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#1 2009-02-05 11:53:09

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Cauchy sequences of rational numbers

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#2 2009-02-05 11:54:02

Daniel123
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Registered: 2007-05-23
Posts: 663

Re: Cauchy sequences of rational numbers

What's the black square for Jane?

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#3 2009-02-05 11:56:46

LuisRodg
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Registered: 2007-10-23
Posts: 322

Re: Cauchy sequences of rational numbers

I believe it means "proof done".

Just like "Q.E.D"

Last edited by LuisRodg (2009-02-05 12:00:17)

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#4 2009-02-05 12:12:24

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

Yeah, that’s what it means. smile


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

   

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#5 2009-02-05 12:48:59

JaneFairfax
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Registered: 2007-02-23
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Re: Cauchy sequences of rational numbers

For this, I will need to assume the result (which I shall prove in a moment) that all Cauchy sequences are bounded.

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Last edited by JaneFairfax (2009-02-05 13:05:49)


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#6 2009-02-05 13:01:35

JaneFairfax
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Registered: 2007-02-23
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Re: Cauchy sequences of rational numbers

Last edited by JaneFairfax (2009-02-05 13:03:24)


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#7 2009-02-06 00:16:41

JaneFairfax
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Registered: 2007-02-23
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Re: Cauchy sequences of rational numbers


So term-by-term multiplication of Cauchy sequences also gives rise to Cauchy sequences. Hence multiplication in
can also be validly defined.

   

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#8 2009-02-06 07:14:21

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Cauchy sequences of rational numbers

I 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..."

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#9 2009-02-06 07:58:02

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

I’m building up to the climax. big_smile But first …

   


http://z8.invisionfree.com/DYK/index.php?showtopic=192

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#10 2009-02-06 08:55:06

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

In fact, as we shall see, it’s more than just a field. But I’m going to build up the pieces slowly. wink

tongue

   

   

http://www.mathisfunforum.com/viewtopic.php?id=10480 big_smile

Did you remember that thread? Never forget anything I post – you never know when it may prove useful one day. lol

smile

Last edited by JaneFairfax (2009-02-08 14:46:42)


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#11 2009-02-08 01:43:27

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

We now define

to be the set of all non-null 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. smile

Last edited by JaneFairfax (2009-02-08 01:47:30)


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#12 2009-02-08 12:15:28

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

smile


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#13 2009-02-08 13:59:36

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Cauchy sequences of rational numbers

Just 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..."

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#14 2009-02-08 23:33:14

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

The hat notation is used in Sutherland’s Introduction to Metric and Topological Spaces. I use it myself because I think it’s cute. tongue


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. cool


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#15 2009-02-09 00:39:41

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

This 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. smile


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#16 2009-02-13 11:38:53

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

I AM VERY SORRY, FOLKS. I JUST NOTICED A GAPING HOLE IN MY PROOF OF THEOREM 6 WHICH I MUST PLUG RIGHT AWAY.

JaneFairfax wrote:

I NEED TO CONFIRM THAT THIS SEQUENCE IS CAUCHY (which I didn’t do)!! yikes

is essentially a sequence of the form
. In view of Theorem 3, one just needs to prove that
is Cauchy. Note that this is not true for all Cauchy sequences
– only for non-null sequences
. And to prove that the sequence of reciprocal terms is Cauchy, I shall need to use the result of Theorem 7 – which means that I ought to have presented Theorem 7 before Theorem 6. neutral

Watch this space.

Last edited by JaneFairfax (2010-12-11 02:25:08)


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#17 2009-02-13 12:15:40

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

smile


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#18 2010-09-10 14:46:18

George,Y
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Registered: 2006-03-12
Posts: 1,306

Re: Cauchy sequences of rational numbers

Great manuveur


X'(y-Xβ)=0

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#19 2010-12-11 02:40:45

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Cauchy sequences of rational numbers

Every 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


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