实数系的连续性和完备性的若干等价定理

Some Equivalence Theorem of the Continuity and Completeness of the System of Real Numbers

以十进制小数表示作为出发点,给出实数定义,并以此为基础证明了单调收敛定理。总结了描述实数系连续性和完备性的若干等价定理,即:单调收敛定理,上(下)确界定理,边界点定理,戴德金分割定理,辛钦定理,区间套定理,聚点原理,有限覆盖定理,致密性定理,柯西收敛准则。

The definition of real numbers based on taking decimal as the jumping-off place, and on this basis, the monotone convergence theorem was proved. Some theorems describing the continuity and completeness of the system of real numbers were summarized, such as monotone convergence theorem, supremum （infimum） theorem, boundary point theorem, Dedekind cut theorem, ＂Khinchin theorem, theorem of nested interval, principle of the point of accumulation, Heine-Borel theorem, Bolzano-Weierstrass theorem, Canchy convergence criterion.