闭区间套定理是实数完备性理论中的一个重要定理，理解该定理，并掌握其证明其他结论的关键技巧和方法对于理解实数理论都有着重要意义。首先，论文给出了闭区间套定理在实数完备性理论中六大基本定理的等价性证明和闭区间上连续函数性质证明中的应用。其次，论文分析和总结了应用闭区间套定理证明数学命题的特点：基于直接法或反证法，选择适当的“遗传性质”并构造闭区间套，将闭区间上的整体性质“遗传”到每一个闭子区间，使得在所“套出的点”的局部区域上保持该性质，进而证明或者反证相关的命题。论文的研究结论对理解闭区间套定理和实数完备性理论，掌握应用该定理证明数学分析问题的方法，以及教学都具有较好的参考价值。
As a matter of fact, nested intervals theorem is one of the important theorems in the theory of the completeness of real numbers. It is showed in this article how to use the nested intervals theorem to prove the results in mathematical analysis. We think the key point in proof is to choose the appropriate hereditary property. Through building proper nested intervals, the overall property can be transferred to the local, and then the result can be proved.
