ILL 1989 Problem 12

Let $P(x)$ be a polynomial such that the following inequalities are satisfied: $$P(0) > 0;$$$$P(1) > P(0);$$$$P(2) > 2P(1) - P(0);$$$$P(3) > 3P(2) - 3P(1) + P(0);$$ and also for every natural number $n,$ $$P(n+4) > 4P(n+3) - 6P(n+2)+4P(n + 1) - P(n).$$ Prove that for every positive natural number $n,$ $P(n)$ is positive.