Interval nested theorem is a very important theorem in mathematical analysis. Together with supremum principle, monotone boundedness principle, convergence point theorem, Cauchy convergence criterion and finite covering theorem, it is called six basic theorems of real number completeness. Because of the equivalence of these theorems, a proposition that can be proved by one theorem can be proved by other theorems in principle, but the difficulty of proof is often very different.

This paper analyzes the characteristics of interval set theorem, and many important conclusions in mathematical analysis, especially those involving the whole to the part, can often be proved by interval set theorem.

It is not difficult to see that the interval set theorem says that large intervals are nested between cells and cells are nested between cells. In this way, a common point is finally nested, which is characterized by obtaining the local properties of a point from the overall properties of the point set. Therefore, when it comes to the proposition from the whole to the local, especially when it is necessary to prove the existence of points with certain properties under certain conditions, it is often suitable to prove [2-] with interval set theorem.