Segre's Theorem

For any Real Number $r\geq 0$, an Irrational number $\alpha$ can be approximated by infinitely many Rational fractions $p/q$ in such a way that

$$-{1\over\sqrt{1+4r}\,q^2} < {p\over q}-\alpha < {r\over \sqrt{1+4r}\,q^2}.$$

If $r=1$, this becomes Hurwitz's Irrational Number Theorem.

See also Hurwitz's Irrational Number Theorem