5
$\begingroup$

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.

  • 0
    ok. but is there a way of doing this without invoking complex numbers?2011-11-16
  • 4
    Hi, Colin, welcome to math.SE! Probably this is a terrible start for you to get a question closed that quickly after posting it, sorry about that, usually it takes much longer. Anyway, the question you ask should be answered in the thread linked to at the top of your post. If there's anything unclear, please leave a comment here and I will try to answer.2011-11-16
  • 0
    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

0 Answers 0