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MAT302 (MAT 302) - MAT302 Analysis II End of Semester Examination - 2024 Exam - University of Buea
- 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.
