We define a distance in like the following:
A distance is defined by the following properties:
1. d(x,y) >= 0
2. d(x,y) = d(y,x)
3. d(x,y) =< d(x,z) + d(z,y)
I managed to prove the two first (it was rather easy). But I'm stuck on the third. Does anybody know how to do? I tried various things from expanding the first part to reducing the second part with the triangular inegality or even the inequation of Cauchy-Schwartz. I ended up always in something that I couldn't prove anymore.
I would be thankful for any idea.