Gramin
Adarsha Multiple Campus
Tarkeshwar-11, Nepaltar Kathmandu
First Terminal Exam 2079
Level/Year:
B. Ed. Second Year Full
Marks:50
Subject:
Real Analysis (Math. Ed.423) Pass
Marks: 20
Time:
1:30 hrs
Candidates are required to write
the answers in their own words as far as practicable. The figures in the margin
indicate full marks.
Attempt
all questions.
Group A 10 ´ 1 = 10
Tick ( √ ) the best answer.
1. Which
of the following set is bounded?
a) {3, 7, 12, 20} b) The set
of whole numbers.
c)
{x: x > 1, xÎ R} d) The set of real numbers.
2. A
real number a is non-negative if
a)
a Î
R b) a Î {0} È
R- c) a Î {0} È
R+ d)
a Ï
R+ È
R-
3. Which
of the following statement is false?
a)
every finite set is closed b) an empty set is an open
c)
interior of a set S is an open subset of S.
d) union of open sets is not
open.
4. The
sequence < 1 + (-
1)n ) has
a)
no limit points b) 0 as a limit point
c)
2 as a limit point d) 0 and 2 a limit points
5. The
sequence < un> is said to be convergent to a number l if for any e > 0 $ m Î N such that
a)
|un - l|
< e "
n ³
m b) |un -
l| £ e "
n ³
m.
c)
|un - l|
> e "
n ³
m d) |un
- l| ³ e
"
n ³
m.
6. Which of the following is an oscillatory sequence?
a) < +1> b) < n > c) <(-1/n)n> d) All of above
7. The value of lim n 1/n is
a) 1 b)
0 c) ¥ d) e
8. Which of the
following is not true sentence?
a) Every sequence has a
monotonies
b) Every sequence has a
bounded subsequence
c) Every bounded sequence
has a convergent setsquare
d) Every subsequence of a
divergent sequence divergent to the same
limit
9. Which of the following is true?
a)
Every bounded sequence has at least one limit point
b)
The set of limit point of a bounded sequence is bonded
c)
The limit point e of every sequence is a closed set
d)
All of above
10.
Which of the following is an open set?
a) The set R of real
numbers b) the set Q
of rationales numbers
c) The closed interval numbers d) {1/n:nÎN}
Group B 4 ´
7 = 28
1. State
and prove Archimedean Property.
OR Prove that the supremum and infimum of a set
if exist are unique
2. Prove
that the union of any finite number of closed set is closed. By taking suitable
example prove that the union of infinite number of closed sets is not closed.
3. Define
limit point of a sequence. Prove that a sequence cannot converge to more than
one limit point.
4. Define monotonic sequence with example.
Prove that a monotonic sequence is convergent if and only if it is bounded.
Group C 1
´
12 = 12
5. (a) Define convergent sequence. Prove that a
sequence cannot to more than one limit point.
(b) Prove that: a sequence u is convergent if and only if to each given e > 0 $ m Î N such that
| un + p – un½< e
"
n ³
m, Ù p
³
0.
Best
of Luck
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