I see that you gave the person 5 warnings - I feel 3 times is less and this is exactly what I would do. Its so much better to give people more chances rather than less and then letting them go instead of letting an ego or something similar come in detween earlier.
Please do not take number 5 so literally. It's the process that matters. You do 2, 3, 4, 5, 6, 7, 8, 9 or 10 chances whatever suits your needs. The actual point is you don't fire people on the first mistake unless you're an asshole.
