I don't believe there is an error in the logic of his argument, but would need to see more precisely where you are seeing one.
The problem (I wouldn't call it the error in logic) is existence = caused. Ergo, the uncaused does not exist.
I can see holding that view; I can see also saying "it just is." But if I'm playing in the reason/logic sphere, I don't see a more compelling argument that existence ≠ caused.