Suppose we have two series $\sum_{k=0}^{\infty} a_k$ and $\sum_{k=0}^{\infty} b_k$. Both of them have nonnegative terms. If $\sum_{k=0}^{\infty} a_k = \infty$ and $\sum_{k=0}^{\infty} a_k b_k < \infty$, then $\liminf_{k \to \infty} b_k = 0 $.
I have an idea of showing the statement by contradiction. Suppose $\liminf_{k \to \infty} b_k = \beta > 0$. By definition, there are at most finitely many $b_k's$ with $b_k \le (\beta - \beta/2)$. Let $I = \{ k \in \mathbb{N}: b_k \le \beta/2 \}$. Then \begin{align*} \sum_{k=0}^{\infty} a_k b_k &\ge \sum_{k \in (\mathbb{N} \setminus I)} a_k b_k \\ & \ge (\beta/2) \sum_{k \in (\mathbb{N} \setminus I)} a_k \\ & > \infty. \end{align*} I am wondering whether there exists more direct method to show the statement. Thanks.