$a_n$ and $b_n$ are sequences of natural numbers, $\lim_{n->\infty}\frac{a_n}{b_n} = x$, where $x$ is irrational. Prove that $a_n$ and $b_n$ diverge to infinity.
I've proved that if a sequence of natural numbers converges to some number, then this sequence has to be constant for big $n$.
If $(a_n)$ does not tend to $\infty$ then some subsequence of $(a_n)$ is bounded. But if a sequence of integers is bounded it will have a constant subsequence. The corresponding subsequence of $(b_n)$ is also constant because the ratio is convergent. It follows that $x=\frac p q$ for some integers $p$ and $q$ which is a contradiction. Use a similar argument to show that $(b_n)$ also must tend to $\infty$.