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I'm trying to prove the following:

Suppose that given $\epsilon>0$, there is a $\delta>0$ such that if $P$ is any partition of $Q$ (a rectangle in $\mathbb{R}^n$) of mesh less than $\delta$, and if, for each subrectangle $R$ determined by $P$, $\mathbf{x_R}$ is a point of $R$, the condition: $$\left | \sum_R f(\mathbf{x_R})v(R)-\int_Qf \right |<\epsilon$$ holds (here $v(R)$ is the volume of $R$), then $f$ is Darboux integrable (clearly $f$ is bounded). The converse is not that difficult, the proof makes use of the Riemann condition (i.e. $U(f,P)-L(f,P)<\epsilon)$. So, if only someone gave me a hint, it would be appreciated.

user227802
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Let $\epsilon>0$ be given and let $P=\{R_1, \ldots, R_n\}$ be a partition of $Q$ of mesh less than $\delta(\epsilon/2)$. We know that for every $i$ there exists an $x_i \in R_i$ such that $$\sup_{x\in R_i}\{f(x)\}-f(x_i)<\frac{\epsilon}{2v(Q)}$$ Let $S(f,P,\{x_i\})$ be the Riemann sum corresponding to this choice of points.

What can you say about $U(f,P)-S(f,P,\{x_i\})$ and how can you use this to prove that $$|U(f,P)-\int_Qf|<\epsilon$$

DGRasines
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