As I read through Willard's General Topology, the problem 12 D-5 states that "the net based on a free ultrafilter is a nontrivial ultranet. Hence, assuming the axiom of choice, there are nontrivial ultranets." First of all, I don't quite understand the meaning of a nontrivial ultranet, since the book itself does not give a precise definition. Second of all, if "the net based on a free ultrafilter is a nontrivial ultranet" is already given, why do we still need axiom of choice to prove the existence of nontrivial ultranet? Help needed!