Come up with a way to produce the set A⊂ℚ (then B=ℚ∖A) out of the sequence of (rational) numbers {a_n}.

Question

Let {a_n} be a Cauchy sequence. The equivalence class of this sequence is a real number. Construct the Dedekind cut (A|B) which presents this real number.

Specifically, you need to come up with a way to produce the set A⊂ℚ (then B=ℚ∖A) out of the sequence of (rational) numbers {a_n}.

Answer 1

Let's build off the definition of a Cauchy sequence: given any ε>0, we know there's an N∈ℕ where ∀n>N, |a_n-a_N|<ε.

Envision, then, some cut that's too low down up: there's a fixed distance d between here and the limit of our sequence; we can pick an ε<d and see that there is some N∈ℕ after which, since all terms of our sequence will be within ε of each other and thus certainly within ε of our limit, they are all greater than the too-low cut. There are at most N elements in our sequence less than or equal to this too-low point.

This gives a mechanism for cutting: construct A:={a_n|there are finitely many m∈ℕ with a_m≤a_n}

(we could make this more formal by using the ∃M∈ℕ where ∀m>M sort of construction again)

I hope this is helpful! Let me know if you have follow-up questions :)