zoukankan      html  css  js  c++  java
  • 6 theorems relating to real number's comleteness

    note:all the 6 theorems are applicable only over real number field, other than rational umber. cause they are
    incorrect in it

    NO.1 theorem of closed nested intervals

    DEFINITION of "CLOSED NESTED INTERVAL"
    suppose the series of closed intervals as ({[a_{n},b_{n}]}) has such qulities as below:
    (i)([a_{n},b_{n}]supset [a_{n+1},b_{n+1}],n=1,2,3cdotcdotcdot)
    (ii)(lim_{n oinfty}(b_{n}-a_{n})=0)
    we call ({[a_{n},b_{n}]}) "a closed nested interval".

    from the chart above, we can infer that:
    (a_{1}leqslant a_{2}leqslant a_{3}cdotcdotleqslant a_{n}leqslant a_{n+1}cdotcdotleqslant b_{n+1}leqslant b_{n}leqslant b_{n-1}cdotcdotleqslant b_{2} leqslant b_{1}quadquadquad(1))

    theorem of "closed nested intervals":
    if %{[a_{n},b_{n}]}% is a closed nested intervals, then there is only one point (xi), with (xiin[a_{n},b_{n}]),n=1,2,3,...
    ie: (a_{n}leqslant xi ,n=1,2,cdotcdotcdot.)

    prove:

  • 相关阅读:
    Git引用
    如何查看Git对象
    Git是如何存储对象的
    图形化的Git
    git中找回丢失的对象
    Git的Patch功能
    ES查看配置和查看全部配置
    增删改查
    Elasticsearch增、删、改、查操作深入详解
    ES博客链接
  • 原文地址:https://www.cnblogs.com/strongdady/p/13581273.html
Copyright © 2011-2022 走看看