MATH 210 Exam 1, Fall 2016

Math 210, Introduction to Analysis- Fall 2016-2017 Quiz 1 , October 8. Time 70 minutes Question 1. (15 pts.) (i) Let X be a metric space. Define what it means for a set A ⊂ X to be compact. (ii) Prove using the definition of limit(using ϵ and n0) that lim n√ 5 = 1 as n −→ ∞ in R with the usual metric. (iii) Let A = {1, 5, √ 5, 2√ 5, . . . , n√ 5, . . . }. Prove that A is compact in R using the defini- tion. Quiestion 2. (10 pts) Find with justification lim sup and lim inf of the sequence defined by xn = (−1)n + n√ 3. Question 3. (15 pts) a) Define what it means for an ordered set S to have the least upper bound property. b) Let A = [0, √ 7] ∩ Q. Find, if it exists, sup A in Q, and in R. You need to justify your answers, whether the sup exists or not. c) Does Q have the least upper bound property? Why? Question 4. (15 pts) Find in R2 the limit of lim n−→∞ �sin(n2) + (−1)nn n3 + 1 , 2√n − 3 √n + 4 � (You may use Calculus to find the limit).Prove using the definition (ϵ, n0) that this is indeed the limit in (R2, d∞) . Question 5. (15 pts). a) For a general metric space (X, d), define the following (i) x is a Limit point of a set E ⊂ X. (ii) The closure of E (i.e E). b) In (R2, d1), let E = {(x, y) ∈ R2 s.t x < 1}. Find E with justification . 1