MAT 302 ANALYSIS II CA 2024

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MAT 302: Test II for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, 31 May 2024. Exercise 1: definitions of continuity/uniform continuity/convergence notions and statements of the Uniform Cauchy Criterion, Weierstrass M-Test and Contraction Mapping Principle. Exercise 2: proving Lipschitz implies uniformly continuous, completeness of [1,infinity), and a Lipschitz-constant analysis of f(x)=x+1/x+1/x^3 on [1,infinity) (the source's literal claim of a 1-Lipschitz bound is shown by explicit counterexample to be false near x=1; the correct Lipschitz constant L=3 via the Mean Value Theorem is derived instead, with the discrepancy documented). Exercise 3: uniform convergence of a series via the Weierstrass M-Test and pointwise/non-uniform convergence of x^2/(x^2+n).