Let A and B be non-empty subsets of Y, and let . Then

.

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 , where (a,b) is in AxB? Because then I run into a problem when I assume that

because I'm using the same element , which I don't think I can assume.

Can anyone give me some help with this?