In classical logic, using the ZF axioms, one can show that there is an isomorphism between the Dedekind cuts in the rationals, and the equivalence classes of Cauchy sequences. One can also (without AC) choose a representative from each equivalence class of Cauchy sequences.
The fundamental theorem of algebra (every non constant polynomial with real or complex coefficients has a zero in the complex numbers) can be proved without AC. In fact, there is a quite explicit algorithm that computes, for each irreducible polynomial (with leading coefficient 1, to be on the safe side), the roots of this polynomial continuously from the coefficients.
In constructivist or intuitionistic logic, the story may be different.