Let $I=I_1\times\cdots\times I_n$ be a $n-$rectangle in $\mathbb R^n$ and $P=P_1\times\cdots\times P_n$ be a partition of $I$, where $P_i$ is a partition of $I_i$. How can one find a uniform partition $P_m$ of $I$, which devides each edge $I_i$ into $m$ subinterval with the same length, and also is a refinement of $P$ ?
My question is because I'm doing a problem as follow : if $f:\, I\longrightarrow\mathbb R$ be Darboux integrable, which means $$\forall\epsilon>0,\ \exists P\in\mathcal P(I):\ U(f,P)-L(f,P)<\epsilon $$ then for $P_m$ be an uniform partition and for any way of choosing the representative point $x_R\in R\in H(P_m)$, we have $$\intop_If=|I|\lim_{m\to\infty}\left[\frac{1}{m^n}\sum_{i=1}^{m^n}f(x_{R_i})\right]=:\lim_{m\to\infty}S\big(f,P_m,x_R\big) $$ which basically a special case of Riemannian integrable for uniform partition.
So my idea is if I can find an uniform refinement $P_m$ of $P$, then I should get \begin{align} 0\leq S\big(f,P_m,x_R\big)-L(f,P_m)\leq U(f,P_m)-L(f,P_m)\leq U(f,P)-L(f,P)<\epsilon \end{align} which will show what I desire. By the way, I have been tried some different ways, calculated it out over 20 pages of draft, it's still not worked and I'm so struggling.
Could anyone give me some hints for the problem ? I really appreciate.