实数完备性六个等价命题的推广

The generalization of six equivalent theorems on completeness of real number

在R2上定义了极小上界、极大下界及单调点列,给出了极小上界、极大下界原理和单调有界定理.论证了极小上界、极大下界原理,单调有界定理,闭域套定理,有限覆盖定理,聚点定理,柯西收敛准则六个命题的等价性.最后将之推广到n维欧氏空间Rn上.

In this paper, we define a kind of minimum upper bound and maximal lower bound and monotony sequence in R2. We discuss the equivalence of the principle of minimum upper bound and maximal lower bound, the convergence principle for monotone and bounded sequence, finitely covered principle, interval contraction principle, principle of the accumulation point, Cauchy＇ s principle of convergence in R2. Finally we generalize this in R2.