Let A and B be non-empty subsets of Y, and let $\displaystyle z \in Y$. Then

$\displaystyle d(A,B) \leq d(A,z)+d(z,B)$.

Obviously, this is a restatement of the triangle inequality, but I'm having a hard time putting the proof into words. Should I begin by assuming $\displaystyle d(A,B)=t_1=|a-b|$, where (a,b) is in AxB? Because then I run into a problem when I assume that

$\displaystyle d(A,z)=t_2=|a-z|$ because I'm using the same element $\displaystyle a \in A$, which I don't think I can assume.

Can anyone give me some help with this?