For any bounded closed interval sequences I_n=[a_n,b_n], (-\infty<a_n\leq b_n<\infty) , if I_n\supset I_{n+1}, (n\in\mathbb{N}) and |a_n-b_n|\rightarrow 0 as n\rightarrow \infty, there exists a real number c such that
\bigcap_{n=1}^{\infty}I_n = c
In other words, if the real number sequence a_1,a_2,\cdots is monotonic increasing bounded above, the bounded real number sequence b_1,b_2,\cdots is monotonic decreasing bounded below and a_n\leq b_n for any n , there is a real number c such that a_n\leq c\leq b_n . Furthermore, if |a_n-b_n|\rightarrow 0 as n\rightarrow \infty, c is a point. That is to say, \lim_{n\rightarrow\infty}a_n=\lim_{n\rightarrow\infty}b_n=c .
In real number system, as a bounded monotone real number sequence has a limit number, we put \lim_{n\rightarrow\infty}a_n=a,\quad \lim_{n\rightarrow\infty}b_n=b . Hence, for any n ,
a_n\leq a\leq b\leq b_n
and
I_n\supset [a,b]
It means \bigcap_{n=1}^{\infty}I_n\ne \phi . There is a real number c\in [a,b] .
As a_n\leq c\leq b_n for any n , |a_n-c|\leq b_n-a_n . Therefore a=c, because |a_n-b_n|\rightarrow 0 . Similarly b=c .
It is called the method of nested intervals. By the method we are able to get a root of some kinds of equations computationally.
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