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, dated 18 April (the year digit in the source scan is partly obscured/overwritten and is therefore omitted rather than guessed; the paper's own year field states 2022 as a best estimate for cataloguing). Exercise 1: definitions of open ball, open set, neighbourhood, metric subspace, convergent/Cauchy sequence, completeness, whether a neighbourhood must itself be open, and fundamental properties of open sets. Exercise 2: the ell-p and ell-infinity sequence spaces are shown to be metric spaces, with ell-2 verified as a metric subspace of ell-infinity. Exercise 3: the equivalence d_infinity <= d_p <= n^(1/p) d_infinity on R^n, and using it to relate convergence/Cauchy/completeness between (R^n,d_p) and (R^n,d_infinity).