Exam2021Proposed solution included

MAT 302 ANALYSIS II Exam 2021

MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, Instructor Dr. Ndambomve P., 2 July 2021. Exercise 1: definitions of Lipschitz function, fixed point, convergent/Cauchy sequence, completeness, continuity, uniform continuity, pointwise/uniform convergence and compactness; statements (without proof) of the Contraction Mapping Principle, Weierstrass M-Test and Uniform Cauchy Criterion. Exercise 2: proving distance-to-a-set properties (x in closure of K iff distance zero; the distance function is 1-Lipschitz hence uniformly continuous); continuous images of compact sets are compact; closed subsets of compact sets are compact. Exercise 3: showing the uniform limit of continuous functions is integrable and that the antiderivative sequence converges uniformly; analysing pointwise and uniform convergence of fn(x)=(2n+sin(nx))/(3n+sin^2(nx)) and computing a limiting integral; proving uniform convergence of a series via the Weierstrass M-Test.

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University of Buea

Faculty of Science - FS

Mathematics - MAT

MAT3022021

ANALYSIS II

Exam · Second Semester · Level 300

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1.
Carefully define the following notions in the framework of metric spaces: a) Lipschitz function, b) fixed point of a map, c) convergent sequence, d) Cauchy sequence, e) complete metric space, f) continuous functions, g) uniformly continuous function, h) pointwise convergence of sequences of functions, i) uniform convergence of sequences of functions, j) compact set. (1×8 marks)
3.
Show that a closed subset of a compact set is compact. (4 marks)