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C Roettger, Spring ’16 Name (please print): . . . . . . . . . Math 201 – Practice Questions for Final Show all work. State clearly what your result is. No cellphones, notes or books. Time is 120 minutes. Problem 1 Complete the definition. ”Let S be a set of real numbers. The real number a is the supremum of S if a) . . . . . . . . . and b) . . . . . . . . . Problem 2 Give an example of a subset S of R which has an infimum but no minimum. Problem 3 Consider the set S= 2 + (−1)n : n∈N n Find the infimum inf S and a sequence in S converging to inf S. Problem 4 Let (an ) be the sequence defined by a0 = 2 and an+1 = 2an − n2 + n. Prove that for all n ∈ N, an = n 2 + n + 2 Problem 5 A subset A of R is called open if for every x ∈ A, there exists an open interval around x which is entirely contained in A. Let f : R → R be a continuous function. Prove that the preimage of every open set if open. Problem 6 Let A = [0, 1] and B = [0, 1] ∪ {2}. Find a bijection from A to B. Problem 7 Define a sequence recursively by setting x0 = 1 and √ xn xn+1 = 2 . Prove that the sequence (xn ) converges to 2. Hint – use induction to prove that it is increasing and bounded by 2. Then use the Mean Value Theorem to prove that it indeed converges to 2.