We show that compactness fails for $ p = \infty $. Let $ f_n = \chi_{(0,\frac{1}{n})} $. We have that $ ||f_n||_{\infty} = 1 \ \forall n $, so we have a bounded sequence in $ L^{\infty} $. But there can be no convergent subsequence, since $ ||f_n - f_m||_{\infty} = 1, n\neq m $. $ \square $
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