MATH 210 Final, Spring 2013

Le American University of Beirut Math 210, Final Exam, May 17, 2013 Duration: 120 minutes (10 points) a) State the definitions of “an isolated point”, “a countable set” and “the closure of a set”. Give an example on each. . b) Suppose X and Y be metric spaces with Y a subset of X. In addition, suppose Kis a subset of Y. Show that K is compact relative to X iff K is compact relative to Y. (10 points) Let K be a subset of real numbers consisting of 0 and the duribers — , for n n=1,2,3,... Since K is closed and bounded, K is compact by the Heine-Borel Theorem. Use this result to construct a compact set of real numbers whose limit points form a countable set. (15 points) The sequence of Fibonacci numbers is defined by A, = 0,4, =1,4,,9 =A, + Ay. - Ant for every n in fN , show that a n a) Writing x, = Xan SX2nv1 S%2n42 <%an- 1 (Hing x, =14-—) x n-| b) ForninN, let J, =[*,,).%,]. i) Using part a), verify that /,,, is a subset of /, foreach ninNV . li) Show that diamI, =0. no c) Find limx,. no (Hint: Using part b), what can you say about { )/, 2) n=l 4) (10 points) Let a, =1+ : +..+ he logn. Prove that lima, exists. n APD (Hint: Consider a,,,—a, and represent it as an integral.) (please turn the page over) 5) (10 points) Suppose De" has.radius of convergence 1 and that Sa n=0 n=0 : «oO converges to a. Show that "a,x" converges to aas x->]-. =0 Ls 6) (10 points ) a) Explain why the Riemann-Stieltjes integral is exp(x" )dar(x) at exists for any increasing function @: [-11] OR. b) Evaluate the above integral in the special case C it x= a(x)= x41 iF So 7) (10 points) Let f :[0,00) —>[0, 00) be defined by f(x) = Vx. Show that a) ve is unbounded on (0,1). b) ff is uniformly continuous on (0,1). c) fis uniformly continuous on [1,). 1+2cos?(nx) He Show that { f, (x)} converges to 0 uniformly onR. 8) (15 points) a) Let f(x) = b) Let g,(x) = * | Foreach xinR compute g(x) = lim g(x). : n Show that { g, (x) } converges to g uniformly on [0,1] but not on[0,). i i =0 for i i 1] satisfies | x" f(x)dx =0 9) (10 points) Show that a continuous function f on [0,1] J all n=0,1,2,.. iff f vanishes everywhere on [0,1]. = <eé on ([0,1)). (Hint: For each ¢ > 0 there is a polynomial P such that | f(x)-P)|