Let be a nonempty set of real numbers which is bounded below. Let . Prove that .

So I want to show that (i) and (ii) . Now exists because and is bounded below. Also, exists since is non-empty and bounded above.

Now and . I took to have a lower bound greater than . You can apply the same argument if you set to have a lower bound less than (e.g. break it into cases)? From these inequalities, we can conclude that ? Adding them we get .