Let
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
be integers. Show that there exists an interval
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
of length
![1/q](/media/m/8/a/a/8aafa49a9ea94b2da7490b419304c632.png)
and a polynomial
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
with integral coefficients such that
![\left|P(x)-\frac pq \right| < \frac{1}{q^2}](/media/m/a/d/a/adaddcc8687d88a8d190e6cd76370c36.png)
for all
%V0
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
$$\left|P(x)-\frac pq \right| < \frac{1}{q^2}$$
for all $x \in I.$