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## #1 2009-02-06 10:53:09

JaneFairfax
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### Cauchy sequences of rational numbers

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #2 2009-02-06 10:54:02

Daniel123
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### Re: Cauchy sequences of rational numbers

What's the black square for Jane?

## #3 2009-02-06 10:56:46

LuisRodg
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### Re: Cauchy sequences of rational numbers

I believe it means "proof done".

Just like "Q.E.D"

Last edited by LuisRodg (2009-02-06 11:00:17)

## #4 2009-02-06 11:12:24

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

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #5 2009-02-06 11:48:59

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 (2009-02-06 12:05:49)

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## #6 2009-02-06 12:01:35

JaneFairfax
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### Re: Cauchy sequences of rational numbers

Last edited by JaneFairfax (2009-02-06 12:03:24)

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## #7 2009-02-06 23:16:41

JaneFairfax
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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.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #8 2009-02-07 06:14:21

Ricky
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### 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..."

## #9 2009-02-07 06:58:02

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

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #10 2009-02-07 07:55:06

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 (2009-02-09 13:46:42)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #11 2009-02-09 00:43:27

JaneFairfax
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### 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 (2009-02-09 00:47:30)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #12 2009-02-09 11:15:28

JaneFairfax
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### Re: Cauchy sequences of rational numbers

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #13 2009-02-09 12:59:36

Ricky
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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..."

## #14 2009-02-09 22:33:14

JaneFairfax
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### 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.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #15 2009-02-09 23:39:41

JaneFairfax
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### 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.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #16 2009-02-14 10:38:53

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.

#### JaneFairfax wrote:

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 (2010-12-12 01:25:08)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #17 2009-02-14 11:15:40

JaneFairfax
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### Re: Cauchy sequences of rational numbers

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

George,Y
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Great manuveur

X'(y-Xβ)=0

## #19 2010-12-12 01:40:45

JaneFairfax
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### 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

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?