I am just finding that I have a couple questions about the definitions of supremum and infimum, that I am having a hard time answering. Perhaps it is a possiblity that I am just overthinking these things a bit...

The definition of supremum is the least upper bound of a set, or it is the least element that is greater than or equal to every other element of the set. Why is it necessary to consider the "least element"? Is it possible to have multiple different upper bounds of a set? And if that is true, then why look for the least one and not the greatest? The way I understand this concept is that the supremum of a set, say {1,2,4} would be 4. I am just having difficulty even trying to come up with a set that there would be confusion in the upper bound...

And another question that I have... is it possible to have multiple distinct suprema for a set?

Of course it is easy to consider a set such as {1,2,4,4} where the supremum (4) occurs multiple times. But my interpretation of the definition of supremum, that is, the supremum is the least upper bound (so there might be multiple upper bounds), leads me to ponder whether there could be multiple suprema for one set.

Or am I possibly just overthinking this?