Total No. of Questions 10] [Total No. of Pages 03
M.Sc. (Previous) DEGREE EXAMINATION, MAY 2018
First Year
MATHEMATICS
Analysis
Time 3 Hours Maximum Marks :70
Answer any five of the following questions.
All questions carry equal marks.
Q1) Let A be the set of all sequences whose elements are the digits 0 and 1.
Prove that this set A is uncountable.
Prove that every k-cell is compact.
Q2) Prove that if a set E in has one of the following three properties, then it
has the other two:
i. E is closed and bounded.
ii. E is compact.
iii. Every infinite subset of E has a limit point in E.
Prove that a subset E of the real line is connected if and only if it has the
following property:
If x∈E, y∈E, and x z then z ∈E .
Q3) Prove that lim
n→∞
1
1
n
n


e and that e is irrational.
Let n Σa be a series of real numbers which converges, but not absolutely.
Suppose
−∞ β ∞ .
Then prove that there exists a rearrangement n Σa′ with partial sums n s′
such that liminf n
n
s α
→∞
′ limsup n
n
s β
→∞
′ .
Q4) Let f be a continuous mapping of a compact metric space X into a metric
space Y. Then prove that f is uniformly continuous on X.
(DM02)
Let E be a nonempty subset of a metric space define the distance from x
in X to E by
E
E
P inf
z
x d x z
∈

Prove that PE 0 if and only if x∈E .
ii) Prove that E P is a uniformly continuous function on by showing
that E E P − P for all x∈X, y∈X.
Q5) Define Riemann Stieltjes integral. Prove that if f is bounded f
has only finitely many points of discontinuity on and α is continuous
at every point at which f is discontinuous then f ∈R(α .
Suppose f f is continuous on and 0
b
a
f x dx . Prove that
f 0 for all .
Q6) Suppose 0 n c for n …, n Σc converges, n s is a sequence of distinct
points in and
1
n n
n
α x c x s
∞

− where I is the unit step function. Let
f be continuous on then prove that
1

b
n n
a
n
f dα c f s
∞

.
Assume that f 0 and that f decreases monotonically on . Prove
that
1
f
∞
converges if and only
1

n
f n
∞
Σ
converges.
Q7) If n f is a sequence of continuous functions on a subset E of a metric space
and if n f → f uniformly on E then prove that f is continuous on E.
Suppose n f is a sequence of functions, differentiable on and such that
0 n f x converges for some point x0 on . If n f ′ converges uniformly
on then prove that n f converges uniformly on to a function f
and lim n
n
f x f x a x b
→∞
′ ′ .
Q8) Prove that if n f is a pointwise bounded sequence of complex functions on
a countable set E then n f has a subsequence nk f such that
nk f x
converges for every x in E.
State and prove Weierstrass approximation theorem.
(DM02)
Q9) Let f and g be measurable real-valued functions defined on the measurable
space let F be a real and continuous on and put
f x∈X. Then prove that h is measurable.
State and prove Lebesgue's monotone convergence theorem.
Q10) State and prove Fatou's theorem.
Prove that 2 L is a complete metric space.