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I don't understand this step in the proof about L infinity

  1. Nov 15, 2009 #1
    I'm learning the proof that [tex]L_{\infty}[/tex] is complete. I do not understand one of the steps.

    Let [tex]f_n[/tex] be a cauchy sequence in [tex]L_{\infty}(E)[/tex] then there exists a subset A in E such that [tex]f_n[/tex] is "uniformly cauchy" on E\A. For m,n choose A so that

    [tex]|f_n-f_m| \leq ||f_m - f_n||_{\infty}[/tex] for all x in E\A. Take the union of all such As and then [tex]f_n[/tex] converges uniformly on E without the As.

    Define f to be [tex]f(x) = \mathop{\lim}\limits_{n \to \infty} f_n(x)[/tex] for x in E without the As, and let it be o otherwise. F is bounded and measurable now all we need is to show that [tex]||f_n - f||_{\infty} \rightarrow 0[/tex] so we know f is in [tex]L_{\infty}[/tex]. We know [tex]m(As)=0[/tex]

    This next bit is where the proof makes a leap that I don't understand.

    [tex]||f_n - f||_{\infty} \leq \sup_{x \in E-A} |f_n -f|[/tex]

    Then is says as n --> infinity we have [tex]\sup_{x \in E-A} |f_n -f| \rightarrow 0[/tex]. But, I have no idea where that inequality came from? What theorem? Please help!
     
  2. jcsd
  3. Nov 16, 2009 #2
    *a little bump*

    I know it's not a simple proof I'll see what my prof. says in office hours. But, hmmm maybe it has something to do with the l infinity norm being the esssup?

    I'm still not used to the esssup and I have a test in 4 days...
     
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