Prove that limits in distribution are unique. That is, suppose X, Y, X1, X2,... are all real valued random variables and (X.):=1 converges in distribution to both X and Y. Show X and Y have the same distribution, i.e. Px = Py. You may use that a cumulative distribution function has at most countably many discontinuities (as noted in the proof of Proposition 1.2 in Lecture 25). You may also use the fact that if S C R is countable, then any 8 € S is the limit of a decreasing sequence (cn)CR\S. Hint: Start by showing that the CDFs of X and Y agree.