Question 2
Let $f(x)$ be a real polynomial of degree at least one. Prove that for every $c\gt 0$ there is a positive integer $n_0$ such that, for every monic real polynomial $P(x)$ of degree at least $n_0$, the number of integers $x$ satisfying $|f(P(x))|\le c$ is at most the degree of $P(x)$.