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June 19th, 2017, 10:57 AM   #11
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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.
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June 19th, 2017, 11:20 AM   #12
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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.
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June 20th, 2017, 07:57 AM   #13
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Quote:
Originally Posted by Dattier View Post
Hi,

The solution :

Let $\displaystyle a<x<y<b$, then

$\displaystyle \forall n\in \mathbb N, {-M_{a-1}-M_{a}}\leq f_n(a)-f_n(a-1)\leq \frac{f_n(x)-f_n(y)}{x-y} \leq f_n(b+1)-f_n(b)\leq M_{b+1}+M_b $

with the theorem of Ascoli we can conlued.

Cordially.
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.
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June 20th, 2017, 10:11 AM   #14
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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.
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June 20th, 2017, 10:32 AM   #15
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For your "doubt"

https://fr.wikipedia.org/wiki/Foncti...nction_convexe
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June 20th, 2017, 10:57 AM   #16
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Quote:
Originally Posted by Dattier View Post

That simply defines a convex function (wicki translation) and its properties, and is unrelated to your incomplete proof, wherever it came from.
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June 20th, 2017, 11:01 AM   #17
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I don't have give a proof, but an explanation.
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