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by Consider the sequence of continuous functions {f} on [0, 1] given { nx 0

by Consider the sequence of continuous functions {f} on [0, 1] given { nx 0

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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by Consider the sequence of continuous functions {f} on [0, 1] given n£ 0<a<l/n 1/n ≤ x ≤ 1 fn(2):= 0 Show that {f} has no subsequence which is convergent in uniform norm. (Hint: Show that every subsequence of {f} converges pointwise to some function. Can the subsequences converge uniformly?) (Remark: This is an example of a sequence of continuous functions bounded in the uniform norm which has no convergent subsequence.)
Transcribed Image Text:by Consider the sequence of continuous functions {f} on [0, 1] given n£ 0<a<l/n 1/n ≤ x ≤ 1 fn(2):= 0 Show that {f} has no subsequence which is convergent in uniform norm. (Hint: Show that every subsequence of {f} converges pointwise to some function. Can the subsequences converge uniformly?) (Remark: This is an example of a sequence of continuous functions bounded in the uniform norm which has no convergent subsequence.)
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