What is the difference between bounded above and bounded below?

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Nathaniel Hogan

Mathematics

20 March, 19:06

What is the difference between bounded above and bounded below?

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Lassie · Answer 1

The set EE is said to be bounded above if and only if there is an M∈R M∈R such that a≤M a≤M for all a∈E a∈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≤M s≤M for all upper bounds MM of EE (In this case we shall say that EE has a finite supremum ss and write s=supE s=supE.

Let E⊂R E⊂R be nonempty.

the set EE is said to be bounded below if and only if there is an m∈R m∈R such that a≥m a≥m for all a∈E a∈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≥m t≥m for all lower bounds mm of EE. In this case we shall say that EE has an infimum tt and write t=infE