LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 29

FIFTH SEMESTER – April 2009

MT 5505 / 5501 – REAL ANALYSIS

 

 

 

Date & Time: 16/04/2009 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks

 

 

SECTION-A

 

Answer all questions:                                                                        (10 x 2=20)

 

1. Define an order – complete set and give an example of it.
2. When do you say that two sets are similar?
3. Define discrete metric space.
4. Give an example of a perfect set in real numbers.
5. Define open map and closed map.
6. When do you say that a function is uniformly continuous?
7. If a function f is differentiable at c, show that it is continuous.
8. Define “total variation” of a function f on [a,b].
9. Give an example of a sequence {a_n}whose lim inf and lim Sup exist, but the sequence is not convergent.
10. Give an example of a function which is not Riemann Stieltjes integrable.

 

SECTION-B

 

Answer any five questions:                                                              (5 x 8=40)

 

1. Let a,b be two integers such that (a,b)=d. Show that there exists integers  such that .
2. Show that the set of all real numbers is uncountable.
3. Let E be a subset of a metric space X. Show that  is the smallest closed set containing E.
4. State and prove Heire Borel theorem.
5. Show that in a metric space every convergent sequence is Cauchy, but not conversely.
6. Let f, g be differentiable at Show that f  g  is differentiable at c and if is also differentiable at c.
7. State and prove Lagrange’s mean valve theorem.
8. Suppose {a_n}is a real sequence. Show that lim Supa_n=l if and only if for every ,

• there exists a positive integer N such that for all and
• given any positive integer m, there exists an integer such that .

 

 

 

 

 

 

SECTION-C

 

Answer any two questions:                                                                          (2 x 20=40)    

 

1. (a) State and prove Minkowski’s inequality.

(b) Show that there is a rational number between any two distinct real numbers.

(c) If show that e is irrational.

1. (a) If F is a family of open intervals that covers a closed interval [a,b], show that a finite sub family of F also covers [a,b].

(b) Let SÌRⁿ. If every infinite subset of S has an accumulation point in S, show that S is closed and bounded.

1.  (a) Let (X,d₁), (Y, d₂) be metric spaces and . Show that f is continuous at if and only if for every sequence in X that converges to  the sequence converges to .

(b) State and prove Bolzano theorem.

 

1. (a) State and prove Taylor’s theorem.

(b) Suppose on [a,b]. Show that on [a,b] and

 

 

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