Bounds and Boundedness

Let's say that we have a set A containing some whatever numbers from the set of natural numbers N.

Let's assume we have the following values:There are values in N that come before A: 1,2,3,4
But there are also an infinite amount of values that come after A: 9,10,11,12....

We can now conclude that there are points that are "bounds" of this set A.
Any and all values bigger than all elements of A that are outside of A, so, 9,10,11,12 and up to infinity, are all upper bounds of A.

Any and all values lower than all elements of A, that are outside of A, so, 1,2,3,4 are all lower bounds of A.

You'll see why this is important as we explain the next concepts:
The Supremum and Infimum