Here is an outline that may help you get some ideas on your proof.
Now suppose that , then .
By definition of inf, .
But that means that , a contradiction.
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 .