June 19th, 2017, 11:57 AM  #11 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
Ascoli Theorem â€œBy definition, a sequence {â€‰fnâ€‰}nâˆˆN of continuous functions on an interval I = [a, b] is uniformly bounded if there is a number M such that fn(x) $\displaystyle \leq$ M for every function â€‰fnâ€‰ belonging to the sequence, and every x âˆˆ [a, b]. (Here, M must be independent of n and x.) The sequence is said to be equicontinuous if, for every Îµ > 0, there exists Î´ > 0 such that 1) fn(x)fn(y) < $\displaystyle \epsilon$ whenever x âˆ’ y < Î´â€‰ for all functions â€‰fnâ€‰ in the sequence. (Here, Î´ may depend on Îµ but not x, y or n.) One version of the theorem can be stated as follows: Consider a sequence of realvalued continuous functions {â€‰fnâ€‰}n âˆˆ N defined on a closed and bounded interval [a, b] of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence {â€‰fnkâ€‰}k âˆˆ N that converges uniformly. wicki So you are showing that your convex functions satiefy the conditions of Ascoli theorem. Why didnâ€™t you say so?. Originally you said fn(x) was defined on R and result applied for all R. As for your proof, your inequalities use that a convex function on [a,b] has a monotonically increasing or decrasing slope, which implies 1) if all the slopes at a and b are bounded. But you donâ€™t know that. 
June 19th, 2017, 12:20 PM  #12 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
You applied this rÃ©sult for [n,n] with $\displaystyle n \in \mathbb N$ "Why didnâ€™t you say so?." Your question, is it a joke ? It is the very principle of an enigma, we don't give the solution, with the enigma. 
June 20th, 2017, 08:57 AM  #13  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
Note: these are not the conditions specified in OP which remains unanswered.* The OP should really read, if fn(x) are convex and bounded on [a,b], do they satisfy conditions for Ascoli's theorem. Yes: your theorem under underlined conditions. *I doubt that there is such a thing as a bounded convex function with monotonically increasing (decreasing) slope on R. Cordially.  
June 20th, 2017, 11:11 AM  #14 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
Dear Zylo, It suffices to have the sequence bounded simply and not uniformly. Bye Last edited by Dattier; June 20th, 2017 at 11:19 AM. 
June 20th, 2017, 11:32 AM  #15 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1  
June 20th, 2017, 11:57 AM  #16  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote: That simply defines a convex function (wicki translation) and its properties, and is unrelated to your incomplete proof, wherever it came from.  
June 20th, 2017, 12:01 PM  #17 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
I don't have give a proof, but an explanation.


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