MAT1325 : Homework #2 Prof : Monica Nevins Due Thursday, February 4, 2016 by 9pm at the Department of Mathematics and Statistics (KED) Staplers are not available at the Department. Instructions : sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m (1) Read each question carefully, and prepare your solutions on a rough draft rst. Your nal solution should be well-written, convincing, readable and elegant, even though the work it took to get there was anything but. Pictures are very useful, and often essential for solving a problem, but the picture itself is not a proof. (2) Remember that your solution should be a convincing explanation of why your answer is correct. (3) It must be possible to read what you have written out loud, as complete sentences. A proof should start with the hypothesis and conclude with the conclusion. Interesting steps (such as using a denition or result proven in class) should be justied. (4) You may use any result proved in class or in DGD, but indicate clearly when you are doing so. For example \".... by the triangle inequality\" or \"... by a lemma proved in class\" or \"... by an exercise from the DGD\". (5) You do not need to justify routine algebraic manipulations, but if you multiply an inequality by y, you must justify if y > 0. (6) Reread your proof, to be sure you said all that was in your head! A common error is to forget to say what your variables mean (eg, n Z, n > 0 or x R) or to forget to say whether you mean every x R or a specic x R. (7) The bonus question is optional; you may earn more than 100% on this homework. Th (8) Remember: although you may discuss the homework with friends, you must submit your own work. So for example, if someone explained the solution to you, you must write up your own solution without reference to any notes. To do otherwise is to cheat yourself your goal is to succeed on your own! Last name: First name: Student number: The following table is just for the grader. https://www.coursehero.com/file/13251934/HW2pdf/ 1 2 Question Max Your grade 1 6 2 4 3 6 4 4 5 6 2b &6 (Bonus) 4 1. (6 points) Find the supremum of each of the following sets, if it exists. Prove your assertions. (a) S = [1, 2) [3, 7) (b) T = { 2r|r Q} (0, 1) (c) U = {n4 + n2 + 4|n N} sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m 2. Let S, T R be nonempty sets which are bounded above. (a) (4 points) Prove that sup(S T ) max(sup S, sup T ). (b) (Bonus, 2 points) Prove that sup(S T ) max(sup S, sup T ), which shows these values are equal. 3. (6 points) Prove that the following statements are equivalent: (1) x > 0, n N such that n > x. (2) x > 0, n N such that nx > 1. (3) x, y R, x > 0 n N such that nx > y. Hint: Prove that (1) (2) (3) (1). 4. (4 points) The Fibonacci sequence is dened by f1 = f2 = 1, and fn = fn1 + fn2 for n 3. Note that the induction proofs here require TWO base cases (for n = 1, 2), and that you prove that if the result holds for both n and n + 1 then it holds for n + 2. You must do (a) before even reading (b)! (a) Prove that n 1, fn N. (b) Prove that n 1, fn = n 1+ 5 2 5 n 1 5 2 . Th 5. (6 points) Using the denition of the limit, prove the following: (a) 1 lim 3 = 0. n n n2 (b) n2 + (1)n 2 lim = 1. n n2 Your solution should start with \"Let > 0.\" Make sure that your rough work is kept distinct from your nal solution, which should be well-organized and written in a logical order. 6. (Bonus) Prove that for n 0, 2n k=1 https://www.coursehero.com/file/13251934/HW2pdf/ Powered by TCPDF (www.tcpdf.org) 1 n 1+ . k 2