Cauchy real and dedekind real are equivalent or isomorphic

[sevenlite]

Hiya, I am looking for the proof for cauchy real and dedekind real are equal (isomorphic). I know they are not equal (CR \= DR) but I need to prove them point to the same real number or mapping from CR -> DR, DR -> CR. I have looked at the textbooks on number system, real analysis and calculus. and i can't find any. what I know is that its a classic proof. Can anyone piont out any book contains this proof or some hints about how to prove it? Thank you

 

[Hurkyl]

Intuitively, how does a Cauchy sequence specify a real number? And how does a Dedekind cut specify a real number?

The bijection should be nearly obvious -- once you know what it is, you just have to go through the motions of writing it down, showing it really is well-defined, a bijection, et cetera. (it's a lot of motions, though)

 

[Hurkyl]

Of course, you could just prove them both isomorphic to the decimal numbers. Or prove that all complete ordered fields are isomorphic, and both happen to be complete ordered fields. You don't have to compare them directly to each other.

 

[sevenlite]

Intuitively, how does a Cauchy sequence specify a real number? And how does a Dedekind cut specify a real number?

The bijection should be nearly obvious -- once you know what it is, you just have to go through the motions of writing it down, showing it really is well-defined, a bijection, et cetera. (it's a lot of motions, though)

I know how to define to CR and DR , I've been told that both surjections Q -> R exist. would u show me just one how its bijection exists? i don't have much knowledge on this. thank you

 

[sevenlite]

Of course, you could just prove them both isomorphic to the decimal numbers. Or prove that all complete ordered fields are isomorphic, and both happen to be complete ordered fields. You don't have to compare them directly to each other.

the same problem, lack of knowledage. would u point out some books that contain one is complete ordered filed? I can look at. thank you

 

[qspeechc]

Chapter 1 of Pugh's Real Mathematical Analysis constructs the real numbers from Dedekind cuts, and then proves that complete ordered fields are unique up to an isomorphism. Later on, in chapter 2, he constructs the real numbers from Cauchy sequences, and then because complete ordered fields are unique we get that the two constructions give essentially the same thing. So you might be interested in geting Pugh's book from the library.

 

[sevenlite]

Chapter 1 of Pugh's Real Mathematical Analysis constructs the real numbers from Dedekind cuts, and then proves that complete ordered fields are unique up to an isomorphism. Later on, in chapter 2, he constructs the real numbers from Cauchy sequences, and then because complete ordered fields are unique we get that the two constructions give essentially the same thing. So you might be interested in geting Pugh's book from the library.

Ive found this book, thank you!

 

[lavinia]

Ive found this book, thank you!

This method of proof is not direct. Why not try a line of reasoning that recognizes that each Dedekind cut determines an equivalence class of Cauchy sequences.

Conversely, each equivalence class of Cauchy sequences either has a limit or determines a Dedekind cut.

 

[Hurkyl]

I don't think a "direct" proof simplifies things any.