If $\vert n\sqrt{2005} - m \vert < \frac{1}{90n}$ then $\vert n\sqrt{2005}+m \vert < 2(\sqrt{2005}+1/90)n < 90n$. But we have $\vert 2005n^2 - m^2 \vert \ge 1$. So $\vert n\sqrt{2005}-m \vert \cdot \vert n\sqrt{2005} + m \vert \ge 1$. Contradiction.