Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2018
University of Bamenda
Faculty of Science
Department of Mathematics and Computer Science
First Semester Final Examination
February 2018.
Course Title: Analysis I.
Course Code : MATHS2101.
Duration: 3 hrs.
Course Instructor: Birne Markdonal G
Instructions: Answer all questions. You are reminded of the necessity for
good English and orderly presentation of your answers.
1. (a) (4 Marks) Let S ϲ ℝ. Define the following with respect to S
(i) Limit point: (ii) An interior point.
(iii) Least upper bound (iv) Greatest lower bound
(b) (3 Marks) Is it true that if a is a limit point of S then a S?
Justify.
(c) (1 Mark) When is a set said to be bounded?
1. (4 Marks) Is the set S = {x
ϵ
ℝ : x
2
< 2r} bounded above
(below)? If so, find its least upper bound (greatest lower bound).
ii. (2 Marks) Does S having
a
maximum element? Justify.
(d) (1 Mark) Give an example of a set with no limit points.
Total =15 Marks
2. Let
ℕ
be a sequence of real numbers and a
ϵ
ℝ.
(a) (2 Marks) What does it mean to say
ℕ
converges to a?,
(b) (4 +4 + 5 Marks) Using your definition in 2a. Justify the following:
i) lim
→
= 2
ii) lim
→
= 0
iii. lim
→
+
√
(c) (5 Marks) Show that if lim
→
= and lim
→
=
then, =
Total=20 Harks
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