LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
DEGREE EXAMINATION – MATHEMATICS
FIFTH SEMESTER – APRIL 2007
MT 5501 – REAL ANALYSIS
Date & Time: 28/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
Answer all the questions: 10 x 2 = 10
- Define an inductive set with an example.
- Prove that every positive integer n (except 1) is either a prime or a product of primes.
- State and prove Euler’s theorem for real numbers.
- Define a Metric space.
- State Cantor’s intersection theorem for closed sets.
- Define an interior point and an open set.
7.Give an example of a uniformly continuous function.
- Define a Cauchy sequence.
- Suppose f and g are defined on (a, b) and are both differentiable at c Î (a, b), then prove
that the function fg is also differentiable at c.
- Define total variation of a function f on .
Answer any five questions: 5 x 8=40
- Prove that the set R of all real numbers is uncountable.
- State and prove Bolzano-Weirstass theorem for R.
- Prove that every compact subset of a metric space is complete.
- Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. If X is compact, then prove that f (X) is a compact subset of Y.
- Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. Then show that a map f: X Y is continuous on X if and only if f -1 (G) is open in X for every open set G in Y.
16 Prove that in a metric space (X, d)
( i ) Arbitrary union of open sets in X is open in X
( ii) Arbitrary intersection of closed sets in X is closed in X.
- Let f: R and f have a local maximum or a local minimum at a point c.
Then prove that f ’(c) = 0.
- Let f be of bounded variation onand xÎ (a, b) Define V: R as follows:
V (a) = 0
V (x) =Vf , a < x ≤ b.
Then show that the functions V and V – f are both increasing functions on.
Answer any two questions: 2 x 20 = 40
19 State and prove Intermediate value theorem for continuous functions.
- State and prove Lagrange’s theorem for a function f : R
21.(a) Suppose c Î (a ,b) and two of the three integrals f da ,f da , and f da
exists. Then show that the third also exists andf da =f da +f da.
(b) When do we say f is Riemann-Stieltjes integrable?
- (a) State and prove Unique factorization theorem for real numbers.
(b) If F is a countable family of countable sets then show that is also countable.