This is an indirect way. Of course it assumes you've defined closed set and limit point. If not, you should be able to see a more direct way.
By writing down two definitions the above statement follows quite easily:
Def 1 x is interior point of S <=> there exists ε>0 and
Def 2 S is open <=> for all ,x: xεS => there exists ε>0 and .
And using the two definitions we have:
S is open <=> (for all ,x : xεS => there exists ε>0 and ) <=> (for all ,x : xεS => x is interior of S) <=> (each point of S is interior)