Question: 5. A metric space (X, d) is called separable if it contains a countable dense subset, that is, if there exists a countable subset E

A metric space (X, d) is called separable if it contains a countable dense subset, that is, if there exists a countable subset E Q X such that E = X. Prove that every compact metric space is separable. Hint: For each n E N, consider an open cover consisting of neighborboods of radius 1 n ' Important metric space. Let ( denote the collection of all sequences inJEN in R which are bounded; i.e. (:= {{annEN : In ER Vn, {an} is a bounded sequence Define doo : to x (0 -> [0, 0) as do ({an)nEN, {Un ]nEN) = sup an - yn. nEN (i) Prove that ((, do ) is a metric space. (ii) Is ( compact? Support your answer with a proof. (iii) Recall the definition of a separable metric space from Question 5. Prove that ( is not separable

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