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