Suppose that (f) is a sequence of functions bounded on the interval / and converging uniformly on I to f. a. Prove that f is bounded on /. b. Prove that there exists a constant A. such that, for all x €/ and neN. If(x)| ≤A and fn(x)| ≤A

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 72E
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Course : Real Analysis

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Suppose that (f.) is a sequence of functions bounded on the interval / and converging uniformly on I to f.
a. Prove that f is bounded on /.
b. Prove that there exists a constant A. such that, for all x E/ and nEN
\f(x)|<A and fn(x)|<A
Transcribed Image Text:Review Questions Suppose that (f.) is a sequence of functions bounded on the interval / and converging uniformly on I to f. a. Prove that f is bounded on /. b. Prove that there exists a constant A. such that, for all x E/ and nEN \f(x)|<A and fn(x)|<A
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