03.04.2004
9.00 - 12.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
FIRST SEMESTER – APRIL 2004
ST 1800/S 715 - ANALYSIS
Max:100 marks
SECTION - A
(10  2 = 20 marks)
Answer ALL questions
1.
2.
3.
4.
5.
Define a bijective function.
Define a metric.
Is the set (0,1) complete? How?
Define the symbols Big O and small o.
Let f(x) = 1 if x is rational
0 if x is irrational, 0  x  1
Is the function Riemann integrable over [0,1]?
6. Define lim inf and lim sup of a sequence xn.
7. Define the linear derivative of a function f: X  Rn; where X  Rm.
mn
8. Find the double limit of xmn =
and lim  lim xn  .
mn
n  m

9. Define uniform convergence of a sequence of functions.
x2 y
lim
10. Let f(x,y) = 4
be defined on R2 - {(0,0)}. Show that
f(x,y) does
2
( x, y )  (0, 0)
x y
not exist.
SECTION - B
(5  8 = 40 marks)
Answer any FIVE questions.
11. State and prove Cauchy's Inequality.
12. Show that R' is complete.
13. Show that any collection of open sets is open and any collection of closed sets is closed.
14. State and prove Banach's fixed point theorem.
15. Let {fn} be a sequence of real functions integrable over the finite interval [a, b]. If fn  f
uniformly on [a, b] then show that i) f is integrable over [a, b] and ii)  f n   f .
16. State and prove Weierstrass M-Test.
17. Show that A is the upper limit of the sequence {xn} if and only if, given  > 0
xn <   for all sufficiently large n
xn >    for infinitely many n
18. Show that if f  R [ g; a, b] then f  R [g; a,b] and
 f dg  
If f is R.S integrable, can you say f R.S. integrable? Justify.
f dg .
(3+3+2)
1
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
19. a) State and prove Cauchy's root test.
b) Discuss the convergence of the infinite series whose nth terms are
(n  a) n
e tn
i)
ii)
n n a
n2
(8+6+6)
20. a) Define a compact metric space. Show that a compact set in a metric space is also
complete.
(5)
b) State and prove Heine - Borel theorem.
(15)
21. a) State and prove a necessary and sufficient condition that the function f is Riemann –
Stieltjes interable.
b) If f is continuous then show that f  R [g; a,b]
c) If f1, f2  R [g; a,b] then show that f1 f2  R [g; a,b]
(6+6+8]
22. a) Let (X, ) and Y, ) be metric spaces. Show that the following condition is necessary
and sufficient for the function f: X  Y to be continuous on X: whenever G is open in
Y, then f-1 (G) is open on X.
b) Let V,W be normed vector spaces. If the function f: V  W is linear, then show that
the following three statements are equivalent.
i)
f is continuous on V
ii)
There is a point xo  V at which f is continuous.
iii)
f ( x)
x
is bounded for x  V - {0}.

2