利用致密性定理及Cauthy准则证明其它基本定理

Proof of Other Principal Theorems with Bolzano-Weierstrass Theorem and Cauchy Convergence Principle

以致密性定理为基础,证明闭区域套定理、有限覆盖定理、聚点定理、以及Cauthy准则这4个基本定理,首先要构造一个有界点列,然后利用致密性定理找到一个收敛子列及其极限,研究这个极限,得出矛盾或要证的结论;利用Cauthy准则证明闭区域套定理、有限覆盖定理、聚点定理、致密性定理这4个定理,首先要构造一个基本点列,利用Cauthy准则找到其极限,通过研究这个极限,得出矛盾或要证的结论.

With Bolzano-Weierstrass Theorem, this paper is first to prove Nested Closed Region Theorem, Finite Covering Principle, Accumulation Principle and Cauchy Convergence Principle, then to prove Nested Closed Region Theorem, Finite Covering Principle, Accumulation Principle and Bolzano-Weierstrass Theorem by using Cauchy Convergence Principle. In the first proof, a limited sequence is established, and then a convergent subsequence and its limit are found by using Bolzano-Weierstrass Theorem. The conclusion is reached based on the result of the study of this limit. In the second proof, a fundamental sequence is established, and then its limit is found by using Cauchy Convergence Principle. The conclusion is reached based on the result of the study of this limit.