Question

If \(\{a_n\}, \{b_n\}\) are increasing sequences with terms that are positive integers, then show that \(x = \lim\limits_{n\to\infty}\dfrac{a_n}{b_n}\) is irrational :

\[\large 0 \lt |x-a_n| \lt \dfrac{1}{b_n}\]

Answer 1

if it helps, the proof is easy if you get the correct start...