Continuous Preference Relation Imply Continuous Utility Fn Existence
I think you're overthinking this proof. MWG only say that such a compact set exists because, by monotonicity, you can always find $\alpha_0$ and $\alpha_1$ such that $\alpha_0*e&lt;\alpha(x')*e&lt;\alpha_1*e$ (with $&lt;$ I mean strictly preferred). Remember that, by monotonicity, if $y&gt;&gt;x$, then $y$ is strictly preferred to $x$. Therefore, we only need to find some $\alpha_0$ and $\alpha_1$ such that $\alpha_0&lt;\alpha(x')&lt;\alpha_1$ for all $x'$. 

This is because the set $\{x':||x'-x||\le\epsilon\}$ is bounded and $x'∼\alpha(x')*e$. 

(Note that with local non-satiation the result does not immediately follow).

You're right, $\alpha(x')*e$  is on the diagonal line Z and $x'$ is not (note that $\alpha(x')$ is mutiplied by $e$, a vector filled by 1's). However, working with $\alpha(x')$ reduces the complexity of the problem. Instead of working with two (or more) dimensions, we just work with one (the diagonal line Z or one axis). We can do that because for any element $x\in \Bbb R_+$, there is $x∼\alpha(x)*e$ which is in the diagonal line.