Example of a function satisfying specific limit condition

Question

Let $\epsilon >0$. I am trying to find a function $f(n):\mathbb{N}\rightarrow \mathbb{R^+}$ such that \begin{equation} \lim_{n \rightarrow \infty} f(n)=0 \end{equation} and \begin{equation} \lim_{n \rightarrow \infty} \frac{\log n}{n^{\epsilon^2} f(n)}=0. \end{equation} The fuction must not depend on $\epsilon$. It would be better if $f(n):\mathbb{N}\rightarrow (0,1]$, but is not as important since I am interested mostly in the asymptotic behavior of the sequence. Could anyone provide an example of such a function? Thank you very much!