MATH 210 Exam 1, Spring 2017

Math 210, I > nt : . roduction to Analysis- Spring 2016-2017 Quiz 1 , February 295. Time 50 minutes s aoe that d is a metric on etermine B 1 ; c) i. Define a 9)» Ba(0, 2) and Ba(1, 2). 1p what it means for (X,d) to be a complete metric space: - Prove that (R, d) is complete. \ Questi | , : Q ion 1. (8 pts.) Define he) dq RR > Re by da.) = Quiestion 2. (4 pts) S ES Pane {x,} C R, define limsup Zp and liminf In- ) ind with justification lim sup and lim inf of the sequence defined by eS - _ (1)tn | sin n| In = + OD Byte Vip & Question 3. (6 pts) Find in R? the limit of lim (7 vnt*) n—?0o iar Ml to find the limit).Prove using the (You may use Calculus definition (€, no) that this is indeed the limit in (R’, a1) - Question 4. (7 pts). s for an ordered set 5 to have the greatest lower bound property. neea O justily _ Find, if it exists, inf A in Q, and in R. You