Putnam 1949 B1

Each rational number $\frac{p}{q}$ with $p,q$ coprime of 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 $\frac{\sqrt{2}}{2}$ is not covered by any of the above closed intervals.