MAT 302 Analysis II Exam 2023

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MAT 302: End of Semester Examination, University of Buea, Faculty of Science, Department of Mathematics, Course MAT302 Analysis II, 14-06-2023. Exercise 1 (10 marks): definitions of open/closed sets, fixed point, convergent/Cauchy sequence, complete metric space, continuity, uniform continuity, pointwise/uniform convergence, and compact 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 (19 marks): statement of the Contraction Mapping Principle, showing a map whose n-th iterate is a contraction has a unique fixed point, proving every contraction is uniformly continuous, and analysis of f(x)=x+1/x+1/x^3 on [1,infinity) -- showing the claimed non-expansion bound |f(x)-f(y)|<=|x-y| actually fails near x=1 (explicit counterexample), that f has no fixed point, and why this does not contradict the Contraction Mapping Principle since f is not a contraction. Exercise 4 (21 marks): statements of the Double Limit Theorem and Alternating Series Test, and analysis of pointwise/uniform convergence for f_n(x)=n/(n+e^x) on R and on half-lines ]-infinity,a], plus uniform convergence of an alternating series involving f_n(x)=1/(n+x) on [0,infinity).