I am searching for an explicit example of a sequence of real numbers $(u_n)$ that satisfies the following properties:
(1) for any $\alpha>0$, $n^{\alpha}\notin \mathcal{O}(u_n)$,
(2) There exists $\beta>0$ such that $u_n \notin \mathcal{O}(n^\beta)$.
(3) [Edited after comments] $(u_n)$ is non-decreasing
It might be easy, but I didn't find any of such $(u_n)$. For instance $\log(n)$ satisfies (1,3) but not (2). Could it be that (1,3) implies not (2) ?
Thanks