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Practice Problems, Math 220 1 Prove that 52n − 1 is divisible by 8 for all n ∈ N. 2 Prove that iff a = b. 1 (a 2 + b) 2 ≤ 1 2 (a2 + b2 ) for all a, b ∈ R with equality 3 Let S = {1/n − 1/m : n, m ∈ N} . Find inf S and supS. 4 Show that a nonempty finite set contains its supremum. 1 5 ], Jn = (0, n1 ) and Kn = (n, ∞). Find n T∞Let In =T[0, ∞ n=1 Jn and n=1 Kn . T∞ n=1 In , 6 Prove that S 0 is a closed set. 7 If S is a compact subset of R and T is a closed subset of S, prove that T is compact using (a) the definition of compactness; and (b) the Heine-Borel Theorem. 1