2. Let X be the set of all bounded sequences r r in R. Define d...

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2. Let X be the set of all bounded sequences r r in R. Define d X x XR by d(r, y supENn-yn Verify for yourself that d is a metric on X, but do not hand in your reasoning for this (a) Prove that (X, d) is complete. (b) For Ne N let EN E X I, 0 for all n N}, and define E =U-1EN. Prove that the closure of E is the set of sequences re X with rn 0. (c) Let 0 X denote the sequence with all entries equal to zero, and let Bz E X d(0, ) 1 be the closed unit ball in X, centred at 0. Prove that B is not compact. Hint: consider the sequence e E X where e, has its nth entry equal to 1 and all others equal to 0. 2. Let X be the set of all bounded sequences r r in R. Define d X x XR by d(r, y supENn-yn Verify for yourself that d is a metric on X, but do not hand in your reasoning for this (a) Prove that (X, d) is complete. (b) For Ne N let EN E X I, 0 for all n N}, and define E =U-1EN. Prove that the closure of E is the set of sequences re X with rn 0. (c) Let 0 X denote the sequence with all entries equal to zero, and let Bz E X d(0, ) 1 be the closed unit ball in X, centred at 0. Prove that B is not compact. Hint: consider the sequence e E X where e, has its nth entry equal to 1 and all others equal to 0.