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%232 Let S be a closed n-cube with v(S) > 0 and let f, g : S → R be two bounded functions. (a) Prove that for any two partitions P,P' of S, we have (f +g) < U(f,P) + U(g,P'), and L(f,P) + L(g, P') < / (G +9).

%232 Let S be a closed n-cube with v(S) > 0 and let f, g : S → R be two bounded functions. (a) Prove that for any two partitions P,P' of S, we have (f +g) < U(f,P) + U(g,P'), and L(f,P) + L(g, P') < / (G +9).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve 2(a)

#2 Let S be a closed n-cube with v(S) > 0 and let f, g : S →R be two bounded functions. (a) Prove that for any two partitions P,P' of S, we have (f +g) <U(f,P) +U(9,P'), and L(f,P) + L(g, P') < / (f+g). S (b) Prove that if f, g are integrable on S, then so is ƒ+ 9, and +g) = +| 9. (6- (First use part (a) to prove that fs(f +g) < S sƒ + J gø and [(f +9) > L,f +L_9)
Transcribed Image Text:#2 Let S be a closed n-cube with v(S) > 0 and let f, g : S →R be two bounded functions. (a) Prove that for any two partitions P,P' of S, we have (f +g) <U(f,P) +U(9,P'), and L(f,P) + L(g, P') < / (f+g). S (b) Prove that if f, g are integrable on S, then so is ƒ+ 9, and +g) = +| 9. (6- (First use part (a) to prove that fs(f +g) < S sƒ + J gø and [(f +9) > L,f +L_9)
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