To prove God is to prove a negative; to prove god is to define god so that there is no god.
Since coming out of fundamentalism, I would rather combine these too. I do not have faith in God as there is no need for me too. I have faith in the Christian Tradition, however, and a hope. But, it requires proof from time to time.
I would rather go on about this, but I am ill at the moment.
What are your thoughts?
By the way, Jim West has differing thoughts.
I think Jim is right in calling the picture a false dichotomy. But I also think you are right in stating that proving God would render God falsifiable. Both positions are compatible.