Analysis 1 (MATS2101)
BSc, Physics - PHYS
Semester: First Semester
Level: 200
Year: 2018
UNIVERSITY OF BAMENDA
Continuous Assessment
FACULTY: SCIENCE INSTRUCTOR: BIME MARDONAL
DEPARTMENT: MATHEMATICS DATE: 2019 C
OURSE CODE: MATTS2101 DURATION:
hrs
COURSE TITLE: Analysis I.
INSTRUCTIONS: Answer all questions. You are reminded of the necessity for good English and orderly
presentation of your answers. Do not unnecessarily smile at the person sitting next to you, they may also
not know the answer, moreover exam hall is not the right place for networking.
1. ( a ) ( l + l + l + l Marks) Let S ⊂ ℝ. 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.
(b) (1 + 1 + 1 Marks) Give examples of the following sets:
1 A non-empty set with no limit points.
ii. A non-empty set with an upper bound but no maximum element.
iii. A non-empty, bounded set with a maximum clement but no minimum element.
2. (a) (Statement: 2 + 2 Marks, Proof: 5 Marks ) State the following and prove 2(a)i
i. The Archimedean principle
ii. The principle of Mathematical Induction
(b) (6 Marks) Prove that
√
+
√
+
√
+ ⋯ +
√
≥
√
, ∀ ∈ ℕ
(c) (1 Mark) Let 5 c l . what does it mean to say S is dense in ℝ
(d) (5 Marks) Let Q denote the set of rational numbers. Show that ℚ is dense in ℝ
3. (2 + 2 + 2 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
4. (3 + 3 + 3 + 3 Marks) Determine and carefully justify the truth value of each of the following
statements (ie with a counter example if the statement is false or a logically correct proof if the
statement is true)
(a) If l is a limit point of S ⊂ ℝ . then l most belong to S.
(b) The set ℚ of rational numbers satisfy the completeness axiom.
(c) The union of two intervals is also an interval
(d) The intersection of two intervals is also an interval
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