We need something more than the axioms of ZFC to prove the Dedekind completeness?

Solution 1:

The point here is that "a bounded collection of Dedekind cuts" is just a set of Dedekind cuts. The union of any set is guaranteed by the axiom of union, and since the set is not empty, the intersection of this set is guaranteed to exist by the axiom [schema] of separation.

So indeed, to prove that $\Bbb R$ [exists and that it] is complete we need far less than $\sf ZFC$. We needed extensionality, power set, infinity, union and separation.

This can be pushed even lower by restricting the power set axiom (to say something along the lines "the set of natural numbers has only two power sets", or some other finite number of iterations that is needed here), and to restrict which separation axioms we use. But let's not get ahead of ourselves.