Question

B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit example of a sequence (n) in C, such that dsup(1) 1 for all while di(fa 1)0 (You do not need to prove your chosen fn are continuous, but you should show that they satisfy the other conditions.) 3 (b) (i) Give the definition of a Cauchy sequence in a metric space (E,d). Hence, say what it means for a metric space to be complete Cauchy sequence is bounded. every bounded sequence in R has a convergent subsequence.) ii) Say what it means for a sequence in a metric space to be bounded. Show that every (iii) Prove that R, with its usual metric, is complete. (Hint: you may use the fact that (iv) Is (0, 1], with the usual metric inherited from R, complete? Justify your answer.4 please turn over 3

Answer 1

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