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

FIFTH SEMESTER – November 2008
MT 5501 – REAL ANALYSIS
Date : 051108 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
SECTIONA
Answer ALL questions: (10 x 2=20 marks)
 State principle of induction.
 Show that the set Z is similar to N.
 Define isolated point of a set in a metric space.
 “Arbitrary intersection of open sets in open” True or False. Justify your answer.
 If {x_{n}} is a sequence in a metricspace, show that { x_{n}} converges to a unique point.
 Define complete metric space and give an example of a space which is not complete.
 When do you say that a function has a right hand derivative at ?
 Define (i) Strictly increasing function
(ii) Strictly decreasing function
 When do you say that a partition is a refinement of another partition? Illustrate by an example.
 Define limit superior and limit inferior of a sequence.
SECTIONB
Answer any FIVE questions: (5 x 8=40 marks)
 Show that is an irrational number.
 Show that collection of all sequences whose terms are 0 and 1 is uncountable.
 Let Y be a subspace of a metric space (X,d). Show that a subset A of Y is open in Y if an only if for some open set G in X.
 Show that every compact subset of a metric space is complete.
 Show that every compact set is closed and bounded in a metric space.
 Let be differentiable at c and g be a function such that where I is some open interval containing the range of f. If g is differentiable at f(a), show that g_{o}f is differentiable at c and (g_{o}f)^{’}(c)=g^{’}(f(o).f^{’}(c).
 If f is of bounded variation on [a,b] and if f is also of bounded variation on [a,c] and [c,b] for , show that .
 Show that lim inf(a_{n}) if and only if for .
 there exists a positive integer N such that for all and
 given any positive integer m, there exists such that .
SECTIONC
Answer any TWO questions: (2 x 20=40 marks)
 (a) State and prove Unique factrization theorem for integers.
(b) If S is an infinite set, show that S contains a countably infinite set.
(c) Given a countable family F of sets, show that we can find a countable family G of pairwise disjoint sets such that .
 (a) State and prove Heine theorem.
(b) If S, T be subsets of a metric space M,
show that (i)
(ii)
Illustrate by an example that
 (a) Let X be a compact metric space and be continuous on X.
Show that is a compact subset of Y.
(b) Show that on R is continuous but not uniformly continuous.
 (a) Let f be of bounded variation on [a,b] and V be the variation of f. Show that V is continuous
from the right at if and only if f is continuous from the right at c.
(b) on [a,b] and g is strictly increasing function defined on [c,d] such that
g ([c,a])=[a,b]. Let h (y) = f (g (y)) and .
Show that .
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