Exam2024Proposed solution included

MAT 302 ANALYSIS II Exam 2024

MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, Instructor Dr. Ndambomve P., June/July 2024. Exercise 1: definitions of ten core metric-space notions. Exercise 2: proofs that a closed ball is closed, that a closed subset of a complete space is complete, that 'not closed' does not imply 'open' (counterexample [0,1)), that continuity does not imply uniform continuity (counterexample 1/x on (0,1)), and that a closed subset of a compact set is compact. Exercise 3: statement of the Contraction Mapping Principle; analysis of f(x)=(1/8)(x+8) on [0,1] showing completeness of [0,1], that f is a contraction with constant 1/8, that f has no fixed point in [0,1], and why this does not contradict the theorem (f does not map [0,1] into itself); plus a proof that continuous images of connected sets are connected. Exercise 4: statements of the Double Limit Theorem and uniform Alternating Series Test; pointwise/uniform convergence analysis of f_n(x)=n^2/(2n^2+x^2) on R versus on a bounded interval [-a,a], with a limit-interchange integral computation; and a uniform-convergence proof for an alternating series.

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University of Buea

Faculty of Science - FS

Mathematics - MAT

MAT3022024

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.
2.
Show that every closed subset of a complete metric space is complete. [3 marks]

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