Exam Details
| Subject | analysis | |
| Paper | ||
| Exam / Course | m.sc. mathematics | |
| Department | ||
| Organization | acharya nagarjuna university-distance education | |
| Position | ||
| Exam Date | May, 2017 | |
| City, State | new delhi, new delhi |
Question Paper
Total No. of Questions 10] [Total No. of Pages 02
M.Sc. (Previous) DEGREE EXAMINATION, MAY 2017
First Year
MATHEMATICS
Analysis
Time 3 Hours Maximum Marks: 70
Answer any five questions.
All questions carry equal marks.
Q1) Let n … be a sequence of countable sets and put S
Then show that S is countable.
Prove that closed subsets of compact sets are compact.
Q2) If a set E in Rk has one of the following three properties, then show that it has
the other two:
E is closed and bounded
ii) E is compact
iii) Every infinite subset of E has a limit point in E.
Suppose Y X . Prove that a subset E of Y is open relative to Y if and only
if E Y ∩G for some open subset G of X.
Q3) Suppose is monotonic. Then prove that converges if and only if it
is bounded.
Show that the product of two convergent series need not converge and may
actually leverage.
Q4) Suppose f is a continuous mapping of a compact metric space X into a metric
space Y. Then prove that f is compact.
Define f on R1 by
Show that f is continuous at every irrational point and it has a simple
discontinuity at rational points.
Q5) State and prove a necessary and sufficient condition for a bounded function f
to be R S integrable on b].
Suppose that f on m f is continuous on and
f on b]. Then show that on b].
Q6) Let f maps into Rk and suppose that f for some monotonically
increasing function α on b]. Then show that f ∈R(α and
b b
a a
fdα f dα .
State and prove the fundamental theorem of integral calculus.
Q7) Prove that every uniformly convergent sequence of bounded functions is
uniformly bounded.
If K is compact, n f for n … and if is pointwise
bounded and equicontinuous on K then prove that
fn is uniformly bounded on K.
ii) contains a uniformly convergent subsequence.
Q8) State and prove the Weirstrass approximation theorem.
Q9) State and prove the Lebesque's monotone convergence theorem.
If is a sequence of measurable functions then prove that the set of points
x at which n f x converges is measurable.
Q10) Suppose f f1 f2 where i f ∈α u on E(i 2). Then show that f ∈α
on E and 1 2
E E E
f f f .
If f on then show that f ∈L on and that
b b
a a
f dx R∫ f dx
M.Sc. (Previous) DEGREE EXAMINATION, MAY 2017
First Year
MATHEMATICS
Analysis
Time 3 Hours Maximum Marks: 70
Answer any five questions.
All questions carry equal marks.
Q1) Let n … be a sequence of countable sets and put S
Then show that S is countable.
Prove that closed subsets of compact sets are compact.
Q2) If a set E in Rk has one of the following three properties, then show that it has
the other two:
E is closed and bounded
ii) E is compact
iii) Every infinite subset of E has a limit point in E.
Suppose Y X . Prove that a subset E of Y is open relative to Y if and only
if E Y ∩G for some open subset G of X.
Q3) Suppose is monotonic. Then prove that converges if and only if it
is bounded.
Show that the product of two convergent series need not converge and may
actually leverage.
Q4) Suppose f is a continuous mapping of a compact metric space X into a metric
space Y. Then prove that f is compact.
Define f on R1 by
Show that f is continuous at every irrational point and it has a simple
discontinuity at rational points.
Q5) State and prove a necessary and sufficient condition for a bounded function f
to be R S integrable on b].
Suppose that f on m f is continuous on and
f on b]. Then show that on b].
Q6) Let f maps into Rk and suppose that f for some monotonically
increasing function α on b]. Then show that f ∈R(α and
b b
a a
fdα f dα .
State and prove the fundamental theorem of integral calculus.
Q7) Prove that every uniformly convergent sequence of bounded functions is
uniformly bounded.
If K is compact, n f for n … and if is pointwise
bounded and equicontinuous on K then prove that
fn is uniformly bounded on K.
ii) contains a uniformly convergent subsequence.
Q8) State and prove the Weirstrass approximation theorem.
Q9) State and prove the Lebesque's monotone convergence theorem.
If is a sequence of measurable functions then prove that the set of points
x at which n f x converges is measurable.
Q10) Suppose f f1 f2 where i f ∈α u on E(i 2). Then show that f ∈α
on E and 1 2
E E E
f f f .
If f on then show that f ∈L on and that
b b
a a
f dx R∫ f dx