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I hope I'm not making some stupid mistake. Suppose $r$ is integral. The distance from the point $[1,r]$ to $C(r)$ is $\sqrt(1+r^2)-r$ which goes to zero as $r$ goes to infinity. For nonintegral $r$ the point $[0,r]$ is even closer to $C(r)$ and hence we have an upper bound $\beta(r) < \frac{1}{2r}$ for sufficiently big $r$.