Sup stands for supremum, which is equal to the maximum value of a finite number set. For an infinite number set, the supremum is the smallest number greater than all the numbers in the set. For example, the supremum of the set (0, 1) is 1.

The supremum is the minimum upper bound of a set and the most basic concept in mathematical analysis. Consider a set of real numbers M. If there is a real number S, so that no number in M exceeds S, then S is called an upper bound of M. If there is a minimum upper bound among all those upper bounds, it is called an supremum of M. A bounded number set has countless upper and lower bounds, but there is only one supremum.

Proof of supremum: every x ∈ X satisfies the inequality x ≤ m; ; For any ε > 0, there is x' ∈ X, so that X' >；; M-ε, what's the number? M = sup{x} is called the supremum of set X.

In the general textbook of mathematical analysis, in the chapter of real number theory, there are a series of theorems to illustrate the continuity of real numbers. Textbooks of the former Soviet Union with rigorous theory are generally based on Dydykin's division theorem to prove other equivalence theorems. In order to simplify textbooks in China, many of them are proved from the starting point of definite theorem. Other theorems that explain the continuity of real numbers include interval set theorem, finite covering theorem and so on.