a, b]. Then we have m(b − a) ≤ L(f) ≤ U(ƒ) ≤ M(b − a) You will need to use Proposition 7.7 and Exercise 3 above to show that L(f) ≤U(f)
Transcribed Image Text:Exercise 3 Suppose A and B are non-empty subsets of the real numbers such that for any x EA and y B, we have x ≤y. Prove that sup(A) ≤ inf(B). Exercise 5 Complete all the details of Proposition 7.9. Let f [a, b] R be a bounded function; say m≤ f(x) ≤ M for all x [a, b]. Then we have m(b − a) ≤ L(f) ≤ U(ƒ) ≤ M(b − a) You will need to use Proposition 7.7 and Exercise 3 above to show that L(f) ≤U(f)
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