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
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.
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.
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)|