Uniform Boundedness of Derivative Sequence of C^1([a,b],R) Functions

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Homework Help Overview

The discussion revolves around the properties of sequences of functions in the space C^1([a,b],R), specifically focusing on the uniform boundedness of the derivative sequence given a uniformly bounded sequence of functions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of uniform boundedness on the derivatives of C^1 functions and question the definitions of bounded sequences in this context.

Discussion Status

There is an ongoing examination of the conditions under which the derivative sequence may be uniformly bounded. Some participants have raised concerns about the assumptions necessary for the conclusions being drawn, particularly regarding equicontinuity and the existence of convergent subsequences.

Contextual Notes

Participants note that the definition of a 'bounded sequence' in C^1 may require additional premises to ensure the desired properties, highlighting the need for careful consideration of the underlying assumptions in the problem.

johnson12
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Im having trouble showing that given a sequence of uniformly bounded C^1([a,b],R) functions,
the derivative sequence is uniformly bounded.
Any suggestions are helpfull
 
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If by 'uniformly bounded', you mean |f_n(x)|<M for some constant M, it's not true that the derivatives are necessarily bounded.
 
Youre right, the reason I ask is b/c I am trying to prove that every bounded sequence in C^1 has a convergent subsequence, Arzela Ascoli type problem.
 
What do you mean by a 'bounded sequence' in C^1? sin(n*x) is a bounded sequence in C^1 (in the sense |f_n|<=1). But it has no convergent subsequence. You need some sort of premise to get the equicontinuity from.
 

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