7. Let fn [a, b] → R be Riemann integrable and suppose that fn → funiformly 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 € N so that S(f, P) ≤ S(ƒN, P) + €. (b) Prove that lim m. [*J₁ (2) "fu(2)de = $* f(2) dz. fn dx n→∞
7. Let fn [a, b] → R be Riemann integrable and suppose that fn → funiformly 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 € N so that S(f, P) ≤ S(ƒN, P) + €. (b) Prove that lim m. [*J₁ (2) "fu(2)de = $* f(2) dz. fn dx n→∞
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 \( f_n : [a, b] \to \mathbb{R} \) be Riemann integrable and suppose that \( f_n \to f \) uniformly on \([a, b]\).
(a) Prove that \( f \) is Riemann integrable. **Hint:** Let \(\epsilon > 0\) and let \( P \) be any partition of \([a,b]\). Use the definition of the uniform convergence of \( f_n \) to \( f \) to show that there exists an \( N \in \mathbb{N} \) so that \( S(f, P) \leq S(f_N, P) + \epsilon \).
(b) Prove that
\[
\lim_{n \to \infty} \int_a^b f_n(x) \, dx = \int_a^b f(x) \, dx.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2Fd22ae42b-4230-4b9a-8c3a-9c6a15a57acd%2Fk0szy5_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let \( f_n : [a, b] \to \mathbb{R} \) be Riemann integrable and suppose that \( f_n \to f \) uniformly on \([a, b]\).
(a) Prove that \( f \) is Riemann integrable. **Hint:** Let \(\epsilon > 0\) and let \( P \) be any partition of \([a,b]\). Use the definition of the uniform convergence of \( f_n \) to \( f \) to show that there exists an \( N \in \mathbb{N} \) so that \( S(f, P) \leq S(f_N, P) + \epsilon \).
(b) Prove that
\[
\lim_{n \to \infty} \int_a^b f_n(x) \, dx = \int_a^b f(x) \, dx.
\]
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