Let X be a metric space and A be a non-empty bounded subset of X. We define the diameter of A as diam(A)=sup{d(x,y): x, y in A}. Suppose that X is a compact metric space and that is a collection of nonempty closed subsets of X. Show that

a) If is decreasing ( i.e., for all n natural, ), then it is satisfied that

b) If it is also true that the sequence of diameters (diam(F_n)) converges to 0 in , then consists of a single point.