advertisement

Student Name: Exam 1 - Math 4320/5320 - Modern Analysis II Dr. Radu C. Cascaval, Spring 2016 Take Home - Due Tuesday, March 8 before class time You may consult the textbook or other printed materials (use of internet resources is highly discouraged), but you cannot get help from any breathing source (like humans, monkeys, dogs etc).! 1. Given a metric space (M, d) and a set nonempty set A, the distance from a point p ∈ M to A is defined as dA (p) = inf d(p, a) a∈A (a) Prove that if A is compact, then the infimum is actually achieve (so is a minimum) and that the function dA : M → R is continuous. (b) Give an example of a set A where the infimum in (a) is not achieved. (c) [Math 5320 only] Show that the property of A being closed is not enough to guarantee the infimum in (a) is achieved (so compactness is essential). 2. (a) Consider the sequence of functions fn (x) = x nx + 1 Prove that {fn } is uniformly convergent on [0, 1]. (b) Consider the sequence of functions gn (x) = 1 nx + 1 Prove that {gn } is convergent pointwise on [0, 1], but it is not uniformly convergent on (0, 1). 3. Let {an } be a bounded sequence of real numbers. For each n ≥ 0, define fn (x) = a0 + a1 x + a2 x 2 an xn + ··· + 2! n! Prove that for each r > 0, the sequence of functions {fn : [−r, r] → R is uniformly convergent on [−r, r]. Is it true that fn converges uniformly on R?