LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – April 2009
MT 5505 / 5501 – REAL ANALYSIS
Date & Time: 16/04/2009 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
SECTIONA
Answer all questions: (10 x 2=20)
 Define an order – complete set and give an example of it.
 When do you say that two sets are similar?
 Define discrete metric space.
 Give an example of a perfect set in real numbers.
 Define open map and closed map.
 When do you say that a function is uniformly continuous?
 If a function f is differentiable at c, show that it is continuous.
 Define “total variation” of a function f on [a,b].
 Give an example of a sequence {a_{n}}whose lim inf and lim Sup exist, but the sequence is not convergent.
 Give an example of a function which is not Riemann Stieltjes integrable.
SECTIONB
Answer any five questions: (5 x 8=40)
 Let a,b be two integers such that (a,b)=d. Show that there exists integers such that .
 Show that the set of all real numbers is uncountable.
 Let E be a subset of a metric space X. Show that is the smallest closed set containing E.
 State and prove Heire Borel theorem.
 Show that in a metric space every convergent sequence is Cauchy, but not conversely.
 Let f, g be differentiable at Show that f g is differentiable at c and if is also differentiable at c.
 State and prove Lagrange’s mean valve theorem.
 Suppose {a_{n}}is a real sequence. Show that lim Supa_{n}=l if and only if for every ,
 there exists a positive integer N such that for all and
 given any positive integer m, there exists an integer such that .
SECTIONC
Answer any two questions: (2 x 20=40)
 (a) State and prove Minkowski’s inequality.
(b) Show that there is a rational number between any two distinct real numbers.
(c) If show that e is irrational.
 (a) If F is a family of open intervals that covers a closed interval [a,b], show that a finite sub family of F also covers [a,b].
(b) Let SÌR^{n}. If every infinite subset of S has an accumulation point in S, show that S is closed and bounded.
 (a) Let (X,d_{1}), (Y, d_{2}) be metric spaces and . Show that f is continuous at if and only if for every sequence in X that converges to the sequence converges to .
(b) State and prove Bolzano theorem.
 (a) State and prove Taylor’s theorem.
(b) Suppose on [a,b]. Show that on [a,b] and
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