theorem

[seema283k]

if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way

[Tac-Tics]

if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way

First, are you talking about an open-and-closed subset of R? If you are, then the empty set is also open-and-closed in R, so you have to specify that A is nonempty.

My approach for this proof would be to consider this.

An open set contains none of its boundary points.
A closed set contains all its boundary points.

The only way for this to be possible is for A to have NO boundary points at all. Show how {} and R are the only two sets that have this property.