MAT 302 Analysis 2 Exam 2026

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MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, 02 July 2026, instructor Dr. Ndambomve P., two hours. Exercise 1 (10 marks): carefully define ten metric-space notions (open/closed sets, fixed point, convergent sequence, Cauchy sequence, complete metric space, continuous functions, uniformly continuous functions, pointwise and uniform convergence of sequences of functions, compact set). Exercise 2 (20 marks): for a metric space (X,d) and A subset X with distance function delta(p,A)=inf over a in A of d(p,a), prove the closure of A equals the set of points at distance zero from A, deduce a closedness criterion, and prove three monotonicity/closure properties relating delta for subsets A subset B. Exercise 3 (17 marks): on the product metric space E=E1 x E2 with d((x1,x2),(y1,y2))=d1(x1,y1)+d2(x2,y2), prove it is a metric, characterize convergence componentwise, and prove completeness of the product iff both factors are complete; then for f(x)=(1/x)(x+8) on [0,1], show [0,1] is complete, examine whether f is a contraction, and show f has no fixed point in [0,1], discussing whether this contradicts the Contraction Mapping Principle. Exercise 4 (23 marks): state without proof the Double Limit Theorem and the Alternating Series Test Theorem for uniform convergence; for f_n(x)=n^2/(2n^2+e^x) show pointwise convergence to 1/2, show non-uniform convergence on R, show uniform convergence on (-infinity,a], and compute a limit of an integral; for f_n(x)=1/(n^2+x) on [0,infinity), show the alternating series sum (-1)^n f_n(x) converges uniformly.