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Cauchy sequence

Cauchy sequence

Advanced Engineering Mathematics
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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Let Y = R([a, b]) be the set (actually a vector space) of bounded, real-valued, Riemann integrable functions on the closed and bounded interval [a, b], and let X = C([a, b]), be the subset (actually a vector subspace) of continuous, real-valued functions on [a, b]. For f, g = Y, define d(f,g) = sup |f(x) — g(x)|. x= [a,b] Does every uniformly Cauchy sequence in R([a, b]) converge uni- formly to a function in R([a, b])?
Transcribed Image Text:Let Y = R([a, b]) be the set (actually a vector space) of bounded, real-valued, Riemann integrable functions on the closed and bounded interval [a, b], and let X = C([a, b]), be the subset (actually a vector subspace) of continuous, real-valued functions on [a, b]. For f, g = Y, define d(f,g) = sup |f(x) — g(x)|. x= [a,b] Does every uniformly Cauchy sequence in R([a, b]) converge uni- formly to a function in R([a, b])?
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