M.Sc. (Semester (CBCS) Examination Oct/Nov-2017
Statistics
REAL ANALYSIS
Day Date: Thursday, 16-11-2017 Max. Marks: 70
Time: 10.30 AM to 01.00 PM
Instructions: Q.1 and Q.2 are compulsory.
Attempt any three questions from Q. 3 to 7.
Figures to the right indicate full marks.
Q.1 Select the correct alternative: 05
The set of interior points of the set is:


Superset of countable set is always
Countable Uncountable
Partially countable None of these
The set of integers in is
Countable Uncountable
Both and None of these
Every infinite and set has a limit point.
Unbounded Bounded
Bounded above Bounded below
A sequence of constant term always
Converges Diverges
Oscillates None of these
Q.1 Fill in the blanks: 05
A set is compact if and only if it is closed and
The limit of sequence
1
∈ is
A set is open if every point of the set is it's point.
Greatest lower bound is also called as
Union of finite number is closed sets is
Q.1 State true and false: 04
Every monotonic non-decreasing sequence converges.
The set is open.
A convergent sequence can have more than one limit.
Riemann-Stieltje's integral is a particular case of Riemann integral.
Q.2 Define and illustrate following: 06
Closed set
Dense set
Write short notes on the following: 08
Taylor's theorem
Mean Value Theorem
Q.3 Define countable and uncountable set. Examine the countability of 07
Define open set. Is countable union of open sets always open? Justify. 07
Q.4 Discuss the convergence of series 1
. 07
Examine the convergence of following series.
Q.5 Discuss the Riemann integration. 07
Find Riemann Stieltje's integral: 3 2
Q.6 Explain Lagrange's method of undetermined multipliers. Hence, minimize
subject to the constraint 8.
07
Prove integration by parts rule using Riemann -Stieltje's integral. 07
Q.7 Discuss Power series and its convergence. 07
Prove: Countable union of countable sets is countable. 07