Analysis 1 (MATS2101)
BSc, Physics - PHYS
Semester: Resit
Level: 200
Year: 2018
UNIVERSITY OF BAMENDA
2017/2018 Resit SESSION
FACULTY: SCIENCE INSTRUCTOR: BIME MARDONAL
DEPARTMENT: Mathematics DATE:
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
1.
(a) (4 Marks) Let S c R and define -S = {x € R : —x € S}. Carefully explain what you understand by the
following:
(i) A limit point of S (ii) An interior point of S
(iii)The Infimum of S (iv) The Supremum of S
(b)
(6 Marks) If S ≠ ∅ and bounded below, show that the Infimum of S, inf S exists and inf S = -sup(-S).
(c)
(2 Marks) State without proof the Archimedean principle.
(d)
(6 Marks) Show that for any interval [a, b] in R, there exists a rational number r E (a, b).
(e)
(2 Marks) Give an example of a non-empty set with no limit point.
Total: 20 marks.
2.
Let (a
n
)
nEN
be a sequence of real numbers and a ∈ ℝ.
(a)
(2 Marks) Carefully explain what it means to say that (a
n
)
n£
N converges to a?
(b)
(3 x 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 lim
→∞
+
√
= 0
Total= 22 Marks.
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