Can the Sequence of Upper Darboux Sums Converge to the Upper Darboux Integral?

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Yoni V
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Homework Statement


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]\}##.

Homework Equations

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
 

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