A metric space is a set X together with a function d (called a metric or "distance function") which assigns a real number d ( x , y ) to every pair x , y such that X satisfies the properties (or axioms ):

1. d ( x , y ) >= 0 and d ( x , y ) = 0 <=> x = y ,
2. d ( x , y ) = d ( y , x ),
3. d ( x , y ) + d ( y , z ) >= d ( x , z )

Now replace X by R .You will get your required definition