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 $n$ ?
Best Answer
Here's a similar condition that is sufficient: there exists a sequence of integers $u_n, v_n \to\infty$ such that $(u_n, v_n) = 1$ and $$\lim_{n\to\infty} v_n a - u_n = 0.$$ (Alternatively, we could just require that $a$ not be exactly equal to any $u_n/v_n$ rather than $(u_n,v_n)=1$.)