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: