Boundedness for Infinite Sets

The most obvious difference between finite and infinite sets when it comes to boundedness is that infinite sets don't have to be bounded.

Another more interesting difference is that infinite sets may not have a maximum, even if they do have an upper bound.

But, even though $1$ isn't a maximum, it still feels very important for the set $\{1/2, 2/3, 3/4, 4/5, 5/6, \ldots\}$; after all, it is an upper bound and it's also "tight"; you can't move the upper bound any closer to the set without making it not an upper bound anymore.

This motivates us to define the supremum of bounded infinite sets (which we've already seen when we were defining real numbers).

Definition: The supremum of a set that's bounded from above is the smallest possible upper bound of the set.

You can verify that if a set does have a maximum, then the supremum of the set is just the same as the maximum.

Also recall that a supremum does exist for every set bounded from above when we're working with real numbers. In fact, this is one of the fundamental reasons we use real numbers in calculus, rather than integers or rational numbers.

The corresponding term for the minimum is the infimum, which is equal to the minimum when that exists:

Definition: The infimum of a set that's bounded from below is the largest possible lower bound of the set.