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

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 $\displaystyle X$ up to 3 elements as well, then then $\displaystyle C(B) = C*(B, >)$ for all $\displaystyle B$. Then it's just a matter of showing that $\displaystyle C*(B,>)$ is nonempty, which will in turn show that $\displaystyle C(B)$ is nonempty. However, we don't know if we have all subsets of $\displaystyle X$ up to 3 elements or not, so the proof cannot be done this way.

Any tips? Thanks!