What is the Construction of R using Equivalence Classes of Cauchy Sequences?

[autre]

This proof I think is related to our construction of R in class using equivalences classes of Cauchy sequences:

Let a\inR, then there exists a sequence b s.t. {b_n}\inQ for all n\inN and lim {b_n} = x.

Here's my attempt:

Let [{b_n}]\inR be the equivalence class of all Cauchy sequences that converge to x. Thus, [{b_n}] = x. Let {b_n}\in[{b_n}]. Then, \existsr\inQ+ and n\inN s.t. \foralln>N, |b_n -x|<r. Thus, lim {b_n} = x.

I feel like I'm missing something, as if I missed a step somewhere. Ideas?

 

[chiro]

This proof I think is related to our construction of R in class using equivalences classes of Cauchy sequences:

Let a\inR, then there exists a sequence b s.t. {b_n}\inQ for all n\inN and lim {b_n} = x.

Here's my attempt:

Let [{b_n}]\inR be the equivalence class of all Cauchy sequences that converge to x. Thus, [{b_n}] = x. Let {b_n}\in[{b_n}]. Then, \existsr\inQ+ and n\inN s.t. \foralln>N, |b_n -x|<r. Thus, lim {b_n} = x.

I feel like I'm missing something, as if I missed a step somewhere. Ideas?

Hello autre.

Since you are talking about Cauchy sequences, all the convergence theorems automatically are implied (they can be proven for a general Cauchy sequence).

Maybe what you could do is look at the actual proof that all Cauchy sequences converge in the delta-epsilon setting.

Unfortunately the proof that I have is from material I took in a Wavelets course and I can not distribute the content freely, but I'm sure there has to be some analysis book out there that does your standard delta-epsilon proof for these sequences, and based on that you could either use it directly, or use the proof to suggest a valid epsilon for your problem.

I'm sorry I can't be more specific at this time, but again if the sequence is a valid Cauchy sequence, then there really should not be more to do.

Hopefully someone more well versed than myself can give you more specific advice (and correct me if I am wrong).

 

[HallsofIvy]

How are you defining such things as "a- x" (a rational) and "|x|" for x an equivalence class of Cauchy sequences?

 

[autre]

How are you defining such things as "a−x" (a rational) and "|x|" for x an equivalence class of Cauchy sequences?

Good question. That didn't really make sense. Maybe I should have:

Let {b_n}∈[{b_n}] and {a_n}}∈x. Then, ∃r∈Q+ and n∈N s.t. ∀n>N, |b_n -a_n|<r. Thus, lim {b_n} = x.

Would that work?