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## Main Question or Discussion Point

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?

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?