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 June 19th, 2017, 10:57 AM #11 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Ascoli Theorem “By definition, a sequence { fn }n∈N 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  fn  belonging to the sequence, and every x ∈ [a, b]. (Here, M must be independent of n and x.) The sequence is said to be equicontinuous if, for every ε > 0, there exists δ > 0 such that 1) |fn(x)-fn(y)| < $\displaystyle \epsilon$ whenever |x − y| < δ  for all functions  fn  in the sequence. (Here, δ may depend on ε but not x, y or n.) One version of the theorem can be stated as follows: Consider a sequence of real-valued continuous functions { fn }n ∈ 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 { fnk }k ∈ N that converges uniformly.- wicki So you are showing that your convex functions satiefy the conditions of Ascoli theorem. Why didn’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 don’t know that. June 19th, 2017, 11:20 AM #12 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 You applied this résult for [-n,n] with $\displaystyle n \in \mathbb N$ "Why didn’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, 07:57 AM   #13
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 Originally Posted by Dattier Hi, The solution : Let \$\displaystyle a
I was simply pointing out that if fn(x) is convex and bounded on [a,b] with monotonically increasing (decreasing) slope bounded at a and b, THEN your proof shows that fn(x) satisfies conditions of Ascoli's theorwem. Under these conditions it is a nice proof.

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, 10:11 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 10:19 AM. June 20th, 2017, 10:32 AM #15 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 For your "doubt" https://fr.wikipedia.org/wiki/Foncti...nction_convexe June 20th, 2017, 10:57 AM   #16
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 Originally Posted by Dattier For your "doubt" https://fr.wikipedia.org/wiki/Foncti...nction_convexe

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, 11:01 AM #17 Member   Joined: May 2017 From: France Posts: 57 Thanks: 1 I don't have give a proof, but an explanation. Tags beautiful, convex, functions, result Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Eureka Art 9 December 12th, 2015 02:10 AM Vasily Applied Math 1 June 23rd, 2012 01:11 PM Rak Real Analysis 1 December 1st, 2009 08:45 AM vikram Applied Math 0 February 4th, 2009 02:27 PM

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