Real Analysis Code – 3732 Test Examination

Govt. M M College, Jessore

Department of Mathematics

3rd Year B. Sc. Honours Test Examination-2007

Subject – Real Analysis         Code – 3732

     Time – 4 hours             Full Marks – 100

 

1(a) Define supremum and infimum of a set of real numbers, state and prove the supremum and infimum principles for R and show that they are equivalent.

(b) Prove that if and, then there is a rational number such that

. If A= then find sup A, inf A, max A and min A.

2 (a) Define open set and closed set of real number. Show that the union of two open

sets is open.

(b) Define limit point and accumulation point of a set of real number. State and prove the Bolzano–weierstrass theorem. If A ={(-1)n} then show that A have two accumulation points but has no limit point .

 

3 (a) Define countable set and uncountable set. Show that the interval [0, 1] is not countable.

(b) Define interior, exterior and boundary points of a set of real numbers. Prove that a subset is closed if it contains all of its limit points.

 

4.(a) Define sequence and convergent sequence. Show that every convergent sequence is bounded. But the converse is not always true.

(b) Define limit of a sequence, show that every convergent sequence has a unique limit.

5.(a) Define Cauchy sequence, state and prove Cauchy’s general principle of convergence for a sequence.

(b) Prove that the sequence < xn > is define by , converges to the positive roots of the equation converges the positive not of the equation

 

6.(a) Define convergent series and absolute convergent series. State and prove the Cauchy’s root test for convergence of infinite series.

(b) Test the convergence of (any two)

 

(i) ,     (ii) ,   (iii) , .

7 (a) Define uniformly continuous function. Show that is uniformly continuous on [0,1] but is not uniformly continuous on [0,1].

(b) State and prove the intermediate value theorem for continuous function.

 

8.(a) Define Riemann –integral. Write the necessary and sufficient condition for a function defined in the interval [a , b] is Riemann integrable and prove its sufficiency. .

(b) Prove that every monotonic function is Remann integrable. Show that in the interval [0,1] the function .

is Riemann integrable .

 

9.(a) Define point-wise convergent sequence and uniform convergence sequence of functions. State and prove weierstrass M-test for uniform convergence of a series of a function.

(b) If , where and then show that convergence point-wise but not uniformly convergence on [0, 1].

 

10.(a) Define metric and norm on R. Let a function be defined on a normed vector space .show that d is a metric on V.

(b) Define compact and complete metric space with examples. Show that Rn is a metric space defined by.




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