I'm studying advanced micro from the Mas-Colell book (exercise 16.C.1)
I was wondering if anyone can help me to solve the following exercise. I have no idea how to deal with it
Show that if a consumption set $X_i \subset \mathbb{R}^{L}$ is nonempty, closed, and bounded and the preference relation $\succeq_i$ on $X_i$ is continuous, then $\succeq_i$ cannot be locally nonsatiated. [Hint: Show that the continuous utility function representing $\succeq_i$ must have a maximum on $X_i$]
Continuous functions attain a minimum and a maximum value over a compact (closed and bounded set). This is a well known theorem.
Since the utility function $u_i$ representing $\succeq_i$ is continuous and $X_i$ is closed and bounded, then $u_i$ must attain a maximum value at some point $x = (x_1,\dots,x_L)$ in $X_i$.
Since $u_i$ has a maximum value at $x$, there is no $y \in X_i : y \succ_i x$.
Therefore, $\succeq_i$ is not locally non-satiable.
The definition of locally non-satiable I use is $\forall x \in X \forall \epsilon > 0 \exists y \in X : ||y-x|| \leq \epsilon$ and $y \succ x$.
Note this definition implies the non-existence of maximum values for the corresponding utility function, if the preferences are indeed representable by a utility function (which happens when the preferences are complete, transitive and continuous).
