Question 5
Let $a_1$, $a_2$, … be an infinite sequence of positive integers, and $N\gt1$ an integer. Suppose that for all integers $n \ge N$, the expression $$
\frac{a_1}{a_2} + \frac{a_2}{a_3} + \dots
+ \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}
$$ is an integer. Prove that $(a_n)$ is eventually constant.
\frac{a_1}{a_2} + \frac{a_2}{a_3} + \dots
+ \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}
$$ is an integer. Prove that $(a_n)$ is eventually constant.