Regardless of the size of X (set of all possible objects), if a preference relation which is complete and transitive is defined on it, then the corresponding choice function generated by it will satisfy the property of finite nonemptiness.
The proof in the text is by induction on the set A which in turn is an element of the collection of all possible subsets of X. But the proof assumes finiteness of A. I am not very sure about this assumption. Can anyone please help me understand this?