M.Sc.(Statistics)(Semester (CBCS) Examination, 2017
REAL ANALYSIS
Day Date: Thursday, 20-04-2017 Max. Marks: 70
Time: 10:30 AM to 01.00 PM
N.B. Attempt five questions.
Q. No. and Q. No are compulsory.
Attempt any three from Q. No. to Q. No.
Figures to the right indicate full marks.
Q.1 Choose correct alternative: 05
The compliment of open set is
Always closed Always open
May or may not be closed None of these
2 The limit of the set*

is
a+1 a 0
3 The set of irrationals is
Countably finite Countably infinite
Uncountably infinite None of these
4 The sequence can have maximum of limit/s
0 1 2 Infinity
5 For the convergence of series Σ
is
Necessary condition Sufficient condition
Both necessary and sufficient
condition
None of these
Fill in the blanks. 05
Subset of uncountable set always
Sepermum of a set is also called as upper bound.
Countable union of countable set is always.
The set of limit points of the set is
A set is open iff its compliment is
State the following sentence are True or False: 04
A countable set is always closed.
A bounded infinite set may or may not have a limit point.
A series of positive terms is always convergent.
Every countinuous function is Riemann integrable.
Page 2 of 2
6+8
Q.2 Prove or disprove: Arbitrary intersection of closed sets is
closed.
Show that arbitrary intersection of open sets or may not be
open.
Write short notes on the following:
Cauchy criterion for convergence of a series.
Geometric series and its convergence.
Q.3 Explain limit superior and limit inferior of a sequence. 07
Define convergence of a sequence of real numbers. Show that
convergence limit is unique, whenever exists.
07
Q.4 Discuss the convergence of geometric series. 07
Discuss the convergence of the seriesΣ
07
Q.5 Discuss any two tests of convergence for series. 07
Explain the concept of 'Riemann integral. 07
Q.6 Explain Lagrange's method of undetermined multipliers to
maximize a function.
07
Find the stationary value of subject to the condition
07
Q.7 Find the radius of convergence of the following series:


07
State the following: 07
Bolzano-Weistrauss theorem
Heine Borel therroem
Taylor's theorem