On Spivak "Calculus on Manifolds" p59 he provides a proof of Fubini theorem but there is a party i don't fully understand. We have in the hypothesis that $A\subset\mathbb{R}^n$ and $B\subset\mathbb{R}^m$ are closet rectangles and that $f:A\times B\to\mathbb{R}$ and we define the function $g_x:B\to\mathbb{R}^n$ by $g_x(y)=f(x,y)$. Also we define the Lower and Upper integrals of $g_x$$$\mathscr{L}(x)=\sup_{\mathcal{P}_B}L(g_x,\mathcal{P}_B)$$ and $$\mathscr{U}(x)=\inf_{\mathcal{P}_B}U(g_x,\mathcal{P}_B),$$ where the sup and inf are taken over all the partitions $\mathcal{P}_B$ of $B$. He first prove that we have the following inequalities; $$L(f,\mathcal{P})\leq L(\mathscr{L},\mathcal{P}_A)\leq U(\mathscr{L},\mathcal{P}_A)\leq U(f,\mathcal{P})$$ for an arbitrary partition $\mathcal{P}$ obtained from a partition $\mathcal{P}_A$ of $A$ and a partition $\mathcal{P}_B$ of $B$ (we just take the cartesian product of the rectangles on each one). Now this is the part of the proof that i dont get; he says that since we have $\sup L(f,\mathcal{P})=\inf U(f,\mathcal{P})$, we have by the above inequalities that $$\sup L(\mathscr{L},\mathcal{P}_A)= \inf U(\mathscr{L},\mathcal{P}_A).$$ This seems intuitive, but why is this the case? Where does this implication come from?
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