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Prove Upper Darboux Integral

  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Let ##f## be a bounded function on ##[0,1]##. Let ##P_n## be a partition of ##[0,1]## such that ##P_n = (0,\frac{1}{n},\frac{2}{n},...,1)##. Finally, we define ##\alpha=\inf\{U(f,P_n):n\geq1\}##, where ##U(f,P)## is the upper Darboux sum of ##f## with partition ##P##.

    Show that ##\alpha = U_f ##, with ##U_f## being the upper Darboux integral of ##f##, i.e. ##U_f = inf\{U(f,P):P\;is\;a\;partition\;of\;[0,1]\}##.

    2. Relevant equations


    3. The attempt at a solution
    We are also given the hint, which we showed in class, that for every partition ##P##,
    $$U(f,P\cup \{s \})-(M_f-m_f) \delta (P) \geq U(f,P)$$ where ##s \in [0,1]##, ##M_f=max(f)## on [0,1] (and correspondingly ##m_f##) and ##\delta(P)## is the norm of P.

    I really can't make the first step. I was thinking of using a theorem about the upper integral ##\bar I=\lim_{\delta(p) \rightarrow0}U(f,P)##, but I can't get it to take me anywhere...

    Thanks
     
  2. jcsd
  3. Apr 3, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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