Exercise 6.2.1 (There are infinitely many $(a,b$ such that $|bx -a| < (\sqrt{5} b)^{-1}$)

Proof. By Theorem 6.1, there exist infinitely many different rational numbers $a∕b$ (where $a\wedge b=1$, $b>0$) such that

If $a∕b$ and $a^{\prime }∕b^{\prime }$ are two such distinct fractions, then $(a,b)\ne (a^{\prime },b^{\prime })$, so there are infinitely many ordered pairs of integers $(a,b)\in \mathbb{Z}\times {\mathbb{N}}^{\text{∗}}$ satisfying (1). Multiplying (1) by $b>0$, this shows that there are infinitely many ordered pairs of integers $(a,b)\in \mathbb{Z}\times {\mathbb{N}}^{\text{∗}}$, such that

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