Analysis 1 (MATS2101)
BSc, Physics - PHYS
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 (1pm-3pm)
COURSE TITLE: Analysis I.
INSTRUCTIONS:
Answer all questions. You are reminded of the necessity for good English and orderly
presentation of your answers.
1. (a) (7 Marks) ) Let S . What do you understand by the following?
(i) A limit point of S (ii) An interior point of S (iii) Least upper bound (iv) Greatest lower
bound (v) The derived set of S. (vi) S is an open set. (vii) S is a closed set
(b) (3 + 2 + 3 Marks) Let S = { 1 ,
,
} . State the following:
(i) The inf and sup of S. (ii) The derived set of S (iii) Is the set S closed? Please justify
(c) (Statement: 2 + 2 Marks, Proof: 5 Marks ) State the following and prove l(c)i
i. The Archimedean principle
ii. The principle of Mathematical Induction
(c) (6 Marks) Prove that
,
Total=30 Marks
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 2(a)i show that
(b) (7 Marks) Show that if
is increasing and bounded above then it converges to
sup{a
n
: n }.
(c) (5 Marks) Suppose a
n
0, n . Show that if
a
n
= a then 0.
(d) (3 Marks) If a
n
> 0 n and
a
n
= . Is it always true that a > 0? Justify.
(e) (7 + 5+ 3 + 5 Marks) Consider the sequence define by a
n
!
"
n !
#
i. Show that 0 < a
n
< 2, n
ii. Show that a
n+1
> a
n
, n .
iii. Deduce that
converges.
iv. Calculate
a
n
Total=40 Marks
Good Luck
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