LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 19

FIFTH SEMESTER – November 2008

MT 5501 – REAL ANALYSIS

 

 

 

Date : 05-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION-A

Answer ALL questions:                                                                                           (10 x 2=20 marks)

 

1. State principle of induction.
2. Show that the set Z is similar to N.
3. Define isolated point of a set in a metric space.
4. “Arbitrary intersection of open sets in open”  True or False. Justify your answer.
5. If {x_n} is a sequence in a metricspace, show that { x_n} converges to a unique point.
6. Define complete metric space and give an example of a space which is not complete.
7. When do you say that a function  has a right hand derivative at ?
8. Define (i) Strictly increasing function

(ii) Strictly decreasing function

1. When do you say that a partition is a refinement of another partition? Illustrate by an example.
2. Define limit superior and limit inferior of a sequence.

 

SECTION-B

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

 

1. Show that is an irrational number.
2. Show that collection of all sequences whose terms are 0 and 1 is uncountable.
3. Let Y be a subspace of a metric space (X,d). Show that a subset A of Y is open in Y if an only if  for some open set G in X.
4. Show that every compact subset of a metric space is complete.
5. Show that every compact set is closed and bounded in a metric space.
6. Let be differentiable at c and g be a function such that where I is some open interval containing the range of f. If g is differentiable at f(a), show that g_of is differentiable at c and (g_of)^’(c)=g^’(f(o).f^’(c).

 

1. If f is of bounded variation on [a,b] and if f is also of bounded variation on [a,c] and [c,b] for , show that .
2. Show that lim inf(a_n) if and only if for .

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

 

 

 

 

 

SECTION-C

Answer any TWO questions:                                                                                   (2 x 20=40 marks)     

1. (a) State and prove Unique factrization theorem for integers.

(b) If S is an infinite set, show that S contains a countably infinite set.

(c) Given a countable family F of sets, show that we can find a countable family G of pairwise disjoint sets such that .

 

1. (a) State and prove Heine theorem.

 

(b) If S, T be subsets of a metric space M,

show that        (i)

(ii)

Illustrate by an example that

 

1. (a) Let X be a compact metric space and  be continuous on X.

Show that is a compact subset of Y.

(b) Show that  on R is continuous but not uniformly continuous.

 

1. (a) Let f be of bounded variation on [a,b] and V be the variation of f. Show that V is continuous

from the right at if and only if f is continuous from the right at c.

(b) on [a,b] and g is strictly increasing function defined on [c,d] such that

g ([c,a])=[a,b]. Let h (y) = f (g (y)) and .

Show that  .

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