October 21, 2005 12:18masterSheet number 74 Page number 5858Lecture FiveProof:Once again, we could use the fact that the preferences have a contin-uous utility representation and apply a standard “maximum theo-rem.” (If the functionf(x,a)is continuous, then the functionh(a)=argmaxxf(x,a)is continuous.) However, I prefer to present a proofthat does not use the notion of a utility function:Assume not.Then, there is a sequence of price vectorspncon-verging top∗such thatx(p∗,w)=x∗, andx(pn,w)does not convergetox∗. Thus, we can assume that(pn)is a sequence converging top∗such that for allnthe distanced(x(pn,w),x∗) > εfor some positiveε.All numberspnkare greater than some positive numberm. There-fore, all vectorsx(pn,w)belong to some compact set (the hypercubeof bundles with no quantity abovew/m) and thus, without loss ofgenerality, we can assume thatx(pn,w)→y∗for somey∗=x∗.
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