LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
FIRST SEMESTER – NOVEMBER 2003
ST-1800/S715 - ANALYSIS
Max:100 marks
04.11.2003
1.00 - 4.00
SECTION-A
Answer ALL questions.
(10x2=20marks)
1. Let Z be the set of all integers. Construct a function form Z to Z which is not one to one
and also not onto.
2. Define a metric on a non-empty set x.
x2
3. The real valued function f on R - (0, 0) is defined by f (x, y) = 2
. Show that
x  y2
2
lim f (x,y) does not exist as (x, y)  (0, 0) .
4. State weirstrass's approximation theorem.
5. If x n  is a convergent sequence in a metric space (X, P) then prove that it is a cauchy
sequence.
6. If Un = O (1/nk-2), for what value of k
u
n
converges?
7. Define the upper limit and lower limit of a sequence.


8. Find lim lim x mn and also the double limit of xmn as m,n   where xmn =
m  n 
mn
.
mn
9. Let f: Rm  R n . Define the linear derivative of f at  .
10. From the infinite series 1  x  x 2  x 3  x 4  ....... where x  1 obtain the expansion for
log (1+x).
SECTION-B
Answer any FIVE questions.
(5x8=40marks)
11. Show that the space R' is complete.
12. State and prove cauchy's inequality.
13. Prove that the union of any collection of open sets is open and the intersection of any
collection of closed sets is closed.
nx
14. a) Show that fn (x) =
x  0 is not uniformly convergent
1 n2 x2
b) Let ( X , P ) and (Y ,  ) be metric spaces. Let the sequence fn : X  Y converge to f
uniformly on x. If C is a point at which each fn is continuous, then show that f is
continuous at C.
15. Let V, W be normed vector spaces. If the function f : V  W is linear, then show that
the following statements are equivalent.
i)
f is continuous on V
ii)
there is a point x  V at which f is continuous.
iii)
f ( x)
x
is bounded for x V  0
16. Examine for convergence of
u
n
if
(n  a) n
n na
i)
un 
ii)
un = x n (1  x) n , x  [0,1]
(a real )
17. Let ( X , P ) be a metric space and let f1, f2, …..fn be functions on X to R'. The function
f = (f1, f2, …..fn) : X  R n is given by f(x) = (f1(x) … fn (x). Prove that f is continuous
at x0 if and only if f1, f2,…..fn are continuous.
18. If f : X  R n ( X C R m ) is differentiable at  , then prove that the linear derivative of f
at  is unique.
SECTION-C
Answer any TWO questions.
(2x20=40marks)
19. a) Let ( X , P ) and (Y ,  ) be metric spaces. Prove 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 in X.
P ( x, y )
b) Show that if  is a metric on x then so is  given by  (x, y) =
and P and
1  P( x, y )
 are equivalent.
(12+8)
20. a) State and prove Banach's fixed point theorem.
b) State and prove Heine - Borel theorem.
21. a) State and prove d' alembert's ratio test
b) Discuss the convergence of
u
n
where u n 
(10+10)
(n!) (3n)! n
10
(4n)!
1 1 1 1 1
     .........
2 3 4 5 6
(8+8+4)
22. a) Show that a necessary and sufficient condition that f  R ( g i a, b) is that, given   0,
c) Discuss the convergence and absolute convergence of 1 
there is a dissection D of [a, b] such that S (D, f, g) - s (D, f, g) <  .
b) If fI, f2  R [g i a, b] then prove that f1 f2  R [g i a, b]
c) If f  R [ g i a, b] then show that f  R [ g i a, b] and

b
b
a
a
 f dg  
f dg .
(7+7+6)