Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). Show that $a_n$ converges, and find its limit.
We can show it converges as follows: suppose $a_n = p_n/q_n = k_n + r_n/q_n$, where $p_n = k_nq_n + r_n$, $0 \leq r_n < q_n$. Then $$a_{n+1} = \frac{p_n/q_n}{1-r_n/q_n} = \frac{p_n}{q_n - r_n},$$so the denominator will keep decreasing until it is a divisor of $p_0$ (maybe 1). Also, note we may take $p_n = p_0$ for all $n$.
Further, the limit will be $\leq \frac{p_0}{\operatorname{gcf}{(p_0,q_0)}}$, because if $f \mid p_0$ and $f\mid q_n$, then $f\mid (p_0 - k_nq_n)=r_n$, so $f \mid q_n - r_n = q_{n+1}$. But the limit may be strictly smaller; for instance, $a_0 = 30/7$ converges right away to 6.
Can we say anything else about the limit of a sequence starting with $a_0$? This was a problem on a qualifier, so I suspect there is more to the answer, but maybe not.