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Problem 5 Let c 0 be the Banach space of real sequences (x n) such that x n!0 as n!1with the supnorm k(x n)k= sup n2N jx njIs the closed unit ball B= f(x n) 2c 0 k(x n)k 1g compact?Solution The closed unit ball in c 0 is not compact For example, let e k= ( nk) 1 n=1 nk= 1 if n= k 0 if n6=kProof that 9x(P(x) _Q(x)) (= 9xP(x) _9xQ(x) Suppose 9xP(x) _9xQ(x) Breaking up by cases, if 9xP(x) is true (case 1), then P(c) is true for some c, and thus P(c) _Q(c) is true for that c If 9xQ(x) is true (case 2), then Q(d) is true for some d, and thus P(d) _Q(d) is true for that d In either case 9x(P(x) _Q(x)) is true
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