Every Rational Number is Approximable to no Order Greater Than 1

Proof

Since every real number is approximable to order one, we just need to prove that each rational number is approximable to no higher order. Suppose $\alpha =\frac{a}{b}$ is a rational number, with $a,b\in \mathbb{Z}$ and $b>0$. Then, for each $p,q\in \mathbb{Z}$ with $q>0$ consider that

If we assume $\alpha \ne \frac{p}{q}$, then the numerator $aq-bp$ is non-zero. Since it is a positive integer, it must be at least $1$ and therefore

Now, assume that $\alpha$ is approximable to some order $s$. Thus, using the above inequality as well, there exists a $c$ such that, for pairs $(p,q)$ with arbitrarily large $q$ we have

Therefore

Since neither $b$ nor $c$ depend on $q$, the only way this can be true for arbitrarily large $q$ is if $s-1\le 0$, or equivalently that $s\le 1$. Thus, $\alpha$ is approximable to no order greater than $1$.