Math 131A Homework 3 Due: February 3rd, 2017 This assignment is due on February 3rd, 2017 at the beginning of lecture. Provide complete well-written solutions to the following exercises. Please note that none of the concepts appearing in Exercises 3 and 4 will be covered on Midterm 1. Exercise 1. Let (sn )nN be a sequence of non-negative numbers such that limn sn = 0. Prove that the sequence ( sn )nN converges to 0, i.e., limn sn = 0. Exercise 2. Let (an ) and (bn ) be sequences of real numbers. Suppose that limn an = a and limn bn = b for real numbers a, b R. Using the limit theorems (of Section 9 of Elementary Analysis by Ross), prove the following statements. Justify your steps. \u0011 \u0010 3 a +4a 1. The sequence nb2 +1n converges and n a3n + 4an a3 + 4a = . n b2 b2 + 1 n+1 lim 2. If bn > 0 for all n N and b > 0, then the sequence lim n \u0010 3an bn bn \u0011 converges and 3an bn 3a b . = bn b Exercise 3. Let (sn ) and (tn ) be sequences of real numbers. Suppose that there exists N0 N such that sn tn for all n N0 . Prove the following statements. 1. If limn tn and limn sn exist, then limn sn limn tn . 2. If limn sn = , then limn tn = . 3. If limn tn = , then limn sn = . Exercise 4. Let (an ) and (bn ) be sequences of real numbers. Prove the following statements. 1. If the set {bn }nN is bounded below, i.e., inf{bn : n N} > , and limn an = , then lim (an + bn ) = . n 2. If limn an = and k < 0, then limn kan = . 3. limn an = if and only if limn (an ) = . 1