Does someone know if the following is true:
Let $\mathbb{X}$ be some arbitrary Banach space. $\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$.
Now, if the series
$$\sum_{k=1}^\infty \|x_k\|_X$$
converges, would the "more generic" series
$$\sum_{k=1}^\infty x_k$$
also converge?
If yes, could you please give the proof (or just mark the proof steps) ?
Thank you.