I'm trying to understand a paper called "Almost no points on a Cantor set are very well approximable". In the proof the author uses the Borel-Cantelli Lemma (in the eighth line at the beginning of the proof on the second page). There are more than one version of it, but I think that the following is used:
It seems the lemma is used when we consider $(A_q:q\in\mathbb{Z}_+)$ given by $A_q=B(p/q,\psi(q)/q)$, however $p$ is not fixed. So could someone explain how is the Borel-Cantelli lemma used in this proof?
Thanks.
Just reading (5) one sees that the authors consider the family $(B(p/q,\psi(q)/q))_{p,q}$ where $(p,q)$ runs over every pair such that $p\in\mathbb Z$, $q\in\mathbb Z_+$, $(p,q)=1$, $p/q\in I$. Since this collection of pairs is (at most) countable, one can order it as $\{B(p/q,\psi(q)/q)\mid p,q\}=\{A_n\mid n\geqslant1\}$ and apply simple Borel-Cantelli lemma to the sequence $(A_n)_{n\geqslant1}$.