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

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.1: Numerical Integration

Problem 21E

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