MATH 3A03 Assignment #2 Due: Friday, October 2, in class 1. Let A and B be nonempty bounded subsets of R. Dene A + B to be the set {a + b : a A and b B}. Show that A + B is upper bounded and that sup (A + B) = sup A + sup B. (Note that A + B is also lower bounded and inf(A + B) = inf A + inf B.) 2. Use the formal denition of a limit to prove that the following sequences converge: (a) sn = 5 4n n+4 (b) sn = 2 3n + 11 3. Use the formal denition of a limit to prove that the following sequences diverge: (a) sn = 3 n n (b) sn = cos 3 4. Supppose that lim sn = 0 and that the sequence {tn } is bounded (but n not necessarily convergent). Prove that lim (sn tn ) = 0. Provide a n counterexample to show that the assumption that the sequence {tn } is bounded is necessary. 1 = 0. Does the converse hold, n sn 5. Prove that if lim sn = then lim n 1 i.e., if lim = 0 then does lim sn = ? n sn n 6. Suppose that lim sn = and {tn } is a sequence such that for all n n N, sn tn . Prove that lim tn = . n 1 Supplementary problems from the textbook (not to be handed in) 2.4.2, 2.4.3, 2.4.5, 2.4.11, 2.5.3, 2.5.5, 2.5.6, 2.6.1, 2.6.2, 2.7.4, 2.8.1, 2.8.5, 2.8.7, 2.8.9. 2