SHOW that every open interval is uncountable

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Prove this theorem same thing to do as above,here just set xn=((a+b)/2).an1an2an3.....ann..... and y=((a+b)/2).y1y2y3.....yn.....

We will prove this theorem by contradiction.

Assume that A is open and A=(a,b).A is countable also.

Since,B=[a,b] is a uncountable set and we know that If B is an uncountable set and A is a countable set,then (B-A) is uncountable set.

Here,B-A=[a,b]-(a,b)={a,b},which is consists of two elements and is finite,so countable,which is contradiction.

This, If A is countable then A can not be open set.