[red_trumpet]

At which point does the construction of the reals require the axiom of choice (AC)? I'm not familiar what exactly happens without AC, but defining R as the set of rational Cauchy-sequences modulo zero-sequences does not seem to use it?

[heinrich5991]

The moment where you want a Cauchy sequence of real numbers to converge.

You have a Cauchy sequence of real numbers (a_i)_i. You pick a representative (b_ij)_j (a Cauchy sequence of rational numbers) for each sequence member a_i (axiom of choice!) and then produce the diagonal sequence (b_ii)_i which is the constructed limit of the Cauchy sequence (a_i)_i.