We showed in class that there is no uniform probability distribution on the natural mumbers. There are other notions of the size of set of naturals A CN. One example is d(A) = lim sup 4u [n]| nEN where [n] = {0,1...n 1}. Observe that d({i}) = 0 for all i and d(N) = 1. (a) Call a set A eventually periodic with period T if there is some N eN such that for all n > N, nE A if and only if n+TE A. Rigorously compute d(A) and d(A). (b) Probability measures are countably additive. Show that d is not even finitely additive by finding a set BCN such that d(B)+ d(B") # d(N). Hnt: Think about how the different behavior of A and A from the previous part before and after N affects the limiting argument and construct B in such a way to prevent that limiting argument from working.