Let be bounded above and . Prove .
Since is bounded below, it has a greatest lower bound . To show that , we need to show that ? It this all there is to it?
Suppose that .
If then by definition of supremum and infimum is not an upper bound of A, so but c is not a lower bound of B. That means , but that means d is a upper bound of A which is impossible because . That means that .
This time suppose that which implies .
But this means that is an upper bound for A but it cannot belong to B. That is a contradiction. Thus .