. For n a non-zero natural number, consider the set: o fit (o) w2 Define M,, = sup A,, and m,, = inf A,,. (a) Find M, (b) Find m,, () Find inf{M, : n 0} (d) Find sup{m, :n 0} Be sure to prove that those are indeed the supremums and infimums of the given sets. . For a set A, the power set of A, denoted P(A), is the set of all subsets of A. Prove that for any set A, [P(A)| # |A|. . Provide an example of sets of real numbers satisfying the following statements. If such an example does not exist then prove that the statement is false. (a) Two sets A and B with ANB =0, supA=supB,supA A and supB B. (b) A sequence of nested open intervals Jy 2 J; 2 J; 2 -+ - with a non empty intersec- tion Npendn # that contains a finite number of elements. (e) A sequence of nested unbounded closed intervals Ly, O L, 2 L, 2 --- with an empty intersection N,exL, = 0. (d) A sequence of closed, bounded, intervals Iy, I, I, . .. with the property that N_,I,, # 0 for all N N, but Nyenl, = 0. . Prove the following Theorem: If A C B and B is countable, then A is either countable or finite. . Prove that Z is countably infinite by proving that the function f : N Z given below is bijective: 2 n even _ 2 ) f(n) = {n 22 nodd 2