# Proof of Proposition 1.4 from Microfoundations I by Kreps

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?

• I don't know the book, but doesn't finiteness of A follow from the property of "finite" nonemptiness? My guess is that the property does not say the choice function will return a nonempty set, only that it will do so if A is finite. – Giskard May 30 '15 at 14:54
• The property of "finite nonemptyness" says here that $C_\succeq(A)\neq\emptyset$ when $A$ is a finite and nonempty subset of $X$. Since the property applies only to finite sets (and fails in general otherwise) one assumes that the set $A$ is finite when proving this. – Michael Greinecker May 30 '15 at 15:07