I came across a Tricki article with the above title, but it was a stub that didn't say how to do that particular trick.
I've run into this issue a few times before (can't put my finger on which contexts at the moment), and I was wondering how you do it in the sense the author meant.
Let $n$ be a fixed positive integer > 1. Then, the interval $$ I_n = \left [ \frac{n-1}n, 1 \right ) $$ contains rationals with denominator $> n$.
Proof: Suppose $\frac{a}b \in n$ and $b < n.$ Clearly $a < b$ so
$$\frac{a}{b} \le \frac{b-1}{b} = 1 - \frac{1}b \le1 - \frac{1}{n-1} = \frac{n-2}{n-1} < \frac{n-1}{n} $$ which cannot happen.