Real Analysis

Govt. M M college, Jessore

Dept. of Mathematics

3rd Year Test Examination-2011

Sub-Real Analysis    Code-3732

Full Marks-100, Time-4 hours

[All questions are of equal value. Answer any six questions ]

1.(a) State and prove Archimedean property of real numbers.

(b) Prove that between any two distinct real numbers, there is always  an irrational numbers and therefore infinitely many irrational numbers .

2.(a) State and prove the Heine Borel Theorem.

(b) Define:- (i) open set.  (ii) closed set  (iii) limit point of a set . (iv)  Prove that a set is closed if and only if its complement is open.

3. (a) Define monotonic sequence. Prove that a monotonic increasing sequence which is bounded above converges to its least upper bound.

(b) State and prove Cauchy’s general Principle of Convergence of a sequence.

4. (a) Define:- (i) Infinite series (ii) alternating series

(iii) convergent series. If a series  is convergent, then .

(b) Test the convergence of any two:-  (i)

(ii) (iii)

5. (a) State and prove the first mean value Theorem of differential calculus.

(b) Verify Rolle’s theorem to the functions (i)  in [1,3]

(ii)  in [1,3]

6. (a) Define (i) point wise convergence of sequence (ii) uniform convergence of a sequence

State and prove Weierstrass’s M-test theorem for uniform convergence of series of functions.

(b) Test for uniform convergence, the sequence < > defined by  on [0, 1]

7. (a) Prove that if a function f(x) is Continuous on [a, b] then prove that the function attains its supermum and infimum at least once in

[a, b].

(b) State and prove the intermediate value Theorem for Continuous function.

8. (a) Write out necessary and sufficient Condition for the Riemann integrability of a function defined on [a, b]  and prove its sufficiency.

(b) If  is Riemann integral on [a, b], then prove that

(i)  is also Riemann integrable (ii)

9. (a) Define point wise convergence and uniform convergence of  sequence of functions.

prove Mn -test for uniform convergence of sequence.

(b) If  , where  and  .

Show that Converges point wise but not uniformly on [0,1].

10. (a) Define:-  (i) Euclidean n-space , (ii) Melrice space, Prove that Euclidean n-space  is Complete.

(b) Let d:  be a function defined by where

show that is metric space for .

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