Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: Resit
Level: 200
Year: 2019
UNIVERSITY OF BAMENDA
2018/2019 Resit SESSION
FACULTY: SCIENCE INSTRUCTOR: BIME MARDONAL
DEPARTMENT: MATHEMATICS DATE:
COURSE CODE: MATS2101 DURATION: 2 hrs
COURSE TITLE: Analysis
INSTRUCTIONS: Answer all questions. You are reminded of the necessity for good English
and orderly presentation of your answers.
1. (a) (4 Marks) Let and define
. Carefully explain what you
understand by the following:
(i) A limit point of (ii) An interior point of
(iii) The Infimum of (iv) The Supremum of
(b) (6 Marks) If and bounded below, show that the Infimum of , inf exist an
inf sup.
(c) (2 Marks) State without proof the Archimedean principle.
(d) (6 Marks) Show that for any internal [a,b] in , there exist a rational number .
(e) (2 Marks) Give an example of a non-empty set with no limit point.
Total=20 Marks
___________________________________________________________________________
2. Let
be a sequence of real numbers and .
(a) (2 Marks) Carefully explain what it means to say that
converges to ?
(b) (3 5 = 15 Marks) Prove the following:
i. A convergent sequence can have at most one limit.
ii. Any convergent sequence is bounded.
iii. If a sequence of real numbers is bounded above and increasing then it converges.
(c) (5 Marks) Using your definition in 2a, show that
!
"
#
$
%
.
Total=22 Marks
___________________________________________________________________________
3. (a) (2 Marks) Carefully state without proof the Sandwich theorem for limit of sequences.
(b) (3 + 4 Marks) Show that:
(i)
& #
$
'
(
)
(ii)
*
(
)
(c) (1 Mark) Define a Cauchy sequence.
(d) (5 Marks) Show that every convergent sequence of real numbers is Cauchy.
(e) (8 + 2 Marks) Let
be a sequence of real numbers defined by
$
+
and
,+
$
+
,$
#
* -./
i. Prove that
is Cauchy.
ii. Deduce that
converges
(f) (3 Marks) Find the domain of the function derivative determine by the formula
0
%!12
3
2
3
1!
Total=28 Marks
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