a. an upper bound of the set is any element such that if you take any element , then

b. a lower bound of the set is any element such that if you take any element , then

note that upper bounds and lower bounds are not necessarily in .

c. the least upper bound of , or we call thesupremum of Sis an upper bound of such that if you take any upper bound of , . In other words, is the smallest among all the upper bounds.

d. the greatest lower bound of , or we call theinfimum of Sis a lower bound of such that if you take any lower bound of , . In other words, is the biggest (greatest) among all the lower bounds.

Example:

Consider the set

these are some upper bounds.. 5, 5.1, 6, 7, 6.5,.. in fact, any number greater than or equal to 5 is an upper bound. While the supremum of the set if 5 since if you take all the upper bounds, 5 is the smallest.. Hope you can tell what the lower bounds are, and what the infimum is.

another example:

you should see that 1 is the supremum of and 0 is its infimum.