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You are not logged in. #1 20090206 10:53:09#2 20090206 10:54:02
Re: Cauchy sequences of rational numbersWhat's the black square for Jane? #3 20090206 10:56:46
Re: Cauchy sequences of rational numbersI believe it means "proof done". Last edited by LuisRodg (20090206 11:00:17) #4 20090206 11:12:24
Re: Cauchy sequences of rational numbersYeah, that’s what it means. Theorem 1 says that two Cauchy sequences can be added term by term and the result is another Cauchy sequence. In view of this, it makes sense to define addition on by #5 20090206 11:48:59
Re: Cauchy sequences of rational numbersFor this, I will need to assume the result (which I shall prove in a moment) that all Cauchy sequences are bounded. Last edited by JaneFairfax (20090206 12:05:49) #6 20090206 12:01:35
Re: Cauchy sequences of rational numbersLast edited by JaneFairfax (20090206 12:03:24) #7 20090206 23:16:41
Re: Cauchy sequences of rational numbersSo termbyterm multiplication of Cauchy sequences also gives rise to Cauchy sequences. Hence multiplication in can also be validly defined. #8 20090207 06:14:21
Re: Cauchy sequences of rational numbersI suppose multiplicative inverses come next, that's a fun proof. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #9 20090207 06:58:02#10 20090207 07:55:06
Re: Cauchy sequences of rational numbersIn fact, as we shall see, it’s more than just a field. But I’m going to build up the pieces slowly. http://www.mathisfunforum.com/viewtopic.php?id=10480 Did you remember that thread? Never forget anything I post – you never know when it may prove useful one day. Last edited by JaneFairfax (20090209 13:46:42) #11 20090209 00:43:27
Re: Cauchy sequences of rational numbersWe now define to be the set of all nonnull Cauchy sequences in satisfying property in the theorem above. Certainly is nonempty since . is thought of as the set of all “positive” Cauchy sequences of rational numbers. Last edited by JaneFairfax (20090209 00:47:30) #12 20090209 11:15:28#13 20090209 12:59:36
Re: Cauchy sequences of rational numbersJust wanted to note, I believe bar notation for coset representatives is much more standard than hats. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #14 20090209 22:33:14
Re: Cauchy sequences of rational numbersThe hat notation is used in Sutherland’s Introduction to Metric and Topological Spaces. I use it myself because I think it’s cute. Sutherland also writes for the sequence . However I choose not to drop the peripheral adjuncts, and write as a reminder that I am talking of the whole Cauchy sequence, not just the th term. #15 20090209 23:39:41
Re: Cauchy sequences of rational numbersThis is an important result. It means that we can unambiguously define an order relation in by The corollary to Theorem 8 says that this order relation is well defined. #16 20090214 10:38:53
Re: Cauchy sequences of rational numbersI AM VERY SORRY, FOLKS. I JUST NOTICED A GAPING HOLE IN MY PROOF OF THEOREM 6 WHICH I MUST PLUG RIGHT AWAY.
I NEED TO CONFIRM THAT THIS SEQUENCE IS CAUCHY (which I didn’t do)!! Watch this space. Last edited by JaneFairfax (20101212 01:25:08) #17 20090214 11:15:40#19 20101212 01:40:45
Re: Cauchy sequences of rational numbersEvery metric space can be “completed” in a way similar to the construction of the real numbers from the rationals by equivalence classes of Cauchy sequences. http://z8.invisionfree.com/DYK/index.php?showtopic=193 