I'm just second-marking some exam scripts, and I wanted to leap on a question and made the following pedantic remark concerning the model answers: "if the metric space is empty then this proof doesn't work because something which is supposed to be finite is $-\infty$. Hence this proof is incomplete -- it's missing the line "If the space is empty then the result is trivial".
But then another question made me wonder whether in fact the lecturer of the course had actually put as part of the definition of metric space, that it be non-empty. A quick trip to Wikipedia revealed that there also the definition required the space to be non-empty.
I certainly don't want to require that a topological space be non-empty, for example. There is presumably some sensible reason why the general convention for topological spaces has been to allow the empty set (this I understand!) but the general convention for metric spaces appears to be not to allow it...