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- #1

let [tex] {f_n} [/tex] be a Cauchy sequence in the normed space Z. We know that [tex] |f_n(x) - f_m(x)| \leq |f_n - f_m|_u [/tex]. So [tex] {f_{n}(x)} [/tex] is a Cauchy sequence in [tex] \mathbb{C} [/tex] which is complete so f_n(x) converges to f(x) for every x. Letting n approach infinite on both sides of the inequality, we get [tex] |f(x) - f_n(x)| \leq lim \inf |f_n - f_m|_u [/tex].

my question is where did that lim inf come from?