Hi,

Consider a bounded sequence $\displaystyle (a_n)_{n \in \mathbb{N}}$ whose terms are eventually "outgrown" by other terms of the sequence, i.e., for any index $\displaystyle n$ there exists an $\displaystyle n' > n$ such that $\displaystyle a_{n'} > a_n$. Does this property imply that $\displaystyle a_n$ is eventually monotonic, i.e., that there exists an index $\displaystyle n_0$ such that $\displaystyle a_n$ is monotonically increasing from index $\displaystyle n_0$ onwards? I can't think of any counterexample. If true, do you know if this a known theorem which I can reference from some standard textbook?

NB: theboundedness assumptionis essential, otherwise you can construct simple counterexamples like $\displaystyle a_n = n + (-1)^n$

Thanks,

jens