MAT 302 ANALYSIS II Exam 2024

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MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, Instructor Dr. Ndambomve P., June 2024. Exercise 1: definitions of ten core metric-space notions (including compact set, in place of connected set as in the sibling June-2024/2025 papers). Exercise 2: proofs that a closed ball is closed, a closed subset of a complete space is complete, 'not closed' does not imply 'open', continuity does not imply uniform continuity, and a closed subset of a compact set is compact. Exercise 3: statement of the Contraction Mapping Principle; a proof that if f^n is a contraction then f has a unique fixed point; a proof that every contraction is uniformly continuous; and analysis of f(x)=x+1/x on [1,infinity), showing completeness, the non-strict Lipschitz bound |f(x)-f(y)|<=|x-y|, the absence of a fixed point, and why this does not contradict the Contraction Mapping Principle (k=1, not strictly less than 1). Exercise 4: statements of the Double Limit Theorem and uniform Alternating Series Test; pointwise/uniform convergence analysis of f_n(x)=n/(n+e^x) on R versus on a left-unbounded interval ]-infinity,a]; and a uniform-convergence proof for an alternating series.