Quote by cyborg6060
My issue is that I don't see what prevents a spaceship throwing off fuel at a constant rate A kg/s, so that:
[itex]\frac{dp}{dt} = \frac{GMm}{r^2}[/itex]
[itex]A v = \frac{GM(m  At)}{r^2}[/itex].
What prevents this system (classically, so long as v is not too large) from escaping the event horizon?
My initial thought would be that v would have to be greater than the speed of light, but I can't quite justify that mathematically.
Any thoughts would be incredibly helpful.

In a Newtonian sense, you're right. But black holes are not Newtonian objects, and although you get the right answer for the Schwarzschild radius, it's completely by incorrect reasoning. It's not actually an issue with special relativity  but with the much larger theory of General Relativity.
If you imagine yourself going towards the event horizon of a real Einsteinian black hole, you would need to accelerate infinitely fast as you fell down towards the event horizon
just to stay at the same radius! Curiously, once you've fallen past the event horizon, any amount of blasting your rockets you do will only hasten your eventual demise! If you want to live as long as possible, the best bet is to sit there and do nothing.