let be nonempty and bounded above, then is also nonempty and bounded above, so both suprema exist. Let's denote .
Is an upper bound of ? Yes, because any can be written as for some and we have and is positive, so .
Is the least upper bound? If it were not, there would be some such that and is an upper bound of . Then is an upper bound of , because for any we have so and . Furthermore, . But this would mean that is not the least upper bound of , which is contradiction.
We conclude that .