Statement
If $Q$ is a rectangle of $\Bbb{R}^n$ and if $f:Q\rightarrow\Bbb{R}$ then $$ L(f,P)=\inf\left\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\right\}\,\,\,\text{and}\,\,\,U(f,P)=\sup\left\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\right\} $$ for any fixed partition $P$ of $Q$.
Unfortunately I can't prove the statement. So could someone help me, please?
Lemma
If $Q$ is a rectangle of $\Bbb{R}^n$ and if $f:Q\rightarrow\Bbb{R}$ then $$ L(f,P)=\inf\left\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\right\}\,\,\,\text{and}\,\,\,U(f,P)=\sup\left\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\right\} $$ for any fixed partition $P$ of $Q$.
Proof. Clearly $$ m_R(f)\le f(x_R)\le M_R(f)$$ for any subrectangle $R$ and for any $x_R\in R$ so that $$ L(f,P)\le\sum_{R\in P}f(x_R)\cdot v(R)\le U(f,P) $$ from which we conclude that $L(f,P)$ and $U(f,P)$ are respectively a lower bound and an upper bound of the collection $\mathcal{S}:=\{\sum_{R\in P}f(x_R)\cdot v(R)$ for any $x_R\in R$} thus we only have to prove that they are respectively the maxium lower bound and minimum upper bound. So for convenience we define $$ h:=\sum_{R\in P}v(R) $$ for any partition $P$ of $Q$. So by the properties of infimum and supremum we know that for any $\epsilon>0$ there exist $\underline{x}_R,\overline{x}_R\in R$ such that $$ f(\underline{x_R})<m_R(f)+\frac{\epsilon}h\,\,\,\text{and}\,\,\,M_R(f)-\frac{\epsilon}h<f(\overline{x}_R) $$ so that multiplying for $v(R)$ and summing over all subrectangle of $P$ we obtain that $$ \sum_{R\in P}f(\underline{x_R})\cdot v(R)<L(f,P)+\epsilon\,\,\,\text{and}\,\,\,U(f,P)-\epsilon<\sum_{R\in P}f(\overline{x}_R)\cdot v(R) $$ and so by the arbitrariness of $\epsilon>0$ the statement holds.