Question 10
(FIN 1) Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
$$\left|P(x)-\frac pq\right|\lt \frac1{q^2}$$
for all $x\in I$.
$$\left|P(x)-\frac pq\right|\lt \frac1{q^2}$$
for all $x\in I$.