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)

- State the least upper bound axiom.

- Prove that any infinite set contains a countable subset.

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

- Distinguish between adherent and accumulation points.

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

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

- State Rolle’s theorem.

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

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

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

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**SECTION B**

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

- State and prove Cauchy-Schwartz inequality.

- Prove that the Cantor set is uncountable.

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

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

- State and prove Lagrange’s mean value theorem.

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

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

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

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**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) |