Exam2025Proposed solution included

MAT 302 ANALYSIS II Exam 2025

MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, Instructor Dr. Ndambomve P., June 2025. Exercise 1: definitions of ten core metric-space notions (open/closed sets, fixed point, convergent/Cauchy sequences, completeness, continuity, uniform continuity, pointwise/uniform convergence, connectedness). Exercise 2: proofs about the distance function delta(p,A) to a set, closure characterisation, and monotonicity/closure-invariance properties of delta. Exercise 3: the sup-metric on a product of two metric spaces, limits/accumulation points of product sequences, completeness of the product space, and continuity preserving compactness (f(K) compact). Exercise 4: statements of the Double Limit Theorem and the uniform Alternating Series Test; detailed pointwise/uniform convergence analysis of f_n(x)=n^2/(2n^2+e^x) including a limit-interchange integral computation; and a uniform-convergence proof for an alternating series via the uniform Alternating Series Test.

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University of Buea

Faculty of Science - FS

Mathematics - MAT

MAT3022025

ANALYSIS II

Exam · Second Semester · Level 300

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Let (X, d) be a metric space. Carefully define the following notions in the framework of metric spaces: a) Open and closed sets, 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) Connected set.
1.
Carefully state, without proof, the Double Limit Theorem and the Alternating Series Test Theorem, for uniform convergence of sequences of functions and series of functions. (6 marks)

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