Problem 2. A metric space {X, d) is called sepamble if it contains a countable dense subset, i.e. there exists a countable set S C X with g = X. {a} Show that the real line with the usual {Euclidean} metric is separable but the real line with a discrete metric is not. (b) (Phillips, Section 4.1 #3) Suppose (X, (1] is totally bounded: that is, for ever},r E > I] there exists a cover of X by niter many open E-balls. Prove that (X,d) is separable. Hint: For each n E N, let E = 1,171 and take 513ml; . . . :En,j(n] such that n} X = U B(:r:mhc). i=1 Show that the countable set 82 {End : n E N13: 11"'1j(n)} isdenseinX