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

Login

Username

Password

Not registered yet?

#1 2009-02-06 10:53:09

JaneFairfax
Legendary Member

Offline

Cauchy sequences of rational numbers













smile


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

A: Click here for answer.
 

#2 2009-02-06 10:54:02

Daniel123
Power Member

Offline

Re: Cauchy sequences of rational numbers

What's the black square for Jane?

 

#3 2009-02-06 10:56:46

LuisRodg
Real Member

Offline

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
Legendary Member

Offline

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

   










smile


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

A: Click here for answer.
 

#5 2009-02-06 11:48:59

JaneFairfax
Legendary Member

Offline

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.





smile

Last edited by JaneFairfax (2009-02-06 12:05:49)


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

A: Click here for answer.
 

#6 2009-02-06 12:01:35

JaneFairfax
Legendary Member

Offline

Re: Cauchy sequences of rational numbers







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


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

A: Click here for answer.
 

#7 2009-02-06 23:16:41

JaneFairfax
Legendary Member

Offline

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.

   








smile


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

A: Click here for answer.
 

#8 2009-02-07 06:14:21

Ricky
Moderator

Offline

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
Legendary Member

Offline

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































smile


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

A: Click here for answer.
 

#10 2009-02-07 07:55:06

JaneFairfax
Legendary Member

Offline

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


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

A: Click here for answer.
 

#11 2009-02-09 00:43:27

JaneFairfax
Legendary Member

Offline

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


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

A: Click here for answer.
 

#12 2009-02-09 11:15:28

JaneFairfax
Legendary Member

Offline

Re: Cauchy sequences of rational numbers



















smile


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

A: Click here for answer.
 

#13 2009-02-09 12:59:36

Ricky
Moderator

Offline

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
Legendary Member

Offline

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


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

A: Click here for answer.
 

#15 2009-02-09 23:39:41

JaneFairfax
Legendary Member

Offline

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


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

A: Click here for answer.
 

#16 2009-02-14 10:38:53

JaneFairfax
Legendary Member

Offline

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


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

A: Click here for answer.
 

#17 2009-02-14 11:15:40

JaneFairfax
Legendary Member

Offline

Re: Cauchy sequences of rational numbers





















smile


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

A: Click here for answer.
 

#18 2010-09-11 12:46:18

George,Y
Super Member

Offline

Re: Cauchy sequences of rational numbers

Great manuveur


X'(y-Xβ)=0
 

#19 2010-12-12 01:40:45

JaneFairfax
Legendary Member

Offline

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?

A: Click here for answer.
 

Board footer

Powered by FluxBB