# Loyola College B.Sc. Mathematics April 2011 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

# MT 5505/MT 5501 – REAL ANALYSIS

Date : 11-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 9:00 – 12:00

SECTION  A

Answer ALL questions.                                    (10 x 2 = 20)

1. State the least upper bound axiom.

1. Prove that any infinite set contains a countable subset.

1. Prove that the intersection of an arbitrary collection of open sets need not be open.

1. Distinguish between adherent and accumulation points.

1. Prove that any polynomial function is continuous at each point in .

1. Give an example of a continuous function which is not uniformly continuous.

1. State Rolle’s theorem.

1. If a real-valued function has a derivative at , prove that is continuous at .

1. Give an example of a sequence of real numbers whose limit inferior and limit superior exist, but the sequence is not convergent.

1. Give an example of a function which is not Riemann-Stieltjes integrable.

SECTION  B

Answer ANY FIVE questions.                                     (5 x 8 = 40)

1. State and prove Cauchy-Schwartz inequality.

1. Prove that the Cantor set is uncountable.

1. Prove that a subset E of a metric space is closed if and only if it contains all its adherent points.

1. Prove that a closed subset of a complete metric space is also complete.

1. State and prove Lagrange’s mean value theorem.

1. If a real-valued function is monotonic on , prove that the set of discontinuities of is countable.

1. If a real-valued function is continuous on , and if exists and is bounded in , prove that  is of bounded variation on .

1. State and prove integration by parts formula concerning Riemann-Stieltjes integration.

SECTION  C

Answer ANY TWO questions.                                     (2 x 20 = 40)

 19. (a) Prove that the set of rational numbers is not order-complete. (b) Prove that the set of all rational numbers is countable. (c) State and prove Minkowski’s inequality.                                                          (10+5+5) 20. (a) Prove that every bounded and infinite subset of  has at least one accumulation point. (b) State and prove the Heine-Borel theorem.                                                       (16+4) 21. (a) Let  and  be metric spaces and . Show that  is continuous at  if and only if for every sequence  in X that converges to , the sequence  converges to . (b) Prove that a continuous function defined on a compact metric space is uniformly continuous.                                                                                                        (10 + 10) 22. (a) State and prove Taylor’s theorem. (b) Prove that a monotonic sequence of real numbers is convergent if and only if it is bounded.                                                                                                                (12+8)

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