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Math 221:Real Analysis 1 March 2009 Question Paper

Math 221:Real Analysis 1 March 2009

Course:Bachelor Of Science In Economics And Mathematics

Institution: Kabarak University question papers

Exam Year:2009

KABARAK UNIVERSITY
EXAMINATIONS
2008/2009 ACADEMIC YEAR
FOR THE DEGREE OF BACHELOR OF SCIENCE IN
ECONOMICS AND MATHEMATICS
COURSE CODE: MATH 221

INSTRUCTIONS:
Attempt QUESTION ONE and ANY OTHER TWO questions.
Question One (30mks)
(a) Prove that between any two distinct rationals there is an irrational number.
(6mks)
(b) Suppose that ?, > , < +
then show that ? (5mks)
(c) By use of all the possible axioms and properties of and ordered field show that
< ff
<
given > 0, > 0 (7mks)
(d) Show that if a sequence has a limit then this limit is unique. (6mks)
(e) State prove the Archimedean property. (6mks)
Question Two (20mks)
(a) Show that the interval ( , ) contains both rational and irrational numbers. (7mks)
(b) State and prove the Sandwich theorem. (8mks)
(c) State a consequence of Sandwich theorem and prove it. (5mks)

Question Three (20mks)
(a) Define a Cauchy sequence then
show that any convergent sequence is a Cauchy sequence. (6mks)
(b) Prove that any Cauchy sequence is bounded and hence show that Cauchy sequence
is convergent. (14mks)
Question Four (20mks)
(a) Define a metric space hen show that ( , ) = s |
?

|

is a metric space.
(7mks)
(b) Prove that
(i) If E ? F then
?

(3mks)
(ii)

is always an open set (2mks)
(iii)

is the largest open subset of X contained in E. (3mks)
(iv) E is open iff E =

(5mks)
Question Five (20mks)
(a) State and prove the intermediate value theorem (6mks)
(b) Let {

} and {

} be two sequences converging to L and h respectively as ???
show that

?
? Lh (9mks)
(c) Show that the sequence {(?1)

} is bounded by 1 but is divergent. (5mks)

Math 221:Real Analysis 1 March 2009 Question Paper

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