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

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

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

Offline

**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

What's the black square for Jane?

Offline

**LuisRodg****Real Member**- Registered: 2007-10-23
- Posts: 322

I believe it means "proof done".

Just like "Q.E.D"

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

Offline

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

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

Offline

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

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)*

Offline

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

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

Offline

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

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

Offline

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

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..."

Offline

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

Offline

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

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)*

Offline

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

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)*

Offline

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

Offline

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

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..."

Offline

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

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.

Offline

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

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.

Offline

**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)*

Offline

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

Offline

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

Great manuveur

**X'(y-Xβ)=0**

Offline

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

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

Offline

Pages: **1**