Question

What is the difference between bounded above and bounded below?

Answer 1

The set EE is said to be bounded above if and only if there is an M∈RM∈R such that a≤Ma≤Mfor all a∈Ea∈E, in which case MM is called an upper bound of EE.A number ss is called a supremum of the set EE if and only if ss is an upper bbound of EE and s≤Ms≤M for all upper bounds MM of EE (In this case we shall say that EE has a finite supremum ss and write s=supEs=supE.

Let E⊂RE⊂R be nonempty.

the set EE is said to be bounded below if and only if there is an m∈Rm∈R such that a≥ma≥m for all a∈Ea∈E, in which case mm is called a lower bound of the set EE.A number tt is called an infimum of the set EE if and only if tt is a lower bound of EE and t≥mt≥m for all lower bounds mm of EE. In this case we shall say that EE has an infimum tt and write t=infE