CA2021Proposed solution included

MAT 302 ANALYSIS II CA 2021

MAT 302: Test I for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, 21 May 2021. Question 1: definitions of metric space, closed set, closure, convergent/Cauchy sequence, completeness. Question 2: examining whether two candidate functions on C1[0,1] are metrics -- rho2(f,g)=sup|f'-g'| is shown NOT to be a metric (fails identity of indiscernibles, counterexample f(x)=x vs g(x)=x+1), while rho1(f,g)=sup|f'-g'|+|f+g| is analyzed with a documented caveat about its literal '+' term's triangle-inequality behaviour versus the standard '-' form; the discrete 0/2 metric is also proven a metric with its distance-1/less-than-1/equal-to-2 sets described. Question 3: d(x,K)=0 iff x in closure(K), d(x,K)=d(x,closure(K)), completeness of closed subsets, and that complete subsets of a metric space are closed.

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University of Buea

Faculty of Science - FS

Mathematics - MAT

MAT3022021

ANALYSIS II

CA · Second Semester · Level 300

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Let X be a set. Define the following notions: (i) Metric space on X, (ii) closed set, (iii) closure of a set, (iv) convergent sequence, (v) Cauchy sequence, (vi) Complete metric space.
2.
Let F ⊂ X be complete. Show that F is closed. (3 marks)

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