  In this paper we consider the probabilistic theory of Diophantine
approximation in projective space over a completion of Q. Using the projective
metric studied by Bombieri, van der Poorten, and Vaaler we prove the analogue
of Khintchine's Theorem in projective space. For finite places and in higher
dimension, we are able to completely remove the condition of monotonicity and
establish the analogue of the Duffin-Schaeffer conjecture.
