Riemann integrability implies Darboux integrability I'm trying to prove the following:

Suppose that given $\epsilon&gt;0$, there is a $\delta&gt;0$ 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:
$$\left | \sum_R f(\mathbf{x_R})v(R)-\int_Qf \right |&lt;\epsilon$$
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. $U(f,P)-L(f,P)&lt;\epsilon)$. So, if only someone gave me a hint, it would be appreciated.
Let $\epsilon&gt;0$ 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 $$\sup_{x\in R_i}\{f(x)\}-f(x_i)&lt;\frac{\epsilon}{2v(Q)}$$ 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 $$|U(f,P)-\int_Qf|&lt;\epsilon$$
Let $\epsilon&gt;0$ 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 $$\sup_{x\in R_i}\{f(x)\}-f(x_i)&lt;\frac{\epsilon}{2v(Q)}$$ 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 $$|U(f,P)-\int_Qf|&lt;\epsilon$$