LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

MT 5505/MT 5501 – REAL ANALYSIS

 

 

 

Date : 25-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

 

Answer ALL questions:                                                                                                        (10 x 2 = 20)

 

1. State and prove the triangular inequality.

 

2. Prove that the sets Z and N are similar.

 

3. Prove that the union of an arbitrary collection of closed sets is not necessarily closed.

 

4. Prove that every neighbourhood of an accumulation point of a subset E of a metric space contains infinitely many points of the set E.

 

5. Show that every convergent sequence is a Cauchy sequence.

 

6. Define the term “complete metric space” with an example.

 

7. State Rolle’s theorem.

 

8. Prove that every function defined and monotonic on is of bounded variation on .

 

9. State the linearity property of Riemann-Stieltjes integral.

 

10. State the conditions under which Riemann-Stieltjes integral reduces to Riemann integral.

 

PART – B

 

Answer ANY FIVE questions:                                                                                 (5 x 8 = 40 marks)

 

11. State and prove Cauchy-Schwartz inequality.

 

12. Prove that the interval is uncountable.

 

13. State and prove the Heine-Borel theorem.

 

14. State and prove the intermediate value theorem for continuous functions.

 

15. Let and be metric spaces and . If  is compact and  is continuous on , prove that  is uniformly continuous on .

 

16. State and prove the intermediate value theorem for derivatives.

 

 

 

17. Suppose on . Prove that on  and that

.

 

18. a) Let be a real sequence. Prove that (a) converges to L if and only if

 

(b)  diverges to  if and only if .

 

PART – C

 

 

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

 

19.	(a) Prove that every subset of a countable set is countable.
	(b) Prove that countable union of countable sets is countable.
	(c) State and prove Minkowski’s inequality.                                                            (8+7+5)
20.	(a) Prove that the only sets in R that are both open and closed are the empty set and the set R itself.
	(b) Let E be a subset of a metric space . Show that the closure  of E is the smallest closed set containing E.
	(c) Prove that a closed subset of a compact metric space is compact.                     (4+8+8)
21.	(a) Let  and  be metric spaces and . Prove that  is continuous on X if and only if  is open in X for every open set G in Y.
	(b) Explain the classification of discontinuities of real-valued functions with examples. (12+8)
22.	(a) State and prove Lagrange’s mean value theorem.
	(b) Suppose  on  and  for every  that is monotonic on . Prove that  must be constant on .
	(c) Prove that a bounded monotonic sequence of real numbers is convergent.       (8+4+8)

 

 

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