Suppose we have a finite quantity $a$, which we would like to prove to be irrational, supposing that it is indeed irrational.
Then, would it be enough to show that $$a=\lim_{n\to\infty}\frac{u_n}{v_n},$$ for some positive integers $u_n,v_n$, where $u_n,v_n\to\infty$ as $n\to\infty$. If so, then would there have to be some divisor properties between the denominator and numerator so that cancellation does not produce an integer or rational number as $n\to\infty$, e.g. suppose $(u_n,v_n)=1$ for all $n$ ?