MAT 302 Analysis II Exam 2024

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MAT 302: End of Semester Examination, University of Buea, Faculty of Science, Department of Mathematics, Course MAT302 Analysis II, 03-07-2024. Exercise 1 (10 marks): definitions of open/closed sets, fixed point, convergent/Cauchy sequence, complete metric space, continuity, uniform continuity, pointwise/uniform convergence, and connected set. Exercise 2 (20 marks): closed balls are closed, closed subsets of complete spaces are complete, whether not-closed implies open, whether continuous implies uniformly continuous, and closed subsets of compact sets are compact. Exercise 3 (17 marks): statement of the Contraction Mapping Principle and analysis of f(x)=(x+8)/9 on [0,1] -- completeness of the domain, the contraction property, and a trick question asking to show f has no fixed point (false: f(1)=1, confirming rather than contradicting the principle) -- plus continuity preserving connectedness. Exercise 4 (23 marks): statements of the Double Limit Theorem and Alternating Series Test, and analysis of pointwise/uniform convergence for f_n(x)=n^2/(2n^2+x^2) on R and on bounded intervals, including a limit-integral interchange, and uniform convergence of an alternating series involving f_n(x)=1/(n^2+x) on [0,infinity).