When is a sequence that is bounded below by an unbounded and monotone increasing also monotone increasing?

Question

Let $(a_n)$ be an unbounded and monotone increasing sequence of positive integers and let $(b_n)$ be another sequence of positive integers such that for all $n \in \mathbb{N}$, $a_n \leq b_n$. Evidently, $(b_n)$ is unbounded. Is it the case that $(b_n)$ is also monotone increasing? If not, is there a "$(b_n)$ independent" condition that I can put on $(a_n)$ that will imply $(b_n)$ is monotone increasing?

Answer 1

$(b_n)$ may not be monotonically increasing, and there isn't a condition you can force on $(a_n)$ to make $(b_n)$ increasing.

For every $(a_n)$ you could set, for example,

$$ b_n= \begin{cases} a_n, & \text{if } n \text { is odd,}\\ a_{n+1}+1 & \text{if } n \text { is even.}\\ \end{cases} $$

Then $(b_n)$ will be decreasing from each even index, when it is $a_{n+1}+1$ to the next odd index, where it is $a_{n+1}$.