If $\sum_j \frac 1 j a_j $ is summable where $\{a_k\}$ is a positive sequence, then is $\liminf a_k = 0$?

Question

Suppose $\sum_{j} \frac 1 j a_j < \infty$ and $a_j$ is a positive sequence, is it that $\liminf_{k \to \infty} a_k = 0$?

If not, suppose $\liminf a_k = 2\varepsilon > 0$, for sufficiently large $k$, we have $a_k \ge 2\varepsilon - \varepsilon = \varepsilon$. This leads to that $\sum_j \frac{1} {j} a_j $ is not summable. Is this argument correct?