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Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE (30 MARKS)
a) Define the following terms as used in metric space (x,d). i. Open set. (2 Marks) ii. Closed set (2 Marks) b) Let S C R be defined by ?? = ?? ? ??;2 = ?? = 7 determine the infimum, supremum, minimal and maximal elements of S if they exist. (4 Marks) c) Show that the set Z of integers is countable. (3 Marks) d) Let (x,d) be a metric space and define:??0 ?? × ?? ? R by ??0 = ??(??,??) 2+??(??,??) . Show that ??0is a metric on X. (5 Marks) e) Determine the convergence or divergence of the series (-1)??(??-1) 2??(??+3)! 8 ??=1 . (3 Marks) f) Show that an arbitrary intersection of open sets need not be open using the family. ???? = 2 + 1 ?? ,4 - 1 ?? :?? ? N . (3 Marks) g) Use the root test to investigate the convergence of the series 3????-?? 8 ??=1 . (4 Marks) h) Prove that the intersection of a finite collection of open sets us open. (4Marks)
QUESTION TWO (20 MARKS)
a) Let S be a subset of the set of real numbers R. Define a limit point of S. (1 Mark) b) Prove that P is a limit point S C R if and only if every neighbourhood of P contains infinitely many points of S. (4 Marks) c) Determine the interior of the sets; i. ?? = 3,7 (2 Marks) ii. ?? = ?? 3 : ?? ? N (3 Marks) d) Prove that every finite subset of R is compact. (5 Marks)
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e) Show that if ?? = (?? + 1) - (?? - 1), for any integer ?? > 1, then r is irrational. (5 Marks)
QUESTION THREE (20 MARKS)
a) Differentiate between continuity an uniform continuity of a function ??:R ? R. (4 Marks) b) Prove that the function ??:R ? R defined by ?? ?? = 4?? - 7 is continuous on any interval [??,??],?? > ??. (5 Marks) c) Show that the sequence 1 2?? is a Cauchy sequence. (5 Marks) d) Show that the sequence defined by ???? = 1,???? ?? ???? ???????? 0,???? ?? ???? ?????? diverges. (3 Marks) e) Show that the function ?? ?? = 1 ?? is not uniformly continuous on 0,8 . (3 Marks)
QUESTION FOUR (20 MARKS)
a) Use the integral test to test for the convergence of the series 2?? ??2+1 8 ??=1 . (3 Marks) b) Define conditional convergence of a series, then determine whether (1)?? ??2+1 converges conditionally or not. (4 Marks) c) Prove that every convergent sequence in a Cauchy sequence. (4 Marks) d) Use the sequence (????) = 1 ?? in x where ?? = 0,1 ??R to show that not every Cauchy sequence is convergent. (3 Marks) e) Prove that the union of countable set is countable. (6 Marks)
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