I these cases the direct approach is seldom constructive. It's tempting to start constructing logical examples that show they must be wrong, but this is much more a matter of psychology than of logic.
Basically, the stronger you argue, the harder they will dig their heels in. With your example you may actually force a situation where they have no other way out than to admit that they're wrong, but in most scenarios they'll find a way to stick to their guns, and the more you push the issue, the harder it becomes to admit they were wrong all along.
So the trick here is small commitments. Don't ask them to accept the whole thing in one go, but get them to accept a small part of it, something they can agree on without admitting defeat. Then just stand back and let them come round on their own.
My favourite example of this principle is trying to convince a rabid anti-Apple consumer to buy an apple laptop instead of a regular PC. If you argue on quality of hardware, or value for money, or user experience, it won't do you any good (regardless of whether those are valid arguments). What you should do is buy them something very cheap and tiny, like an iPod nano, for their birthday. Once they commit to that, the anti-Apple stance is no longer a point of principle, but they're somewhere in the grey area, free to move around.
It's all about breaking down the principled stance and the ties to their pride. Only once you've got rid of that should you come in with the logical arguments.