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)$[a_{n},b_{n}]\supset [a_{n+1},b_{n+1}],n=1,2,3\cdot\cdot\cdot$
(ii)$lim_{n\to\infty}(b_{n}-a_{n})=0$
we call ${[a_{n},b_{n}]}$ "a closed nested interval".

from the chart above, we can infer that:
$a_{1}\leqslant a_{2}\leqslant a_{3}\cdot\cdot\leqslant a_{n}\leqslant a_{n+1}\cdot\cdot\leqslant b_{n+1}\leqslant b_{n}\leqslant b_{n-1}\cdot\cdot\leqslant b_{2} \leqslant b_{1}\quad\quad\quad(1)$

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

prove: