MAT 302 ANALYSIS II CA 2022

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MAT 302: Test I for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, 22 April 2022. Question 1: definitions of metric, closed set, closure, convergent/Cauchy sequence, completeness; the discrete 0/2-valued metric is proven to be a metric and its distance-1/less-than-1/equal-to-2 point sets described. Question 2: proving the d_infinity <= d_p <= n^(1/p) d_infinity equivalence on R^n and its consequences for convergence, Cauchy sequences and completeness between (R^n,d_p) and (R^n,d_infinity). Question 3: proving d(x,K)=0 iff x is in the closure of K, that the distance function is uniformly continuous, that every complete subset is closed, and that closed subsets of a complete space are complete.