oblixps
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in the proof of showing that the vector space of all complex valued functions with the norm |f|_u = sup(|f(x)|) over all x in the domain is complete, there was a step that was confusing:
let {f_n} be a Cauchy sequence in the normed space Z. We know that |f_n(x) - f_m(x)| \leq |f_n - f_m|_u. So {f_{n}(x)} is a Cauchy sequence in \mathbb{C} 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 |f(x) - f_n(x)| \leq lim \inf |f_n - f_m|_u.
my question is where did that lim inf come from?
let {f_n} be a Cauchy sequence in the normed space Z. We know that |f_n(x) - f_m(x)| \leq |f_n - f_m|_u. So {f_{n}(x)} is a Cauchy sequence in \mathbb{C} 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 |f(x) - f_n(x)| \leq lim \inf |f_n - f_m|_u.
my question is where did that lim inf come from?