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Possible Duplicate:
Understanding proof of completeness of $L^{\infty}$

Most books I've been reading say that showing $L^\infty$ is complete is easy, but I've been struggling with it and I need help. I know I have to show that every Cauchy sequence, $\{f_n\}\in L^\infty$ converges to some $f\in L^\infty$.
So, this is all I have thus far.

Given $\varepsilon > 0$, $\exists~ N$ such that $\inf \{M : \mu\{t: |f_n(t)-f_m(t)|\gt M\}=0\}=\Vert f_n-f_m\Vert_\infty \lt \varepsilon$ for $n,m \geqslant N$.

This is where I gut stuck.

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    Concerning your comment: The complex numbers are completely immaterial for the argument. Just replace every occurrence of $\mathbb{C}$ by $\mathbb{R}$.2011-11-16

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