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You don't need to use any fancy theorem, the trick is to disentangle the definitions. Everything follows directly from the definitions. $x^0=x^*(p^0,w)$ means that $p^0x^0\leq w$ and that $u(x)\leq u(x^0)$ for all $x$ such that $p^0x\leq w$. In words $x_0$ is affordable and at least as good as every other affordable bundle (in many cases, better and not ...


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