Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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**JaneFairfax****Member**- Registered: 2007-02-23
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**Daniel123****Member**- Registered: 2007-05-23
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What's the black square for Jane?

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**LuisRodg****Real Member**- Registered: 2007-10-23
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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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**JaneFairfax****Member**- Registered: 2007-02-23
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Yeah, thats 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

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**JaneFairfax****Member**- Registered: 2007-02-23
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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)*

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**JaneFairfax****Member**- Registered: 2007-02-23
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*Last edited by JaneFairfax (2009-02-05 13:03:24)*

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**JaneFairfax****Member**- Registered: 2007-02-23
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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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**Ricky****Moderator**- Registered: 2005-12-04
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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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**JaneFairfax****Member**- Registered: 2007-02-23
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**JaneFairfax****Member**- Registered: 2007-02-23
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In fact, as we shall see, its more than just a field. But Im 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)*

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**JaneFairfax****Member**- Registered: 2007-02-23
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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)*

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**JaneFairfax****Member**- Registered: 2007-02-23
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**Ricky****Moderator**- Registered: 2005-12-04
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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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**JaneFairfax****Member**- Registered: 2007-02-23
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The hat notation is used in Sutherlands *Introduction to Metric and Topological Spaces*. I use it myself because I think its 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.

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**JaneFairfax****Member**- Registered: 2007-02-23
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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.

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

**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 didnt 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)*

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**JaneFairfax****Member**- Registered: 2007-02-23
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**George,Y****Member**- Registered: 2006-03-12
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Great manuveur

**X'(y-Xβ)=0**

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**JaneFairfax****Member**- Registered: 2007-02-23
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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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