Give an example of a perfect set in $\mathbb R$ that does not contain any of the rationals.
I found out a proof here using continued fractions. I have trying to understand the proof.
In continued fraction, I studied (from Burton's Elementary Number Theory) that if $\frac{p_n}{q_n}$ is the $n$th convergent i.e., if $\frac{p_n}{q_n} =[a_0;, a_1,a_2,\cdots,a_n]$ convergent to the irrational number $x$ then $$\Big|\frac{p_n}{q_n}-x\Big|<\frac{1}{q_{n+1} q_n}$$
But in the proof (in the given link), the author has written $\Big|x_n-x\Big|<\frac{1}{q_{n-1} q_n}$.
How to construct $x_n$? what is the relation between $x_n$ and $[a_0;, a_1,a_2,\cdots,a_n]$ in this context?
I did not understand it. Please help me.
If $x_n = p_n/q_n$ then $$ \left| x_n - x \right| < \frac{1}{q_n q_{n+1}} $$