LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

CV 13

DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5501 – REAL ANALYSIS

 

 

Date & Time: 28/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

 

 

Answer all the questions:                                                                           10 x 2 = 10

 

1. Define an inductive set with an example.

 

2. Prove that every positive integer n (except 1) is either a prime or a product of primes.

 

3. State and prove Euler’s theorem for real numbers.

 

4. Define a Metric space.

 

5. State Cantor’s intersection theorem for closed sets.

 

6. Define an interior point and an open set.

 

7.Give an example of a uniformly continuous function.

 

8. Define a Cauchy sequence.

 

9. Suppose f and g are defined on (a, b) and are both differentiable at c Î (a, b), then prove

 

that the function fg is also differentiable at c.

 

10. Define total variation of a function f on .

 

Answer any five questions:                                                                                         5 x 8=40

 

11. Prove that the set R of all real numbers is uncountable.

 

12. State and prove Bolzano-Weirstass theorem for R.

 

13. Prove that every compact subset of a metric space is complete.

 

14. Let (X, d₁) and (Y, d₂) be metric spaces and f: X Y be continuous on X. If X is compact, then prove that f (X) is a compact subset of Y.

 

15. Let (X, d₁) and (Y, d₂) be metric spaces and f: X Y be continuous on X. Then show that a map f: X Y is continuous on X if and only if f ^-1 (G) is open in X for every open set G in Y.

 

     16    Prove that in a metric space (X, d)

( i ) Arbitrary union of open sets in X is open in X

( ii) Arbitrary intersection of closed sets in X  is closed in X.

 

17. Let f: R and f have a local maximum or a local minimum at a point c.

Then prove that f ’(c) = 0.

 

18. Let f be of bounded variation onand xÎ (a, b) Define V:  R as   follows:

V (a) = 0

V (x) =V_f , a <  x ≤ b.

Then show that the functions V and V – f are both increasing functions on.

 

Answer any two questions:                                                                                                      2 x 20 = 40

                                                                                      

19   State and prove Intermediate value theorem for continuous functions.

 

20.   State and prove Lagrange’s theorem for a function f :  R       

21.(a) Suppose c Î (a ,b) and two of the three integrals f da ,f da , and f da

exists. Then show that the third also exists andf da =f da +f da.

 

(b) When do we say f is Riemann-Stieltjes integrable?

 

22. (a) State and prove Unique factorization theorem for real numbers.

 

(b) If F is a countable family of countable sets then show that  is also countable.

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