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 c 2 0, define the function u : X - R by letting us(x) = ul(2) + cuz(x). We might say z puts "much more weight" on ~1 than on ~2. 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 CL. [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.2 Must there exist some c 2 0 such that u is a utility representation for CL