I'll only try to give intuition for the supremum, and in . The infimum is pretty much the same deal

Imagine you have a set in . It is legitimate to ask whether this set has a maximum element or not, that is: is there an element ,belongingto , such that for all other ? This element may or may not exist.

Example 1: , where has a maximum element: .

Example 2: , does not have a maximum element. You can get as close to as you wish, but there is no "end point".

Example 3: does not have a maximum element.

Now let's take a closer look at example 2. If we were to define a "generalized maximum" concept, it would be extremely tempting to say the "generalized maximum" of is , since, as we've stated, we can get as close to as we wish... we're just... missing the point itself!

What property then, does have, that makes us want to choose it as the "generalized maximum"? And, since we want to generalize the notion of maximum, we want to define this new concept in such a way that is also a "generalized maximum" in example 1.

Instead of thinking of the intervals and , we then think of the set of all upper bounds of and respectively, and observe that, in both cases, is the minimum of the corresponding sets of upper bounds. We take that as a definition: we say that an element is asupremumof a set if is the least upper bound of .

And of course, we may think of the supremum of as a value that is bigger than all elements of , and is "infinitely" close to . It may, or may not belong to itself. If it does belong, we are in the case of example 1: the set has a maximum. If it doesn't, we're in the case of example 2.

Finally observe that example 3 does not have a supremum either. There is no fixed value that is greater than all others of the interval.

Let me know if it was of any help ^^