Math 3IA3 * Assignment 2 Due in Crowdmark by 11:59pm on Friday, 9 February 2024 You will receive a link that will lead you to the place where you will submit your assignment solutions. Solution to each question is submitted separately. You can upload files in pdf, jpg, or png formats. Late assignment penalty is a 15% deduction in the grade. Assignments handed in more than 72 hours after the due date will not be accepted. 1. [4] (a) Consider the following nested sequence of intervals: n=(1-10,1+10) where n = 1,2,3,.... Find their intersection N2, I;, and justify your answer. (b) Consider the following nested sequence of intervals: Jo=(1,1+10) o0 where n = 1,2.3,.... Find their intersection N,_,.J, and justify your answer. Does your answer contradict the Nested Intervals Property? Why or why not? 2. [4] Use the formal definition of a limit to prove that each sequence converges, and find its limit: (EL) an = :; . T an 11 by, = (b) b 2n?+n 3. [4] Use the formal definition of a limit to show that the sequence a,, = 4, diverges. 4. [4] Suppose that &1_1}310 sp = 0 and that the sequence (t,,) is bounded (but not necessarily convergent). (a) Prove that lim (snt,) =0 (b) Provide a counterexample to show that the assumption that the sequence (t,,) is bounded is necessary. 5. [4] Suppose that M is a lower bound for the set S. Suppose also that there is a sequence sp 5 such that nllin:}c snp = M. Prove that M = inf S. 6. [4] Suppose that nll_{l;Q an, = a, and b b for all but finitely many n. 7. [4] Use the formal definition of a limit to show that lim /2 + 2 V2. S | =