7. Let fn [a, b] R be Riemann integrable and suppose that fn →f uniformly on [a, b]. (a) Prove that f is Riemann integrable. Hint: Let € > 0 and let P be any partition of [a.b]. Use the definition of the uniform convergence of fn to f to show that there exists an N EN so that S(f, P) ≤ S(ƒn, P) + €. (b) Prove that lim [fu(x) dx = [ f(x) dr. n→∞ a a
7. Let fn [a, b] R be Riemann integrable and suppose that fn →f uniformly on [a, b]. (a) Prove that f is Riemann integrable. Hint: Let € > 0 and let P be any partition of [a.b]. Use the definition of the uniform convergence of fn to f to show that there exists an N EN so that S(f, P) ≤ S(ƒn, P) + €. (b) Prove that lim [fu(x) dx = [ f(x) dr. n→∞ a a
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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![7. Let fn [a, b] → R be Riemann integrable and suppose that fn →f uniformly
on [a, b].
(a) Prove that f is Riemann integrable. Hint: Let € > 0 and let P be any
partition of [a.b]. Use the definition of the uniform convergence of fn to
f to show that there exists an N EN so that S(f, P) ≤ S(ƒn, P) + €.
(b) Prove that
lim. ["fa(x) dx = [* f(x) dr.
a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F0aa3ffda-75f7-4238-800b-c79fa5e052c9%2Fxlbtgi2r_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let fn [a, b] → R be Riemann integrable and suppose that fn →f uniformly
on [a, b].
(a) Prove that f is Riemann integrable. Hint: Let € > 0 and let P be any
partition of [a.b]. Use the definition of the uniform convergence of fn to
f to show that there exists an N EN so that S(f, P) ≤ S(ƒn, P) + €.
(b) Prove that
lim. ["fa(x) dx = [* f(x) dr.
a
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