Proving a negative--it can be done, and people do it routinely.

The other term for proving a negative is ruling things out as possible.  That's what's going on with the Zimmerman trial as we speak: prosecution trying to prove that what Zimmerman claims happened could not possibly have happened, and citing events and conditions which prove that particular negative.  The defense's business is not only to prove what happened actually did happen the way it was described, but also how what the prosecution claims happened couldn't really have happened. It's a question I often ask when encountering people who claim certain events in history happened as they were taught, or as they read, but couldn't have, given the circumstances of those times.

If you have passed any upper level mathematics classes involving algebra, you too have proven negatives in the process of solving for an unknown variable, because you have to rule out what the variable couldn't be in order to arrive at what it actually is.  You have doubtless encountered the Venn Diagram, which graphically depicts sets (families) of data in order to determine where the sets overlap and where they don't.  If you're about the business of determining where the sets have something in common, what you've done with the non-intersecting parts is rule them out, and in this manner prove a negative in their case.

Negatives can be proven, and in court cases MUST be proven. Also true of the methodology of solving sudoku puzzles.  If you cannot rule out all the numbers from 1 to 9 except one number, then you're making a mistake of writing in your best guess.  Guess at sudoku, and you fail nearly all the time, especially on puzzles of greater difficulty than the beginner level puzzle.  You MUST prove that all other numbers except one number do NOT belong inside the cell.

So the next time you get into an argument with an evangelist who proclaims that nobody can prove or disprove the existence of God, and you're an atheist, simply point out that what you've done is rule him out.  It's doable.