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$a_n$ 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 $n$.

user4201961
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    Diverge to $\infty$ and converge aren't the only two things a sequence can do. Better than split into cases here is probably to use directly, via the definition, that $c_n$ converges. – Arthur Nov 06 '18 at 07:35

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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 $\infty$.

Kavi Rama Murthy
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