If (X,d) is a metric space (not defined), prove that (X,p) is a metric space defined by p(x,y) = d(x,y)/(1+d(x,y)).

I'm having trouble proving that p has the triangle equality property.

1st attempt:

I wrote d(x,z)/(1+d(x,z)) <= d(x,y)/(1+d(x,y)) + d(y,z)/(1+d(y,z)), multiplied the whole thing by (1+d(x,z))(1+d(x,y))(1+d(y,z)), then cancelled out as many terms as possible, but I only got d(x,z) <= d(x,y) + d(y,z) + 2d(x,y)d(y,z) + d(x,y)d(x,z)d(y,z).

I think it will be simpler to solve this problem by solving an analogous problem, that x <= y + z implies that x/(1+x) <= y/(1+y) + z/(1+z).