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a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R. Prove that f is integrable.

a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R. Prove that f is integrable.

Elements Of Modern Algebra
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 7E: For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={...
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a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R. Prove that f is integrable. Hint: Show that for every partition P, |L(f, P) – L(fn, P)| < ||f – fn|| (b – a), and similarly for the upper sum. Combine this with Riemann's criterion. b) Give an example of a sequence g„: [0, 1] → R of integrable functions converging pointwise to the non-integrable function (Dirichlet's function) g(x) = { O 1 :x irrational :x rational
Transcribed Image Text:a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R. Prove that f is integrable. Hint: Show that for every partition P, |L(f, P) – L(fn, P)| < ||f – fn|| (b – a), and similarly for the upper sum. Combine this with Riemann's criterion. b) Give an example of a sequence g„: [0, 1] → R of integrable functions converging pointwise to the non-integrable function (Dirichlet's function) g(x) = { O 1 :x irrational :x rational
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