The basic theorem of real number system is also called the completeness theorem and continuity theorem of real number system. These theorems are fixed boundary existence theorem, monotone bounded theorem, finite covering theorem, convergence point theorem, compactness theorem, closed interval set theorem and Cauchy convergence criterion. They are equivalent to each other and describe the continuity of real numbers in different forms. They are also important tools to solve some theoretical problems in mathematical analysis. Among various theorems of calculus, the equivalence of seven basic theorems does not mean that they are all true, but that they are all true or not. This requires a more basic theorem to prove that one of them is true, that is, they are both true at the same time. The introduced method is mainly to admit Dai Dejin's axiom, and then prove that these seven basic theorems are equivalent, thus starting to establish a series of concepts and theorems of calculus. There are some new equivalence theorems in some papers, but these seven theorems are common basic theorems in teaching.

First, the principle of supremum (supremum)

A nonempty set with an upper (lower) bound must have an upper (lower) supremum.

Second, the monotonous definition.

Monotone bounded sequence must have a limit. Specifically:

Monotone increasing (decreasing) sequences with upper (lower) bounds must converge.

3. Closed interval nesting theorem (Cauchy-Cantor theorem)

For any set of closed intervals, there must be a common point belonging to all closed intervals. If the interval length tends to zero, then this point is the only common point.

Fourthly, the finite covering theorem (borell-Leberg theorem, Heine-Porel theorem).

Any open covering on a closed interval must have a finite sub-covering. In other words, any open covering on a closed interval must take out a finite number of open intervals to cover this closed interval.

Five, the limit point theorem (Polchano-Weisstras theorem, convergence point theorem)

Bounded infinite point sets must have convergence points. In other words, every infinitely bounded set has at least one limit point.

6. Sequential compactness of bounded closed intervals (compactness theorem)

Bounded sequences must have convergent subsequences.

Seven. Completeness (Cauchy convergence criterion)

The necessary and sufficient condition for a sequence to converge is that it is a Cauchy sequence. Or: Cauchy will converge and the convergence sequence is Cauchy.

Note: Only propositions with sufficient and necessary conditions can be called "criteria", otherwise they cannot be called "criteria".

The above seven propositions are called the basic theorems of real number system. The seven basic theorems of real number system describe the continuity of real numbers in different forms, and they are equivalent. In the proof, we can prove their equivalence by one-cycle proof. The proof of their equivalence can be found in Notes on Mathematical Analysis.

The basic theorem of real number system is a very important tool in proving the properties of continuous functions on closed intervals, but the equivalence between them cannot prove that they are all true. There must be a more basic theorem to prove that one of them is true, so that all the above propositions are true. After careful consideration, the problem boils down to the introduction of real numbers. For example, in Fichkingolz's Calculus Course, the definite theorem can be derived from the continuity of real numbers, while in Mathematical Analysis (Volume I) (Fourth Edition) compiled by the Department of Mathematics of East China Normal University, the definite theorem is derived from the decimal form of real numbers, which also shows the importance of establishing a strict definition of real numbers. Logically, real numbers should be established first, and then the basic theorem of real number system can be obtained, so that strict limit theory can be established in real number domain, and finally strict calculus theory can be obtained. However, the development of the history of mathematics is just the opposite. The strict limit theory was first established in1at the beginning of the 9th century. 18 after the basic theorem of real number system is basically formed.