Proposition: If one vampire were to appear on Earth, humanity would go extinct too fast for a stable equilibrium to develop.
Assumption 1: Each vampire must drain the blood of one human every 24 hours.
Assumption 2: A human who has had their blood sucked by a vampire will become a vampire.
Current population of Earth: 6,782,504,562 (Source; checked 00:31 GMT on September 7th, 2009)
Let vn be the number of vampires n days after the appearance of the first vampire. When vn = 6,782,504,562, humanity will be extinct.
If each vampire sucks the blood of one human in a 24-hour period, then in each 24-hour period, the population of vampires will increase by vn. In other words:
vn+1 = vn
=>vn = 2vn-1
Proposition: vn = 2nv0
Prove true for n = 1
v1 = 21v0
v1 = 2v0, which is true, because it agrees with vn+1 = vn.
Assume true for n=k
vk = 2kv0
Prove true for n=k+1
vk+1 = 2k+1v0
vk+1 = 2vk
=>2vk = 2k21v0
=>vk = 2kv0, which has already been assumed to be true.
Thus, for all k, the equation is true for k+1.
=> for all n, the equation is true for n+1. Since it has been proven true for n=1, it follows that the equation is true for all values of n.
Thus, it has been proved by induction that vn = 2nv0
When vn ≥ 6,782,504,562, humanity will have gone extinct.
2nv0 = 6,782,504,562
Let v0 = 1.
2n(1) = 6,782,504,562
2n = 6,782,504,562
=>n = log26,782,504,562
=>n ≈ 3.322log106,782,504,562
=>n ≈ 32.66 days.
Thus, if you were to start with a single vampire, humanity would be extinct within 33 days. That's why authors should only have their vampires drink a little bit of blood, every so often.