LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc.. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – SUPPLEMENTARY – JUNE 2012
MT 5505/5501- REAL ANALYSIS
Date : 29-06-2012
Time : 2:00 - 5:00
Dept. No.
Max. : 100 Marks
SECTION A
Answer ALL questions.
(10 x 2 = 20)
1. State and prove the triangular inequality.
2. Prove that every infinite set has a countable subset.
3. Prove that intersection of an arbitrary collection of closed sets is a closed set.
4. Prove that every neighbourhood of an accumulation point of a subset E of a metric space contains infinitely
many points of E.
5. Show that a Cauchy sequence need not be convergent.
6. Give an example of a real-valued function which is discontinuous everywhere in R.
7. If the function f is continuous on  a, b  and f (x)  0 in  a,b  , prove that the function f is strictly
increasing in  a, b  .
8. Prove that every function defined and monotonic on [a, b] is of bounded variation on [a, b] .

9. Determine the limit inferior and limit superior of the sequence  n .
n 1
10. Give an example of a discontinuous function which is Riemann-Stieltjes integrable.
SECTION B
Answer ANY FIVE questions.
11. State and prove Minkowski’s inequality.
12. Prove that the set R of real numbers is uncountable.
13. State and prove the Heine-Borel theorem.
14. Prove that the Euclidean space R n is a complete metric space.
(5 x 8 = 40)
15. Prove that every continuous function defined on a closed interval assumes every value between its bounds
at least once.
16. State and prove the Rolle’s theorem.
17. State and prove integration by parts formula concerning Riemann-Stieltjes integration.
18. Prove that a bounded monotonic sequence of real numbers is convergent.
SECTION C
Answer ANY TWO questions.
(2 x 20 = 40)
19. (a) State and prove the necessary and sufficient condition for a real number to be the
supremum of a bounded above set.
(b) If  is a countable collection of pairwise disjoint countable sets, prove that the set
F is countable.
(12+8)
FF
20. (a) If  is a family of open intervals that covers a closed interval [a, b] , show that a
finite subfamily of  also covers [a, b] .
(b) If every infinite subset of S  R n has an accumulation point in S, show that
the set S is closed and bounded.
(12+8)
21. (a) Let  X,d1  and  Y,d 2  be metric spaces and f : X  Y . Prove that the function
f is continuous on X if and only if f 1(C) is closed in X for every closed set C in Y.
1

, if x  0

(b) Discuss the continuity of the function f given by f (x)   1  e  1 x

0
, if x  0

at the point x  0 .
(c) Prove that a continuous function defined on a compact metric space is uniformly
continuous.
(5+5+10)
22.
x2 x4 x6
(a) Prove that log(sec x) 


     .
2 12 45
(b) Let
a n 
be a sequence of real numbers. Prove that
if and only if liminf a n  limsupa n  L .
a n 
converges to L
(c) Let f () on [a, b] and g be a strictly increasing function defined on [c,d] such
that g [c,d]  [a,b] . Let h and  be the composite functions given by
h(y)  f (g(y)) and (y)  (g(y)) . Prove that h () on [c,d] and that
b
d
 f d   h d  .
a
(5+5+10)
c
$$$$$$$