Supremum vs Maximum

If you have a finite set of elements then it is easy to tell which element is the maximum of all because you can exhaust the set in finite time.
But if you have to find  maximum of an infinite set how do you point out that element?
Tricky?
Yes in this case you know the maximum value but can't say which element has it.
So, we name it supremum.
For example, you can easily tell which one is the maximum among (1,2,3,...9) but
If I give you an open interval  (2, 3) ,you know that 3 is the maximum but it is not inside the open interval (2, 3). Here, we say it is the supremum instead of maximum.