$\def\hl#1#2{\bbox[#1,1px]{#2}} \def\box#1#2#3#4#5{\color{#2}{\bbox[0px, border: 2px solid #2]{\hl{#3}{\color{white}{\color{#3}{\boxed{\underline{\large\color{#1}{\text{#4}}}\\\color{#1}{#5}\\}}}}}}} \def\verts#1{\left\vert#1\right\vert} \def\R{\mathbb{R}}$ $\box{black}{black}{} {Question} {\text{Suppose that $R$ is a rectangle in $\R^2$.}\\ \text{Prove that a function $f:R\to\R$ is integrable if and only if}\\ \begin{align} \inf\left\{\iint_RgdA:g(x)\ge f(x)\text{ for all }x\in R\right\} =\sup\left\{\iint_RhdA:h(x)\le f(x)\text{ for all }x\in R\right\} \end{align}\\~\\ \text{Prove that if $f:R\to\R$ is integrable, then}\\ \text{$ \begin{align} \iint_R fdA=&\inf\left\{\iint_RgdA:g(x)\ge f(x)\text{ for all }x\in R\right\}\\ =&\sup\left\{\iint_RhdA:h(x)\le f(x)\text{ for all }x\in R\right\} \end{align}$}}$
$~$(The question is from this online note)
My thought
For the first direction of the first part, we assume that $f$ is integrable.
From the definition that means $f$ is bounded and we have \begin{align} &\iint_Rf(x)dA\\ =&\underline{\iint_R}f(x)dA&&=\overline{\iint_R}f(x)dA\\ =&\sup_PL_Pf&&=\inf_PU_Pf\\ =&\sup_P\sum_{i=1}^{JK}m_i\text{area}(R_i)&&=\inf_P\sum_{i=1}^{JK}M_i\text{area}(R_i)\\ =&\sup_P\sum_{i=1}^{JK}\inf\{f(x):x\in R_i\}\text{area}(R_i)&&=\inf_P\sum_{i=1}^{JK}\sup\{f(x):x\in R_i\}\text{area}(R_i)\\ =&\sup_P\sum_{i=1}^{JK}\inf\{f(x):x\in R_i\}(x_j-x_{j-1})(y_k-y_{k-1})&&=\inf_P\sum_{i=1}^{JK}\sup\{f(x):x\in R_i\}(x_j-x_{j-1})(y_k-y_{k-1}) \end{align} However, I still can't tell how to conclude $$\inf\left\{\iint_RgdA:g(x)\ge f(x)\text{ for all }x\in R\right\} =\sup\left\{\iint_RhdA:h(x)\le f(x)\text{ for all }x\in R\right\}$$ from this condition, and I don't get another direction too, could someone help me.