Question 171
Every rational number $\frac{p}{q}$ from the open interval $(0, 1)$ is covered by the closed interval
$$
\left[ \frac{p}{q} - \frac{1}{4q^2}, \frac{p}{q} + \frac{1}{4q^2} \right].
$$
Prove that the number $\frac{\sqrt{2}}{2}$ is not covered by any of these intervals.
$$
\left[ \frac{p}{q} - \frac{1}{4q^2}, \frac{p}{q} + \frac{1}{4q^2} \right].
$$
Prove that the number $\frac{\sqrt{2}}{2}$ is not covered by any of these intervals.