MAT 2125 Assignment 4 Due 4:30pm, 24-March, 2015. Instructor: Barry Jessup Family Name: 1 First Name: 2 Student number: 3 4 5 (For the marker’s use only !) Total PLEASE READ THESE INSTRUCTIONS CAREFULLY. 1. Only certain parts of the questions will be graded; you will learn which parts the day the before the assignment is due. Do not wait until then to begin! 2. Read each question carefully, and to help the marker, please answer questions in the space after each question. You may use the backs of pages if necessary, but be sure to indicate to the marker that you have done this. 3. Part marks can be earned. The correct answers require justification written legibly and logically: you must convince us that you know why your solution is correct. You may use any result proven in class without proof, but if the theorem or property has a name, cite it when you use it. 4. You may submit this assignment to me after your DGD, after class, by slipping it under my door, or by putting it into the box marked “MAT2125” in the cabinet in the foyer of the math building (KED) on the due date, by 4:30pm. 2 1. For n 2 N \ {0}, define open intervals Vn on the real line by Vn = ( 1 1 , ). 2n n a) Explain briefly why (0, 1) is not compact. b) Prove that C = {Vn | n 1} is an open cover of (0, 1). c) Without using the Heine-Borel theorem, prove that no finite subset of C covers (0, 1). 1 1 d) Note that C is also an open cover of [ 12 , 8 ]. Find an explicit finite sub-cover of C which covers 1 1 [ 12 , 8 ]. 3 2. Suppose that f : A ! R is uniformly continuous on A, and let A¯ denote the closure of A. a) Prove that if {an }n 1 ✓ A is Cauchy, so is {f (an )}n 1. b) Prove that there exists a continuous function g : A¯ ! R such that 8x 2 A, g(x) = f (x). c) Prove that there exists a unique continuous function g : A¯ ! R such that 8x 2 A, g(x) = f (x). d) Give an example of A ⇢ R, and f : A ! R which is not uniformly continuous on A for which the conclusion of (a) fails. 4 3. A function f : A ! Rp is Lipschitz continuous on A if 9M 0 such that 8 x, y 2 A, kf (x) f (y)k  M kx yk. (We then say f is Lipschitz continuous with constant M on A.) a) Prove that if a function f is Lipschitz continuous on A, then f is uniformly continuous on A. b) Show that f (x) = p x is not Lipschitz continuous on [0, 1]. c) Give an example where f is uniformly continuous on A but is not Lipschitz on A. 5 4. For each of the following sequences of functions {fn }n 1, (i) decide if f = lim fn exists, and n!1 (ii) if f = lim fn exists, decide whether the convergence is only pointwise or is uniform on the indicated n!1 domain. a) fn : [0, 1] ! R defined by fn (x) = p n x. x . 1+n+x q c) fn : [1, 1) ! R defined by fn (x) = x2 + n12 . b) fn : [0, 1) ! R defined by fn (x) = q d) fn : [0, 1) ! R defined by fn (x) = x2 + p prove, then use: 3 > a > 0 ) 1 + a > a3 .) 1 n2 . (Hint: Consider |f fn | when 0 6= |x| is small; 6 5. For the following series, find the largest subset A ⇢ R where the series converges pointwise (and find the limit function in part (a)). Then find largest subset B ⇢ R where the series converges uniformly. a) X n 1 x (1 + x)n X n!xn 1 n b) (You may assume the results of Q.33 on the list. Denote lim (1 + ) by the symbol e.) n!1 nn n n 1 X xn c) n2n n 1