谢楼上和楼上楼上的

我原是用楼上所说的方法证明的，但是后来又被否决了，因为可能lub不能取到

下面给出老师写的证明，有兴趣的可以看看哦

In a more general case, we do not assume differentiability of the utility function.
Use the method of contradiction, suppose sup(u) exists.
Then we can find the corresponding bundle x∗ that give max(u) （or sup(u) ）.
This is because the superior set is closed, i.e., the preference is continuous. The
optimal bundle shall give the smallest e that achieves max(u) . This is easy to prove
because the range of the e function is bounded below by zero and closed. As the
prices >> 0, hence the set of the range contains the smallest number that is
e(p,max(u)).
Then since u is strictly increasing, the preference is strictly monotone, the consumer
can always find a x′ that is >> x∗ and gives larger u.
Contradiction.