Rational Approximation

If $r$ is any number and $n$ is any Integer, then there is a Rational Number $m/n$ for which

$$0\leq r-{m\over n} < {1\over n}.$$ (1)

If $r$ is Irrational and $k$ is any Whole Number, there is a Fraction $m/n$ with $n\leq k$ and for which

$$0\leq r-{m\over n} < {1\over nk}.$$ (2)

Furthermore, there are an infinite number of Fractions $m/n$ for which

$$0\leq r-{m\over n} < {1\over n^2}.$$ (3)

Hurwitz has shown that for an Irrational Number $\zeta$

$$\left\vert{\zeta-{h\over k}}\right\vert < {1\over ck^2},$$ (4)

there are infinitely Rational Numbers $h/k$ if $0<c\leq \sqrt{5}$, but if $c>\sqrt{5}$, there are some $\zeta$ for which this approximation holds for only finitely many $h/k$.