A special pair of sequences in $c_0$ This could be a rather trivial question but I've spent some time on it and I can't come up with anything:
Do there exist sequences of positive numbers, say $(\alpha_n),(\beta_n)$ in $c_0$, i.e. $\alpha_n\to0$ and $\beta_n\to0$ such that the following two conditions are satisfied:

*

*$\sum_{n=1}^\infty n\beta_n<\infty$


*there exists $\delta>0$ and a subsequence of indices $n_1<n_2<n_3<\dots$ such that $\beta_{n_k}\cdot\sum_{j=1}^{n_k}\alpha_j\ge\delta$, i.e. the sequence $\{\beta_n\sum_{j=1}^n\alpha_j\}_{n=1}^\infty$ does not converge to $0$.
All I have tried is to play around with some standard sequences, but once I get the one condition satisfied, the other one breaks down. I would appreciate any help!
A comment: in order for condition 2 to be satisfied, one needs to take $\alpha_n$ to be a sequence that is not in $\ell^1$. On the other hand, for condition (1) to be satisfied $(\beta_n)$ not only has to be in $\ell^1$, but it has to converge "fast enough".

No, there do not exist such sequences.  Since $\alpha_n=o(1)$, we must have $$\sum_{n=1}^N{\alpha_n}=\sum_{n=1}^N{o(1)}=o(N)$$
Thus $$\sum_{N=1}^{\infty}{\left(\beta_N\sum_{n=1}^N{\alpha_n}\right)}=\sum_{N=1}^{\infty}{o(N)\beta_N}\lesssim\sum_{N=1}^{\infty}{N\beta_N}<\infty$$ where ${\lesssim}$ indicates dropping a constant prefactor.  Since $\sum_{N=1}^{\infty}{\left(\beta_N\sum_{n=1}^N{\alpha_n}\right)}$ is summable, its terms must tend to $0$, whence the claim.
