MATH 347 Date: 12/12/2016 FINAL EXAM NAME: Problem 1 (10pts) Problem 2 (10pts) Problem 3 (10pts) Problem 4 (12pts) Problem 5 (8pts) Problem 6 (10pts) Problem 7 (10pts) Problem 8 (10pts) 1 Problem 1. (a) Let {an }nN be the sequence given by declaring an = 1 for all n N. Show that {an }nN does not converge to 2. Use only the definition of convergence. (b) Let {an }nN be the sequence given by declaring an = (1)n for all n N. Show that {an }nN does not converge to 1. Use only the definition of convergence. 2 Problem 2. Suppose that {ai } i=0 is a sequence of real numbers that converges to a real number . (a) [3 pts] Show that limi (ai ) = 0. (b) [7 pts] Show that limi (ai ai+k ) = 0 for any k N. 3 Problem 3. (a) Define the infimum of a set A R. b) Prove the following or give a counterexample: If A R has an infimum then that infimum is an element of A. 4 Problem 4. Let O be the set of odd integers and E be the set of even integers. (a) Give an injection f : O E (show that it is an injection) (b) Give a surjection h : O E (show that it is a surjection) (c) Give a bijection g : O E (show that it is a bijection) 5 Problem 5. (a) Show that n X i=0 i= n(n + 1) 2 6 for all n N. Problem 6. Let A Z. Suppose that (1) 0 A (2) if n A then n + 1 and n 1 are both elements of A. Show that every integer is in A. 7 Problem 7. (a) Show that if n Z is divisible by 5 then n + 3 is not divisible by 5. (b) Show that if m Z is divisible by 2 and m = ab for a, b Z then either a is divisible by 2 or b is divisible by 2. 8 Problem 8. (a) Let A be a set and suppose that / A. Show that [A {}]2 = A2 [A {}] [{} A] {(, )}. (b) Let A be a finite set with n elements. Show that A2 has n2 elements. Apply induction to n. 9 Problem 9. Let A, B, C, D be sets. a) [5 pts] Prove or give a counterexample to the following: (A B) (C D) = (A C) (B D). b) [5 pts] Prove or give a counterexample to the following: (A B) (C D) = (A C) (B D). 10 Additional page for work 11