8. The following theorem was given in the lecture notes: "Theorem. If > is a preference relation on a finite set X, then > has a utility representation with values being natural numbers." Give an alternative proof of the theorem as follows (a) show first that a maximal element always exists in a finite set with a preference relation on its elements, 3 (b) then proceed by starting with the whole set and its maximal element, (c) and proceed by induction so that finally you have a singleton set left. (d) Finally, think about how this proof can be helpful when designing experiments to elicit preference orderings over alternatives and summarize your thoughts ( maxi- mum of 50 words)