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Unformatted text preview: McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 255-001 HONOURS ANALYSIS 2 Examiner: Professor K. GowriSankaran Date: Wednesday April 12, 2006 Associate Examiner: Professor S. Drury Time: 2:00 pm — 5:00 pm INSTRUCTIONS (a) Answer questions in the exam booklets provided. (b) All questions count equally. (c). This is a closed book exam. No computers, notes or text books are permitted. (d) Calculators are not permitted. ‘ (e) Use of a regular and or translation dictionary is not permitted. This exam comprises of the cover page, and 2 pages of 6 questions. McGill University MATH 255 FINAL EXAMINATION No Calculators Answer all questions. All questions count equally. 1. Decide if the following statements are true or false. 00 00 (a) 2 non converges =# 2 an converges 1 1 (h) f (as) := sin(1/a:) is Riemann - Darboux integrable on [0, 11 (c) Suppose f is continous on [1, 00) and K” f is finite then f tends zero as n —> oo. 2. Justify your conclusions. (a) fn := nx(1 — $2)" for each 71,95 6 [0, 1]. Prove that (fn) converges pointwise but the convergence is not uniform. ' 00 (b) Suppose (an) is a bounded sequence of real numbers such that 2a,. diverges. n=0 CD Show that 2 ans” has radius of convergence 1. 0 n 3. (3.) Find k; k2 + n2 by usmg the definltlon of Riemann — Darboux integral of an appropriate continuous function. (b) Let : [0, 1] ——> R be continuous. Prove that the Cauchy-Reimann integral 1 ~ f (m) . - . —-—.— da: 15 finlte. f0 \/ 1 - m2 4. (a) SupposeiP and Q are polynomial functions of degree p and q respectively. Suppose °° P It further that Whatever be k E N, Q(k) 7E 0. Prove that 23-4),c+1 ( ) ' Ic=1 1s conver exit 6209) I g if and only if p < q. 00 1 (b) Show that Z ——3— < 2 + . 1 n5 5. (a) Let K1 and K2 be two non—void compact subsets of R such that -K1 0 K2 :: (Z). Prove that inf{|a: — y] :2: E K], y 6 K2} is > 0. page 1 MATH 255 — FINAL EXAMINATION April 2006 (b) Let (fn) be a sequence of continuous functions on R such that V x 6 [0,1], f,,(a:) S f,._1 Vn 2 2. Suppose Va: 6 [0, 1], fn(x) ——> 0. Prove that (fn) converges to 0 uniformly. [HintzVe > 0, prove that {Vm} is an open cover of [0, 1] if Vm = {:1: : fm(x) < 6}] 6. (3) Define/ Explain the following concepts. (i) d is a metric on a set X (ii) 3:" E X and y 6 X,mn ——> y in the metric d (b) Let F be a non—void closed subset of a space X with a metric d and‘let MaF9=mfiflamiy€Fl Prove that p is a continuous function on X such that {:17 : p($, F) := 0} = F. (c) Let F1 C X, F2 C X be two disjoint non-void closed subsets of the metric space _ M— (X,d). Prove that f(z) — We, F1) + pm F2) Use the f in (c) above and show that there are open sets V,- D Fj such that mn%=0 is a continuous function X —+ [0, 1]. page 2 ...
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