I have a strictly increasing sequence of positive integers $\left\{ v_{n}\right\} _{n\geq1}$ for which $\sum_{n=1}^{\infty}\frac{1}{v_{n}}=\infty$ and $\liminf_{n\rightarrow\infty}\frac{v_{n}}{n}=\infty$. I would like to show that no such sequence can exist. By using the converse of the limit comparison test on the fact that the $v_{n}$s are not reciprocal-summable, I obtain: $$\liminf_{n\rightarrow\infty}\frac{v_{n}}{n^{1+\delta}}=0,\textrm{ }\forall\delta>0$$
Is this sufficient to reach the desired contradiction? In other words, can I conclude that: $$0=\lim_{\delta\downarrow0}\liminf_{n\rightarrow\infty}\frac{v_{n}}{n^{1+\delta}}=\liminf_{n\rightarrow\infty}\left(\lim_{\delta\downarrow0}\frac{v_{n}}{n^{1+\delta}}\right)=\liminf_{n\rightarrow\infty}\frac{v_{n}}{n}=\infty$$
Or is there some arcane detail which prevents me from making this conclusion?