5. Let X be a set, and let ~1 and ~2 be two preference relations on X, and let CL be the lexicographic preference relation as defined in question 3. Suppose u1 : X - R is a utility representation for 1, and u2 : X - R is a utility representation for ~2. For every number 6 2 0, define the function u : X - R by letting u (x) = ul(x) + Eu2(x). [We might say Zz puts "much more weight" on 1 than on 22. This question gives a sense in which that interpretation is valid, but also shows that this interpretation can be a bit misleading.] (a) Consider the special case in which X is finite. Show that, for some e > 0, the function u is a utility representation for Zz. [Hint: parts (a,c) of the previous question might be helpful.] (b) [Hard, optional question. You don't need to know this, but you can think about it if you would find it fun.] Suppose X is countably infinite. Must there exist some e 2 0 such that u is a utility representation for CI