LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – APRIL 2012
MT 5505/MT 5501 – REAL ANALYSIS
Date : 25042012 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
PART – A
Answer ALL questions: (10 x 2 = 20)
 State and prove the triangular inequality.
 Prove that the sets Z and N are similar.
 Prove that the union of an arbitrary collection of closed sets is not necessarily closed.
 Prove that every neighbourhood of an accumulation point of a subset E of a metric space contains infinitely many points of the set E.
 Show that every convergent sequence is a Cauchy sequence.
 Define the term “complete metric space” with an example.
 State Rolle’s theorem.
 Prove that every function defined and monotonic on is of bounded variation on .
 State the linearity property of RiemannStieltjes integral.
 State the conditions under which RiemannStieltjes integral reduces to Riemann integral.
PART – B
Answer ANY FIVE questions: (5 x 8 = 40 marks)
 State and prove CauchySchwartz inequality.
 Prove that the interval is uncountable.
 State and prove the HeineBorel theorem.
 State and prove the intermediate value theorem for continuous functions.
 Let and be metric spaces and . If is compact and is continuous on , prove that is uniformly continuous on .
 State and prove the intermediate value theorem for derivatives.
 Suppose on . Prove that on and that
.
 a) Let be a real sequence. Prove that (a) converges to L if and only if
(b) diverges to if and only if .
PART – C
Answer ANY TWO questions: (2 x 20 = 40)
19.  (a) Prove that every subset of a countable set is countable. 
(b) Prove that countable union of countable sets is countable.  
(c) State and prove Minkowski’s inequality. (8+7+5)  
20.  (a) Prove that the only sets in R that are both open and closed are the empty set and the set R itself. 
(b) Let E be a subset of a metric space . Show that the closure of E is the smallest closed set containing E.  
(c) Prove that a closed subset of a compact metric space is compact. (4+8+8)  
21.  (a) Let and be metric spaces and . Prove that is continuous on X if and only if is open in X for every open set G in Y. 
(b) Explain the classification of discontinuities of realvalued functions with examples.
(12+8) 

22.  (a) State and prove Lagrange’s mean value theorem. 
(b) Suppose on and for every that is monotonic on . Prove that must be constant on .  
(c) Prove that a bounded monotonic sequence of real numbers is convergent. (8+4+8) 
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