MA250 - Homework Assignment #3 (Fall 2016) Due: Friday Oct. 7, 2016 (at the beginning of class) Instructor: Dr. Chester Weatherby Instructions: Complete the following problems and hand in a neatly written and stapled assignment with your solutions/proofs. If any work is not legible, it will be given a score of zero. Please do not attach any cover page (including this question page) to your work. 1) Prove using the definition of limit that: \u0013 \u0012 2 2 2n n 14 = a) lim 2 7n + n + 101 7 \u0012 b) lim 1 7n2 + 3n 4 \u0013 =0 2) BS 3.1.10 Prove that if lim(xn ) = x and if x > 0, then there exists a natural number M such that xn > 0 for all n M . 3)Prove directly, using the definition of limit (so don't quote a theorem from class or the book) that for a convergent sequence (xn ) x and any c R we have that lim(cxn ) = c lim(xn ). 4) Let X = (xn ) and Y = yn be sequences, where yn is never 0. Suppose also that both Y and X/Y converge. Prove the X also converges. 5) BS 3.2.7 If (bn ) is a bounded sequence and lim(an ) = 0, show that lim(an bn ) = 0. Explain why Theorem 3.2.3 cannot be used