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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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