3. Assume≿is locally nonsatiated. Then, for everyx∈𝑋, there exists somey∈𝑋such thaty≻x. Therefore, there is no best element.1.234Assume otherwise, that is assume thatx∗≿xfor everyx∈𝐵(p, 𝑚) but that∑𝑛𝑖=1𝑝𝑖𝑥𝑖< 𝑚. Let𝑟=𝑚−∑𝑛𝑖=1𝑝𝑖𝑥𝑖be the unspent income. Spending the residualon good 1, the commodity bundlex=x∗+𝑟𝑝1e1is affordable𝑛∑𝑖=1𝑝𝑖𝑥𝑖=𝑛∑𝑖=1𝑝𝑖𝑥∗𝑖+𝑝1𝑟𝑝1=𝑚Moreover, sincex≩x∗,x≻x∗, which contradicts the assumption thatx∗is the bestelement in𝑋(p, 𝑚).1.235Assume otherwise, that is assume thatx∗≿xfor everyx∈𝐵(p, 𝑚) but that∑𝑛𝑖=1𝑝𝑖𝑥∗𝑖< 𝑚. This implies thatx∗∈int𝑋(p, 𝑚). Therefore, there exists a neigh-borhood𝑁ofx∗with𝑁⊆𝑋(p, 𝑚).Within this neighborhood, there exists somex∈𝑁⊆𝑋(p, 𝑚) withx≻x
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Open set, Z0, x∗, Michael Carter, open neighborhood, Foundations of Mathematical Economics