Paper-III
Real Analysis & Group Theory
Real Numbers :The completeness properties of R-Exact bounds-Completeness axiomNeighbourhood-opensets-closed sets. ( No question is to be set from this portion)
Unit-I: Sequences and Infinite series
Sequences: Definition of Sequence-Method of defining sequences-Range-Boundedness-Convergent
sequences-Algebra of convergent sequences—divergent sequence-Monotonic sequences-Limit points
of a sequence-Bolzano-Weierstrass theorem-Cauchy sequences-Cauchy’s general principle of
convergence.
Infinite Series: Convergence and divergence of series
Test of series (Only Problems)-Geometric series-p-series-Comparison test-D’Alembert’s testCauchy’s nth root test-Alternating series--Leibnitz test(Excluding Limit form,Raabe’s Test)
Unit-II: Limits and Continuity
Limits: Definition of Left and Right limits-Limit of a function-Algebra of limits-Limits of
standard functions
Continuity: Continuity and discontinuity of a function – Algebra of Continuous functionsproperties of continuous functions-Uniform continuity
Theorems:-1) Borel’s theorem 2) Every continuous function is bounded
3) If f:[a,b] →R is continuous on [a,b] then f attains its exact bounds at least once in [a,b].
4)Bolzano’s theorem
Unit-III : Groups - Sub Groups – Cosets - Legrange’s Theorem-Normal sub groups
Unit-IV : –Homomorphism , Isomorphism of Groups- Permutation Groups-Cyclic Groups
PRESCRIBED TEXT BOOK:
A Text Book of B.Sc. MATHEMATICS Vol-II , V.Venkateswara Rao and Krishna Murthy…
S.Chand and Company Ltd.-2012 Revised editions. Unit-I (Page No. 326 to 375 , 378 to407 & 413)
Unit-II( Page No.427 to 445 & 450 to 484), Unit-III,IV (Page No. 37 to 175),
Reference Book
Introduction to Abstract algebra By John.B.Fraleigh(Narosa publications), Elements of Real Analysis by
Shantinarayana &Dr.M.D.Raisinghania(S.Chand&CO)
Model Question Papers
Sem-III - Paper-III
Max Marks : 75
Section – A
Answer any five questions
( 5 x 5 = 25 Marks )
1. Prove that the sequence {sn} ,where sn =
1
1
1
is convergent

 ...... 
n 1 n  2
nn

1
n
n 1 2  3
2. Test for convergence 
3. Prove that lim lim
𝑛→∞
n
3x  x
7x  5 x
does not exists
1
4. F:R--- >R be a function such that f(x)=
1
ex e x
1
x
e e
1
x
, if x≠0 and f(0)=1, discuss the continuity at x = 0
5. Show that the fourth roots of unity form an abelian group with respect to multiplication
6. G is an abelian group .If a, b € G such that O(a) = m. O(b) = n and (m,n) = 1 then show that
O(ab) = mn
7. If f = ( 1 2 3 4 5 8 7 6 ), g = ( 4 1 5 6 7 3 2 8) are cyclic permutations then show
that (fg)-1 = g-1f-1
8. Show that the group (G = { 1,2,3,4,5,6},X7) is cyclic group and find the generators.
Section – B
Answer any five questions
( 5 x 10 = 50 Marks )
9. Prove that a monotonic sequence is convergent iff it is bounded
1.3.5.7........(2n  1) n 1
x , ( x  0)
2.4.6..........2n
n 1

10. Test for convergence

11. If f: [a,b] →R is continous on [a,b] then prove that f is bounded on [a,b].
12. If f: [a,b] →R is continous on [a,b] then show that f attains its bounds.
13. The union of two subgroups of a group is also a subgroup iff one is contained in other.
14. State and prove the Legrange’s theorem for groups.
15. State and prove fundamental theorem of homomorphism of groups.
16. Show that the order of a cyclic group is eual to the order of its generator.