Let $a_n$ and $b_n$ be 2 positive and increasing sequence. Then is it true that either the sequence $a_n/b_n $ or sequence $ b_n/a_n $ is bounded ?
At first i thought, if $a_n/b_n $ is unbounded, then $b_n/a_n $ is bounded and vice versa. This is very intuitive to me since it's like saying: if $a_n$ grows faster than $b_n$ then $b_n $ grows slower then $ a_n$, then $b_n/a_n$ should be bounded given that they are all positive.
But then, i think the above is wrong, $a_n/b_n$ is unbounded then the limit law $lim \frac {a_n} {b_n}\times lim \frac{b_n}{a_n} $ doesn't apply and $b_n/a_n$ can also be unbounded.
Can someone give me an example that both $a_n/b_n$ and $b_n/a_n$ are unbounded and explain why they can both grow unbounded ?