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3) Let X = Z and d(x,x) = 0 or d(x,y) = 2n, 1/2/2 1 if x # y, where 2n is the largest power of 2 dividing x- - y. Prove that the following two series are Cauchy. One of them is convergent (find its sum) while the other not (explain). 80 n=0 2n n=0 (-2)n - 00 4) Let Cn be a sequence of positive numbers. Prove that liminf n Cn+1 liminf n n n Cn. Cn

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