# Prove Upper Darboux Integral

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1. Mar 29, 2016

### Yoni V

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. Apr 3, 2016