User talk:Totalcynic

Sequence

Dear Totalcynic, thank you for your edits of the sequence page. I have some comments though. You inserted your text in the middle of the definiton of the sequence, right before the intuitive definiton of a sequence, and after the rigurous defintion. I would suggest that you put your text at the very bottom, preferably in a new section, which has to be named appropriately. What do you think? Thanks. --Olegalexandrov 15:52, 15 Dec 2004 (UTC)

Olegalexandrov - you are no doubt correct. I'd not completely considered the organization of the page. I like the work you've done categorizing it - it's much more comprehensible. I'm quite new at this - I apologize for my ineptitude in this regard. The sequence page still needs quite a bit of work - I think monotonicity still needs to be convered *somewhere*, as that is fundamental in analysis. Additionally, there is no distinction between infinite and finite sequences - something that is certainly of importance to areas of symbolic computation and thermodynamics. Thanks! Josh

You are right, one needs to mention finite sequences and montone sequences. I will think of this tomorrow. Cheers, --Oleg Alexandrov 04:17, 19 Dec 2004 (UTC)

Just as a quick note - I added in a few of the old revision sentences for monotonicity and finite/infinite sequences. It still needs quite a bit of work - the definition of finite/infinite is definitely unclear, but maybe it's a placeholder for future thought. -- Josh

Great! I am still not perfecty happy about some things with that paragraph. I feel it goes into too much detail. Some of that stuff belongs to the real analysis page maybe. Or maybe not. Besides, I looked at the Bolzano-Weierstrass theorem. It does not use monotonicity in proving the theorem. What if we keep just the definition of monotone sequence, and nothing more? There is a paragraph below which alludes to uses of sequences in real analysis. That is maybe enough. If we want to mention how the sequences get used in real analysis, we will write many pages, because uses are many. What do you think? --Oleg Alexandrov 22:31, 19 Dec 2004 (UTC)

I think you're probably right. I will disagree up to a point on the Bolzano-Weierstrass theorem. I just looked at the page myself, and the "sketch" of the proof is incomplete, at best and downright wrong at worst. The B.W. theorem relies fundamentally on the property of sequences that allow us to always find a monotonic subsequence (using peak indices) and bound the sequence on a finite interval. This isn't clear at all in the current article. In my opinion, B.W. illustrates a fundamental property of sequences. However, diatribe aside, I will certainly agree with you that if we open the "real analysis" can of worms we'll be writing for a very long time. As such, I think you're correct in your assessment that we remove the stuff about real analysis and limit the rest of the properties of sequences solely to the basics. I've probably just obligated myself to tackle the Bolzano-Weierstrass issue separately <sigh>. Maybe we could make a section for "Important theorems about sequences" and just provide links to some of these things? Thanks for your patience with me on this one! -- Josh

I think the Bolzano-Weierstrass theorem is correct the way it is. To make the proof complete, one needs to prove that the subsequence obtained after doing all that halving of intervals is indeed convergent, which is easy, and intuitively clear. I know the other proof of the Bolzano-Weierstrass theorem using monotone seqences, but the proof given in the Bolzano-Weierstrass theorem wikipedia page does not need monotone sequences. However, you could add your proof as an alternative proof. If you want, putting a page of theorems in Real analysis which use sequences could be a good idea. --Oleg Alexandrov 17:21, 20 Dec 2004 (UTC) By the way, Wikipedia allowes one to leave one's name and modification time (like a signature if you wish) on a page, the way I did it above. You need to click on the second button from the right on the toolbar above the text area when you edit text.