Possible proof of Gabriel's Theorem?
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From: Jason on 2 Mar 2005 10:15 > > However, since both n and s tend to infinity, the limit is taken > > once over the entire sum. With n and s coinciding, we get > > gabriel's proof. > > > > This seems to work but is this last step legal? > _What_ last step? David, The last step being that since n coincides with s and the limits are both taken over infinity, the inner limit falls away to be replaced by the outer limit. What do you think? Jason
From: Jason on 2 Mar 2005 10:31 Denis, The original post included everything. It then became unmanageable as cyber thugs kept posting garbage to the thread. You can find the details at: http://www.geocities.com/john_gabriel In response to your assumptions about someone trying to discredit newton, cauchy and all the others: I don't see how you can say this, because even gabriel acknowledges newton's work. The above result is not something you can prove using riemann sums. Please be kind enough to prove the result in full using riemann sums as you claimso that we can all see. Have never seen gabriel's theorem stated before, nor a connection between f(x+w)-f(x) and the integral in the way gabriel shows it. If I can prove it, then I intend to use it as a teaching aid because it requires nothing else, i.e. no knowledge of real analysis or deep properties of the real number system. Jason
From: Jason on 2 Mar 2005 10:35 Which is quite correct. I believe this agrees with the classic definition. f'(0) is undefined for f(x) =abs(x)
From: denis feldmann on 2 Mar 2005 17:02 Jason a ýcrit : > Denis, > > The original post included everything. It then became unmanageable as > cyber thugs kept posting garbage to the thread. > > You can find the details at: http://www.geocities.com/john_gabriel > > In response to your assumptions about someone trying to discredit > newton, cauchy and all the others: I don't see how you can say this, > because even gabriel acknowledges newton's work. By saying poor Newton was obliged to illegal contorsions and infinitesimals, as he could not see the truth... The above result is > not something you can prove using riemann sums. Please be kind enough > to prove the result in full using riemann sums as you claimso that we > can all see. For the intervall [x,x+w], the fonction t->f'(t) is Riemann integrable,the associated Riemann sum is w/n sum(f'(x+kw/n),k=0..n-1)), with limit (n->+oo) = integral (f'(t), t=x..x+w) = f(x+w)-f(x). Done. As this is really trivial , I conclude you are a troll. Please be kind enough to prove otherwise. > Have never seen gabriel's theorem stated before, nor a connection > between f(x+w)-f(x) and the integral in the way gabriel > shows it. If I can prove it, then I intend to use it as a teaching aid > because it requires nothing else, i.e. no knowledge of > real analysis or deep properties of the real number system. Ha! How do you prove it *without * any definition of anything, like derivatie or Riemannsum? > > Jason >
From: David C. Ullrich on 2 Mar 2005 18:38
On 2 Mar 2005 07:15:30 -0800, "Jason" <logamath(a)yahoo.com> wrote: >> > However, since both n and s tend to infinity, the limit is taken >> > once over the entire sum. With n and s coinciding, we get >> > gabriel's proof. >> > >> > This seems to work but is this last step legal? > >> _What_ last step? > >David, > >The last step being that since n coincides with s and the limits are >both taken over infinity, the inner limit falls away to be replaced by >the outer limit. > >What do you think? I think two things: First, I think that you think I know what you're talking about. There are no inner and outer limits in any of the posts in this thread, nor any sums visible - if you think people are going to try to make sense of your corrections to the incoherent things we can find elsewhere you're wrong - you should post the entire proof in a coherent form. Also, I _know_ that in general interchanging limits is the hard part when you're proving almost anything in analysis - if you just assume that that works you're usually sweeping the entire proof under the rug. >Jason ************************ David C. Ullrich |