$\DeclareMathOperator{\graph}{graph}$I want to show that for $\Omega\subset\mathbb{R}^2$ an open bounded regular domain with a $C^2$ function $u:\overline{\Omega}\to\mathbb{R}$ that the graph $\graph(u):=\{(x,y,u(x,y))\,:\,(x,y)\in\overline{\Omega}\}\subset\mathbb{R}^3$ of $u$ is minimal and satisfies the following inequality:
For $\Sigma\subset \Omega\times\mathbb{R}$ a compact surface with $\partial\Sigma=\partial\graph(u)$ the divergence theorem (on suitable domains) yields
$$\DeclareMathOperator{\area}{area}
\area(\graph(u))\leq \area(\Sigma).
$$
What I have thought of so far is the following: If $\Sigma$ and $\graph(u)$ partially intersect apart from their boundary, the application could become more complicated. To avoid this, consider the following:
We can apply the divergence theorem on the connected component of $\Omega \times [-M,\infty) \backslash \graph(u)$ containing $\Omega \times \{-M\}$, where $M$ is sufficiently large such that $\graph(u)$ and $\Sigma$ are contained in $\Omega \times [-M,\infty).$ Then we apply again the divergence theorem but in the connected component of $\Omega \times [-M,\infty) \backslash \Sigma$ containing $\Omega \times \{-M\}$.
My problem consists of how to apply the divergence theorem here exactly since we did not cover it in the lectures. Could anyone help me on how to apply it to show the desired property?