Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2019
UNIVERSITY OF BAMENDA
Continuous Assessment
FACULTY: SCIENCE INSTRUCTOR: BIME MARDONAL
DEPARTMENT: MATHEMATICS DATE: 2019
COURSE CODE: MATS2101 DURATION:
hrs
COURSE TITLE: Analysis 1.
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...
NB: Not all questions of equal marks are of equal difficulties.
1. (a) (1 + 1 + 1 + 1 Marks) Let ⊂ ℝ. What do you understand by the following?
(i) A limit point of (ii) An interior point of (iii) Least upper bound (iv) Greatest
lower bound.
(b) (1 + 1 + 1 Marks) Give examples of the following element.
i. 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 element but no minimum element.
2. (a) (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
(b) (6 Marks) Prove the
√
+
√
+
√
+ ⋯ +
√
≥
√
, ∀ ∈ ℕ.
(c) (1 Mark) Let ⊂ ℝ. What does it mean to say is dense in ℝ.
(d) (5 Marks) Let ℚdenote the set of rational numbers. Show that ℚ is dense in ℝ.
3. (2 + 2 + 2 Marks ) Let =
1,
,
,
…
. State the following;
(i) The inf and sup of . (ii) The derived set of (iii) Is the set 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 is a limit point of ⊂ ℝ then most belong to .
(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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