The specific meaning is as follows:

We assume that e is a nonempty subset in R. If there is a real number β∈R, the following two conditions are satisfied:

1) For any x∈E, there is x≤β. β is the upper bound of e.

2) for any α > 0, there must be at least one x∈E, so that x > β-α, that is, any number β-α less than β must not be the upper bound of e.

Then we say that β is the supremum of e.

Write β = sup E

Similarly, there is a definition that is symmetrical with the supremum called the supremum, which is expressed by inf, which is the abbreviation of infimum in English.