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JaneFairfax

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Registered: 2007-02-23

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Cauchy sequences of rational numbers

Daniel123

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Re: Cauchy sequences of rational numbers

What's the black square for Jane?

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)

JaneFairfax

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Re: Cauchy sequences of rational numbers

Yeah, 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

   

JaneFairfax

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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.

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

JaneFairfax

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Re: Cauchy sequences of rational numbers

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

JaneFairfax

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Registered: 2007-02-23

Posts: 6,868

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.

   

Ricky

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Registered: 2005-12-04

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Re: Cauchy sequences of rational numbers

I suppose multiplicative inverses come next, that's a fun proof.

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"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..."

JaneFairfax

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Re: Cauchy sequences of rational numbers

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

   

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

JaneFairfax

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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.

   

   

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 (2009-02-08 14:46:42)

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.

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

JaneFairfax

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Registered: 2007-02-23

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Re: Cauchy sequences of rational numbers

Ricky

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Registered: 2005-12-04

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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..."

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.

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.

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.

JaneFairfax

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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.

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

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.

Watch this space.

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

JaneFairfax

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Re: Cauchy sequences of rational numbers

George,Y

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Posts: 1,379

Re: Cauchy sequences of rational numbers

Great manuveur

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X'(y-Xβ)=0

JaneFairfax

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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