Approximation of a rational number with values of polynomial

Question

Given two positive integers $p,\space q$, prove that there exist a polynomial $P(x)\in\mathbb{Z}[x]$ and interval $I\subseteq\mathbb{R}$ of length $\frac{1}{q}$ such that $\forall x\in I\space\space\space|P(x)-\frac{p}{q}|<\frac{1}{q^2}$.