# Construction of R question

• autre
In summary, the proof presented relates to the construction of R using equivalence classes of Cauchy sequences. It states that for any a in R, there exists a sequence b in Q such that the limit of b is x. This is proven by showing that for any b in the equivalence class of Cauchy sequences that converge to x, there exists an r in Q+ and n in N such that for all n greater than N, the absolute value of b_n minus x is less than r. This implies that the limit of b is indeed x. The process of proving this is similar to the standard delta-epsilon proof for Cauchy sequences.

#### autre

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

Let a$\in$R, then there exists a sequence b s.t. {b_n}$\in$Q for all n$\in$N and lim {b_n} = x.

Here's my attempt:

Let [{b_n}]$\in$R be the equivalence class of all Cauchy sequences that converge to x. Thus, [{b_n}] = x. Let {b_n}$\in$[{b_n}]. Then, $\exists$r$\in$Q+ and n$\in$N s.t. $\forall$n>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?

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

Let a$\in$R, then there exists a sequence b s.t. {b_n}$\in$Q for all n$\in$N and lim {b_n} = x.

Here's my attempt:

Let [{b_n}]$\in$R be the equivalence class of all Cauchy sequences that converge to x. Thus, [{b_n}] = x. Let {b_n}$\in$[{b_n}]. Then, $\exists$r$\in$Q+ and n$\in$N s.t. $\forall$n>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).

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

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?