Non-Borel set

The example

Every irrational number has a unique representation by a continued fraction

$x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}$

where $a_{0}\,$ is some integer and all the other numbers $a_{k}\,$ are positive integers. Let $A\,$ be the set of all irrational numbers that correspond to sequences $(a_{0},a_{1},\dots )\,$ with the following property: there exists an infinite subsequence $(a_{k_{0}},a_{k_{1}},\dots )\,$ such that each element is a divisor of the next element. This set $A\,$ is not Borel. For more details see descriptive set theory and the book by Kechris, especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.