Self-studying Gelfand pre-calculus (Functions and graphs, pg.92). Are there any more ways to prove that there is an integral point at a distance of less than $1/1000$ from the straight line $ y=\sqrt{3}x $ given the value of $\sqrt{ 3}$ is provided.(Wanted to understand elegant ways to prove this)
One possible approach: Assuming $\sqrt3 = 1.73205080757$, for a distance of $1/10000 < 1/1000$ (given distance in the question) from the line, suppose we calculate a point $(x,y)$ as $(y/x) = \sqrt3 - (1/10000) =17320/10000$ (approx.) which is the integral point $(10000,17320)$, then the value of $17320/ \sqrt3 = 9999.70666236$. On considering the value $9999.707$ (nearby round off in the x-axis, since $9999.70666236 * \sqrt3 - 17320 = 0$ ), $(\sqrt3*9999.707- 17320)$ turns out to be $0.00058480215$ which is less than a distance of $(1/1000)$. Hence there exists such integral points like $(10000,17320)$.