The following theorem and proof are taken from the Lecture Notes of Ariel Rubinstein.
Assume that the demand function is derived from maximizing a preference relation $\succeq$ which satisfies monotonicity. Then:
Theorem:
The hyperplane $H= \{ (p, w) | px(p^{\ast}, w^{\ast})=w \}$ is tangent to the $\succeq^{\ast}$-indifference curve at $(p^{\ast}, w^{\ast})$.
Proof:
By the monotonicity of the preferences $(p^{\ast},w^{\ast}) \in H$. For any $(p,w)\in H$, the bundle $x(p^{\ast},w^{\ast}) \in B(p,w)$. Hence, $x(p,w) \succeq (p^{\ast},w^{\ast})$ and thus $(p,w) \succeq^{\ast} (p^{\ast},w^{\ast})$.
However, though I believe that I figured out each proposition in the proof separately, I couldn't grasp the proof itself. Can someone explain the reasoning here?
Thanks.
