1. Prove that the series 7n (a) converges uniformly on every bounded interval (b) does not converge absolutely for any value of r 2. Suppose f is a real continuous function on R, n (t) (nt) and { n :n N) is equicontinuous on 3. Suppose is an equicontinuous sequence of functions on a compact set K, and {fn) converges . Let K be a compact metric space, and S CC(K), where C (K) is the space of all complex continuous 0,1. What conclusion can you draw about f? pointwise on K. Show that {fn converges uniformly functions on K, equipped with the metric d ( .9) = max EK | (x)-g(z) . Suppose that s is closed and pointwise bounded. Show that S is compact 5. Let {fn be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and put 岔 Fn (x)/fn (t) dt for a s sb 0 Show that there exists a subsequence {Fn^1 which converges uniformly on [a, b

1. Prove that the series 7n (a) converges uniformly on every bounded interval (b) does not converge absolutely for any value of r 2. Suppose f is a real continuous function on R, n (t) (nt) and { n :n N) is equicontinuous on 3. Suppose is an equicontinuous sequence of functions on a compact set K, and {fn) converges . Let K be a compact metric space, and S CC(K), where C (K) is the space of all complex continuous 0,1. What conclusion can you draw about f? pointwise on K. Show that {fn converges uniformly functions on K, equipped with the metric d ( .9) = max EK | (x)-g(z) . Suppose that s is closed and pointwise bounded. Show that S is compact 5. Let {fn be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and put 岔 Fn (x)/fn (t) dt for a s sb 0 Show that there exists a subsequence {Fn^1 which converges uniformly on [a, b

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 74E

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