6 theorems relating to real number's comleteness

摘要:
注:所有6个选项仅适用于实数字段,但不适用于其他原因,因为它们是正确的。第1个选项为间隔中的关闭定义“关闭间隔”假设一系列关闭力

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".
6 theorems relating to real number's comleteness第1张
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:

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上篇prove of supremum and infimum principle using Cauchy convergence criterium用实数域的闭区间套定理证明确界原理下篇

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