Govt. M M college, Jessore
Dept. of Mathematics
3rd Year Test Examination-2008
Sub-Real Analysis Code-3732
Full Marks-100 Time-4 hours
[All questions are of equal value. Answer any six questions ]1(a)Define completeness axiom and Dedekind’s axiom in Â. Show that the two axioms are equivalent.
(b)Define Supermen and infamous of a set of real numbers. Let AÎÂ :A f and A is bounded. If B A : B f, then show that Inf A Inf B Sup B Sup A.2(a)What do yow mean by accumulation point of a set. State and prove the Heine Borel Theorem.
(b)Define closed set. Show that a set is closed iff its complement is open.3(a)Define a convergent sequence. Show that every convergent sequence is bounded. But the converge is not always true. Give example.
(b)Show that the sequence < > converge to the limit 1.4(a)Define monotonic sequence? Prove that a monotonic increasing sequence which is bounded above converges to its least upper bound.
(b)Let =1 and , n . Is < > bounded? Is it monotonic? Is it convergent? Give reasons for your answers. Find if < > is convergent.5(a)Define:- (i) Infinite series (ii) Divergent series . If a series is convergent, then . Is the converse true?
(b)Test the convergence of any two:-
(i) (ii)
(iii)6(a)Prove that if a function is continuous in a closed interval then it is bounded there in and attains its bounds
(b)Show that is uniformly continuous on [0,1] but is not uniformly continuous on that interval.7(a)State and prove Lagrange’s Mean Value Theorem. What is its geometric interpretation? Is there any relation between Cauchy’s Mean Value Theorem and Lagrange’s Mean Value Theorem?
(b)Verify Rolle’s theorem to the functions(i) in [1,3] (ii) in [1,3]8(a)Define (i) Partition (ii) norm of a Partition (iii) Lower and upper Riemann sum. Prove that a function continuous on [a, b] is R-integrable on [a, b].
(b)If the function is defined on [0, 1] by then show that f is Riemann integrable and find9(a)Define (i)sequence (ii) point wise convergence of sequence (iii) uniform convergence of a sequenceState and prove Weierstrass’s M-test theorem for uniform convergence of series of functions.
(b)Test for uniform convergence, the sequence < > defined byon [0, 1]10(a)Prove that for any bounded function f, the lower Riemann integral can not exceed the upper Riemann integral.(b)If f(x) be defined on (0, 2) as follows
then evaluate the upper and lower Riemann integrals in (0, 2).
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Real Analysis Test Exam
Real Analysis
একাদশ গনিত ১ম পত্র
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