I said that d is not a map into R. And this is because if you take f(x)=x and g(x)=0, then d(f,g)=sup{|x|:x in R} is not an element of R. Indeed, the real number sup{|x|:x in R}, if it exists, is an upper bound for the set {|x|:x in R}. But this set is unbounded from above and hence possesses no upper bound so sup{|f(x)|:x in R} does not exists.