LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc., DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – NOVEMBER 2012
MT 5505/5501- REAL ANALYSIS
Date : 01-11-2012
Time : 9.00 – 12.00
Dept. No.
Max. : 100 Marks
PART – A
(10  2 = 20)
Answer ALL the questions:
1. Define a countable set.
2. When do you say that a set is order complete?
3. Define (a) adherent point, (b) accumulation point.
4. Is infinite union of closed sets closed? Justify.
5. Show that every convergent sequence is Cauchy.
6. Give an example of a function which is continuous but not uniformly continuous.
7. State the chain rule.
8. State Lagrange’s mean value theorem.
9. Define the limit superior of a sequence.
10. When do you say that a function
is Riemann integrable?
PART – B
(5  8 = 40)
Answer any FIVE questions:
11. State and prove Cauchy Schwarz inequality.
12. Let F be a collection of pairwise disjoint countable sets. Show that U FF F is countable.
13. Show that a closed subset of a compact metric space is compact.
14. Let ( X , d ) be a metric space. Show that (a) the union of an arbitrary collection of open sets
in X is open in X , and (b) the intersection of finite collection of open sets in X is open in X .
15. Show that the Euclidean space
is complete.
16. State and prove intermediate value theorem for derivatives.
17. Let
be a real sequence.
.
1
x  x
 log n  
dx  1.

x2
k 1 k
1
n
18. Show that
n
Show that
converges to
if and only if
PART – C
(2  20 = 40)
Answer any TWO questions:
19. Let
on
,
be differentiable on
that the Riemann integral
and
exists and
be continuous on
. Show
.
20. State and prove the Bolzano-Weierstrass theorem.
21. (a) Prove that in a compact metric space
, every sequence in has a subsequence
which converges in .
(b) Show that a continuous function defined on a compact metric space is uniformly
continuous.
22. (a) State and prove Taylor’s theorem.
(b) Does continuity implies bounded variation? Justify.
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