Question: show that if X is finite, then any rational preference relation generates a nonempty choice rule; that is, C(B) \neq \emptyset for any B \subset X with B \neq \emptyset

We know that if a preference relation is rational, then it is both transitive and complete. Then the choice structured generated satisfies the weak axiom. If we have all subsets of X up to 3 elements as well, then then C(B) = C*(B, >) for all B. Then it's just a matter of showing that C*(B,>) is nonempty, which will in turn show that C(B) is nonempty. However, we don't know if we have all subsets of X up to 3 elements or not, so the proof cannot be done this way.

Any tips? Thanks!