LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 10

FIFTH SEMESTER – November 2008

MT 5505 – REAL ANALYSIS

 

 

 

Date : 03-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer  ALL  Questions                                                                                           (10 x 2=20 marks)

1. Define similar sets with an example.
2. If a and b are any real numbers such that

 

3. Define a metric space.
4. Give an example of a set E in which every interior point of E is also an accumulation point of E but

not conversely.

5. Define a convergent sequence.
6. State intermediate value theorem for continuous functions.
7. Define Open ball and Closure of set E.
8. When is a sequence {a_n} said to be Monotonic increasing and decreasing?
9. State the linearity property of Riemann- Stieltjes integral.
10. Define limit superior of a real sequence.

SECTION – B

Answer  ANY FIVE Questions.                                                                                (5 x 8=40 marks)

 

11. State and prove Minkowski’s inequality.
12. If n is any positive integer, then prove that Nⁿ is countably infinite.
13. Let(X, d ) be a metric space. Then prove that
14. i) the union of an arbitrary collection of open sets in X is open in X.
15. ii) the intersection of an arbitrary collection of closed sets in X is closed in
16. Prove that a closed subset of a compact metric space is compact.
17. Prove that every compact subset of a metric space is complete.
18. Let f : ( X, d ) → R^k be continuous on X. If X is compact, then prove that f is

bounded on X.

17. State and prove Rolle’s theorem.
18. Let {a_n} be a real sequence. Then prove that

(i) {a_n} converges to l  if and only if  lim inf a_n = lim sup a_n = l

(ii) {a_n} diverges to + ∞ if and only if  lim inf a_n = + ∞

SECTION C

Answer  ANY TWO  Questions.                                                                              (2 x 20 = 40 marks)

 

19. (a) Prove that the set R is uncountable. (10 marks)

(b) State and prove Cauchy –Schwartz inequality.                                                          (10 marks)

 

20. (a) State and prove Bolzano- Weierstrass theorem. (18 marks)

(b) Give an example of a metric space in which a closed ball

is not the  closure of the open ball B(a ; r ).                                                                 (2 marks)

 

21. (a) Let ( X, d₁) and (Y, d₂) be metric spaces and f : X→Y. If x₀X, then prove that f is

continuous at x₀ if and only if for every sequence {x_n} in X that converges to x₀,

the sequence { f (x_n ) } converges to f (x₀).                                                                  (12marks)

(b) Prove that Euclidean space ^k is complete.                                                                ( 8 marks)

 

 

1. b) State and prove Taylor’s theorem.                                                                  (10 marks)

 

 

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