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Math 221:Real Analysis 1 April 2011 Question Paper

Math 221:Real Analysis 1 April 2011

Course:Bachelor Of Education Science

Institution: Kabarak University question papers

Exam Year:2011

KABARAK UNIVERSITY
UNIVERSITY EXAMINATIONS
2010/2011 ACADEMIC YEAR
FOR THE DEGREE OF BACHELOR OF EDUCATION
SCIENCE
COURSE CODE: MATH 221

INSTRUCTIONS:
Attempt question ONE and any other TWO questions.

QUESTION ONE (30 MARKS)
(a) What do you understand by the following terminologies
(i) Continum property (2 marks)
(ii) Suprenum (2 marks)
(iii) Infimum (2 marks)
(iv) Extended real number system. (2 marks)
(b) Show that ( , ) contains both rational and irrational numbers. (5 marks)
(c) Show that if and are positive real numbers then <
<
. (6 marks)
(d) State and prove the Sandwich theorem for sequences. (6 marks)
(e) Show that a set of Â is bounded ? a real number such that | | ? . (5 marks)

QUESTION TWO (20 MARKS)
(a) Prove that the limit of a sequence is unique. (7 marks)
(b) Show that an interval ( , ) contains both rational and irrational numbers. (7 marks)
(c) Prove that ?8 is irrational. (6 marks)

QUESTION THREE (20 MARKS)
(a) Prove that if a sequence {

} is convergent then its image is bounded. ( 7 marks)
(b) Show that a sequence {??1)
} is bounded by 1 but is divergent. ( 7 marks)
(c) Show that lim ?
?
??? (6 marks)

QUESTION FOUR (20 MARKS)
(a) State and prove Cauchy’s criterion for limits of functions. (10 marks)
(b) Prove the mean value theorem of functions. (10 marks)

Math 221:Real Analysis 1 April 2011 Question Paper

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