How will you prove that d(x,y)>=0 in a metric space.
That is part of the definition of a metric space, not something to be proved about metric spaces.
Unless you've been given a specific set and distance function on that set, and need to prove that the distance function is non-negative, in which case we'd need to see the details before we can help.
We can define a metric on a set as a map satisfying:
From and it is easy to prove:
For example, to prove take .
Fernando Revilla