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

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.