Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2019
UNIVERSITY OF BAMENDA
FIRST SEMESTER EXAMINATION, 2018/2019 SESSION
FACULTY: SCIENCE INSTRUCTOR: BIME MARDONAL
DEPARTMENT: MATHEMATICS DATE: 14-03-2019
COURSE CODE: MATS2101 DURATION: 2 hrs (1 pm-3pm)
COURSE TITLE: Analysis
INSTRUCTIONS: Answer all questions. You are reminded of the necessity for good English
and orderly presentation of your answers.
1. (a) (7 Marks) Let What do you understand by the following?
(i) A limit point of (ii) An interior point of . (vi) Least upper bound (iv) Greatest
lower bound. (v) The derived set of . (vi) is an open set. (vii) is a closed set
(b) (3 + 2 + 3 Marks) Let
. State the following;
(i) The inf and sup of . (ii) The derived set of (iii) Is the set closed? Please justify
(c) (Statement: 2 + 2 Marks, Proof: 5 Marks) State the following and prove 1(c)i
i. The Archimedean principle
ii. The principle of Mathematical Induction
(d) (6 Marks) Prove the
Total=30 Marks
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2. (a) (2 + 5 Marks) Let
be a sequence of real numbers and .
i. What does it mean to say
converges to ?
ii. Using your definition in
i show that !"
#$
%&
.
(b) (7 Marks) Show that if
is increasing and bounded above then it converges to
sup
'
.
(c) (5 Marks) Suppose
( and !"
#$
then (.
(d) (3 Marks) If
) ( and !"
#$
. Is it always true that ) (? Justify.
(e) (7 + 5 + 3 + 5 Marks) Consider the sequence define by
*
+
i. Show that ( ,
-
ii. Show that
%
iii. Deduce that
converges.
iv. Calculate !"
#$
.
Total=40 Marks
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