$a_i$, $b_i$, $i=0,\ldots,\infty$, are two integer sequences with $\gcd(a_i,b_i)=1$ for all $i$.
Is then the limit $\frac{a_\infty}{b\infty}$ irrational if not $\frac{a_\infty}{b\infty}=\frac{A}{B}$ for some $\gcd(A,B)=1$ (assuming the limit exists at all)?