Construction of R question

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

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

Here's my attempt:

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

Answers and Replies

  • #2
chiro
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This proof I think is related to our construction of R in class using equivalences classes of Cauchy sequences:

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

Here's my attempt:

Let [{b_n}][itex]\in[/itex]R be the equivalence class of all Cauchy sequences that converge to x. Thus, [{b_n}] = x. Let {b_n}[itex]\in[/itex][{b_n}]. Then, [itex]\exists[/itex]r[itex]\in[/itex]Q+ and n[itex]\in[/itex]N s.t. [itex]\forall[/itex]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).
 
  • #3
HallsofIvy
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How are you defining such things as "[itex]a- x[/itex]" (a rational) and "|x|" for x an equivalence class of Cauchy sequences?
 
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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?
 

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