some definitions related to open sets 4 (a interval is connected)

We reached in the preceding post to a proposition which insisted that a connected set on the real number line was a interval. So, we shall give a brief proof of the proposition.

Suppose that $I$  is a connected set on the real number line
and $I=A\cup B$ , where $A,B\subset \mathbb{R}$ , $A\cap B=\phi$ 
and both $A$  and $B$  are closed and not empty.

We are able to choose two points $a_1$  from $A$  and $b_1$  from $B$ , and put $a_1<b_1$ .
Dividing the new interval $(a_1,b_1)$ in half, the one must contain points of $A$  and $B$ .

Let the one be a small interval $(a_2,b_2)$  and we repeat the same operation.
Then, we get the sequence of the interval $(a_n,b_n)$  where $a_n\in A$  and $b_n\in B$ 
and $(a_n,b_n)\supset (a_{n+1},b_{n+1})$ .

If the operation is repeated infinitely, as the new interval becomes smaller and smaller, the real number sequence $a_n$  and $b_n$  have a same limit point $c$ .

As $A$  and $B$  are closed, $c$ is a intersection point of two sets. However, it is in contradiction to the precondition.